NeuroMathCompfocuses on the exploration of the brain from the mathematical and computational perspectives.

We want to unveil the principles that govern the functioning of neurons and assemblies thereof and to use our results to bridge the gap between biological and computational vision.

Our work is quite mathematical but we make heavy use of computers for numerical experiments and simulations. We have close ties with several top groups in biological neuroscience. We are pursuing the idea that the "unreasonable effectiveness of mathematics" can be brought, as it has been in physics, to bear on neuroscience.

Computational neuroscience attempts to build models of
neurons at a variety of levels, microscopic, i.e., the single
neuron, the minicolumn containing of the order of one hundred
or so neurons, mesoscopic, i.e., the macrocolumn containing
of the order of
10
^{4}-10
^{5}neurons, and macroscopic, i.e., a cortical
area such as the primary visual area V1.

Modeling such assemblies of neurons and simulating their behaviour involves putting together a mixture of the most recent results in neurophysiology with such advanced mathematical methods as dynamic systems theory, bifurcation theory, probability theory, stochastic calculus, theoretical physics and statistics, as well as the use of simulation tools.

We conduct research in the following three main areas.

Modeling and simulating single neurons.

Modeling and simulating assemblies of neurons.

Visual perception modeling.

Understanding the principles of information processing in the brain is challenging, theoretically and experimentally.

Modeling such assemblies of neurons and simulating their behaviour involves putting together a mixture of the most recent results in neurophysiology with such advanced mathematical methods as dynamical systems theory, bifurcation theory, probability theory, stochastic calculus, theoretical physics, and statistics, as well as the use of simulation tools

In order to test the validity of the models we rely
heavily on experimental data. These data come from single or
multi electrode recordings and optical imaging and are
provided by our collaborations with neurophysiology
laboratories such as the UNIC
http://

The NeuroMathComp team works at the three levels. We proposed a realistic model of single neurons by making use of physiological data and the theory of dynamical systems and bifurcations. At this level of analysis we also proposed a variety of theoretical tools from the theory of stochastic calculus and solved an open problem of determining the probability law of the spike intervals for a simple but realistic neuron model, the leaky integrate and fire with exponentially decaying synaptic currents . We also provided a mathematical analysis , through bifurcation theory, of the behaviour of a particular mesoscopic model, the one due to Jansen and Rit . We also proposed mean-field techniques for bridging the gap between the microscopic and mesoscopic scales . Finally we studied in detail several models for the statistics of spike trains in networks of neurons , .

We also started some efforts at the macroscopic level, in particular for modeling visual areas, see Section .

Another scientific focus of the team is the combined study of computer and biological vision. We think that a more detailed knowledge of the visual perception in humans and primates can have a potential impact on algorithm design and performance. Thus, we develop so-called bio-inspired approaches to model visual tasks. This work is multidisciplinary: It involves knowledge from neuroscience and physiology, and tries to reproduce what psychophysical experiments reveal. As a final goal, we want to compete with recent computer vision approaches (see, e.g. for a presentation of variational approaches in computer vision).

The models that we develop will be bio-inspired with two main goals:

Reproduce the functional architecture of the brain. Considering a given scale of observation of the brain's activity, this will define the proper mathematical formalism to consider. For example, at the microscopic scale, the nervous system uses spikes as a way to emit and code the information. So we need to define systems able to generate relevant spiking patterns with spiking neuron models (see, e.g., the Virtual Retina simulator ) but also statistical tools to analyse these spikes. At the macroscopic scale, we will be only interested on the brain's activity at a population level, so that the neural field formalism will be useful (see, e.g., a neural field model for motion estimation presented in ). .

Reproduce the percept and biological results. Validation of these models is crucial. Since we claim that our models are bio-inspired, our goal is also to validate them thanks to biology. For example, we showed that Virtual Retina reproduces closely single cell measurements acquired on cat ganglion cells. At the perceptual level, our models should also be able to reproduce a percept, which may be non trivial to reproduce with standard computer vision approaches. For example, we compared the results of the bio-inspired motion model and more generally, we proposed in a new way to evaluate motion estimation based on human performance.

Computer vision is another way to benchmark our approaches and prove their efficiency on real applications, and it is one of our goal to show how such bio-inspired approaches can compete with state-of-the-art computer vision approaches. For example, this has been successfully accomplished for the problems of action recognition , and motion estimation .

We developed a retina simulation software which transforms a video into spike trains (see also Adrien Wohrer PhD ). This project was developped in the scope of the EC project FACETS. Our goal was twofold: Allow large scale simulations (up to 100,000 neurons) in reasonable processing times and keep a strong biological plausibility (see for a review), taking into account implementation constraints. The underlying model includes a linear model of filtering in the Outer Plexiform Layer, a well-posed shunting feedback at the level of bipolar cells accounting for rapid contrast gain control , and a spike generation process modeling ganglion cells. We proved the pertinence of our software by reproducing several experimental measurements from single ganglion cells such as cat X and Y cells. This software is an evolutionary tool for neuroscientists that need realistic large-scale input spike trains in subsequent treatments, and for educational purposes. We also developed a web service, so that one may test directly the main software on his own data, without any installation. Virtual Retina was distributed in 2007 with an open-source licence (CeCILL C). Up to our knowledge, this is the only free retina simulator that allows to reproduce a spiking output from a video stream, which has been confronted to fine biological data.

