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Romanyshyn Y.M.^1),2), Pavlysh V.A.¹⁾, Korzh R.O.¹⁾, Pukish S.R.¹⁾,

Kokhalevych Y.R.¹⁾

¹⁾ Lviv Polytechnic National University

²⁾ University of Warmia and Mazury in Olsztyn

VOLTERRA SERIES FOR THE FITZHUGH-NAGUMO NEURON MODEL

Introduction. The FitzHugh-Nagumo neuron model [1] represents, unlike a system of four first order ordinary nonlinear differential equations of Hodgkin-Huxley model, a similar system of two first order differential equations (one of which is nonlinear) in Cauchy’s form and is a significant simplification of Hodgkin-Huxley model, however, retains the basic property of neuron - returning to its original state after the formation of neural impulse (or a series of impulses) as the result of external signal action. One of the methods of constructing nonlinear dynamic objects mathematical models is their representation as Volterra series, kernels of which can be considered as the generalization of linear systems impulse response to nonlinear ones. The features of Volterra series construction and calculation of its kernels for Hodgkin-Huxley neuron model were considered in [2, 3]. However, despite the expediency of such representation of FitzHugh-Nagumo neuron model, the representation of this model by Volterra series was not investigated, that specifies the relevance of the paper.

The purpose of this paper is to analyze the peculiarities of the representation of FitzHugh-Nagumo model of neuron by Volterra series, calculation of series kernels, particularly the first order kernels in the spectral and time form.

Target setting of representation of FitzHugh-Nagumo neuron model by Volterra series. The method of representation of Hodgkin-Huxley neuron model by Volterra series, developed in [2] and used in [3], is quite universal and can be applied also, with appropriate modifications, for constructing of Volterra series for FitzHugh-Nagumo neuron model. The version of FitzHugh-Nagumo neuron model, represented in [1], we represent, by analogy with [2, 3], as the system of equations:

; ;

; , (1)

where ; - density of the external current of the neuron activation, Acm^-2; 100 mVcm²A^-1msec^-1; - voltage on the neuron membrane, mV; ; - time, msec; – internal function of time and membrane voltage, mVmsec; 0.01 mV²msec; -0.1 mV; 1 mV; 1 mV²×msec^-1; 0.5 msec^-1.

We look for the representation of functions through input signals in the form of the following Volterra series:

, (2)

where ; ; - kernels of Volterra series for the -th variable of appropriate order; .

Calculation of kernels of Volterra series. It is more convenient to carry out calculation of Volterra kernels in the frequency domain [2]. Let carry out the integral Fourier transform of Volterra series:

, (3)

where - spectra of functions ; - spectra of kernels ; - spectrum of the function .

It can be shown (by changing the sequence of integration, using the formula for the spectrum of two signals product and the spectrum of the signal, displaced in time) that this transformation is put to the form:

. (4)

The features of FitzHugh-Nagumo neuron model is a polynomial pattern of functions and (in particular, the function is the third degree polynomial relative to and the function is a linear one) and therefore there is no need to expand these functions by multiple McLoren series, as it was done in [2, 3]. However, from the point of view the results, obtained in [2, 3], it seems appropriate to use the representation of functions and by McLoren series of general view, but with a finite number of components (obviously ):

. (5)

Obviously, that:

; ; ;

; ; . (6)

All other derivatives are equal to zero.

Let make Fourier transform of the system of equations (1) taking into account the expression (5):

, (7)

where - the Kronecker symbol.

After substitution of the expression (4) in the expression (7) we get the equations, that connect the spectrum of input signal , spectra of Volterra series kernels , the partial derivatives of functions by and integrals of consecutive repetition factors containing these spectra. Since these equations should be true for any function the equations for calculation of kernels spectra can be obtained by equating the corresponding expressions.

From equating the coefficients at we get:

; . (8)

This is a system of two linear equations with two unknown spectra of first order kernels. For FitzHugh-Nagumo neuron model this system of equations takes the form:

. (9)

The solutions of this system of equations are:

; . (10)

As a result, the values are represented by fractional rational expressions regarding , denominators of expressions are the second degree polynomial, numerator of - the first degree polynomial and numerator of - constant. Module (spectrum of the first order kernel) is shown in Fig. 1, a). The inverse transformation of spectrum is carried out in a symbolic form using Symbolic Math Toolbox of MATLAB. Fig. 1, b) represents a diagram of the first order kernel calculated on the basis of the obtained expression. These characteristics are qualitatively consistent with the corresponding diagrams for the first order kernel of Volterra series of Hodgkin-Huxley model [3], although different by numerical parameters, since both models use different normalization. The diagrams in Fig. 1 for the module of the first order kernel spectrum and the kernel itself, as for the first order kernel of Hodgkin-Huxley model, are close in pattern to the frequency and impulse characteristics of energy model of neuron, obtained in [4].

kOhm cm^2

msec^-1

cm^2/F

, msec

a) b)

Fig. 1. Module of the spectrum and the first order kernel of Volterra series

From equating the coefficients at in the integrals we obtain:

;

. (11)

This system of equations can be reduced to the form:

. (12)

Similarly to the previous, for the spectra of the second order kernels we obtain the system of linear algebraic equations with a matrix of the same structure as for the first order kernels. For the FitzHugh-Nagumo neuron model this system of equations takes the form:

. (13)

The solutions of this system are:

; (14)

. (15)

From equating the double integrals and validity of this equality at arbitrary function we obtain the condition:

. (16)

Consequently we get the system of equations for FitzHugh-Nagumo model:

;

. (17)

The solution of this system is:

;

. (18)

Similarly, although with much more complicated transformations, one can get the systems of algebraic equations for Volterra kernels of higher orders.

Conclusion. The models of nonlinear systems as Volterra series, which are the generalization of the convolution integral for linear systems, allow to express the output signal through the input one in the form of components that correspond to the linear part and nonlinear ones of higher orders. The FitzHugh-Nagumo neuron model is represented by the system of two first order differential equations (one of which is nonlinear), which connect input signal (current density), output signal (voltage on the membrane), internal variable and model parameters. For elimination of the internal variable the reduction of the model to representation as a Volterra series the kernels of which are the generalization of impulse response of the linear system for nonlinear ones can be used. Using this model, despite the complexity of the Volterra series, allows us to consider different approximations of neuron model - linear, nonlinear models of different orders. Calculation of Volterra kernels spectra is reduced to solving single-type systems of two linear algebraic equations with two unknowns.

References:

1. Gerstner W., Kistler W.M. Spiking Neuron Models. Single Neurons, Populations, Plasticity. - Cambridge University Press, 2002. - 5,26 MB.

2. Kistler W., Gerstner W., Leo van Hemmen J. Reduction of the Hodgkin-Huxley Equations to a Single-Variable Threshold Model // Neural Computation 9(5). - P. 1015-1045.

3. Romanyshyn Y.M., Kokhalevych Y.R., Pukish S.R. Volterra series for the Hodgkin-Huxley neuron model // Simulation and informational technologies. Collection of scientific works of Pukhov Institute for Modelling in Energy Engineering of National Academy of Sciences of Ukraine. Issue 56. – Kyiv. – 2010. - P. 156-163 [in Ukrainian].

4. Smerdov A.A., Romanyshyn Y.M. Electric model of neuron at single excitation // Problems of cybernetics: Biomedinformatics and its applications. – Moscow.: Academy of Sciences of the USSR. – 1988. – P. 168-174 [in Russian].