Wave-number spectrum of electrocorticographic signals

Theory of corticothalamic brain activity in a spherical geometry: Spectra, coherence, and correlation

Corticothalamic neural field theory is applied to a spherical geometry to better model neural activity in the human brain and is also compared with planar approximations. The frequency power spectrum, correlation, and coherence functions are computed analytically and numerically. The effects of cortical boundary conditions and resulting modal aspects of spherical corticothalamic dynamics are explored, showing that the results of spherical and finite planar geometries converge to those for the infinite planar geometry in the limit of large brain size. Estimates are made of the point at which modal series can be truncated and it is found that for physiologically plausible parameters only the lowest few spatial eigenmodes are needed for an accurate representation of macroscopic brain activity. A difference between the geometries is that there is a low-frequency 1/f spectrum in the infinite planar geometry, whereas in the spherical geometry it is 1/f^{2}. Another difference is that the alpha peak in the spherical geometry is sharper and stronger than in the planar geometry. Cortical modal effects can lead to a double alpha peak structure in the power spectrum, although the main determinant of the alpha peak is corticothalamic feedback. In the spherical geometry, the cross spectrum between two points is found to only depend on their relative distance apart. At small spatial separations the low-frequency cross spectrum is stronger than for an infinite planar geometry and the alpha peak is sharper and stronger due to the partitioning of the energy into discrete modes. In the spherical geometry, the coherence function between points decays monotonically as their separation increases at a fixed frequency, but persists further at resonant frequencies. The correlation between two points is found to be positive, regardless of the time lag and spatial separation, but decays monotonically as the separation increases at fixed time lag. At fixed distance the correlation has peaks at multiples of the period of the dominant frequency of system activity.

Generalized seizures in a neural field model with bursting dynamics

The mechanisms underlying generalized seizures are explored with neural field theory. A corticothalamic neural field model that has accounted for multiple brain activity phenomena and states is used to explore changes leading to pathological seizure states. It is found that absence seizures arise from instabilities in the system and replicate experimental studies in numerous animal models and clinical studies.

Neural field theory with variance dynamics

Previous neural field models have mostly been concerned with prediction of mean neural activity and with second order quantities such as its variance, but without feedback of second order quantities on the dynamics. Here the effects of feedback of the variance on the steady states and adiabatic dynamics of neural systems are calculated using linear neural field theory to estimate the neural voltage variance, then including this quantity in the total variance parameter of the nonlinear firing rate-voltage response function, and thus into determination of the fixed points and the variance itself. The general results further clarify the limits of validity of approaches with and without inclusion of variance dynamics. Specific applications show that stability against a saddle-node bifurcation is reduced in a purely cortical system, but can be either increased or decreased in the corticothalamic case, depending on the initial state. Estimates of critical variance scalings near saddle-node bifurcation are also found, including physiologically based normalizations and new scalings for mean firing rate and the position of the bifurcation.

Generation and control of cortical gamma: findings from simulation at two scales

A continuum model of electrocortical activity was applied separately at centimetric and macrocolumnar scales, permitting analysis of interaction between scales. State equations included effects of retrograde action potential propagation in dendritic trees, and kinetics of AMPA, GABA and NMDA receptors. Parameter values were provided from independent physiological and anatomical estimates. Realistic field potentials and pulse rates were obtained, including resonances in the alpha/theta and gamma ranges, 1/f(2) background activity, and autonomous gamma activity. Zero-lag synchrony and travelling waves occurred as complementary aspects of cortical transmission, and lead/lag relations between excitatory and inhibitory cell populations varied systematically around transition to autonomous gamma oscillation. Properties of the simulations can account for generation and control of gamma activity. All factors acting on excitatory/inhibitory balance controlled the onset and offset of gamma oscillation. Autonomous gamma was initiated by focal excitation of excitatory cells, and suppressed by laterally spreading trans-cortical excitation, which acted on both excitatory and inhibitory cell populations. Consequently, although spatially extensive non-specific reticular activation tended to suppress autonomous gamma, spatial variation of reticular activation could preferentially select fields of synchrony.

