Theoretical electroencephalogram stationary spectrum for a white-noise-driven cortex: evidence for a general anesthetic-induced phase transition

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.

Gap junctions modulate seizures in a mean-field model of general anesthesia for the cortex

During slow-wave sleep, general anesthesia, and generalized seizures, there is an absence of consciousness. These states are characterized by low-frequency large-amplitude traveling waves in scalp electroencephalogram. Therefore the oscillatory state might be an indication of failure to form coherent neuronal assemblies necessary for consciousness. A generalized seizure event is a pathological brain state that is the clearest manifestation of waves of synchronized neuronal activity. Since gap junctions provide a direct electrical connection between adjoining neurons, thus enhancing synchronous behavior, reducing gap-junction conductance should suppress seizures; however there is no clear experimental evidence for this. Here we report theoretical predictions for a physiologically-based cortical model that describes the general anesthetic phase transition from consciousness to coma, and includes both chemical synaptic and direct electrotonic synapses. The model dynamics exhibits both Hopf (temporal) and Turing (spatial) instabilities; the Hopf instability corresponds to the slow (≲8 Hz) oscillatory states similar to those seen in slow-wave sleep, general anesthesia, and seizures. We argue that a delicately balanced interplay between Hopf and Turing modes provides a canonical mechanism for the default non-cognitive rest state of the brain. We show that the Turing mode, set by gap-junction diffusion, is generally protective against entering oscillatory modes; and that weakening the Turing mode by reducing gap conduction can release an uncontrolled Hopf oscillation and hence an increased propensity for seizure and simultaneously an increased sensitivity to GABAergic anesthesia.

Spike, rate, field, and hybrid methods for treating neuronal dynamics and interactions

Spike-, rate-, and field-based approaches to neural dynamics are adapted and hybridized to provide new methods of analyzing dynamics of single neurons and large neuronal systems, to elucidate the relationships and intermediate forms between these limiting cases, and to enable faster simulations with reduced memory requirements. At the single-neuron level, the new approaches involve reformulation of dynamics in synapses, dendrites, cell bodies, and axons to enable new types of analysis, longer numerical timesteps, and demonstration that rate-based methods can predict spike times. In multineuron systems, hybrids and intermediates between spike-based and field-based coupling between neurons are used to provide stepping stones between descriptions based on pairwise spike-based interactions between neurons and ones based on neural field-based interactions within and between populations, including arbitrary spatial structure and temporal delays in the connections in general. In particular, a new neuron-in-cell approach is introduced that is a hybrid between neural field theory and spiking-neuron models in analogy to particle-in-cell methods in plasma physics. This approach enables large speedups in computations while preserving spike shapes and times. Various approaches are illustrated numerically for specific cases.

Corticothalamic dynamics: structure of parameter space, spectra, instabilities, and reduced model

Linear instabilities are analyzed in a physiologically based mean-field corticothalamic model and a reduced-parameter model derived from it. In both models, the stable zone corresponding to normal arousal states is bounded by a series of surfaces demarcating the onsets of instabilities. The stable zone is found to depend on delay and rate parameters, whose values have a simple relationship to the number of instabilities and dominant frequencies on the stable zone's boundary. The dominant frequencies of linear activity inside the stable zone are found to lie in clearly delineated regions, each corresponding to an instability surface on its boundary and having approximately the same dominant frequency. These regions are ordered in parameter space according to their dominant frequencies, and an instability associated with the intrathalamic loop is shown to have the highest frequency that can become unstable. This reveals an important role for the thalamus in controlling the stability and bandwidth of dynamics in the corticothalamic system as a whole. The reduced model is found to agree well with the full model in a wide region of parameter space and, thus, is a useful guide to the full model's dynamics.

