TWI550529B - Dynamical event neuron and synapse models for learning spiking neural networks - Google Patents

Description

Dynamic event neurons and synaptic models for learning spiking neural networks [Priority Request under the Patent Law]

This patent application claims the rights of the provisional application No. 61/728,409 filed on November 20, 2012 and the provisional application No. 61/759,181 filed on January 31, 2013, the provisional application It was assigned to the assignee of the case and is hereby explicitly incorporated by reference.

Certain aspects of the present case are generally related to neural networks, and in particular to event-based continuous time models for neurons and synapses.

An artificial neural network that can include a group of interconnected artificial neurons (i.e., a neuron model) is a computing device or a method that will be performed by a computing device. Artificial neural networks may have corresponding structures and/or functions in a biological neural network. However, artificial neural networks can provide innovative and useful computing techniques for certain applications where traditional computing techniques are cumbersome, impractical, or incompetent. Since artificial neural networks can infer functions from observations, such networks are designed by conventional techniques due to the complexity of tasks or data. This feature is particularly useful in applications that are more cumbersome.

One type of artificial neural network is a spiking neural network Road, the spiking neural network incorporates time concepts and neuronal and synaptic states into the spiking neural network working model, thereby providing a rich set of behaviors from which the computational functions can emerge from the neural network. . The spiking neural network is based on the concept that a neuron fires or "sends a spike" at one or more specific times based on the state of the neuron, and that time is important for neuronal function. When a neuron is excited, the neuron produces a spike that travels to other neurons, which in turn can adjust the state of the neuron based on the time at which the spike was received. In other words, the information can be encoded in the relative or absolute timing of the spikes in the neural network.

Some aspects of the present invention provide a method for updating the state of an artificial neuron. The method generally includes determining a first state of an artificial neuron, wherein the neuron model of the artificial neuron has a closed form solution in continuous time and wherein the state dynamics of the neuron model is divided into two or more states Determining an operational phase of the artificial neuron from the two or more phases based at least in part on the first state; and based at least in part on the first state of the artificial neuron and the determined operational state Update the state of the artificial neuron.

Some aspects of the present invention provide means for updating the state of an artificial neuron. The apparatus generally includes a processing system configured to: determine a first state of an artificial neuron, wherein the neuron model of the artificial neuron has a closed form solution in a continuous time and wherein the state of the neuron model Dynamically being divided into two or more states; determining an operational phase of the artificial neuron from the two or more phases based at least in part on the first state; and based at least in part on the artificial neuron The first state and the determined operational state update the state of the artificial neuron.

Some aspects of the case provide a state for updating artificial neurons device of. The apparatus generally includes means for determining a first state of an artificial neuron, wherein the neuron model of the artificial neuron has a closed form solution in continuous time and wherein the state dynamics of the neuron model are divided into two or a plurality of states; means for determining an operational phase of the artificial neuron from the two or more states based at least in part on the first state; and for at least partially based on the artificial neuron A means of updating the state of the artificial neuron in response to a state and a determined operational state.

Some aspects of the case provide a state for updating artificial neurons Computer program product. The computer program product generally includes a computer readable medium having code for: determining a first state of an artificial neuron, wherein the neuron model of the artificial neuron has a closed form solution in a continuous time and wherein the nerve The state dynamics of the metamodel is divided into two or more states; the operating phase of the artificial neuron is determined from the two or more states based at least in part on the first state; and based at least in part on the The first state of the artificial neuron and the determined working state are used to update the state of the artificial neuron.

Some aspects of the case provide nerves for generating artificial neurons The method of behavior. The method generally includes determining a first state of an artificial neuron, wherein the neuron model of the artificial neuron has a closed shape in continuous time And the linear dynamics of the neuron model are divided into two or more states; the working phase of the artificial neurons is determined from the two or more states based at least in part on the first state; Updating the state of the artificial neuron based in part on the first state of the artificial neuron and the determined operational state phase; and generating a plurality of neural behaviors of the artificial neuron by utilizing linear dynamics of the neuron model.

Some aspects of the case provide nerves for generating artificial neurons The device of behavior. The apparatus generally includes a processing system configured to: determine a first state of an artificial neuron, wherein the neuron model of the artificial neuron has a closed form solution in continuous time and wherein the linear dynamics of the neuron model are divided Determining an operational phase of the artificial neuron from at least a portion of the two or more states based at least in part on the first state; based at least in part on the first state of the artificial neuron and Determining the state of the working state to update the state of the artificial neuron; and generating a plurality of neural behaviors of the artificial neuron by utilizing the linear dynamics of the neuron model.

Some aspects of the case provide nerves for generating artificial neurons Behavioral equipment. The apparatus generally includes means for determining a first state of an artificial neuron, wherein the neuron model of the artificial neuron has a closed form solution in continuous time and wherein the linear dynamics of the neuron model are divided into two or a plurality of states; means for determining an operational phase of the artificial neuron from the two or more states based at least in part on the first state; for first, based at least in part on the artificial neuron Means of updating the state of the artificial neuron by the state and the determined operational state; and for generating a plurality of neural behaviors of the artificial neuron by utilizing linear dynamics of the neuron model means.

Some aspects of the case provide nerves for generating artificial neurons The behavior of computer program products. The computer program product generally includes a computer readable medium having code for: determining a first state of an artificial neuron, wherein the neuron model of the artificial neuron has a closed form solution in a continuous time and wherein the nerve The linear dynamics of the metamodel are divided into two or more states; the working phase of the artificial neurons is determined from the two or more states based at least in part on the first state; based at least in part on the artificial The first state of the neuron and the determined working state are used to update the state of the artificial neuron; and various neural behaviors of the artificial neuron are generated by utilizing the linear dynamics of the neuron model.

Some aspects of the case provide a state for updating artificial neurons Methods. The method generally includes updating a state of an artificial neuron based at least in part on an event, wherein the neuron model of the artificial neuron has a closed form solution in continuous time and wherein the state dynamics of the neuron model are divided into two or More states; updating the state of the artificial neuron at intervals; and updating the state of the artificial neuron if an event occurs at or between times.

Some aspects of the case provide a state for updating artificial neurons s method. The means generally includes a processing system configured to: update a state of an artificial neuron based at least in part on an event, wherein the neuron model of the artificial neuron has a closed form solution in continuous time and wherein the neuron model State dynamics are divided into two or more states; the state of the artificial neuron is updated at time intervals; and if at time or between times When an event occurs, the state of the artificial neuron is updated.

Some aspects of the case provide a state for updating artificial neurons device of. The apparatus generally includes means for updating a state of an artificial neuron based at least in part on an event, wherein the neuron model of the artificial neuron has a closed form solution in continuous time and wherein the state dynamics of the neuron model are divided Two or more states; means for updating the state of the artificial neuron at time intervals; and means for updating the state of the artificial neuron if an event occurs at or between times .

Some aspects of the case provide a state for updating artificial neurons Computer program product. The computer program product generally includes computer readable media having code for updating a state of an artificial neuron based at least in part on an event, wherein the neuron model of the artificial neuron has a closed form solution in continuous time And wherein the state dynamics of the neuron model is divided into two or more states; the state of the artificial neuron is updated at time intervals; and if an event occurs at a time or between times, the artificial nerve is updated The state of the yuan.

100‧‧‧Nervous system

102‧‧‧ neuron

104‧‧‧Synaptic connection network

106‧‧‧ neurons

108‧‧‧ signal

110‧‧‧ Output spikes

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2041‧‧‧ signal

204i‧‧‧ signal

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2061‧‧ ‧ synaptic weight

206i‧‧‧ synaptic weight

206N‧‧‧ synaptic weight

208‧‧‧ output signal

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306‧‧‧Crossover

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2104‧‧‧current zero inclination

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5502‧‧‧ phase short pulse behavior

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5602‧‧‧ Mixed mode behavior

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5702‧‧‧ spike

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5802‧‧‧ Suppression

5804‧‧‧Excitatory input

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5902‧‧‧ State track

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6002‧‧‧ suppression

6004‧‧‧Excitatory input

6006‧‧‧short pulse

6300‧‧‧ operation

6300A‧‧‧ components

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6302A‧‧‧ means

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6500A‧‧‧ components

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6602‧‧‧General Processor

6604‧‧‧ memory block

6606‧‧‧Program memory

6700‧‧‧ realized

6702‧‧‧ memory

6704‧‧‧Internet

6706‧‧‧Processing unit

6800‧‧‧ realized

6802‧‧‧ memory group

6804‧‧‧Processing unit

6900‧‧‧Neural Network

6902‧‧‧Local Processing Unit

6904‧‧‧Local state memory

6906‧‧‧Local parameter memory

6908‧‧‧ memory

6910‧‧‧ memory

6912‧‧‧Locally connected memory

Unit 6914‧‧‧

6916‧‧‧ components

To more clearly understand the manner in which the above-described features of the present disclosure are used, the above briefly summarized aspects may be more specifically described with reference to the various aspects, some of which are illustrated in the drawings. It should be noted, however, that the drawings are only illustrative of certain aspects of the present invention and are not to be considered as limiting.

FIG. 1 illustrates an example network of neurons in accordance with certain aspects of the present disclosure.

Figure 2 illustrates a computing network (neural system) according to certain aspects of the present case Or an instance of a processing unit (neuron) of a neural network.

3 illustrates an example of a spike timing dependent plasticity (STDP) curve in accordance with certain aspects of the present disclosure.

4 illustrates an example of a normal phase and a negative phase for defining the behavior of a neuron model, in accordance with certain aspects of the present disclosure.

FIG. 5 illustrates an example abstract event state machine in accordance with certain aspects of the present disclosure.

Figure 6 illustrates an example of an event that triggers a dynamic change in a neuron model in accordance with certain aspects of the present disclosure.

Figure 7 illustrates an example of a situation in which consistency with a Leaky Integral Excitation (LIF) neuron model can be detected in accordance with certain aspects of the present disclosure.

Figure 8 illustrates an example of a situation in which certain aspects of the present case are not/difficult to detect consistency with other neuron models.

Figure 9 illustrates an example of a monotonically decreasing state difference of a LIF neuron model in accordance with certain aspects of the present disclosure.

Figure 10 illustrates an example of a state difference in a LIF neuron model according to certain aspects of the present disclosure that is not monotonically decreasing.

Figure 11 illustrates an example of a situation in which certain sub-threshold input timings cannot be distinguished in accordance with certain aspects of the present disclosure.

Figure 12 illustrates an example of a situation in which it is possible to distinguish between on-threshold input timings in accordance with certain aspects of the present disclosure.

Figure 13 illustrates an example of a situation in which certain sub-threshold input timings can be distinguished, in accordance with certain aspects of the present disclosure.

Figure 14 illustrates an instant coupling at an event in accordance with certain aspects of the present disclosure. And examples of decoupling between events.

Figure 15 illustrates an example comparison of dynamics in a normal phase and a negative phase in accordance with certain aspects of the present disclosure.

16 illustrates an example of state dynamics independent of inter-event updates between events according to certain aspects of the present disclosure.

Figure 17 illustrates an example of a human event that affects dynamics, although there is no input or output, in accordance with certain aspects of the present disclosure.

Figure 18 illustrates an example comparison of basic dynamics with and without periodic human events in accordance with certain aspects of the present disclosure.

Figure 19 illustrates an example of a typical zero-tilt of a Hunzinger Cold neuron model in accordance with certain aspects of the present disclosure.

20 illustrates example spike results for a repetitive computational simple model and a dynamic event model in accordance with certain aspects of the present disclosure.

21 illustrates an example of a state trajectory across a zero tilt line in a negative phase in accordance with certain aspects of the present disclosure.

Figure 22 illustrates an example of state trajectory geometry in accordance with certain aspects of the present disclosure.

Figure 23 illustrates an example of damped oscillations in accordance with certain aspects of the present disclosure.

Figure 24 illustrates an example of a track change in accordance with certain aspects of the present disclosure.

Figure 25 illustrates an example of the relationship between inter-event dynamics and zero-tilt according to certain aspects of the present disclosure.

Figure 26 illustrates an example of oscillations due to inter-event intervals in accordance with certain aspects of the present disclosure.

Figure 27 illustrates the creation of an interval between events according to certain aspects of the present invention. An example of a track.

28 illustrates an example of voltage derivative acceleration of a trans-state phase threshold in accordance with certain aspects of the present disclosure.

29 illustrates an example of voltage derivative deceleration of a trans-state phase threshold in accordance with certain aspects of the present disclosure.

Figure 30 illustrates an example of a limit cycle due to a positive finite return phase in accordance with certain aspects of the present disclosure.

Figure 31 illustrates an example of behavior in a negative finite return phase according to certain aspects of the present disclosure.

Figure 32 illustrates an example of a vector field state phase of a Hunzinger Cold neuron model in accordance with certain aspects of the present disclosure.

Figure 33 illustrates an example of spike latency in accordance with certain aspects of the present disclosure.

Figure 34 illustrates an example of a spike latency state trajectory in accordance with certain aspects of the present disclosure.

Figure 35 illustrates an example of spike latency in the presence and absence of coupling in accordance with certain aspects of the present disclosure.

FIG. 36 illustrates an example resonator in accordance with certain aspects of the present disclosure.

Figure 37 illustrates an example of a resonator state trajectory in accordance with certain aspects of the present disclosure.

FIG. 38 illustrates an example integrator in accordance with certain aspects of the present disclosure.

Figure 39 illustrates an example of threshold variability in accordance with certain aspects of the present disclosure.

Figure 40 illustrates an example rebound spike and back according to certain aspects of the present case. Short pulses.

Figure 41 illustrates an example of a rebound state trajectory in accordance with certain aspects of the present disclosure.

Figure 42 illustrates an example of bistable state in accordance with certain aspects of the present disclosure.

Figure 43 illustrates an example of a bistable state trajectory in accordance with certain aspects of the present disclosure.

Figure 44 illustrates an example of sub-threshold oscillations in accordance with certain aspects of the present disclosure.

Figure 45 illustrates an example of post-peak sub-threshold oscillations in accordance with certain aspects of the present disclosure.

Figure 46 illustrates an example of a resting orbital path before and after a spike, in accordance with certain aspects of the present disclosure.

Figure 47 illustrates an example of post-potential depolarization in accordance with certain aspects of the present disclosure.

Figure 48 illustrates an example of a suitable aspect in accordance with certain aspects of the present disclosure.

Figure 49 illustrates an example of a accommodative state trajectory in accordance with certain aspects of the present disclosure.

Figure 50 illustrates an example of class 1 excitement in accordance with certain aspects of the present disclosure.

Figure 51 illustrates an example of class 2 excitement in accordance with certain aspects of the present disclosure.

Figure 52 illustrates an example of a tonic spike in accordance with certain aspects of the present disclosure.

Figure 53 illustrates an example of phase-type spikes in accordance with certain aspects of the present disclosure.

Figure 54 illustrates an example of a tonic short pulse in accordance with certain aspects of the present disclosure.

Figure 55 illustrates the phase-type short pulse in accordance with certain aspects of the present disclosure. example.

Figure 56 illustrates mixed mode behavior in accordance with certain aspects of the present disclosure. Example.

Figure 57 illustrates peak frequency self-adjustment according to certain aspects of the present disclosure. An example.

Figure 58 illustrates spikes caused by inhibition in accordance with certain aspects of the present disclosure An example.

Figure 59 illustrates spikes caused by inhibition according to certain aspects of the present disclosure An example of a state track.

Figure 60 illustrates a short pulse caused by inhibition according to certain aspects of the present disclosure An example of a rushed state trajectory.

Figure 61 illustrates a summary of nominal example behaviors in accordance with certain aspects of the present disclosure. total.

Figure 62 illustrates the Hunzinger Cold god according to certain aspects of the present case. The meta model behavior example parameters.

Figure 63 illustrates the use of certain aspects of the present invention for updating artificial gods. An example operation of the state of the meta.

FIG. 63A illustrates example components capable of performing the operations illustrated in FIG. 63. .

Figure 64 illustrates example operations for generating various neural behaviors of artificial neurons in accordance with certain aspects of the present disclosure.

FIG. 64A illustrates example components capable of performing the operations illustrated in FIG.

Figure 65 illustrates an update of an artificial god in accordance with certain aspects of the present disclosure. Other example operations of the state of the meta.

FIG. 65A illustrates example components capable of performing the operations illustrated in FIG. 65. .

Figure 66 illustrates the use of a general purpose processor in accordance with certain aspects of the present disclosure. An example block diagram of a method to update the state of an artificial neuron.

Figure 67 illustrates the use of certain aspects of the present invention for updating artificial gods. An example block diagram of a method of transiting a state in which a memory can interface with an individual decentralized processing unit.

Figure 68 illustrates a decentralized version in accordance with certain aspects of the present disclosure. An example block diagram of a method of memory and decentralized processing unit to update the state of an artificial neuron.

Figure 69 illustrates an example of a neural network in accordance with certain aspects of the present disclosure. Now.

Various aspects of the present invention are described more fully hereinafter with reference to the accompanying drawings. However, the present invention may be embodied in many different forms and should not be construed as being limited to any specific structure or function. Rather, these aspects are provided so that this disclosure will be thorough and complete, and the scope of the invention will be fully conveyed by those skilled in the art. Based on the teachings herein, those skilled in the art will appreciate that the scope of the present invention is intended to cover any aspect of the present disclosure as disclosed herein, regardless of whether it is implemented independently or in combination with any other aspect of the present invention. For example, any number of aspects set forth herein can be used to implement an apparatus or a method of practice. In addition, the scope of the case is intended to cover the use as Such an apparatus or method of practicing the various aspects of the present invention as set forth herein, or other structural, functional, or structural and functional aspects of the present invention. It should be understood that any aspect of the present disclosure disclosed herein may be implemented by one or more elements of the scope of the patent application.

The wording "exemplary" is used in this document to mean "used as an example. Sub or graphic". Any aspect described herein as "exemplary" is not necessarily to be construed as preferred or advantageous.

Although specific aspects are described herein, many variations of such aspects Body and replacement fall within the scope of this case. Although some benefits and advantages of the preferred aspects are mentioned, the scope of the present invention is not intended to be limited to a particular benefit, use, or objective. Rather, the various aspects of the present invention are intended to be broadly applied to various techniques, system configurations, networks, and protocols, some of which are illustrated in the drawings and the detailed description of the preferred aspects. The detailed description and drawings are merely illustrative of the present invention, and the scope of the present invention is defined by the scope of the appended claims.

Example nervous system, training and operation

FIG. 1 illustrates an example nervous system 100 having multiple levels of neurons in accordance with certain aspects of the present disclosure. The nervous system 100 can include a neuron level 102 that is coupled to another level of neurons 106 by a synaptic connection network 104 (ie, a feedforward connection). For simplicity, only two levels of neurons are illustrated in Figure 1, but fewer or more levels of neurons may be present in a typical nervous system. It should be noted that some neurons may be connected to other neurons in the same layer via lateral connections. In addition, some neurons may be connected back to neurons in the previous layer via a feedback connection.

As illustrated in Figure 1, each neuron in stage 102 can receive input Signal 108, which may be generated by a plurality of neurons of a previous stage (not shown in FIG. 1). Signal 108 may represent the input current of the neurons of stage 102. This current can accumulate on the neuron membrane to charge the membrane potential. When the membrane potential reaches the membrane potential threshold, the neuron can excite and produce an output spike that will be passed to the inferior neuron (eg, stage 106). Such behavior can be simulated or mimicked in hardware and/or software, including analog and digital implementations.

In biological neurons, the output tip produced when a neuron is excited The peak is called the action potential. The electrical signal is a relatively rapid, transient, all or none of the nerve pulses having an amplitude of about 100 mV and a duration of about 1 ms. In a particular embodiment of a nervous system having a series of connected neurons (e.g., a peak is transferred from a first order neuron in FIG. 1 to another stage), each action potential has substantially the same amplitude and duration, thus The information in the signal is represented only by the frequency and number of spikes or the time of the peak, not by the amplitude. The information carried by the action potential is determined by the peak, the spiked neuron, and the time of the spike relative to one or more other spikes.

The transfer of spikes from primary neurons to another level of neurons can be achieved via a synaptic connection (or simply "synaptic") network 104, as illustrated in FIG. Synapse 104 can receive an output signal (ie, a spike) from stage 102 neurons (presynaptic neurons relative to synapse 104), and according to adjustable synaptic weights ,..., (where P is the total number of synaptic connections between neurons of stage 102 and stage 106) to scale their signals. Moreover, the scaled signals can be combined to be an input signal for each neuron in stage 106 (post-synaptic neuron relative to synapse 104). Each of the stages 106 neurons can generate an output spike 110 based on the corresponding combined input signal. The output spikes 110 can then be passed to another level of neurons using another synaptic connection network (not shown in Figure 1).

