Modelling the spatial effects of the anaesthetic-induced phase-transition in the cerebral cortex

In this thesis I investigate the theoretical stochastic behaviour of a one-dimensional model of the cerebral cortex, exposed to varying concentrations of a general anaesthetic agent. The model is that of Steyn-Ross et al. (2003). Theirs is a continuum theory based on the electrical response of a neural mass known as the macrocolum. The model predicts that as anaethetic concentration is increased the cortex will undergo a sudden electrical phase transition corresponding to loss of consciousness (LOC). Similarly, at return of consciousness (ROC) a second distinct phase transition is predicted. Spatial variability is incorporated into the original homogeneous cortical model of Steyn-Ross et al. (1999). This is done by including the possibility of spatial variation in distant excitatory and inhibitory inputs. By modelling the cortex in this way, we hope to gain an understanding of how the cortex functions, and how anaethestic agents “shut-down” the brain. I simulate the one-dimensional system numerically in order to verify analytical predictions. Both analytical and numerical results show an increase in the coherence (spatial-correlation) of the electrical activity along the one-dimensional rod on approach to both LOC and ROC. Theory and simulations also show that the electrical fluctuations in the unconscious cortex should have a larger correlation length than for the cortex in the conscious state, suggesting that the unconscious state is the more ordered. I derive the theoretical power spectrum and discuss some of its properties. By expanding the model to include spatial variability, we discover the possibility of self-organized structures forming spontaneously in the one-dimensional cortex. These “Turing” or dissipative structures are stationary in time, showing giant DC voltage variations along the cortical rod. Although the dissipative structures can from a rich variety of pseudo-periodic patterns, the physiological significance of such stationary neural structures is not yet clear