A method for constructing biologically heterogeneous neural networks

Abstract

This invention provides a method for constructing biological heterogeneous neural networks. S1: Based on the ion channel function abstraction method, the constants in the activation and deactivation kinetic functions of ion channels in the first dimensionality-upgrading model are used as undetermined kinetic coefficients P(Θ). S2: Based on the number of ion channels in the target model, the corresponding ion channels in the second dimensionality-upgrading model are optimized. S3: Based on the random parameter search method, all undetermined coefficients P(Θ) and their corresponding conductance values ​​G in the ion channel kinetic equations of the second dimensionality-upgrading model are searched to obtain candidate solutions. S4: The target model is processed based on polar coordinate projection. S5. Obtain a degeneracy feature model among ion channel currents; S6. Perform minimum mean square error calibration on all candidate solutions in the second dimensionality-upgrading model to obtain a third dimensionality-upgrading model; S7. Based on polar coordinate projection, take the intersection points of each current in the third dimensionality-upgrading model as degeneracy features, and select the candidate solution most similar to the degeneracy feature model among ion channel currents as the optimal solution to construct a biological heterogeneous neuronal network; This invention obtains model parameters and structures that are more consistent with biological reality by randomly searching for ion channel dynamic parameters with high degrees of freedom and combining non-time-scale channel degeneracy comparison.

Description

Technical Field

[0001] This invention relates to neuron model simulation technology in the field of computational neuroscience, and more particularly to a method for constructing biologically heterogeneous neuronal networks. Background Technology

[0002] Neuron models are digital tools that quantitatively describe the electrophysiological characteristics of biological neurons through mathematical equations and computational algorithms. Their core components include dynamic equations of cell membrane potential, ion channel conduction mechanisms, and rules for synaptic information transmission.

[0003] In the field of computational neuroscience, neuron models, as the basic units for studying neural information processing, have significant application value in areas such as brain-computer interface development, research on the mechanisms of nervous system diseases, and the construction of brain-like intelligent systems. Existing neuron models are mainly divided into two categories: biophysical models (such as the Hodgkin-Huxley model) and simplified computational models (such as the Integrate-and-Fire model). The former aims at simulation and has developed many types of models to simulate the action potentials of neurons in different organisms and with different characteristics, almost all of which use the Hodgkin-Huxley model and its ion channel differential equations as their basic framework. Although these models can accurately reproduce the characteristics of various electrical activities (such as specific discharge frequencies, specific maximum / minimum voltage values, subthreshold potentials, burst discharges, etc.), the correlation coefficients of their channel conductance values ​​deviate from biologically measured data, seriously affecting the biological reliability of the models. This error may originate from two sources: first, insufficient model dimensionality, resulting in a mismatch between the model's actual number of ion channels and the real-world situation, leading to the dimensionality reduction of certain channels of the same type, thus affecting the correlation coefficients between channels; second, although the model can generate the target action potential, the kinetic parameters of its ion channels deviate from the actual parameters. Furthermore, in biological neural networks, neurons exhibit significant heterogeneity, such as the diversity in the types and numbers of ion channels and the differences in electrophysiological properties. However, existing neuronal models typically assume neurons to be homogeneous units, making it difficult to accurately simulate the complex behavior of heterogeneous neural networks, which presents new challenges for constructing reliable biological neural networks. Summary of the Invention

[0004] To address the shortcomings of existing models in terms of biological reliability and heterogeneity, this invention provides a method for constructing biologically heterogeneous neural networks. This method employs a high-degree-of-freedom ion channel dynamics parameter search process and a novel measure of inter-channel degeneracy to screen model parameters. Degeneracy refers to the phenomenon that the currents of different ion channels may exhibit similar behaviors under specific conditions; that is, combinations of multiple ion channels can produce the same electrophysiological characteristics. This phenomenon is prevalent in biological neurons, reflecting the functional redundancy and diversity among ion channels, and is an important source of complexity and robustness in biological neural networks. This method can provide inter-channel conductance correlation coefficients that conform to biological realities while preserving the action potential characteristics of the target model, and can simulate the electrophysiological characteristics of heterogeneous neurons, providing a new approach for constructing biologically reliable heterogeneous neural networks.

