Numerous macroscale models of arteries have been developed, comprised of populations of discrete coupled Endothelial Cells (EC) and Smooth Muscle Cells (SMC) cells, an example of which is the model of Shaikh et al. (2012), which simulates the complex biochemical processes responsible for the observed propagating waves of Ca2+ observed in experiments. In a 'homogenised' model however, the length scale of each cell is assumed infinitely small while the population of cells are assumed infinitely large, so that the microscopic spatial dynamics of individual cells are unaccounted for.

We wish to show in our study, our hypothesis that the homogenised modelling approach for a particular system can be used to replicate observations of the discrete modelling approach for the same system. We may do this by deriving a homogenised model based on Goldbeter et al. (1990), the simplest possible physiological system, and comparing its results with those of the discrete Shaikh et al. (2012), which have already been validated with experimental findings. We will then analyse the mathematical dynamics of our homogenised model to gain a better understanding of how its system parameters influence the behaviour of its solutions. All our homogenised models are essentially formulated as partial differential equations (PDE), specifically they are of type reaction diffusion PDEs. Therefore before we begin developing the homogenised Goldbeter et al. (1990), we will first analyse the Brusselator PDE with the goal that it will help us to understand reaction diffusion systems better. The Brusselator is a suitable preliminary study as it shares two common properties with reaction diffusion equations: oscillatory solutions and a diffusion term.