Section: New Results

Numerical and theoretical studies of slow-fast systems with complex oscillations

Canard-Mediated (De)Synchronization in Coupled Phantom Bursters

Participants : Elif Köksal Ersöz, Mathieu Desroches, Maciej Krupa, Frédérique Clément.

In [32] , we study canard-mediated transitions in mutually coupled phantom bursters. We extend a multiple-timescale model which provides a sequence of dynamic events, i.e. transition from a frequency modulated relaxation cycle to a quasi-steady state and resumption of the relaxation regime through small amplitude oscillations. Folded singularities and associated canard solutions have a particular impact on the dynamics of the original system, which consists of two feedforward coupled FitzHugh-Nagumo oscillators, where the slow subsystem (regulator) controls the periodic behavior of the fast subsystem (secretor). We first investigate the variability in the dynamics depending on the canard mechanism that occurs near the folded singularities of the 4D secretor- regulator configuration. Then, we introduce a second secretor and focus on the slow-fast transitions in the presence of a linear coupling between the secretors. In particular, we explore the impact of the relationship between the canard structures and the coupling on patterns of synchronization and desynchronization of the collective dynamics of the resulting 6D system. We identify two different sources of desynchronization induced by canards, near a folded-saddle singularity and a folded-node singularity, respectively.

Part of these results have also been presented as posters at the SIAM Conference on Applications of Dynamical Systems (Snowbird, May 17-21, 2015) and 1st International Conference on Mathematical Neuroscience (Antibes Juan les Pins, June 8-10-2015).

Mixed-Mode Oscillations in a piecewise linear system with multiple time scale coupling

Participants : Soledad Fernández García, Maciej Krupa, Frédérique Clément.

We analyze a four dimensional slow-fast piecewise linear system with three time scales presenting Mixed-Mode Oscillations. The system possesses an attractive limit cycle along which oscillations of three different amplitudes and frequencies can appear, namely, small oscillations, pulses (medium amplitude) and one surge (largest amplitude). In addition to proving the existence and attractiveness of the limit cycle, we focus our attention on the canard phenomena underlying the changes in the number of small oscillations and pulses. We analyze locally the existence of secondary canards leading to the addition or subtraction of one small oscillation and describe how this change is globally compensated for or not with the addition or subtraction of one pulse.

Noise-induced canard and mixed-mode oscillations in large stochastic networks with multiple timescales

Participants : Jonathan Touboul, Maciej Krupa, Mathieu Desroches.

We investigate in [28] the dynamics of large stochastic networks with different timescales and nonlinear mean-field interactions. After deriving the limit equations for a general class of network models, we apply our results to the celebrated Wilson-Cowan system with two populations with or without slow adaptation, paradigmatic example of nonlinear mean-field network. This system has the property that the dynamics of the mean of the solution exactly satisfies an ODE. This reduction allows to show that in the mean-field limit and in multiple populations with multiple timescales, noise induces canard explosions and Mixed-Mode Oscillations on the mean of the solution. This sheds new light on the qualitative effects of noise and sensitivity to precise noise values in large stochastic networks. We further investigate finite-sized networks and show that systematic differences with the mean-field limits arise in bistable regimes (where random switches between different attractors occur) or in mixed-mode oscillations, were the finite-size effects induce early jumps due to the sensitivity of the attractor.

Canard explosion in delayed equations with multiple timescales, applications to the delayed Fitzhugh-Nagumo system

Participants : Maciej Krupa, Jonathan Touboul.

In two contributions, we investigated theoretically the presence of canard explosions of delayed differential equations, and have applied these results to the FitzHugh-Nagumo neuronal model.

• In [21] we analyze canard explosions in delayed differential equations with a one-dimensional slow manifold. This study is applied to explore the dynamics of the van der Pol slow-fast system with delayed self-coupling. In the absence of delays, this system provides a canonical example of a canard explosion. We show that as the delay is increased a family of `classical' canard explosions ends as a Bogdanov-Takens bifurcation occurs at the folds points of the S-shaped critical manifold.

• Motivated by the dynamics of neuronal responses, we analyze in [21] the dynamics of the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. Beyond the regime of small delays, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov-Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canard-induced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation.

Canard-induced loss of stability across a homoclinic bifurcation

Participants : Mathieu Desroches, Jean-Pierre Françoise, Lucile Megret.

In [16] , we investigate the possibility of bifurcations which display a dramatic change in the phase portrait in a very small (on the order of ${10}^{-7}$ in the example presented here) change of a parameter. We provide evidence of existence of such a very rapid loss of stability on a specific example of a singular perturbation setting. This example is strongly inspired of the explosion of canard cycles first discovered and studied by E. Benoît, J.-L. Callot, F. Diener and M. Diener. After some presentation of the integrable case to be perturbed, we present the numerical evidences for this rapid loss of stability using numerical continuation. We discuss then the possibility to estimate accurately the value of the parameter for which this bifurcation occurs.

Analysis of Interspike-Intervals for the General Class of Integrate-and-Fire Models with Periodic Drive

Participant : Justyna Signerska-Rynkowska.

In [27] , we study one-dimensional integrate-and-fire models of the general type $\dot{x}=F(t,x)$ and analyze properties of the firing map which iterations recover consecutive spike timings. We impose very week constraints for the regularity of the function $F(t,x)$ e.g. often it suffices to assume that F is continuous. If additionally $F$ is periodic in $t$, using mathematical study of the displacement sequence of an orientation preserving circle homeomorphism, we provide a detailed description of the regularity properties of the sequence of interspike-intervals and behaviour of the interspike-interval distribution.

A geometric mechanism for mixed-mode bursting oscillations in a hybrid neuron model

Participants : Justyna Signerska-Rynkowska, Jonathan Touboul, Alexandre Vidal.

In [35] , we exhibit and investigate a new type of mechanism for generating complex oscillations featuring an alternation of small oscillations with spikes (MMOs) or bursts (MMBOs) in a class of hybrid dynamical systems modeling neuronal activity. These dynamical systems, called nonlinear adaptive integrate-and-fire neurons, combine nonlinear dynamics modeling input integration in a nerve cell with discrete resets modeling the emission of an action potential and the subsequent return to reversal potential. We show that presence of complex oscillations in these models relies on a fundamentally hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts. The decomposition into these two mechanisms leads us to propose a purely geometrical interpretation of these complex trajectories, and this relative simplicity allows to finely characterize the MMO patterns through the study of iterates of the adaptation map associated with the hybrid system. This map is however singular: it is discontinuous and has unbounded left- and right-derivatives. We apply and develop rotation theory of circle maps for this class of adaptation maps to precisely characterize the trajectories with respect to the parameters of the system. In contrast to more classical frameworks in which MM(B)Os were evidenced, the present geometric mechanism neither requires no more than two dimensions, does not necessitate to have separation of timescales nor complex return mechanisms.

Part of these results have also been presented as posters at the SIAM Conference on Applications of Dynamical Systems (Snowbird, May 17-21, 2015) and 1st International Conference on Mathematical Neuroscience (Antibes Juan les Pins, June 8-10-2015).