Ряди Вольтерри для моделей Ходжкіна–Хакслі та ФітцХ’ю-Нагумо нейрона

Abstract

Розглянуто представлення моделей Ходжкіна–Хакслі та ФітцХ’ю–Нагумо нейрона рядами Вольтерри та особливості обчислення ядер Вольтерри для цих моделей. Отримано системи лінійних алгебраїчних рівнянь для спектрів ядер Вольтерри та їх розв’язки. Для реалізації оберненого перетворення Фур’є використано пакет прикладних програм Symbolic Math Toolbox системи MATLAB. Наведено графіки модуля спектра та ядра першого порядку ряду Вольтерри для обох моделей нейрона. The peculiarity of Volterra series use as to comparison with classical models of nonlinear dynamical systems in the form of nonlinear differential equations is that Volterra series directly evaluates output signal of the system by multidimensional convolution of input signal. Kernels of Volterra series can be considered as generalization of impulse response of linear systems to nonlinear ones. The choice of certain components of Volterra series enables to take into account nonlinear peculiarities of different orders. The representation of Hodgkin-Huxley and FitzHugh-Nagumo models of neuron by Volterra series and features of calculation of Volterra kernels for these neuron models are considered. The systems of nonlinear differential equations of the first order in the form of Cauchy of Hodgkin-Huxley and FitzHugh-Nagumo neuron models with continuous analytical functions at the right side of equations give the possibility to reduce the problem of determination Volterra kernels to solving of algebraic equations systems, which are represented in analytical form. For the determination of Volterra kernels the conversion into frequency domain by integral Fourier transform is used. For the Hodgkin-Huxley model the multiple Maclaurin series expansions of the functions at the right side of system of differential equations are used, for the FitzHugh-Nagumo model the expressions at the right side of equations are polynomial. The systems of four (for Hodgkin-Huxley model) and two (for FitzHugh-Nagumo model) linear algebraic equations for spectra of Volterra kernels and their solutions are obtained. Symbolic Math Toolbox of MATLAB system for inverse Fourier transforms is used. Diagrams of spectrum module and first order kernel of Volterra series are given for both neuron models. Constructed Volterra series models can be used for research of dynamic properties of neural networks with Hodgkin-Huxley and FitzHugh-Nagumo models of neuron using.