On a boundary value problem for a system of differential equations 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 u ̇_i (t)=-αv_i (t)+∑_(j=1)^n▒〖w_ji f_δ (v_j (t-τ_ji))+I_i (t),i=(1,n) ̅,t≥0.〗 is under discussion. The model parameters are assumed to be given: α>0, τ_ii=0, w_ii=0, τ_ji≥0, and w_ji>0 at i≠j, I_i (t)≥0 at t≥0. Activation function f_δ (δ — the time of the transition of a neuron to the state of activity) is considered of two types: 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.)┤ For the system of differential equations under consideration, a boundary value problem with the conditions v_i (0)-v_i (T)=γ_i, i=(1,n) ̅ is studied. In both cases δ=0 (discontinuous function f_0) and δ>0 (f_0 continuous function), a solution exists, and if δ>(T|W|_(R^n→R^n ))/(1-e^(-αT) ),где W=(w_ij )_(n×n), the problem has a unique solution. The work also provides estimates for the solution and its derivative. Theorems on fixed points of continuous mappings of metric and normed spaces and on fixed points of monotonic mappings of partially ordered spaces are used. The results obtained are applied to the study of periodic solutions of the differential system under consideration.