Modelife Workshop: Non-linear dynamics in neuroscience

Dynamical neuroscience describes non-linear dynamics at the different levels of the brain, from single neural cells to the behavior of a single neuron in large-scale neuronal networks, or from neuronal physiology to cognitive processes. The goal of this workshop is to update a broad audience with different modeling approaches to resolve fundamental questions in neuroscience, where non-linear dynamics are embedded in systems with several space and time scales.

Invited speakers :

• Anmar Khadra, McGill University, Montreal, Cerebellar stellate cell excitability: Runup, first-spike latency, switching and non-sequential spike adding Abstract
  Cerebellar stellate cells (CSC), involved in motor control, are inhibitory interneurons that synapse onto Purkinje cells. They are spontaneously active neurons that tonically fire action potentials in the absence of pharmacological agents and burst in the presence of 4AP and Cd2+. Our group has extensively studied the electrophysiological properties of CSCs, including their temporal increase in excitability (termed runup), non-monotonic first-spike latency, transient single spiking, switching in responsiveness and bursting. We developed a revised data-based Hodgkin-Huxley (HH) type model and applied a two-point boundary value problem combined with the continuation method in AUTO as well as slow-fast analysis to decipher the underlying dynamics of these key properties of CSCs. Our finding indicates that CSC- HH model possesses type I excitability with a SNIC bifurcation and that the ghost of the SNIC, the stable manifold of the saddle along with the period doubling bifurcations are responsible for producing the key properties of CSCs. In this talk, I will provide an overview of these results and explain the implication of such studies on understanding neuronal excitability.
• Jürgen Reingruber, IBENS, Paris, Non-linear ion dynamics in the confined ciliary space of olfactory receptor neurons Abstract
  Olfactory receptor neurons embedded in the olfactory epithelium in the nasal cavity are the principal neurons underlying olfaction. They have long and slender cilia where odor molecules bind to receptors on the ciliary membrane. This initiates a signal transduction cascade that activates ion channels leading to large inward currents that depolarize the cell. We develop a spatio-temporal model based on electrodiffusion to study how such large currents flowing into the constrained ciliary space affect the coupled dynamics of $Na^+$, $Ca^{2+}$, $K^+$ and $Cl^-$ concentrations in the cilia. Our analysis reveals that the degree of ion concentration fluctuations during an olfactory response strongly depend on the ion species that carries the current. This provides a putative explanation why the major part of the depolarizing current is carried by $Cl^-$ in olfactory receptor neurons contrary to $Na^+$ in classical neurons.
• Mathieu Desroches, INRIA, Nice, Slow-fast analysis of neural bursters: old and new Abstract
  In this talk, I will present recent work on multiple-timescale dynamical systems displaying complex oscillations with both slow and fast components. After a brief review of bursting oscillations and the role of so-called spike-adding transitions in square-wave bursters, I will introduce a four-dimensional extension of this scenario which creates small-amplitude slow (sub-threshold) oscillations in between bursts, mediated by so-called canard solutions. In the second half of the talk, I will revisit another type of four-dimensional bursting scenario with two slow variables, namely parabolic bursting, and provide explanations on how the spike-adding mechanism in such bursters is also organised by canards but of a different type than before. This will be showcased on several examples of parabolic bursters, both biophysical ones like the Plant model, and simplified ones like theta models. Finally, I will show how the burst-excitable structure of networks of theta model may persist across scales up to some mean-field limit. [This is based on joint papers with D Avitabile (Amsterdam), GB Ermentrout (Pittsburgh), TJ Kaper (Boston), M Krupa (Nice) and S Rodrigues (Bilbao)]
• Romain Veltz, INRIA, Nice, Analysis of a neural field model for color perception unifying assimilation and contrast Abstract
  We present a recent model [Song et al. 2019] of color perception unifying assimilation and contrast. This model, which relies on the notion of color opponency introduced by Hering, has been tuned to reproduce some nontrivial behaviors of the color shifts observed in experiments. Next, we perform bifurcation analysis, based on the properties of Wiener-Hopf operators, of this planar model to predict visual hallucinations. Numerical simulations are provided to assess the global stability of the predicted visual hallucinations.

Programme