Control properties of neural activity in brain networks with multiple outlets: modelling via stochastic neutral FitzHugh–Nagumo inclusion

This paper investigates the study of stochastic neutral FitzHugh–Nagumo inclusion with non-instantaneous impulses governed by the Rosenblatt process. The FitzHugh–Nagumo model, a simplified version of the Hodgkin–Huxley model, gives a useful framework for understanding neural dynamics. It effectively captures the key features of neuronal excitability and action potential generation. The study applies the nonlinear alternative of the Leray–Schauder fixed-point theorem along with the measure of non-compactness to establish the existence of solutions under stochastic framework. Additionally, this paper explores the neural system's optimal control with the aid of Balder's theorem. Optimal control methods allow for the precise regulation of the neuron's excitability and action potential generation in the presence of stochastic influences, offering insights into managing neural dynamics. An example is given to validate the theoretical results for controlling neural activity.