Global Dynamics and Synchronization of Hodgkin-Huxley-Wilson Neural Networks

TL;DR

This paper introduces and analyzes a new Hodgkin-Huxley-Wilson neural network model, demonstrating its global dissipative dynamics and conditions for exponential synchronization, including extensions to fractional memristive networks.

Contribution

The paper proposes a novel Hodgkin-Huxley-Wilson neural network model and rigorously proves its global dynamics and synchronization properties, extending to fractional memristive cases.

Findings

Global solution dynamics are robustly dissipative with a sharp ultimate bound.

Complete synchronization occurs at an exponential rate under explicit coupling strength conditions.

Results extend to Caputo fractional memristive Hodgkin-Huxley-Wilson neural networks.

Abstract

Hodgkin-Huxley equations as a monumental breakthrough in biological and physiological theory of the 20th century had been distilled into many simplified models to study, typically FitzHugh-Nagumo equations and Hindmarsh-Rose equations, but the model itself not being fully investigated in terms of global and asymptotic dynamics due to its strong nonlinearity and higher dimensionality. In this paper a new model called Hodgkin-Huxley-Wilson neural networks is proposed and investigated. This model captured the essential features of the nonlinearity and the conductances of two dominant ionic current channels of sodium and potassium coupled with the membrane equation in the original Hodgkin-Huxley model. Through uniform and sharp \emph{a priori} estimates by hard analysis on the solutions of the model equations and the interneuron differencing equations, It is rigorously proved that global solution dynamics are robustly dissipative with a sharp ultimate bound and that complete synchronization of the Hodgkin-Huxley-Wilson neural networks at an exponential convergence rate occurs if the interneuron coupling strength satisfies an explicitly computable threshold condition. The main results are further extended to Caputo fractional memristive Hodgkin-Huxley-Wilson neural networks.