Modeling long-range synchronization in default and cognitive  states of the brain

I present a mean-field model of the cortex comprised of interacting populations of excitatory and inhibitory neurons which communicate via chemical (neurotransmitter-controlled) and electrical (gap junction) synapses. The model consists of closed set of stochastic differential equations that describe the spatially-averaged behaviour of the firing rates of neurons. When the soma response is slow relative to dendritic events, the model predicts ultra-slow spatiotemporal oscillations in cortical activity reminiscent of "default" BOLD patterns seen in fMRI. This slow oscillation arises from a subharmonic interaction between Turing (spatial) and Hopf (temporal) symmetry-breaking instabilities. If the soma response to dendritic input is prompt, a gamma-band (~40-Hz) standing-wave instability emerges, entraining the cortex into long- range synchronous oscillations consistent with the cognitive state of the cortex.