The main motivation of this study is to develop a better understanding of anaesthetic drug effects on brain dynamics including the paradoxical enhancement of seizure activity by some anaesthetic drugs. This thesis investigates two mean-field descriptions for the effect of general anaesthetic agents on brain activity: the extended Waikato cortical model (WM) and the Hindriks and van Putten (HvP) thalamocortical model. In the standard Waikato model, the population-average neuron voltage is determined by incoming activity at both electrical (gap-junction) and chemical synapses, the latter mediated by AMPA (excitatory) and GABAA (inhibitory) receptors. Here we extend the standard WM by including NMDA (excitatory) and GABAB (inhibitory) synapses. GABAergic anaesthetics, such as propofol, boost cortical inhibition by prolonging the tail of the unitary IPSP (inhibitory postsynaptic potential) at GABAA receptors, while increasing the synaptic gain at the slower-acting GABAB receptors. Dissociative anaesthetics act on NMDA receptors to give a voltage-dependent alteration of excitatory synaptic gain. We find that increasing GABAB or NMDA effect can alter the spatiotemporal dynamics of the standard WM, tending to suppress spatial (Turing) patterns in favour of temporal (Hopf) oscillations. The extended WM predicts increased susceptibility to seizure when GABAB effect is increased, particularly if the GABAergic agent reduces gap-junction diffusion. We tested these WM predictions with two biological experiments. We found that potentiation of GABAB receptors in slices of mouse cortical tissue tended to enhance seizure-like activity. However, our in vivo investigation of the effect of closure of gap junctions did not reveal any seizure patterns in mouse EEG signals. In the second part of this thesis, we present a detailed analysis of the HvP thalamocortical mean-field model for propofol anaesthesia. While we were able to confirm the Hindriks and van Putten predictions of increases in delta and alpha power at low levels of anaesthetic sedation, we find that for deeper anaesthetic effect, the model jumps from the low-firing state to an extremely high-firing stable state (~250 spikes/s), and remains stuck there even at GABAA prolongations as high as 300% which would be expected to induce full comatose suppression of all firing activity. To overcome this pathological behaviour, we tested two possible modifications: first, eliminating the population-dependent anaesthetic sensitivity (efficacy) of the HvP model; second, incorporating reversal potentials and tuning the excitatory sigmoid parameters defining the mapping from voltage to firing rate. The first modification removes the pathological state, but predicts de-creasing alpha and delta power as drug concentration increases. The second modification predicts induction-emergence hysteresis (drug concentration is higher at induction than at emergence), but the alpha rhythm is lost, being replaced by a dominant delta-band oscillation.