[0005] To address the problems of the existing technology, the present invention adopts the following technical solution:

[0006] A method for constructing biologically heterogeneous neural networks, the method being based on neural networks, a first dimensionality-upgrading model, a second dimensionality-upgrading model, a third dimensionality-upgrading model, and a target model, comprising the following steps:

[0007] S1. Based on the ion channel function abstraction method, the constants in the activation and deactivation kinetic functions of the ion channel in the first-dimensional model are taken as undetermined kinetic coefficients P(Θ), that is:

[0008] α n =f1(V,P(Θ1)),β n =f2(V,P(Θ2))

[0009] α n =θ1*(V+θ2) / (1-exp(-(V+θ2) / θ3)),θ1,θ2,θ3∈P(Θ1)

[0010] β n =θ4*exp(-(V+θ5) / θ6),θ4,θ5,θ6∈P(Θ2)

[0011] Where P(Θ) represents the undetermined coefficients in the ion channel kinetics equation;

[0012] S2. The first dimensionality-upgrading model constructs a second dimensionality-upgrading model with matching ion channel numbers based on the number of ion channels in the target model, and constructs independent undetermined kinetic coefficients P(Θ) and corresponding conductivity values ​​G for each ion channel in the second dimensionality-upgrading model.

[0013] S3. Based on the random parameter search method, search for all undetermined coefficients P(Θ) and corresponding conductance values ​​G in the ion channel kinetic equation of the second dimension-upgrading model. If the undetermined kinetic coefficients P(Θ) and corresponding conductance values ​​G in a certain search process are within the confidence interval of the features of the second dimension-upgrading model that match the corresponding features of the target model, then the undetermined kinetic coefficients P(Θ) and corresponding conductance values ​​G are taken as a set of candidate solutions.

[0014] S4. Based on polar coordinate projection, process the currents of each ion channel in the target model to obtain the degeneracy characteristic model among the currents of the ion channels.

[0015] S5. The ion channel currents of all candidate solutions in the second dimension-upgrading model are calibrated according to the minimum mean square error between the current curves to obtain the third dimension-upgrading model.

[0016] S6. Based on polar coordinate projection, the intersection points of each current in the third-dimensional model are taken as degeneracy features. The candidate solution that is most similar to the degeneracy feature model between ion channel currents is selected as the optimal solution, and then a biological heterogeneous neuronal network is constructed.

[0017] Further, in step S4, the ion channel currents of the target model are processed based on polar coordinate projection to obtain a degeneracy feature model among ion channel currents; including:

[0018] Multiple spike potentials within a continuous time window are decomposed into combinations of multiple single discharges, and the start sequence value and end sequence value of each single discharge are recorded.

[0019] The current of each channel is decomposed into multiple groups according to the start sequence value and the end sequence value. The linear interpolation method is used to divide each group of current into multiples of 2π with an accuracy greater than the minimum time step.

[0020] The absolute value of each set of current data is processed using the natural logarithm, and then a constant bias C is added.

[0021] I log =log(abs(I)+max{abs(I),0}+C)

[0022] Among them, I log It is the processed current data, and I is the current data between a certain set of start sequence values ​​and end sequence values;

[0023] By I log Mapped to polar coordinates within the angular range of (0, 2π);

[0024] I log =[I1,I2,…,I N ]

[0025]

[0026] Among them, I log This is the processed current data, with a length of N, where n is the index of the current point, n∈[1,N]; r n It is the distance in polar coordinates, θ n It is an angle in polar coordinates; I n This is the value of the current point; for the entire current data I log Its polar coordinate mapping can be expressed as:

[0027]

[0028] Further, in step S5, the ion channel currents of all candidate solutions in the second dimensionality-upgrading model are calibrated according to the minimum mean square error between the current curves to obtain the third dimensionality-upgrading model; wherein:

[0029] The formula for calculating the minimum mean square error is as follows:

[0030]

[0031] I1 = [I1] (1) ,I1 (2) ,I1 (3) ,...I1 (n) ], I2=[I2 (1) I2 (2) I2 (3) ,...I2 (n) ]

[0032] Among them, I1 (i) I² represents the current value at the i-th point in the first current sequence. (i) represents the current value at the i-th point in the second current sequence, and n represents the length of the current sequence.