Interrelating anatomical, effective, and functional brain connectivity using propagators and neural field theory

It is shown how to compute effective and functional connection matrices (eCMs and fCMs) from anatomical CMs (aCMs) and corresponding strength-of-connection matrices (sCMs) using propagator methods in which neural interactions play the role of scatterings. This analysis demonstrates how network effects dress the bare propagators (the sCMs) to yield effective propagators (the eCMs) that can be used to compute the covariances customarily used to define fCMs. The results incorporate excitatory and inhibitory connections, multiple structures and populations, asymmetries, time delays, and measurement effects. They can also be postprocessed in the same manner as experimental measurements for direct comparison with data and thereby give insights into the role of coarse-graining, thresholding, and other effects in determining the structure of CMs. The spatiotemporal results show how to generalize CMs to include time delays and how natural network modes give rise to long-range coherence at resonant frequencies. The results are demonstrated using tractable analytic cases via neural field theory of cortical and corticothalamic systems. These also demonstrate close connections between the structure of CMs and proximity to critical points of the system, highlight the importance of indirect links between brain regions and raise the possibility of imaging specific levels of indirect connectivity. Aside from the results presented explicitly here, the expression of the connections among aCMs, sCMs, eCMs, and fCMs in terms of propagators opens the way for propagator theory to be further applied to analysis of connectivity.

The population firing rate in the presence of GABAergic tonic inhibition in single neurons and application to general anaesthesia

Tonic inhibition has been found experimentally in single neurons and affects the activity of neural populations. This kind of inhibition is supposed to set the background or resting level of neural activity and plays a role in the brains arousal system, e.g. during general anaesthesia. The work shows how to involve tonic inhibition in population rate-coding models by deriving a novel transfer function. The analytical and numerical study of the novel transfer function reveals the impact of tonic inhibition on the population firing rate. Finally, a first application to a recent neural field model for general anaesthesia discusses the origin of the loss of consciousness during anaesthesia.

Dynamic causal models and physiological inference: a validation study using isoflurane anaesthesia in rodents

Generative models of neuroimaging and electrophysiological data present new opportunities for accessing hidden or latent brain states. Dynamic causal modeling (DCM) uses Bayesian model inversion and selection to infer the synaptic mechanisms underlying empirically observed brain responses. DCM for electrophysiological data, in particular, aims to estimate the relative strength of synaptic transmission at different cell types and via specific neurotransmitters. Here, we report a DCM validation study concerning inference on excitatory and inhibitory synaptic transmission, using different doses of a volatile anaesthetic agent (isoflurane) to parametrically modify excitatory and inhibitory synaptic processing while recording local field potentials (LFPs) from primary auditory cortex (A1) and the posterior auditory field (PAF) in the auditory belt region in rodents. We test whether DCM can infer, from the LFP measurements, the expected drug-induced changes in synaptic transmission mediated via fast ionotropic receptors; i.e., excitatory (glutamatergic) AMPA and inhibitory GABA(A) receptors. Cross- and auto-spectra from the two regions were used to optimise three DCMs based on biologically plausible neural mass models and specific network architectures. Consistent with known extrinsic connectivity patterns in sensory hierarchies, we found that a model comprising forward connections from A1 to PAF and backward connections from PAF to A1 outperformed a model with forward connections from PAF to A1 and backward connections from A1 to PAF and a model with reciprocal lateral connections. The parameter estimates from the most plausible model indicated that the amplitude of fast glutamatergic excitatory postsynaptic potentials (EPSPs) and inhibitory postsynaptic potentials (IPSPs) behaved as predicted by previous neurophysiological studies. Specifically, with increasing levels of anaesthesia, glutamatergic EPSPs decreased linearly, whereas fast GABAergic IPSPs displayed a nonlinear (saturating) increase. The consistency of our model-based in vivo results with experimental in vitro results lends further validity to the capacity of DCM to infer on synaptic processes using macroscopic neurophysiological data.

Neural field theory of synaptic plasticity

Plasticity is crucial to neural development, learning, and memory. In the common in vivo situation where postsynaptic neural activity results from multiple presynaptic inputs, it is shown that a widely used class of correlation-dependent and spike-timing dependent plasticity rules can be written in a form that can be incorporated into neural field theory, which enables their system-level dynamics to be investigated. It is shown that the resulting plasticity dynamics depends strongly on the stimulus spectrum via overall system frequency responses. In the case of perturbations that are approximately linear, explicit formulas are found for the dynamics in terms of stimulus spectra via system transfer functions. The resulting theory is applied to a simple model system to reveal how collective effects, especially resonances, can drastically modify system-level plasticity dynamics from that implied by single-neuron analyses. The simplified model illustrates the potential relevance of these effects in applications to brain stimulation, synaptic homeostasis, and epilepsy.