Physiologically based calculation of steady-state evoked potentials and cortical wave velocities

Steady-state evoked potentials (SSEPs) elicited by sinusoidal stimuli are predicted from a physiologically-based model, including bielectrode and volume conduction effects. Comparison with visual SSEPs yields constraints on phase and latency of the retinothalamic transfer function that are consistent with experiment. Predictions of phase velocities measured as SSEPs cross the cortex are consistent with low values measured for slow waves in sleep, while resonant behavior induced by corticothalamic loops, especially near the alpha peak, contributes to wide scatter in waking-state phase velocity measurements comparable to effects from volume conduction. The common use of bielectrode derivations to compensate for volume conduction effects is examined and shown to be incomplete, tending to lead to underestimates of phase velocity, especially at low frequencies and near the alpha peak, due to incorrect elimination of true long-wavelength contributions to the SSEP.

Visual gamma oscillations: waves, correlations, and other phenomena, including comparison with experimental data

Mean-field theory of brain dynamics is applied to explain the properties of gamma (> or approximately 30 Hz) oscillations of cortical activity often seen during vision experiments. It is shown that mm-scale patchy connections in the primary visual cortex can support collective gamma oscillations with the correct frequencies and spatial structure, even when driven by uncorrelated inputs. This occurs via resonances associated with the the periodic modulation of the network connections, rather than being due to single-cell properties alone. Near-resonant gamma waves are shown to obey the Schrödinger equation, which enables techniques and insights from quantum theory to be used in exploring these classical oscillations. Resulting predictions for gamma responses to stimuli account in a unified way for a wide range of experimental results, including why oscillations and zero-lag synchrony are associated, and variations in correlation functions with time delay, intercellular distance, and stimulus features. They also imply that gamma oscillations may enable a form of frequency multiplexing of neural signals. Most importantly, it is shown that correlations reproduce experimental results that show maximal correlations between cells that respond to related features, but little correlation with other cells, an effect that has been argued to be associated with segmentation of a scene into separate objects. Consistency with infill of missing contours and increase in response with length of bar-shaped stimuli are discussed. Background correlations expected in the absence of stimulation are also calculated and shown to be consistent in form with experimental measurements and similar to stimulus-induced correlations in structure. Finally, possible links of gamma instabilities to certain classes of photically induced seizures and visual hallucinations are discussed.

Patchy propagators, brain dynamics, and the generation of spatially structured gamma oscillations

Propagator theory of brain dynamics is generalized to incorporate a new class of patchy propagators that enable treatment of approximately periodic structures such as are seen in the visual cortex. Complex response fields are also incorporated to allow for features such as orientation preference and wave-number selectivity. The results are applied to the corticothalamic system associated with the primary visual cortex. It is found that this system can generate gamma ( > or = 30 Hz) oscillations during stimulation, whose properties are consistent with experimental findings on gamma frequency and bandwidth, and existence of fine-scale spatial structure. It is found that a potential resonance is associated with each reciprocal lattice vector corresponding to periodic modulations of the propagators. It is found that the lowest resonances are the most likely to give rise to noticeable spectral peaks and increases of correlation amplitude, length, and time, and that these aspects are prominent only if the system is close to marginal stability, in accord with previous measurements and discussions of cortical stability. These features also enable gamma resonances to be stimulus-evoked, with substantial resonance sharpening for relatively small changes in mean neural firing rate. The results also imply dependence of gamma frequency on stimulus features.

Multiscale brain modelling

A central difficulty of brain modelling is to span the range of spatio-temporal scales from synapses to the whole brain. This paper overviews results from a recent model of the generation of brain electrical activity that incorporates both basic microscopic neurophysiology and large-scale brain anatomy to predict brain electrical activity at scales from a few tenths of a millimetre to the whole brain. This model incorporates synaptic and dendritic dynamics, nonlinearity of the firing response, axonal propagation and corticocortical and corticothalamic pathways. Its relatively few parameters measure quantities such as synaptic strengths, corticothalamic delays, synaptic and dendritic time constants, and axonal ranges, and are all constrained by independent physiological measurements. It reproduces quantitative forms of electroencephalograms seen in various states of arousal, evoked response potentials, coherence functions, seizure dynamics and other phenomena. Fitting model predictions to experimental data enables underlying physiological parameters to be inferred, giving a new non-invasive window into brain function that complements slower, but finer-resolution, techniques such as fMRI. Because the parameters measure physiological quantities relating to multiple scales, and probe deep structures such as the thalamus, this will permit the testing of a range of hypotheses about vigilance, cognition, drug action and brain function. In addition, referencing to a standardized database of subjects adds strength and specificity to characterizations obtained.

