Improved energy-adaptive coupling for synchronization of neurons with nonlinear and memristive membranes

Synaptic connections in neural systems can be adaptively regulated through the exchange of field energy between neurons. This paper investigates the energy-based adaptive coupling mechanism in the context of two neuron models: a nonlinear membrane model and a memristive membrane model. Both models are examined under various external and intrinsic conditions, revealing rich dynamical behaviors including periodic, quasi-periodic, and chaotic firing patterns, as well as multistability. An energy-based adaptive coupling strategy, based on a threshold-triggered adjustment of coupling intensity driven by energy diversity, has been previously introduced for reaching synchronization and energy balance in neurons. Here, we enhance this adaptive coupling to incorporate the hyperbolic tangent of the energy difference relative to a threshold. This smooth, bounded function allows the coupling intensity to evolve more robustly and precisely. Synchronization is analyzed for both models using both the original and proposed adaptive coupling strategy by computing the synchronization factor across parameter sets. Comparative simulations demonstrate that the proposed coupling strategy yields improved synchronization performance in both pairs of neurons and ring network configurations. The enhanced coupling consistently achieves higher synchronization factors, faster convergence, and greater robustness across complex dynamical regimes.