Multiscale models of neural tissue

The brain is a complex system which contains a large number of neurons. This complex nature of the brain can be simplified by mathematical models. Mathematical models are almost equally as important as biological experiments and represent some part of the real world by mathematical terms. They help us to understand mathematical neuroscience. The functional dynamics of brain and the behaviour of neurons have become popular recently. Hence, mathematical neuroscience becomes an attractive prospective to study because of a large availability of biological and computation data. In the first part of the thesis, we outline a literature review covering background material for this thesis. Neuroimaging techniques are used to understand the structure and dynamics of the brain. Then, we analyse the Liley and Janset-Rit model as the neural mass models. After a brief and selective description of neurobiology, our study demonstrates the relationship between functional connectivity (FC) and structural connectivity (SC) via using Wilson-Cowan model. The bifurcation diagram of Wilson-Cowan model is plotted in XPP package programme. While SC is calculated from CoComac database which are taken from the macaque brain, FC is calculated by correlation. Then, the relationship between FC and SC is measured by the Pearson and Spearman correlation. A greater similarity is seen along the border of the Hopf bifurcation. Hence, phase response curves are calculated by the adjoint method, and then phase interaction functions are produced in Matlab in order to examine the stability of the synchronised solutions. Following this, one question of interest is the relationship between epileptic discharges as measured by EEG and the haemodynamic response as measured by fMRI. The haemodynamic model is discussed and its outputs as BOLD signals are computed. Then, the haemodynamic model is coupled to the Wilson-Cowan model and the Liley model. Eventually the BOLD signals are calculated after coupling. The system has BOLD signal dynamics including an initial dip, positive response and poststimulus undershoot