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

Abstract

In a series of two papers, we investigate the mechanisms by which complex oscillations are generated in a class of nonlinear dynamical systems with resets modeling the voltage and adaptation of neurons. This first paper presents mathematical analysis showing that the system can support bursts of any period as a function of model parameters. In continuous dynamical systems with resets, period-incrementing structures are complex to analyze. In the present context, we use the fact that bursting patterns correspond to periodic orbits of the adaptation map that governs the sequence of values of the adaptation variable at the resets. Using a slow-fast approach, we show that this map converges towards a piecewise linear discontinuous map whose orbits are exactly characterized. That map shows a period-incrementing structure with instantaneous transitions. We show that the period-incrementing structure persists for the full system with non-constant adaptation, but the transitions are more complex. We investigate the presence of chaos at the transitions.

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

“…Introduction. In this paper, we continue our study of hybrid integrate-andfire neuronal models from [49], turning our attention to the analysis of mixed-mode oscillations (MMOs). MMOs are trajectories exhibiting small or subthreshold oscillations alternating with one or more large amplitude oscillations or spikes.…”

“…From the biological viewpoint, neuronal activity patterns, including MMOs, rely on ionic and biochemical mechanisms that are accurately described by nonlinear dynamical systems of relatively high complexity, such as variants on the celebrated Hodgkin-Huxley model [17,50]. As described in our companion paper [49], in contrast with detailed biophysical models, integrate-and-fire models are abstractions of the voltage dynamics in which differential equations describing the dynamics of membrane depolarization are combined with a discrete reset corresponding to the emission of an action potential (a spike) and subsequent hyperpolarization. These models, first introduced more than a century ago [30], have evolved to incorporate nonlinearities to model the fast dynamics of spike initiation [8,10] and additional variables modeling adaptation [19], synaptic dynamics [38] or resonant properties [22].…”

“…We show that these can exhibit a wide variety of MMO patterns when the subthreshold dynamics features two unstable fixed points, a saddle and an unstable focus. We investigate spike patterns through iterates of a discrete map, the so-called adaptation map introduced in [60] (see the companion paper [49] for more details on the construction and use of this map). While previous works have considered settings in which the adaptation map is continuous, here, in the presence of an unstable focus, we will show that the adaptation map is singular: it may be undefined on a countable set of values at which the map has well-defined and finite left and right limits and infinite one-sided derivatives.…”

“…with γ ≤ 1 and d ≥ 0; in this work, we allow γ < 1, which arises in accounting for spike duration (see [49] for details). Numerical simulations performed in the present article correspond to the case of the quartic model F (v) = v 4 + av with, unless otherwise stated,…”

“…As discussed in the companion paper [49], the 1-dimensional adaptation map Φ can be defined based on the orbits of the system in the phase plane. Specifically, if…”