Propagator theory of brain dynamics

A physiologically based continuum model of brain dynamics is extended to incorporate arbitrary numbers of structures and neural populations, multiple outgoing fields of activity from a single population of neurons to various targets, improved treatment of converging or diverging projections and mesoscopic structure, and generalized connections to quantities observable via electroencephalography and other methods. The results are applied to study the corticothalamic system, predicting an intracortical resonance that leads to enhancements of electroencephalographic activity in the gamma (>30 Hz) range. This resonance involves feedback loops incorporating slow, short-range inhibitory fibers.

Analysis of the electroencephalographic activity associated with thalamic tumors

A physiologically based model of corticothalamic dynamics is used to investigate the electroencephalographic (EEG) activity associated with tumors of the thalamus. Tumor activity is modeled by introducing localized two-dimensional spatial non-uniformities into the model parameters, and calculating the resulting activity via the coupling of spatial eigenmodes. The model is able to reproduce various qualitative features typical of waking eyes-closed EEGs in the presence of a thalamic tumor, such as the appearance of abnormal peaks at theta ( approximately 3Hz) and spindle ( approximately 12Hz) frequencies, the attenuation of normal eyes-closed background rhythms, and the onset of epileptic activity, as well as the relatively normal EEGs often observed. The results indicate that the abnormal activity at theta and spindle frequencies arises when a small portion of the brain is forced into an over-inhibited state due to the tumor, in which there is an increase in the firing of (inhibitory) thalamic reticular neurons. The effect is heightened when there is a concurrent decrease in the firing of (excitatory) thalamic relay neurons, which are in any case inhibited by the reticular ones. This is likely due to a decrease in the responsiveness of the peritumoral region to cholinergic inputs from the brainstem, and a corresponding depolarization of thalamic reticular neurons, and hyperpolarization of thalamic relay neurons, similar to the mechanism active during slow-wave sleep. The results indicate that disruption of normal thalamic activity is essential to generate these spectral peaks. Furthermore, the present work indicates that high-voltage and epileptiform EEGs are caused by a tumor-induced local over-excitation of the thalamus, which propagates to the cortex. Experimental findings relating to local over-inhibition and over-excitation are discussed. It is also confirmed that increasing the size of the tumor leads to greater abnormalities in the observable EEG. The usefulness of EEG for localizing the tumor is investigated.

Unifying and interpreting the spectral wavenumber content of EEGs, ECoGs, and ERPs

A biological model of corticothalamic dynamics is used to investigate the spatial power spectrum (wavenumber spectrum) of electrical activity in the brain. The model provides a single framework for unifying different aspects of activity. Comparisons of the predicted spectra with published electrocorticographic, electroencephalographic, and evoked response potential data enable physiology and anatomy to be inferred, producing results which are complementary to those obtained from comparisons in the frequency domain; the inferred quantities are consistent with, and complementary to, direct physiological and anatomical measurements. We also use the model to quantify the interdependence of the wavenumber and frequency domains, and deduce that further experiments that cover large wavenumber and frequency ranges simultaneously would greatly increase our knowledge of brain function. We conclude that both the frequency and wavenumber domains should be studied in order to build the fullest picture of brain dynamics: the two domains are both complementary and interdependent.