Virtual Retina allows to generate simulated retinal
responses which are precise for
*individual*ganglion cells and artificial stimuli.
However, one crucial question in the community is now to
analyse
*collective*ganglion cells responses to both articial
and natural stimuli. To do so, the development of Virtual
Retina has to follow to take into account this new objective.
This will be also possible thanks to the statistical tools to
analyse spike trains which are jointly developped in the team
by Bruno Cessac etal. (see section
)

Website:
`http://
www-sop.
inria.
fr/
odyssee/
software/
virtualretina/
`

Virtual Retina is under CeCILL C licence.

*APP logiciel Virtual Retina:
IDDN.FR.OO1.210034.000.S.P.2007.000.31235*

This is a library providing numerical tools for the simulation of neural networks and the analysis of spike trains either coming from neural simulators or from biological experiments.

It is designed mainly as

An existing simulator plug-in (e.g. MVASpike or other simulators via the NeuralEnsemble meta-simulation platform),

Additional modules for computations with neural unit assembly on standard platforms (e.g. Python or the Scilab platform).

Original modules for the analysis of spike train statistics intended, in the mid term, to be used by the neuroscientists community.

Achievements include:

Spike trains statistical analysis via Gibbs distributions. They are based on the estimation of a parametric Gibbs potential optimaly characterizing the statistics of empirical spike trains (by minimisation of the Kullback-Leibler divergence between the empirical measure and the Gibbs measure). From this, classical statistical indicators such as firing rate, correlations, higher order moments and statistical entropy are obtained. Also, the form of the Gibbs potential provides essential informations on the underlying neural network and its structure. This method does not only allow us to estimate the spikes statistics but also to compare different models, thus answering such questions about the neural code as: are correlations (or time synchrony or a given set of spike patterns,. . . ) significant with respect to rate coding?

Spiking network programing for exact event's sequence restitution;

Siscrete neural field parameters algorithmic adjustments and time-constrained event-based network simulation reconciling clock and event based simulation methods.

Compared to existing libraries Enas offers new
computational methods taking into account time constraints in
neural networks (such as memory effects), based on
theoretical methods rooted in statistical physics and applied
mathematics. The algorithms used are based on linear
programming, nonlinear parameter estimations, statistical
methods. The C/C++ code has been organized as “bean java” to
ease its use by programmers non specialized in advanced
object programming. As a consequence the code is distributed
in the form of an include source for the lightest and the
most universal integration into users codes. The standard
algorithms are based on the best free libraries in the domain
such as gsl
http://

Event neural assembly simulation is developed in gForge. It is under CeCILL C licence

*APP logiciel Enas:
IDDN.FR.OO1.360008.000.S.P.2009.000.10600*.

Its development as a friendly software designed for the neuroscience community is our next purpose (ADT proposal).

Website:
`http://
enas.
gforge.
inria.
fr/
`

Bidimensional spiking models currently gather a lot of attention for their simplicity and their ability to reproduce various spiking patterns of cortical neurons. They are particularly used for large network simulations. These models describe the dynamics of the membrane potential by a nonlinear differential equation that blows up in finite time, coupled to a second equation for adaptation. Spikes are emitted when the membrane potential blows up or reaches a threshold . The precise simulation of the spike times and of the adaptation variable is critical for it governs the spike pattern produced, and is hard to compute accurately because of the exploding nature of the system at the spike times. We thoroughly studied the precision of fixed time-step integration schemes for this type of models and demonstrated that these methods produce systematic errors that are unbounded, as the cutoff value is increased, in the evaluation of the two crucial quantities: the spike time and the value of the adaptation variable at this time. Precise evaluation of these quantities therefore involve very small time steps and long simulation times. In order to achieve a fixed absolute precision in a reasonable computational time, we proposed here a new algorithm to simulate these systems based on a variable integration step method that either integrates the original ordinary differential equation or the equation of the orbits in the phase plane, and compared this algorithm with fixed time-step Euler scheme and other more accurate simulation algorithms.

This work has been accepted for publication in Neural Computation , and code is provided in Matlab and C.

Processing and encoding of information in neural dynamics
is a very active research field, although still much of the
role of neural assemblies and their internal interactions
remains unknown. With the advent of new Multi-Electrod Arrays
(MEA) techniques, the simultaneously recording of the
activity of groups of neurons (up to several hundreds) over a
dense configuration, supply today a critical database to
unravel the role of specific neural assemblies. The goal of
the present project is to propose to the neuroscientists
community statistical methods and numerical tools to
analysing spike train statistics. Our work is grounded on
theoretical results on Gibbs distributions in neural networks
which as lead us to propose a C/C++ library of algorithms
freely available at
http://

We provide rigorous and exact results characterizing the statistics of spike trains in a network of leaky integrate and fire neurons, where time is discrete and where neurons are exposed to noise, without restriction on the synaptic weights. Using methods in chains with complete connections we show the existence and uniqueness of an invariant measure of Gibbs type and discuss its properties. We also discuss Markovian approximations and relate them to the approaches currently used in computational neuroscience to analyse experimental spike trains statistics. This establishes a rigorous ground for the current investigations attempting to characterize real spike trains data with Gibbs distributions, such as the Ising-like distribution, using the maximal entropy principle. In this setting, Ising model appears as the "next step" after the Bernoulli model (independent neurons) since it introduces spatial pairwise correlations, but not time correlations. The range of validity of this approximation is discussed together with possible approaches allowing to introduce time correlations, with algorithmic extensions.

This work has been accepted for publication in Journal of Mathematical Biology .

