CN113344191B - Continuous Rulkov electronic neuron circuit with super multi-stability - Google Patents

Continuous Rulkov electronic neuron circuit with super multi-stability Download PDF

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CN113344191B
CN113344191B CN202110544138.8A CN202110544138A CN113344191B CN 113344191 B CN113344191 B CN 113344191B CN 202110544138 A CN202110544138 A CN 202110544138A CN 113344191 B CN113344191 B CN 113344191B
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徐权
居朱涛
刘通
周杰
陈墨
武花干
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Abstract

The invention relates to the technical field of electronic neurons, in particular to a super-multistability neuronA continuous Rulkov electronic neuron circuit, comprising: the memristor equivalent circuit comprises a memristor equivalent circuit and a Rulkov neuron main circuit, wherein the output end of the memristor equivalent circuit is electrically connected with the input end of the Rulkov neuron main circuit, and the memristor equivalent circuit comprises an operational amplifier U which is electrically connected in sequence1Multiplier M1、M2Resistance R1、R2、R3And a capacitor C1The Rulkov neuron main circuit comprises an integration channel I and an integration channel II which are electrically connected. The discrete-time Rulkov neuron model is converted into the continuous-time Rulkov neuron model by adopting a forward difference algorithm, and the sinusoidal excitation and memristor electromagnetic induction effects are introduced, so that more flexibility is provided for the calculation of the nerve morphology, an analog circuit of the continuous-time Rulkov neuron model is designed, and reference values are provided for the continuous research of the discrete neuron model and the hardware realization of the discrete neuron model.

Description

Continuous Rulkov electronic neuron circuit with super multi-stability
Technical Field
The invention relates to the technical field of electronic neurons, in particular to a continuous Rulkov electronic neuron circuit with super multi-stability.
Background
With the development of science and technology, the study of the neuron by people is more specific and deeper, and researchers continuously improve and adjust the neuron on the basis of the original model by combining the experimental data and the requirement of practical study work, for example, to solve the complex problems of high dimensional number, multi-parameter, etc. faced by ODEs-based neuron models in qualitative and quantitative research, some physicists propose a relatively simplified single neuron model, for example, a Rulkov mapping model, an Izhikevich model and the like, the Rulkov neuron model based on the mapping and the Izhikevich neuron model dispersed by utilizing the Euler method are used as important forms of the neuron model based on ODEs, so that the discharging modes such as rapid peak discharging, normal peak discharging, internal cluster discharging and the like similar to a Hodgkin-Huxley model can be generated, and the mapping-based neuron network can also simulate the biological behavior patterns of a large number of real neuron clusters, and a Rulkov neural network model is proposed in such a background.
The chaotic Rulkov neuron model is a mapping-based neuron model proposed by Rulkov in 2001, and the model describes the discharge mode of a single biological neuron by utilizing a two-dimensional iterative mapping; because the mapping-based neuron model has obvious superiority in the aspects of computing time, transparency of a computing algorithm, computing resources, data storage and the like, the chaotic Rulkov neuron model is widely applied to different fields, particularly the field of computational neuroscience.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the method adopts a forward difference algorithm to convert the model into a continuous time Rulkov neuron model, introduces sinusoidal excitation and memristor electromagnetic induction action, provides more flexibility for the calculation of the nerve morphology, designs an analog circuit of the continuous time Rulkov neuron model, and provides reference values for the research of the continuity of the discrete neuron model and the realization of hardware thereof.
