Thierry Viéville @ Inria

Are biological neurons and networks that vicious ? Or only their model ?.
A link is built between a biologically plausible generalized integrate and fire (GIF) neuron model with conductance-based dynamics and a discrete time neural network model with spiking neurons, for which rigorous results on the spontaneous dynamics has been obtained. More precisely the following has been shown.
i) Occurrence of periodic orbits is the generic regime of activity, with a bounded period in the presence of spike-time dependence plasticity, and arbitrary large periods at the edge of chaos (such regime is indistinguishable from chaos in numerical experiments, explaining what is obtained in),

ii) the dynamics of membrane potential has a one to one correspondence with sequences of spikes patterns ("raster plots")

This allows a better insight into the possible neural coding in such a network and provides a deep understanding, at the network level, of the system behavior. Moreover, though the dynamics is generically periodic, it has a weak form of initial conditions sensitivity due to the presence of the sharp spiking threshold.

A step further, constructive conditions are derived, allowing to properly implement visual functions on such networks. The time discretization has been carefully conducted avoiding usual bias induced by e.g. Euler methods and taking into account a rather complex GIF model for which the usual arbitrary discontinuities are discussed in details. The effects of the discretization approximation has been analytically and experimentally analyzed.

With this new point of view, we also reconsider some ``biological'' results obtained on ``models'' with biologically implausible discontinuities. It allows us to reduce the bio-physical membrane equation to a very simple but powerful gIF numerical model. This also dramatically reduces the algorithmic complexity of event-based network simulations.

key-words. Cortical maps; Spiking neural networks; Edge of chaos.

Co-authors. Bruno Cessac.
Links.
Specifying biological networks computations with variational approaches.
We revisit here the links between (i) high-level specification of how the brain represents and categorizes the causes of its sensory input and (ii) related analog or spiking neural networks.

Focusing on visual processing, we show -for a rather general class of cortical map computations- how it is possible to directly rely ``what is to be done'' (perceptual task) with ``how to do it'' (neural network calculation).

More precisely, in computer vision, efficient computations using implementations of regularization processes allows to obtain well-defined and powerful estimations. Having a biologically plausible representation of such mechanisms is thus a challenging issue.

Reviewing recent ideas about brain activity representation, we show how the key idea is to (i) represent what is to be done as an optimization problem, (ii) considering regularization mechanisms (implemented using so-called partial-differential-equations) and (iii) ``compiling'' the related analog or spiking neural network parameters.

The basement of the present contribution is the development of an unbiased approximation of a diffusion operator used in regularization mechanisms (introducing a summation property). A direct link between continuous formulation and the related sampled implementation is obtained. It also provides convergence properties for the related network.

Not only analog networks but also deterministic spiking neural network can implement the present specifications, using here piece-wise approximations in the so called Spike Response Model, also leading to a fast event-based simulation of such networks.

key-words. Cortical maps; Diffusion operator; Visual function; Cortical columns.

Co-authors. Bruno Cessac, Horacio Rostro, Sandrine Chemla, Pierre Kornprobst.
Links.
One step towards an abstract view of computation in spiking neural-networks.
Neural network information is mainly conveyed through (i) event-based quanta, spikes, whereas high-level representation of the related processing is almost always modeled in (ii) some continuous framework. Here, we propose a link between (i) and (ii) which allows to derive the spiking network parameters given a continuous processing and also obtain an abstract interpretation of the related processing.

In event based neural network models, the output of a neuron is entirely characterized by the sequence of spike firing times and the Gerstner and Kistler Spike Response Model of a biological neuron defines the state of a neuron via a single variable. At the computational level, using piece-wise linear profiles yields a closed-form calculation of the spiking events, thus allows to obtain an efficient and exact implementation of (1) in event-based massive neuronal simulators such as MVASPIKE.

Using this model and following Maas and Natchslager, we represent the signal as the last spike delay with respect to a given temporal reference, consider piece-wise linear response profiles (as approximations of Hodhkins-Huxley related profiles), introduce a temporal discretization of the input current, and obtain a direct link with continuous representation of neural map computation:
- the resistive coefficient being proportional to the spiking threshold

- the variational approach diffusion being in direct relation with the synaptic weights

- the corrective term being controlled by the axonal delay

- the input gain being controlled by the input resistance
with closed-form correspondence allowing to explicitly calculate the neural network parameters given an abstract continuous representation.

This relationship is valid only in a given temporal window, with saturation outside, as for analog networks. Here it appears that fast adaptive delays (as observed in recent intra-cellular experiments of e.g. Fregnac et al.) is a crucial element in this model. In the derivation a constraint coherent with S.T.D.P. adaptation rules (yielding the same constraint) as derived by, e.g., Guyonneau. It also corresponds to what is obtained from a variational framework relating the neuronal weights to a continuous diffusion operator, as introduced by Cottet. This last formulation is in direct relation with a sub-class of Cohen-Grossgerg dynamical systems.

We illustrate the previous derivation with an event-based implementation of an early-vision processing layer, for a 1D spiking neural network, correspond to an edge-preserving smoothing of the input, using a non-linear diffusion operator.

key-words. Cortical maps; Diffusion operator; Visual function; Spiking neural networks.

