Wild oscillations in a nonlinear neuron model with resets: (II) Mixed-mode oscillations - Inria - Institut national de recherche en sciences et technologies du numérique Accéder directement au contenu
Article Dans Une Revue Discrete and Continuous Dynamical Systems - Series B Année : 2017

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

Résumé

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 a daptation 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.

Dates et versions

hal-01256368 , version 1 (14-01-2016)

Identifiants

Citer

Justyna Signerska-Rynkowska, Jonathan Touboul, Alexandre Vidal, Jonathan Rubin. Wild oscillations in a nonlinear neuron model with resets: (II) Mixed-mode oscillations. Discrete and Continuous Dynamical Systems - Series B, 2017, 22 (10), pp.4003-4039. ⟨10.3934/dcdsb.2017205⟩. ⟨hal-01256368⟩
414 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More