In the vertebrate brain excitatory synaptic contacts typically occur on the tiny
evaginations of neuron dendritic surface known as dendritic spines. There is clear
evidence that spine heads are endowed with voltage dependent excitable channels
and that action potentials invade spines. Computational models are being increasingly
used to gain insight into the functional significance for a spine with excitable
membrane. The spike-diffuse-spike (SDS) model is one such model that admits to
a relatively straightforward mathematical analysis. In this paper we demonstrate
that not only can the SDS model support solitary travelling pulses, already observed
numerically in more detailed biophysical models, but that it has periodic travelling
wave solutions. The exact mathematical treatment of periodic travelling waves in
the SDS model is used, within a kinematic framework, to predict the existence of
connections between two periodic spike trains of different interspike interval. The
associated wave front in the sequence of interspike intervals travels with a constant
velocity without degradation of shape, and might therefore be used for robust
encoding of information.
This is a pre-print. The definitive version: COOMBES, S., 2001. From periodic travelling waves to travelling fronts in the spike-diffuse-spike model of dendritic waves. Mathematical Biosciences, 170(2), pp. 155-172.