Calcium dynamics and wave propagation in coupled cells.

Intercellular waves of calcium (Ca2+) are an important signalling mechanism in a wide variety of cells within the body, crucial for cellular coordination and control. In particular the Ca2+ concentration within smooth muscle cells (SMCs) lining the blood vessel walls controls the cell dilation and contraction and thus the vessel radius. The process of functional hyperaemia by which neuronal activity results in a localised response of increased blood ow via the dilation of SMCs is associated with multiple pathologies such as cortical spreading depression (CSD). This process can be modelled by a `neurovascular unit (NVU)' containing a neuron, astrocyte, and the SMC and endothelial cell (EC) within the vessel wall. Our research consists of modelling the Ca2+ dynamics of a both a single SMC and two coupled SMCs (via an intercellular Ca2+ ux) mainly with the minimal nonspatial Goldbeter et al. (1990) cell model. This is compared with the more complex model of a SMC/EC `unit' which also includes the in uence of neuronal stimulation on the SMC. The Ca2+ dynamics of both models are found to be similar in structure: the system will be either excitable, nonexcitable or oscillatory depending on a model dependent parameter controlling the rate of inotisol trisphosphate (IP3) induced Ca2+ release into the cell. However the SMC/EC model also produces small amplitude oscillations and bistability when neuronal stimulation is high and the model parameter is low. The behaviour of a coupled cell system is seemingly model independent: in particular an excitable coupled with an oscillatory or two nonidentical coupled oscillatory cells will exhibit qualitatively di erent behaviour when weakly coupled such as variable amplitude oscillations. The formation and propagation of Ca2+ waves are simulated by the Goldbeter et al. (1990) model in a two dimensional (2D) spatial medium; spatial curvature is then introduced by simulating the model on a torus. When the local dynamics of the medium are spatially constant a new wave solution in the form of a stable wave segment when there is some gradient in Gaussian curvature. When the local dynamics of the medium are spatially varied, spiral waves or apparent spatiotemporal chaos are produced when the rate of di usion is low and either the surface is strongly curved or the initial conditions (ICs) of the medium are su ciently inhomogeneous. Based on the similarities in the nonspatial results the spatial Goldbeter et al. (1990) model could provide insight into the behaviour of the corresponding complex spatial SMC/EC model