Solitary waves in a model of dendritic cable with active spines

We consider a continuum model of dendritic spines with active membrane dyna mics 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 sy stems for excitable tissue, both of which may be used to model the active p rocesses in spine-heads. In the rst case the coupling of the spine-head dyn amics to a passive dendritic cable via a direct electrical connection yield s a model that may be regarded as a simplification of the Baer and Rinzel c able theory of excitable spiny nerve tissue [J. Neurophysiology, 65 (1991), pp. 874-890]. This model is computationally simple with few free parameter s. Importantly, as in the full model, numerical simulation illustrates the possibility of a traveling wave. We present a systematic numerical investig ation of the speed and stability of the wave as a function of physiological ly important parameters. A further reduction of this model suggests that ac tive spine-head dynamics may be modeled by an all-or-none type process whic h we take to be of the integrate-and-fire (IF) type. The model is analytica lly tractable allowing the explicit construction of the shape of traveling waves as well as the calculation of wave speed as a function of system para meters. In general a slow and fast wave ar found to coexist. The behavior o f the fast wave is found to closely reproduce the behavior of th wave seen in simulations of the more detailed model. Importantly a linear stability t heory is presented showing that it is the faster of the two solutions that are stable. Beyond a critical value the speed of the stable wave is found t o decrease as a function of spine density. Moreover, the speed of this wave is found to decrease as a function of the strength of the electrical resis tor coupling the spine-head and the cable, such that beyond some critical v alue there is propagation failure. Finally, we discuss the importance of a model of passive electrical cable coupled to a system of IF units for physi ological studies of branching dendritic tissue with active spines.