A spatially continuous mean field theory of electrocortical activity

Abstract

A set of nonlinear continuum field equations is presented which describes the dynamics of neural activity in cortex. These take into account the most pertinent anatomical and physiological features found in cortex with all parameter values obtainable from independent experiment. Derivation of a white noise fluctuation spectrum from a linearized set of equations shows the presence of strong resonances that correspond to electroencephalographically observed 0.3-4 Hz (mammalian delta), 4-8 Hz (mammalian theta), 8-13 Hz (mammalian alpha) and >13 Hz (mammalian beta) activity. Numerical solutions of a full set of one-dimensional nonlinear equations include properties analogous to cortical evoked potentials, travelling waves at experimentally observed velocities, threshold type spike activity and limit cycle, chaotic and noise driven oscillations at the frequency of the mammalian alpha rhythm. All these types of behaviour are generated with parameters that are within ranges reported experimentally. The strong dependence of the phenomena observed on inhibitory-inhibitory interactions is demonstrated. These results suggest that the classically described alpha may be instantiated in a number of qualitatively distinct dynamical regimes, all of which depend on the integrity of inhibitoryinhibitory population interactions.

Stabilization of bimanual coordination due to active interhemispheric inhibition: a dynamical account

“…To formalize the crude, qualitative interpretation of the neural underpinnings of bimanual coordination forwarded in the preceding discussion, we build on earlier theoretical studies of spatiotemporal patterning in the brain (e.g., Winfree 1967;Wilson and Cowan 1972;Freeman 1975;Ermentrout and Cowan 1979;Nunez 1995;Jirsa and Haken 1996;Liley et al 1999;Frank et al 2000;Wright et al 2001). Emanating from large ensembles of mutually interacting neurons, averaging methods like mean-field approaches result in weakly nonlinear mappings from action potentials to dendritic currents.…”

Modelling general anaesthesia as a first-order phase transition in the cortex

“…The Liley et al (1999) model consists of eight coupled partial differential equations (PDEs) that describe the interacting behaviours of the excitatory and inhibitory neural populations within a cortical macrocolumn. The cortex is represented as a 1-D continuum of macrocolumn mass (i.e., a "cortical rod").…”

“…To investigate this possibility, we adopted the Liley et al (1999) continuum model of the cortex, and incorporated the effect of a general anaesthetic agent into the model by assuming that the effectiveness of inhibitory synaptic events increases as the concentration of the anaesthetic increases. We found that, for certain ranges of anaesthetic concentration, the model predicts the existence of multiple steady states for brain activity, leading to the possibility that, at a critical level of anaesthetic, there would be a sudden switch-over from normal levels of cortical activity to a quiescent, low-firing state.…”

Effects of the anesthetic agent propofol on neural populations

“…The latter theoretical studies (Steyn-Ross and SteynRoss 1999; Bojak and Liley 2005;Molaee-Ardekani et al 2007) are based on the model of Liley et al (1999), whose basic elements we discuss briefly in the following. The model considers a continuous spatial mean-field of neurons in one or two spatial dimensions, synapses and axonal connections; the synapses and neurons may be excitatory and inhibitory.…”