CN116402105B - Method for establishing fractional order Chay neuron model with complex discharge characteristic - Google Patents
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- 210000002569 neuron Anatomy 0.000 title claims abstract description 62
- 238000000034 method Methods 0.000 title claims abstract description 21
- 229910001424 calcium ion Inorganic materials 0.000 claims description 24
- BHPQYMZQTOCNFJ-UHFFFAOYSA-N Calcium cation Chemical compound [Ca+2] BHPQYMZQTOCNFJ-UHFFFAOYSA-N 0.000 claims description 18
- 210000000170 cell membrane Anatomy 0.000 claims description 18
- 102000004257 Potassium Channel Human genes 0.000 claims description 15
- 239000012528 membrane Substances 0.000 claims description 15
- 108020001213 potassium channel Proteins 0.000 claims description 15
- 238000010586 diagram Methods 0.000 claims description 11
- 238000012546 transfer Methods 0.000 claims description 11
- 102000004310 Ion Channels Human genes 0.000 claims description 9
- 108090000862 Ion Channels Proteins 0.000 claims description 9
- DGAQECJNVWCQMB-PUAWFVPOSA-M Ilexoside XXIX Chemical compound C[C@@H]1CC[C@@]2(CC[C@@]3(C(=CC[C@H]4[C@]3(CC[C@@H]5[C@@]4(CC[C@@H](C5(C)C)OS(=O)(=O)[O-])C)C)[C@@H]2[C@]1(C)O)C)C(=O)O[C@H]6[C@@H]([C@H]([C@@H]([C@H](O6)CO)O)O)O.[Na+] DGAQECJNVWCQMB-PUAWFVPOSA-M 0.000 claims description 6
- 125000003275 alpha amino acid group Chemical group 0.000 claims description 6
- 230000008859 change Effects 0.000 claims description 6
- 150000001875 compounds Chemical class 0.000 claims description 6
- 230000001419 dependent effect Effects 0.000 claims description 6
- 230000003834 intracellular effect Effects 0.000 claims description 6
- 239000011734 sodium Substances 0.000 claims description 6
- 229910052708 sodium Inorganic materials 0.000 claims description 6
- 108091006146 Channels Proteins 0.000 claims description 3
- 230000003213 activating effect Effects 0.000 claims description 3
- 230000004913 activation Effects 0.000 claims description 3
- 230000001174 ascending effect Effects 0.000 claims description 3
- 230000009849 deactivation Effects 0.000 claims description 3
- 102000008396 voltage-gated potassium channel activity proteins Human genes 0.000 claims description 3
- 108040002559 voltage-gated potassium channel activity proteins Proteins 0.000 claims description 3
- 238000006243 chemical reaction Methods 0.000 claims description 2
- 230000010354 integration Effects 0.000 claims description 2
- 230000005611 electricity Effects 0.000 claims 1
- 230000000638 stimulation Effects 0.000 abstract 1
- 208000028659 discharge Diseases 0.000 description 50
- 238000002474 experimental method Methods 0.000 description 19
- 238000010587 phase diagram Methods 0.000 description 15
- 238000012795 verification Methods 0.000 description 13
- 230000000739 chaotic effect Effects 0.000 description 10
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- 238000004088 simulation Methods 0.000 description 6
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- 108090000312 Calcium Channels Proteins 0.000 description 2
- 102000003922 Calcium Channels Human genes 0.000 description 2
- 230000009286 beneficial effect Effects 0.000 description 2
- 238000013473 artificial intelligence Methods 0.000 description 1
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- 239000000470 constituent Substances 0.000 description 1
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- 238000010168 coupling process Methods 0.000 description 1
- 238000005859 coupling reaction Methods 0.000 description 1
- 238000007599 discharging Methods 0.000 description 1
- 239000003814 drug Substances 0.000 description 1
- 210000000653 nervous system Anatomy 0.000 description 1
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
- G06N3/00—Computing arrangements based on biological models
- G06N3/02—Neural networks
- G06N3/06—Physical realisation, i.e. hardware implementation of neural networks, neurons or parts of neurons
- G06N3/063—Physical realisation, i.e. hardware implementation of neural networks, neurons or parts of neurons using electronic means
Abstract
The invention discloses a fractional order Chay neuron model with complex discharge characteristics, which adopts Caputo definition to study fractional derivatives, uses a set of differential equations to describe the discharge behavior of neurons, only comprises three dynamic variables and has no external current stimulation, and introduces fractional order q on the basis of the integer order Chay neuron model to obtain the expression of the fractional order Chay neuron model. The fractional order Chay model established by the method is more complex in discharge characteristic than the integer order Chay model, and the discharge characteristic is true and accurate.