We propose a generalization of the existing maximum
entropy models used for spike trains statistics analysis.
We bring a simple method to estimate Gibbs distributions,
generalizing existing approaches based on Ising model or
one step Markov chains to arbitrary parametric potentials.
Our method enables one to take into account memory effects
in dynamics. It provides directly the free-energy density
and the Kullback-Leibler divergence between the empirical
statistics and the statistical model. It does not assume a
specific Gibbs potential form and does not require the
assumption of detailed balance. Furthermore, it allows the
comparison of different statistical models and offers a
control of finite-size sampling effects, inherent to
empirical statistics, by using large deviations results. A
numerical validation of the method is proposed and the
perspectives regarding spike-train code analysis are also
discussed. The corresponding numerical algoritms have been
included in the Enas library developed at INRIA gforge
under Cecill-C licence
http://

This work has been submitted to the Journal of Computational Neuroscience , presented at the Neurocomp 2010 conference and published in the proceedings , .

We apply the formalism of Gibbs distributions to characterizing the statistics of multi-unit spike trains in bio-inspired simulated data from Virtual Retina, but also experimental data obtained from Multi-Electrode- Array (MEA) recordings from retina provided by Adrian Palacios (Centro de Neurociencia Valparaiso).

This work has been presented at the Neurocomp 2010 conference and published in the proceedings .

The study of multidimensional stochastic processes involves complex computations in intricate functional spaces. In particular, the diffusion processes, which include the practically important Gauss-Markov processes, are ordinarily defined through the theory of stochastic integration. Inspired by the Lévy-Cieselski construction of the Wiener process, we proposed an alternative representation of multidimensional Gauss-Markov processes as expansions on well-chosen Schauder bases, with independent random coefficients of normal law with zero mean and unitary variance. We thereby offered a natural multi-resolution description of Gauss-Markov processes as limits of the finite-dimensional partial sums of the expansion, that are strongly almost-surely convergent. Moreover, such finite-dimensional random processes constitute an optimal approximation of the process, in the sense of minimizing the associated Dirichlet energy under interpolating constraints. This approach allows simpler treatment in many applied and theoretical fields and we provide a short overview of applications we are currently developing.

This contribution is currently submitted and corresponds to the arXiv preprint .

Our goal here is a better understanding of the extent to which computing and modeling with spiking neuron networks might be biologically plausible and computationally efficient. Based on a thorough characterization of the main constraints on spiking neural networks dynamics this has lead us to propose new algorithms to infer the structure of the network from its spike trains and to propose an FPGA implentation of spiking neural networks.

Restricting to deterministic spiking neuron networks, we obtained original results on: (i) time constraints, (ii) links between continuous signals and spike trains, (iii) spiking neuron networks parameter adjustment. When implementing spiking neuron networks, for biological simulation or computational purpose, it is important to take into account these constraints, preventing one from implementing unrealistic (e.g. violating physical time constraints) or unecessary (artificially introducing spikes when continuous calculations would be sufficient) mechanisms in neural networks models. This provides fundamental keys to implementing large-scale spiking neuron networks.

This work has been published in Journal of Physiology, Paris .

We consider the deterministic evolution of a time-discretized spiking network of neurons with connection weights having delays, modeled as a discretized neural network of the generalized integrate and fire (gIF) type. The purpose is to study a class of algorithmic methods allowing to calculate the proper parameters (synaptic weights) to reproduce exactly a given spike train generated by an hidden (unknown) neural network. This problem is linear (L) if the membrane potentials are observed and LP (Linear-Programming) if only spike times are observed, in the context of gIF models. The L or LP adjustment mechanism is local to each unit and has the same structure as an "Hebbian" rule. This paradigm is easily generalizable to the design of input-output spike train transformations. This means that we have a practical method to "program" a spiking network, i.e. find a set of parameters allowing us to exactly reproduce the network output, given an input

This work has been published in an INRIA research report and submitted in Journal of Computational Neuroscience, 2010 , .

Spiking neural networks are able to perform very powerful computations with precise timed spikes. We are developing an FPGA (Field Programmable Gate Array) reconfigurable platform that enables the simulation of in silico models of spiking neural networks. Since the model is directly mapped into a FPGA device, the neural processing is accelerated and the time consumed reduced. We use VHDL and Handel-C to design the reconfigurable architecture of a discrete time Integrate-and-Fire model , coded in CUDA, running on GPU.

This work has been presented in the conference Neurocomp 2010 and published in the proceedings .

The brain activity is characterized by the interaction of a very large number of neurons that are strongly affected by noise. However, signals often arise at macroscopic scales integrating the effect of many neurons into a reliable pattern of activity. In order to study such large neuronal assemblies, one is often led to derive mean-field limits summarizing the effect of the interaction of a large number of neurons into an effective signal.

In this section we address the problem of simplifying the description of a large populations of neurons interacting in a network. The techniques we use fall in the category of what is known in statistical physics and mathematics as mean-field methods.

In
, we proposed a method based on
to derive the mean-field
equations of a network of completely connected neural
masses falling into
Ppopulations. Each neural mass is described by a
Wilson and Cowan model perturbed by Brownian noise. The
connectivity weights are modelled as independent Gaussian
random variables, independent of the Brownians. We showed
that when the number of neural masses grew to infinity the
activity of the network could be represented by a set of
Pmean-field equations. These stochastic equations are
non-Markov, involving a special process called the
interaction process. Their solution is a Gaussian process
whose mean can be described by an ordinary differential
equation, coupled to a fixed-point equation for the
covariance. They can be thought of as describing the law of
the limit activity of the network as the number of neurons
increases whatever the (frozen) synaptic weights drawn from
their law. The resulting equations turned out to be very
difficult to analyse.

