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A recent neurophysical model of propagation of electrical waves in the cortex is extended to include a physiologically motivated subcortical feedback loop via the thalamus. The electroencephalographic spectrum when the system is driven by white noise is then calculated analytically in terms of physiological parameters, including the effects of filtering of signals by the cerebrospinal fluid, skull, and scalp. The spectral power at low frequencies is found to vary as f(-1) when awake and f(-3) when asleep, with a breakpoint to a steeper power-law tail at frequencies above about 20 Hz in both cases; the f(-1) range concurs with recent magnetoencephalographic observations of such a regime. Parameter sensitivities are explored, enabling a model with fewer free parameters to be proposed, and showing that spectra predicted for physiologically reasonable parameter values strongly resemble those observed in the laboratory. Alpha and beta peaks seen near 10 Hz and twice that frequency, respectively, in the relaxed wakeful state are generated via subcortical feedback in this model, thereby leading to predictions of their frequencies in terms of physiological parameters, and of correlations in their occurrence. Subcortical feedback is also predicted to be responsible for production of anticorrelated peaks in deep sleep states that correspond to the occurrence of theta rhythm at around half the alpha frequency and sleep spindles at 3/2 times the alpha frequency. An additional positively correlated waking peak near three times the alpha frequency is also predicted and tentatively observed, as are two new types of sleep spindle near 5/2 and 7/2 times the alpha frequency, and anticorrelated with alpha. These results provide a theoretical basis for the conventional division of EEG spectra into frequency bands, but imply that the exact bounds of these bands depend on the individual. Three types of potential instability are found: one at zero frequency, another in the theta band at around half the alpha frequency, and a third at the alpha frequency itself.

Evoked potentials -- the brain's transient electrical responses to discrete stimuli -- are modeled as impulse responses using a continuum model of brain electrical activity. Previous models of ongoing brain activity are refined by adding an improved model of thalamic connectivity and modulation, and by allowing for two populations of excitatory cortical neurons distinguished by their axonal ranges. Evoked potentials are shown to be modelable as an impulse response that is a sum of component responses. The component occurring about 100 ms poststimulus is attributed to sensory activation, and this, together with positive and negative feedback pathways between the cortex and thalamus, results in subsequent peaks and troughs that semiquantitatively reproduce those of observed evoked potentials. Modulation of the strengths of positive and negative feedback, in ways consistent with psychological theories of attentional focus, results in distinct responses resembling those seen in experiments involving attentional changes. The modeled impulse responses reproduce key features of typical experimental evoked response potentials: timing, relative amplitude, and number of peaks. The same model, with further modulation of feedback, also reproduces experimental spectra. Together, these results mean that a broad range of ongoing and transient electrocortical activity can be understood within a common framework, which is parameterized by values that are directly related to physiological and anatomical quantities.

There is some complementarity of models for the origin of the electroencephalogram (EEG) and neural network models for information storage in brainlike systems. From the EEG models of Freeman, of Nunez, and of the authors' group we argue that the wavelike processes revealed in the EEG exhibit linear and near-equilibrium dynamics at macroscopic scale, despite extremely nonlinear – probably chaotic – dynamics at microscopic scale. Simulations of cortical neuronal interactions at global and microscopic scales are then presented. The simulations depend on anatomical and physiological estimates of synaptic densities, coupling symmetries, synaptic gain, dendritic time constants, and axonal delays. It is shown that the frequency content, wave velocities, frequency/wavenumber spectra and response to cortical activation of the electrocorticogram (ECoG) can be reproduced by a “lumped” simulation treating small cortical areas as single-function units. The corresponding cellular neural network simulation has properties that include those of attractor neural networks proposed by Amit and by Parisi. Within the simulations at both scales, sharp transitions occur between low and high cell firing rates. These transitions may form a basis for neural interactions across scale. To maintain overall cortical dynamics in the normal low firing-rate range, interactions between the cortex and the subcortical systems are required to prevent runaway global excitation. Thus, the interaction of cortex and subcortex via corticostriatal and related pathways may partly regulate global dynamics by a principle analogous to adiabatic control of artificial neural networks.

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The spectral analysis of multichannel magnetoencephalographic data is presented. This analysis revealed a local similarity regime in brain activity ͑in more than two decades of frequencies͒ and provided new parameters for noninvasive experimental studies of the brain.

This paper generalizes an existing continuum model of the large-scale electrical activity of the brain by incorporating local feedback. The model is first reformulated by parametrizing neuronal function in more detail and we then show that only four broad classes of local feedback are consistent with general physiological constraints. The corresponding linear dispersion relations are found to contain additional modes as a result of feedback. It is shown that each feedback class can produce lightly damped or undamped oscillatory modes at frequencies similar to those seen in electroencephalography, although not all combinations of type and speed of feedback result in oscillatory behavior. Three kinds of lightly damped resonances are found, each with distinctive frequency characteristics. The effect of feedback on other roots is also described.

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