Summary: 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 wave front. The author studies 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.
He analyses 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 two active sites and exact stationary solutions, respectively. He then explores 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, he extends the study to more detailed models including the reaction kinetics of signalling, and demonstrates the same features of wave propagation. He discusses how such waves may initiate pattern formation, and the role of such mechanisms in morphogenesis.