Differential Non-autonomous Representation of the Integrative Activity of a Neural Population by a Bilinear Second-Order Model with Delay

Abstract

For neuromorphic processes specified by the behavior of a local neural population (for example, processes induced by a “brain-machine” interface platform of the Neuralink type), we study the solvability of the problem of the existence of a differential realization of these processes in the class of bilinear nonstationary ordinary differential equations of the second order (with delay) in a separable Hilbert space. This formulation belongs to the type of inverse problems for an additive combination of nonstationary linear and bilinear operators of evolutionary equations in an infinite-dimensional Hilbert space. The metalanguage of the theory being developed is the constructions of tensor products of Hilbert spaces, orthocomplemented lattice structures, the functional means of the nonlinear Rayleigh-Ritz operator, and the principle of maximum entropy. It is shown that the property of sublinearity of this operator permits one to obtain conditions for the existence of such differential realizations; concurrently, metric conditions of the continuity of the projectivization of this operator are substantiated with the calculation of the fundamental group of its compact image. This work was financially supported by the Russian Foundation for Basic Research (project no. 19-01-00301).