Neuronal activity evokes a localised increase in cerebral blood flow (CBF) in a response known as neurovascular coupling (NVC). This response is achieved through communication between neurons, astrocytes and vascular cells, together comprising a neurovascular unit (NVU). Impaired NVC is associated with several pathological conditions such as Alzheimer’s disease, stroke, and cortical spreading depression (CSD). Although NVC has been widely studied the inner workings and signalling pathways of the brain have yet to be fully elucidated. Mathematical models provide an alternative and complementary method of research, working in tandem with physical experiments and providing a deeper understanding of experimental results as well as guiding future experiments.

This thesis details the extension of a foundational biophysical model of a single NVU with multiple new pathways and compartments allowing for the simulation of both normal and impaired NVC. The single NVU model was first extended with the nitric oxide (NO) pathway, the glutamate induced astrocytic calcium (Ca²⁺) pathway with epoxyeicosatrienoic acid (EET) signalling, and the stretch dependent transient receptor potential vanniloid-related 4 (TRPV4) Ca²⁺ channel on the astrocytic endfoot. It was found that the potassium (K⁺) pathway governs the fast onset of vasodilation while the NO pathway has a delayed response. Increases in astrocytic Ca²⁺ concentration to levels consistent with experimental data were insufficient for inducing either vasodilation or constriction, in contrast to a number of experimental results. However astrocytic Ca²⁺ was shown to strengthen K⁺ induced NVC by further opening the big potassium (BK) channel on the astrocytic endfoot.

The model was then extended with a complex neuron submodel and the blood-oxygen-level dependent (BOLD) response, allowing for model validation through comparison with experimental data. The change in CBF due to neural activity in the model showed good agreement with experimental data obtained from the rat barrel cortex. When using an experimental neural input profile, the model compared well for short stimuli but was unable to replicate the double maximum of the experimental CBF profile for longer periods. An additional pathway through the locus coeruleus (LC) (on a purely phenomenological basis) provided a better comparison with experimental data, further showing that there exist numerous signalling pathways in the NVC process.

The NVU model was then embedded in a large two dimensional (2D) cerebral tissue slice model with a coupled vascular tree, solved in a parallel environment using high performance computing. The tissue slice model was first extended with extracellular Fickian K⁺ diffusion, allowing for direct communication between adjacent NVUs. It was found that a localised neuronal stimulation resulted in vasodilation with a decreasing gradient in vessel radius from the stimulated to non-stimulated area. During vasomotion there was emergent behaviour in the form of waves of increased vessel radius moving towards the stimulated area. This indicates that communication within the tissue via extracellular ion diffusion has a much greater effect than communication solely through the vascular tree.

Extracellular ion electrodiffusion and an astrocytic gap junction network were added to allow further communication throughout the tissue. It was found that under pathological CSD conditions, K⁺ waves could propagate radially outwards from a stimulated area with a wave of vasoconstriction followed by slight vasodilation comparing well with murine experiments. Extracellular Fickian diffusion was insufficient for allowing a wave to propagate. An astrocytic gap junction network reduced the duration of the vasoconstrictive wave and the area initially affected. This indicates that communication throughout the extracellular space (ECS) is necessary in the model for allowing a CSD wave to propagate, whereas the communication through the astrocytic gap junction network regulates the vascular response.

The flat 2D tissue slice model could simulate the smooth cortex of murine animals but could not take into account the highly folded nature of the human cortex. The model was extended with a spatial Gaussian curvature mapping which allowed for investigation into how the surface curvature affects the propagation of CSD waves throughout the tissue. It was found that for a surface with spatially varied curvature comparable to a section of human cortex, areas of positive Gaussian curvature inhibited wave propagation due to decreased extracellular diffusion rate, whereas areas of negative curvature promoted propagation. CSD was observed travelling as wave segments (as opposed to radial waves on a flat surface), providing some insight into the differences seen between human and animal experiments.