Asymptotic stability proof and port-Hamiltonian physics-informed neural network approach to chaotic synchronization in Hindmarsh-Rose neurons

Abstract: We study chaotic synchronization in a five-dimensional Hindmarsh-Rose neuron model augmented with electromagnetic induction and a switchable memristive autapse. For two diffusively coupled neurons, we derive the linearized error dynamics and prove global asymptotic stability of the synchronization manifold via a quadratic Lyapunov function. Verifiable sufficient conditions follow from Sylvester's criterion on the leading principal minors, and convergence is established using Barbalat's lemma. Leveraging Helmholtz's decomposition, we separate the error field into conservative and dissipative parts and obtain a closed-form expression for the synchronization energy, along with its dissipation law, providing a quantitative measure of the energetic cost of synchrony. Numerical simulations confirm complete synchronization, overall decay of the synchronization energy, and close agreement between Lyapunov and Hamiltonian diagnostics across parameter sweeps. Building on these results, we introduce a port-Hamiltonian physics-informed neural network that embeds the conservative/dissipative structure in training through physically motivated losses and structural priors. The learned Hamiltonian and energy-rate match analytical benchmarks. The framework narrows the gap between dynamical systems theory and data-driven discovery, and provides a template for energy-aware modeling and control of nonlinear neuronal synchronization.