On the electro-sensing of weakly electric fish

In this paper we are interested in electro-sensing inverse problems. Our objective is to understand the electro-perception mechanism of weakly electric fish. These species of fish have the ability to recognize the environment around them in complete darkness by generating a weak electrical field at different frequencies, and perceiving the transdermal potential perturbation. Assuming that the target has a known conductivity profile, the electro-sensing inverse problem consists in recovering the shape and location of the target from measurements of the electric potential over the skin. Using an original spectral decomposition of the solution to the direct problem in terms of Poincaré variational eigenfunctions, we retrieve the Cauchy data of the electric potential over the fish skin corresponding to the case where we substitute the target by a perfect conductor with the same shape and position. We then identify the target from the recovered Cauchy data. We derive uniqueness and stability estimates to the considered electro-sensing inverse problem. The numerical validation of our theoretical approach is realized by reconstructing different targets using synthetic data in dimension two. The numerical experiments are conducted using gradient descent optimization algorithms.