[0033] Beneficial effects

[0034] Compared with traditional technical solutions, the beneficial effects of this invention are:

[0035] This invention proposes a method for constructing a complex neural network model with heterogeneity based on the Hodgkin-Huxley model of increased dimensionality. Compared with general neuronal biophysical models, this method can obtain model parameters and structures that are more consistent with biological reality by randomly searching for ion channel dynamic parameters with high degrees of freedom and combining non-time-scale channel degeneracy comparison, which can effectively improve the biological credibility of the model. Attached Figure Description

[0036] Figure 1 Degeneracy diagram of potassium and sodium ion channels in the HH model under polar coordinate projection;

[0037] Figure 2 Degeneracy comparison of the nine candidate solutions and the corresponding ion channels of the target model (CS model) (marked as the most similar group);

[0038] Figure 3 Correlation coefficients of channel conductance in the target model (CS model) and correlation coefficients under the optimal solution; Detailed Implementation

[0039] The following is in conjunction with the appendix Figure 1 - Appendix Figure 3 The technical solution of the present invention will be further described below.

[0040] This invention addresses the shortcomings of existing models where the correlation coefficients of ion channel conductance do not conform to biological realities and have poor biological reliability. It provides a method for constructing biologically heterogeneous neural networks. The method, based on a neural network, a first-dimensionality upgrade model, a second-dimensionality upgrade model, a third-dimensionality upgrade model, and a target model, includes the following steps:

[0041] S1. Regarding the abstraction of ion channel functions, this invention uses the constants within the activation and deactivation kinetic functions of potassium and sodium ion channels in the first-dimensional model as undetermined coefficients P(Θ). While maintaining the framework of its ordinary differential equations, the overall differential equations are abstracted, maximizing the freedom of the ion channel kinetic equations. This allows the second-dimensional model to simulate the action potential characteristics of the most complex models possible, based on the first-dimensional model. Detailed analysis of the ion channel function abstraction process follows:

[0042] Ion channel function abstraction involves setting all constants in the ordinary differential kinetic equations of each ion channel in the normal first-dimensional model as undetermined parameters, taking the potassium ion channel as an example:

[0043] α n =0.01*(V+55) / (1-exp(-(V+55) / 10))

[0044] β n =0.125*exp(-(V+65) / 80)

[0045]

[0046] I K =g K *n 4 *(VE K )

[0047] Where: n represents the potassium ion channel activation gate, α n and β nThe state of the potassium ion channel activation gate at each time step is determined by V, which represents the membrane voltage value of the previous state, and E. K Represents the potassium ion reversal potential, g K This represents the conductivity value corresponding to the potassium ion channel; the potassium ion channel contains an activation gate n, which is activated by α. n With β n The state is updated, and the current activation gate state is determined by combining the activation gate state from the previous moment. The current potassium ion current is determined by four variables: the potassium ion channel conductance, the current activation gate state, and the current membrane voltage. All constant terms are set as undetermined coefficients.

[0048] α n =f1(V,P(Θ1)),β n =f2(V,P(Θ2))

[0049] α n =θ1*(V+θ2) / (1-exp(-(V+θ2) / θ3)),θ1,θ2,θ3∈P(Θ1)

[0050] β n =θ4*exp(-(V+θ5) / θ6),θ4,θ5,θ6∈P(Θ2)

[0051] Where P(Θ) represents the undetermined coefficients in the ion channel kinetic equation; the solution to this ordinary differential equation requires the following inputs from the original HH model: initial membrane voltage V, initial potassium ion channel activation gate n, α n Initial value, β n Initial value, potassium ion channel conductivity g K Potassium ion reversal potential E K The parameter value now requires an additional input of α. n With β n Each of their own P(Θ).

[0052] S2, taking the number of each ion channel in the target model as a reference, the corresponding ion channels in the second-dimensional model are upgraded to make each ion channel in the target model correspond one-to-one, ensuring that it has similar complexity to the target model to realize various characteristics of the action potential. At this time, the dynamic equations and conductance values ​​of each ion channel in the second-dimensional model are independent of each other.

