## Section: Scientific Foundations

Keywords : dynamic systems, stochastic calculus, neuron models, neuronal networks.

### Computational Neurosciences

- optical imaging
A usually invasive technique that allows to display a visual correlate of the activity of the cortex. One distinguishes intrinsic and extrinsic optical imaging.

Understanding the principles of information processing in the brain is challenging, theoretically and experimentally. Computational neuroscience attempts to build models of neurons at a variety of levels, microscopic, i.e. 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 mathematics as dynamic systems theory, bifurcation theory, probability theory, stochastic calculus, and statistics, as well as the use of simulation tools [4]

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://www.unic.cnrs-gif.fr/ or the INCM http://www.incm.cnrs-mrs.fr/ . Other sources of measurements such as functional MRI, MEG and EEG, see 3.3 below, are either obtained within Odyssée as it is the case for EEG, or from other collaborative efforts such as the one with La Timone, see 2.2 above, or Neurospin.

The Odyssee team works at the three levels. We have proposed two realistic models of single neurons [3] , [15] by making use of physiological data and the theory of dynamic systems and bifurcations. At this level of analysis we have also proposed a variety of theoretical tools from the theory of stochastic calculus [14] 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 [13] . We have also provided a mathematical analysis, through bifurcation theory, of the behaviour of a particular mesoscopic model [8] , the one due to Jansen and Rit [102] .

We have also started some efforts at the macroscopic level, in particular for modeling visual areas, see 3.4 below. For this particular level, information about the anatomical connectivity such as the one provided by diffusion imaging techniques is of fundamental importance, see 3.2 below.