Biological synapses can be classified as electrical or chemical. Electrical synapse Used to send excitatory signals, while chemical synapses regulate excitatory or inhibitory (hyperpolarized) actions in postsynaptic neurons and can also be used to amplify neuronal signals. The excitatory signal typically depolarizes the membrane potential (i.e., increases the membrane potential relative to the resting potential). An action potential occurs in a post-synaptic neuron if a sufficient excitatory signal is received during a certain period of time to depolarize the membrane potential above a threshold. In contrast, inhibitory signals generally hyperpolarize the membrane potential (i.e., decrease membrane potential). If the inhibitory signal is strong enough, it cancels out the sum of the excitatory signals and prevents the membrane potential from reaching the threshold. In addition to counteracting synaptic excitability, synaptic inhibition can also exert strong control over spontaneously active neurons. Spontaneous activity neurons are neurons that emit spikes without further input (eg, due to neuronal dynamics or feedback). By suppressing the spontaneous production of action potentials in these neurons, synaptic inhibition can shape the excitation pattern in neurons, which is commonly referred to as engraving. The various synapses 104 can act as any combination of excitatory or inhibitory synapses, depending on the desired behavior.

The nervous system 100 can be implemented by a general purpose processor and a digital signal processor ( DSP), Special Application Integrated Circuit (ASIC), Field Programmable Gate Array (FPGA) or other programmable logic (PLD), individual gate or transistor logic, individual hardware components, executed by the processor The software module or any combination thereof is used for simulation. The nervous system 100 can be used in a wide range of applications, such as Image and pattern recognition, machine learning, motor control and similar applications. Each neuron in the nervous system 100 can be implemented as a neuron circuit. A neuron membrane that is charged to a threshold that initiates an output spike can be implemented, for example, as a capacitor that integrates the current flowing through the neuron membrane.

In the aspect, the capacitance of the current integrator as a neuron circuit The device can be removed and replaced with a smaller memristor element. This approach can be applied to neuron circuits, as well as to a variety of other applications where bulk capacitors are used as current integrators. Additionally, each synapse 104 can be implemented based on a memristor element, where synaptic weight changes can be related to changes in memristor resistance. The use of nanometer-sized memristors significantly reduces the area of neuronal circuits and synapses, which makes it possible to implement extremely large-scale neural system hardware implementations.

The functionality of the neural processor that simulates the nervous system 100 can be Depending on the weight of the synaptic connections, these weights control the strength of the connections between the neurons. Synaptic weights can be stored in non-volatile memory to preserve the functionality of the processor after power down. In the aspect, the synaptic weight memory can be implemented on an external wafer separate from the main nerve processor wafer. The synaptic weight memory can be packaged as a replaceable memory card separately from the neural processor chip. This can provide a variety of functionality to the neural processor, where the particular functionality can be based on the synaptic weights stored in the memory card currently attached to the neural processor.

2 illustrates an example 200 of a processing unit (eg, a neuron or neuron circuit) 202 of a computing network (eg, a nervous system or neural network) in accordance with certain aspects of the present disclosure. For example, neuron 202 may correspond to any of the neurons from stages 102 and 106 of FIG. The neuron 202 can receive a plurality of input signals 204 _{1 -} 204 _N (x _{1 -} x _N ), which can be signals external to the nervous system or signals generated by other neurons of the same nervous system or Both. The input signal can be a real or complex value of current or voltage. The input signal can include a value having a fixed point or floating point representation. The input signals can be delivered to neurons 202 by synaptic connections that scale the signals according to adjustable synaptic weights 206 _{1 -} 206 _N (w _1- w _N ), wherein N can be the total number of input connections for neuron 202.

Neuron 202 can combine the scaled input signals and And using the combined scaled input to produce an output signal 208 (ie, signal y). Output signal 208 can be a real or complex value current or voltage. The output signal can include a value having a fixed point or floating point representation. The output signal 208 can then be passed as an input signal to other neurons of the same nervous system or as an input signal to the same neuron 202 or as an output of the nervous system.

Processing unit (neuron) 202 can be simulated by circuitry and The input and output connections of the circuit can be simulated by a synapse circuit using wires. The input and output connections of the processing unit 202 and the processing unit 202 can also be simulated by software code. Processing unit 202 can also be simulated by circuitry, while circuit input and output connections can be simulated by software code. In an aspect, processing unit 202 in the computing network can include an analog circuit. In another aspect, processing unit 202 can include a digital circuit. In yet another aspect, processing unit 202 can include a mixed signal circuit having both analog and digital components. The computing network can include any of the aforementioned forms of processing units. Computational networks (neural systems or neural networks) using such processing units can be used in a wide range of applications, such as image and pattern recognition, machines Learning, motor control, etc.

Synaptic weights during training procedures of neural networks (eg, weights from Figure 1) ,..., And/or the weights 206 _{1 -} 206 _N from Figure 2 can be initialized with random values and increased or decreased according to learning rules. Some examples of learning rules are peak timing dependent plasticity (STDP) learning rules, Hebb rules, Oja rules, Bienenstock-Copper-Munro (BCM) rules, and the like. Many times, these weights can be stabilized to one of two values (ie, a bimodal distribution of weights). This effect can be used to reduce the number of bits per synaptic weight, increase the speed of reading and writing from/to memory that stores synaptic weights, and reduce the power consumption of synaptic memory.

Synaptic type

In hardware and software models of neural networks, the processing of synaptic related functions can be based on synaptic types. Synaptic types can include non-plastic synapses (no change in weight and delay), plastic synapses (weights can be changed), structured delay plastic synapses (weight and delay can be changed), all plastic synapses (weights, delays, and Connectivity can vary) and variants based thereon (eg, the delay can change, but there is no change in weight or connectivity). The advantage of this is that the processing can be subdivided. For example, a non-plastic synapse does not require a plasticity function to be performed (or wait for such a function to complete). Similarly, delay and weight plasticity can be subdivided into operations that can operate together or separately, sequentially or in parallel. Different types of synapses may have different look-up tables or formulas and parameters for each of the different types of plasticity that are applicable. Therefore, the methods will access the relevant tables for the type of synapse.

It further involves the following facts: peak timing-dependent structuring Plasticity can be performed independently of synaptic plasticity. Structural plasticity can be performed even if the weight magnitude does not change (for example, if the weight has reached a minimum or maximum value or if it has not changed for some other reason) because of structural plasticity (ie, delay-changing The amount) can be a direct function of the front-to-back spike time difference. Alternatively, the structural plasticity may be set as a function of the amount of weight change or may be set based on conditions related to the weight or the limit of the weight change. For example, synaptic delays may only change when a weight change occurs or when the weight reaches zero, but do not change when the weight reaches a maximum limit. However, it may be advantageous to have independent functions such that the programs can be parallelized to reduce the number and overlap of memory accesses.

Synaptic plasticity decision

Neuronal plasticity (or simply "plasticity") is the ability of neurons and neural networks in the brain to alter neuronal and neural network synaptic connections and behaviors in response to new information, sensory stimuli, development, damage, or dysfunction. . Plasticity is important for learning and memory in biology and for computing neuron science and neural networks. Various forms of plasticity have been investigated, such as synaptic plasticity (eg, according to Herb theory), spike-time-dependent plasticity (STDP), non-synaptic plasticity, mobility-dependent plasticity, structural plasticity, and homeostasis plasticity.

STDP is a learning program that regulates the strength of synaptic connections between neurons. The connection strength is adjusted based on the relative timing of the output of a particular neuron and the received input spike (ie, the action potential). Under the STDP procedure, long-term enhancement (LTP) can occur if the input spike to a neuron tends to occur on average just before the output spike of the neuron. So make this Specific inputs are somewhat stronger. On the other hand, long-term suppression (LTD) can occur if the input spike average tends to occur immediately after the output spike. This makes the particular input somewhat weaker and is hence called "spike timing dependent plasticity." Thus, inputs that may be the cause of post-synaptic neuronal excitation are even more likely to contribute in the future, making inputs that are not the cause of post-synaptic spikes less likely to contribute in the future. The program continues until a subset of the initial connection set is retained, while the impact of all other connections is mitigated to zero or close to zero.

Because neurons generally occur in many of their inputs in a short period of time ( That is, an output spike is generated when accumulated enough to cause an output, so the input subset that is typically retained includes those inputs that tend to be correlated in time. In addition, since the input that occurs before the output spike is boosted, the input that provides the earliest sufficient cumulative indication of the correlation will eventually become the last input to the neuron.

The STDP learning rule can be effectively adapted by the time difference between the spike time t _pre of the presynaptic neuron and the spike time t _{post of the} postsynaptic neuron (ie, t = t _post - t _pre ) The presynaptic neuron is connected to the synaptic weight of the synapse of the postsynaptic neuron. A typical formulation of STDP is to increase the synaptic weight (ie, to enhance the synapse if the time difference is positive (pre-synaptic neurons are excited before the postsynaptic neuron), and if the time difference is negative (synapse) Post-neurons are stimulated before presynaptic neurons) to reduce synaptic weights (ie, suppress the synapse).

In STDP procedures, changes in synaptic weight over time can often be achieved using exponential decay, as provided by: Where k ₊ and k _- are time constants for positive and negative time differences, respectively, a ₊ and a _- are corresponding scaled magnitudes, and μ is an offset applicable to positive time differences and/or negative time differences.

Figure 3 illustrates that synaptic weight changes to presynaptic (pre) according to STDP An example diagram that changes with the relative timing of post-synaptic spikes represents a graph 300. If the presynaptic neurons are excited before the postsynaptic neurons, the corresponding synaptic weights can be increased, as illustrated in section 302 of graph 300. This weight increase can be referred to as the synaptic LTP. As can be observed from chart section 302, the amount of LTP can decrease substantially exponentially as a function of the difference between pre- and post-synaptic spike times. The opposite firing order may reduce synaptic weights, as illustrated in section 304 of graph 300, resulting in a LTD of the synapse.

As illustrated in graph 300 of FIG. 3, the negative offset μ can be applied to the LTP (causality) portion 302 of the STDP chart. The x-axis crossing point 306 (y = 0) can be configured to coincide with the maximum time lag for considering the correlation of the causal input from layer i-1. In the case of frame-based input (ie, the input is in the form of a frame of a particular duration including spikes or pulses), the offset value μ can be calculated to reflect the frame boundary. The first input spike (pulse) in the frame can be considered to either decay as a function of the post-synaptic potential directly or in a state of influence on the neural state over time. If the second input spike (pulse) in the frame is considered to be related or related to a specific time frame, the relevant time before the frame and the relevant time after the frame may be in the time frame. The boundaries are separated and the values at the relevant times may be different by shifting one or more portions of the STDP curve (eg, for times greater than one frame and negative for less than one frame) Positive) to treat the frame differently in terms of plasticity. For example, a negative offset μ can be set to offset LTP such that the curve actually goes below zero at a pre (post)-post (post) time greater than the frame time and the curve is therefore part of the LTD rather than the LTP portion.

Neuron model and operation

There are some general principles for designing useful spike neuron models. A good neuron model can have a rich potential behavior in two computational states: consistency detection and functional computation. Moreover, a good neuron model should have two elements to allow time coding: the arrival time of the input affects the output time, and the consistency detection can have a narrow time window. Finally, in order to be computationally attractive, a good neuron model can have closed-form solutions in continuous time and have stable behavior, including near attractors and saddle points. In other words, useful neuron models are neuron models that are practicable and can be used to model rich, realistic, and biologically consistent behaviors and can be used to engineer and reverse engineer neural circuits.

The neuron model can depend on events, such as input arrivals, output spikes, or other events, whether the events are internal or external. In order to achieve a rich set of behavioral skills, a state machine that can exhibit complex behaviors may be desirable. If the event itself occurs in the case of an input contribution (if any) that affects the state machine and constrains the dynamics after the event, the future state of the system is not just a function of state and input, but a state, an event, and Input function.

In the aspect, neuron n can be modeled as a spike-leaked integral excitation (LIF) neuron, and the neuron membrane voltage v _n ( t ) is governed by the following dynamics: Wherein α and β are parameters, W _{m, n} is connected to a presynaptic neuron m postsynaptic neuron synapse weight of the synapse weight of n, and y _m (t) m is the peak output neurons, nerve The element m can reach the cell body of the neuron n according to Δ t _m,n delayed by dendritic or axonal delay.

It should be noted that there is a delay from the time when sufficient input to the postsynaptic neuron is established until the time when the postsynaptic neuron actually fires. In the dynamic model of neuron spikes (such as a simple model Izhikevich) there between when depolarization threshold _t v v _Peak peak spike voltage difference, a time delay may be caused. For example, in this simple model, neuronal cell dynamics can be governed by pairs of differential equations about voltage and recovery, namely: Wherein v is the membrane potential, u is a membrane recovery variable, k is the parameters describing the time scale membrane potential v, a is a parameter describes restoration variables u time scale, b is described recovery variable u fluctuation of the threshold membrane potential v. The sensitivity parameter, v _r is the membrane resting potential, I is the synaptic current, and C is the membrane capacitance. According to this model, neurons are defined to issue spikes when v > v _peak .

Hunzinger Cold model

The Hunzinger Cold neuron model is capable of reproducing a rich variety of The minimum bimodal phase peak linear dynamic model of the neural behavior. The one- or two-dimensional linear dynamics of the model can have two phases, where the time constant (and coupling) can depend on the phase. In the subliminal phase, the time constant (which is negative by convention) represents the leakage channel dynamics, which generally acts to return the cells to rest in a biologically consistent linear manner. The time constant in the upper-threshold phase (positive by convention) reflects the dynamics of the anti-leakage channel, which typically drives the cell to issue spikes while simultaneously causing latency in spike generation.

As shown in Figure 4, the dynamics of the model can be divided into two (or more) states. The isomorphic phase may be referred to as a negative phase 402 (also interchangeably referred to as a Leaked Integral Excitation (LIF) phase, not to be confused with a LIF neuron model) and a normal phase 404 (also interchangeably referred to as an anti-interference) Leak-integrated excitation (ALIF) phase, not to be confused with the ALIF neuron model). In the negative phase 402, the state tends to rest ( v _- ) at a time of future events. In this negative phase, the model generally exhibits time input detection properties and other subliminal behaviors. In the normal phase 404, the state issuing tend to spike events (v _s). In this normal phase, the model exhibits computational properties, such as latency that causes spikes to be issued depending on subsequent input events. The formulation of dynamics in terms of events and the division of dynamics into these two states are the fundamental characteristics of the model.

Linear two-state phase two-dimensional dynamics (for states v and u ) can be defined by convention as: Where q _ρ and r are linear transformation variables for coupling.

The symbol ρ is used herein to indicate the dynamic phase. When discussing or expressing the relationship of the specific phase, the symbol ρ is replaced by the symbol "-" or "+" for the negative phase and the normal phase, respectively.

The model state is defined by the membrane potential (voltage) v and the recovery current u . In the basic form, the phase is essentially determined by the state of the model. There are some subtle but important aspects of this precise and general definition, but it is currently considered that the model is in the normal phase 404 if the voltage v is above the threshold ( v ₊ ), otherwise it is in the negative phase 402.

The phase dependent time constants include a negative phase time constant τ _- and a normal phase time constant τ ₊ . The recovery current time constant τ _u is usually independent of the state. For convenience, the negative phase time constant τ _- is usually specified to reflect the negative of the decay, so that the same equation for voltage evolution can be used for the normal phase. In the normal phase, the exponent and τ ₊ will generally be Positive, just like τ _u .

The dynamics of these two state elements can be coupled by shifting the state away from its null-cline transformation, where the transformation variables are: q _ρ =- τ _ρ βu - v _ρ (7)

r = δ ( v + ε ) (8) where δ is the coupling conductivity-time parameter, ε is the coupling offset voltage, and β is the coupling resistance. The two values of v _ρ are the cardinality of the reference voltages of the two states. The parameter v _- is the base voltage of the negative phase, and the membrane potential will generally deviate towards v _- in the negative phase. The parameter v ₊ is the base voltage of the normal phase, and the membrane potential will generally tend to deviate from v ₊ in the normal phase.

The zero inclinations of v and u are provided by the negative of the transformation variables q _ρ and r , respectively. The parameter δ is a scaling factor that controls the slope of the u- zero tilt. The parameter ε is usually set equal to -v _- . The parameter β is the resistance value that controls the slope of the v- zero inclination in the two states. The τ _ρ time constant parameter not only controls the exponential decay, but also controls the zero tilt slope in each phase separately.

This model is defined as a spike that is issued when the voltage v reaches the value v _S (spike voltage). Subsequently, the state is usually reset on the occasion of a rescheduled event (the rescheduled event can be technically identical to the spike event):

u = u +△ u (10) where It is the reset voltage, and Δ u is the reset current reset offset. Re-voltage Usually set to v _- .

According to the principle of transient coupling, closed-form solutions are not only possible for states (and have a single exponential term), but are also possible for the time required to reach a particular state. The closed form state solution is:

Thus, the model state can be updated only when an event occurs, such as based on an input (pre-synaptic spike) or an output (post-synaptic spike). You can also perform operations at any given time, with or without input or output.

Moreover, according to the principle of transient coupling, the time of the post-synaptic spike can be predicted, so the time to reach a particular state can be determined in advance without the need for repeated arithmetic techniques or numerical methods (eg, Euler numerical methods). The time delay before the previous voltage state v _{0 is} given until the voltage state v _{f is} reached is provided by:

If the spike is defined as the time at which the voltage state v reaches v _S , the closed form solution from the time the voltage is at a given state v until the peak occurs or the relative delay is: among them It is a phase threshold and is usually set to the parameter v ₊ , but other variations may be possible.

The above definition of model dynamics depends on whether the model is in the normal or negative phase. As mentioned, the coupling and phase ρ can be calculated based on the event. For the purpose of state propagation, the phase and coupling (transform) variables can be defined based on the state of the time of the previous (previous) event. For the purpose of subsequently estimating the peak output time, the phase and coupling variables can be defined based on the state of the time of the next (current) event.

Existence of the Cold model, execution of simulations, simulations, or Several possible implementations of the model. This includes, for example, event-updates, step-and-event updates, and step-and-update modes. An event update is an update in which the status is updated based on an event or "event update" (at a specific time). A step update is an update of the model that is updated at intervals (eg, 1 ms). This does not necessarily require repeated arithmetic methods or numerical methods. Event-based implementation with limited time resolution in a step-based simulator by updating the model only when the event occurs at or near the step or by "step-event" update It is also possible.

The next step in this case will be about the Hunzinger Cold model and possible More details of the implementation.

Neural coding

Useful neural network models, such as neural network models including the artificial neurons 102, 106 of Figure 1, can be encoded via any of a variety of suitable neural coding schemes, such as coherency coding, time coding, or rate coding. News. In coherent coding, information is encoded in the consistency (or temporal proximity) of the action potentials (spike release activity) of the neuron cluster. In time coding, neurons encode information by precise timing of action potentials (ie, spikes), whether in absolute time or relative time. The information can thus be encoded in a relative spike between a group of neurons. In contrast, rate coding involves encoding neural information in the excitation rate or cluster excitation rate.

If the neuron model can perform time coding, the neuron model can also perform rate coding (because the rate is exactly a function of timing or inter-peak spacing). In order to provide time coding, a good neuron model should have two elements: (1) the arrival time of the input affects the output time; and (2) the consistency detection can have a narrow time window. Connection delay provides a means of extending consistency detection to temporal mode decoding because the elements are allowed to achieve timing consistency by appropriately delaying the elements of the time pattern.

arrival time

In a good neuron model, the arrival time of the input should have an effect on the output time. Synaptic input - whether it is the Dirac △ function or the shaped post-synaptic potential (PSP), whether excitatory (EPSP) or inhibitory (IPSP) - has an arrival time (eg, the time of the delta function) Or the time of the start or peak of a step or other input function), the arrival time can be called loss Into the time. The neuron output (ie, the spike) has an occurrence time (regardless of where the arrival time is (eg at the cell body, at the point along the axon or at the end of the axon)), the arrival time can be called For output time. The output time can be the peak time of the spike, the beginning of the spike, or any other time associated with the output waveform. The universal principle is that the output time depends on the input time.

At first glance, it may seem that all neuron models follow this principle. But generally it is not the case. For example, rate-based models do not have this feature. Many spike models generally do not follow this. The Leaky Integral Excitation (LIF) model does not excite faster when there is extra input (beyond the threshold). Furthermore, models that may follow this point in the case of modeling with very high timing resolution will typically not follow this point when timing resolution is limited, such as limited to 1 ms steps.

Input

The input to the neuron model may include a Dirac delta function, such as an input in the form of a current or an input based on a conductivity. In the latter case, the contribution to the state of the neuron can be continuous or state dependent.

General principles of neuron models

Certain aspects of the present case provide general principles for designing useful spike neuron models. Useful neuron models are neuron models that are practicable and can be used to model rich, realistic, and biologically consistent behaviors. In addition, useful neuron models can be used to engineer and reverse engineer (interpret) neural circuits in the sense of defining a complete stable computational relationship.

Behavior from the event

Natural cells seem to exhibit a variety of behaviors, including tonic (tonic And phasic spikes and short pulses, integrating inputs, adapting to input, sub-threshold oscillations, resonance, springback, adaptive input characteristics, and more. Different behaviors can often be elicited in the same cell by inputs with different characteristics. In the present case, such actions may be considered from the point of view of the events leading to such acts.

Correspondingly, cells can be viewed as abstract event state machines, where things Cell dynamics between pieces are determined by machine state, and the transition between machine states is determined by events. In this context, the machine state is the meta state that governs the dynamic characteristics, not the model state variables themselves. Model state variables evolve based on dynamics in the meta state.