There are two criticisms of this approach that can be formulated. The first one is the fact that we are considering a network of neural masses, not of individual spiking neurons. The second is that the model of the uncertainty on the synaptic weights is a bit strange: we draw them at random and then assume they are constant over time for each sample. It would seem more realistic to assume that they fluctuate over time. To address these two criticisms we developed a new approach based on the work of Tanaka, McKean and Sznitman. We consider realistic models of spiking neurons such as the one pioneered by Hodgkin and Huxley, assume that the synaptic weights are the sum of a constant and a white noise, therefore fluctuating in time, and prove that one can derive mean-field equations which are well-posed, enjoy a unique solution and such that the average solution of the network converges toward this solution when the network size grows to infinity. We also establish fluctuation results describing the finite size effect.

In the case of the network of neural masses studied in , the method described in section leads to a set mean-field equations whose solution is also a Gaussian random process. The equation for the mean is identical to the one found in but the one describing the variance, the only quantity related to the covariance function appearing in the equation for the mean, is considerably simpler, leading to an ordinary differential equation. Mean and variance can therefore be described by a set of coupled ordinary differential equations whose solutions can be studied for example as functions of the parameters describing the synaptic weights and the inputs. The bifurcations of these solutions with respect to these parameters can also be studied leading to a complete understanding of how the dynamics of the network varies when one progressively increases the variance on the synaptic weights. Non-trivial synchronization or stabilization effects appear as the external or the synaptic noise intensity are varied, in accordance with the simulations of finite-size networks.

In the case of a network of spiking neurons described, e.g., by Fitzhugh Nagumo or Hodgkin Huxley models we introduce a realistic chemical synaptic model based on and derive the mean-field equations using the method described in . Simulation of these equations show that phenomena similar to those described in section , i.e. synchronization and desynchronization, do appear. Unlike the previous case we cannot in general obtain a closed set of equations for the moments of the limit law and we must rely on the analysis of the corresponding Fokker-Planck equation and the bifurcations of its solutions.

This work was partially funded by the FACETS-ITN grant and the ERC advanced grant NerVi number 227747.

Department of Mathematics, University of Pittsburgh, USA

Classical mean-field approaches consider the evolution of a deterministic variable, the mean activity, thus neglecting the stochastic nature of neural behavior. In this article, we build upon a recent approach that includes correlations and higher order moments in mean-field equations, and study how these stochastic effects influence the solutions of the mean-field equations, both in the limit of an infinite number of neurons and for large yet finite networks. We show that, though the solutions of the deterministic mean-field equation constitute uncorrelated solutions of the new mean-field equations, the stability properties of limit cycles are modified by the presence of correlations, and additional non-trivial behaviors including periodic orbits appear when there were none in the mean field. The origin of all these behaviors is then explored in finite-size networks where interesting mesoscopic scale effects appear. This study leads us to show that the infinite-size system appears as a singular limit of the network equations, and for any finite network, the system will differ from the infinite system.

This work is currently submitted and corresponds to the arXiv preprint .

We propose and explore a general framework for studying neural mass models given by ordinary differential equations and link their behaviors with typical cortical behaviors, with a special focus on epileptic patterns, by the study of the (codimension 2 and 3) bifurcations of the vector field and of the stochastic properties of the solutions of these equations. This systematic approach based on a dictionary linking vector field bifurcations and epilepsy-related phenomena is then applied to a simple model of choice, the Jansen and Rit model. This model features most phenomena of interest. A more complex recent model (featuring the same behaviors), the Wendling and Chauvel model, can also be studied with these tools. These studies allow to test hypothesis and investigate the effect of medication in a model-based approach and to test physiological and anatomical facts and predictions on the origin of pathological cortical behaviors.

This work is submitted and corresponds to the arxiv preprint .

The presence of self-organized criticality in biology is often evidenced by a power-law scaling of event size distributions, which can be measured by linear regression on logarithmic axes. We show that such a procedure does not necessarily mean that the system exhibits self-organized criticality. We first provide an analysis of multisite local field potential (LFP) recordings of brain activity and show that event size distributions defined as negative LFP peaks can be close to power-law distributions. However, this result is not robust to change in detection threshold, or when tested using more rigorous statistical analyses such as the Kolmogorov–Smirnov test. Similar power-law scaling is observed for surrogate signals, suggesting that power-law scaling may be a generic property of thresholded stochastic processes. We next investigate this problem analytically, and show that, indeed, stochastic processes can produce spurious power-law scaling without the presence of underlying self-organized criticality. However, this power-law is only apparent in logarithmic representations, and does not survive more rigorous analysis such as the Kolmogorov–Smirnov test. The same analysis was also performed on an artificial network known to display self-organized criticality. In this case, both the graphical representations and the rigorous statistical analysis reveal with no ambiguity that the avalanche size is distributed as a power-law. We conclude that logarithmic representations can lead to spurious power-law scaling induced by the stochastic nature of the phenomenon. This apparent power-law scaling does not constitute a proof of self-organized criticality, which should be demonstrated by more stringent statistical tests.

This work has appeared in PloS ONE, .

