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Section: New Results

Slow-Fast Dynamics in Neural Models

From Canards of Folded Singularities to Torus Canards in a Forced van der Pol Equation

Participants : John Burke [Boston University, USA] , Mathieu Desroches, Albert Granados [Technical University of Denmark, Lyngby, Denmark] , Tasso Kaper [Boston University, USA] , Maciej Krupa, Theodore Vo [Boston University, USA] .

In this work, 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 forcing regime, we demonstrate the existence of primary maximal canards induced by folded saddle nodes of type I and establish explicit formulas for the parameter values at which the primary maximal canards and their folds exist. Then, we turn to the intermediate- and high-frequency forcing regimes and show that the forced van der Pol possesses torus canards instead. These torus canards consist of long segments near families of attracting and repelling limit cycles of the fast system, in alternation. We also derive explicit formulas for the parameter values at which the maximal torus canards and their folds exist. Primary maximal canards and maximal torus canards correspond geometrically to the situation in which the persistent manifolds near the family of attracting limit cycles coincide to all orders with the persistent manifolds that lie near the family of repelling limit cycles. The formulas derived for the folds of maximal canards in all three frequency regimes turn out to be representations of a single formula in the appropriate parameter regimes, and this unification confirms the central numerical observation that the folds of the maximal canards created in the low-frequency regime continue directly into the folds of the maximal torus canards that exist in the intermediate- and high-frequency regimes. In addition, we study the secondary canards induced by the folded singularities in the low-frequency regime and find that the fold curves of the secondary canards turn around in the intermediate-frequency regime, instead of continuing into the high-frequency regime. Also, we identify the mechanism responsible for this turning. Finally, we show that the forced van der Pol equation is a normal form-type equation for a class of single-frequency periodically driven slow/fast systems with two fast variables and one slow variable which possess a non-degenerate fold of limit cycles. The analytic techniques used herein rely on geometric desingularisation, invariant manifold theory, Melnikov theory, and normal form methods. The numerical methods used herein were developed in Desroches et al. (SIAM J Appl Dyn Syst 7:1131–1162, 2008, Nonlinearity 23:739–765 2010).

This work has been published in J. Nonlinear Sci. and is available as [19] .

Extending the zero-derivative principle for slow-fast dynamical systems

Participants : Eric Benoît [Université de La Rochelle, France] , Morten Brøns [Technical University of Denmark, Lyngby, Denmark] , Mathieu Desroches, Maciej Krupa.

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 work, we discuss methods of approximating slow manifolds using the so-called zero-derivative principle. We demonstrate several test functions that work for systems with explicit time scale separation including ones that can be generalized to systems without explicit timescale separation. We also discuss the possible spurious solutions, known as ghosts, as well as treat the Templator system as an example.

This work has been published in ZAMP and is available as [17] .

Canards, folded nodes and mixed-mode oscillations in piecewise-linear slow-fast systems

Participants : Mathieu Desroches, Antoni Guillamon [Polytechnic University of Catalunya, Barcelona, Spain] , Enrique Ponce [University of Sevilla, Spain] , Rafel Prohens [University of the Balearic Islands, Palma, Spain] , Serafim Rodrigues [Plymouth University, UK] , Antonio Teruel [University of the Balearic Islands, Palma, Spain] .

Canard-induced phenomena have been extensively studied in the last three decades, both from the mathematical and from the application viewpoints. Canards in slow-fast systems with (at least) two slow variables, especially near folded-node singularities, give an essential generating mechanism for Mixed-Mode oscillations (MMOs) in the framework of smooth multiple timescale systems. There is a wealth of literature on such slow-fast dynamical systems and many models displaying canard-induced MMOs, in particular in neuroscience. In parallel, since the late 1990s several papers have shown that the canard phenomenon can be faithfully reproduced with piecewise-linear (PWL) systems in two dimensions although very few results are available in the three-dimensional case. This work aims to bridge this gap by analyzing canonical PWL systems that display folded singularities, primary and secondary canards, with a similar control of the maximal winding number as in the smooth case. We also show that the singular phase portraits are compatible in both frameworks. Finally, we show on an example how to construct a (linear) global return and obtain robust PWL MMOs.

