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

A. S. Lanina   ; E. A. Pluzhnikova

Vestnik rossijskih universitetov. Matematika, Tome 27 (2022) no. 139, pp. 270-283

Résumé

The Hopfield-type model of the dynamics of the electrical activity of the brain, which is a system of differential equations of the form \begin{equation*} \dot{v}_{i}= -\alpha v_{i}+\sum_{j=1}^{n}w_{ji}f_{\delta}(v_{j})+I_{i}( t), \ \ \, i=\overline{1,n}, \ \ \, t\geq 0, \end{equation*} is investigated. The model parameters are assumed to be given: $\alpha>0,$ $w_{ji}>0$ for $i\neq j$ and $w_{ii}=0,$ $I_{i}(t)\geq 0.$ The activation function $f_{\delta}$ ($\delta$ is the time of the neuron transition to the state of activity) of two types is considered: $$ \delta=0 \ \Rightarrow f_{0}(v)=\left\{ \begin{array}{ll} 0, &v\leq\theta,\\ 1, &v>\theta; \end{array}\right. \ \ \ \ \ \ \delta> 0 \ \Rightarrow \ f_{\delta}(v)=\left\{ \begin{array}{ll} 0, & v\leq \theta,\\ {\delta}^{-1}( v-\theta), & \theta < v \leq \theta+\delta,\\ 1, &v>\theta+\delta. \end{array}\right.$$ In the case of $\delta> 0$ (the function $f_{\delta}$ 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 $\delta= 0$ (the function $f_{0}$ is discontinuous at the point $\theta$), 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.

Détail

Keywords: neural network, differential equation with discontinuous right-hand side, delay, Cauchy problem, upper and lower solutions, mappings of partially ordered spaces.
Mots-clés : existence of solution

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