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 active processes in spine-heads. In the rst 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 reduction of this model suggests that active spine-head dynamics may be modeled by an all-or-none type process which we take to be of the integrate-and-fire (IF) type. The model is analytically tractable allowing the explicit construction of the shape of traveling waves as well as the calculation of wave speed as a function of system parameters. In general a slow and fast wave ar found to coexist. The behavior of 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 theory 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 to 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 resistor coupling the spine-head and the cable, such that beyond some critical value there is propagation failure. Finally, we discuss the importance of a model of passive electrical cable coupled to a system of IF units for physiological studies of branching dendritic tissue with active spines.