Noise-induced escape in an excitable system

Abstract

We consider the stochastic dynamics of escape in an excitable system, the FitzHugh-Nagumo (FHN) neuronal model, for different classes of excitability. We discuss, first, the threshold structure of the FHN model as an example of a system without a saddle state. We then develop a nonlinear (nonlocal) stability approach based on the theory of large fluctuations, including a finite-noise correction, to describe noise-induced escape in the excitable regime. We show that the threshold structure is revealed via patterns of most probable (optimal) fluctuational paths. The approach allows us to estimate the escape rate and the exit location distribution. We compare the responses of a monostable resonator and monostable integrator to stochastic input signals and to a mixture of periodic and stochastic stimuli. Unlike the commonly used local analysis of the stable state, our nonlocal approach based on optimal paths yields results that are in good agreement with direct numerical simulations of the Langevin equation.

Spike-Threshold Variability Originated from Separatrix-Crossing in Neuronal Dynamics

“…However, because that the threshold is instantaneous, it is possible to extend the framework to including the noise or fluctuating inputs, which cause instantaneous random variation of the threshold and state. Khovanov et al 35. and Franović et al 3637.…”

Activation process in excitable systems with multiple noise sources: One and two interacting units

“…This has been achieved by introducing an appropriate terminating boundary set, given by the spiking branch of the cubic nullcline. Note that our problem setup is different from the earlier numerical studies on pulse triggering, which have introduced the terminating boundary as a fixed threshold [16,17], as well as the recent study on a single F HN unit [15], where the ghost separatrix has been used as a terminating boundary within the generalized escape problem. The advantages of our approach lie in that the terminating boundary is analytically tractable, while the adopted formulation further facilitates an immediate generalization of the activation problem from the case of a single unit to different scenarios with two interacting units.…”

“…As announced earlier, the discussion is put into a broader perspective by comparing the M P AP s to the trajectories generated by the effective Hamiltonian system (6) according to the prescription provided in Subsection III A. Taking the case r x : r y = 0 : 1 as an example, we first demonstrate how the problem of first pulse emission is different from the extension of the escape problem to an excitable F HN unit addressed in [15]. Apart from the ghost separatrix, denoted by the dotted line, Fig.…”

“…The latter further implies that the boundary constitutes an invariant set. Boundary layer may still be perceived as a single line, a kind of "ghost separatrix" [15], because at distances d >> ǫ from the fold point (1, 2/3), all the constituent trajectories become virtually indistinguishable.…”

“…To explain the differences, one first invokes a general remark that for excitable systems, two possible types of response derive from the proximity to bifurcation rather than the bistable dynamics, so that the threshold-like behavior is not associated to a genuine separatrix (saddle structure), but rests on the coaction of nonlinearity and the sharp separation between the system's characteristic time scales. Thus, resolving the origin and character of threshold-like behavior for excitable elements may be linked to an unstable fixed point or, as in case of a F HN unit, to a quasi-separatrix ("ghost separatrix") [15]. This is apparently different from the typical escape problem scenario.…”