In this view, the event mobilizes cell dynamics until the next thing Pieces. The dynamics between the previous event and the next event are defined by the state of the machine, which is determined based on previous events. However, it is important to discern that this does not prevent the state from being determined at any time between events, but rather only prevents the dynamic definition from changing. This abstract event state machine concept allows for dynamic definitions to vary across machine states. Dynamics can thus be defined or parameterized in one way for a subset of machine states or states, and dynamically defined in another way for another subset of machine states. This means that the dynamics in different machine states or states can be defined independently. In the case, the behavior does not need to be different for all machine states, and it is not necessary for each event to result in a dynamic significant change or even any change.

Figure 5 illustrates an abstract event state machine in accordance with certain aspects of the present disclosure. Example 500 of the reading. The machine state is illustrated as circle 502, while within the machine state The dynamics of the model state variables are represented by arrows 504 within circle 502. The transition between machine states occurs at the event as indicated by arrow 506 between circles 502.

In the aspect, three consecutive events can be considered, the first and The last event includes an immediate input that significantly changes the state of the cell as it occurs. For simplicity, the events may be a Dirac Δ function added to a voltage state that represents some equivalent integrated current or conductivity-based input that affects the membrane potential of the cell. However, a transition event between the first and last input events can be considered to have no associated input or output. According to some aspects of the present case, a hypothetical neuron model can be considered, wherein the behavior of the model is defined to vary depending on whether the transition event is included or omitted, even if the event does not have an associated input or output.

In order to achieve a rich collection of behaviors that represent biological cells ( Behavioral repertoire) will require an event state machine that exhibits complex behavior. Moreover, a well-defined, well-behaved, low-complexity model is also a desirable goal. If the event itself occurs in any case where any contribution to the state variable (such as input, if any) can affect the state machine and change the dynamic constraints on the event, then the future state of the system is not only the previous state. , time, and input function, but state, time, input, and function of the event. Given more dependencies, you can more freely design individual model dependencies and dynamics in different meta-states or states. Thus, model elements can potentially be less complex and still produce an overall model that can achieve a richer collection of behaviors.

In addition, the model can be theoretically defined in the form of events. due to The machine is subject to event-based processing rather than continuous time processing, so the model can then be implemented to be theoretically defined rather than implemented by the approximation theory. The case provides a new harmonious approach that includes events as the basis of the model rather than treating the event as an inconvenient practice constraint.

Abstract principles expressed above may be defined by univariate or multivariate mathematically state S, S at time t ₀ and t _f between in the absence of any event in accordance with the previous state and some function of the elapsed time evolution. The state at any time t between the event at time t ₀ and the event at time t _f > t can be expressed as: S ( t ) = f ( t - t ₀ , S ( t ₀ )). (15)

By definition, a state has a closed form of expression as it evolves between events. However, if an event exists at the time t between two events, the state at time t _f> at t can _{be: S (t f) = f} (t f - t, f (t - t 0, S ( t ₀ ))) ≠ f ( t _f - t ₀ , S ( t ₀ )). (16)

The importance of this is that the dynamics of the model can be routed along different paths regardless of how or even if there is input, simply because of the existence of the event. FIG. 6 illustrates an example 600 of the particular situation, ie, instances 602, 604 of events that cause dynamic changes. In instance 602, there is no transition event between times t ₀ and t _f . In instance 604, there are transition events that are not input or output. However, the trajectory of state 606 may change at the transition event at time t .

A neuron model with multiple variables is not fundamentally incompatible with this concept. Counter-intuitively, dynamics can be defined independently for state variables. If the dynamics are decoupled, the question is what are the advantages of the multivariate state. . In an aspect, the multivariate state may provide an opportunity to transition to a new meta state based on the event, wherein the dynamics in the new meta state are determined based on the state at the time of the event. In other words, the multivariate state can provide another dimension for determining behavioral changes at the time of the event.

This event-based view can be provided for design with smaller calculations A framework of neuronal models with abundant behavioral capabilities. The framework can be the basis for further principles for shaping the design of spiked neuron models.

Consistency detection and plasticity

The ability of a spiked neuron model to detect fine temporal consistency may depend on memory characteristics. A leak-integrated-excited neuron model with exponential decay can detect temporal consistency to some extent depending on the rate of leakage (or decay), and the rate of leakage (or decay) determines how quickly the neuron forgets the previous input. Such models may be limited in other computational properties or biological behaviors that may be reproduced. However, while considering alternative or more complex models, it may be beneficial to retain the advantages of the leaky integrated excitation behavior, including the desired memory characteristics.

The basic exponential type with leakage integral excitation model can be defined as a state in which, with no input, the Δ t decays toward rest (the rest can be defined as 0 without loss) according to the following formula: v ( t + Δ t )= v ( t ) e ^{−Δ t / τ} , (17) where v is a state (eg, voltage) and τ is a time constant. The rate of decay is decelerated such that the difference in the decay curve for the two different starting states is greatest at the starting point and converges thereafter. Therefore, any change in the input amplitude at a particular time (the input amplitude will change state at that time) has a monotonically decreasing effect on the memory of the cell.

Figure 7 illustrates an example behavior 700 of a leaky integrated excitation model with exponential decay in two overlapping scenarios. For example, there are two input events where the input contribution is modeled as a Dirac △ function that is added to the state variable at the time of the corresponding event. The first input event 702 can occur at time t ₀ , thereby causing the status trace 704 to increase by one step. The second input 706 can occur at a later time, the second input 706 having sufficient input amplitude such that if the two inputs are added together, the state will be brought above a threshold 708 to cause the model to be issued peak. However, whether the second input occurs at time t ₁ or t ₁ 'affects whether the state rises above the threshold. If the second input event 706 occurs sufficiently early after the first input event 702, the threshold 708 may be exceeded. Otherwise, the memory of the first input 702 has decayed and the second input 706 may not be sufficient to trigger a spike. Thus, there is a means of detecting whether the first event 702 and the second event 706 are temporally close or identical.

Figure 8 illustrates the behavior 800 of two alternative models demonstrating a counterexample (which is not or difficult to detect consistency). The integral excitation model illustrated on chart 802 cannot detect temporal consistency because the model is not forgotten. The relatively more complex model illustrated on chart 804, such as a model with a quadratic expression on state derivatives, may not have sophisticated timing consistency detection capabilities at all. The quadratic expression on the state derivative is in the form of dv / dt = v ( v - v _threshold ) / τ , similar to the Izhievich simple model. Although the parameters of more complex models can be varied to reduce the effective time constant, the same applies to models with leakage integral excitation. The underlying dynamics (shapes) are preserved as a fundamental property, regardless of the time scale.

Figure 9 illustrates the situation where the input amplitude changes from value 902 to value 904. An example 900 of a monotonically decreasing difference in the state of the leaky integrated excitation model. As shown in Figure 9, the model forgets the difference between the inputs over time.

FIG. 10 illustrates a quadratic model based counterexample 1000 when the input magnitude changes from a value of 1002 to a value of 1004. The difference in state decay is non-monotonic for the changed input amplitude. As shown in Figure 10, the states in both cases first diverge, causing the difference to increase first and then decrease. Therefore, the model remembers a substantially larger difference (at the input time t ₀ ) before forgetting the difference between the inputs.

These memory behaviors are relevant in learning because the input amplitude is The learning period is usually variable due to the self-adjustment applied to the weight (intensity) of the input. Peak timing dependent plasticity typically involves adjusting the synapse (connection) weight (intensity) by an amount that depends on the time difference between the input event and the spike output event. In the case of monotonically decreasing memory behavior, weight adjustment has a stable influence in the sense that the influence decreases over time. However, in the case of non-monotonic memory behavior, weight adjustment will have an unstable effect in the sense that the weight adjustment depends on the latency between the input event and the future event may have a greater or lesser impact on future events.

Still, consider the broad expectations expected in a particular environment. Sexual detection capabilities. Rather than modifying the neuron model dynamics to fit such targets, an input or input filter can be defined to suit this environment. There are many ways to shape an input into a wider form. However, it will be difficult to cause the cells to detect a narrower consistency than is permitted by the dynamics of the damping. Time to passivate the neuron model The ability to detect is limited to the resolution of all inputs.

Excitatory or inhibitory postsynaptic potentials (input effects) can be modeled as a linear combination of indices:

In general, the difference between the two indices provides a fully realistic waveform . In an aspect, there may be one or more events associated with the input. For example, an input event can be considered to coincide with the peak or start of the waveform (or any other arbitrary reference point or points). Since the input itself can be opened in time, the time window in which the consistency detection can be performed can be controlled by input definition or filtering.

An important specificity is that the abstract event state machine concept is not The model is prevented from receiving a wider input than the Dirac △ function. Whether the input is defined in discrete or continuous form, the equivalent input can be applied to the model at the time of one or more input events.

Preserving fine timing consistency detection capabilities in neuron models Both narrow and wide consistency detection can be allowed, as supplied by the leak-integrated behavior. However, the time consistency detection capability provides a framework for decoding information encoded in time and outputting information encoded in the rate (spikes or no spikes). The basic band leakage integral excitation model is fired when the final element of the input arrives (or a fixed waiting time thereafter). Thus, the output time can be constrained by the final contribution input. This is why the output information is encoded in the rate rather than the difference in encoding in time. All other things being equal, the information that can be encoded in the spike timing mode is more than the information that can be encoded in the spike rate. The information encoded in time is more general. Although the spike rate is characteristic of the spike time mode, for any given spike There is no unique spike time mode for rate. Therefore, the spike timing mode yields a rate, but the reverse is not true.

Time calculation

Spike neuron models are often defined in the form of thresholds. Typically, when a threshold is exceeded, the model issues a spike (may have some wait time). The problem that arises is the latency-inspired concept is computationally useful.

The usefulness may be limited if a particular conditional cell is to be fired immediately or even with a specific waiting time, as a fixed delay can be achieved by other means, such as by axonal, synaptic or dendritic procedures. Nonetheless, there may be advantages if the relative delay between the event (enough to make the input of the machine state transition) and the spike event (output) is variable. In particular, if the relative delay depends on the input or event after (or before or before) the initiating event, there may be significant advantages. Such general computing power will indeed be useful, especially if the output timing is a well-defined function, because well-defined functions will not only provide a framework for engineering systems with spiked neurons, but also provide A framework for understanding what the spiked neural network is computing.

Specifically, if the delay of the output spike is a well-defined function of the relative delay from the reference event and any transition events (including inputs), the relative output time Δ t _k,r is defined as the relative input time Δ The function of the function of t _i,r : Where I _k is the input set to cell k . Essentially, postsynaptic spike may be considered in this form as compared to a reference time t spike relative time △ t _r of the encoded information. The reference time can be any or some previous spike time, such as the time of the first presynaptic spike or the last post-synaptic spike.

Encoding information presents certain challenges in time. Cells may have limited memory for input, and timing may only be consistent over a particular limited resolution. This means that it takes more time to code more information. However, the more important the information, the faster the information should be conveyed. Accordingly, certain aspects of the case support without loss of generality in the relative time information is treated as normalized x has range [0,1], wherein the means for communicating delay time of the information may be expressed as: △ t =- α log x , (20) The larger the value x (more significant information), the shorter the time delay (response). A value of 1 corresponds to an immediate response (no delay), while a value of 0 (no significance) corresponds to an infinite delay (no input or never a spike). Viewing both input and output timing in this context can reveal a dynamic in which the faster the input occurs or the more significant the input, the faster the output occurs.

Variable output wait relative to the last contributing input or throw event Time can be achieved with anti-leakage dynamics (i.e., where the state moves toward a spike condition or has a positive feedback). According to the aforementioned information time coding principle, it may be desirable to have dynamics where the earlier input has a large impact on the output time. This is different from the dynamics in which the feedback is negative (as in the case of leaky integration). However, instead of trying to combine the leak-integrated behavior with the anti-leakage integral behavior, the abstract event state machine concept provides a framework for defining multiple separate behavioral phase or meta-state sets, thus in different machine states. Dynamics can be defined or parameterized independently.

In a simple example, the behavior with leakage integral can be in a machine state The phase is defined in the phase, and the anti-leakage integral behavior can be defined in another machine state phase. These can be referred to as subthreshold and suprathreshold behavior. However, the term threshold may be misleading in the context of an abstract event state machine because the transition between machine states occurs based on the event rather than on the state condition itself. As discussed above, the state condition may be independent of the machine state transition unless there is an event.

It can be observed that the model with leakage integral excitation is insufficient. For example, a model with a leaky integral excitation is not sufficiently flexible. In an aspect, the input can be expressed as a linear combination of discrete input quantum, such as a Dirac △ function. Two input events can be considered to occur at times t ₀ and t ₁ , and the two input events alone are not sufficient to bring the leaked integral excitation model above the threshold. However, if the input amplitudes are summed together, the voltage change will bring the model above the threshold with the leaky integral excitation model. Figure 11 illustrates an example 1100 of voltage state dynamics over time for two situations, where the only difference is the time of the first input event, time t ₀ versus time t ₀ '. However, it should be noted that regardless of whether the first input occurs at t ₀ or at t ₀ ', the cells exceed the threshold 1102 at the same time t ₁ , thereby making it impossible to distinguish between the two input scenarios based on the output of the cell.

In another aspect, the input is not a discrete Dirac △ function. In abstraction In the event state machine, the input contributes to the machine state transition only at the time of the event. Cumulative or equivalent integral inputs can only be applied to make machine state transitions at the time of the event. Therefore, the same principle applies to the input amplitude as well as the input timing.

The basic exponential type anti-leakage integral excitation model can be defined in the sense of conventional modeling as having a higher than threshold value (which can be defined as 0 without loss of generality), without any input, according to the following formula Δ t The state of increasing toward the peak: v ( t + Δ t ) = v ( t ) e ^{Δ t / τ} . (twenty one)

The term "preferred" refers to a formulation that is not strictly based on the proposed abstract event state machine framework, which is intended to demonstrate the basic principles. This model can be considered as a demonstration to make the timing behavior of the reference value a priority. With this type of anti-leakage integration excitation model, timing differences can be distinguished because the time to peak can depend on the input timing.

Figure 12 illustrates an example comparison 1200 of two scenarios demonstrating this point in accordance with certain aspects of the present disclosure. In both cases, the state at time t ₀ and 1202 is above the threshold starts to rise. However, in the first case, the first input event 1204 occurs at time t ₁ and the rise toward the peak is accelerated. In the second case, the same input event 1206 occurs with a similar input contribution (amplitude) at a later time t ₁ '. However, the output time changes as a result because the earlier input accelerates the anti-leakage dynamics. If the input is substantially changed (increased) at time t ₁ ', the same output time can be achieved. However, this is not desirable. In this case, the input amplitudes will be different, not the same. If the magnitude of the input at time t ₁ changes, the trend from time t ₁ will also change. The goal here is the mechanism used to distinguish timing.

Related to this, the magnitude range of synaptic weights (connection strength) is usually constrained. Thus, without loss of generality, weights can be defined as having symbols (excitatory or inhibitory) and normalized ranges (eg, range [0, 1]). Moreover, the plasticity rules for learning can tend to lead to weighted bimodal clustering. (ie, close to 0 and close to 1).

Some aspects of the case are based on defining two states in phase The behavior is combined with the leaky integral excitation model and the anti-leakage integral excitation model to support the conceptual model in the spirit of the abstract event state machine. The term "threshold" can be used loosely to reflect the conditions applied when an event transitions from one phase to another (ie, a transition event that links states before and after the event).

Figure 13 illustrates two input timing diagrams 1302 and 1304 in the context of neuron model construction, in which both band leakage and anti-leakage behavior are exhibited. It is worth noting that the model construction can distinguish the input timing differences in any phase in the sense of output time. Figure 12 illustrates the differentiation of input timing in a positive (anti-leakage) phase, where Figure 13 illustrates the input to the negative (leakage) phase when the first input event time changes from t ₀ to t ₀ ' The distinction of timing. As a result, the state at time t ₁ of the second input event (i.e., plot 1302 in Figure 13 is different relative to plot 1304).

A neuron model based on the above concepts is provided in this case. Used to The dynamic and mathematical expressions of the computational functionality described above will be discussed in detail in subsequent sections regarding such models (including time calculations) defined in accordance with the principles described in this section.

An important question is whether the anti-leakage integration stimulating behavior is possible Present in biological cells. Biological cells have regenerative up-stroke dynamics that produce voltage spikes. The A channel is a fast-starting voltage gated transient potassium ion channel that counteracts rapid sodium influx and slows the regenerative upstroke. As a result, it is possible to achieve a very long waiting time (hundreds of milliseconds or longer) from the time when the upstroke is caused until the peak of the spike occurs. Used to trigger the A channel The voltage level can be slightly lower than the "threshold" used for the sodium channel chain reaction, so that the actual timing of events and inputs can also change the latency nature.

Practical theory

It should be noted that there is no ideal theoretical model as a baseline or measure by which all other models or all other models should be judged to be emulated. A model is a construct that its designer has designed for the purpose of its modeler. From a biological modeling point of view, it is desirable to be able to readily determine the parameters of a neuron model to match the behavior of a biological cell over a range of conditions, i.e., the abundance of behavioral capabilities is an advantage.

A good model will allow an example to be designed to represent a particular behavior, such as a biologically observed behavior. Moreover, whenever an attempt is made to design a behavior in a phase, a good model will not cause the dynamics in the other phase to deviate from the biologically expected behavior. Unless at the boundary of the inflection point, the behavior will generally not change drastically due to smaller numerical parameter changes. In addition, a good model will be theoretically defined and may be generally implemented as defined in theory (and facilitated to be generally efficiently implemented as defined in theory) without incurring numerical differences depending on the implementation. In other words, good models will behave the same in different implementations. In addition, a good model can reproduce the biological behavior of the design without having to use numerical artifacts rather than based on theoretically defined dynamics.

It is also expected that a well-defined practice model will provide a stable computational framework for designing cells and networks rather than designing search parameter spaces, and thus understanding existing networks in a computational sense. It is expected that the model implemented will be as generally defined in theory.

Model definition

The goal is a behaviorally rich, biologically consistent, and computationally convenient model of neurons that follows the above principles. Some aspects of the case support the design of a neuron model that achieves these principles. This general neuron model is unique in that the general neuron model is defined by the state at the time of the event and by the operation that governs the change of the state from one event to the next.

Research on large-scale biological spike neural networks has been driven by neuronal models that rely on rich representations of electrophysiological behavior and are mathematically manageable. Recently, a variety of nonlinear models have been shown to reproduce a substantial set of computational properties of characteristic neurons. However, while it has been suggested that a quadratic model may be the simplest model capable of reproducing such properties, adding a maintained subthreshold oscillation to the collection also requires another model. Moreover, due to the coupled nonlinear dynamics, solving such models may require numerical methods. Ensuring stability in the case of plasticity also constrains the modeling parameters. In this case, a new, simple linear peak neuron model was proposed in a different approach. The linear spike neuron model reproduces a rich variety of neural dynamics, including maintained sub-threshold oscillations and a set of behaviors exhibited by nonlinear models. The model has a bimodal phase dominated by subthreshold band leakage integral excitation dynamics and suprathreshold anti-leakage integral excitation dynamics along with linear coupling. The time constants and couplings in each phase can be independent, and thus can be related to physiological properties (eg, the behavior of voltage gate K ⁺ channels; transient A currents) on a per phase basis. Due to the linear nature, the spike timing dependent plasticity (STDP) monotonically changes the postsynaptic potential over time, resulting in time coding stability advantages. A closed form solution provides a future state given a delay and provides a delay for achieving a given future state (spike). The principle of instantaneous coupling can also be applied. This model is uniquely subject to theoretical analysis and event-based hardware analogy for large-scale biological spike networks with arbitrary fine timing. Linear on-threshold dynamics are utilized to indicate that the spike network can calculate general linear equations by encoding information in spike timing.

This case is based on the dynamic principle of neuron calculation to compile the minimum bimodal phase spiked neuron model, and demonstrates that the model can reproduce a rich variety of behaviors, including subliminal features such as maintained oscillations. The one- or two-element linear dynamics of the model can have two states. The time constant (and coupling) can depend on the phase of the phase. In the subliminal phase, the time constant (which is negative by convention) represents the leakage channel dynamics, which generally acts to return the cells to rest in a biologically consistent linear fashion. The time constant of the threshold phase state (as positive by convention) to reflect the anti-leak channel dynamic, dynamic general anti-leak channel drive spiking cells, and because the gate voltage is selected from K ⁺ channels and transient character A current spike is generated in the Inviting waiting time. The biological evidence is used to analyze linear subthreshold and suprathreshold dynamics and compare them to previous nonlinear models. Long-term enhancement (LTP) and long-term depression (LTD) were also analyzed for how to change the shape of the postsynaptic potential (PSP) in the context of time coding. In this new model, the PSP difference for a given change in synaptic strength is monotonically decreasing over time, revealing potential stability advantages during plasticity.

