In the context of cranial cellular modelling, mathematical techniques have been applied to model oscillations of concentration over time. Models were made for different cell types to match data and known chemical reactions within the cell over time. The ability to oscillate the concentration over time provides the necessary environment for waves of concentration to traverse across space and time. Whilst experimental procedures have advanced in recent years, there still exists a gap in knowledge that computational modelling attempts to fill.

Five Simplified Generic Oscillatory Cell Model/s (SGOCM) are described in detail: the Goldbeter model, the Dupont model, the Ermentrout model, the FitzHugh-Nagumo model and the Koenigsberger model. All of these models exhibit either steady state or oscillatory dynamics altered by a bifurcation parameter. They all exhibit periods of oscillation generally increasing alongside the bifurcation parameter towards at least one of the bifurcation points (labelled the lower bifurcation point). Each of these 5 microscale models were converted, via homogenisation, to a spatially orientated macroscale system and applied with a spatially varying stimulus. On the addition of Fickian Diffusion (FD) excursions of high concentration ‘waves’ were seen to propagate through space over time into the previously non-oscillatory low concentration region beyond the lower bifurcation point. These excursions were made the focus of this research.

A hypothesis was made indicating a link between the shape of the concentration oscillation of the variable being diffused over time (‘Wave Shape’) and its ability to produce excursions into the previously non-oscillatory region. It was theorised that a Front Heavy (FH) asymmetric profile was needed in order to produce excursions. This was further quantified via a mathematical derivation as the Front Heavy Score (FH-score) whereby a FH-score> 0 predicted excursions. This theory was then tested, and held true, for all the models and subsequently held true on three additional Toy models designed to produce specific desired outcomes. The ability to predict excursion allows for an aim to modify the single cell dynamics such that this wave movement would or would not occur in the surrounding cells.

Next, a tool was developed, known as the Excitability Profile (E-Profile) to compare across models. The E-Profile was used to view the dependence of the bifurcation parameter and an applied perturbation from the steady state value on the excitability of a system. The E-Profile was used to further understand the aforementioned models’ spatio-temporal results on the addition of diffusion. This tool was then related to the depth of excursions on the spatio-temporal solutions via an Excitability Path (E-Path). This was then used to accurately predict either the diffusion coefficient (with a standard deviation of 1:8% from the true D) given a spatio-temporal solution of a known model or to predict the depth (to ±1.1% of the macroscale length) and maximum concentration over space of the excursions from the single cell dynamics and spatial stimulus profile alone.