  Progress toward characterization of structural and biophysical properties of
neural dendrites together with recent findings emphasizing their role in neural
computation, has propelled growing interest in refining existing theoretical
models of electrical propagation in dendrites while advocating novel analytic
tools. In this paper we focus on the cable equation describing electric
propagation in dendrites with different geometry. When the geometry is
cylindrical we show that the cable equation is invariant under the
Schr\"odinger group and by using the dendrite parameters, a representation of
the Schr\"odinger algebra is provided. Furthermore, when the geometry profile
is parabolic we show that the cable equation is equivalent to the Schr\"odinger
equation for the 1-dimensional free particle, which is invariant under the
Schr\"odinger group. Moreover, we show that there is a family of dendrite
geometries for which the cable equation is equivalent to the Schr\"odinger
equation for the 1-dimensional conformal quantum mechanics.
