Section: New Results
Numerical and theoretical studies of slow-fast systems with complex oscillations
Coupled multiple timescale dynamics in populations of endocrine neurons: Pulsatile and surge patterns of GnRH secretion
Participants : Elif Köksal Ersöz, Alexandre Vidal, Frédérique Clément.
The gonadotropin releasing hormone (GnRH) is secreted by hypothalamic neurons into the pituitary portal blood in a pulsatile manner. The alternation between a frequency-modulated pulsatile regime and the ovulatory surge is the hallmark of the GnRH secretion pattern in ovarian cycles of female mammals. In this work, we aimed at modeling additional features of the GnRH secretion pattern: the possible occurrence of a two-bump surge (“camel surge”) and an episode of partial desynchronization before the surge.
We have proposed a six-dimensional extension of a former four-dimensional model with three timescale and introduced two mutually-coupled, slightly heterogenous GnRH subpopulations (secretors) regulated by the same slow oscillator (regulator). We have considered two types of coupling functions between the secretors, including dynamic state-dependent coupling, and we have used numerical and analytic tools to characterize the coupling parameter values leading to the generation of a two-bump surge in both coupling cases. We have revealed the impact of the slowly varying control exerted by the regulator onto the pulsatile dynamics of the secretors, which leads to dynamic bifurcations and gives rise to desynchronization. To assess the occurrence time of desynchronization during the pulsatile phase, we have introduced asymptotic tools based on quasi-static and geometric approaches, as well as analytic tools based on the H-function derived from phase equation and numerical tracking of period-doubling bifurcations. We discuss the role of coupling parameters in the two-bump surge generation and the speed of desynchronization.
Symmetric coupling of multiple timescale systems with mixed-mode oscillations
Participants : Soledad Fernández García, Alexandre Vidal, Fabrizio de Vico Fallani [EPI Aramis] , Frédérique Clément.
We have analyzed a six-dimensional slow-fast system consisting of two coupled identical oscillators. Each oscillator is a three-dimensional system consisting of a FitzHugh-Nagumo system with an additional variable representing the calcium concentration. Individually, each three-dimensional subsystem possesses an attractive Mixed-Mode oscillations limit cycle, displaying small oscillations due to the presence of a folded saddle-node type II singularity for a certain range of the parameters values. We have considered a linear coupling through the fast variable in the slow equation and study the synchronization patterns of two identical systems with identical coupling parameter. Apart from stable in-phase and stable anti-phase synchronization patterns, the system presents almost-in-phase synchronization, oscillation death of one of the oscillators and total oscillation death, intertwined with complex transitions involving period doubling cascade, period adding phenomena and chaos. We have pointed out the role of Mixed-Mode oscillations in the birth of the different patterns and the transitions from one regime to another.
Part of these results have been presented as a contributed talk to the SIAM conference on life science https://www.siam.org/meetings/ls16/: (A Study of the Synchronization Between Two Coupled Neuron Models Generating Mixed-Mode Oscillations. A. Vidal, S.Fernández García, F. Clément, F. De Vico Fallani). MS48 Applications of Multiple Time Scale Dynamics in Biological Systems.
3D-Explosion of cycles and spike-adding in the Hindmarsh-Rose model
Participants : Lucile Megret, Mathieu Desroches [Sophia] , Jean-Pierre Françoise, Maciej Krupa [Sophia] .
We have considered slow-fast systems that feature bursting oscillations, the minimal configuration being two fast variables and one slow variable. In the Hindmarsh-Rose model, as the slow variable z evolves, the fast dynamics undergoes several bifurcations (two Hopf bifurcations, two homoclinic bifurcations, two focus-node and two saddle-node bifurcations). We have focused on the existence of a sequence of 3D-candidate limit periodic sets of a new type. Numerical simulations have shown that it generates for the full 3D-dynamics and (the small parameter) " small enough a 3D-explosion of cycles. We have discussed the relation between this 3D-explosion and the spike-adding. We have also emphasized another new phenomenon induced by the slow-crossing of a saddle-node bifurcation with solutions which after coming close to the fold point, continue to follow it along its non-hyperbolic center manifold. We have shown how this phenomenon is also involved in the spike-adding mechanism taking place in square-wave bursters such as the Hindmarsh-Rose system.
Part of these results have been presented at the “36e Séminaire de la Sociéte Francophone de Biologie théorique”, St-Flour (France), June 12-15 2016.
Wild oscillations in a nonlinear neuron model with resets
Participants : Jonathan Rubin [University of Pittsburgh] , Justyna Signerska-Rynkowska, Jonathan Touboul, Alexandre Vidal.
In a series of two studies, we have investigated the mechanisms by which complex oscillations are generated in a class of nonlinear dynamical systems with resets modeling the voltage and adaptation of neurons.
The first study  presents a mathematical analysis showing that the system can support bursts of any period as a function of model parameters, and that are organized in a period-incrementing structure. In continuous dynamical systems with resets, such period-incrementing structures are complex to analyze. In the present context, we have used the fact that bursting patterns correspond to periodic orbits of the adaptation map that governs the sequence of values of the adaptation variable at the resets. Using a slow-fast approach, we have shown that this map converges towards a piecewise linear discontinuous map whose orbits are exactly characterized. That map shows a period-incrementing structure with instantaneous transitions. We have further shown that the period-incrementing structure persists for the full system with non-constant adaptation, yet the transitions are more complex. We have also established the presence of chaos at the transitions.
The second study  shows that these neuron models can generically display a form of mixed-mode oscillations (MMOs), which are trajectories featuring an alternation of small oscillations with spikes or bursts (multiple consecutive spikes). The mechanism by which these are generated relies fundamentally on the hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts, contrasting with classical MMO mechanisms in ordinary differential equations involving more than three dimensions and generally relying on a timescale separation. The decomposition of mechanisms reveals the geometrical origin of MMOs, allowing a relatively simple classification of points on the reset manifold associated to specific numbers of small oscillations. We have shown that the MMO pattern can be described through the study of orbits of a discrete adaptation map, which is singular as it features discrete discontinuities with unbounded left- and right-derivatives. We have studied the orbits of the map via rotation theory for circle maps and elucidated in detail complex behaviors arising in the case where MMOs display a single small oscillation per cycle.
Canard Explosions in delay differential equations
Participants : Jonathan Touboul, Maciej Krupa [Sophia] .
We have analyzed in  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 have shown 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.