[0053] The abstracted second-dimensional model needs to be processed by comparing the number of ion channels in the target model. For example, the Connor-Steven model (hereinafter referred to as the CS model) contains two potassium ion channels compared to the HH model. KDR With g A Then it is necessary to adjust the potassium ion channel g in the HH model. K Upgrade to gK1 g K2 To match the complexity of the CS model:

[0054] α n1 =A1*(V+B1) / (1-exp(-(V+B1) / C1))

[0055] α n2 =A2*(V+B2) / (1-exp(-(V+B2)C2)),

[0056] A1, A2, B1, B2, C1, C2 ∈ P(Θ1)

[0057] β n1 =D1*exp(-(V+E1) / F1)

[0058] β n2 =D2*exp(-(V+E2) / F2),D1,D2,E1,E2,F1,F2∈P(Θ2)

[0059]

[0060] I K =(VE K )*(g K1 *n1 4 +g K2 *n2 4 )

[0061] This lays the foundation for the subsequent random search process for parameters.

[0062] S3, using the action potential characteristics of the target model under reference parameter values ​​as constraints, employs a stochastic parameter search process to search for all undetermined coefficients P(Θ) and corresponding conductance values ​​G within the ion channel kinetic equations of the second-dimensional model. If, in a certain search process, P(Θ) and G cause the action potential characteristics of the second-dimensional model to conform to the confidence intervals of the corresponding characteristics of the target model, then this set of P(Θ) and G is considered as a set of candidate solutions. Wherein: stochastic parameter search

[0063] Using the action potential characteristics of the target model under reference parameter values ​​as constraints, a random parameter search process is used to search for all undetermined coefficients P(Θ) and corresponding conductance values ​​G in the ion channel kinetic equation of the HH model. If P(Θ) and G in a certain search process make the action potential characteristics of the HH model conform to the confidence interval of the corresponding characteristics of the target model, then the set of P(Θ) and G is taken as a set of candidate solutions.

[0064] The random parameter search requires using the features of the target model under the reference parameter values ​​as a standard to guide the selection of solutions. First, a sufficiently large number of iterations is determined to generate several sets of random P(Θ) and channel conductance values ​​G. For each set of randomly generated parameters, feature extraction is performed on the action potential obtained after inputting it into the upgraded HH model.

[0065] Taking the CS model as an example, the features to be extracted from its action potential include: discharge frequency, maximum membrane voltage, minimum membrane voltage, subthreshold potential length, depolarization length, and repolarization length. These features are extracted using methods that conform to general definitions. For example, the maximum / minimum membrane voltage is obtained by averaging all maximum / minimum values ​​greater than a certain threshold after the action potential reaches steady state, and adjacent maximum / minimum values ​​can be considered as a complete peak potential. The discharge frequency is obtained by multiplying the difference between the indices of two adjacent extreme values ​​by the time step after the model is in a periodic discharge state, and taking the reciprocal. The subthreshold potential length and depolarization length are obtained by taking the rising branch in each complete peak potential as the standard, taking the average value of the interval from the current peak to the first 1 / 3 of the next peak as the threshold, the rising branch of the voltage interval below this threshold as the subthreshold part, and the rising branch above this threshold as the depolarization part, and calculating the lengths of the two parts separately.

[0066] The complex polarization length is the descent branch length. The above operations are used to obtain the reference range of constraints on the standard CS model and to determine the characteristics of the third-dimensional model in the current solution during the random parameter search process.

[0067] After a sufficient number of iterations, a certain number of candidate solutions that meet the constraints will be obtained. The parameters represented by these solutions will enable the second-dimensional model to generate action potential characteristics that meet the constraints.

[0068] S4. Project the current of each ion channel in the target model into polar coordinates. Use the two adjacent voltage minima of a single peak potential as the start index and end index. Group the current data of each ion channel according to the start index and end index. Use linear interpolation to divide each group of currents into multiples of 2π with an accuracy greater than the minimum time step. After taking the absolute value and natural logarithm of the current value and biasing it, use the processed current value as the radius and the relative index position of the current point as the angle. Project it onto the polar coordinates in the range of [0, 2π] to obtain a non-time-scale degeneracy model of ion channel currents.

