Spike-adding and reset-induced canard cycles in adaptive integrate and fire models

Abstract

We study a class of planar integrate and fire (IF) models called adaptive integrate and fire (AIF) models, which possesses an adaptation variable on top of membrane potential, and whose subthreshold dynamics is piecewise linear (PWL). These AIF models therefore have two reset conditions, which enable bursting dynamics to emerge for suitable parameter values. Such models can be thought of as hybrid dynamical systems. We consider a particular slow dynamics within AIF models and prove the existence of bursting cycles with N resets, for any integer N . Furthermore, we study the transition between N -and (N + 1)-reset cycles upon vanishingly small parameter variations and prove (for N = 2) that such transitions are organised by canard cycles. Finally, using numerical continuation we compute branches of bursting cycles, including canard-explosive branches, in these AIF models, by suitably recasting the periodic problem as a two-point boundary-value problem.

Modeling impairment of ionic regulation with extended Adaptive Exponential integrate-and-fire models

“…Further analysis would be required to follow up on such considerations, which are an interesting topic for future work. For more elements about slow-fast bursting dynamics in IF models, we refer the reader to [Desroches et al, 2021, Desroches et al, 2024.…”

Synchronization of Canards in Coupled Canard-Generating Bonhoeffer-Van Der Pol Oscillators Subject to Weak Periodic Perturbations

“…Canards are among the most important discoveries of the late 20th century, and they have been studied extensively for over four decades [1]- [11]. Here, we consider a van der Pol oscillator of the following form: † The author is with the Graduate School of Regional Development and Creativity, Utsunomiya University, Utsunomiya 321-8585, Japan † † The author is with the Department of Electrical, Electronics and Information Engineering, Nagaoka University of Technology, Nagaoka 940-2188, Japan † † † The authors are with the Graduate School of Electrical and Information Engineering, Shonan Institute of Technology, Fujisawa 251-8511, Japan…”

Canard Mechanism and Rhythm Dynamics of Neuron Models

“…A recent review summarizes the pioneering work of Rinzel and Izhikevich on bursting patterns' classification of excitatory systems and proposes a canard mechanism for folded-node singularities, explaining that conductance-based MMOs consist of subthreshold and superthreshold oscillations [100]. Canards have been the subject of active study since it was proposed in the 1980s, and they have been used to understand the firing patterns of neurons [101][102][103][104]. Canards are closely related to synchronization, spiking-adding, and complex oscillations of excitation networks [105][106][107].…”