Description
Technical Field
The invention belongs to the technical field of information, and relates to a method for establishing a fractional order Chay neuron model with complex discharge characteristics.
Background
Neurons are important constituent units of the nervous system, and each neuron forms a complex neural network through synaptic coupling. Since information is transferred between neurons in the form of firing, it is important to study the firing characteristics of neurons. Among many neurons, the Chay neuron model closely approximates the firing characteristics of real neurons, and is also easier to implement than Hindmarsh-Rose neurons.
In the study of the discharge characteristics of neurons, fractional calculus is taken as a general form of integer calculus, so that the dynamics behavior of the neurons can be described more accurately, and the method becomes a new study direction.
Disclosure of Invention
The invention aims to provide a method for establishing a fractional order Chay neuron model with complex discharge characteristics, which solves the problem that the discharge characteristics of the conventional integer order neuron model are not accurate enough.
The technical scheme adopted by the invention is that the method for establishing the fractional order Chay neuron model with the complex discharge characteristic introduces fractional order on the basis of the integer order Chay neuron modelqThe expression of the fractional Chay neuron model is:
(2)
in the formula (2), the amino acid sequence of the compound,V I is the bias voltage of the sodium ion-calcium ion mixed ion channel;V K is the bias voltage of the potassium ion channel;V L is the bias voltage of the leakage ion channel;V C is the bias voltage of the calcium ion channel;g I is the maximum conductance of the voltage dependent mixing channel;g K.V maximum conductivity of the voltage dependent potassium ion channel;g K.C potassium ion channel maximum conductance depending on calcium ion concentration in cell membrane;g L is the maximum conductance of the leakage path;K C is the rate of intracellular calcium ion outflow;ρis a proportionality constant;V、nis a fast variable, and respectively represents membrane potential and voltage sensitive potassium channel probability;Cis a slow variable representing intracellular calcium ion concentration; d, dn/dtRepresenting the rate of change of the potassium ion channel opening probability depending on the membrane potential; d, dC/dtThe change rate of the calcium ion concentration of the cell membrane is shown, and the right two items respectively show the calcium ions entering and exiting the cell membrane;m ∞ andh ∞ the probability of activation and deactivation of the sodium ion-calcium ion mixed ion channel, respectively;n ∞ is the probability of potassium ion channel openingnIs a stable value of (2);τ n is the relaxation time, the smaller the value is, the steeper the spike is, the expression of each of the four variables is as shown in formula (3):
(3)
in the formula (3), the amino acid sequence of the compound,λ n a time constant that is a fast variable;α m andβ m the speeds of the ascending and descending phases of the activating molecules in the cell membrane are respectively shown;α n andβ n respectively represent K + Speed of entry into and exit from cell membranes;α h andβ h the expression of each of these six variables is as follows, indicating the rate of rise and fall, respectively, of inactive molecule content in the cell membrane:
。(4)
the method has the beneficial effects that the fractional Chay neuron model has complex discharge characteristics, and numerical simulation and hardware experiment verification show that the fractional Chay neuron model is correct and effective, and is beneficial to promoting related research in the fields of biology, medicine, artificial intelligence and the like.
