This thesis examines the critical transitions between distinct neural states associated with the transition to neuron spiking and with the induction of anaesthesia. First, mathematical and electronic models of a single spiking neuron are investigated, focusing on stochastic subthreshold dynamics on close approach to spiking and to depolarisation-blocked quiescence (spiking death) transition points. Theoretical analysis of subthreshold neural behaviour then shifts to the anaesthetic-induced phase transition into unconsciousness using a mean-field model for interacting populations of excitatory and inhibitory neurons. The anaesthetic-induced changes are validated experimentally using published electrophysiological data recorded in anaesthetised rats. The criticality hypothesis associated with brain state change is examined using neuronal avalanches for experimentally recorded rat local field potential (LFP) data and mean-field pseudoLFP simulation data. We compare three different implementations of the FitzHugh--Nagumo single spiking neuron model: a mathematical model by H. R. Wilson, an alternative due to Keener and Sneyd, and an op-amp based nonlinear oscillator circuit. Although all three models can produce nonlinear ``spiking" oscillations, our focus is on the altering characteristics of noise-induced fluctuations near spiking onset and death via Hopf bifurcation. We introduce small-amplitude white noise to enable a linearised stochastic analyses using Ornstein--Uhlenbeck theory to predict variance, power spectrum and correlation of voltage fluctuations during close approach to the critical point, identified as the point at which the real part of the dominant eigenvalue becomes zero. We validate the theoretical predictions with numerical simulations and show that the fluctuations exhibit critical slowing down divergences when approaching the critical point: power-law increases in the variance of the fluctuations simultaneous with prolongation of the system response. We expand the study of stochastic behaviour to two spatial dimensions using the Waikato mean-field model operating near phase transition points controlled by the infusion or elimination of anaesthetic inhibition. Specifically, we investigate close approach to the critical point (CP), and to the points of loss of consciousness (LOC) and recovery of consciousness (ROC). We select the equilibrium states using $\lambda$ anaesthetic inhibition and $\Delta V^{\text{rest}}_e$ cortical excitation as control parameters, then analyse the voltage fluctuations evoked by small-amplitude spatiotemporal white noise. We predict the variance and power spectrum of voltage fluctuations near the marginally stable LOC and ROC transition points, then validate via numerical simulation. The results demonstrate a marked increase in voltage fluctuations and spectral power near transition points. This increased susceptibility to low-intensity white noise stimulation provides an early warning of impending phase transition. Effects of anaesthetic agents on cortical activity are reflected in local field potentials (LFPs) by the variation of amplitude and frequency in voltage fluctuations. To explore these changes, we investigate LFPs acquired from published electrophysiological experiments of anaesthetised rats to extract amplitude distribution, variance and time-correlation statistics. The analysis is broadened by applying detrended fluctuation analysis (DFA) to detect long-range dependencies in the time-series, and we compare DFA results with power spectral density (PSD). We find that the DFA exponent increases with anaesthetic concentration, but is always close to 1. The penultimate chapter investigates the evidence of criticality in anaesthetic induced phase-transitions using avalanche analysis. Rat LFP data reveal an avalanche power-law exponent close to $\alpha = 1.5$, but this value depends on both the time-bin width chosen to separate the events and the \textit{z}-score threshold used to detect these events. Power-law behaviour is only evident at lower anaesthetic concentrations; at higher concentrations the avalanche size distribution fails to align with a power-law nature. Criticality behaviour is also indicated in the Waikato mean-field model for anaesthetic-induced phase-transition using avalanches detected from the pseudoLFP time-series, but only at the critical point (CP) and at the secondary phase-transition points of LOC and ROC. In summary, this thesis unveils evidence of characteristic changes near phase transition points using computer-based mathematical modelling and electrophysiological data analysis. We find that noise-driven fluctuations become larger and persist for longer as the critical point is closely approached, with similar properties being seen not only in single-neuron and neural population models, but also in biological LFP signals. These results consistent with an increase of susceptibility to noise perturbations near phase transition point. Identification of neuronal avalanches in rat LFP data for low anaesthetic concentrations provides further support for the criticality hypothesis.