Section: New Results

Slow-fast dynamics in Neuroscience

Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system

Participants : Daniele Avitabile [University of Nottingham, UK] , Mathieu Desroches, Edgar Knobloch [University of California Berkeley, USA] , Martin Krupa [UCA, Inria MathNeuro] .

A subcritical pattern-forming system with nonlinear advection in a bounded domain is recast as a slow-fast system in space and studied using a combination of geometric singular perturbation theory and numerical continuation. Two types of solutions describing the possible location of stationary fronts are identified, whose origin is traced to the onset of convective and absolute instability when the system is unbounded. The former are present only for non-zero upstream boundary conditions and provide a quantitative understanding of noise-sustained structures in systems of this type. The latter correspond to the onset of a global mode and are present even with zero upstream boundary conditions. The role of canard trajectories in the nonlinear transition between these states is clarified and the stability properties of the resulting spatial structures are determined. Front location in the convective regime is highly sensitive to the upstream boundary condition, and its dependence on this boundary condition is studied using a combination of numerical continuation and Monte Carlo simulations of the partial differential equation. Statistical properties of the system subjected to random or stochastic boundary conditions at the inlet are interpreted using the deterministic slow-fast spatial dynamical system.

This work has been published in Proceedings of the Royal Society A and is available as [15].

Canard dynamics and anticipated synchronisation in spiking models

Participants : Elif Köksal Ersöz, Mathieu Desroches, Claudio Mirasso [University of the Balearic Islands, Palma, Spain] , Serafim Rodrigues [Ikerbasque, BCAM, Bilbao, Spain] .

This project is on the phenomenon of anticipated synchronisation, studied theoretically in a number of models of excitable systems over the past fifteen years or so, and observed experimentally in laser systems. The idea is that when coupling two identical excitable system unidirectionally from a “master” system to a “slave” system with a delayed term of the slave's signal in its own differential equation, one may observe that the slave reacts to an external stimulus before the master, and this is referred to as anticipation or anticipated synchronisation. Even though a number of studies have reported and analysed this effect in various systems, its main underpinning mechanisms remain elusive. In the current project, we show that in the case where the systems have an explicit slow-fast nature, then the canard regime can induce anticipation and explain its feature. Our objective is to go beyond the theoretical explanation, on which we are currently preparing an article, and to propose an electrophysiological protocol so as observe this phenomenon in real neurons. This is very much related to the PhD work of Elif Köksal Ersöz on the synchronisation properties of canard oscillators, in particular to the paper [25] (see Section 4.4.5 below). This postdoc is funded by the “tail” of the ERC Advanced Grant NerVi held by Olivier Faugeras.

Spike-adding in a canonical three time scale model: superslow explosion & folded-saddle canards

Participants : Mathieu Desroches, Vivien Kirk [University of Auckland, New-Zealand] .

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 systems with two time scales. However, the details of the process can be much more complex in this three-time-scale context than in two-time-scale systems. In particular, we find that spike-adding can involve canard explosions occurring on two different time scales and is associated with passage near a folded-saddle singularity. We show that the form of spike-adding transition that occurs depends on the geometry of certain singular limit systems, specifically the relative positions of the critical and superslow manifolds. We also show that, unlike the case of spike-adding in two-time-scale systems, the onset of a new spike in our model is not typically associated with a local maximum in the period of the bursting oscillation.

This work has been submitted for publication and is available as [31].

Piecewise-linear (PWL) canard dynamics: Simplifying singular perturbation theory in the canard regime using piecewise-linear systems

Participants : Mathieu Desroches, Soledad Fernández-García [University of Sevilla, Spain] , Martin Krupa [UCA, Inria MathNeuro] , Rafel Prohens [University of the Balearic Islands, Spain] , Antonio Teruel [University of the Balearic Islands, Spain] .

In this chapter we gathered recent results on piecewise-linear (PWL) slow-fast dynamical systems in the canard regime. By focusing on minimal systems in ${ℝ}^2$ (one slow and one fast variables) and ${ℝ}^3$ (two slow and one fast variables), we proved the existence of (maximal) canard solutions and show that the main salient features from smooth systems is preserved. We also highlighted how the PWL setup carries a level of simplification of singular perturbation theory in the canard regime, which makes it more amenable to present it to various audiences at an introductory level. Finally, we presented a PWL version of Fenichel theorems about slow manifolds, which are valid in the normally hyperbolic regime and in any dimension, which also offers a simplified framework for such persistence results.

This work has been accepted for publication as a chapter in a book titled Nonlinear Systems; Vol. 1: Mathematical Theory and Computational Methods (Springer, in press) and is available as [28].

Synchronization of weakly coupled canard oscillators

Participants : Elif Köksal Ersöz, Mathieu Desroches, Martin Krupa [UCA, Inria MathNeuro] .

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, synchronization conditions were established under the weak coupling assumption, leading to a criterion for the existence of synchronous solutions of weakly coupled relaxation oscillators. Previous analysis relies on the fact that the slow nullcline does not intersect the fast nullcline near one of its fold points, where canard solutions can arise. In the present study we use numerical continuation techniques to solve the adjoint equations and we show that synchronization properties of canard cycles are different than those of classical relaxation cycles. In particular, we highlight a new special role of the maximal canard in separating two distinct synchronization regimes: the Hopf regime and the relaxation regime. Phase plane analysis of slow-fast oscillators undergoing a canard explosion provides an explanation for this change of synchronization properties across the maximal canard.

This work has been published in Physica D and is available as [25].