The technical scheme adopted by the invention is as follows: the continuous Rulkov electronic neuron circuit with super multi-stability comprises a memristor equivalent circuit and a Rulkov neuron main circuit, wherein the output end of the memristor equivalent circuit is electrically connected with the input end of the Rulkov neuron main circuit;
further, the memristor equivalent circuit comprises an operational amplifier U1Multiplier M1、M2Resistance R1、R2、R3And a capacitor C1(ii) a Resistance R1One terminal and an operational amplifier U1And a capacitor C1Is connected to a capacitor C1The other end and an operational amplifier U1Output terminal of (1), multiplier M1Are connected to the X and Y inputs of a multiplier M1And multiplier M2The Y input end of the power supply is connected; multiplier M2X input terminal and resistor R1Another terminal, resistance R2Is connected with one end of the connecting rod; multiplier M2Output terminal and resistor R3Is connected to a resistor R3The other end of (1) andresistance R2Is connected at the other end with a resistor R3The other end of the memory resistor is used as the output end of the memristor equivalent circuit;
the Rulkov neuron main circuit comprises an integration channel I and an integration channel II, wherein the integration channel I comprises: operational amplifier U2、U3、U4、U5Capacitor C2Resistance R4、R5、R6、R7、R8、R9、R10、R11、R12、R13、R14、R15Multiplier M3、M4Operational amplifier U2The reverse input ends of the resistors are respectively connected with the resistors R4Resistance R5Resistance R6Resistance R7Capacitor C2One end of (1), the output end of the memristor equivalent circuit, and a capacitor C2And the other end of the same is respectively connected with an operational amplifier U2Output terminal of (1), resistor R7Another terminal of (1), a resistor R8One terminal of (1), resistance R16And a multiplier M3Is connected to the X input terminal of an operational amplifier U3Respectively connected with the resistor R8And the other end of (3) and a resistor R9Is connected to a resistor R9And the other end of the operational amplifier U3Output terminal and multiplier M3Is connected to the Y input terminal of the multiplier M3Output terminal and resistor R10Is connected to an operational amplifier U4Respectively connected with the resistor R10Another terminal of (1), a resistor R11One terminal of (1), resistance R12Is connected to an operational amplifier U4Respectively connected with the resistor R12Another end of (1), multiplier M4Is connected to the X input terminal of the multiplier M4Respectively connected with the resistor R4Another terminal of (1), operational amplifier U5Is connected to the output of the multiplier M4Output end of (3) is connected with a resistor R in series13Then respectively connected with a resistor R14And an operational amplifier U5The reverse input end of the input terminal is connected;
the second integration channel includes: operational amplifier U6、U7Resistance R16、R17、R18Capacitor C3Operational amplifier U6Is connected with the resistor R in series at the reverse input end16The other end of (1), an operational amplifier U6And an operational amplifier U6Is connected with a capacitor C in parallel at the reverse input end3Operational amplifier U6Output end of (3) is connected with a resistor R in series17Postand operational amplifier U7Inverting input terminal connected, operational amplifier U7Inverting input and operational amplifier U7Output end of the resistor R is connected in parallel18；
Operational amplifier U1、U2、U3、U4、U6、U7The non-inverting input terminal of the operational amplifier U is grounded5Is connected with the resistor R in series at the same-direction input end15And then grounded.
Further, the discrete Rulkov neuron model can be expressed as:
Figure GDA0003498251380000031
wherein n represents a discrete time (n ═ 1,2 …); alpha and sigma represent control variables; x is the number ofnRepresenting the fast variation of the transmembrane voltage of the neuron cell in the system; y isnA slow variable representing ion recovery current; eta is a very small parameter to ensure that the second state variable yn+1Is a slow variable.
On the basis, a forward difference algorithm is adopted to convert the discrete model into a continuous Rulkov neuron model for research, wherein the continuous Rulkov neuron model can be expressed as follows:
Figure GDA0003498251380000032
in the formula, x is the neuron membrane potential, y is a recovery variable, and alpha, sigma and beta are control parameters.
On the basis of a continuous Rulkov neuron model, a sinusoidal signal excitation and memristor electromagnetic induction action are introduced into a membrane potential term to form the continuous Rulkov electronic neuron with super multi-stability, wherein the model is expressed as follows:
Figure GDA0003498251380000041
wherein x is the neuronal membrane potential, y is the recovery variable,
Figure GDA0003498251380000042
for memristor internal variables, alpha is a control parameter, a and b represent memristor internal control parameters, sigma represents the control parameter, beta is a constant,
Figure GDA0003498251380000043
the method is characterized in that the method represents memristive magnetic induction current, k represents electromagnetic induction coupling strength, Asin (2 pi F tau) represents an external sinusoidal excitation signal, and F and A represent the frequency and amplitude of the sinusoidal excitation signal respectively.
