Memcapacitor-enabled bursting oscillations in Wilson neurons: A fast-slow dynamics perspective

Conventional Wilson neurons suffer from a fixed membrane capacitance that neglects the intrinsic memory properties of biological membranes, thereby restricting their dynamics to simple firing patterns. In this work, we upgrade the classical Wilson model to a three-dimensional fast–slow system by replacing the static capacitor with a charge-controlled memcapacitor. Fast–slow decomposition and bifurcation analysis reveal that the resulting bursting is of the fold–Hopf type, characterized by an S-shaped equilibrium manifold comprising a stable lower branch, an unstable middle saddle branch, and an upper branch that stabilizes beyond the Hopf threshold. The transitions between quiescent and spiking states are governed by the interplay of saddle-node and Hopf bifurcations. Numerical simulations uncover two distinct stimulus-dependent regulatory phenomena: (i) when the stimulation amplitude exceeds the Hopf threshold, bifurcation delay induces a prolonged near-quiescent window embedded within each bursting episode; and (ii) increasing the stimulation frequency elevates the spiking threshold, reduces the number of spikes per burst, but significantly increases the temporal bursting density, revealing an intrinsic antagonistic trade-off between frequency and burst content. These findings demonstrate that memcapacitive membrane memory effectively enriches the dynamical repertoire of Wilson neurons. The proposed model provides a robust theoretical foundation for investigating memory-augmented neural dynamics and holds promise for applications in neuromorphic computing and bio-inspired hardware systems.