Theoretical predictions for spatial covariance of the electroencephalographic signal during the anesthetic-induced phase transition: Increased correlation length and emergence of spatial self-organization

In a recent series of papers, the authors have developed a stochastic theory to describe the electrical response of a spatially homogeneous cerebral cortex to infusion of a general anesthetic agent. We showed that by modeling the GABAergic (propofol-like) drug effect as a prolongation of the inhibitory postsynaptic impulse response, we obtain a prediction that there will be a hysteretically separated pair of first-order phase transitions in the population-average excitatory soma voltage, the first occurring at the point of induction of unconsciousness, and the second at the point of emergence from unconsciousness. In the present paper we generalize our earlier “zero-dimensional” homogeneous cortex to a one-dimensional (1D) line of cortical “mass,” thus allowing for the possibility of spatial inhomogeneities in neural activity. Following the spirit of our earlier adiabatic (“slow membrane”) philosophy, we impose a spatioadiabatic approximation that permits us to compute analytic expressions for changes in EEG (electroencephalographic) correlation length and EEG spatial covariance as a function of anesthetic effect. We establish that the correlation length of the EEG fluctuations is expected to increase at the approach to the transition points, and this finding is consistent with both the homogeneous-cortex prediction of increased correlation time (“critical slowing down”) near transition, and the recent, comprehensive anesthetic study by John et al.[Conscious. Cogn. 10, 165 (2001)] reporting an increase in EEG coherence near the points of loss and recovery of consciousness. In addition, we find that if the long-range (corticocortical) excitatory-to-inhibitory connectivity in the 1D cortex is stronger than the long-range excitatory-to-excitatory connectivity, then the spatioadiabatic system can organize itself into large-amplitude spatial patterns (“dissipative structures”) consisting of giant stationary quasiperiodic voltage fluctuations distributed along the cortical rod