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Bifurcations analysis by symmetries breaking.

Experimental studies have shown that image representation in the visual cortex involves groups of neurons (« orientation » cortical columns) sensitive to image characteristics such as orientation and spatial frequency. The activity of these columns can be measured by optical imaging or MEG. Neural masses models are often used to mimic the cortex activity at this scale of resolution. Assuming that orientation columns are coupled with symmetries allows some simplifications in the mathematical description of neural masses models dynamics (Bressloff & Cowan [1], Petitot [2]). Thus, when sensitivity to orientation is taken into account it is represented by an angle. Another angle (between [0,pi]) can model the spatial frequency. Therefore, these two parameters define a point on the S² sphere. This leads to consider dynamical equations invariant under rotations on this sphere (symmetries from the rotation group SO(3)). Using a model restricted to columns only sensitive to orientation (but not to spatial frequency), Bressloff et Cowan, associated with M. Golubitsky have been able to describe visual hallucinations observed under the influence of drugs like LSD. This analysis has been made using bifurcations methods and spontaneous symmetries breaking, with a very good agreement with real observations. We want to extend this analysis to the case of columns sensitive to orientation and spatial frequency. 

[1] P.C. Bressloff, J.D. Cowan.SO(3) symmetry breaking for orientation and spatial frequency tuning in the visual cortex. Phys Rev Lett. 88, 7, p. (2002 ou 2003).
[2] J. Petitot, J. Lorenceau. Neurogeometry and Visual Perception, Special Issue of the Journal of Physiology-Paris, 97, 2-3 (2003).
[3] P.C. Bressloff, J.D. Cowan, M. Golubitsky, P.J. Thomas, M.C. Wiener. What geometric visual hallucinations tell us about the visual cortex. Neural Computation 14, 473-491 (2002).