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|Title: ||Waves and propagation failure in discrete space models with nonlinear coupling and feedback|
|Authors: ||Owen, Markus R.|
|Issue Date: ||2002|
|Abstract: ||Many developmental processes involve a wave of initiation of pattern formation,
behind which a uniform layer of discrete cells develops a regular pattern that determines
cell fates. This paper focuses on the initiation of such waves, and then on
the emergence of patterns behind the wavefront. I study waves in discrete space
differential equation models where the coupling between sites is nonlinear. Such
systems represent juxtacrine cell signalling, where cells communicate via membrane
bound molecules binding to their receptors. In this way, the signal at cell j is a
nonlinear function of the average signal on neighbouring cells. Whilst considerable
progress has been made in the analysis of discrete reaction-diffusion systems, this
paper presents a novel and detailed study of waves in juxtacrine systems.
I analyse travelling wave solutions in such systems with a single variable representing
activity in each cell. When there is a single stable homogeneous steady
state, the wave speed is governed by the linearisation ahead of the wave front. Wave
propagation (and failure) is studied when the homogeneous dynamics are bistable.
Simulations show that waves may propagate in either direction, or may be pinned.
A Lyapunov function is used to determine the direction of propagation of travelling
waves. Pinning is studied by calculating the boundaries for propagation failure for
sigmoidal and piecewise linear feedback functions, using analysis of 2 active sites and
exact stationary solutions respectively. I then explore the calculation of travelling
waves as the solution of an associated n-dimensional boundary value problem posed
on [0, 1], using continuation to determine the dependence of speed on model parameters.
This method is shown to be very accurate, by comparison with numerical
simulations. Furthermore, the method is also applicable to other discrete systems
on a regular lattice, such as the discrete bistable reaction-diffusion equation.
Finally, I extend the study to more detailed models including the reaction kinetics
of signalling, and demonstrate the same features of wave propagation. I discuss how
such waves may initiate pattern formation, and the role of such mechanisms in
|Description: ||This pre-print has been submitted, and accepted, to the journal, Physica D - Nonlinear Phenomena [© Elsevier]. The definitive version: OWEN, M.R., 2002. Waves and propagation failure in discrete space models with nonlinear coupling and feedback. Physica D - Nonlinear Phenomena,173(1-2), pp. 59-76, is available at: http://www.sciencedirect.com/science/journal/01672789.|
|Appears in Collections:||Pre-prints (Maths)|
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