In the model, β is considered as 0, 3 equations in 3 inverse integration circuit equivalent implementation formulas (3) are respectively adopted and are recorded as an integration channel one, an integration channel two and an integration channel three, wherein the integration channel one and the integration channel two are implemented through a Rulkov neuron main circuit, the integration channel three is implemented through a memristive equivalent circuit, and the circuit equations corresponding to the 3 integration channels can be expressed as follows according to kirchhoff's law:
Figure GDA0003498251380000044
in the formula, Vx、VyAnd
Figure GDA0003498251380000046
is 3 circuit state variables corresponding to x, y and x in the continuous Rulkov neuron model
Figure GDA0003498251380000045
C1、R1、R2、R3、g1、g2For memory-resistance equivalent circuit element parameters, C2、C3、R4、R5、R6、R7、R10、R11、R12、R16、g3、g4For the parameters of the main circuit element of the Rulkov neuron, Asin (2 pi ft) is a sinusoidal voltage signal excitation source, and the frequency and the amplitude of f and A sinusoidal voltage excitation signals.
The invention has the beneficial effects that:
1. the observed multiple attractors in the experimental circuit are basically consistent with the simulation result, and the correctness of theoretical analysis and numerical analysis is verified.
2. The constructed continuous Rulkov electronic neuron circuit with super multi-stability has scientific theoretical basis, and the designed analog circuit unit provides reference value for the research of discrete neuron model continuity and the hardware realization thereof.
Drawings
FIG. 1 is a schematic diagram of the structure of the main circuit of a Rulkov neuron;
FIG. 2 is a schematic diagram of a memristive equivalent circuit structure;
FIG. 3 is initial memristance values
Figure GDA0003498251380000051
On the abscissa of the branch diagram
Figure GDA0003498251380000052
Variation of (2), ordinate xmaxRepresents the maximum membrane potential;
FIG. 4 shows the numerical simulation results when the initial values are (0,0, -2.2), (0,0,0.5), (0,0,1) and (0,0,1.45), respectively
Figure GDA0003498251380000053
Co-existing multiple attractors generated in a plane;
FIG. 5 is a circuit experimental verification
Figure GDA0003498251380000054
Co-existing multiple attractors captured in a planar experiment.
Detailed Description
The invention will be further described with reference to the accompanying drawings and examples, which are simplified schematic drawings and illustrate only the basic structure of the invention in a schematic manner, and therefore only show the structures relevant to the invention.
The invention provides a continuous Rulkov electronic neuron circuit with super multi-stability, which comprises: the memristive equivalent circuit of fig. 2 and the Rulkov neuron main circuit of fig. 1; the memristor equivalent circuit is used as input and introduced into a Rulkov neuron main circuit to form a continuous Rulkov electronic neuron circuit with super multi-stability.
The memristor equivalent circuit is an integral channel three, and the input voltage is-VxOperational amplifier U1Output voltage of
Figure GDA0003498251380000064
Through a multiplier M1、M2Resistance R1、R2、R3And a capacitor C1And then outputting the dimensionless equivalent voltage.
The Rulkov neuron main circuit comprises an integration channel I and an integration channel II, and a resistor R of the integration channel I11And R14Respectively connected into DC power supply V in neuron 11V and V2＝11V，VxAnd VyAnd
Figure GDA0003498251380000065
the internal output end of the neuron and the external output end under external stimulation are connected with different channels of an oscilloscope for observation, and a sinusoidal signal voltage excitation source VSAsin (2 pi ft) as an external voltage stimulus input, operational amplifier U2The voltage at the output terminal is VxThrough an operational amplifier U3The rear output end is-VxResistance R5Connecting a sinusoidal signal voltage excitation source VS。
The second integration channel realization circuit comprises an operational amplifier U6、U7Operational amplifier U6The voltage at the output terminal is VyIs connected with an operational amplifier U7Operational amplifier U after reverse input end7Output voltage of-Vy。
Mathematical modeling: the invention is based on a continuous Rulkov neuron model, in order to better research the influence of external stimulation on the discharge behavior of the Rulkov neuron, sinusoidal signal excitation and memristor electromagnetic induction are introduced as external stimulation input, and for convenience of analysis and circuit realization, the model can be described as follows by a first-order ordinary differential equation set:
Figure GDA0003498251380000061
wherein x is the neuronal membrane potential, y is the recovery variable,
Figure GDA0003498251380000062
for memristor internal variables, alpha is a control parameter, a and b represent memristor internal control parameters, sigma represents the control parameter, beta is a constant,
Figure GDA0003498251380000063
the method is characterized in that the method represents memristive magnetic induction current, k represents electromagnetic induction coupling strength, Asin (2 pi F tau) represents an external sinusoidal excitation signal, and F and A represent the frequency and amplitude of the sinusoidal excitation signal respectively.