Specifically, neural or cortical fields are continuous assemblies of mesoscopic models, also called neural masses, of neural populations that are fundamental in the modeling of macroscopic parts of the brain. Neural fields are described by nonlinear integro-differential equations. The solutions to these equations represent the state of activity of these populations when submitted to inputs from neighbouring brain areas. Understanding the properties of these solutions is essential in advancing our understanding of the brain.

We studied neural field models with delays which define a useful framework for modeling macroscopic parts of the cortex involving several populations of neurons. Nonlinear delayed integro-differential equations describe the spatio-temporal behavior of these fields. Using methods from the theory of delay differential equations, we show the existence and uniqueness of a solution of these equations. A Lyapunov analysis gives us sufficient conditions for the solutions to be asymptotically stable. We also presented a fairly detailed study of the numerical computation of these solutions. This is, to our knowledge, the first time that a serious analysis of the problem of the existence and uniqueness of a solution of these equations has been performed. Another original contribution of ours is the definition of a Lyapunov functional and the result of stability it implies. We illustrate our numerical schemes on a variety of examples that are relevant to modeling in neuroscience.

This work was presented at the Neurocomp 2010 conference, .

This work has appeared in Physica D, special issue on Mathematical Neuroscience, . This work was partially funded by the ERC advanced grant NerVi number 227747.

We studied the dependency of the stationary solutions of the neural fields equations with respect to the stiffness of the nonlinearity and the contrast of the external inputs. This is done by using degree and bifurcation theories in the context of functional, in particular infinite dimensional, spaces. The joint use of these two theories allowed us to make new detailed predictions about the global and local behaviours of the solutions. We also provided a generic finite dimensional approximation of these equations which allows us to study in great details two models. The first model is a neural mass model of a cortical hypercolumn of orientation sensitive neurons, the ring model . The second model is a general neural field model where the spatial connectivity is described by heterogeneous Gaussian-like functions.

This work has appeared in SIAM Journal on Applied Dynamical Systems, .

This work was partially funded by the ERC advanced grant NerVi number 227747.

This work is submitted in SIAM Journal on Applied Dynamical Systems. Part of this work was presented at the Neurocomp 2010 conference,

This work was partially funded by the ERC advanced grant NerVi number 227747.

Our goal is to understand the dynamics of the cortical states of parts of the primate visual cortex when stationary (independent of time) stimuli are presented. A first step to achieve this goal is to understand the stationary cortical states. These neural mass models have been used to describe the activity of neural populations that are found in the visual cortex of primates. They feature stationary solutions, when submitted to a stationary input. These solutions depend quite sensitively on such parameters of the model as the stiffness of the nonlinearity and the contrast of the external input. The use of bifurcation theory allows us to make new detailed predictions about the global and local behaviours of the solutions. We apply these results to the study of a neural mass model of a cortical hypercolumn of orientation sensitive neurons called the ring model .

This work is submitted in PlosCompBio

This work was partially funded by the ERC advanced grant NerVi number 227747.

Motivated by a model for the perception of textures by
the visual cortex in primates, we analysed the bifurcation
of periodic patterns for nonlinear equations describing the
state of a system defined on the space of structure
tensors, when these equations are further invariant with
respect to the isometries of this space. We showed that the
problem reduces to a bifurcation problem in the hyperbolic
plane
(Poincaré disc). We made use of the concept of
periodic lattice in
to further reduce the problem to one on a compact
Riemann surface
, where
Gis a cocompact, torsion-free Fuchsian group. The
knowledge of the symmetry group of this surface allowed to
carry out the machinery of equivariant bifurcation theory.
Solutions which generically bifurcate are called
"H-planforms", by analogy with the "planforms" introduced
for pattern formation in Euclidean space. This concept is
applied to the case of an octagonal periodic pattern, where
we are able to classify all possible H-planforms satisfying
the hypotheses of the Equivariant Branching Lemma. These
patterns are however not straightforward to compute, even
numerically, and we described a method for computation
illustrated with a selection of images of octagonal
H-planforms.

This work has been accepted for publication in the Journal of Nonlinear Science, .

This work was partially funded by the ERC advanced grant NerVi number 227747.

We consider a spatial two-dimensional network of a
special kind of neural masses, the ring model
studied in section
. Such a network has been
studied for example in
in the context of equivariant
bifurcation theory. Our goal is to solve the network
equations as fast as possible on a GPU architecture without
for arbitrary spatial (N) and angular resolutions (P). A
naive discretization of the problem leads to solving a set
of
N^{2}×
Pcoupled nonlinear ordinary
differential equations. The PG-kernel approach, (for
Pingerle-Goursat) kernel, proposed in
consists in approximating the
operator defined by the spatial connectivity function
between the neural masses by a finite-range operator. This
is obtained by writing the connectivity function as a sum
of
Qproducts of functions and leads to solving a set of
Qcoupled nonlinear ordinary differential equations
*independently*of the spatial and angular resolutions.
If
Q<<
N
^{2}×
Pa significant gain in
computation time is possible.

This work was partially funded by the FACETS-ITN grant and the ERC advanced grant NerVi number 227747.