Drawings
FIG. 1a shows an embodiment of the method of the present inventionqWhen the number of the codes is =1,V C a bifurcation chart of peak-to-peak intervals of different orders when the variable is; also in FIG. 1bq=0.99; in FIG. 1cq=0.95；
FIG. 2a is a timing diagram of the 0.99 th order membrane potential in an embodiment of the method of the invention, whereinV C =90; also in FIG. 2bV C =97; in FIG. 2cV C =99; in FIG. 2dV C =100; in FIG. 2eV C =200; in FIG. 2fV C =400；
FIG. 3a is a 0.99 th order of an embodiment of the method of the present inventionV-CPhase diagram in whichV C =90; also in FIG. 3bV C =97; in FIG. 3cV C =99; in FIG. 3dV C =100; in FIG. 3eV C =200; in FIG. 3fV C =400；
FIG. 4a is a 0.99 th order of an embodiment of the method of the present inventionV-nPhase diagram in whichV C =90; also in FIG. 4bV C =97; in FIG. 4cV C =99; in FIG. 4dV C =100; in FIG. 4eV C =200; in FIG. 4fV C =400；
FIG. 5a is a timing chart of the membrane potential at the initial stage of chaotic discharge verified by hardware experiments; FIG. 5b shows the initial stage of chaotic discharge by hardware experimentsV-CA phase diagram;
FIG. 6a is a timing diagram of the membrane potential at the late stage of chaotic discharge verified by hardware experiments; FIG. 6b shows the hardware experiment to verify the late stage of chaotic dischargeV-CA phase diagram;
FIG. 7a is a timing diagram of the membrane potential corresponding to the discharge of the hardware experiment verification period 5; FIG. 7b is a graph of hardware experiment verification period 5 discharge correspondenceV-CA phase diagram;
FIG. 8a is a timing diagram of the membrane potential corresponding to the discharge in the hardware experiment verification period 2; FIG. 8b is a graph of hardware experiment verification period 2 discharge correspondenceV-CPhase diagram.
Detailed Description
The invention will be described in detail below with reference to the drawings and the detailed description.
The method for establishing the fractional order Chay neuron model is implemented according to the following steps:
using the Caputo definition to study fractional derivatives, the univariate functionf(x) A kind of electronic deviceqThe differential order is defined as formula (1):
(1)
in the formula (1), the components are as follows,qin order of the fractional order of the number,m-1<q≤m；mis not smaller thanqIs the smallest integer of (a);aandtthe upper and lower limits of the integral, respectively; Γ () is a Gamma function;f (m) (x) Is a unitary functionf(x) A kind of electronic devicemAn order derivative;
the Chay neuron model describes the discharging behavior of neurons by a set of differential equations, only comprises three dynamic variables and has no external current stimulus, and according to the fractional order definition of the formula (1), fractional order is introduced on the basis of the integer-order Chay neuron modelqFractional order Chay nerveThe expression of the meta model is:
(2)
in the formula (2), the amino acid sequence of the compound,V I is the bias voltage of the sodium ion-calcium ion mixed ion channel;V K is the bias voltage of the potassium ion channel;V L is the bias voltage of the leakage ion channel;V C is the bias voltage of the calcium ion channel;g I is the maximum conductance of the voltage dependent mixing channel;g K.V maximum conductivity of the voltage dependent potassium ion channel;g K.C potassium ion channel maximum conductance depending on calcium ion concentration in cell membrane;g L is the maximum conductance of the leakage path;K C is the rate of intracellular calcium ion outflow;ρis a proportionality constant;V、nis a fast variable, and respectively represents membrane potential and voltage sensitive potassium channel probability;Cis a slow variable representing intracellular calcium ion concentration; d, dn/dtRepresenting the rate of change of the potassium ion channel opening probability depending on the membrane potential; d, dC/dtThe change rate of the calcium ion concentration of the cell membrane is shown, and the right two items respectively show the calcium ions entering and exiting the cell membrane;m ∞ andh ∞ the probability of activation and deactivation of the sodium ion-calcium ion mixed ion channel, respectively;n ∞ is the probability of potassium ion channel openingnIs a stable value of (2);τ n is the relaxation time, the smaller the value is, the steeper the spike is, the expression of each of the four variables is as shown in formula (3):
