Heriot-Watt Mathematics Report Series
HWM01-7, 12 Mar 2001
Travelling waves in the Baer and Rinzel model of spine studded dendritic tissue
G J Lord and S Coombes
Abstract
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 \textit{connections} to
periodics and \textit{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.
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Publication
G J Lord, S Coombes
, Travelling waves in the Baer and Rinzel model of spine studded dendritic tissue, Physica D, 161, 1-20, (2002).
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