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. Byconsidering a systematic reduction of the Hodgkin-Huxleydy namics that is valid on all but very short time-scales we derive 2 dimensional and 1 dimensional systems for excitable tissue, both of which may be used to model the active processes 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 spinynerv e tissue [3]. This model is computationally simple with few free parameters. Importantly, as in the full model, numerical simulation illustrates the possibilityof 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 mayb e modeled byan 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 are found to co-exist. The behavior of the fast wave is found to closely reproduce the behavior of the 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 is 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 integrate-and-fire units for physiological studies of branching dendritic tissue with active spines.