(3)
in the formula (3), the amino acid sequence of the compound,λ n a time constant that is a fast variable;α m andβ m the speeds of the ascending and descending phases of the activating molecules in the cell membrane are respectively shown;α n andβ n respectively represent K + Speed of entering and exiting cell membranes;α h andβ h the expression of each of these six variables is as follows, indicating the rate of rise and fall, respectively, of inactive molecule content in the cell membrane:
(4)
the fractional order Chay neuron model in the formula (2) is realized in MATLAB/Simulink, a fractional order integrator is constructed by using a time domain-frequency domain conversion method, and fractional order solving is carried out on the model; wherein the order isqThe transfer function of the fractional order integrator in the frequency domain is given by equation (5):
(5)
in the formula (5), 1-s q Representation ofqAn order integration step, wherein,s=jwis a complex number of frequencies, which are the frequencies,qis a positive real number, 0<q≤1。
At a certain frequency band [ ]w L ，w H ) The fractional order integrator is represented by fractional order poles in a given frequency band, see equation (6):
(6)
in the Bode diagram of the transfer function shown in formula (5), -20qSlope of dB/dec through ANDsThe alternating slopes of-20 dB/dec and 0 dB/dec, corresponding to alternating poles and zeros on the negative real axis of the plane, are approximated as shown in equation (7):
(7)
the fractional order integrator is formed byTThe +1 order linear transfer function approximates, if the error between the actual line and the approximate line is specified as'y’dB，p T Is the original transfer function at-3qAngular frequency determined at dB slope, then parametersb、c、p 0 、z 0 AndTthe expression of (2) is as follows:
(8)
(9)
(10)
(11)
(12)
(13)
in the middle ofp i 、z i Respectively the firstiPoles and zeros;bis the ratio of the positions of the zero and the previous pole;cis the ratio of the locations of the pole and the previous zero.
From this, the model establishment of the fractional Chay neuron of the invention is completed.
Examples:
taking 0.99 order as an example, selecting an approximation frequency bandw max Frequency angle =100p T Approximation error of =0.01y=0.1 dB to calculate the approximate transfer function of the 0.99 th order integrator, the expression for this approximate transfer function is given in equation (14):
(14)
based on the approximate transfer function formula (14) corresponding to the 0.99-order integrator obtained by deduction and the fractional-order Chay neuron model proposed by the formula (2), carrying out numerical simulation and hardware experiment verification on the complex discharge characteristic of the 0.99-order Chay neuron model.
And (3) experimental verification:
1) Parameter selection of fractional order Chay neurons
Selecting bias voltage of calcium ion channelV C The values of the other parameters are shown in Table 1 as variables. Simulation experiments are carried out by Matlab software, and a film potential time sequence chart is obtained,V-CA plane phase diagram,V-nThe plane phase diagram and peak-to-peak interval bifurcation diagram (ISI) were analyzed to verify the complex discharge characteristics of fractional-order Chay neurons.
Table 1, 3-dimensional Chay neuron model parameters take values and implications
2) Complex firing characteristics verification of fractional Chay neurons
Under the condition that other parameters remain unchanged, selecting the fractional orderqWhen=1, the model becomes an integer-order model. Respectively take outq=1、q=0.99 sumq=0.95, bias voltage of calcium ion channelV C The peak-to-peak interval bifurcation diagrams when they are variables are shown in FIG. 1a, FIG. 1b, FIG. 1c, FIG. 1aqPeak-to-peak interval bifurcation when=1, fig. 1b isqPeak-to-peak interval bifurcation at=0.99, fig. 1cqPeak-to-peak interval bifurcation diagram at=0.95. Wherein the method comprises the steps ofV C The value range of (a) is [80mV-450 mV ]]。
As is evident from fig. 1a, 1b, 1c, as followsV C The discharge pattern of the Chay neurons takes on a number of different forms and, with fractional orderqNeurons exhibit more complex firing characteristics.
