Equilibrium and Nonequilibrium Phase Transitions in a Continuum Model of an Anesthetized Cortex

In this chapter we investigate a range of dynamic behaviors accessible to a continuum model of the cerebral cortex placed close to the anesthetic phase transition. If the anesthetic transition from the high-firing (conscious) to the low-firing (comatose) state can be modeled as a jump between two equilibrium states of the cortex, then we can draw an analogy with the vapor-to-liquid phase transition of the van der Waals gas of classical thermodynamics. In this analogy, specific volume (inverse density) of the gas maps to cortical activity, with pressure and temperature being the analogs of anesthetic concentration and subcortical excitation. It is well known that at the thermodynamic critical point, large fluctuations in specific volume are observed; we find analogous critically-slowed fluctuations in cortical activity at its critical point. Unlike the van der Waals system, the cortical model can also exhibit nonequilibrium phase transitions in which the homogeneous equilibrium can destabilize in favor of slow global oscillations Oscillations (Hopf temporal instability), stationary structures (Turing spatial instability), and chaotic spatiotemporal activity patterns (Hopf–Turing interactions). We comment on possible physiological and pathological interpretations for these dynamics. In particular, the turbulent state may correspond to the cortical slow oscillation between “up” and “down” states observed in nonREM sleep and clinical anesthesia.