Co-authors. Léonard Gérard, Pierre Kornprobst, Olivier Rochel, et al.
Links.
Neural fields and self-organizing receptive fields using a variational approach.
Self organizing-maps, as e.g. introduced by Kohonen, are artificial neural networks characterized by competitive learning of the processing elements. They thus can be used for the pattern analysis of input signal. Here we consider, as a working example, the self-organization of visual receptive fields, as it occurs in V1 and study to which extend using a variational approach may help specifying such a mechanism.

The goal of the present contribution is thus humble in the sense that we simply would like to explore at a technical level, the advantages and limits of the variational specification in this case, in order to better adjust the parameters.

This formulation includes non-linear formulation of self-organizing maps as proposed by Fort and Pages, while unusual non-linear profiles allowing edge preserving smoothing, via non-linear diffusion are introduced. This criterion also acts as a convergence function (à la Lyapounov) of the underlying dynamical system.

The assumption is that they contribute to regularize the learning process. As such, they likely speed-up the convergence. They however constraint the solution to be taken in a more restrictive space of smoother functions. Experimental results are going to confront these assumptions to the ground truth.

A step ahead, the ``arg max'' rule itself is implemented by a winner-take-all mechanism, given an initial condition, the formulation leading to a local distributed implementation and guaranties the convergence towards a local minimum of the criterion.

A theoretical result predicts a stable interaction though feed-backs between these two layers and an experimental verification is proposed here.

key-words. Cortical maps; Diffusion operator; Spiking neural networks.

Co-authors. Frédéric Alexandre, Jérémie Fix, Axel Hutt, Nicolas Rougier.
Links.
A deterministic biologically plausible classifier.
Regarding biological visual classification, recent series of experiments have enlighten that data classification can be realized in the human visual cortex with latencies of about 100 ms, which, considering the visual pathways latencies, is only compatible with a very specific processing architecture, described by the so-called Thorpe model. Surprisingly enough, this experimental evidence is in coherence with algorithms derived from the statistical learning theory, following the work of Vapnik. More precisely, there is a double link: on one hand, the Vapnik theory offers tools to evaluate and analyze the Thorpe model performances and on the other hand, this model is an interesting front-end for algorithms derived from the Vapnik theory. The present contribution develops this idea and experiments its performances using a tiny sign language recognition experiment.

Biological motion recognition refers to our ability to recognize a scene (motion or movement) based on the evolution of a limited number of points acquired for instance with a motion capture tool. Much work has been done in this direction showing how it is possible to recognize actions based on these points. Following the reference work of Giese and Poggio, we propose an approach to extract such points from a video based on spiking neural networks with rank order coding. Using this estimated set of points, we verify that correct biological motion classification can be perfomed. We use some recent results of Thorpe et al. who claim that the neural information is coded by the relative order in which these neurons fire. This allows to select a limited set of relevant points to be used in the motion classification. Several experiments and comparisons with previous neurological work and models are proposed.
The result of these simulations show that information from early visual processes appears to be sufficient to classify biological motion.

key-words. Neuronal classifier; Supervised learning; Vapnik dimension.

Co-authors. Sylvie Crahay, Yvan Dimov, Pierre Kornprobst, Simon Thorpe.
Links.
Biologically plausible trajectory generators.
Considering the biological or artificial control of a trajectory generation, we propose a biologically plausible model based on harmonic potentials.

Such methods assume that obstacles to avoid (or constraints not to violate) correspond to maxima of the potential, while the goal corresponds to a unique minimum. The corresponding algorithm thus behaves as if one throws a sheet onto this state space, this hyper-surface relief being elevated on obstacles, with a hole at the goal location, so that finding a trajectory reduces to ``roll down'' along this relief towards the minimal height location.

The originality of the present work is to build an harmonic potential (thus without local minimum) as a finite linear combination of elementary harmonic functions. The set of these components samples the border of the admissible domain bounded by obstacles or constraints.

This leads to an internal representation of the problem as a non-topographical map incrementally built during the system exploration and non-linearly linked to the real problem geometry. As such, it provides a biologically plausible quantitative model of some hippocampus mechanisms and of the related cognitive maps, in coherence with usual biological assumptions about such behavior.

key-words. Trajectory generation; Harmonic potential; Biological model.

Co-authors. Cécile Vadot.
Links.
Multi-Model Parametric Estimation.
This work is realized within the scope of a general attempt to understand parametric adaptation, regarding visual perception. The key idea is to analyze how we may use multi-model parametric estimation as a 1st step towards categorization. More generally, the goal is to formalize how the notion of ``objects'' or ``events'' in an application may be reduced to a choice in a hierarchy of parametric models used to estimate the underlying data categorization. These mechanisms are to be linked with what occurs in the cerebral cortex where object recognition corresponds to a parametric neuronal estimation.

We hope to bring here an algorithmic element in relation with the ``grand-ma'' neuron modelization. We thus revisit the problem of parameter estimation in computer vision, presented here as a simple optimization problem, considering (i) non-linear implicit measurement equations and parameter constraints, plus (ii) robust estimation in the presence of outliers and (iii) multi-model comparisons.

Here, (1) a projection algorithm based on generalizations of square-root decompositions allows an efficient and numerically stable local resolution of a set of non-linear equations. On the other hand, (2) a robust estimation module of a hierarchy of non-linear models has been designed and validated. A step ahead, the software architecture of the estimation module is discussed with the goal of being integrated in reactive software environments or within applications with time constraints.

key-words. Non-linear estimation; Robust estimation; Multi-model estimation.

Co-authors. François Gaspard, Diane Lingrand.
Links.