Discussing the engineering of the cusp neural network using this new model And a unique approach to reverse engineering (interpretation). A method of engineering a rate-based network has been proposed. However, regardless of whether or not a spike is issued, such a network Information in the road can be encoded in the excitation rate. Moreover, a wide variety of tuning curves and filters are often required. The result of this model is a closed-form solution for both future states and peak wait times. Unusually, if the relative spike timing is considered to encode a negative logarithmic information value, then in the linear threshold upper phase of the model, the information in the post-synaptic spike wait time can become relative to the pre-synaptic spike timing. A linear function of information. This means that the linear system of equations can actually be calculated instantaneously by the cusp neural network. This is possible even without precise synaptic weights, since the coefficients of the linear system of equations are alternatively represented by synaptic delays. Positive and negative coefficients can be translated into excitement and inhibition, respectively. Conveniently, larger coefficients can be translated into shorter delays, so that a significant response naturally occurs faster (spikes). Accuracy may depend on timing resolution, and the linear model is particularly subject to large-scale event-based modeling of spiked neural networks with arbitrarily fine temporal resolution.

Model dynamics

According to some aspects of the case, the model is designed in the form of an event, where the event is the basis of the defined behavior. The behavior may depend on the event, and the inputs and outputs may occur at the event and the dynamics are coupled at the event.

The dynamics of the neuron model can be divided into two states called the negative phase (also known as the leaky integrated excited state phase) and the normal phase (also known as the anti-leakage integrated excited state phase). Formulating dynamics in the form of events and dividing the dynamics into these two states is the fundamental property of the presented neuron model. In the negative phase, the state generally tends to the target state at a time of future events. In the negative phase, the neuron model can exhibit temporal input detection properties and other Sub-threshold behavior and other more complex dynamics. In the normal phase, the state generally tends to deviate from the anti-target condition. In this normal phase, the neuron model can exhibit computational properties, such as waiting for spikes and other more complex dynamics depending on subsequent input events.

The symbol ρ will be used to indicate the dynamic phase in this case. When discussing or expressing the relationship of the specific phase, the symbol ρ is replaced by the symbol "-" or "+" for the negative phase and the normal phase, respectively. The multivariate model state includes a membrane potential (voltage) v and an abstracted recovery current u . In the basic form, the phase ρ is essentially determined by this state at the time of the event. The neuron model is defined as above a threshold if the previous event at the voltage v v> , then in the normal phase at the next event, otherwise the neuron model is defined as being in the negative phase. Currently, typical settings can be considered, where Is set equal to the constant v ₊ .

The dynamics of the model state are conveniently described in the dynamic form of the transformed state versus { v ', u '}. The state at the time of the event is transformed into: v '= v + q _ρ , (22)

u '= u + r , (23) where q _ρ and r are linear transformation variables. The voltage transformation can depend on the phase ρ of the state. The model dynamics, which also depend on the phase of the phase, are defined by the differential equation from the form of the transformed state pair: As mentioned above, τ _- is the negative phase voltage time constant, τ ₊ is the normal phase voltage time constant, and τ _u is the recovery current time constant. These are constant values, but variable time constants are also possible. For convenience, the negative phase time constant τ _- can be specified as a negative amount so that voltage dynamics can be expressed in the same form for all states. The indices τ ₊ and τ _u are generally positive.

The state dynamics of this neuron model are defined in the event framework. Between events, these dynamics are defined by the above common decoupled difference equations. The dynamics of these two state elements are only coupled at the event by a transformation that deviates the state from its zero-tilt at the time of the event, where the transformation variables are: q _ρ =- τ _ρ βu - v _ρ ', (26)

r = δ ( v + ε ), (27) where δ is the coupling conductivity-time, ε is the coupling offset voltage, and β is the coupling resistance. A more general formulation with β _ρ can also be used (where the transformations in the two states and the time constant are completely independent). Typically, the phase state beta] across all common parameters, but with a β _ρ [rho] may be arranged separately for each state. Similarly, τ _u is usually shared across states. The two values of v _ρ ' are usually the base voltage or reference voltage of the two states, that is, v _ρ '= v _ρ . As previously mentioned, the parameter v - is the base voltage of the negative phase, and the membrane potential will generally tend to v - in the negative phase. As previously mentioned, the parameter v ₊ is the base voltage of the normal phase, and the membrane potential will generally tend to deviate from v ₊ in the normal phase. The parameter ε can usually be set to -v _- .

Given the state at time t {v ', u'}, that the model at time t + △ t evolutionary state has a closed form solution:

Thus, the model state is only needed and is defined to be updated based on the event, such as based on input (pre-synaptic spikes) or output (post-synaptic spikes). This can be generalized because operations can also be performed at the time of a human event (whether there is input or output). By definition, a transformation is defined as being at the event, not between events. This means that q _ρ and r and even ρ should not be recalculated unless there is an event.

Thus, the model is only coupled at the event and the phase (whether positive or negative) is only determined at the event. The model state variables v and u are generally coupled via the variables q _ρ and r as described above, and the variables q _ρ and r are determined only at the event (or step size). The variables q _ρ and r are calculated based on the previous state. These state elements then evolve independently to the next event. In effect, this means that the state variables are "decoupled instantaneously" between events.

Figure 14 illustrates an instant coupling at an event in accordance with certain aspects of the present disclosure. An example 1400 of decoupling and decoupling between events. As illustrated in FIG. 14, pre-synaptic (input) event 1402 is followed by another pre-synaptic event 1404 and post-synaptic (output) event 1406. The state can be evolved from event to event ground (ie, from event 1402 to event 1404 and to event 1406) using decoupled dynamics.

Figure 15 illustrates an example 1500 of dynamic comparisons in a normal phase and a negative phase in accordance with certain aspects of the present disclosure. Figure 15 illustrates that the behavior of the model's voltage (y-axis) between two events depends on the inter-event interval (x-axis). The parameters τ _- = -14.3ms, τ ₊ = 2.6ms, τ _u = 33.3ms, v _- = -60mV, v ₊ = -40mV, ε = - v _- , β = 0.01 and δ = 2 approximately reflect The setting of new cerebral cortical pyramidal cells. The voltage offset is ε ₊ = ε _- =1mV. The initial voltage is the voltage at the first event.

In the first case 1502, the state at the time of the first event is v = v ₊ + ε ₊ and u =0, where ε ₊ is a small positive voltage offset. Assuming a typical setup, the model will be in the normal phase due to v > v ₊ . As a result of the transformation variable q ₊ = - v ₊ , the voltage will increase away from v ₊ . In the second scenario 1504, the state at the time of the first event is v = v ₊ - ε _- and u =0, where ε _- is a small positive voltage offset. As a result of the transformation variable q _- = - v _- , the voltage will decrease towards v _- .

The model is defined as a threshold spiking v _S (i.e., generating an output event) at the voltage v reaches. The state is then typically reset at the retargeting event (the retargeting event can technically be identical to the spike output event):

u = u + △ u , (31) where Δ u is the recovery current re-voltage. Re-voltage Can usually be set to v _- .

In addition, the time of the post-synaptic spike can be predicted so that the time to reach a particular state can be determined in advance in a closed form. Given the previous voltage state v ₀ , the time delay from (from voltage state v ₀ ) until the voltage state v _{f is} reached is provided by:

If the spike is defined as occurring when the voltage state v reaches the value v _S (spike voltage), then the closed-form solution of the amount of time or relative delay measured from the voltage at the given state v until the occurrence of the spike for: among them Is the phase threshold and is usually set to v ₊ . It should be noted that the phase threshold is not zero tilt.

The above definition of model dynamics depends on whether the model is in the normal or negative phase. As mentioned earlier, the coupling and phase ρ are calculated based on the event. For the purpose of state propagation, the phase and coupling (transform) variables are defined based on the state at the time of the previous (previous) event. However, once the state propagates to the current event, the predicted, phase, and coupled variables for the peak output time (next event) are defined based on the state at the time of the current event after the input is included.

For a event or step - COUPLED updated event, t from the event time to the event time t + △ t state evolution may be as described above in formula (11), (12), (28) and (29) in the general definition of . The step-based update can optionally be configured to be equivalent to an event/step-time event (by updating q and r based only on events), or by defining an event/time at each step (calculating q and r) ) to configure differently. The difference is usually negligible. In order to make the calculations efficient, it may be preferable to implement an event update (or step-to-event update if limited to a step-based simulator). In such cases, the above closed form equations (14) and (32) can be used to predict spike events. In the update step, △ t is a constant (simulation step) and, accordingly, the formula (26) - (27) into equation (28) - (29):

Thus, converting an event update or step-event update implementation to a step-update form can include: (i) removing event-based update conditions (updated at each step) and removing predictions of spikes, and ( optionally simplified equation for the case of ii) described above in a given △ t is fixed.

Current and conductivity models can be similarly defined and calculated from event to event and can be incorporated into spike predictions. Alternatively, the input model can be computed in an implementation other than a neuron model (eg, a step-updated conductivity model with a step-event-updated neuron model).

Another property of this linear model is the monotonic difference in memory. This difference in the procedure of the membrane potential decreases over time, so for example in the negative (LIF) domain (ignoring or eliminating the second state for clarity), the difference monotonically decreases: Thus, the effect of peak timing-dependent plasticity is stable over time: changes in synaptic weight due to peak timing-dependent plasticity have a diminishing effect on subsequent behavior of cells over time.

Another property of this linear model is that the relative spike latency can be expressed as a function of the relative input spike timing, which is derived from equations (14) and (32). For the simplification of the conceptual demonstration, consider using a heavy-side with a one-dimensional state β =0 (or no u ) with a threshold at v _ρ '=0 (ie q =0). Simple Dirac current spike input for function θ (weight 1 or 0): Where τ _i is the time from the input spike of presynaptic neuron i , and Δ τ _i is the connection delay from presynaptic neuron i . Therefore, the time from the increment to the voltage film state v ₀ to the spike is:

definition , providing a near linear relationship: among them And Δ w = v ₀ / v _S . For sufficiently small Δ w , the relationship is approximately linear. Therefore, the logarithmic information (the negative logarithm of x _i ) encoded in the relative peak timing τ _i and the connection delay Δ τ _i similarly related to the coefficient w _{i in the} linear domain can be considered: τ _i =-log _b x _i ; Δ τ _i = -log _b ( w _i v _S ). (40)

Thus, the neuron spike latency as a function of the input spike timing is equivalent to a linear system of equations in the negative log domain. The time at which the voltage state reaches v ₀ is used above as a reference for the input and output spike times, but any time reference can be used for either. The accuracy of the calculation of the form of the values in the linear domain will depend on the temporal resolution in the neural spike timing domain.

Input and plasticity

Inputs can be applied to the model only at the event. In a typical formulation, the input can be applied to the model state after the state has progressed from the previous event to the input event. Thus, more generally: Where h _v and h _u are input channel functions. In a simple case, the instantaneous current input can be modeled as a discrete Dirac Δ function applied to the voltage state, ie, i = θ ( t ). In this case, the input to the state of the model is applied at the time of the event, and: h _v ( x , i ) = x + iβ ; h _u ( x , i ) = x , (43) where β is the film resistance.

However, the input is alternatively continuous, such as describing excitability or The sum of the weighted exponential decay of the inhibitory postsynaptic potential, whether current-based or conductivity-based. In this case, the input to the state of the model applies at the time of the event, but is also potentially applicable before or after the event. Thus, the equivalent total integration input contribution from the input to the next event and the input from the past event, as accumulated between the previous event and the next event, may be applied at the time of the next event. Therefore, a closed form solution for continuous input contributions is also an advantage, but is not required.

A continuous excitatory or inhibitory input that decreases exponentially can be defined by: It has a closed form solution and can thus evolve from event to event decoupled from the { v , u } state. From time t to time t + previous event on the total contribution of the next event time period △ t is provided by the following formula: Wherein for the inhibitory contribution g < 0, and for the excitability contribution g < 0, and the same input channel function can be used for the Dirac input. Input and state evolution are decoupled, and this can be compensated for at the time of the event, but is usually unnecessary.

Conductivity-based inputs can also be integrated: Where E _g is the reference voltage for the input and V is the voltage level calculated to compensate for the removal of the ( v - E _g ) term from the integration. The approximation of V = v ( t ) (the voltage at the previous event) is fully true if v varies relatively small events across events or events occur at smaller inter-event intervals. If this is not the case, the more complex formulation of V or the use of artificial events discussed below can be used to achieve the desired effect while adhering to the definition of the model.

By definition, the model does not consider events in peak time estimates. The future input between. The general reason for this is due to the fact that it is expected to have a closed form. Even if a closed form solution may be available for some continuous input formulation, if the rate of the event is sufficiently high, an equivalent effect can be achieved by alternatively defining the input or channel function and otherwise by using an artificial event.

By definition, all forms of plasticity are applied at the event, Just like everything else in the model. Peak timing dependent plasticity is particularly suitable for this because long-term enhancement and long-term suppression can be seen as triggered by pre-synaptic (input) events before or after a post-synaptic (output) event, respectively. Any form of structural plasticity will also be applied at the event as defined. Structural plasticity can be thought of as, for example, modifying, establishing, or deleting synaptic connections. Equivalently, the parameters of abstract synapses, such as delays and weights, can generally be changed as the abstract synapses used to model the deleted synapses are reused to model new synapses with different parameters. Thus, multiple forms of plasticity can be generalized in the form of variable synaptic parameters. This variability can be discrete or continuous, but will evolve from event to event in the model anyway.

Algorithmic solution

The algorithmic solution of the model may include: (i) making the state progress from the previous event to the next event; (ii) giving the input of the next event time (or at the next The state at the next event time is updated when the event applies an input equivalent to the accumulated input between the previous event and the next event; and (iii) when the next event is expected to occur. An event can include an input event, which is generally considered to occur when the synaptic input propagates to the cell body of the neuron. Events can also include output events, which are generally thought to occur when spikes are emitted by the cell body of the neuron and begin to propagate along the axon. Since the closed form solution is available, the model state can also be determined between events if desired, but unless there is an event, the phase and coupling are not updated.

The following algorithms describe the operations that are typically based on events. status It can progress depending on the event. The input can then be progressed and the input applied (if the event or interval between events has an input) and the next event (output) is expected. The operations required to make a state progress may vary depending on whether the event is an output event or another event (for example, an input or a human event).

If the state of progress based on non-output events, you can first decide from time t from the previous event time △ t. The phase can be determined based on the previous state { v , u } at the previous event. Given the phase and previous states, it is possible to decide to transform q _ρ and r . {V ', u'} can be determined by the previous state transition of the event. Advances in the state may be transformed at time t + △ t at the decision. Advances in the state {v, u} may be t + △ determined at time t. If the state progresses based on an output event, the state can be re-determined.

Input progress and application input are only required if there is input during the event or during the time between the previous event and the next event. For input progress and application input, it may be necessary to first adapt the plastic input parameters. Progress of the input time can be determined at t + △ t. Any new input can be included at time t (if any new input events exist). It is possible to determine the equivalent event interval integral input contribution i . The new state { v , u } can be determined by applying the equivalent input i .

If there is any chance that the next output event time has changed, then the time of the next output event should be expected. To predict the next output event, the phase can be determined first based on the new state { v , u }. The updated transform variable q _ρ can be determined and the relative delay Δ t _S up to the next output spike event can be predicted. Whether the event is an input event or an output event or an artificial event, the prediction of the next output spike is generally applicable, as any state update can change the expected time for the spike.

Artificial event

By definition, the model is only updated based on events. A human event represents an event defined to define the dynamic behavior of the model, not the input or output event itself. There are several reasons why a modeler may wish to define an artificial event.

Modelers often want to see traces of voltage and current states with fine time resolution. Their states can be calculated periodically between events without the need to define artificial events. This would require the use of the transformed variables and phase calculated at the previous event (rather than the previous cycle) to calculate the voltage and current. This means that at any time between events, the state can be determined, but there is no coupling, so the state at the time of the next event does not change, whether or not the state is calculated between events.

Figure 16 illustrates this concept, that is, an example 1600 illustrating state dynamics between events that are independent of inter-event updates. As illustrated in Figure 16 Presynaptic (input) event 1602 followed by another presynaptic event 1604 And post-synaptic (output) event 1606. The state may evolve from event to event (ie, from event 1602 to event 1604 to event 1606) using decoupled dynamics independently of the inter-event update (ie, the update at inter-event time 1608).

By definition, a coupled transformation is defined as being at the event, not between events. Transform variables and states are only calculated at the event. This means that q _ρ and r and even ρ should not be recalculated unless there is an event. Technically, a human event can be defined as occurring at the voltage crossover phase threshold, which is a time that can be calculated in advance due to the decoupling between the closed form expression and the events. Thus, assuming that a human event has not been defined, the voltage and current states can be updated between events as desired, but the voltage and current state updates are based on parameters from previous events and offsets from previous events. In other words, by definition, coupling should only occur at the event.

Defining artifacts allows the modeler to change the coupling. May also exist Define the cause of the artificial event at a convenient time. There is nothing wrong with defining or not defining human events, but the modeler should understand that defining artificial events generally changes the behavior of the model.

One reason to define a human event is to use a lower event rate Model instances to achieve behavioral characteristics of model instances with high event rates. If there is a high input event rate, the model dynamics can progress at smaller time intervals. However, if there is a low input event rate, the model dynamics may progress at larger time intervals without human events. In general, unless the time interval is much larger, the difference in behavior may not be significant. even so, Adjustments to parameters such as time constants can also be made to compensate.

However, as an alternative, an artificial event can be defined. in particular If the interval between non-artificial events is large, the artificial event can be defined as occurring between non-artificial events. Technically, this can be done in several ways. For example, an artificial event can be scheduled experimentally with a certain delay after each non-artificial event. If another non-artificial event occurs before the artificial event, the artificial event is rescheduled with a delay after the latest non-artificial event. If the human event does not occur first, the human event is rescheduled again with the same delay. Alternatively, human events may be scheduled periodically. Artificial events can also be defined to occur conditionally, such as depending on the state or spike rate or at a time calculated in advance.

Artificial events for nets with sufficient fan-in or fan-out connectivity Achieving desired behavior on the road is often unnecessary because there is a large amount of input for each cell, and each input contributes an input event to the recipient cell. Therefore, if a specific behavior of a periodic event is desired, it is more than enough for the input event itself to satisfy the role.

Figure 17 illustrates the concept of a human event, that is, illustrated although not There are instances 1700 of inputs or outputs that also affect dynamic human events 1702 and 1704. Artificial events can only change the coupling of the model. Thus, at any inter-event time, such as inter-event time 1706, the state may evolve decoupled from the previous event. Again, regardless of whether the state is determined during the inter-event time, the state at the next event will be the same, as illustrated in FIG.

Figure 18 compares the voltage evolution between two events with no artifacts in between (plot 1806, 1808) and the two events between periodic events with periodic artifacts (plots 1810, 1812). For example, the human events in Figure 18 can occur at 5 ms intervals. In Figure 1802, it can be observed that the differences are small and thus difficult to distinguish. In chart 1804, three changes have occurred to increase the difference. First, the voltage at the first event has been moved very close to v ₊ as this will increase the time delay to a particular voltage level and this will increase the difference. Second, the value of u at the first event is significantly reduced. Third, artificial events occur at shorter intervals. The difference is still very small but may be significant depending on the desired behavior and the density (or rate) of non-artificial events. It can also be observed from these behaviors that changes in the time constant will compensate for these differences.

In general, as described above, the model is suitable for event-based Solve in the simulation. However, the model can also be solved in conventional step-based simulations. In any case, there are basically two ways to do this: (i) no human events; and (ii) artificial events. By definition, there are different instances of the model. Model operations are defined as where an event occurs, whether it is an artificial or other event. Thus, in the absence of a human event, the code executed at each step should be conditional on the occurrence of the event at that time slot (actually, no operation can occur at the step without the event). Alternatively, in the case of a human event, a human event can be defined to occur at each time slot. Since closed-form solutions are available, no numerical methods are needed, regardless of whether the time between events is constant or variable. Another question is whether the time of the event is quantified. Typically, in a step-based simulation, time is quantized as a step. Therefore, these instances will be different (whether significant or not) based on event-based formulation, whether or not artificial events are used.

While periodic artificial events will generally require more computation and therefore will generally be less desirable, there are some potential simplifications. For example, the time Δ t since the previous event may be a constant (time interval). For this reason, the transformed state update can be simplified to a single multiplication for each state element. The infinite impulse response filter is then provided by: v '( t + Δ t ) = c _ρ v '( t ), (47) u '( t + Δ t ) = c _u u '( t ), ( 48) where the constant is defined as with .

Another simplification is expected to peak by checking whether or not at each interval v To replace. Spike conditions are defined as v v _S . However, if the event is defined as occurring artificial quantization of the time interval △ t, the condition may actually reach the peak between the various intervals. As a result, it should be ensured that the spike occurs at the desired time, whether it occurs at the previous event or at a later event.

Basic behavioral control

The basic model behavior is controlled by the aforementioned parameters τ ₊ , τ _- , τ _u , v ₊ , v _- , v _S , β , δ and Δ u . The time constant controls the rate at which the voltage or current is directed toward or away from the zero-thrust decay, and the zero-tilt intersects the current axis at the base voltage of the phase. The coupling at the event is determined by the slope of the transformation equation. For voltage conversion, the slope is provided by the parameters τ ₊ and β , and for current conversion, the slope is provided by the parameter δ . This means that the slope of the zero-thaw line in the normal phase boundary depends on the normal phase time constant, but this can be compensated by β .

In standard parameterization, only one coupling resistance parameter β can be utilized, and the coupling resistance parameter β is common to both phases. However, β _ρ can also be configured separately for each state. By the above derived parameters , Separate control from ε can achieve additional behavioral aspects. However, in general, the preset value can be used based on the above basic parameters. For example, the parameter ε is usually set to -v _- .

