DESROCHES , Mathieu

The results gathered in this thesis deal with multiple time scale dynamical systems near non-hyperbolic points, giving rise to canard-type solutions, in systems of dimension 2, 3 and 4. Bifurcation theory and numerical continuation methods adapted for such systems are used to analyse canard cycles as well as canard-induced complex oscillations...

We examine the origin of complex bursting oscillations in a phenomenological ordinary differential equation model with three time scales. We show that bursting solutions in this model arise from a Hopf bifurcation followed by a sequence of spike-adding transitions, in a manner reminiscent of spike- adding transitions previously observed in...

We revisit elliptic bursting dynamics from the viewpoint of torus canard solutions. We show that at the transition to and from elliptic burstings, classical or mixed-type torus canards can appear, the difference between the two being the fast subsystem bifurcation that they approach, saddle-node of cycles for the former and subcritical Hopf for...

We propose a generalization of the neurotransmitter release model proposed in \emph{Rodrigues et al. (PNAS, 2016)}. We increase the complexity of the underlying slow-fast system by considering a degree-four polynomial as parametrization of the critical manifold. We focus on the possible transient and asymptotic dynamics, exploiting the...

Neuronal excitability manifests itself through a number of key markers of the dynamics and it allows to classify neurons into different groups with identifiable voltage responses to input currents. In particular, two main types of excitability can be defined based on experimental observations, and their underlying mathematical models can be...

Canards are special solutions to ordinary differential equations that follow invariant repelling slow manifolds for long time intervals. In realistic biophysical single-cell models, canards are responsible for several complex neural rhythms observed experimentally, but their existence and role in spatially extended systems is largely...

We show that during the transition from and to elliptic burstings both classical and mixed-type torus canards appear in a Wilson-Cowan type neuronal network model, as well as in its corresponding mean-field framework. We show numerically the overlap between the network and mean-field dynamics. We comment on that mixed-type torus canards result...

The present work develops a new approach to studying parabolic bursting, and also proposes a novel four-dimensional canonical and polynomial-based parabolic burster. In addition to this new polynomial system, we also con- sider the conductance-based model of the Aplysia R15 neuron known as Plant's model, and a reduction of this prototypical...

International audience

Abstract Synchronization is a widespread phenomenon in the brain. Despite numerous studies, the specific parameter configurations of the synaptic network structure and learning rules needed to achieve robust and enduring synchronization in neurons driven by spike-timing-dependent plasticity (STDP) and temporal networks subject to homeostatic...

We revisit the Wendling-Chauvel neural mass model by reducing it to eight ODEs and adding a differential equation that accounts for a dynamic evolution of the slow inhibitory synaptic gain. This allows to generate dynamic transitions in the resulting nine-dimensional model. The output of the extended model can be related to EEG patterns...

Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s...

Neuronal excitability manifests itself through a number of key markers of the dynamics and it allows to classify neurons into different groups with identifiable voltage responses to input currents. In particular, two main types of excitability can be defined based on experimental observations, and their underlying mathematical models can be...

We report a detailed analysis on the emergence of bursting in a recently developed neural mass model that takes short-term synaptic plasticity into account. The one being used here is particularly important, as it represents an exact meanfield limit of synaptically coupled quadratic integrate & fire neurons, a canonical model for type I...

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...

In this article, we present a computational study of the Conductance-Based Adaptive Exponential (CAdEx) integrate-and-fire neuronal model, focusing on its multiple timescale nature, and on how it shapes its main dynamical regimes. In particular, we show that the spiking and so-called delayed bursting regimes of the model are triggered by...

In this article, we present a computational study of the Conductance-Based Adaptive Exponential (CAdEx) integrate-and-fire neuronal model, focusing on its multiple timescale nature, and on how it shapes its main dynamical regimes. In particular, we show that the spiking and so-called delayed bursting regimes of the model are triggered by...

A minimal system for parabolic bursting, whose associated slow flow is integrable, is presented and studied both from the viewpoint of bifurcation theory of slow-fast systems, of the qualitative analysis of its phase portrait and of numerical simulations. We focus the analysis on the spike-adding phenomenon. After a reduction to a periodically...

submitted for publication

From the action potentials of neurons and cardiac cells to the amplification of calcium signals in oocytes, excitability is a hallmark of many biological signalling processes. In recent years, excitability in single cells has been related to multiple-timescale dynamics through canards, special solutions which determine the effective thresholds...

Synchronization has been studied extensively in the context of weakly coupled oscillators using the so-called phase response curve (PRC) which measures how a change of the phase of an oscillator is affected by a small perturbation. This approach was based upon the work of Malkin, and it has been extended to relaxation oscillators. Namely,...

We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an infinite-dimensional dynamical system for the fast variables is coupled to a finite-dimensional dynamical...

Cortical spreading depression (CSD) is a wave of transient intense neuronal firing leading to a long lasting depolarizing block of neuronal activity. It is a proposed pathological mechanism of migraine with aura. Some forms of migraine are associated with a genetic mutation of the Nav1.1 channel, resulting in its gain of function and implying...