Solitary waves in a model of dendritic cable with active spines
File(s)
Published version
Author(s)
Bressloff, PC
Coombes, S
Type
Journal Article
Abstract
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 activeprocesses 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 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 are found to coexist. 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 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.
Date Issued
2000-01
Date Acceptance
2000-01-28
Citation
SIAM Journal on Applied Mathematics, 2000, 61 (2), pp.432-453
ISSN
0036-1399
Publisher
Society for Industrial and Applied Mathematics
Start Page
432
End Page
453
Journal / Book Title
SIAM Journal on Applied Mathematics
Volume
61
Issue
2
Copyright Statement
Copyright © 2000 Society for Industrial and Applied Mathematics.
Identifier
http://dx.doi.org/10.1137/s0036139999356600
Publication Status
Published
Date Publish Online
2000-08-03