Modeling and nonlinear analysis of chaotic wave processes in electrochemically active neuronal media based on matrix decomposition

Abstract

A general model of the origin and evolution of chaotic wave processes in electrochemically active neuronal media based on the proposed method of matrix decomposition of operators of nonlinear systems has been developed. The mathematical models of Hodgkin – Huxley and FitzHugh – Nagumo of an electrochemically active neuronal media are considered. The necessary conditions for self-organization of chaotic self-oscillations in the FitzHugh – Nagumo model are determined. Computer modeling based on the matrix decomposition of chaotic wave processes in electrochemically active neuronal media has shown the interaction of higher-order nonlinear processes leading to stabilization (to a finite value) of the amplitude of the chaotic wave process. Mathematically, this is expressed in the synchronous “counteraction” of nonlinear processes of even and odd orders in the general vector-matrix model of an electrochemically active neuronal media being in a chaotic mode. It is noted that the state of hard self-excitation of nonlinear oscillations in an electrochemically active neuronal media leads to the appearance of a chaotic attractor in the state space. At the same time, the proposed vector-matrix model made it possible to find more general conditions for the appearance and evolution of chaotic wave processes in comparison with the initial Landau turbulence model and, as a result, to explain the occurrence of consistent nonlinear phenomena in an electrochemically active neuronal media.

Keywords

• electrochemically active neuronal media
• fitzhugh – nagumo model
• necessary conditions for self-organization of self-oscillations
• chaotic attractor
• matrix series in state-space
• vector-matrix model of chaotic wave processes
• mode of hard self-excitation of nonlinear oscillations
• stabilization of the amplitude of chaotic process