A Population Model of the Thalamus and its Wave Interactions with the Cortex

In this thesis we construct a population model of the thalamus based on the mean-field cortex developed by Steyn-Ross et al. and Liley et al. We allow interactions between the thalamus and cortex via damped wave equations and show spindle-like oscillations (10-15 Hz) propagating from the thalamic reticular population into the cortical populations. We first consider a mean-field model of a cortex that is isolated from time-dependent sub-cortical inputs. The model is a continuum theory based on the electrical activity of a neural mass called the macrocolumn, containing groups of excitatory and inhibitory neurons. We demonstrate induction of 'unconsciousness' in the macrocoulmn under propofol-like general anaesthetic, and show there is a hysteretic separation between points of 'loss of consciousness' and 'recovery of consciousness'. The thalamus, a sub-cortical structure important to the relay of sensory input into the cortex, is described using a revised set of equations based on the mean-field cortical model. We define two neuron types within the thalamus: the specific group, assumed to be excitatory, and the reticular group, assumed to be inhibitory. A new bimodal mapping function is developed to relate mean membrane potentials to mean firing rates within the thalamic macrocolumn. This bimodal function is generated by combining two sigmoidal functions, and is a representation of the observed increase in the firing rates of reticular neurons at hyperpolarized membrane potentials. In contrast, the mapping function for specific population is modelled by a standard sigmoid. We investigate the isolated dynamics of the mean-field thalamus, using parameters from Robinson et al., and find spindle-like oscillations emerge following a transition from unstable equilibrium states to stable states consistent with the linear stability analysis. A coupling of the isolated cortical and thalamic systems is accomplished using four damped wave equations. We investigate the stationary states of the thalamo-cortical model and find the spindle-like oscillations present in the thalamus propagate into the cortical populations