Saltatory Waves in the Spike-Diffuse-Spike Model of Active Dendritic Spines

In this Letter we present the explicit construction of a saltatory traveling pulse of nonconstant profile in an idealized model of dendritic tissue. Excitable dendritic spine clusters, modeled with integrate-and-fire (IF) units, are connected to a passive dendritic cable at a discrete set of points. The saltatory nature of the wave is directly attributed to the breaking of translation symmetry in the cable. The conditions for propagation failure are presented as a function of cluster separation and IF threshold.

Figure 1

An example of a piece of spine studded dendritic tissue (from rat hippocampal region CA1 stratum radiatum) $\sim 5\mu \mathrm{m}$ in length. Taken with permission from Synapse Web, Boston University, http://synapses.bu.edu

Figure 2

A plot of wave speed $v$ as a function of threshold ${\Gamma }_{\rho }$, as determined by Eq. (13) for the continuum SDS model. Here $\epsilon ={\epsilon }_0={\tau }_R=D=1$. The upper (lower) branch is stable (unstable).

Figure 3

A plot of wave speed $v$ for a saltatory pulse as a function of cluster spacing $d$. Here, ${\Gamma }_n=0.05$ and $\epsilon ={\epsilon }_0={\tau }_R=D=1$. The upper (lower) branch is stable (unstable).

Figure 4

Continuation of the limit point in Fig. 3 showing the region in $(d,{\Gamma }_n)$ where stable saltatory traveling waves exist.

Figure 5

Plot of the analytically obtained saltatory solution $V(x,t)$ in the dendritic cable with parameters as in Fig. 3 and $d=1$. The $x$ axis covers 10 lattice sites and the $t$ axis $10d/v$.