Mathematical Methods for Neurosciences : Research Articles

ENS MathInfo - Master MVA

Olivier Faugeras

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Liste des projets

  1. Bifurcation analysis of a general class of nonlinear integrate-and-fire neurons Jonathan Touboul: In this paper we define a class of formal neuron models being computationally efficient and biologically plausible, i.e., able to reproduce a wide range of behaviors observed in in vivo or in vitro recordings of cortical neurons. This class includes, for instance, two models widely used in computational neuroscience, the Izhikevich and the Brette–Gerstner models. These models consist of a 4-parameter dynamical system. We provide the full local bifurcation diagram of the members of this class and show that they all present the same bifurcations: an Andronov–Hopf bifurcation manifold, a saddle-node bifurcation manifold, a Bogdanov–Takens bifurcation, and possibly a Bautin bifurcation, i.e., all codimension two local bifurcations in a two-dimensional phase space except the cusp. Among other global bifurcations, this system shows a saddle homoclinic bifurcation curve. We show how this bifurcation diagram generates the most prominent cortical neuron behaviors. This study leads us to introduce a new neuron model, the quartic model, able to reproduce among all the behaviors of the Izhikevich and Brette–Gerstner models self-sustained subthreshold oscillations, which are of great interest in neuroscience.
  2. Spiking Dynamics of Bidimensional Integrate-and-Fire Neurons Jonathan Touboul and Romain Brette: Spiking neuron models are hybrid dynamical systems combining differential equations and discrete resets, which generate complex dynamics. Several two-dimensional spiking models have been recently
    introduced, modeling the membrane potential and an additional variable, and where spikes are defined by the divergence of the membrane potential variable to infinity. These simple models reproduce a large number of electrophysiological features displayed by real neurons, such as spike frequency adaptation and bursting. The patterns of spikes, which are the discontinuity points of the hybrid dynamical system, have been studied mainly numerically. Here we show that the spike patterns are related to orbits under a discrete map, the adaptation map, and we study its dynamics and bifurcations. Regular spiking corresponds to fixed points of the adaptation map, while bursting corresponds to periodic orbits. We find that the models undergo a transition to chaos via a cascade
    of period adding bifurcations. Finally, we discuss the physiological relevance of our results with regard to electrophysiological classes.
  3. Neural excitability, spiking and bursting Eugene M. Izhikevich: Bifurcation mechanisms involved in the generation of action potentials (spikes) by neurons are reviewed here, and the author shows how the type of bifurcation determines the neuro-computational properties of the cells. He also describes the phenomenon of neural bursting using geometric bifurcation theory to extend the existing classification of bursters, including many new types.
  4. Effects of noise in excitable systems B. Lindner et al.: We review the behavior of theoretical models of excitable systems driven by Gaussian white noise. We focus mainly on those general properties of such systems that are due to noise, and present several applications of our findings in biophysics and lasers. As prototypes of excitable stochastic dynamics we consider the FitzHugh–Nagumo and the leaky integrate-and-fire model, as well as cellular automata and phase models. In these systems, taken as individual units or as networks of globally or locally coupled elements, we study various phenomena due to noise, such as
    noise-induced oscillations, stochastic resonance, stochastic synchronization, noise-induced phase transitions and noise-induced pulse and spiral dynamics.
    Our approach is based on stochastic differential equations and their corresponding Fokker–Planck equations, treated by both analytical calculations and/or numerical simulations. We calculate and/or measure the rate and diffusion coefficient of the excitation process, as well as spectral quantities like power spectra and degree of coherence. Combined with a multiparametric bifurcation analysis of the corresponding cumulant equations, these approaches provide a comprehensive picture of the multifaceted dynamical behaviour of noisy excitable systems.
  5. Mean field theory for a balanced hypercolumn model of orientation selectivity in primary visual cortex A Lerchner, G Sterner, J Hertz and M Ahmadi: The authors present a complete mean field theory for a balanced state of a simple model of an orientation hypercolumn. The theory is complemented by a description of a numerical procedure for solving the mean-field equations quantitatively.
  6. Stochastic population dynamics of spiking neurons Paolo Del Giudice and Maurizio Mattia : Review of some developments in the use of the theory of stochastic processes and nonlinear dynamics in the study of large scale dynamical models of interacting spiking neurons.
