I have therefore several different activities, the common denominators being signal and physics.

My main concerns today are :

- the study of networks of nonlinear systems, with main application to the neural system

- the possible positive influence of noise in nonlinear processings, with again as a main concern the neural system

- the theory of signals for complex systems (self-similarity and broken version, generalized stationarities)

- application of signal processing in physics

Networks of nonlinear systems

This part of my activities is at itS beginning from a theoretical point of view. But i'm collaborationg since 2006 with Catherine Villard, a physicist from Institut Néel,

Grenoble, France, on a project which aims at the study of real neural networks which geometry is controlled. Furthermore, we want to perfectly couple the neurons with

a measuring device called MicroElectrode Array. Ultimatly this will provide us an experimental apparatus to study the function of structured networks (see a poster presented at the 2006 MEA meeting).

At the moment, we validate the experiment in the study of small networks of three coupled neurons (see a poster presented in december 2007 at the TAUC symposium on neuroscience)! More information may be found in conference papers [62,69].

I came to this topic after having studying noise-enhaced processing in some types of networks that mimicneural networks. This is described in the following section. Now, I want to use these models to investigate information flows in neural networks.

Noise-enhanced processing

I've begun to work on this subject by the study of stochastic resonance. I developped with Steeve Zozor the theory of stochastic resonance in discrete time systems.

Stochastic resonance has a very precise meaning : it corresponds to a synchronisation between two time scales that may co-exist in a nonlinear system attacked by a periodic signal (first time scale) and a noise (second time scale for some nonlinear systems, e.g. quartic potential). The second time scale typically correspond to a mean escape time. If we study the output of the nonlinear system, there is a preferred noise power for which the periodic signal will be maximally seen. Thus, if we plot for example the signal-to-noise ratio at the first harmonic in the output as a function of the input noise power, we obtain a bell shaped curve, indicating that here exists an optimal noise power for the transmission of the periodical signal.

Steeve applied this to the problem of detecting a sine wave in noise. If noise is not Gaussian, he designed ad-hoc networks of nonlinearities, each processing the input with different parameters, mimicking therefore stochastic resonance.A max operator selects the best noninearity as the best detector. For Gaussian noise, the procedure approaches the optimality of matched filtering, and for other type of noise, performance may almost perfect (square ROC curves). In particular case, the detection procedure can be termed noise-enhanced detection and use explicitly the concept of stochastic resonance. But there are other instances of noise enhanced detection that we have studied.

You can have more information by reading a few of our papers, e.g. published papers [9,14,16,21]

From an engineering point of view, noise-enhanced processing seems to be a no sense. Indeed adding noise to a signals to improve its processing seems very strange. However, there are situations in which noise is inherent in the system that process some system. This is the case of the brain! It it well recognize now that neurons are noisy. Indeed, the basic processes for the coupling between neurons (synaptic couplings) are stochastic by nature. When looking at the response of a neuron (or a neural network) to some sitmulus, it is striking to see the stochastic variability of this response. I have then turned to the study of noise-enhanced processing in the living system, by working on very simple models. For example, with Steeve Zozor, we have study on very simple models of the retina if microsaccads (micro- erratic movements of the eyes) do provide a noise-enhanced processing (see submitted papers [2], conference papers [70,71]). My main other concern at the moment is the study of pooling networks, networks of neurons that process a same information in parallel, the output of each being pooled to create the output of the networks. In these networks, the neurons are noisy. Therefore, each neuron can be view as highly non reliable processor, but the pooling creates a reliable output. These kind of neural networks has been shown to provide almost optimal processing of information (see Ma et. al., Nature Neuroscience, 2006, 9, 1432-1438). I work with M. McDonnell from The University of South Australia and N. Stocks from Warwick University on different aspects of these networks. Information may be found in conference papers [60,63,68].

Signals for complex systems

There is no well accepted definition of a comlex system, but some basic ingredients are necessary for a system to be qualified as complex. The system should be composed of many components in non trivial interaction, it may exchange with the outside (energy or information). Typical complex systems are as different as a sandpile or the www. When measuring some variables in complex system, we are often faced with signals that do not follow the usual good properties of a signal: stationarity, mixing, Gaussianity. Often, one of the preceding property is violated. If mixing is not present, then we are in front with long range dependent signals. If Gaussianity is not present, we may have signals with heavy tailed pdf. In both case, we are often faced with power laws. One of the more striking feature of a pure power law is the absence of any caracteristic scale. This property is inherited from self-similarity. A stochastic signal is self-similar if is equal to a time dilated version of itself up to some amplitude renormalization. Equality here has to be understood in the probability distribution sense. Mathematically, self-similarity is well defined and well documented now. However, from a physical perspective, strict self-similarity cannot exists, because at least of the existence of cutoffs in the frequency domain. We say that the symmetry obeyed by the signal (self-similarity) is broken.

I've worked on this subject (broken symmetries) in collaboration with P. Flandrin and P. Borgnat, from ENSLyon. We have studied several subjects, from Discrete Scale Invariance (the self-similarity is true only for a discrete set of ratio) to generalized self-similarity using general groups (however isomorphic to the reals with the addition). Information may be found in published papers [18,23]. I also work on the use of random Iterated Function System for the design of fractal random signals, see [submitted 3, conference 67,72].

Application of Signal Processing in Physics

I have worked on many different applications of signal processing in physics (see published papers [10,12,13,19,25,28,30,31,32]). In the more recent works, i collaborate with a group of physicists on the study of the Extreme Ultra Violet solar spectrum. We especially work with the data collected by SEE on-board TIMED, an american satellite. Each day since 2002, SEE collects the solar radiation in the EUV range (from about 20 to 200 nm) with a good resolution. After some processing (calibration, elimination of flares--solar big eruptions--), the data are in form of a spectrum, composed of the emission lines of many elements (and different configuration of each) present in the Sun. Further, the variability of the Sun is then reflected by the daily measurement of this spectrum. We have applied on these data a Bayesian approach to source separation nly imposing the constraint of positivity of the sources (indeed, they are spectra). With such a approach we are able to separate the solar EUV specrtum into the superposition of 3 spectra which are physically interpretable (see submitted paper [4]).