[0069] S5. The third dimensional model is obtained by taking the minimum mean square error between the corresponding current curves of each ion channel current in the second dimensional model.

[0070] S6. Based on polar coordinate projection, the intersection points of each current within the third dimension are used as degeneracy features. The candidate solution most similar to the degeneracy model between ion channel currents on a non-timescale scale is selected as the optimal solution. Finally, a dimension-upgraded model with ion channel conductance correlation coefficients conforming to biological realities is obtained, providing a model parameter basis for heterogeneous neural networks with high biological reliability. Polar coordinate projection and degeneracy comparison process:

[0071] To obtain the optimal solution from the candidate solutions, it is necessary to further project the current of each ion channel onto polar coordinates to compare the degeneracy between channels.

[0072] First, determine the range of each projection: take the indices of two adjacent lowest voltage points in the action potential, i.e., the start index and the end index, as a range of a single spike potential. Then, multiplying the index by the time step represents the time of each voltage point, and its value represents the membrane voltage value at the corresponding time. Divide the current sequences of all ion channels according to the start and end indices, and then use linear interpolation to divide each current group into multiples of 2π with an accuracy greater than the minimum time step. That is, if the time step is 0.01s, the difference between the start and end indices is approximately 1000. Therefore, linear interpolation is needed to expand the current sequence to a length of 360(2π)*3 = 1080 so that it can be projected onto polar coordinates.

[0073] Secondly, the current sequence needs to be processed by taking its absolute value, its natural logarithm, and then biased (e.g., by adding a constant 1 to all processed current values) to minimize the influence of conductance on channel degeneracy. The processed current values ​​are used as the radius, and the relative index position of the current point is used as the angle, projected onto polar coordinates in the range [0, 2π] to obtain the non-time-scale degeneracy of ion channel currents.

[0074]

[0075] Where I is the processed current data with a length of N, and n is the index of the current point, n∈[1,N]; r n It is the distance (radius) in polar coordinates, θ n It is an angle in polar coordinates; I n This is the value of the current at the current point. For the CS model and the normal HH model, the polar coordinate projection of the current is as follows: Figure 1 As shown.

[0076] Figure 1 The degeneracy between the ion channel currents is shown intuitively, while the influence of conductivity changes and the time axis is eliminated as much as possible.

[0077] Finally, for all candidate solutions, the solution with the minimum sum of mean square errors between the current curves corresponding to the target model and the highest degeneracy similarity at the intersection points of the currents within the model is selected as the optimal solution. The specific number of optimal solutions depends on the required degree of heterogeneity for the neural network. The formula for calculating the minimum mean square error is as follows:

[0078]

[0079] I1 = [I1] (1) ,I1 (2) ,I1 (3) ,...I1 (n) ], I2=[I2 (1) I2 (2) I2 (3) ,...I2 (n) ]

[0080] Among them, I1 (i) I² represents the current value at the i-th point in the first current sequence. (i) This represents the current value at the i-th point in the second current sequence, and n represents the length of the current sequence (the two sequences have the same length). Figure 2 The figure shows the polar coordinate projections of the nine optimal solutions that minimize the mean square error between the current curves from a sufficiently large set of candidate solutions.

[0081] Figure 2 The figure above shows the current polar coordinates of the CS model (Figure 1) and the current polar coordinate degeneracy of the nine optimal solutions. Figure 2 The set of optimal solutions circled in the diagram has current-to-current intersection points in its polar coordinates that are most similar to the CS model. Specific ion channel correlation coefficients for this set of solutions can be found in [link to relevant documentation]. Figure 3 .

[0082] Figure 3 Compared to the CS model (top figure), the correlation coefficient of inter-channel conductance in the upgraded HH model (bottom figure) under the optimal solution is significantly more consistent with biological reality (experimental data show that the expression levels of each ion channel in real neurons do not show a negative correlation).