Numerical simulation: using the MATLAB ODE23 algorithm to develop numerical studies on continuous Rulkov electronic neurons with initial memristance values as system parameters, where α is 11, σ is 6, a is 2, F is 1, a is 1, b is 2, and k is 0.1, and fig. 3 shows plotted internal memristance variables plotted
Figure GDA0003498251380000071
Taking initial value of memristor
Figure GDA0003498251380000072
The time-varying bifurcation diagram can find that the maximum value of the membrane potential changes with the initial value of the memristor to generate bifurcations of a multiple period and a reverse multiple period, and the maximum value of the membrane potential contains rich dynamic behaviors, so that an infinite number of attractors coexist.
Shown in FIG. 4 as an example of a limited number
Figure GDA0003498251380000073
The numerical simulation results of the coexisting attractors generated in the plane are respectively
Figure GDA0003498251380000074
A cycle 2 limit cycle with an initial value of (0,0, -2.2), a chaotic attractor with an initial value of (0,0,0.5), a cycle 2 limit cycle with an initial value of (0,0,1), and a cycle 3 limit cycle with an initial value of (0,0, 1.45).
Circuit simulation and experimental verification: in fig. 1 and 2, the super multi-stability continuous Rulkov electronic neuron circuit has three integrating channels for implementing a first, a second and a third equation of equation (1), wherein the first and the second equations are implemented by a Rulkov neuron main circuit, the third equation is implemented by a memristive equivalent circuit, and the circuit equations shown in fig. 1 and 2 can be written according to kirchhoff's circuit law and the electrical characteristics of circuit components:
Figure GDA0003498251380000075
wherein, Vx、VyAnd
Figure GDA0003498251380000076
are three circuit variables that correspond to x, y and x in a continuous Rulkov neuron model
Figure GDA0003498251380000077
-VxIs a variable VxOutput variable, -V after inverting amplifieryIs a variable VyAnd (4) outputting the variable after passing through the inverting amplifier.
The time precision is taken to be 0.1ms, namely R is 10k omega and C1＝C2＝C3By comparing formula (3) and formula (4), 10nF, f 1/RC yields:
Figure GDA0003498251380000081
the design adopts MULTISI 12.0 to complete the circuit construction and simulation operation work, the discrete device adopts an AD711JN operational amplifier and an AD633JN multiplier with the power supply voltage of +/-15V working voltage, the discrete element adopts a resistor and a capacitor, the circuit element parameters are typical circuit parameter values in the formula (5), and because different required initial capacitor voltages are difficult to accurately distribute in the experiment, the behavior of only a limited number of coexisting attractors is verified through repeatedly switching a power supply and randomly sensing, the experimental result is captured by a Tektronix four-channel oscilloscope, and the obtained coexisting attractor result is shown in figure 5.
By comparing the experimental result with the numerical simulation result, the observed multiple attractors in the experimental circuit can be proved to be basically consistent with the simulation result, and the correctness of theoretical analysis and numerical analysis can be verified; therefore, the continuous Rulkov electronic neuron with super multi-stability constructed by the invention has scientific theoretical basis, and the designed analog circuit unit provides reference value for the research of discrete neuron model continuity and hardware realization thereof.
