Appl Math Optim (2008) 57: 1–16
Shape Minimization of Dendritic Attenuation
Antoine Henrot · Yannick Privat
Published online: 1 September 2007
© Springer Science+Business Media, LLC 2007
Abstract What is the optimal shape of a dendrite? Of course, optimality refers to
some particular criterion. In this paper, we look at the case of a dendrite sealed at
one end and connected at the other end to a soma. The electrical potential in the ﬁber
follows the classical cable equations as established by W. Rall. We are interested
in the shape of the dendrite which minimizes either the attenuation in time of the
potential or the attenuation in space. In both cases, we prove that the cylindrical
shape is optimal.
Keywords Optimal shape · Cable equation · Dendrite · Eigenvalue problem
Is Nature always looking for optimum for living organisms? In particular, are the
organs designed to optimize some criterion? Complete answers to these questions
are likely never to be discovered. Nevertheless, assuming that Nature proceeds in the
most efﬁcient way, can lead to a better understanding of the modeling of an organ and
the underlying physical or chemical phenomena. This is this idea of inverse modeling
that we had in mind when we began this work. Roughly speaking, it can be described
by the following steps:
(i) Let us consider a given organ of a living body.
(ii) Write a mathematical model which describes the behavior of this organ.
(iii) Imagine a (numerical) criterion that Nature would like to optimize for this organ.
A. Henrot (
) · Y. Privat
Institut Élie Cartan de Nancy, UMR 7502 Nancy-Université-CNRS-INRIA, B.P. 239,
Vandœuvre-lès-Nancy Cedex, France