Mathieu Desroches

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This paper is concerned with the geometry of slow manifolds of a dynamical system with two slow and one fast variable. Specifically, we study the dynamics near a folded node singularity, which is known to give rise to so-called canard solutions. Geometrically, canards are intersection curves of two-dimensional attracting and repelling slow manifolds, and(More)
Biochemical energy is the fundamental element that maintains both the adequate turnover of the biomolecular structures and the functional metabolic viability of unicellular organisms. The levels of ATP, ADP and AMP reflect roughly the energetic status of the cell, and a precise ratio relating them was proposed by Atkinson as the adenylate energy charge(More)
Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast(More)
We present a numerical strategy to compute one-parameter families of isolas of equilibrium solutions in ODEs. Isolas are solution branches closed in parameter space. Numerical continuation is required to compute one single isola since it contains at least one unstable segment. We show how to use pseudo-arclength predictor-corrector schemes in order to(More)
Short-term synaptic plasticity strongly affects the neural dynamics of cortical networks. The Tsodyks and Markram (TM) model for short-term synaptic plasticity accurately accounts for a wide range of physiological responses at different types of cortical synapses. Here, we report a route to chaotic behavior via a Shilnikov homoclinic bifurcation that(More)
This article concerns the phenomenon of Mixed-Mode Bursting Oscillations (MMBOs). These are solutions of fast-slow systems of ordinary differential equations that exhibit both small-amplitude oscillations (SAOs) and bursts consisting of one or multiple large-amplitude oscillations (LAOs). The name MMBO is given in analogy to Mixed-Mode Oscillations, which(More)
The well-studied Hindmarsh-Rose model of neural action potential is revisited from the point of view of global bifurcation analysis. This slow-fast system of three paremeterised differential equations is arguably the simplest reduction of Hodgkin-Huxley models capable of exhibiting all qualitatively important distinct kinds of spiking and bursting(More)
Communication between neurons at chemical synapses is regulated by hundreds of different proteins that control the release of neurotransmitter that is packaged in vesicles, transported to an active zone, and released when an input spike occurs. Neurotransmitter can also be released asynchronously, that is, after a delay following the spike, or spontaneously(More)