When (when)q=0.99,V C When 90, 97, 99, 100, 200 and 400 are taken, respectively, the film potential timing charts corresponding to the peak-to-peak period bifurcation diagrams are shown in FIG. 2a, FIG. 2b, FIG. 2c, FIG. 2d, FIG. 2e and FIG. 2f,V-Cthe plane phase diagrams are shown in fig. 3a, 3b, 3c, 3d, 3e and 3f,V-nthe plane phase diagrams are shown in fig. 4a, 4b, 4c, 4d, 4e and 4 f. It can be seen that the 0.99 th order Chay neurons are inV C Occurs when =97The double period branches, and the discharge mode is changed from the period 1 discharge to the period 2 discharge. When (when)V C At=99, the discharge pattern of the neuron evolves into a chaotic discharge form. In the initial stage of chaotic discharge, 0.99 th order Chay neurons show the form of peak discharge. However, whenV C At=100, the discharge pattern of the model evolves into the form of a cluster discharge. Subsequently, the evolution law of the 0.99 order neuron discharge form is similar to that of the integer order neuron discharge form, in thatV C Reverse addition cycle bifurcation occurs when=103, whenV C When the values of (1) are 120, 130, 150, 160, and 200, respectively, the discharge cycles of the neurons become cycle 9, cycle 8, cycle 7, cycle 6, and cycle 5 in orderV C When=400, the neuron takes on a period 2 discharge form, the overall discharge trend is more complex than that of the integer-order Chay neuron, the discharge cycle number is increased when the period discharges under the same parameters, and the chaotic discharge phenomenon occursV C Take smaller values.
Comparing the 0.99 th order Chay neurons to integer order Chay neurons, we found: first, the 0.99 th order Chay neurons followV C Is increased, and the double period branching occurs earlier and enters a chaotic discharge state. Second, the periodic firing pattern of 0.99 th order Chay neurons is more complex. Under the condition that the rest parameters are kept unchanged, after reverse addition period bifurcation occurs on the 0.99-order neuron, the period number of periodic discharge is obviously more than the integer order, and the transition between the periodic discharge forms is more frequent. Meanwhile, the interval width corresponding to the periodic discharge of the 0.99-order Chay neuron is much smaller than that of the integral-order Chay neuron, and finallyV C An increase to a larger value is required to cause the discharge form to switch back to the peak discharge of the single cycle.
3) The complex discharge characteristic of the 0.99-order Chay neuron obtained based on Matlab numerical simulation is verified by adopting a hardware implementation mode. FIG. 5a is a film potential timing chart of a hardware experiment verifying the initial stage of chaotic discharge, and FIG. 5b is a graph of a hardware experiment verifying the initial stage of chaotic dischargeV-CA phase diagram; FIG. 6a is a hardware experiment verification chaosFIG. 6b is a film potential timing chart of the late discharge stage, and the hardware experiment shows that the film potential timing chart corresponds to the late discharge stageV-CA phase diagram; FIG. 7a is a timing chart of the membrane potential corresponding to the discharge of the hardware experiment verification period 5, and FIG. 7b is a timing chart of the membrane potential corresponding to the discharge of the hardware experiment verification period 5V-CA phase diagram; FIG. 8a is a timing chart of the membrane potential corresponding to the discharge of the hardware experiment verification period 2, and FIG. 8b is a timing chart of the membrane potential corresponding to the discharge of the hardware experiment verification period 2V-CPhase diagram. The discharge form and the numerical simulation obtained by the hardware experiments and the numerical simulation shown in the eight small diagrams of fig. 5a to 8bV-CThe phase diagram correspondence between the variables fully demonstrates the correctness of the complex discharge characteristics of the 0.99 th-order Chay neurons in the embodiment of the invention.