A continuous mapping of sleep states through association of EEG with a mesoscale cortical model

Here we show that a mathematical model of the human sleep cycle can be used to obtain a detailed description of electroencephalogram (EEG) sleep stages, and we discuss how this analysis may aid in the prediction and prevention of seizures during sleep. The association between EEG data and the cortical model is found via locally linear embedding (LLE), a method of dimensionality reduction. We first show that LLE can distinguish between traditional sleep stages when applied to EEG data. It reliably separates REM and non-REM sleep and maps the EEG data to a low-dimensional output space where the sleep state changes smoothly over time. We also incorporate the concept of strongly connected components and use this as a method of automatic outlier rejection for EEG data. Then, by using LLE on a hybrid data set containing both sleep EEG and signals generated from the mesoscale cortical model, we quantify the relationship between the data and the mathematical model. This enables us to take any sample of sleep EEG data and associate it with a position among the continuous range of sleep states provided by the model; we can thus infer a trajectory of states as the subject sleeps. Lastly, we show that this method gives consistent results for various subjects over a full night of sleep and can be done in real time.

Effects of the anesthetic agent propofol on neural populations

The neuronal mechanisms of general anesthesia are still poorly understood. Besides several characteristic features of anesthesia observed in experiments, a prominent effect is the bi-phasic change of power in the observed electroencephalogram (EEG), i.e. the initial increase and subsequent decrease of the EEG-power in several frequency bands while increasing the concentration of the anaesthetic agent. The present work aims to derive analytical conditions for this bi-phasic spectral behavior by the study of a neural population model. This model describes mathematically the effective membrane potential and involves excitatory and inhibitory synapses, excitatory and inhibitory cells, nonlocal spatial interactions and a finite axonal conduction speed. The work derives conditions for synaptic time constants based on experimental results and gives conditions on the resting state stability. Further the power spectrum of Local Field Potentials and EEG generated by the neural activity is derived analytically and allow for the detailed study of bi-spectral power changes. We find bi-phasic power changes both in monostable and bistable system regime, affirming the omnipresence of bi-spectral power changes in anesthesia. Further the work gives conditions for the strong increase of power in the δ-frequency band for large propofol concentrations as observed in experiments.

A model of feedback control for the charge-balanced suppression of epileptic seizures

Here we present several refinements to a model of feedback control for the suppression of epileptic seizures. We utilize a stochastic partial differential equation (SPDE) model of the human cortex. First, we verify the strong convergence of numerical solutions to this model, paying special attention to the sharp spatial changes that occur at electrode edges. This allows us to choose appropriate step sizes for our simulations; because the spatial step size must be small relative to the size of an electrode in order to resolve its electrical behavior, we are able to include a more detailed electrode profile in the simulation. Then, based on evidence that the mean soma potential is not the variable most closely related to the measurement of a cortical surface electrode, we develop a new model for this. The model is based on the currents flowing in the cortex and is used for a simulation of feedback control. The simulation utilizes a new control algorithm incorporating the total integral of the applied electrical potential. Not only does this succeed in suppressing the seizure-like oscillations, but it guarantees that the applied signal will be charge-balanced and therefore unlikely to cause cortical damage.

Axonal velocity distributions in neural field equations

By modelling the average activity of large neuronal populations, continuum mean field models (MFMs) have become an increasingly important theoretical tool for understanding the emergent activity of cortical tissue. In order to be computationally tractable, long-range propagation of activity in MFMs is often approximated with partial differential equations (PDEs). However, PDE approximations in current use correspond to underlying axonal velocity distributions incompatible with experimental measurements. In order to rectify this deficiency, we here introduce novel propagation PDEs that give rise to smooth unimodal distributions of axonal conduction velocities. We also argue that velocities estimated from fibre diameters in slice and from latency measurements, respectively, relate quite differently to such distributions, a significant point for any phenomenological description. Our PDEs are then successfully fit to fibre diameter data from human corpus callosum and rat subcortical white matter. This allows for the first time to simulate long-range conduction in the mammalian brain with realistic, convenient PDEs. Furthermore, the obtained results suggest that the propagation of activity in rat and human differs significantly beyond mere scaling. The dynamical consequences of our new formulation are investigated in the context of a well known neural field model. On the basis of Turing instability analyses, we conclude that pattern formation is more easily initiated using our more realistic propagator. By increasing characteristic conduction velocities, a smooth transition can occur from self-sustaining bulk oscillations to travelling waves of various wavelengths, which may influence axonal growth during development. Our analytic results are also corroborated numerically using simulations on a large spatial grid. Thus we provide here a comprehensive analysis of empirically constrained activity propagation in the context of MFMs, which will allow more realistic studies of mammalian brain activity in the future.