A typical setting for the phase threshold provided above is . However, the voltage zero tilt is not a vertical line in the { v , u } state space. This feature may be advantageous because it allows for richer behavior. Additionally, alternatively, the phase threshold may be defined as a zero-tilt that borders the normal.

Given the state transition, the zero inclinations of v and u are provided by the negative of the transformation variables q _ρ and r , respectively: v =- q _ρ and u =- r .

v = τ _ρ βu + v _ρ or u =( v - v _ρ )/ τ _ρ β , (49)

u =- δ ( v + ε ) or v = - u / δ - ε . (50)

The parameter δ is a scaling factor that controls the slope of the u- zero tilt, which may be positive (for δ <0) or negative (for δ >0). Assuming a preset setting, the u zero tilt line crosses the x-axis at v _- . The parameter β is the resistance value that controls the slope of the v- zero tilt in the two states. The τ _ρ time constant parameter not only controls the exponential decay, but also controls the zero tilt slope in each phase separately.

Figure 19 illustrates two examples 1902 and 1904 of zero tilt of voltage and current for negative and positive u zero tilt slopes. Other situations may occur depending on the parameters, but such examples are exemplary. Attention should be paid to the discontinuity at the phase zero threshold 1906 at the phase threshold ( v ₊ in this exemplary case). It should be recalled that in the normal phase v > v ₊ , the voltage tends to deviate from the zero inclination - q ₊ (plot 1908), while in the negative phase, the voltage tends to zero the inclination - q _- (plot 1910 ). In these two states, if τ _u >0, the current tends to zero inclination -r (plot 1912, 1914).

In the aspect of the case, the transformations control the temporal behavior of the model. The voltage transformation offset variable q _{ρ is} defined by a linear equation that depends on the recovery current. When the current is zero, the transformation is entirely due to the offset v _ρ . For this reason, the voltage state can be translated in the transformed state such that the base voltage state is zero. Revisiting the formula used to estimate the peak time, we can observe that the logarithm at u =0 is:

Thus, the transformation of state x shifts and normalizes the state model to produce a time delay between 0 and 1 in the normal phase. The information in the spike is in the form of its relative timing. From an information theory point of view, the normalization time can be thought of as encoding the information value (or state) with the range [0, 1] as: So that the larger the value, the shorter the time delay (response), and the value 0 corresponds to an infinite delay (never a spike is issued). Therefore, the parameterization of the model allows control of the information representation in the peak timing. This aspect is particularly advantageous for computational design purposes.

Finally, it should be noted that the coupling to the current state may be reduced or eliminated. This coupling is controlled by the parameter β , so if β = 0, no u variable is needed. Similarly, if the u variable is not dynamic, ie Δ u =0 and τ _u → ∞, no u variable is needed. The coupling from the voltage is dominated by the parameter δ , where a smaller or δ of 0 can also reduce or eliminate the effect of the voltage state on the current state. If due to the small dynamic coupled β, △ u _u [tau] or greater without significant, variable u may be omitted. However, u variables provide a richer opportunity for model dynamics.

Cold neuron-dynamic event model

Fundamentally important biological neurobehavioral behaviors may not be simulated or predicted with typical spike neuron models because they do not capture: (i) fine (continuous) timing, or (ii) continuous time dynamics . Even in a row Models expressed in the form of time differential equations also do not have closed-form solutions, and therefore tend to be approximated by repeated operations, such as using the Euler method. The problem of repeating the computational model is to observe how the spike timing may change dramatically (by, for example, tens of milliseconds or more) by simply changing the time step resolution and changing only a small amount (eg, from 1 ms to 0.1 ms). More) The time is already obvious. Although attempting to approximate such a model with a fine time step may be computationally cumbersome, it is more important that a fine time step generally does not account for continuous time dynamics, especially if the model has multiple interdependencies. State variables (such as voltage and current) and multiple attractors. Therefore, the goal is a continuous-time dynamic neuron model that can capture biologically realistic temporal effects.

As described above, the dynamic event neuron model has a state defined by variables { v , u } representing the membrane potential (voltage) and the recovery current, respectively. Given no further input, the following conditional prediction rules can be used to anticipate a closed event (to any desired time accuracy) based on the current state: Where q _C =1/ α _C (- βu _C + γ _C ), (54) and β , v _peak , a ₊ , γ ₊ are the parameters described above. This rule is actually an anti-leakage integral excitation (ALIF) model. The value u _C is a conditional parameter and can be constant or set to a function of u . When a spike occurs, the state is updated as follows: v = v _{pos t} and u = u + u _post . A more complicated version involves replacing the condition v > v _i with a condition ( v - v _t )( v - v _r ) > u / k which also depends on the recovery current u , where v _t and v _r are respectively depolarized and Resting potential.

The future state of the model can be determined using independent conditional update rules. These rules include the independent propagation of state variables:

u ( t +Δ t )=( u ( t )+ r ) e ^{- at} - r , (56) where r = δ ( v _- + ε ), (57) and δ , ε , a are parameters. Again, these are continuous time closed form equations that can be calculated to any desired time accuracy. The value v _- is a conditional parameter and can be a constant or a function of v . If v > v _t , the factor c = +, otherwise c = -. In the latter case, the calculation of q _- requires the parameters α _- , γ _- . At c = +, the neurons are in the depolarized region or in the ALIF mode. When c = -, the neuron is in LIF mode, returning to a resting state unless an input is supplied.

When there is an input, the state should be updated to the time of the input event by using state propagation and then updated to account for the input v = v + i . State variables should also generally be bounded.

Algorithm

The event-based algorithm used to calculate the model involves processing two events: a synaptic input event and an expected spike event.

At the time of the synaptic input event, the following steps can be performed. It may be determined since the last status update from time △ t. If the voltage condition is met, let c = +, otherwise let c = -, formula (54) can be used to calculate q _C , and equation (55) can be used to update the voltage. Subsequently, equation (57) can be used to calculate r , and equation (56) can be used to update the recovery current. Thereafter, the input can be added to the voltage. If the voltage condition is met, let c = +, otherwise let c = -, use equation (54) to recalculate q _C , and use equation (53) to recalculate the estimated spike time. Finally, the peak event can be rescheduled.

It should be noted that in order to recalculate q _C , the amount i to be added to the potential may be different from the discrete time simulation operation because there is no time step. The i value that will be used for the discrete iteration simulation can be scaled by the time step to obtain an equivalent value for event-based modeling. Optionally, if there is no input, the algorithm steps can be performed at least every predefined period T.

The following steps can be performed as the peak event is expected. The status can be reset or updated (optional / as applicable). If the voltage condition is satisfied, so c = +, else let c = -, q _C may be recalculated using the formula (54), and may use the formula (53) to recalculate the expected spike time. Thereafter, the peak event can be rescheduled.

In order to configure the model parameters, consider the dynamic neuron parameters described by Izhikevich: membrane capacitance C , threshold potential , Resting potential v _r, spike peak level v _peak, depending on neuronal factor k base strength and the input resistance, and B, the re-voltage c after the spike, the recovery time constant a and the net current d during a spike.

For the dynamic event model, set: α _C = k (△ v _C ) / C , (58)

β =1/ C , (59)

γ _C =- v _C α _C . (60)

It should be noted that γ _C depends on α _C and v _C , where α _C depends on Δ v _C . The choice of Δ v _C may depend on the condition, ie, whether it is c = + or c = -. In general, if c = +, then: otherwise:

However, it is not necessary to use the current value = v to calculate the above parameters. As an alternative, for simplicity A constant value is used, ie for c = +, the mean of v _peak and v _t can be used. For c = -, for example, the mean of v _r and v _t can be used.

The dynamic event model is a true continuous time model because there is no Dependence on time step or time resolution. On the other hand, accurately calculating the Izhikevich simple model in continuous time is problematic because it is impossible to find closed-form solutions of two interdependent differential equations. A typical implementation of the Izhikevich simple model is based on discrete inverse arithmetic simulations (or Euler numerical methods). Although quantitative access to the data sheet can be used by definition, this is not continuous time.

In an aspect, the dynamic event model can be configured to produce a theoretically continuous time simple model of behavior. Instant dynamics can be reviewed in continuous time. Equation (53) can be rearranged into the following form: If you use the conversion of voltage v '= v + q ₊ , then:

Taking the Laplace transform for the time t = Δ t _spike , manipulating and taking the Laplace inverse transform produces: Replace v ' and then replace q _{+ to} provide: Further substitution parameters provide a differential equation for the potential element according to the Izhikevich simple model (assuming no input):

Therefore, in a timely manner, continuous time behavior can be achieved. It can be shown that the same situation is true for the potential at c = - and for the recovery current. Figure 20 illustrates the similarity of the peak results of the repeated operation simple model and the dynamic event model, that is, the spike result 2002 relative to the spike result 2004.

In a dynamic model, the voltage state is only present in events (inputs or pointers) The peaks are updated, and the repeated simple model has updates every millisecond. Both experience the same exact spike input. This instance is executed without a minimum update time. By using the minimum update time, the similarity can be increased even more, and the shorter the update time, the more similar.

Important, although the same or similar behavior (results) can be achieved It is understood that these models are completely different. In the dynamic event model, state variables are independent and the behavior is conditional. In essence, a simple model cannot be solved (executed) in continuous time. On the other hand, the dynamic event model is a fully continuous time model.

Model dynamics

The dynamics of the model will be explained in this case in a way that enables a good understanding of the behavior. The richness of the model's behavioral capabilities stems from event-based formulation, coupling at events, and decoupling between events and moving The state is divided into states. This generalized model is unique in these aspects and the resulting behavioral aspects reflect this.

Resting track

21 illustrates an example 2100 of a state trajectory across a zero-tilt in a negative phase of the model, in accordance with certain aspects of the present disclosure. At any point along the voltage zero-tilt 2102, the current state variable evolution can be non-zero and thus the state can be dragged away from the voltage zero-tilt 2102. Likewise, at any point along the current zero-tilt 2104, the voltage state variable evolution can be non-zero and thus the state can be dragged away from the current zero-tilt 2104. Although the state at the zero-tilt has a zero derivative for a state variable corresponding to a zero-tilt, the derivative of the other state variable at the other points may be non-zero. As a result, the oscillation phase is possible in the negative state in which the state of about <v _-, 0> state rotation.

The immediate state trajectory is provided by the difference equation. Instead of transforming variables, the state trajectory in the negative phase is: The state position vector relative to the resting origin < v _- , 0 > is: The angle between the state trajectory and the position vector is given by: Figure 22 illustrates an example 2200 of state trajectory geometry provided by equations (68), (69), and (70).

The projection of the state trajectory on the position vector provides a component of the trajectory towards rest, or a decay component. The dot product produces this component scaled by the magnitude of the position vector:

The left term in equation (71) is negative indicating the direction toward the resting origin. A simple conceptual case for visualization is if τ _u = - τ _- . Then, the term in the left square bracket in equation (71) is equal to . However, since the left side term is scaled by the time constant 1 / τ _- , it can be observed that the decay towards the origin is possible. FIG. 23 illustrates an example 2300 of a state trajectory 2302 around a resting origin in a state space. The parameters are v _- = -70mV, τ _- = -20ms, τ _u = 20ms, β = 5, δ = -0.26, ε = - v _- , and the parameters have an initial state <-75 mV , 0>, at every 0.1 In the case of ms, there is an event of 1000 ms plotted. Since the interval between events is small, this behavior approximates the behavior provided by the continuous time state derivative.

In order to avoid fading towards the origin, the second term on the right side of equation (71) may need to be positive. In order to achieve a positive u- zero tilt slope, it is known that δ should be negative. However, as long as β > - δ / τ _u , the component β + δ / τ _u is positive. However, u ( v - v _- ) is negative only if one of u or ( v - v _- ) is negative (with respect to the second or fourth quadrant of the resting origin). Therefore, in order to counter the decline towards the origin, there are several possibilities: (i) increasing the magnitude of τ _- ; (ii) increasing β + δ / τ _u ; (iii) moving the orbit inward toward the resting origin Move, whereby the term ( v - v _- ) ² + u ² can be less than u ( v - v _- ); and (iv) change the interval between events.

Therefore, it is possible to not only reduce the recession so that the orbit does not degenerate, but also establish an oscillation that circulates outward and emits a spike. For example, adjusting δ produces the state path illustrated in FIG. 24, namely the maintained track 2402 and the outward spiral 2404. The parameter δ is changed, for example, from -0.26 to -4 to maintain the orbit (see plot 2402), and to -5 to circle outward (see plot 2404).

It should be noted that solving =0 generation: It is a straight line in the state space. However, this does not prevent a declining, sustained or explosive orbit from escaping because Can be changed to positive and negative values.

In order to understand the above item (iv) and why the interval between events is critical, this case has thoroughly studied the transient coupling problem at the event.

Instant coupling

Events can affect the behavior of the model, whether or not the event has input. The reason for this is that the model dynamic variables are coupled at the event and decoupled between events, as defined above. The above analysis of typical parameterization and space, where without significant input, voltage and current evolution changes almost imperceptibly over a short period of time, regardless of when the event occurred.

In the negative phase, under typical settings, the voltage and current state variables can decay toward the corresponding zero tilt. The individual state will arrive at the separate zero-tilt after an infinite interval of events. However, since both of these state variables change, the intercept point on the zero-tilt line changes. This means that although the voltage decays towards some q _- ( t ₀ ) defined at the previous event, the zero-tilt has changed to q _- ( t ₁ ) at the time of the next event.

FIG. 25 illustrates such a concept, that is, an example 2500 illustrating the relationship between inter-event dynamics and zero-tilt. In example 2500, voltage zero-tilt 2502 is to the right of first state 2504 (at the first event or event 0), so the voltage moves toward the zero-tilt 2502. During a certain inter-event interval t , the voltage reaches the point indicated by the horizontal arrow 2506 (approaching a zero-tilt). However, at the same decoupling interval, the current moves toward the zero-tilt 2508. Thus, at the second event 2510 (event 1), the state will be in the upper right of the state space. However, if the slope of the voltage zero tilt is negative, the second state 2510 can be to the right of the zero tilt 2502 (crossing the zero tilt 2502).

As a result, in the case where the model is completely in the negative phase, oscillation is possible only due to sufficient inter-event intervals. Oscillation can occur in individual state variables or even in two state variables (if the first and second states span the two zero-tilt). Such oscillations may occur even with short inter-event intervals and without parameters that are beneficial for orbital execution. FIG. 26 conceptually illustrates an example 2600 of the damped dual state variable oscillations in accordance with certain aspects of the present disclosure. In this scenario, the slopes of the zero-tilt lines in the negative phase are both negative, thus intersecting at the resting point 2602 (the resting point 2602 is u =0 and v = v _- ). This oscillation is damped because the interval between events is finite (ignoring the possibility that the time constant is equal to zero). This oscillation can also be described as a state that jumps back and forth across the zero tilt.

However, the interval between events is also a more familiar track as discussed in the previous section. The shape of the road contributes. The root cause is the same as above, except that when combined with the intrinsic orbital effect, a much smaller inter-event interval effect can be sufficient to modify the behavior of the model to achieve desired properties, such as sub-threshold oscillations.

Considering inter-event interval, the track may be based on the inter-event interval △ t is defined change of state: It is subtly but importantly different from (68), and therefore: among them

Since τ _- <0 and τ _u >0, then Δ t >0, τ _u > <0 and l _u <0, and in equation (74) The first term Δ t >0 of the upper projection is thus negative, causing the orbit to decay toward the resting origin. A simple conceptual case for visualization is if τ _u = - τ _- = τ . Then, l _{_} = l _u = l , and:

It can be observed from equation (76) that if βτ + δ is sufficiently large, the second term u ( v - v _- )( βτ + δ ) cancels out the first term (( v - v _- ) ² + u ² ) . Also it observed, since l can be extracted as a common factor, so △ t control the angle θ between the vector and the position vector trajectory. The larger △ t, l becomes more negative, and the track in the radial direction component greater. As a result, when the second term u ( v - v _- )( βτ + δ ) has a sign opposite to the first term (( v - v _- ) ² + u ² ), the larger interval between events Δ t The state of the recession can be pushed back into the orbit, or if it is sufficiently large, the state of the recession can be pushed outwardly.

The first case is illustrated in FIG. The same parameters as in Fig. 23 were used except for the interval between events Δ t = 1.55 ms instead of 0.1 ms. In FIG. 27, the state has a decaying track 2702 at some inter-event intervals, but is restored in other inter-event intervals in which the second term u ( v - v _- ) is large, and this is sufficiently large between the events. The situation is significant. This is why the track plot 2702 in Figure 27 appears to be thicker because the trajectories from the event to the event have slightly different corresponding states on each cycle and overlap or cross each other.

It should be noted that solving =0 generation: It is still a straight line in the state space. However, this does not prevent a declining, sustained or explosive orbit from escaping because Can be changed to positive and negative values.

Limited return

At first glance it may be assumed that without further input, the model will return towards rest when in the negative phase and will eventually issue a spike in the normal phase. However, this is generally not true. In fact, even without further input, the model can also issue spikes from the negative phase and return to the rest from the normal phase.

The behavior of this model in restoring current dynamics is straightforward. The recovery current can decrease when it is above the current zero-tilt- r and can increase when it is below it (assuming τ _u >0, or vice versa if τ _u <0). However, the voltage dynamics are more complicated because the voltage dynamics depend on the phase of the phase and thus on the phase threshold. However, the phase threshold is generally not the same as the voltage zero tilt (although it can be so defined).

A typical definition of a phase threshold is: Where v ₊ is a constant. It can be assumed that v _ρ '= v _ρ in the transformation variable (zero-limb expression). Accordingly, the voltage at zero line for the positive zero displacement line - q ₊ = v ₊ at u = at 0 crossover v _+, and the negative zero displacement line _- q ₊ = v - at _{u = (v + - v -} ) / τ _- β is the more v ₊ . These two zero-tilt lines draw two sub-states with specific properties. In these sub-states (one of the sub-states is in the normal phase and one of the sub-states is in the negative phase), the voltage returns to the phase threshold for a limited time (rather than in infinite time). Furthermore, without any subsequent inputs, the voltage can actually cross v ₊ into the opposite phase, either from the normal phase to the negative phase or vice versa. In the case of bordering the finite states, at the threshold voltage, the voltage derivative is discontinuous (although the sign may remain the same). For these reasons, the two phases can be labeled as a finite return phase.

Positive finite return phase

In the normal phase, given τ ₊ >0, the voltage tends to deviate from the normal phase zero-tilt. However, the voltage tends to deviate from the normal phase zero tilt line different from the voltage back state phase threshold v ₊ . Specifically, there is only a specific region of u in which the above two are equivalent under the definition of a typical phase threshold.

Positive return theorem

Under the definition of a typical phase threshold, there is a positive finite return phase as a subspace of the normal phase, where the state tends to a negative phase.

prove

In the normal phase v > v ₊ and in order to return the state towards the negative phase, dv / dt <0 or ie: v <- q _{+ is required} . (79) Thus, v < τ ₊ βu + v ₊ . (80) These two conditions together define a region within the normal phase in which the voltage state tends to a negative phase rather than a spike.

Since τ ₊ >0 and β >0, u >( v - v ₊ )/ τ ₊ β , (81) is a straight line with a slope of 1/ τ ₊ β and an offset of - v ₊ / τ ₊ β To define a non-zero region (as long as | τ ₊ β | > 0).

For example, if τ ₊ = 1ms, β = 1 and v ₊ = -40mV, then u = v +40. Thus, at v = -30 mV (i.e., above v ₊ ), any state with u > 10 can produce a decay toward the negative phase. For example, at u = 20, - q ₊ = -20.

Finite positive return theorem

In a positive finite return phase, the state tends to enter a negative phase in a finite time.

prove

In the positive finite return phase, v ₊ < v <- q ₊ . In order to arrive at a negative phase state state _{v f, v f <v +} <v. Thus, v _f + q ₊ < v + q ₊ <0. (82) As a result, for finite v _f and finite non-zero τ ₊ ,

Following the above example, q ₊ = - u +40. At u = 5, in order to reach v _f = -45 mV in the negative phase from v = -35 mV in the normal phase, Δ t 1.1ms.

Positive finite return limit cycle

On the normal phase side of the threshold, v = v ₊ + ε ₊ , where ε ₊ is a positive voltage offset, and any state with sufficient u > 0 decays into the negative phase at time Δ t ₊ :

On the negative phase side of the threshold, v = v ₊ - ε _- , where ε _- is a positive voltage offset, and any state with sufficient u further decays into the negative phase at time Δ t _- , ie:

Thus, in the direction of the derivative, the difference at the discontinuity is: among them And Δ v _± = ( v ₊ - v _- ). It should be noted that for the parameters considered, Δ v _± and k _ρ are always positive. With ε _ρ → 0, Δ v _- - Δ v ₊ = - Δ v _± k _{_} + βu ( τ _- k _- + τ ₊ k ₊ ). (88) In order to remove discontinuities will require:

However, it is of interest to discontinuities. The discontinuity is a negative amount unless the βu term is sufficiently large and ( τ _- k _- + τ ₊ k ₊ ) is a positive amount. Normally, ( τ _- k _- + τ ₊ k ₊ ) is positive, even if τ ₊ is usually greater than τ _- because for significant Δ t _ρ , k _- is usually much smaller than k ₊ . Negative step discontinuity means that the voltage drop across the phase threshold is accelerated. Figure 28 illustrates the voltage derivative 2800 in such a situation. And FIG. 15 are the same parameters, where △ t ρ ₌ 1ms and u = 100. Any constant input sufficient to overcome the voltage leakage in the negative phase will be able to overcome the leakage in the normal phase.

