Homogenization and Concentration of Capacity in the Rod Outer Segment with Incisures

We present a quantitative model of the spatio-temporal dynamics of second messengers mediating phototransduction in retinal rods. The spatial domain (the rod outer segment) has a quite complex geometry, involving different ‘‘thin’’ domains, whose thickness is three orders of magnitude smaller than the other dimensions. The model relies on a ‘‘pointwise’’ application of first principles leading to a system of evolution equations set in such a structured geometry. Then, exploiting an idea first presented in [Andreucci, D., Bisegna, P. and DiBenedetto, E., 2002, Homogenization and concentrated capacity in reticular almost disconnected structures. Comptes Rendus Mathematique. Academie des Sciences. Paris, Se´ries I, 335, 329–332], the diffusion problem is reduced to one with a simpler geometry, still preserving the essential features of the original one. This is achieved by an homogenization and concentration limit. However, here we take into account for the first time the presence of ‘‘incisures’’, which are important for phototransduction, and introduce new mathematical features mainly in the concentration limit.