Global Dynamics and Synchronization of 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, 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 captures the essential features of the nonlinearity and the conductances of two dominant ionic current channels of sodium and potassium coupled with the membrane voltage in gated firing functionality of biological neural networks by the original Hodgkin-Huxley model. Through uniform and sharp \emph{a priori} estimates purely by mathematical hard analysis on the solutions of the model equations and the derived 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. Synchronization result with fractional-power convergence rate instead is also proved for fractional memristive Hodgkin-Huxley-Wilson neural networks.