However, because a sufficiently large βu, large inter-event interval △ t _ρ, u or higher long time constant τ ₊ (or shorter time constant τ _-), may be a discontinuity in the direction of the voltage derivative Positive. Figure 29 illustrates the voltage derivative 2900 in such a situation. The parameters are the same as in Figure 15, except for β = 0.02, Δ t _ρ = 5 ms, τ _- = -50 ms and u = 100.

A constant excitatory input sufficient to overcome voltage leakage in the negative phase may not be sufficient to overcome leakage in the normal phase. Thus, there is a potential for voltage cycling under constant excitability inputs. If the u zero inclination intersects the area of the cycle, there is even a potential for a limit cycle.

Figure 30 illustrates a constant excitatory input (applied at the event) The lower limit cycle 3000. At the previous event (event 0), the state is in the negative phase. However, the input at the first event (Event 1) is sufficient to overcome the recession. At the second event (Event 2), the input can bring the state into the normal phase. The input at the third event (Event 3) is not sufficient to overcome the stronger decay in the normal phase, and thus the state moves back towards the negative phase and has entered the negative phase before the time of the next event.

Negative finite return phase

In the negative phase, given τ _- <0, the voltage tends to a negative phase zero-tilt. However, the voltage tends to be negative for the phase zero and the other is different from the falling phase threshold. Specifically, there is only a specific region of u in which the above two are equivalent under the definition of a typical phase threshold.

Negative return theorem

There is a negative finite return phase of the subspace as a negative phase, in which if τ _- β is negative, the state tends to a normal phase.

prove

For negative-phase state, v <v ₊ and the negative phase state claim dv / dt> 0. According to this definition, v <- q _- . In order to have a region where -q _- > v ₊ , τ _- βu > ( v ₊ - v _- ). (90) If τ _- β < 0, then: u <( v ₊ - v _- ) / τ _- β , (91) is a non-zero region as long as τ _- β > _- 。. This region is defined by a zero slope straight line crossing the voltage threshold v ₊ at ( v ₊ - v _- ) / τ _- β .

The derivative of the voltage in the negative finite return phase will thus be positive. However, simply because the state tends to the normal phase does not mean that the state will reach the normal phase. Figure 31 illustrates an example 3100 of behavior in a negative finite return phase. Trajectories 3102 and 3104 are illustrated for two initial states, both of which initially tend to the normal phase but only one arrives at the normal phase, i.e., trace 3104. For τ _- = -10ms, β = 1 and v ₊ = -40mV and v _- = -60mV, the negative finite return phase is defined by v < v ₊ and u <-2. At u <-2, -q _- > v ₊ , since v < v ₊ , the derivative is positive and tends to be a normal phase. These two initial states < v , u > are <-45, -4> and <-45, -5>.

On the negative phase side of the threshold, v = v ₊ - Δ v _- , any state with sufficient u tends to enter the normal phase, its voltage derivative: On the normal phase side of the threshold, v = v ₊ + Δ v ₊ ,

The acceleration of the voltage across the barrier in the direction of the derivative is: g (Δ v ) = τ ₊ Δ v ₊ - τ _- ( v ₊ - v _- - Δ v _- ). (94) For τ _- <0, g (Δ v ) < 0 at the limits Δ v _- → 0 and Δ v ₊ → 0, so that the state generally accelerates toward v _S .

Finite negative return theorem

In the negative finite return phase, for a finite negative τ _- , the state tends to enter the normal phase in a finite time.

prove

In the negative finite return phase, v < v ₊ <- q _- . In order to reach the state v _f in the normal phase, v _f > v ₊ > v . Thus, v + q _- < v _f + q _- . (95)

For the negative phase, v + q _- <0, but v _f < q _- can be chosen such that as a result, for a negative finite non-zero τ _- , the following holds: Then the above example, at the u = -10, for v = -45mV and v _f = -35mV, △ t 0.5ms.

Advanced control

32 illustrates an example 3200 of a normal phase and a negative phase having a positive finite return phase and a negative finite return phase (regions 3202 and 3204, respectively), zero tilt 3206, 3208, and for δ <0 , Concept vector field for a typical case of = v ₊ and ε = - v _- .

State threshold

However, as noted above, there may be alternative definitions for the phase threshold. For example, the state threshold can be defined as a zero tilt along the voltage above u =0, or ie =max( v ₊ , q ₊ ). This will eliminate the positive finite return phase, but will not eliminate the negative finite return phase. The replacement definition will be the setting =max( q _- , q ₊ ), so that the states are divided at the rightmost voltage zero. However, given a typical slope, the positive and negative voltage zeros are crossed below u = 0, and thus the voltage differential will still generally not be continuous. Specifically, this situation will be the opposite of the above case because the positive voltage zero-tilt will be further to the left of the phase threshold when u = 0 or less. There are various other alternatives that the modeler can define, and it should be noted that the general analysis methods discussed above can also be applied to dynamics derived from other or atypical settings.

The finite return phase is the result of a phase threshold different from the voltage zero. In order to remove the positive return phase, the phase threshold can be made dependent on the second state variable, ie equal to the normal phase zero tilt (when the zero tilt is applied), otherwise equal to the preset value:

However, this still leaves a negative finite return phase. Although both of these finite return phase phases may be useful, removing the negative finite return phase phase will prevent cell excitation for any large voltage state if the current state is sufficiently negative. The modification of the phase threshold is:

This is due to the negative voltage zero tilt across the voltage zero tilt discontinuity at v ₊ '. However, this can also be controlled. In order to remove the two finite return phase phases,

Decoupling voltage zero tilt slope and time constant

Further design freedom can be obtained by using a generalized version defined by the voltage zero-tilt. It should be recalled that the standard model is defined by transformation: q _ρ =- τ _ρ βu - v _ρ ', (100)

r = δ ( v + ε ). (101)

However, since the voltage dynamics are governed by the following formula, This parameterization may therefore be limiting in the sense that the voltage zero-tilt slopes are related by the slope factor β . To remove this limitation, a more general definition is: q _ρ =- β _ρ u - v _ρ ', (103) where there is a subtle change from βτ _ρ to β _ρ . This generalized definition is related to the standard definition by the following settings: β _ρ = τ _ρ β . (104) This generalization can be used with or without the finite return phase control discussed above.

One-dimensional equivalent

This model inherits the computational benefits of the Anti-Leak Integral Stimulation (ALIF) neuron model. It has been shown in the present case that for some specific parameterizations with low coupling (small β and δ ) and high second state time constants (large τ _u ), the voltage time constant is scaled to compensate for the second state zero tilt. The effect of the line attractor on voltage decay and rise time, the second state variable ( u ) can be omitted while retaining nearly equivalent behavior.

The reason for this is that the solution to the dynamics of the model (if continuous coupling is assumed) is of the form: among them , b =( a - τ _ρ + τ _u )/2 τ _u τ _ρ , c =(- a - τ _ρ + τ _u )/2 τ _u τ _ρ , v '= v - v _ρ ' and u '= u + δε . If the coupling is weak and τ _u is large, after the approximation, may propose a τ _u and b 1/ τ _ρ , and c 0. Correspondingly,

The time to reach this state (such as issuing a spike or rest or any arbitrary v ( t )) is provided by:

In order to achieve substantially the same effect with only one variable (ie, only v , not v and u ), the time constant τ _ρ can be adjusted to τ _ρ ' to obtain an equivalent time t , or ie:

It can be recalled that, by default, ε = - v _- , and thus for δ < 0, the denominator in the logarithm of the right side of equation (108) is larger, thereby making the resulting score smaller and the logarithm smaller. For this reason, τ _ρ ' needs to be less than (faster than) τ _ρ to account for the removal of the second state, which originally slows down the effective behavior time response.

Some aspects of the case support model definitions and basic model dynamics derived from the negative (with leakage integral excitation) phase and the positive (anti-leakage integral excitation) phase and the instantaneous coupling at the event. The case also provides additional dynamic features, including driving the fundamentals derived from the oscillatory or resonant behavior of the model definition including transient coupling. In the following, it will be described in this context how these dynamics can produce a wide variety of biologically consistent cellular behaviors.

Model behavior

In this section, the case provides the dynamic aspects and classes described above. How dynamics allow the model to be designed to exhibit an explanation of the rich variety of behaviors commonly used to characterize the type of biological cell behavior, including responses to spike timing patterns, sustained or step inputs, ramps, and other inputs. Specific parameterized examples will also be provided. However, these are merely examples, as there are generally various ways to design a model to exhibit specific behavioral characteristics. However, specific examples may be illustrative in demonstrating how certain parameters govern or influence behavioral differences in accordance with the dynamic principles described above. Given the flexibility of the model, it would be impractical to consider all combinations and situations, so the case focuses on a set of demonstrative behaviors and a subset of illustrative means of achieving each of their behaviors. Similarly, the behavior is not limited to the specific dynamics described in the previous sections, but also includes similar or related dynamics in accordance with the general principles discussed herein.

Arbitrary reference cell

For convenience, nominal (arbitrary) cell parameterization will be used as a baseline or baseline. In order to demonstrate various responses to the input, examples will usually be used. , τ _- <0, τ ₊ >0, and τ _u >0 and have parameterization and configuration of artificial events that occur at nominal periodicity. The nominal setting without loss of generality can be: τ _- = -20ms, τ ₊ = 2ms, τ _u = 10ms, v ₊ = -60mV, v _- = -70mV, v _S = 30mV, β =1, δ = -0.25 and Δ u =0, and event periodicity of 1 ms can be considered. However, there are no specific reasons for choosing their nominal parameters or settings other than the other nominal parameters or settings. Other parameterizations can be used to achieve similar or even the same behavior. The purpose of selecting a specific nominal instance is merely to show how various actions can be achieved with or without changes to specific parameters. In other words, the purpose of selecting this particular nominal instance is to resolve whether cells with the same parameters can demonstrate multiple behaviors and whether they can be varied in multiple/different ways (eg, by a parameter or a replacement parameter or a combination of parameters or by input) Change) to achieve the problem of behavior change.

Model behavior can be generated on any desired time scale because The time unit in the definition of the model is arbitrary. Therefore, without loss of generality, the behavior on the abstract time scale can be discussed.

The case review responds to various input contexts as well as graphical representations. A demonstration of the behavioral set of design aspects. These behaviors are based on synchronization or timing patterns, tolerance or excitability, peak delivery modes at maintained inputs, or actual tonicity and phase-type spikes and short pulses and statics at sustained inputs. Organize or reorient the dynamics of the orbit.

Synchronization and spike timing mode

Since the input is applied at the event, the Dirac △ function can be used as the fundamental basis for the input. Multiple delta function inputs generally together form a peak timing mode (whether the input to the postsynaptic cell is from the same synapse or multiple synapses). Examine the representative set of behaviors that respond to such patterns. Although this is characterized by such input, such behavior is also relevant in other input contexts. The △ function input is only a convenient means of element analysis in discrete time.

Peak waiting time

The normal phase dynamics of the model incur a spike waiting time. The normal phase time constant τ ₊ determines the nominal latency in the absence of coupling. In the case of coupling, the current dynamics depend on whether the current is above or below zero. The only requirement for a spike waiting time is to move the state into the normal phase at the event (ie, to issue a spike). In the case with a Dirac function △ input, the process proceeds with normal resting state is only required from the input amplitude voltage △ V Contribution entry satisfies _{△ v> v + - v -} . In general, the smaller the initial amount of state shifting into the normal phase, the longer the peak wait time.

FIG. 33 illustrates an example 3300 of spike latency in the event of an input event 3302 occurring. Graph 3304 illustrates how it is possible to incur substantial spike waiting times. The parameters are the same as in the nominal setting, and Δ v =11 mV, that is, 1 mV greater than the value sufficient to reach the threshold of the phase. It should be noted that this input drives the voltage state deeper into the normal phase, and the spike wait time is shorter. The latency in this case can be affected (increased) by periodic coupling (due to non-zero β ).

Figure 34 illustrates this effect in the state space (the x-axis is the voltage state and the y-axis is the current state). In Figure 34, the two phase reference voltages v _- and v ₊ are plotted as vertical dashed lines, while the current u and voltage v zero tilt lines are plotted as solid lines over a limited field. The state trajectory is illustrated by curve 3402.

For comparison purposes, the spike and state trajectories are illustrated in Figure 35 for the following two scenarios: coupled (spike trace 3502 and state trajectory 3504) and no coupling (spike trace 3506 and state trajectory 3508). The parameters are the same as before, but β = 0. In the case of coupling, the state trajectory that deviates from the phase threshold has a positive current component and thereby brings the voltage closer to the voltage zero tilt, rather than being more polar in the horizontal, and thus increasing the wait Time (compared to no coupling).

Thus, the degree of coupling affects the peak wait time. Reducing the coupling will reduce the peak wait time. However, reducing the coupling period will also have the effect of reducing latency. The reduced latency can also be achieved simply by shortening the normal phase time constant τ ₊ or by reducing the input amount such that the state initially enters the normal phase by a small amount. In summary, there are multiple ways to achieve a specific wait time, and even if some elements are constrained by other design goals, there is freedom to design the model for the desired behavior.

This flexibility in the model is an advantage because cell behavior can be used The next dimensions are designed: inputs, cellular parameters, and events (including human events). In addition, this flexibility applies not only to peak wait times, but also to model behavior and dynamics. As mentioned earlier, these dimensions are all elements that can be controlled independently. Not only is the input and cell parameters different, the event itself (whether or not there is an associated input or output), the control coupling is also a distinct dimension, and thus affects the dynamics.

Subliminal oscillation

Sub-threshold oscillations can occur after the spike due to the return state. The post-peak oscillations can be mobilized as a result of a recondition (where the voltage and current states are reset after a spike) or as a result of the state trajectory reaching a spike condition (due to a reorientation of the current state) or a combination of the two. As discussed above, the reset condition may result in convergence to a resting state (resting origin) or a resting orbit that diverge therefrom.

This dynamic is the underlying feature that implements the resonator behavior exhibited in the example 3600 of Figure 36. The parameters are the same as the nominal settings, and Δ v = 10 mV, that is, alone is not sufficient to induce a transition to the normal phase. The first pair of peaks 3602-3604 are separated by, for example, 20 ms, and the second pair of peaks 3606-3608 are separated by, for example, 40 ms. A key aspect of resonator behavior is that the input coincides with/synchronizes with sub-threshold oscillations to reinforce this effect. Thus, the timing of the inputs relative to one another is a distinguishable input characteristic in the sense of output, as illustrated by resonator behavior 3610 after the second pair of peaks 3606-3608 in FIG. In other words, cells with resonant behavior can distinguish inputs with one periodicity (frequency) from inputs with another periodicity (frequency).

A decay threshold oscillation that drives the behavior of the resonator can be seen in the state trajectory near the rest point. Figure 37 illustrates such dynamics. The same convention applies to all state space trajectories, ie, the x-axis is the voltage state and the y-axis is the current state; the two reference voltages v _- and v ₊ are plotted as vertical dashed lines, while the current u and voltage v are zero-dip Lines are plotted as solid lines in a limited area. The state trajectory is illustrated by curve 3702. The dynamics of the decaying counterclockwise orbit near the resting point (both before and after the peak) are directly related to the restoring orbits above the intersection of the voltage and current zero-tilt in the negative phase. Equal dynamics. Current and conductivity inputs are applied to the voltage state. If the input occurs periodically but there is an offset at the point of subsequent input and to the left of the rest point on the orbit in the state space, the excitatory effect is damped. If instead, the input occurs when the state is at a point on the right side of the resting point in the orbit, then the excitatory effect is enhanced.

In contrast, the integral behavior illustrated in the example 3800 of FIG. 38 does not have this discriminating power because there is no oscillation that would cause a portion of its orbit to the left of the resting voltage. Integral behavior can be achieved in a variety of ways, including decoupling current dynamics. The parameters are the same as before except δ =0 and Δ v = 8 mV. As can be observed from the graph 3802 in Fig. 38, as a result, there is no oscillation.

The underlying coupling feature can also be implemented due to the suppression input. The behavior of "significant variability in depolarization thresholds". Suppressive input is like excitement Sexual input can be used to perturb the model state so that it moves into subliminal oscillations. If the subsequent excitatory input occurs synchronously (consistently) with the oscillation, the effect can be enhanced and provide an impression that the "threshold" of the cell has changed.

Figure 39 illustrates the direct result of this underlying feature, where the excitatory input alone may not result in a spike (assuming the cell is at rest), followed by an inhibitory input followed by an excitatory input (with a specific inter-peak interval) How the spike pattern can cause spikes. The parameters are the same as the nominal settings. The inter-spike interval is 20 ms and has Δ v = 10 mV for each input. This behavior can be understood in the context of Figure 37. The difference is that instead of using excitatory input 3902 to oscillate at the point on the track to the right of the resting point, the suppression input 3904 is followed by excitatory input 3906 to oscillate at the point on the track to the left of the resting point. Since it follows the track, it can be observed in Figure 39 that voltage 3908 oscillates (beyond excitability input 3906 exceeds the resting point). These spike timing related behaviors result from synchronization or out of synchronization of the input timing with the current resonant frequency of the cell.

The influence of the input amplitude and input timing on the state and thus the effect on the spike behavior can be designed by designing the orbit in the space of < v , u > space (for example, circular or elliptical) and controlling the time constant of the trajectory rate. To control. The former affects the invariance with respect to timing (bandwidth) and amplitude, while the latter affects the main resonant center frequency.

Threshold oscillation

Above, the oscillation corresponds to the sub-threshold state trajectory. However, the trajectory is not constrained to the negative phase. If the rebound from the inhibitory input is sufficient, a spike may be triggered. Depending on the parameters, this may require a larger suppression input or multiple suppression inputs to move the state into the outward spiral path. Short pulses can be designed by increasing the current dynamics (the slope of the current zero tilt). FIG. 40 illustrates an example 4002 of a rebound spike 4004 resulting from an input event 4006. For the rebound spike instance 4002, the parameters are the same as the nominal settings, and the inhibitory input 4006 is applied over 4 ms (at a manual event interval of 1 ms). FIG. 40 illustrates an example 4008 of a rebound short pulse 4010 caused by the inhibitory input 4012. For the rebound short pulse instance 4008, the parameters are the same as the nominal settings except for δ = -0.5.

Figure 41 illustrates the situation in the case of the rebound spike instance 4002 of Figure 40. An instance 4100 of the state space trajectory 4102. The inhibitory input moves the state deeper into the negative phase and moves to a negative current state from which the state follows a partial orbit (counterclockwise) around the resting origin. However, due to the relatively large radius of the orbit, the path intersects the boundary of the state and causes a spike. This is the difference between the subthreshold oscillation and the rebound spike/short pulse. It should be noted that there are many ways to influence whether the trajectory crosses the phase boundary, that is, by controlling the trajectory, moving the reference, changing the input, or designing the event.

In addition, the same underlying features enable bistable behavior. Once the short pulse is in progress, the state dynamics are in the orbit (the track passes through the re-mode). In order to stop the short pulse, an input may be required to push the state out of the track. In the aspect, either type of input (excitability or inhibition) may be suitable. FIG. 42 illustrates an example 4200 of bistable behavior with excitatory inputs 4204, 4206 (graph 4202). The parameters are the same as the nominal settings except that the input is a single event but larger, such as Δ v =20mV.

43 illustrates the state for the bistable instance 4200 from FIG. Example 4300 of spatial trajectory 4302. The features to be noted are twofold. First of all, The initial excitability input 4204 results in a spike and the re-parameter brings the state into the orbit, resulting in further spikes (see, for example, chart 4202). Second, the final excitability input 4206 moves the state from the point on the track back to the vicinity of the resting voltage, which results in a decaying spiral back to the resting origin. It should be noted that the timing of the final excitatory input is important.

In the case of this case, it can also be based on the aforementioned analysis in the previous chapter. To modify the parameters to counter the decay and maintain the orbit in the negative phase. In addition, two or more of the methods described in this section can be used in combination.

Coupling, event, and intersection between oscillations

The above provides a variety of demo dynamics of the model, including recession, maintenance, and explosive orbits. Due to coupling or events (whether input, output, or artificial events), cells can transition between these types of cycles. For example, above the threshold, the oscillation is demonstrated as after the spike (output event) and the decaying track (to rest) is demonstrated as after the excitatory input (input event). In general, if an oscillation already exists, the change in dynamic behavior may also depend on the oscillation. The present case first demonstrates applying one of the foregoing principles to design an instance of oscillation by coupling (unlike events, inputs, or other parameters), and then demonstrates how the oscillation can be designed to continue even after an event.

Figure 44 illustrates an example 4400 of sub-threshold oscillations 4402, i.e., how the amplitudes of the coupling parameters β and δ are increased enough to counter orbital decay. The parameters are the same as the nominal settings except for β = 2 and δ = -0.725. Input 4404 is set to Δ v = 10 mV, which is insufficient to induce a transition to the normal phase. In such a case, the input is insufficient to enter the positive domain, and thus the system behavior illustrated by graph 4402 is associated with sub-threshold oscillations.

