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Solitary Waves in a Model of Dendritic Cable with Active Spines

We consider a continuum model of dendritic spines with active membrane dynamics uniformly distributed along a passive dendritic cable. By considering a systematic reduction of the Hodgkin--Huxley dynamics that is valid on all but very short time-scales we derive two-dimensional and one-dimensional systems for excitable tissue, both of which may be used to model the activeprocesses in spine-heads. In the first case the coupling of the spine-head dynamics to a passive dendritic cable via a direct electrical connection yields a model that may be regarded as a simplification of the Baer and Rinzel cable theory of excitable spiny nerve tissue [J. Neurophysiology, 65 (1991), pp. 874--890]. This model is computationally simple with few free parameters. Importantly, as in the full model, numerical simulation illustrates the possibility of a traveling wave. We present a systematic numerical investigation of the speed and stability of the wave as a function of physiologically important parameters. A further reducti...