We review and extend the previous work where a model was introduced for Hopfield-type neural networks, which allows for the existence of heteroclinic dynamics between steady patterns. This dynamics is a mathematical model of periodic or aperiodic switching between stored information items in the brain, in particular, in the context of...
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February 7, 2017 (v1)Journal articleUploaded on: February 28, 2023
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September 15, 2016 (v1)Journal article
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...
Uploaded on: March 25, 2023 -
May 5, 2017 (v1)Journal article
This survey article is concerned with the study of bifurcations of discontinuous piecewise-smooth maps, with a special focus on the one-dimensional case. We review the literature on circle maps and quasi-contractions and provide paths through this literature to prove sufficient conditions for the occurrence of two types of bifurcation scenarios...
Uploaded on: February 28, 2023 -
November 18, 2016 (v1)Journal article
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...
Uploaded on: March 25, 2023 -
March 2020 (v1)Journal article
Gamma rhythm (20–100Hz) plays a key role in numerous cognitive tasks: working memory, sensory processing and in routing of information across neural circuits. Incomparison with lower frequency oscillations in the brain, gamma-rhythm associated firing of the individual neurons is sparse and the activity is locally distributed in the cortex. Such...
Uploaded on: December 4, 2022 -
May 28, 2019 (v1)Journal article
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...
Uploaded on: December 4, 2022 -
July 18, 2016 (v1)Journal article
submitted for publication
Uploaded on: March 25, 2023 -
February 7, 2017 (v1)Journal article
International audience
Uploaded on: February 28, 2023 -
June 15, 2017 (v1)Journal article
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,...
Uploaded on: March 25, 2023 -
October 17, 2019 (v1)Journal article
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...
Uploaded on: February 28, 2023 -
October 11, 2017 (v1)Journal article
submitted for publication
Uploaded on: March 25, 2023 -
June 18, 2018 (v1)Journal article
We analyzed a generic relaxation oscillator under moderately strong forcing at a frequency much greater that the natural intrinsic frequency of the oscillator. Additionally, the forcing is of the same sign and, thus, has a nonzero average, matching neuroscience applications. We found that, first, the transition to high-frequency synchronous...
Uploaded on: December 4, 2022 -
July 9, 2015 (v1)Journal article
Slow–fast systems often possess slow manifolds, that is invariant or locally invariant sub-manifolds on which the dynamics evolves on the slow time scale. For systems with explicit timescale separation, the existence of slow manifolds is due to Fenichel theory, and asymptotic expansions of such manifolds are easily obtained. In this paper, we...
Uploaded on: March 25, 2023 -
June 2023 (v1)Journal article
NaV1.1 (SCN1A) is a voltage-gated sodium channel mainly expressed in GABAergic neurons. Loss of function mutations of NaV1.1 lead to epileptic disorders, while gain of function mutations cause a migraine in which cortical spreading depolarizations (CSDs) are involved. It is still debated how these opposite effects initiate two different...
Uploaded on: June 2, 2023 -
September 7, 2020 (v1)Journal article
Specific kinds of physical and biological systems exhibit complex Mixed-Mode Oscillations mediated by folded-singularity canards in the context of slow-fast models. The present manuscript revisits these systems, specifically by analysing the dynamics near a folded singularity from the viewpoint of inflection sets of the flow. Originally, the...
Uploaded on: December 4, 2022 -
September 4, 2015 (v1)Journal article
In this work, we analyze the existence and stability of canard solutions in a class of planar piecewise linear systems with three zones, using a singular perturbation theory approach. To this aim, we follow the analysis of the classical canard phenomenon in smooth planar slow–fast systems and adapt it to the piecewise-linear framework. We first...
Uploaded on: March 25, 2023 -
August 10, 2022 (v1)Journal article
An important function of the brain is to predict which stimulus is likely to occur based on the perceived cues. The present research studied the branching behavior of a computational network model of populations of excitatory and inhibitory neurons, both analytically and through simulations. Results show how synaptic efficacy, retroactive...
Uploaded on: December 3, 2022 -
December 26, 2019 (v1)Publication
24 pages, 11 figures
Uploaded on: December 4, 2022 -
September 16, 2018 (v1)Book section
In this chapter we gather recent results on piecewise-linear (PWL) slow-fast dynamical systems in the canard regime. By focusing on minimal systems in $\mathbb{R}^2$ (one slow and one fast variables) and $\mathbb{R}^3$ (two slow and one fast variables), we prove the existence of (maximal) canard solutions and show that the main salient features...
Uploaded on: February 28, 2023 -
July 27, 2021 (v1)Journal article
Loss of function mutations of SCN1A, the gene coding for the voltage-gated sodium channel NaV1.1, cause different types of epilepsy, whereas gain of function mutations cause sporadic and familial hemiplegic migraine type 3 (FHM-3). However, it is not clear yet how these opposite effects can induce paroxysmal pathological activities involving...
Uploaded on: December 4, 2022 -
2020 (v1)Journal article
In this article we present a biologically inspired model of activation of memory items in a sequence. Our model produces two types of sequences, corresponding to two different types of cerebral functions activation of regular or irregular sequences. The switch between the two types of activation occurs through the modulation of biological...
Uploaded on: December 4, 2022 -
February 23, 2016 (v1)Journal article
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...
Uploaded on: March 25, 2023 -
April 6, 2016 (v1)Journal article
In this article, we study canard solutions of the forced van der Pol equation in the relaxation limit for low-, intermediate-, and high-frequency periodic forcing. A central numerical observation made herein is that there are two branches of canards in parameter space which extend across all positive forcing frequencies. In the low-frequency...
Uploaded on: March 25, 2023 -
November 1, 2021 (v1)Journal article
Spreading depolarizations (SDs) are involved in migraine, epilepsy, stroke, traumatic brain injury, and subarachnoid hemorrhage. However, the cellular origin and specific differential mechanisms are not clear. Increased glutamatergic activity is thought to be the key factor for generating cortical spreading depression (CSD), a pathological...
Uploaded on: December 3, 2022