
 
Neuroscience and its applications are greatly developing worldwide and Europe is one of the
important contributors to the advancement of this discipline.
Because of the variety of topics that it has to address, it is characterized by a very broad interdisciplinarity
and requires the cooperation of actors in several fields of knowledge.
In this context, the need for developing new theoretical, mathematical, and computational tools can be clearly
identified and must be addressed.
The purpose of this semester is to present some of the relevant modern mathematical tools through
short courses and to explore several facets of the current research through workshops.

Program committee: 

P. Bressloff (University of Utah), N. Brunel (Université Paris 5), P. Chossat (CNRS  INRIA), O. Faugeras (ENS  INRIA), W. Gerstner (EPFL), V. Jirsa (CNRS  Université de la Méditerranée). 

WORKSHOPS
0307/10/2011
 Organizers : N. Brunel, O. Faugeras (Schedule  presentations  Portfolio pdf)
Meanfield methods and multiscale analysis of neuronal populations
2428/10/2011
 Organizers : P. Bressloff, S. Coombes
Spatiotemporal evolution equations and neural fields
0711/11/2011  Organizers : N. Brunel, W. Gerstner,
J. Sjostrom, H. Markram
Learning and Plasticity
59/12/2011 Organizers : V. Jirsa, G. Deco
Mathematical Models of Cognitive Architectures
COURSES
10/1012/10  Denis Talay (INRIA), Samuel Herrmann (Ecole des Mines  INRIA), Etienne Tanré (INRIA)
Stochastic models and simulations in neuroscience
Lecture 1: 
Diffusion processes hitting times. Application to spike trains modelling.

Lecture 2: 
Stochatic differential equations with nonLipschitz coefficients: analysis and simulation,
applications to models in Neuroscience. 
Lecture 3: 
Links between Partial Differential Equations (PDEs) and Stochatic Differential Equations (SDEs). 
Lecture 4: 
Analysis of stochastic numerical methods for SDEs and PDEs. Applications in Neuroscience.

Lecture 5: 
Probabilistic numerical methods for diffusion processes hitting times and spike trains. 
13/1014/10 & 17/1019/10  Fred Wolf (MPI Göttingen)
Selforganisation in the development of visual
cortical circuits
Lecture 1: 
Overview 

Singularities in the Brain: Understanding Network SelfOrganization from Homotopy, Symmetries, and Fields. 
Lecture 2: 
Pattern selection of aperiodic solutions 

Visual Cortical Architecture: A Living PomeauNewelTurbulentCristal. 
Lecture 3: 
Pattern selection of aperiodic solutions: 

Quantitative Universality in Brain Evolution. 
Lecture 4: 
Structure and Function: 

Euclidean Symmetry Correctly Predicts the Strategic Positioning of Orientation Columns. 
Lecture 5: 
Guiding in vivo experiments by theory 

Moving the Wheels: Testing Massive Multi Stability of Circuit SelfOrganization. 
17/1021/10  Nils Berglund (Université d’Orléans)
Bifurcations in stochastic systems with multiple timescales
Lecture 1: 
Deterministic fastslow systems


Examples from neuroscience  conductionbased models; slowmanifolds, 

Fenichel theory; dynamic bifurcations: dynamic saddle
node,
pitchfork
and Hopf bifurcation; example: canards in the
FitzhughNagumo equations

Lecture 2: 
Onedimensional slowly timedependent
stochasticsystems


Dynamics near stable equilibrium branches; dynamic saddlenode bifurcation; dynamic pitchfork bifurcation; example:
stochastic resonance

Lecture 3: 
Multidimensional fastslow systems with noise


Dynamics near stable slow manifolds; dynamic Hopf
bifurcation; example: elliptic bursting

Lecture 4: 
Excitable systems


Excitability of type I: SNIC bifurcation, MorrisLecar model, interspike time statistics; excitability of type II: singular Hopf
bifurcation, FitzhughNagumo model, interspike time statistics

Lecture 5: 
Mixedmode oscillations


Deterministic models: folded node singularity, canards; stochastic case: effect of noise on mixedmode patterns; HodgkinHuxley model

02/1104/11  Stephen Coombes (Université de Nottingham)
Neural field modelling
Lecture 1:

Tissue level firing rate models with axodendritic connections I


Turing instability analysis

Lecture 2:

Tissue level firing rate models with axodendritic connections II


Amplitude equations


Brain wave equations

Lecture 3:

Travelling waves and localised states


Construction and stability (Evans functions)


Interface dynamics

Lecture 4:

Waves in random neural media

Lecture 5:

Tissue level spiking models: the dynamics of the continuum Lighthouse
model

14/1118/11  Pascal Chossat (CNRS  INRIA)
Dynamical systems in the presence of symmetry in the biological context
Lecture 1: 
Introduction to the general concepts of
symmetry
and equivariant bifurcations

Lecture 2: 
Symmetry breaking bifurcations, a non trivial
example: Turing patterns

Lecture 3: 
Equivariant Hopf bifurcation, example:
oscillatory patterns in coupled oscillators

Lecture 4: 
Symmetryinduced intermittency: the concept of heteroclinic cycles, example: multispecies competition

Lecture 5: 
Symmetryinduced unsteady patterns in spatially
extended system, example: spiral waves.

28/1102/12  Jean Petitot (Ecole Polytechnique  CREA)
Neurogeometry of functional architectures for visual perception
Lecture 1: 
Functional architecture of V1

Lecture 2: 
Implementing the contact structure of the 1jet
space of planar curves

Lecture 3: 
The E(2) group of planar isometries, subRiemannian geometry and illusory contours

Lecture 4: 
Noncommutative harmonic analysis

Lecture 5: 
Open problems in neurogeometry





