Gated triadic coupling in a Hopfield neural network: cooperative-competitive dynamics and synchronization

Inspired by the tripartite synapse, where a third element dynamically modulates the effective coupling between two neurons, this study abstracts that idea into a minimal three-neuron Hopfield neural network with multiplicative gated coupling. The third neuron acts as a dynamic modulator, adjusting the synaptic strength between the other two units without invoking auxiliary astrocytic dynamics. Through nonlinear analyses including bifurcation diagrams, Lyapunov exponent spectra, and two-parameter dynamical maps, we show that the three gated coupling strengths engage in both cooperative and competitive interactions, collectively governing transitions among chaotic, periodic, and globally coordinated firing patterns. Phase synchronization analysis shows that the three coupling strengths differentially regulate the transition from chaotic to phase-coordinated states. Circuit design and simulations validate the theoretical predictions and confirm the physical realizability of the model. This work elucidates the regulatory functions of gated triadic coupling in a minimal three-neuron motif and provides a circuit-realizable framework for developing tunable neuromorphic systems.