Global Dynamics and Synchronization of Hodgkin-Huxley-Wilson Neural Networks
Abstract
Hodgkin-Huxley equations as a monumental breakthrough in neuroscience theory had been distilled into many simplified models to study, but this mathematical model itself was not 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 coupled with the neuron membrane voltage in gated firing and bursting functionality of biological neural networks in the original Hodgkin-Huxley model. Through the uniform and sharp \emph{a priori} estimates purely by mathematical hard analysis on the solutions of the model equations and then the interneuron differencing equations of gap functions, it is rigorously proved that global solution dynamics are robustly dissipative with a sharp ultimate bound and that exponential synchronization of the Hodgkin-Huxley-Wilson neural networks at a uniform convergence rate occurs if the interneuron coupling strength satisfies an explicitly computable threshold condition. The synchronization result with fractional-power convergence rate is also proved for time-fractional memristive Hodgkin-Huxley-Wilson neural networks.
Related
- High-Order Unconditionally Stable IMEX BDF--FEM Schemes for the Hodgkin--Huxley Model on a Branched Axon
- Global synchronization in time-varying directed higher-order networks
- Stochastic dynamics and synchronization in motif-based neuronal networks
- Simple Cuts, Simple Landscapes: Global Synchronization and Rank-Width-One Matrices
- Predefined-Time Synchronization of Fractional-Order Delayed Memristor-Based Fuzzy Cellular Neural Networks
BibTeX
@misc{you2026global,
title = {{Global Dynamics and Synchronization of Hodgkin-Huxley-Wilson Neural Networks}},
author = {Yuncheng You},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.07803},
eprint = {2605.07803},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.07803},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global Dynamics and Synchronization of Hodgkin-Huxley-Wilson Neural Networks
AU - Yuncheng You
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2605.07803
UR - https://arxiv.org/abs/2605.07803
AB - Hodgkin-Huxley equations as a monumental breakthrough in neuroscience theory had been distilled into many simplified models to study, but this mathematical model itself was not 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 coupled with the neuron membrane voltage in gated firing and bursting functionality of biological neural networks in the original Hodgkin-Huxley model. Through the uniform and sharp \emph{a priori} estimates purely by mathematical hard analysis on the solutions of the model equations and then the interneuron differencing equations of gap functions, it is rigorously proved that global solution dynamics are robustly dissipative with a sharp ultimate bound and that exponential synchronization of the Hodgkin-Huxley-Wilson neural networks at a uniform convergence rate occurs if the interneuron coupling strength satisfies an explicitly computable threshold condition. The synchronization result with fractional-power convergence rate is also proved for time-fractional memristive Hodgkin-Huxley-Wilson neural networks.
ER -