After spike designed to maintain the oscillation of the cells can only re-configured by a current offset △ u state to be achieved into the track area. In general, this can be configured by Δ u , Or both to achieve the re-determined state on the orbital path.

Figure 45 illustrates an example 4500 of post-peak sub-threshold oscillations, i.e., how the same cells can oscillate under the threshold before and after the spikes (before and after input 4504) (see diagram 4502 in Figure 45). The parameters are the same as above (no change) except that Δ u changes from 0 to -11, which is sufficient to move the return state into the track path.

Figure 46 illustrates an example 4600 of a resting orbital path before and after a spike, in accordance with certain aspects of the present disclosure. Figure 46 illustrates how sub-threshold oscillations after the spike (solid line 4602) generally return to the orbital path as before the spike (dashed line 4604). The X axis is the voltage state v and the y axis is the current state u . These traces correspond to the results of the state path overlap of Figures 44-45. It should be noted that the width of the oscillating region can be designed to be arbitrarily large relative to the contribution input.

Oscillating phase

It should be noted that the behavior of the cells after the spike is issued generally includes a negative voltage swing (the oscillation begins to decrease). However, depolarization after the spike is also possible. FIG 47 illustrates an example 4700, after which the potential (input after 4704) depolarization (see graph 4702) is reset by adjusting the current to achieve the offset △ u. The parameters are the same as above except for Δ u =-7.5. Effectively, this places the oscillations in different phases at reset.

Multiple behavior

The vast majority of the above behaviors are without changing the model parameters. Achieved. All behaviors are achieved without substantially changing the parameters (up to one or two adjustments). In addition, adjusting specific parameters to change some behavioral aspects does not interfere with other behavioral aspects to be retained. This advantage can be observed in behaviors characterized in other ways than the spike timing mode.

Adaptability and excitability

Cellular behavior is often characterized in a manner that has other input waveform shapes that dictate the excitability of the cells, which can excite cells and excite cells. One of the more interesting input waveform shapes is the ramp because the input will evolve. In particular, such characterization may involve supplying a gradual increase in current to the cells. The ramp can be characterized by the slope (increasing rate) and duration. Increasing (or otherwise changing) the input is of interest when compared to how cell dynamics change during the input and whether cell dynamics are offset or enhanced input changes.

Tolerance

The tolerance is one in which the dynamics of the cell will be higher than the increasing input as long as the rate of increase of the input is relatively low to recover from the increasing input without issuing a spike. However, if the rate of increase is faster, the cells may not recover and prevent spikes. FIG. 48 illustrates an example 4800 of an adaptive principle. The parameters are the same as the nominal settings except τ _- =-5ms, τ ₊ =1ms, τ _u =50ms, δ =-1. The duration of the first ramp 4802 is 100 ms, and the duration of the second ramp 4804 is 5 ms. The first ramp 4802 has a slope of 8 mV/ms and the second ramp 4804 has a slope of 3/4 of the first ramp. It should be noted that the second ramp is shorter and smaller than the first ramp (lower peak and total integral input). Nonetheless, the spike 4806 originates from the second ramp rather than the first ramp.

Figure 49 illustrates the empty state for the accessible example 4800 from Figure 48. Example 4900 of an inter trajectory 4902. It should be noted that the input is applied to the voltage state but is applied at a rate that is slow enough to allow the current state to bring the trajectory into the decay trajectory toward the rest of the rest. Conversely, if the slope (input rate) increases, the dynamics do not respond quickly enough to prevent spikes.

Spike and short pulse mode

For increasing excitatory inputs, varying peak rate behavior can also be achieved. By reducing both the negative phase voltage time constant and the current time constant, a class 1 behavior, or a gradually increasing peak rate, can be achieved. This emphasizes the effect of increasing the instantaneous magnitude of the input (rather than the integral or cumulative input). As long as the re-parameters bring the state back to a similar trajectory each time, the behavior change (inter-peak interval) will be greatly affected by the instantaneous input amplitude.

FIG. 50 illustrates an example 5000 of an increased spike release rate 5002 as a result of the rising excitability input 5004. The parameters are the same as the nominal settings except that the time constant is shortened to: τ _- = -5ms, τ _u = 5ms and δ = -0.1 to slightly reduce the coupling compared to the nominal. It should be recalled that for the nominal reset parameter, Δ u =0 and = v _- .

If the parameter is re-set to move the state so that the decay towards the rest is faster after each spike, so that as the excitability input ramps up, the inter-peak spacing can remain relatively constant rather than decrease, resulting in a relatively more constant spike The release rate, which is the class 2 behavior. Thus, by balancing the re-conditioning of the current (and/or voltage), a class 2 behavior or a phase of excitatory input for the rise can be obtained. For a constant rate of excitation.

Figure 51 illustrates an example 5100 of dispensing a spike 5102 at a relatively constant rate (fixed interval) despite the rising excitability input 5104. The parameters are the same as for Class 1, except that Δ u = 10 to balance (compensate) the increasing input and δ = -0.5 to increase the coupling. It should be noted that the excitability input ramps for both Class 1 and Class 2 instances are the same, for example, 140 ms in length, equivalent to 25 mV at peak/end. It should be noted that here as in other voltage trace plots, the voltage at the peak does not appear to reach the spike threshold. However, this is only because when the voltage reaches the spike threshold, the voltage is immediately reset, so that the voltage plotted at the peak time is actually a re-set voltage, so in these plots, the spike does not always appear to be high. . This is just an artifact of what the recorded voltage is.

Tonic and phase type spikes and short pulses

The application of step input currents has often been used as a pilot protocol for biological cell characterization, including tonic and phase-type spikes and short pulses. These behaviors are quite straightforward, because the focus is less on timing and input, and more on the subsequent output pattern, especially in the case of tonic behavior.

Tonic and phase spikes

The step transition itself is temporarily ignored, and the sustain input thereafter can also be considered as a rough approximation of the summing effect of many discrete inputs occurring over a period of time (possibly with substantial filtering effects). At the maintained excitatory input, the ankylosing spike is derived from the behavior of the input repeatedly driving the voltage into depolarization. Depending on the state variable coupling and activity, as a direct result, The rate of spike release can vary. FIG. 52 illustrates an example 5200 of a tonic spike release behavior 5202 as a result of step 5204. The parameters do not change compared to the nominal settings. Behavior 5202 is generated by input step 5204 (maintained excitability input). However, this behavior can be obtained with a variety of other parameterizations, including by merely increasing the input amount with their parameterization subject to phase-type spikes as described below.

The difference in phase-type spike behavior is that the spike stops after the transition. However, the difference may not be as big as it seems at first glance. If the reset condition places the trajectory on the non-spike path after the first spike (although there is a maintained input), or if the trajectory itself during the first spike shifts the state into a path that will decay rather than re-issue the spike after re-setting, The bit phase behavior will occur. The difference between the former and the latter is what mechanism (or combination of mechanisms) drives the state to the non-spike state. Figure 53 illustrates an example 5300 of the latter mechanism separately in accordance with certain aspects of the present disclosure. It can be observed that the phase type spike 5302 stops after the transition of the input step 5304. The parameters are the same as the nominal, except that the current decay time constant increases to τ _u = 40 ms, and the current decay time constant affects where the state ends after the reset. It should also be noted that various other parameterizations or inputs may result in the same or similar behavior.

Tonic and phase short pulses

Tonic and phase short pulses are similar to tonic and phase spikes. However, in order to generate short pulses, the only fundamental change that may be required (as compared to the nominal) to parameterize is to increase the re-set voltage or adjust the re-set current offset so that the cells continue to issue spikes. In addition, the current time constant will also affect short pulse characteristics, such as how long a short pulse lasts.

FIG. 54 illustrates an example 5400 of a tonic short pulse 5402 that is primarily achieved by adjusting a reset condition. The parameters are the same as the tonic and phase type spikes except that the reset condition is changed to Δ u = 1.5 and = -45 mV to establish a short pulse after input step 5404.

Phase-type short pulse behavior is similar and can be achieved using the same mechanism. The difference (decrease) in the input turns the tonicity into a phase type because the rebound can be reduced and not enough to get rid of the next short pulse. This can be demonstrated by adjusting the relative phase phase reference voltage difference to adjust the sensitivity of cell behavior to exact input levels. FIG. 55 illustrates an example 5500 of phase-type short pulse behavior 5502 at input step 5504. The parameters are the same as the tonic short pulses except that the current reset condition is not changed compared to the nominal, ie, Δ u =0, and by setting =-50mV to change the phase reference difference.

Self-adjusting variant

Mixed mode behavior is similar, if not often difficult to distinguish from tonic peaks. However, there are other ways of producing mixed mode behavior, including, for example, parameterization for tonic or phase short pulses. This is due to two reasons. First, the short pulse is only a sequence of spikes that are close in time, and the necessary condition to be regarded as a short pulse "closeness" is merely a time scale or a defined problem (often arbitrary). Second, a single spike is essentially a short pulse with only one spike. Thus, the mixing behavior can be produced by elements of a tonic or phase-type short pulse (one or more spikes).

FIG. 56 illustrates an example 5600 of how the tonic/phase-type short pulse parameterization can also result in the mixed mode behavior 5602 initiated by the input step 5604. The parameters are the same as the phase-type short pulses except for a small change to the reset condition: Δ u =1 and =-53mV. This is just another example of how one type of behavior can be derived from various combinations of parameterization or input formulation.

The same general principles can be applied to spike frequency self-adjustment behavior, as illustrated in example 5700 of FIG. The parameters are the same as above, except for minor changes to the reset condition: Δ u = 0.5 and return to nominal =-70mV to show how shared parameters explain various behaviors. The frequency of the spike 5702 in the "maintained short pulse" initiated by the input step 5704 can be controlled by various parameters including, for example, a reset condition or a time constant.

Various direct behaviors that respond to step excitatory inputs can be borrowed It is explained by the reset condition and the evolution of the current state as determined by the current state decay time constant. However, various acts may be made by alternative parameterizations and mechanisms (or combinations) and the examples provided are illustrative and are not meant to be exhaustive or preferred. These examples are chosen to demonstrate a particular extreme parameter difference in changing behavior or to explain a particular relationship without necessarily having any intention of minimizing parameter changes.

Zero tilt saddle

The current time constant τ _u is usually positive, so that the state tends to zero tilt. Furthermore, the coupling parameter δ is generally negative, so that the zero inclination has a positive slope. However, if the sign of τ _u is inverted, the transformation variable and the zero inclination r become saddle points rather than attractors. This will result in a spike-free loop with no input because: (i) above the zero-tilt, the state is pushed up and dragged to the negative/deviated positive voltage; (ii) in the fully aggressive (small) τ _- In this case, the state is dragged across the zero tilt line; and (iii) once below the zero tilt line, the state is pushed down and pulled outward toward the spike voltage. As the state is retimed, the loop repeats and the cells continue to issue spikes.

However, cells can be stopped by issuing excitatory inputs. peak. This may require adding enough excitatory inputs to shift the voltage derivative at the larger negative voltage. As a result, the negative voltage derivative becomes positive. If the negative voltage derivative is sufficiently positive, the state trajectory can traverse back to zero tilt and loop back (threshold sub-cycle) and decay to a stable point even under the desired conditions. However, if the excitatory input is removed (or offset by inhibition), the cells can re-enter the spike release cycle.

Some additional atypical parametric aspects can be considered when demonstrating these principles below. For example, the state threshold It can be set to a value other than the normal phase reference voltage v ₊ . This means that the state trajectory can be designed using one of the voltage zero tilts that exceeds the typical category (reference voltage). For example, by reducing the phase threshold to = v _- , the normal phase zero tilt effectively extends back to v _- and dominates the dynamics between v _- and v ₊ and above v ₊ . This can be used to achieve various desired effects, including pushing the voltage state rather than pulling the voltage state.

In the example 5800 of FIG. 58, the suppression 5802 is added to shift (eliminate) the constant excitability input 5804, resulting in a spike 5806 caused by the suppression. The parameters modified compared to the nominal instance are: τ _u = -25, δ = +0.9, and = v _- and τ _- = -10. The reset condition is also set to Δ u =10 and =-60mV. When the inhibition eliminates the excitatory input, the cell begins to excite and continues to do so because the reset condition brings the cell back to an unstable condition without input. When the suppression is removed, the net excitability input places the state above the current saddle point, which pushes the current state back to zero instead of exploding.

Figure 59 illustrates an example of a spike caused by inhibition from Figure 58 Example 5900 of state track 5902 of 5800. It should be noted that removing the excitement (by eliminating it by suppression) leads to a spike firing cycle. Reintroducing the excitatory input places the trajectory in a decaying orbit around the raised origin due to the input offset.

Short pulses can be achieved by applying one or more of the principles discussed above. For example, the resizing mode can be changed to place the state in a faster track path. Figure 60 illustrates an example 6000 of short pulses caused by suppression. Suppression 6002 is added to shift (eliminate) the constant excitability input 6004, resulting in a short pulse 6006 caused by the suppression. The parameters are the same as above except △ u =0 and =-48mV.

In summary, the model is flexible because even atypical parameters The scope can also be used to model rich biologically realistic behaviors.

Cell template

For practical purposes, a limited set of examples is discussed in this case. Accordingly, these examples demonstrate a limited set of behaviors, each of which is produced in one of many possible ways. Model designers should be encouraged to apply these principles to achieve such and related or similar desired behaviors and interactions between dynamics, and it is believed that multiple design combinations (parameterization, events, and inputs) are generally possible. The foregoing behavior is reproduced in FIG.

Figure 62 summarizes the nominal instance parameter settings and the specific variations of the parameters demonstrated in the above examples. This should not give the actual value of the demonstration or the importance of a specific variant. Rather, these examples can be used as a conceptual paradigm for relative parameterization (how the parameters relate to each other) and behavioral aspects.

Review view 62 reveals the same parameters (or similar parameters, such as How cells with only a single parameter change can exhibit multiple behaviors. In the aspect Modifying a parameter can prevent a subset of behaviors, allow new behaviors, change the realm in which the behavior is exhibited, or some combination of the described. When looking at the limited set of example parameters illustrated in Figure 62, it may be attractive to assume that two behaviors with different parameters require the difference to achieve behavioral differences. However, other means of establishing an action may have parameters that are equivalent or similar to another behavior. It should be recalled that these examples were chosen to demonstrate the principles of a particular design modeling. It should also be noted that events affect dynamics, and input formulation is another dimension of design.

It should be noted that in Figure 62, the nominal examples are arbitrary. Phase A sharper input with a stronger input can also result in a strong straight spike. Mixed mode behavior can be difficult to distinguish from tonic spikes. Different parameters are used in the mixed mode only to demonstrate that the behavior can be generated or interpreted in different ways.

Figure 63 illustrates the use of certain aspects of the present invention for updating artificial gods. An example operation 6300 of the state of the warp. At 6302, a first state of the artificial neuron can be determined, wherein the neuron model of the artificial neuron can have a closed form solution in continuous time and wherein the state dynamics of the neuron model can be divided into two or more State. At 6304, an operational phase of the artificial neuron can be determined from the two or more states based at least in part on the first state. At 6306, the state of the artificial neuron can be updated based at least in part on the first state of the artificial neuron and the determined operational phase.

Figure 64 illustrates the use of certain aspects of the present invention for generating artificial gods An example operation 6400 of various neurobehavioral behaviors. At 6402, a first state of the artificial neuron can be determined, wherein the neuron model of the artificial neuron has a closed form solution in continuous time and wherein the linear dynamics of the neuron model are Divided into two or more states. At 6404, an operational phase of the artificial neuron can be determined from the two or more states based at least in part on the first state. At 6406, the state of the artificial neuron can be updated based at least in part on the first state of the artificial neuron and the determined operational phase. At 6408, various neural behaviors of the artificial neuron can be generated by utilizing the linear dynamics of the neuron model.

Figure 65 illustrates an update of an artificial god in accordance with certain aspects of the present disclosure. An example operation 6500 of the state of the warp. At 6502, the state of the artificial neuron can be updated based at least in part on an event, wherein the neuron model of the artificial neuron has a closed form solution in continuous time and wherein the state dynamics of the neuron model are divided into two or more Multiple states. At 6504, the state of the artificial neuron can be updated at time intervals. At 6506, if an event occurs at a time or between times, the state of the artificial neuron can be updated.

Figure 66 illustrates the use of a general purpose processor in accordance with certain aspects of the present disclosure. 6602 An example implementation 6600 of the foregoing method for updating a state of an artificial neuron and generating various neural behaviors of an artificial neuron. Variables (neural signals), synaptic weights, and system parameters associated with the computing network (neural network) may be stored in memory block 6604, while instructions executed at general purpose processor 6602 may be from program memory 6606. Loaded in. In the aspect of the present invention, the instructions loaded into the general purpose processor 6602 can include: code for determining a first state of the artificial neuron, wherein the neuron model of the artificial neuron has a closed form solution in continuous time And wherein the state (linear) dynamics of the neuron model is divided into two or more states; for determining the work of the artificial neuron from the two or more states based at least in part on the first state State code a code for updating a state of the artificial neuron based at least in part on the first state of the artificial neuron and the determined operational state phase; and for generating the artificial nerve by utilizing linear dynamics of the neuron model A variety of neurobehavioral codes.

In another aspect of the present case, loading into the general purpose processor 6602 The instructions may include code for updating a state of the artificial neuron based on the event, wherein the neuron model of the artificial neuron has a closed form solution in continuous time and the state dynamics of the neuron model is divided into two or more a plurality of states; a code for updating a state of the artificial neuron at intervals; and a code for updating a state of the artificial neuron if an event occurs at or between times.

Figure 67 illustrates the foregoing for updating a person in accordance with certain aspects of the present disclosure. An example implementation 6700 of a state of a working neuron and a method for generating various neural behaviors of an artificial neuron, wherein the memory 6702 can be connected to an individual (distributed) processing unit of the computing network (neural network) via the interconnection network 6704 ( The neuro processor) 6706 is docked. The variables (neural signals), synaptic weights, and system parameters associated with the computing network (neural network) can be stored in memory 6702 and can be loaded from memory 6702 via the connection of interconnect network 6704 to each Processing unit (neural processor) 6706. In the aspect of the present invention, the processing unit 6406 can be configured to: determine a first state of the artificial neuron, wherein the neuron model of the artificial neuron has a closed form solution in continuous time and the state of the neuron model (linear Dynamically being divided into two or more states; determining an operational phase of the artificial neuron from the two or more phases based at least in part on the first state; based at least in part on the artificial neuron First A state and a determined operational state are used to update the state of the artificial neuron; and a plurality of neural behaviors of the artificial neuron are generated by utilizing linear dynamics of the neuron model.

In another aspect of the present disclosure, processing unit 6706 can be configured to: Updating the state of the artificial neuron in an event, wherein the neuron model of the artificial neuron has a closed form solution in continuous time and wherein the state dynamics of the neuron model is divided into two or more states; The state of the artificial neuron is updated at intervals; and if an event occurs at a time or between times, the state of the artificial neuron is updated.

Figure 68 illustrates a decentralized memory based on certain aspects of the present disclosure. The body 6802 and the decentralized processing unit (neural processor) 6804 implement an example 6800 of the aforementioned method for updating the state of the artificial neurons and generating various neural behaviors of the artificial neurons. As illustrated in FIG. 68, a memory bank 6802 can directly interface with a processing unit 6804 of a computing network (neural network), wherein the memory bank 6802 can be stored with the processing unit (neural processor) 6804. Associated variables (neural signals), synaptic weights, and system parameters. In the aspect of the present invention, the processing unit 6804 can be configured to: determine a first state of the artificial neuron, wherein the neuron model of the artificial neuron has a closed form solution in continuous time and the state of the neuron model (linear Dynamically being divided into two or more states; determining an operational phase of the artificial neuron from the two or more phases based at least in part on the first state; based at least in part on the artificial neuron The first state and the determined operational state update the state of the artificial neuron; and generate a plurality of neural behaviors of the artificial neuron by utilizing linear dynamics of the neuron model.

In another aspect of the present disclosure, processing unit 6804 can be configured to: Updating the state of the artificial neuron based on the event, wherein the neuron model of the artificial neuron has a closed form solution in continuous time and wherein the state dynamics of the neuron model is divided into two or more states; by time The state of the artificial neuron is updated at intervals; and if an event occurs at a time or between times, the state of the artificial neuron is updated.

Figure 69 illustrates an illustration of a neural network 6900 in accordance with certain aspects of the present disclosure. Example implementation. As illustrated in FIG. 69, neural network 6900 can include a plurality of local processing units 6902 that can perform various operations of the methods described above. Each processing unit 6902 can include local state memory 6904 and local parameter memory 6906 that store parameters of the neural network. In addition, the processing unit 6902 can include a memory 6908 having a local (neuron) model program, a memory 6910 having a local learning program, and a local connection memory 6912. In addition, as illustrated in FIG. 69, each local processing unit 6902 can interface with a unit 6914 for configuration processing and interface with routing connection processing component 6916, which can provide configuration of local memory for the local processing unit. Element 6916 provides routing between local processing units 6902.