This work has been accepted for publication in SIAM Review and is available as [46] .

Canard solutions in planar piecewise linear systems with three zones

Participants : Soledad Fernández-García [Inria Paris-Rocquencourt, France] , Mathieu Desroches, Maciej Krupa, Antonio Teruel [University of the Balearic Islands, Palma, Spain] .

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 prove the existence of an intersection between repelling and attracting slow manifolds, which defines a maximal canard, in a non-generic system of the class having a continuum of periodic orbits. Then, we perturb this situation and prove the persistence of the maximal canard solution, as well as the existence of a family of canard limit cycles in this class of systems. Similarities and differences between the piecewise linear case and the smooth one are highlighted.

This work has been published Dynam. Syst. and is available as [24] .

Spike-adding mechanism in parabolic bursters: the role of folded-saddle canards

Participants : Mathieu Desroches, Maciej Krupa, Serafim Rodrigues [Plymouth University, UK] .

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 consider the conductance-based model of the Aplysia R15 neuron known as Plant's model, and a reduction of this prototypical biophysical parabolic burster to three variables, including one phase variable, namely Rinzel's theta model. Revisiting these models from the perspective of slow-fast dynamics reveals that the number of spikes per burst may vary upon parameter changes, however the spike-adding process occurs in a brutal (explosive) fashion that involves special solutions called canards. This spike-adding canard explosion phenomenon is analysed by using tools from geometric singular perturbation theory in tandem with numerical bifurcation techniques. We find that the bifurcation structure persists across both parabolic bursters, that is, spikes within the burst are incremented via the crossing of an excitability threshold given by a particular type of canard orbit, namely the strong canard of a folded-saddle singularity. Using these findings, we construct a new polynomial approximation of Plant's model, which retains all the key elements for parabolic bursting, including the canard mediated spike-adding transitions. Finally, we briefly investigate the presence of spike-adding via canards in planar phase models of parabolic bursting, namely the theta model by Ermentrout and Kopell.

This work has been submitted for publication and is available as [47] .

Canards and spike-adding transitions in a minimal piecewise-linear Hindmarsh-Rose square-wave burster

Participants : Mathieu Desroches, Soledad Fernández-García [Inria Paris-Rocquencourt, France] , Maciej Krupa.

We construct a piecewise-linear (PWL) approximation of the Hindmarsh-Rose (HR) neurone model that is minimal, in the sense that the vector field has the least number of pieces, in order to reproduce all the dynamics present in the original HR model with the classical parameter values. This includes spiking, square-wave bursting, and also special trajectories called canards, which possess long repelling segments and organise the transition between stable bursting patterns with n and n+1 spikes. This is the spike-adding canard explosion. We propose a first approximation of the smooth bursting model, using a continuous PWL system, and show that its fast subsystem cannot possess a homoclinic bifurcation, which is necessary to obtain proper square-wave bursting. We then relax the assumption of continuity of the vector field across all zones and show that we can obtain a homoclinic bifurcation in the fast subsystem. We use the recently developed canard theory for PWL systems in order to reproduce the spike-adding canard explosion feature of the HR model as studied, e.g., in  [66] .

This work has been submitted for publication and is available as [45] .

Ducks in space

Participants : Daniele Avitabile [University of Nottingham, UK] , Mathieu Desroches, Edgar Knobloch [University of California at Berkeley, USA] , Maciej Krupa.

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, one of which is present for all values of the bifurcation parameter while the other is present for zero or sufficiently small inlet boundary conditions but only when the bifurcation parameter is large enough. For slightly larger inlet boundary condition a continuous transition from one type to the other takes place as the bifurcation parameter increases. The origin of the two solution types is traced to the onset of convective and absolute instability on the real line. The role of canard trajectories in the transitions 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 are interpreted using the deterministic slow-fast spatial-dynamical system.

This work has been submitted for publication and is available as [43] .