Wild oscillations in a nonlinear neuron model with resets: (Ⅱ) Mixed-mode oscillations

Abstract

This work continues the analysis of complex dynamics in a class of bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal voltage dynamics with adaptation and spike emission. We show that these models can generically display a form of mixed-mode oscillations (MMOs), which are trajectories featuring an alternation of small oscillations with spikes or bursts (multiple consecutive spikes). The mechanism by which these are generated relies fundamentally on the hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts, contrasting with classical MMO mechanisms in ordinary differential equations involving more than three dimensions and generally relying on a timescale separation. The decomposition of mechanisms reveals the geometrical origin of MMOs, allowing a relatively simple classification of points on the reset manifold associated to specific numbers of small oscillations. We show that the MMO pattern can be described through the study of orbits of a discrete adaptation map, which is singular as it features discrete discontinuities with unbounded left-and right-derivatives. We study orbits of the map via rotation theory for discontinuous circle maps and elucidate in detail complex behaviors arising in the case where MMOs display at most one small oscillation between each consecutive pair of spikes.

Wild oscillations in a nonlinear neuron model with resets: (Ⅰ) Bursting, spike-adding and chaos

“…Our analysis is based on an adaptation map, which can be defined on the whole real line in this situation. In the companion paper [57], we will switch gears and investigate the case where the subthreshold system has two unstable fixed points, a spiral and a saddle, with a heteroclinic orbit from the former to the latter. In that setting, we obtain and study the corresponding, distinctive forms of dynamics that arise, which feature alternations of small oscillations and spikes or bursts, and are known as mixed-mode oscillations.…”

“…In [32], a generalized linear integrate-and-fire system was investigated via a similar map that is locally contractive, either globally or in a piecewise manner; conditions for spiking and bursting dynamics were established and bifurcations underlying transitions between solution patterns were studied. In the accompanying paper [57], we address the question of the dynamics of the system in the presence of two unstable equilibria of the subthreshold dynamics; in that case, the map Φ is no longer continuous, and the rotation theory for discontinuous maps is applied to characterize the dynamics. The present study focuses on characterizing a sequence of period-incrementing bifurcations associated with spike-adding transitions in bursting solutions.…”

Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models

“…A general framework to study the dynamics of hybrid systems becomes difficult to obtain, even when the input currents are assumed to be constant (the system remains autonomous), mainly because they are discontinuous due to the reset condition. One of the most common strategies in the nonsmooth literature (see [ML12,dBBCK08]) is to smooth the dynamics by considering the so-called impact map (also known in neuroscience as firing phase map or adaptation map) defined on the threshold where the reset condition is applied [TB09, RSRTV17,CTW12]. However, the impact map does not allow one to study itineraries or trajectories that do not hit the threshold and has some domain restrictions.…”

Farey–Lorenz Permutations for Interval Maps

“…The motivation for this paper comes from the paper [3], where the authors consider a mathematical model of a neuron, and after some simplifications they get an interval map. Under certain assumptions, this map is Lorenz-like.…”