In light of the foregoing description of the preferred embodiment of the present invention, many modifications and variations will be apparent to those skilled in the art without departing from the spirit and scope of the invention. The technical scope of the present invention is not limited to the contents of the specification, and must be determined according to the scope of the claims.

Claims (2)

1. Continuous Rulkov electronic neuron circuit with super multi-stability, which is characterized in that: the memristor equivalent circuit comprises a memristor equivalent circuit and a Rulkov neuron main circuit, wherein the output end of the memristor equivalent circuit is electrically connected with the input end of the Rulkov neuron main circuit; the memristor equivalent circuit comprises an operational amplifier U1Multiplier M1、M2Resistance R1、R2、R3And a capacitor C1(ii) a The resistor R1One end of the operational amplifier U1And a capacitor C1Is connected to the capacitor C1The other end of the operational amplifier U1The output terminal of (1), the multiplier M1Is connected to the X, Y input terminal of the multiplier M1And the multiplier M2The Y input end of the power supply is connected; the multiplier M2And the resistor R1The other end, the resistor R2Is connected with one end of the connecting rod; the multiplier M2And the output end of the resistor R3Is connected to one end of the resistor R3And the other end of (2) and the resistor R2The other end of the memristor equivalent circuit is connected and used as the output end of the memristor equivalent circuit;
the Rulkov neuron main circuit comprises an integration channel I and an integration channel II, wherein the integration channel I comprises: operational amplifier U2、U3、U4、U5Capacitor C2Resistance R4、R5、R6、R7、R8、R9、R10、R11、R12、R13、R14、R15Multiplier M3、M4The operational amplifier U2Are respectively connected with the resistors R4The resistor R5The resistor R6The resistor R7The capacitor C2And the output end of the memristor equivalent circuit, the capacitor C2And the other end of each of the first and second transistors is connected to the operational amplifier U2The output terminal of (1), the resistor R7Another terminal of (3), the resistor R8And said multiplier M3Is connected to the X input terminal of the operational amplifier U3Respectively with said resistor R8And the other end of (2) and the resistor R9Is connected to one end of the resistor R9And the other end of (1) and the operational amplifier U3An output terminal and the multiplier M3Is connected to the Y input terminal of the multiplier M3And the output end of the resistor R10Is connected to said operational amplifier U4Respectively with said resistor R10Another terminal of (3), the resistor R11One end of, the resistor R12Is connected to said operational amplifier U4Respectively with the resistor R12Another end of (3), said multiplier M4Is connected to the X input terminal of the multiplier M4Respectively with the resistors R4Another terminal of (1), the operational amplifier U5Is connected to the output of the multiplier M4Is connected in series with the resistor R13Then respectively connected with the resistors R14And said operational amplifier U5The reverse input end of the input terminal is connected;
the second integration channel comprises: operational amplifier U6、U7Resistance R16、R17、R18Capacitor C3The operational amplifier U6Is connected in series with the resistor R16The operational amplifier U6And said operational amplifier U6Is connected in parallel with the capacitor C3The operational amplifier U6Is connected in series with the resistor R17post-AND the operational amplifier U7Reverse input connection, the operational amplifier U7Inverting input and said operational amplifier U7Is connected in parallel with the resistor R18；
The operational amplifier U1、U2、U3、U4、U6、U7Is grounded, the operational amplifier U5The same-direction input end of the resistor R is connected in series with the resistor R15And then grounded.
2. A continuous Rulkov electronic neuron circuit with super multi-stability according to claim 1, wherein the memristive equivalent circuit and the Rulkov neuron main circuit can be expressed by circuit equations corresponding to 3 integration channels according to kirchhoff's law as follows:
Figure FDA0003498251370000021
in the formula, Vx、VyAnd
Figure FDA0003498251370000022
is 3 circuit state variables, C1、R2、R3、g1、g2For memory-resistance equivalent circuit element parameters, C2、C3、R4、R5、R6、R7、R10、R11、R12、R16、g3、g4For the parameters of the main circuit element of the Rulkov neuron, Asin (2 pi ft) is a sinusoidal voltage signal excitation source, and the frequency and the amplitude of f and A sinusoidal voltage excitation signals.
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