Our simulator Virtual Retina
http://

The dynamics of motion integration show striking similarities when observed at neuronal, psychophysical, and oculomotor levels. Based on the inter-relation and complementary insights given by those dynamics, our goal was to test how basic mechanisms of dynamical cortical processing can be incorporated in a dynamical model to solve several aspects of 2D motion integration and segmentation. In , , we proposed a model inspired by the hierarchical processing stages of the primate visual cortex: we describe the interactions between several layers processing local motion and form information through feedforward, feedback, and inhibitive lateral connections. Also, following perceptual studies concerning contour integration and physiological studies of receptive fields, we postulate that motion estimation takes advantage of another low-level cue, which is luminance smoothness along edges or surfaces, in order to gate recurrent motion diffusion. With such a model, we successfully reproduced the temporal dynamics of motion integration on a wide range of simple motion stimuli: line segments, rotating ellipses, plaids, and barber poles. Furthermore, we showed that the proposed computational rule of luminance-gated diffusion of motion information is sufficient to explain a large set of contextual modulations of motion integration and segmentation in more elaborated stimuli such as chopstick illusions, simulated aperture problems, or rotating diamonds. As a whole, in we proposed a new basal luminance-driven motion integration mechanism as an alternative to less parsimonious models, we carefully investigated the dynamics of motion integration, and we established a distinction between simple and complex stimuli according to the kind of information required to solve their ambiguities.

This work was partially supported by the EC IP project FP6-015879 (FACETS), the EC ICT project No. 215866 (SEARISE) and the Région Provence-Alpes-Côte d'Azur.

Our goal is to identify the key biological features to reach a good compromise between bio-inspiration and computational efficiency for the motion estimation problem. Motion estimation is a task performed very well by the primate visual cortex. But motion estimation is also a key feature for many vision and robotic applications, and it is an active field of research in the computer vision community. So the question is: From modelling to evaluation, can biology be a source of inspiration?

This work can be seen as an extension of , but the only relation comes from the general structure based on two recurently connected layers (V1 and MT). However, the scope and content of this contribution is very different. More precisely, distinguishing results are:

The model: we choose the neural field formalism which provides a sound mathematical framework to describe the model at a macroscopic scale. Within this framework we defined the cortical activity as coupled integro-differential equations and proved the well-posedness of the model.

The results: We showed how our model performs on some classical computer vision videos, and compared its behaviour against the visual system on a simple classical video used in psychophysics (while in , we only presented results on psychophysical experiments, focusing on the dynamics of motion estimation).

The proposal of bio-inspired benchmark (see also , , ), which is a novel challenge for modelers in computer vision. We showed how it is possible to benchmark an algorithm with respect to human performance (i.e., based on results obtained in psychophysics).

This work is submitted in IJCV, available as an INRIA research report

This work was partially supported by the EC IP project FP6-015879 (FACETS), the EC ICT project No. 215866 (SEARISE) and the Région Provence-Alpes-Côte d'Azur, and the ERC advanced grant NerVi number 227747.

Analog-to-digital signal conversion (ADC) is a crucial step in a signal processing chain. Although a wide-range of algorithms is already implemented, the raising interest towards the solutions provided by biological systems to engineering problems motivated our study on how these biological systems tackle the issue of ADC. Interestingly, the mammalians retina offers an appropriate study environment. Indeed, the neural code of the retina consists of a series of uniformly-shaped electrical impulses, the spikes, yielding a binary code. In , , we base our work on a biologically realistic model of the mammalians retina, and focus on its behavior during the first 30ms of the visual stimuli processing. This leads us to the specification of a novel bio-inspired quantization algorithm. We proved that, under the model assumption, our new bio-inspired quantizer emphasizes high magnitude signals rather than high probability ones. Besides, our system gradually changes across time from a coarse and quasi-uniform quantizer to a refined and highly non-linear one. This yields an original time depending bio-inspired quantization system. The proposed bio-inspired quantizer offers several interesting features as scalability, and introduces time parametrizing in the coding system which opens perspectives for the conception of an implicit bit-allocation mechanism.

SEARISE is a three-year project started in March 2008. It involves the following academic partners: Fraunhofer-Gesellschaft (Germany), University of Genoa (Italy), Ulm University (Germany) University of Bangor (Wales). Two industrial partners are also involved: TrackMen Ltd. and LTU Arena.

The SEARISE project develops a trinocular active cognitive vision system, the Smart-Eyes, for detection, tracking and categorization of salient events and behaviours. Unlike other approaches in video surveillance, the system will have human-like capability to learn continuously from the visual input, self-adjust to ever changing visual environment, fixate salient events and follow their motion, categorize salient events depending on the context. Inspired by the human visual system, a cyclopean camera will perform wide range monitoring of the visual field while active binocular stereo cameras will fixate and track salient objects, mimicking a focus of attention that switches between different interesting locations.

The core of this artificial cognitive visual system will be a dynamic hierarchical neural architecture – a computational model of visual processing in the brain. Information processing in Smart-Eyes will be highly efficient due to a multi-scale design: Controlled by the cortically plausible neural model, the active cameras will provide a multi-scale video record of salient events. The processing will self-organize to adapt to scale variations and to assign the majority of computational resources to the informative parts of the scene.

The Smart-Eyes system will be tested in real-life scenarios featuring the activity of people in different scales. In a long-range distance scenario, the system will be monitoring crowd behaviour of sport fans in a football arena. In a short range scenario, the system will be monitoring the behaviour of small groups of people and single individuals. The system’s capability for self-adaptation will be specifically demonstrated and quantified compared to systems with ‘classical’ architecture that are trained once and then used on a set of test scenes.

