On properties of solutions to differential systems modeling the electrical activity of the brain

Abstract

The Hopfield-type model of the dynamics of the electrical activity of the brain, which is a system of differential equations of the form v ̇_i=-αv_i+∑_(j=1)^n▒〖w_ji f_δ (v_j )+I_i (t), i=(1,n) ̅ 〗, t≥0. is investigated. The model parameters are assumed to be given: α>0, w_ji>0 for i≠j and w_ii=0, I_i (t)≥0. The activation function f_δ (δ is the time of the neuron transition to the state of activity) of two types is considered: δ=0⟹f_0 (v)={■(0,&v≤θ,@1,&v>θ;) δ>0⟹f_δ (v)={■(0,&v≤θ,@δ^(-1) (v-θ),&θ<v≤θ+δ,@1,&v>θ+δ.)┤ ┤ In the case of δ>0 (the function f_δ is continuous), the solution of the Cauchy problem for the system under consideration exists, is unique, and is non-negative for non-negative initial values. In the case of δ=0 (the function f_0 is discontinuous at the point θ), it is shown that the set of solutions of the Cauchy problem has the largest and the smallest solutions, estimates for the solutions are obtained, and an example of a system for which the Cauchy problem has an infinite number of solutions is given. In this study, methods of analysis of mappings acting in partially ordered spaces are used. An improved Hopfield model is also investigated. It takes into account the time of movement of an electrical impulse from one neuron to another, and therefore such a model is represented by a system of differential equations with delay. For such a system, both in the case of continuous and in the case of discontinuous activation function, it is shown that the Cauchy problem is uniquely solvable, estimates for the solution are obtained, and an algorithm for analytical finding of solution is described.