Mathematical Methods for Neurosciences : Research Articles
ENS MathInfo - Master MVA
Olivier Faugeras
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Liste des projets
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.