Traveling waves in the Baer and Rinzel model of spine studded dendritic tissue

The Baer and Rinzel model of dendritic spines uniformly distributed along a dendritic cable is shown to admit a variety of regular traveling wave solutions including solitary pulses, multiple pulses and periodic waves. We investigate numerically the speed of these waves and their propagation failure as functions of the system parameters by numerical continuation. Multiple pulse waves are shown to occur close to the primary pulse, except in certain exceptional regions of parameter space, which we identify. The propagation failure of solitary and multiple pulse waves is shown to be associated with the destruction of a saddle-node bifurcation of periodic orbits. The system also supports many types of irregular wave trains. These include waves which may be regarded as connections to periodics and bursting patterns in which pulses can cluster together in well-defined packets. The behavior and properties of both these irregular spike-trains is explained within a kinematic framework that is based on the times of wave pulses. The dispersion curve for periodic waves is important for such a description and is obtained in a straightforward manner using the numerical scheme developed for the study of the speed of a periodic wave. Stability of periodic waves within the kinematic theory is given in terms of the derivative of the dispersion curve and provides a weak form of stability that may be applied to solutions of the traveling wave equations. The kinematic theory correctly predicts the conditions for period doubling bifurcations and the generation of bursting states. Moreover, it also accurately describes the shape and speed of the traveling front that connects waves with two different periods.