Mesoscopic modeling of cortical columns, bifurcations, and imaging 


Brain Imaging techniques such as Optical Imaging or MEGEEG require a precise modeling of the cortex electrical activity at space scales smaller than 1 mm^{2} of cortical matter. It is therefore useful to propose models of this activity, than can be analysed mathematically. In this spirit cortical columns models have been proposed by several authors [Lopesetal, 1976], [JansenRit, 1995]. To model cortical activity at larger scales it is then necessary to consider spatially extended models, like neural masses models [Foliasetal:05, Coombes:05, Blomquist:05]. The mathematical properties of these models can then be analysed using bifurcations techniques [ErmentroutCowan:79a,b, Bressloffetal:01a,01b,03, GrimbertFaugeras; 06].
However, these models are not obtained from the dynamics at a "microscopic" scale (neurons), but instead via a phenomenological approach describing directly the local field at a mesoscopic scale. Thus, in most approaches, equations are obtained by a description of neural population i in terms of an average potential V_{i} and an average frequency rate n_{i}. Moreover, one postulates a relation n_{i }= S_{i} (V_{i}) (where S_{i} is a sigmoidal function) which holds for neurons (microscopic scale). Therefore, one applies at the mesoscopic scale a relation only valid at a microscopic scale. This relies on questionable assumptions, neglecting for example the local fields fluctuations, which may have an important impact on dynamics. As a matter of fact, a natural question is how to generate the phenomenological equations for the mesoscopic dynamics of cortical columns via microscopic dynamics. Methods from statistical physics allows to do this and to make an analysis of these equations.
In this spirit we are developing dynamic meanfield methods to characterize neural networks models describing the activity of P neuronal populations in cortical columns, including non stationary dynamics and field fluctuations (see section "Modeling cortical columns dynamics"). A primary goal is then to be able to characterize mathematically or numerically the dynamical regimes exhibited by a realistic model of a cortical column model, with several neural populations, and with a connectivity scheme based on anatomic data [ThomsonLamy, 07]. This study is the first step toward the characterization of local field potentials at a scale corresponding precisely to the resolution of optical imaging or MEG. Taking into account the fluctuations and correlations of local field should bring, as we checked on simple examples, a richer phenomenology. This requires a detailed analysis of the model dynamics. This last point is addressed by a bifurcation analysis of meanfield equations. Especially, techniques analysing the effects of symmetries and symmetries breaking are currently under investigations. Indeed, because of the regular disposition of cortical columns, a mathematically fruitful simplification assumes invariance properties such permutations between neurons or groups of neurons. This allows to give to the model a structure simplifying its analysis (see section "Bifurcation analysis of cortical columns dynamics.").
In parallel with this mathematical analysis we have developed a new numerical method where one iterates a map acting on bounded times trajectories of the law of the mean field process. This method allows to access non stationary regimes corresponding for example to having time dependent inputs.
The ultimate goal is to compare the dynamical behaviour predicted by our analysis to real experimental data to better understand the link between what is measured and the underlying cortical activity. Thus, to validate part of our theoretical and numerical results, we want to compare them to the cortical activity of the visual system (V1/V2) measured with MEG and intrinsic optical imaging. The MEG is done in collaboration with M. Clerc et T. Papadopoulo (ODYSSEE). The optical imaging part is done in collaboration with DyVa team in INCM, as a continuation of F. Chavanne (INCM) and F. Grimbert (ODYSSEE) work [Grimbertetal, 07, GrimbertChavanne, 07] (see section "Measuring cortical columns dynamics.").
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