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The authors analytically characterize the piecewise-linear McKean neuron model under the combined influence of Gaussian diffusion and Poissonian stochastic resetting applied to the fast voltage variable. By deploying an adiabatic approximation when resetting occurs on the fast timescale, they solve a modified Fokker-Planck equation via parabolic cylinder functions to derive an averaged slow dynamic governed by a tractable, parameter-dependent fixed point. Furthermore, they reveal that shifting stochastic resetting to the slow timescale fundamentally alters the phase space, driving noisy oscillations via nullcline branch-switching.
Stochastic resetting completely flips the behavior of excitable neural systems from stable fixed-point convergence to continuous noisy oscillations depending solely on whether interrupts act on fast or slow timescales.
In this paper we investigate the combined effects of stochastic resetting and diffusion on a slow--fast dynamical system given by the piecewise-linear McKean model. That is, the fast variable $v$ is subject to Gaussian white noise with effective diffusivity $D$ and is reset to a fixed value $v_r$ at a random sequence of times generated from a Poisson process with rate $r$. Assuming that resetting occurs on the fast timescale, we freeze the slow variable \(w\) under an adiabatic approximation and determine the resulting non-equilibrium stationary state (NESS) of the fast variable as the solution of a modified Fokker--Planck equation. We show that the NESS can be expressed in terms of parabolic cylinder functions, whose asymptotic behavior allows us to recover the corresponding NESS without diffusion in the small-diffusion limit. The NESS is used to derive an averaged equation for the slow dynamics, whose solution converges to a stable fixed point \(w^*\) that depends on \(D\), \(r\) and \(v_r\). This fixed point effectively determines the long-time behaviour of the full system. Finally, we analyze the regime in which resetting occurs on the same timescale as the slow variable. We show how the slow variable now undergoes noisy oscillations due to resetting-induced switching between branches of the fast nullcline and derive the corresponding NESS for $w$ in the non-diffusive case.