Claims (5)
1. A method for establishing a fractional order Chay neuron model with complex discharge characteristics is characterized in that a Caputo definition is adopted to study fractional derivatives and monobasic functionsf(x) A kind of electronic deviceqThe differential order is defined as formula (1):
(1)
in the formula (1), the components are as follows,qin order of the fractional order of the number,m-1<q≤m；mis not smaller thanqIs the smallest integer of (a);aandtthe upper and lower limits of the integral, respectively; Γ () is a Gamma function;f (m) (x) Is a unitary functionf(x) A kind of electronic devicemAn order derivative;
introducing fractional order according to Caputo definition based on integer order Chay neuron modelqThe expression of the fractional Chay neuron model is:
(2)
in the formula (2), the amino acid sequence of the compound,V I is the bias voltage of the sodium ion-calcium ion mixed ion channel;V K is the bias voltage of the potassium ion channel;V L is the bias voltage of the leakage ion channel;V C is the bias electricity of the calcium ion channelPressing;g I is the maximum conductance of the voltage dependent mixing channel;g K.V maximum conductivity of the voltage dependent potassium ion channel;g K.C potassium ion channel maximum conductance depending on calcium ion concentration in cell membrane;g L is the maximum conductance of the leakage path;K C is the rate of intracellular calcium ion outflow;ρis a proportionality constant;V、nis a fast variable, and respectively represents membrane potential and voltage sensitive potassium channel probability;Cis a slow variable representing intracellular calcium ion concentration; d, dn/dtRepresenting the rate of change of the potassium ion channel opening probability depending on the membrane potential; d, dC/dtThe change rate of the calcium ion concentration of the cell membrane is shown, and the right two items respectively show the calcium ions entering and exiting the cell membrane;m ∞ andh ∞ the probability of activation and deactivation of the sodium ion-calcium ion mixed ion channel, respectively;n ∞ is the probability of potassium ion channel openingnIs a stable value of (2);τ n is the relaxation time, the smaller the value is, the steeper the spike is, the expression of each of the four variables is as shown in formula (3):
(3)
in the formula (3), the amino acid sequence of the compound,λ n a time constant that is a fast variable;α m andβ m the speeds of the ascending and descending phases of the activating molecules in the cell membrane are respectively shown;α n andβ n respectively represent K + Speed of entry into and exit from cell membranes;α h andβ h the expression of each of these six variables is as follows, indicating the rate of rise and fall, respectively, of inactive molecule content in the cell membrane:
。(4)
2. the method for building the fractional-order Chay neuron model with the complex discharge characteristic according to claim 1, wherein in the formula (2), the fractional-order Chay neuron model is implemented in MATLAB/Simulink, a fractional integrator is built by using a time domain-frequency domain conversion method, and the model fractional order is solved.
3. The method for building a model of a Chay neuron with a complex discharge characteristic fractional order as claimed in claim 2, wherein the fractional integrator order isqThe transfer function in the frequency domain is given by equation (5):
(5)
in the formula (5), 1-s q Representation ofqAn order integration step, wherein,s=jwis a complex number of frequencies, which are the frequencies,qis a positive real number, 0<q≤1。
4. The method for building a model of a Chay neuron with a complex discharge characteristic fractional order according to claim 3, wherein the fractional order integrator is arranged in a frequency band ofw L ，w H ) When represented by fractional order poles in a given frequency band, see equation (6):
(6)
in the Bode diagram of the transfer function shown in formula (5), -20q Slope of dB/dec through ANDsThe alternating slopes of-20 dB/dec and 0 dB/dec, corresponding to alternating poles and zeros on the negative real axis of the plane, are approximated as shown in equation (7):
。(7)
5. the method for building a fractional Chay neuron model with complex discharge characteristics according to claim 4, wherein said fractional integratorIs composed ofTThe +1 order linear transfer function approximates, if the error between the actual line and the approximate line is specified as'y’dB，p T Is the original transfer function at-3qAngular frequency determined at dB slope, then parametersb、c、p 0 、z 0 AndTthe expression of (2) is as follows:
(8)
(9)
(10)
(11)
(12)
(13)
in the middle ofp i 、z i Respectively the firstiPoles and zeros;bis the ratio of the positions of the zero and the previous pole;cis the ratio of the locations of the pole and the previous zero.
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