Spiking without Resets: Continuous Integrate-and-Fire Dynamics in Neuronal Circuits

The leaky integrate-and-fire paradigm is widely used to describe spiking dynamics in biological and artificial neurons. However, its implementation typically relies on explicit reset rules or intrinsic mechanisms involving negative differential resistance. Here we address the question: Can spike-like behavior emerge within a fully continuous dynamical framework without such ingredients?. We study a minimal resistive-capacitive circuit coupled to a conductance-activated quasi-linear memristor characterized by a single intrinsic relaxation time scale and an internal state variable. We demonstrate that spiking arises from the nonlinear coupling of the state variable and its voltage-dependent equilibrium value. Rather than being governed by a minimum transition slope in the static current-voltage characteristics, the onset of spiking does not exhibit sharp parametric thresholds but instead depends sensitively on the excitation frequency.