FACETS is an integrated project within the biologically inspired information systems branch of IST-FET. The FACETS project aims to address, with a concerted action of neuroscientists, computer scientists, engineers and physicists, the unsolved question of how the brain computes. It combines a substantial fraction of the European groups working in the field into a consortium of 13 groups from Austria, France, Germany, Hungary, Sweden, Switzerland and the UK. About 80 scientists have joined their efforts over a period of 5 years, starting in September 2005 and ending in August 2010. A project of this dimension has rarely been carried out in the context of brain-science related work in Europe, in particular with such a strong interdisciplinary component.

Olivier Faugeras responded to the 2008 ERC call “IDEAS”. His project, NerVi, submitted to the “Mathematics and Interfaces” panel, has been accepted and obtained a 5 years funding for a total amount of 1.7 Million Euros.

The project is to develop a formal model of information representation and processing in the part of the neocortex that is mostly concerned with visual information. This model will open new horizons in a well-principled way in the fields of artificial and biological vision as well as in computational neuroscience. Specifically the goal is to develop a universally accepted formal framework for describing complex, distributed and hierarchical processes capable of processing seamlessly a continuous flow of images. This framework features notably computational units operating at several spatiotemporal scales on stochastic data arising from natural images. Mean- field theory and stochastic calculus are used to harness the fundamental stochastic nature of the data, functional analysis and bifurcation theory to map the complexity of the behaviours of these assemblies of units. In the absence of such foundations, the development of an understanding of visual information processing in man and machines could be greatly hindered. Although the proposal addresses fundamental problems, its goal is to serve as the basis for ground-breaking future computational development for managing visual data and as a theoretical framework for a scientific understanding of biological vision.

This 'Marie-Curie Initial Training Network' (funded by
the EU) involves 15 groups at European Research
Universities, Research Centers and Industrial Partners in 6
countries. It funds two PhD students in the NeuroMathComp
group. Website:
`http://
facets.
kip.
uni-heidelberg.
de/
ITN/
index.
html`

The BrainScaleS project aims at understanding function and interaction of multiple spatial and temporal scales in brain information processing. The fundamentally new approach of BrainScaleS lies in the in-vivo biological experimentation and computational analysis. Spatial scales range from individual neurons over larger neuron populations to entire functional brain areas. Temporal scales range from milliseconds relevant for event based plasticity mechanisms to hours or days relevant for learning and development. In the project generic theoretical principles will be extracted to enable an artificial synthesis of cortical-like cognitive skills. Both, numerical simulations on petaflop supercomputers and a fundamentally different non-von Neumann hardware architecture will be employed for this purpose. Neurobiological data from the early perceptual visual and somatosensory systems will be combined with data from specifically targeted higher cortical areas. Functional databases as well as novel project-specific experimental tools and protocols will be developed and used. New theoretical concepts and methods will be developed for understanding the computational role of the complex multi-scale dynamics of neural systems in-vivo. Innovative in-vivo experiments will be carried out to guide this analytical understanding. Multiscale architectures will be synthesized into a non-von Neumann computing device realised in custom designed electronic hardware. The proposed Hybrid Multiscale Computing Facility (HMF) combines microscopic neuromorphic physical model circuits with numerically calculated mesoscopic and macroscopic functional units and a virtual environment providing sensory, decision-making and motor interfaces. The project also plans to employ petaflop supercomputing to obtain new insights into the specific properties of the different hardware architectures. A set of demonstration experiments will link multiscale analysis of biological systems with functionally and architecturally equivalent synthetic systems and offer the possibility for quantitative statements on the validity of theories bridging multiple scales. The demonstration experiments will also explore non-von Neumann computing outside the realm of brain-science. BrainScaleS will establish close links with the EU Brain-i-Nets and the Blue Brain project at the EPFL Lausanne. The consortium consists of a core group of 10 partners with 13 individual groups. Together with other projects and groups the BrainScaleS consortium plans to make important contributions to the preparation of a future FET flagship project. This project will address the understanding and exploitation of information processing in the human brain as one of the major intellectual challenges of humanity with vast potential applications.

This project starts on January 1st, 2011 and is funded for four years.

Bruno Cessac is a reviewer for the CONYCIT (Chile) and COFECUB (Brasil) program and for the reviews Physica D, Nonlinearity, Chaos, Journal of Statistical Physics, IEEE Transaction in Neural Networks, Journal of Mathematical Biology, Journal of Computational Neuroscience. He was a lecturer in Valparaiso at the School PRIMAVERA “Computing with spikes: results and challenges” in November 2010. He is in charge of internships organisation in the Master of Computational Biology, Nice.

Pascal Chossat is director of the Centre International de Rencontres Mathématiques (CIRM) in Marseille, an international conference centre in mathematics jointly operated by CNRS and the French Mathematical Society. He stepped down in June 2010. He is deputy scientific director of INSMI, the newly created mathematics institute of CNRS, in charge of the international relations of CNRS in this field. He is the coordinator of a geographic EraNet (EC program) named New Indigo for the development of scientific networks between European member states and India.

Olivier Faugeras is a member of the French Academy of
Sciences, the French Academy of Technology. He is on the
Editorial board of the International Journal of Computer
Vision (IJCV). He is the co-editor in chief of the Journal of
Mathematical neuroscience, Springer: Website:
`http://
www.
mathematical-neuroscience.
com/
`He is a member of the Institut Thématique
Multi-organismes Neurosciences, Sciences cognitives,
Neurologie, Psychiatrie. He is a member of the ERC PE1 panel
for 2010-2011. He was the chairman of the AERES evaluation
committee of the TIMC laboratory in Grenoble (June 2010). He
was a member of the ANR evaluation SVSE4 panel for
neuroscience.

