TITLE:
Stability and Bifurcation in a Hopfield Neuron Model with Delays
AUTHORS:
Suqi Ma, S. J. Hogan
KEYWORDS:
Hopfield Network Model, Distributed Time Delay, Center Manifold, Period Doubling Bifurcation
JOURNAL NAME:
International Journal of Modern Nonlinear Theory and Application,
Vol.14 No.1,
March
20,
2025
ABSTRACT: The Hopfield system is an artificial neuron model that can be applied to neuron memory and information processing. Like synaptic connections of neurons, the inhibition or excitable feedback often incorporate delay effects, which are either discrete or distributed time delays. With delays varying, Hopf bifurcation of distributed time delay is investigated and stability regime is partitioned by Hopf curves on the parameter plane. Lyapunov-Schmidt reduction skills combined with center manifold theory are applied to discuss the stability of bifurcating periodical solutions arising from Hopf points. In addition, DDE-Biftool software significantly provides the numerical computation of stability analysis of periodical solutions appearing in discrete time delay Hopfield system. The period-doubling bifurcation of periodical solutions, which form P2 circles and P4 circles, respectively, by continuous periodical solutions with varying free parameters, is discussed.