  7. The Kuramoto model: a simple paradigm for synchronization phenomena Juan A. Acebrón et al.: Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last years are presented. Relevant applications of the model in different contexts are also included.
  8. Neural networks as spatio-temporal pattern-forming systems Bard Ermentrout: Models of neural networks are developed from a biological point of view. Small networks are analysed using techniques from dynamical systems. The behaviour of spatially and temporally organized neural fields is then discussed from the point of view of pattern formation. Bifurcation methods, analytic solutions and perturbation methods are applied to these models.
  9. Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex Paul C. Bressloff, Jack D. Cowan, Martin Golubitsky, Peter J. Thomas, Matthew C. Wiener : This paper is concerned with a striking visual experience: that of seeing geometric visual hallucinations. It describes a mathematical investigation of their origin based on the assumption that the patterns of connection between retina and striate cortex (henceforth referred to as V1) - the retinocortical map - and of neuronal circuits in V1, both local and lateral, determine their geometry.
  10. A cortical model based of perceptual completion in the roto-translation space G. Citti and A. Sarti: A mathematical model of perception completion and formation of subjective surfaces is presented, which is at the same time inspired by the architecture of the visual cortex, and is the lifting in the 3-dimensional roto translation group of the phenomenological variational models based on elastica functional. The initial image is lifted by the simple cells to a surface in the rototranslation group and the completion process is modelled via a diffusion driven motion by curvature. The convergence of the motion to a minimal surface is proved. Results are presented for modal and amodel completion for the classic Kanizsa images.
  11. Asynchronous States and the Emergence of Synchrony in Large Networks of Interacting Excitatory and Inhibitory Neurons D. Hansel and G. Mato: We investigate theoretically the conditions for the emergence of synchronous activity in large networks, consisting of two populations of extensively connected neurons, one excitatory and one inhibitory. The neurons are modeled with quadratic integrate-and-fire dynamics, which provide a very good approximation for the subthreshold behavior of a large class of neurons. In addition to their synaptic recurrent inputs, the neurons receive a tonic external input that varies from neuron to neuron. Because of its relative simplicity, this model can be studied analytically. We investigate the stability of the asynchronous state (AS) of the network with given average firing rates of the two populations. First, we show that the AS can remain stable even if the synaptic couplings are strong. Then we investigate the conditions under which this state can be destabilized.We show that this can happen in four generic ways. The first is a saddle-node bifurcation, which leads to another state with different average firing rates. This bifurcation, which occurs for strong enough recurrent excitation, does not correspond to the emergence of synchrony. In contrast, in the three other instability mechanisms, Hopf bifurcations, which correspond to the emergence of oscillatory synchronous activity, occur.We show that these mechanisms can be differentiated by the firing patterns they generate and their dependence on the mutual interactions of the inhibitory neurons and cross talk between the two populations. We also show that besides these codimension 1 bifurcations, the system
    can display several codimension 2 bifurcations: Takens-Bogdanov, Gavrielov-Guckenheimer, and double Hopf bifurcations.
  12. The neurogeometry of pinwheels as a sub-Riemannian contact structure Jean Petitot: We present a geometrical model of the functional architecture of the primary visual cortex (V1) and, more precisely, of its pinwheel structure. The problem is to understand from within how the internal ‘‘immanent’’ geometry of the visual cortex can produce the ‘‘transcendent’’ geometry of the external space. We use first the concept of blowing up to model V1 as a discrete
    approximation of a continuous fibration p : R x P -> P with base space the space of the retina R and fiber the projective line P of the orientations of the plane. The core of the paper consists first in showing that the horizontal cortico-cortical connections of V1 implement what the geometers call the contact structure of the fibration p, and secondly in introducing an integrability condition and the integral curves associated with it. The paper develops then three applications: (i) to Field’s, Hayes’, and Hess’ psychophysical concept of association field, (ii) to a variational model of curved modal illusory contours (in the spirit of previous models due to Ullman, Horn, and Mumford), (iii) to Ermentrout’s, Cowan’s, Bressloff’s, Golubitsky’s models of visual hallucinations.