The optimal shape of a nerve fiber

We wonder whether some shapes observed in Nature could follow from the optimization of a criterion. More precisely, we consider an organ or a part of the human body and we try to guess a criterion that Nature could have tried to optimize. Then, we solve the resulting shape optimization problem in order to compare the shape obtained by a theoretical or a numerical way with the real shape of the organ. If these two shapes are similar, it may be deduced that the criterion is relevant.
We consider in this talk the example of a nerve fiber of an axon or a dendrite kind. We propose two criterions to explain its shape. The first one stands for the attenuation throughout the time of the electrical message and the second one stands for the attenuation throughout the space of that message. In our choice of modeling, we distinguish two sorts of nerve fibers: these connected to the nucleus of the cell and these connected with two other fibers. The corresponding problems boil down to the minimization with respect to the domain of the eigenvalues of an elliptic operator and of a transfer function expressed with the trace of the electrical potential in the fiber on the boundary of the domain.