Search

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, Séries 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. [ABSTRACT FROM AUTHOR]

We study a problem set in a finely mixed periodic medium, modelling electrical conduction in biological tissues. The unknown electric potential solves standard elliptic equations set in different conductive regions (the intracellular and extracellular spaces), separated by a dielectric surface (the cell membranes), which exhibits both a capacitive and a nonlinear conductive behaviour. Accordingly, dynamical conditions prevail on the membranes, so that the dependence of the solution on the time variable t is not only of parametric character. As the spatial period of the medium goes to zero, the electric potential approaches in a suitable sense a homogenization limit u0, which keeps the prescribed boundary data, and solves the equation ${\rm div}\, [B^0\nabla_x u_0+\int_0^t A^1(t-\tau)\nabla_x u_0(\tau)d\tau-\mathcal{F}]=0$. This is an elliptic equation containing a term depending on the history of the gradient of u0; the matrices B0, A1 in it depend on the microstructure of the medium. More exactly, we have that, in the limit, the current is still divergence-free, but it depends on the history of the potential gradient, so that memory effects explicitly appear. The limiting equation also contains a term ℱ keeping trace of the initial data. [ABSTRACT FROM AUTHOR]

We study an evolutive model for electrical conduction in biological tissues, where the conductive intra-cellular and extracellular spaces are separated by insulating cell membranes. The mathematical scheme is an elliptic problem, with dynamical boundary conditions on the cell membranes. The problem is set in a finely mixed periodic medium. We show that the homogenization limit u0 of the electric potential, obtained as the period of the microscopic structure approaches zero, solves the equation −div(σ0∇xu0+A0∇xu0+∫0tA1(t−τ)∇xu0(x,τ) dτ−F(x,t))=0 where σ0>0 and the matrices A0, A1 depend on geometric and material properties, while the vector function F keeps trace of the initial data of the original problem. Memory effects explicitly appear here, making this elliptic equation of non standard type. To cite this article: M. Amar et al., C. R. Mecanique 331 (2003). [Copyright &y& Elsevier]