According to some aspects of the present case, operation 6300 illustrated in Figures 63-65 6,400 and 6500 can be executed in hardware, such as by one or more processing units 6902 from FIG.

Various operations of the methods described above may be capable of performing corresponding work Can be executed by any suitable means. Such means may include various hardware and/or software components and/or modules including, but not limited to, circuits, special application integrated circuits (ASICs) or processors. In general, where the operation is illustrated in the drawings, These operations may have corresponding pairing means functional elements with similar numbers. For example, operations 6300, 6400, and 6500 illustrated in Figures 63, 64, and 65 correspond to components 6300A, 6400A, and 6500A illustrated in Figures 63A, 64A, and 65A.

As used herein, the term "decision" covers a wide variety of Kind of action. For example, a "decision" may include calculations, calculations, processing, derivation, research, inspection (eg, viewing in a table, database, or other data structure), detection, and the like. Moreover, "decision" may include receiving (eg, receiving information), accessing (eg, accessing data in memory), and the like. Moreover, "decisions" may also include analysis, selection, selection, establishment, and the like.

As used herein, the phrase "at least one" in a list of items refers to any combination of the items, including the individual members. As an example, " at least one of a , b or c " is intended to cover: a , b , c , a - b , a - c , b - c and a - b - c .

Combining the various illustrative logic blocks, modules, and The circuit may be a general purpose processor, digital signal processor (DSP), special application integrated circuit (ASIC), field programmable gate array signal (FPGA), or other programmable logic device (PLD) designed to perform the functions described herein. ), individual gate or transistor logic, individual hardware components, or any combination of the foregoing to implement or perform. A general purpose processor may be a microprocessor, but in the alternative, the processor may be any commercially available processor, controller, microcontroller, or state machine. The processor may also be implemented as a combination of computing devices, such as a combination of a DSP and a microprocessor, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration.

The steps of the method or algorithm described in this case can be directly on the hardware Implemented in a software module executed by a processor or a combination of the two. The software modules can reside in any form of storage medium known in the art. Some examples of storage media that may be used include random access memory (RAM), read only memory (ROM), flash memory, EPROM memory, EEPROM memory, scratchpad, hard disk, removable magnetic Disc, CD-ROM, etc. The software module can include a single instruction or a majority of instructions, and can be distributed over several different code segments, distributed among different programs, and distributed across multiple storage media. The storage medium can be coupled to the processor such that the processor can read and write information from/to the storage medium. Alternatively, the storage medium can be integrated into the processor.

The methods disclosed herein comprise one or more steps or actions for achieving the methods described. The method steps and/or actions may be interchanged without departing from the scope of the claimed invention. In other words, the order and/or use of specific steps and/or actions may be modified without departing from the scope of the claims.

The functions described can be implemented in hardware, software, firmware, or any combination of the foregoing. If implemented in hardware, the example hardware settings can include a processing system in the device. The processing system can be implemented with a bus architecture. The bus bar can include any number of interconnect bus bars and bridges depending on the particular application of the processing system and overall design constraints. The bus bar connects various circuits including the processor, machine readable media, and bus interface. The bus interface can be used to connect a network interface card or the like to a processing system via a bus bar. The network interface card can be used to implement signal processing functions. For some aspects, the user interface (eg, keypad, display, mouse, joystick, etc.) Can be connected to the bus. The busbars can also be connected to various other circuits (such as timing sources, peripherals, voltage regulators, power management circuits, etc.), which are well known in the art and will not be described again.

The processor is responsible for managing the bus and general processing, including executing software stored on machine readable media. The processor can be implemented with one or more general purpose and/or special purpose processors. Examples include microprocessors, microcontrollers, DSP processors, and other circuitry that can execute software. Software should be interpreted broadly to mean instructions, materials, or any combination of the above, whether referred to as software, firmware, media, software, hardware, or other description. As an example, the machine readable medium may include RAM (random access memory), flash memory, ROM (read only memory), PROM (programmable read only memory), EPROM (erasable program) Design read-only memory), EEPROM (Electrically Erasable Programmable Read Only Memory), scratchpad, magnetic disk, optical disk, hard drive or any other suitable storage medium or any combination thereof. Machine readable media can be implemented in a computer program product. The computer program product can include packaging materials.

In a hardware implementation, the machine readable medium can be part of the processing system separate from the processor. However, as will be readily appreciated by those skilled in the art, the machine readable medium or any portion thereof can be external to the processing system. By way of example, a machine readable medium can include a transmission line, a carrier modulated by the data, and/or a computer product separate from the device, all of which can be accessed by the processor via the bus interface. Alternatively or additionally, the machine readable medium or any portion thereof may be integrated into the processor, such as cache memory and/or general purpose register files.

The processing system can be configured as a general purpose processing system having one or more microprocessors providing processor functionality and external memory providing at least a portion of machine readable media, a microprocessor and external memory The body is connected to other supporting circuits by an external bus structure. Alternatively, the processing system can include one or more neuromorphic processors for implementing the neuron model and the nervous system model described herein. As a further alternative, the processing system may use an ASIC (Special Application Integrated Circuit) with a processor integrated in a single chip, a bus interface, a user interface, a support circuitry, and at least a portion of machine readable media. To implement, either with one or more FPGAs (field programmable gate arrays), PLDs (programmable logic devices), controllers, state machines, gated logic, individual hardware components, or any other suitable circuitry or It can be implemented by any combination of circuits capable of performing the various functions described throughout the present application. Those skilled in the art will recognize how best to implement the functions described with respect to the processing system, depending on the particular application and the overall design constraints imposed on the overall system.

Machine readable media can include several software modules. The software modules include instructions that, when executed by the processor, cause the processing system to perform various functions. The software modules can include a transmission module and a receiving module. Each software module can reside in a single storage device or be distributed across multiple storage devices. As an example, when a trigger event occurs, the software module can be loaded into the RAM from the hard drive. During execution of the software module, the processor can load some instructions into the cache to increase access speed. One or more cache memory lines can then be loaded into the general purpose scratchpad file for execution by the processor. When talking about the functionality of a software module, it will be understood that such functionality is performed on the processor. The instructions from the software module are implemented by the processor.

If implemented in software, each function can be stored as one or more instructions or codes on a computer readable medium or on a computer readable medium. Computer readable media includes both computer storage media and communication media, including any media that facilitates the transfer of a computer program from one location to another. The storage medium can be any available media that can be accessed by the computer. By way of example and not limitation, such computer readable medium may include RAM, ROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage device or can be used to carry or store instructions or data structures. Any other medium that expects code and can be accessed by a computer. Any connection is also properly referred to as computer readable media. For example, if the software is transmitted from a web site, server, or other remote source using coaxial cable, fiber optic cable, twisted pair, digital subscriber line (DSL), or wireless technologies such as infrared (IR), radio, and microwave. The coaxial cable, fiber optic cable, twisted pair, DSL, or wireless technologies (such as infrared, radio, and microwave) are included in the definition of the media. As used herein, a disk (Disk) and disc (Disc), includes compact disc (CD), laser disc, optical disc, digital versatile disc (DVD), floppy disk and Blu-ray ^® disc where disks (Disk) magnetically often The data is reproduced, and the disc uses laser to optically reproduce the data. Thus, in some aspects, computer readable media can include non-transitory computer readable media (eg, tangible media). Additionally, for other aspects, computer readable media can include transient computer readable media (eg, signals). The above combinations should also be included in the scope of computer readable media.

Therefore, some aspects may include operations for performing the operations described herein. Computer program product. For example, such computer program products can include computer readable media having stored thereon (and/or encoded) instructions executable by one or more processors to perform the operations described herein. For some aspects, computer program products may include packaging materials.

In addition, it should be appreciated that modules and/or other suitable means for performing the methods and techniques described herein can be downloaded and/or otherwise obtained by a user terminal and/or base station where applicable. For example, such a device can be coupled to a server to facilitate the transfer of means for performing the methods described herein. Alternatively, the various methods described herein can be provided via storage means (eg, RAM, ROM, physical storage media such as compact discs (CDs) or floppy disks, etc.) such that once the storage means is coupled or provided for use The terminal and/or the base station can obtain various methods by this means. Moreover, any other suitable technique suitable for providing the methods and techniques described herein to a device can be utilized.

It should be understood that the scope of the patent application is not limited to the precise arrangements and elements illustrated. Various modifications, changes and variations can be made in the arrangement, operation and details of the methods and apparatus described above without departing from the scope of the claims.

6300‧‧‧ operation

6302‧‧‧Steps

6304‧‧‧Steps

6306‧‧‧Steps

Claims (43)

A method for updating a state of an artificial neuron, the method comprising the steps of: determining a first state of the artificial neuron, wherein a neuron model of the artificial neuron has a closed form solution in continuous time And wherein the state dynamics of the neuron model are divided into two or more regimes; the artificial neurons are determined from the two or more states based at least in part on the first state An operational phase; and updating the state of the artificial neuron based at least in part on the first state of the artificial neuron and the determined operational phase. The method of claim 1, wherein the two or more states comprise: a first phase and a second phase, and wherein the state dynamics of the neuron model are in the first phase It tends to rest and tends to issue spikes in the second phase. The method of claim 1, wherein the two or more states comprise: a first phase and a second phase, and wherein the state dynamics of the neuron model are in the first phase It tends to a first reference and tends to deviate from a second reference in the second phase. The method of claim 1, wherein the two or more states comprise: a first phase and a second phase, and wherein the states of the neuron model Dynamics exhibits a Leaky Integral Excitation (LIF) behavior in the first phase and an Anti-Leakage Integral Excitation (ALIF) behavior in the second phase. The method of claim 1, the method further comprising the step of determining, based on at least one of the first state and the operational phase, a time at which the neuron model is different from the time of the first state Second state. The method of claim 5, wherein the first state of the neuron model corresponds to a first event, wherein the second state corresponds to a second event, and wherein the second event is after the first event The next event. The method of claim 6, wherein the first event comprises: an input event, an output event, or an artificial event for the neuron model. The method of claim 6, wherein the step of determining the second state comprises the step of: based at least in part on a time between the first event and the second event and the states in the working phase Dynamically determines the second state of the neuron model. The method of claim 8, wherein the first state is defined by a membrane potential element ( v ) and a recovery current ( u ) of the neuron model, and wherein the second state is determined according to the following formula: v '= v + q _ρ , u '= u + r , q _ρ =- τ _ρ βu - v _ρ ', and r = δ ( v + ε ), where Δ t is the first state and the second state Between the best time of death, if v > Then ρ =+, if v Then ρ =-, Is a one-state phase threshold, τ _ρ is a voltage time constant, τ _u is a recovery current time constant, β is a resistance, v _ρ is a base voltage of the working phase, q _ρ and r are state transformation variables, δ Is a scaling factor and ε is an offset voltage. The method of claim 1, the method further comprising the step of determining when the artificial neuron will fire based, at least in part, on the first state or the at least one of the operational phase. The method of claim 10, wherein the first state is defined by a membrane potential element ( v ) and a recovery current ( u ) of the neuron model, and wherein the step of determining when the artificial neuron will be excited comprises The following steps: Use the following rules: Where τ ₊ is a normal phase voltage time constant, v _S is a defined voltage of an output spike of the neuron model, q ₊ is a normal phase state transition variable, Is a state with the threshold, and a △ t _S is the expected time until the artificial neuron will fire before. The method of claim 1, wherein the step of updating the state of the artificial neuron comprises the step of updating the state at a time of an event or updating the state immediately after a time of an event. The method of claim 1, wherein the step of updating the state of the artificial neuron comprises the step of updating the state at the determined time interval. The method of claim 1, wherein the step of updating the state of the artificial neuron comprises the step of updating the state if an event occurs between the determined time interval or the determined time interval. The method of claim 1, wherein the state dynamics are defined by a membrane potential element of the neuron model and a recovery current. The method of claim 1, the method further comprising the step of dynamically applying the state to the state of the neuron model according to the following formula after the first state progresses from a previous event to an input event Input event: Wherein the first state and a recovery current (u) is defined by a membrane potential-membered (v) the neuron model, △ t is the previous event between the input event and a time elapsed, h _v h _u and Is the input channel function, τ _ρ is a voltage time constant, τ _u is a recovery current time constant, and i is a function to model the input event. The method of claim 16, wherein a total contribution of the elapsed time between the previous event and the input event is provided by: Where g is an exponentially decaying excitatory or inhibitory input, and τ _g is a decay time constant. The method of claim 1, the method further comprising the step of: progressing the state of the incoming neuron from a previous event to a next event; giving an input at the next time event The state is updated at the next event time; and an event after the next event is expected to occur. The method of claim 1, the method further comprising the step of applying one or more artificial events to change the coupling of the neuron model, wherein the state of the artificial neuron is after applying the one or more artificial events Decoupled from the last event. The method of claim 9, wherein the film potential ( v ) and the zero inclination of the recovery current ( u ) are respectively provided by the negative of the state transformation variables q _ρ and r , wherein: v = τ _ρ βu + v _ρ or u =( v - v _ρ )/ τ _ρ β , u =- δ ( v + ε ) or v =- u / δ - ε , and wherein the parameter δ is the slope controlling the u zero inclination A scale factor that can be positive (for δ <0) or negative (for δ >0). An apparatus for updating a state of an artificial neuron, comprising a processing system configured to: determine a first state of the artificial neuron, wherein a neuron model of the artificial neuron is in continuous time Has a closed form solution, and wherein the state dynamics of the neuron model is divided into two or more states; at least in part based on the first state from the two or more states Determining an operational phase of the artificial neuron; and updating the state of the artificial neuron based at least in part on the first state of the artificial neuron and the determined operational phase. An apparatus for updating a state of an artificial neuron, comprising: means for determining a first state of the artificial neuron, wherein a neuron model of the artificial neuron has a closed form solution in continuous time And wherein the state dynamics of the neuron model are divided into two or more states; Means for determining an operational phase of the artificial neuron from the two or more phases based at least in part on the first state; and for the first based at least in part on the artificial neuron A means of updating the state of the artificial neuron with the state and the determined working state. A computer program product for updating a state of an artificial neuron, comprising a computer readable medium, the computer readable medium containing code for determining a first state of the artificial neuron, Wherein a neuron model of the artificial neuron has a closed form solution in continuous time, and wherein the state dynamics of the neuron model is divided into two or more states; at least in part based on the Determining an operational phase of the artificial neuron from the two or more states; and updating the at least in part based on the first state of the artificial neuron and the determined operational phase This state of artificial neurons. A method for generating neural behavior of an artificial neuron, the method comprising the steps of: determining a first state of the artificial neuron, wherein a neuron model of the artificial neuron has a closed form solution in continuous time, And wherein the linear dynamics of the neuron model are divided into two or more states; at least in part based on the first state from the two or more states Determining an operational phase of the artificial neuron; updating the state of the artificial neuron based at least in part on the first state of the artificial neuron and the determined operational phase; and utilizing the neuron model These linear dynamics produce a variety of neural behaviors of the artificial neuron. The method of claim 24, wherein the neural behavior comprises: a sub-threshold oscillation of the maintenance of the artificial neuron. The method of claim 25, wherein the neural behavior of the artificial neuron further comprises at least one of: rebound, depolarization after a spike, a tonic peak, a pre-peak and a post-peak oscillation, and a short rebound Pulse, tolerance, phase spike, bistable, class 1 excitation, tonic short pulse, resonance, spike caused by inhibition, class 2 excitation, phase short pulse, short pulse caused by inhibition, frequency self-adjustment or Integral behavior. The method of claim 24, wherein the linear dynamics of the neuron model comprises: a sub-threshold phase having a leak-integrated excitation that acts to return the artificial neuron to rest A dynamic negative time constant, and a threshold upper state phase having a positive time constant representing an anti-leakage integral excitation dynamic that drives the artificial neuron to issue a spike with a waiting time. The method of claim 27, wherein the linear dynamics of the neuron model are defined as: q _ρ =- τ _ρ βu - v _ρ , and r = δ ( v + ε ), where v is a membrane potential element of the neuron model, u is a recovery current of the neuron model, τ _{ρ =+} is A normal phase voltage time constant, τ _{ρ =-} is a negative phase voltage time constant, τ _u is a recovery current time constant, q _ρ and r are state transition variables, β is a coupling resistance, and δ is a coupled conductance Rate-time, ε is a coupled offset voltage, v _{ρ =+} is a normal phase base voltage, and v _{ρ =-} is a negative phase base voltage. A method as recited in claim 28, wherein the closed state solution for the state of the artificial neuron and for a time from another state to the first state is defined as: Where △ t is the v _f of the first state of the time of arrival from the other state v _0. An apparatus for generating neural behavior of an artificial neuron, comprising a processing system configured to: determine a first state of the artificial neuron, wherein a neuron model of the artificial neuron is in continuous time Has a closed form solution, and wherein the linear dynamics of the neuron model are divided into two or more states; at least in part based on the first state from the two or more states Determining an operational phase of the artificial neuron; updating the state of the artificial neuron based at least in part on the first state of the artificial neuron and the determined operational phase; and by utilizing the neuron These linear dynamics of the model produce a variety of neural behaviors of the artificial neuron. An apparatus for generating neural behavior of an artificial neuron, comprising: means for determining a first state of the artificial neuron, wherein a neuron model of the artificial neuron has a closed form solution in continuous time And wherein the linear dynamics of the neuron model are divided into two or more regimes; for determining the artifact from the two or more states based at least in part on the first state Means of a working phase of a neuron; means for updating the state of the artificial neuron based at least in part on the first state of the artificial neuron and the determined operational state; and The linear dynamics of the neuron model are utilized to generate a variety of neural behavioral means of the artificial neuron. A computer program product for generating neural behavior of an artificial neuron, comprising a computer readable medium, the computer readable medium containing code for determining a first state of the artificial neuron, Wherein a neuron model of the artificial neuron has a closed form solution in continuous time, and wherein the linear dynamics of the neuron model are divided into two or more regimes; based at least in part on the Determining an operational phase of the artificial neuron from the two or more states; updating the artificial based at least in part on the first state of the artificial neuron and the determined operational phase This state of the neuron; and the various neural behaviors of the artificial neuron by utilizing the linear dynamics of the neuron model. A method for updating a state of an artificial neuron, the method comprising the steps of: updating the state of the artificial neuron based at least in part on an event, wherein a neuron model of the artificial neuron has a continuous time a closed form solution and wherein the state dynamics of the neuron model are divided into two or more regimes; the state of the artificial neuron is updated at time intervals; and if at time or between times When an event occurs, the state of the artificial neuron is updated. The method according to item 33 of the mentioned request, wherein the event from time t to time t + △ t event to update the status lines by: Where v is a membrane potential element of the neuron model, u is a recovery current of the neuron model, τ _ρ is a one-state voltage time constant, τ _u is a recovery current time constant, and q _ρ and r are states Transform variables. The method as recited in claim 34, wherein the events are expected to occur according to the following formula: Where △ t is from another state v ₀ v _f reaches a state of a time. The method according to item 33 of the mentioned request, wherein in these time intervals t + △ t at the state updating is defined as: Where v is a membrane potential element of the neuron model, u is a recovery current of the neuron model, τ _ρ is a one-phase voltage time constant, τ _u is a recovery current time constant, q _ρ and r are state transitions The variable, β is the coupling resistance, δ is a coupling conductivity-time, ε is a coupling offset voltage, and v _ρ is the one-phase voltage. The method of claim 33, wherein the difference in a membrane potential of the neuron model decreases over time. The method of claim 37, wherein the difference is monotonically decreasing as provided by the following equation in a negative phase of the two or more states: Wherein v ₁ and v ₂ is the membrane potential of the neuron element model, t + △ t is a time of the event, and τ _- is a negative voltage relative to the time constant state. The method of claim 33, wherein a spike wait time of the neuron model is a function of an input spike timing as provided by: Where τ ₊ is a negative normal phase voltage time constant, v _S is a threshold voltage, v ₀ is a voltage film state, τ _i is a time from an input spike of a presynaptic neuron i , and Δ τ _i is a connection delay from the presynaptic neuron i . The method as recited in claim 39, wherein the letter is used as an input spike timing The peak wait time of the number is equivalent to a linear system of equations in a negative logarithmic domain. An apparatus for updating a state of an artificial neuron, comprising a processing system configured to: update the state of the artificial neuron based at least in part on an event, wherein a neuron of the artificial neuron The model has a closed form solution in continuous time and wherein the state dynamics of the neuron model is divided into two or more regimes; the state of the artificial neuron is updated at time intervals; and if When an event occurs at or between times, the state of the artificial neuron is updated. An apparatus for updating a state of an artificial neuron, comprising: means for updating the state of the artificial neuron based at least in part on an event, wherein a neuron model of the artificial neuron has a continuous time a closed form solution and wherein the state dynamics of the neuron model are divided into two or more regimes; means for updating the state of the artificial neuron at time intervals; and for A means of updating the state of the artificial neuron at or after an event occurs. A computer program product for updating the state of an artificial neuron, comprising a computer readable medium, the computer readable medium comprising the following actions The code: updating the state of the artificial neuron based at least in part on an event, wherein a neuron model of the artificial neuron has a closed form solution in continuous time and wherein the state dynamics of the neuron model are divided into Two or more regimes; updating the state of the artificial neuron at time intervals; and updating the state of the artificial neuron if an event occurs at or between times.