Pierre Kornprobst was a member of the comité de suivi
doctoral (CSD) until May. He is the coordinator of the Master
of Science in Computational Biology from Université Nice
Sophia Antipolis (Website:
`http://
www.
computationalbiology.
eu`). The scientific goal of this program is to focus on
the human being from different perspectives (understanding
and modeling functional aspects or interpreting biomedical
signals from various devices) and at different scales (from
molecules to organs and the whole organism).

Bruno Cessac was Maître de conférences in Nice University until September 2010. He is now research director at INRIA. He has been teaching thermodynamics (L2 level), quantum physics (L2 level), probability theory (L3 level), C programming (L3 level), linear algebra (L3 level), statistical physics (M1 level), neural networks dynamics (M2 level, master of physics and master of computational biology), for Physics students. He is still teaching theoretical methods in neuronal dynamics analysis in the Master of Computational Biology, Nice.

Olivier Faugeras teaches at the ENS Paris the course ”Mathematical methods for neuroscience“ part of the Master MVA and the ENS Math/Info section - (27h).

Grégory Faye is a teaching assistant of analysis courses for first year students L1 at UNSA (Université de Nice) (64h).

Hassan Nasser is a teaching assistant at the Ecole Polytechnique Universutaire de Nice. He his teaching microprocessors and numeric electronic (L3, 64h).

Émilien Tlapale teaches "Programming in C++", at EPU (Université de Nice-Sophia Antipolis) in the electronic department (fifth year).

In 2010 Bruno Cessac has been invited:

to the Conference "Stochastic models in neuroscience", Marseille, 18-22 January 2010 at CIRM;

to the "Workshop School Chaos and Dynamics in Biological Networks" Cargese 3-8 May 2010;

to the "Workshop on Spike Train Measures and Their Applications to Neural Coding” 2/3 June 2010 in Plymouth/UK;

to give a lecture in Valparaiso at the School PRIMAVERA “Computing with spikes: results and challenges” in November 2010.

Pascal Chossat gave invited talks at

13e Rencontre du Non Linéaire, organizers M. Lefranc, C. Letellier and L. Pastur, 10- 12/03/2010, “Bifurcations dans l’espace hyperbolique en relation avec un modèle de perception des structures visuelles par le cortex”,

Congrès "8th AIMS Conference on Dynamical Systems, Differential Equations and Applications", Dresdeen 25-28/05/2010, “Bifurcation of patterns in hyperbolic space”,

Colloque "Géométrie, Mécanique et Dynamique", organizers J-P Ortega and J. Marsden, CIRM 12-16/07/2010, “Bifurcation of patterns in hyperbolic space”,

Colloque franco-japonais "Reaction-diffusion systems: experiment, modeling and analysis", organizer D. Hilhorst, Orsay 21-22/10/2010, “Bifurcation of patterns in hyperbolic space in relation with a model of visual cortical perception”,

Colloque WAMCE 2010, organizer C. Schaerer, Asuncion (Paraguay) 25-26/10/2010, “Bifurcation of patterns in hyperbolic space in relation with a model of visual cortical perception”.

Olivier Faugeras gave invited talks at

the CIRM Workshop, “Stochastic models in neuroscience”, Marseille, January, ,

the Ladislav Tauc Conferences and I.T.M.O. Neurosciences Joint Meeting: MULTISCALE ANALYSIS of NEURAL SYSTEMS, Gif-s-Yvette, February, ,

the Workshop "Maths pour l'image", Orléans, April, ,

the AIMS Conference Spatially Structured Oscillations and Waves in Neural Media, Dresden, May, ,

the ITMO Workshop on multiscale analysis in neuroscience, Paris, College de France, July, ,

the OCCAM workshop on Future challenges in mathematical and computational neuroscience, Oxford, September, ,

the Workshop on Progress in neural field theory, Reading, September, ,

the Symposium in honor of Mike Brady, Oxford, September, ,

the Tauc conference, "From Mathematical Image Analysis to Neurogeometry of the Brain", Gif-sur-Yvette, December.

He was the plenary speaker at ECCV2010, the 11th European Conference in Computer Vision, Heraklion, Crete, September, .

He gave a talk on the future
challenges in mathematical neuroscience at OF'60
(Website:
`http://
www-sop.
inria.
fr/
manifestations/
of60/
`), the one-day scientific meeting on Artificial
Vision, Biological Vision and Computational Neuroscience
organized at the INRIA Sophia Antipolis – Méditerranée
research centre to celebrate his 60th birthday.

In 2010 Pierre Kornprobst has been invited:

to the symposium "Multi-scale analysis of neural systems: taking the challenge seriously", Gif-sur-Yvette, 15-16 February 2010;

to Colloque “Méthodes mathématiques pour l’image”, Orléans 1-3 April 2010.

Jonathan Touboul gave an invited talk at OF'60.

We thank Théo Papadopoulo of the Athena Project Team for his careful proofreading of the original version of this document. This has allowed us to produce a better version of this report.

Olivier Faugeras warmly thanks all his
friends and colleagues who made OF'60 (Website:
`http://
www-sop.
inria.
fr/
manifestations/
of60/
`) possible. Special thanks go to Nicholas Ayache,
Jean-Daniel Boissonnat, Rachid Deriche, Gérard Giraudon,
Agnès Cortell, Agnès Bessière and Marc Barret.