In this thesis, I study the maturational changes of the fetal sheep ECoG (electrocorticogram) in its third-trimester of gestation (95-140 days of gestation), investigate three continuum models for electrical behaviour of the cortex, and tune the parameters in one of these models to generate the discontinuous EEG waves in the immature cortex. Visual inspection of the ECoG time-series shows that the third-trimester of fetal sheep is comprised of two stages: early third-trimester characterised by bursting activity separated by silent intervals, and late third-trimester with well-defined SWS (slow wave sleep) and REM (rapid eye movement) sleep states. For the late third-trimester, the results of power, correlation time, and SVD (singular value decomposition) entropy analysis demonstrate that the sleep state change is a cortical phase transition—with SWS-to-REM transition being a first-order transition, and REM-to-SWS second-order. Further analyses by correlation time, SVD entropy, and spectral edge frequency display that the differentiation of the two distinct SWS and REM sleep states occurs at about 125 dGA (day gestational age). Spectral analysis divides the third-trimester into four stages in terms of the frequency and amplitude variations of the major resonances. Spindle-like resonances only occur in the first stage. A power surge is observed immediately prior to the emergence of the two sleep states. Most significant changes of the spectrum occur during the fourth stage for both SWS (in amplitude) and REM (in frequency) sleep states. For the modelling of the immature cortex, different theoretical descriptions of cortical behaviour are investigated, including the ccf (cortical column field) model of J. J. Wright, and the Waikato cortical model. For the ccf model at centimetric scale, the time-series, fluctuation power, power law relation, gamma oscillation, phase relation between excitatory and inhibitory elements, power spectral density, and spatial Fourier spectrum are quantified from numerical simulations. From these simulations, I determined that the physiologically sophisticated ccf model is too large and unwieldy for easy tuning to match the electrical response of the immature cortex. The Waikato near-far fast-soma model is constructed by incorporating the back-propagation effect of the action potential into the Waikato fast-soma model, state equations are listed and stability prediction are performed by varying the gap junction diffusion strength, subcortical drive, and the rate constants of the near- and far-dendritic tree. In the end, I selected the classic and simpler Waikato slow-soma mean-field model to use for my immature cortex simulations. Model parameters are customised based on the physiology of the immature cortex, including GABA (an inhibitory neurotransmitter in adult) excitatory effect, number of synaptic connections, and rate constants of the IPSPs (inhibitory postsynaptic potential). After hyperpolarising the neuron resting voltage sufficiently to cause the immature inhibitory neuron to act as an excitatory agent, I alter the rate constant of the IPSP, and study the stability of the immature cortex. The bursting activity and quiet states of the discontinuous EEG are simulated and the gap junction diffusion effect in the immature cortex is also examined. For a rate constant of 18.6 s-1, slow oscillations in the quiet states are generated, and for rate constant of 25 s-1, a possible cortical network oscillation emerges. As far as I know, this is the first time that the GABA excitatory effect has been integrated into a mean-field cortical model and the discontinuous EEG wave successfully simulated in a qualitative way.