Estimation of neurophysiological parameters from the waking EEG using a biophysical model of brain dynamics

This paper presents the results from using electroencephalographic (EEG) data to estimate the values of key neurophysiological parameters using a detailed biophysical model of brain activity. The model incorporates spatial and temporal aspects of cortical function including axonal transmission delays, synapto-dendritic rates, range-dependent connectivities, excitatory and inhibitory neural populations, and intrathalamic, intracortical, corticocortical and corticothalamic pathways. Parameter estimates were obtained by fitting the model's theoretical spectrum to EEG spectra from each of 100 healthy human subjects. Statistical analysis was used to infer significant parameter variations occurring between eyes-closed and eyes-open states, and a correlation matrix was used to investigate links between the parameter variations and traditional measures of quantitative EEG (qEEG). Accurate fits to all experimental spectra were observed, and both inter-subject and between-state variability were accounted for by the variance in the fitted biophysical parameters, which were in turn consistent with known independent experimental and theoretical estimates. These values thus provide physiological information regarding the state. transitions (eyes-closed vs. eyes-open) and phenomena including cortical idling and alpha desynchronization. The parameters are also consistent with traditional qEEG, but are more informative, since they provide links to underlying physiological processes. To our knowledge, this is the first study where a detailed biophysical model of the brain is used to estimate neurophysiological parameters underlying the transitions in a broad range (0.25-50 Hz) of EEG spectra obtained from a large set of human data.

Estimation of multiscale neurophysiologic parameters by electroencephalographic means

It is shown that new model-based electroencephalographic (EEG) methods can quantify neurophysiologic parameters that underlie EEG generation in ways that are complementary to and consistent with standard physiologic techniques. This is done by isolating parameter ranges that give good matches between model predictions and a variety of experimental EEG-related phenomena simultaneously. Resulting constraints range from the submicrometer synaptic level to length scales of tens of centimeters, and from timescales of around 1 ms to 1 s or more, and are found to be consistent with independent physiologic and anatomic measures. In the process, a new method of obtaining model parameters from the data is developed, including a Monte Carlo implementation for use when not all input data are available. Overall, the approaches used are complementary to other methods, constraining allowable parameter ranges in different ways and leading to much tighter constraints overall. EEG methods often provide the most restrictive individual constraints. This approach opens a new, noninvasive window on quantitative brain analysis, with the ability to monitor temporal changes, and the potential to map spatial variations. Unlike traditional phenomenologic quantitative EEG measures, the methods proposed here are based explicitly on physiology and anatomy.

Spatially uniform and nonuniform analyses of electroencephalographic dynamics,with application to the topography of the alpha rhythm

Corticothalamic dynamics are investigated using a model in which spatial nonuniformities are incorporated via the coupling of spatial eigenmodes. Comparison of spectra generated using the nonuniform analysis with those generated using a uniform one demonstrates that, for most frequencies, local activity is only weakly dependent on activity elsewhere in the cortex; however, dispersion of low-wave-number activity ensures that distant dynamics influence local dynamics at low frequencies (below approximately 2 Hz ), and at the alpha frequency (approximately 10 Hz ), where propagating signals are inherently weakly damped, and wavelengths are large. When certain model parameters have similar spatial profiles, as is expected from physiology, the low-frequency discrepancies tend to cancel, and the uniform analysis with local parameter values is an adequate approximation to the full nonuniform one across the whole spectrum, at least for large-scale nonuniformities. After comparing the uniform and nonuniform analyses, we consider one possible application of the nonuniform analysis: studying the phenomenon of occipital alpha dominance, whereby the alpha frequency and power are greater at the back of the head (occipitally) than at the front. In order to infer realistic nonuniformities in the model parameters, the uniform version of the model is first fitted to data recorded from 98 normal subjects in a waking, eyes-closed state. This yields a set of parameters at each of five electrode sites along the midline. The inferred parameter nonuniformities are consistent with anatomical and physiological constraints. Introducing these spatial profiles into the full nonuniform model then quantitatively reproduces observed site-dependent variations in the alpha power and frequency. The results confirm that the frequency shift is mainly due to a decrease in the corticothalamic propagation delay, but indicate that the delay nonuniformity cannot account for the observed occipital increase in alpha power; the occipital alpha dominance is due to decreased cortical gains and increased thalamic gains in occipital regions compared to frontal ones.