Section: New Results

Slow-fast dynamics in Neuroscience

Local theory for spatio-temporal canards and delayed bifurcations

Participants : Daniele Avitabile [VU Amsterdam, Inria MathNeuro] , Mathieu Desroches, Romain Veltz, Martin Wechselberger [University of Sydney, Australia] .

We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an infinite-dimensional dynamical system for the fast variables is coupled to a finite-dimensional dynamical system for slow variables. We prove the existence of centre-manifolds for generic models of this type, and study the reduced, finite-dimensional dynamics near bifurcations of (possibly) patterned steady states in the layer problem. Theoretical results are complemented with detailed examples and numerical simulations covering systems of local- and nonlocal-reaction diffusion equations, neural field models, and delay-differential equations. We provide analytical foundations for numerical observations recently reported in literature, such as spatio-temporal canards and slow-passages through Hopf bifurcations in spatially-extended systems subject to slow parameter variations. We also provide a theoretical analysis of slow passage through a Turing bifurcation in local and nonlocal models.

This work has been submitted for publication and is available as [37].

Pseudo-plateau bursting and mixed-mode oscillations in a model of developing inner hair cells

Participants : Harun Baldemir, Daniele Avitabile [VU Amsterdam, Inria MathNeuro] , Krasimira Tsaneva-Atanasova [University of Exeter, UK] .

Inner hair cells (IHCs) are excitable sensory cells in the inner ear that encode acoustic information. Before the onset of hearing IHCs fire calcium-based action potentials that trigger transmitter release onto developing spiral ganglion neurones. There is accumulating experimental evidence that these spontaneous firing patterns are associated with maturation of the IHC synapses and hence involved in the development of hearing. The dynamics organising the IHCs' electrical activity are therefore of interest.

Building on our previous modelling work we propose a three-dimensional, reduced IHC model and carry out non-dimensionalisation. We show that there is a significant range of parameter values for which the dynamics of the reduced (three-dimensional) model map well onto the dynamics observed in the original biophysical (four-dimensional) IHC model. By estimating the typical time scales of the variables in the reduced IHC model we demonstrate that this model could be characterised by two fast and one slow or one fast and two slow variables depending on biophysically relevant parameters that control the dynamics. Specifically, we investigate how changes in the conductance of the voltage-gated calcium channels as well as the parameter corresponding to the fraction of free cytosolic calcium concentration in the model affect the oscillatory model bahaviour leading to transition from pseudo-plateau bursting to mixed-mode oscillations. Hence, using fast-slow analysis we are able to further our understanding of this model and reveal a path in the parameter space connecting pseudo-plateau bursting and mixed-mode oscillations by varying a single parameter in the model.

This work has been accepted for publication in Communications in Nonlinear Science and Numerical Simulation and is available as [18].

Parabolic bursting, spike-adding, dips and slices in a minimal model

Participants : Mathieu Desroches, Jean-Pierre Françoise [LJLL, Sorbonne Université, Paris] , Martin Krupa [LJAD, UCA, Inria MathNeuro] .

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 forced one-dimensional system, we uncover the link with the dips and slices first discussed by J.E. Littlewood in his famous articles on the periodically forced van der Pol system.

This work has been published in Mathematical Modelling of Natural Phenomena and is available as [26].

Anticipation via canards in excitable systems

Participants : Elif Köksal Ersöz [INSERM, Rennes, Inria MathNeuro] , Mathieu Desroches, Claudio Mirasso [University of the Balearic Islands, Spain] , Serafim Rodrigues [Ikerbasque & Basque Center for Applied Mathematics, Spain] .

Neurons can anticipate incoming signals by exploiting a physiological mechanism that is not well understood. This article offers a novel explanation on how a receiver neuron can predict the sender's dynamics in a unidirectionally-coupled configuration, in which both sender and receiver follow the evolution of a multi-scale excitable system. We present a novel theoretical viewpoint based on a mathematical object, called canard, to explain anticipation in excitable systems. We provide a numerical approach, which allows to determine the transient effects of canards. To demonstrate the general validity of canard-mediated anticipation in the context of excitable systems, we illustrate our framework in two examples, a multi-scale radio-wave circuit (the van der Pol model) that inspired a caricature neuronal model (the FitzHugh-Nagumo model) and a biophysical neuronal model (a 2-dimensional reduction of the Hodgkin-Huxley model), where canards act as messengers to the senders' prediction. We also propose an experimental paradigm that would enable experimental neuroscientists to validate our predictions. We conclude with an outlook to possible fascinating research avenues to further unfold the mechanisms underpinning anticipation. We envisage that our approach can be employed by a wider class of excitable systems with appropriate theoretical extensions.

This work has been published in Chaos: An Interdisciplinary Journal of Nonlinear Science and is available as [28].

Canard-induced complex oscillations in an excitatory network

Participants : Elif Köksal Ersöz, Mathieu Desroches, Antoni Guillamon [Polytechnic University of Catalunya, Spain] , John Rinzel [Center for Neural Science and Courant Institute of Mathematical Sciences, New York University, USA] , Joel Tabak [University of Exeter, UK] .

In this work we have revisited a rate model that accounts for the spontaneous activity in the developing spinal cord of the chicken embryo [69]. The dynamics is that of a classical square-wave burster, with alternation of silent and active phases. Tabak et al. [69] have proposed two different three-dimensional (3D) models with variables representing average population activity, fast activity-dependent synaptic depression and slow activity-dependent depression of two forms. In [66], [67], [68] various 3D combinations of these four variables have been studied further to reproduce rough experimental observations of spontaneous rhythmic activity. In this work, we have first shown the spike-adding mechanism via canards in one of these 3D models from [69] where the fourth variable was treated as a control parameter. Then we discussed how a canard-mediated slow passage in the 4D model explains the sub-threshold oscillatory behavior which cannot be reproduced by any of the 3D models, giving rise to mixed-mode bursting oscillations (MMBOs); see [10]. Finally, we related the canard-mediated slow passage to the intervals of burst and silent phase which have been linked to the blockade of glutamatergic or GABAergic/glycinergic synapses over a wide range of developmental stages [68].

This work has been submitted for publication and is available as [12].