[0083] Although the present invention has been described above, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many modifications under the guidance of the present invention without departing from the spirit of the present invention, and these modifications are all within the protection scope of the present invention.

Claims

1. A method for constructing biologically heterogeneous neural networks, characterized in that: The method, based on neural networks, a first dimensionality-upgrading model, a second dimensionality-upgrading model, a third dimensionality-upgrading model, and a target model, includes the following steps: S1. Based on the ion channel function abstraction method, the constants in the activation and deactivation kinetic functions of the ion channel in the first-dimensional model are taken as undetermined kinetic coefficients P(Θ), that is: a n =f1(V,P(Θ1)),β n =f2(V,P(Θ2)) α=θ1*(V+θ2) / (1-exp(-(V+θ2) / θ3)),θ1,θ2,θ3∈P(Θ1) b n =θ4*exp(-(V+θ5) / θ6),θ4,θ5,θ6∈P(θ2) Where P(Θ) represents the undetermined coefficients in the ion channel kinetics equation; S2. The first dimensionality-upgrading model constructs a second dimensionality-upgrading model with matching ion channel numbers based on the number of ion channels in the target model, and constructs independent undetermined kinetic coefficients P(Θ) and corresponding conductance values ​​G for each ion channel in the second dimensionality-upgrading model; S3. Based on a random parameter search method, all undetermined coefficients P(Θ) and corresponding conductance values ​​G in the ion channel kinetic equation of the second dimensionality-upgrading model are searched. If the undetermined kinetic coefficients P(Θ) and corresponding conductance values ​​G in a certain search process fall within the confidence interval of the features of the second dimensionality-upgrading model that match the corresponding features of the target model, then the undetermined kinetic coefficients P(Θ) and corresponding conductance values ​​G are taken as a set of candidate solutions; S4. Based on polar coordinate projection, process the currents of each ion channel in the target model to obtain the degeneracy characteristic model among the currents of the ion channels. S5. The ion channel currents of all candidate solutions in the second dimension-upgrading model are calibrated according to the minimum mean square error between the current curves to obtain the third dimension-upgrading model. S6. Based on polar coordinate projection, the intersection points of each current in the third-dimensional model are taken as degeneracy features. The candidate solution that is most similar to the degeneracy feature model between ion channel currents is selected as the optimal solution, and then a biological heterogeneous neuronal network is constructed.

2. The method for constructing a biological heterogeneous neural network according to claim 1, characterized in that: In step S4, the ion channel currents of the target model are processed based on polar coordinate projection to obtain a degeneracy characteristic model among ion channel currents; including: Multiple spike potentials within a continuous time window are decomposed into combinations of multiple single discharges, and the start sequence value and end sequence value of each single discharge are recorded. The current of each channel is decomposed into multiple groups according to the start sequence value and the end sequence value. The linear interpolation method is used to divide each group of current into multiples of 2π with an accuracy greater than the minimum time step. The absolute value of each set of current data is processed using the natural logarithm, and then a constant bias C is added. I log =log(abs(I)+max{abs(I),0}+C) Among them, I log It is the processed current data, and I is the current data between a certain set of start sequence values ​​and end sequence values; By I log Mapped to polar coordinates within the angular range of (0, 2π); I log =[I1,I2,…,I N ] Among them, I log This is the processed current data, with a length of N, where n is the index of the current point, n∈[1,N]; r n It is the distance in polar coordinates, θ n It is an angle in polar coordinates; I n This is the value of the current point; for the entire current data I log Its polar coordinate mapping can be expressed as:

3. The method for constructing a biological heterogeneous neural network according to claim 1, characterized in that: In step S5, the ion channel currents of all candidate solutions in the second dimensionality-upgrading model are calibrated according to the minimum mean square error between the current curves to obtain the third dimensionality-upgrading model; wherein: The formula for calculating the minimum mean square error is as follows: I1=[I1 (1) ,I1 (2) ,I1 (3) ,...I1 (n) ],I2=[I2 (1) ,I2 (2) ,I2 (3) ,...I2 (n) ] Among them, I1 (i) I² represents the current value at the i-th point in the first current sequence. (i) represents the current value at the i-th point in the second current sequence, and n represents the length of the current sequence.

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