Large-scale neural dynamics: simple and complex

We review the use of neural field models for modelling the brain at the large scales necessary for interpreting EEG, fMRI, MEG and optical imaging data. Albeit a framework that is limited to coarse-grained or mean-field activity, neural field models provide a framework for unifying data from different imaging modalities. Starting with a description of neural mass models, we build to spatially extend cortical models of layered two-dimensional sheets with long range axonal connections mediating synaptic interactions. Reformulations of the fundamental non-local mathematical model in terms of more familiar local differential (brain wave) equations are described. Techniques for the analysis of such models, including how to determine the onset of spatio-temporal pattern forming instabilities, are reviewed. Extensions of the basic formalism to treat refractoriness, adaptive feedback and inhomogeneous connectivity are described along with open challenges for the development of multi-scale models that can integrate macroscopic models at large spatial scales with models at the microscopic scale.

The electrocortical effects of enflurane: experiment and theory

BACKGROUND

High concentrations of enflurane will induce a characteristic electroencephalogram pattern consisting of periods of suppression alternating with large short paroxysmal epileptiform discharges (PEDs). In this study, we compared a theoretical computer model of this activity with real local field potential (LFP) data obtained from anesthetized rats.

METHODS

After implantation of a high-density 8 x 8 electrode array in the visual cortex, the patterns of LFP and multiunit spike activity were recorded in rats during 0.5, 1.0, 1.5, and 2.0 minimum alveolar anesthetic concentration (MAC) enflurane anesthesia. These recordings were compared with computer simulations from a mean field model of neocortical dynamics. The neuronal effect of increasing enflurane concentration was simulated by prolonging the decay time constant of the inhibitory postsynaptic potential (IPSP). The amplitude of the excitatory postsynaptic potential (EPSP) was modulated, inverse to the neocortical firing rate.

RESULTS

In the anesthetized rats, increasing enflurane concentrations consistently caused the appearance of suppression pattern (>1.5 MAC) in the LFP recordings. The mean rate of multiunit spike activity decreased from 2.54/s (0.5 MAC) to 0.19/s (2.0 MAC). At high MAC, the majority of the multiunit action potential events became synchronous with the PED. In the theoretical model, prolongation of the IPSP decay time and activity-dependent EPSP modulation resulted in output that was similar in morphology to that obtained from the experimental data. The propensity for rhythmic seizure-like activity in the model could be determined by analysis of the eigenvalues of the equations.

CONCLUSION

It is possible to use a mean field theory of neocortical dynamics to replicate the PED pattern observed in LFPs in rats under enflurane anesthesia. This pattern requires a combination of a moderately increased total area under the IPSP, prolonged IPSP decay time, and also activity-dependent modulation of EPSP amplitude.

EEG generator--a model of potentials in a volume conductor

EEG generator-a model of potentials in a volume conductor. The potential recorded over the cortex electro-corticogram (ECoG) or over the scalp [electroencephalograph (EEG)] derives from the activity of many sources known as "EEG generators." The recorded amplitude is basically a function of the unitary potential of a generator and the statistical relationship between different EEG generators in the recorded population. In this study, we first suggest a new definition of the EEG generator. We use the theory of potentials in a volume conductor and model the contribution of a single synapse activated to the surface potential. We then model the contribution of the generator to the surface potential. Once the generator and its contribution are well defined, we can quantitatively assess the degree of synchronization among generators. The measures obtained by the model for a real life scenario of a group of generators organized in a specific statistical way were consistent with the expected values that were reported experimentally. The study sheds new light on macroscopic modeling approaches which make use of mean soma membrane potential. We showed major contribution of activity of superficial apical synapses to the ECoG signal recorded relative to lower somatic or basal synapses activity.

Modeling brain activation patterns for the default and cognitive states

We argue that spatial patterns of cortical activation observed with EEG, MEG and fMRI might arise from spontaneous self-organisation of interacting populations of excitatory and inhibitory neurons. We examine the dynamical behavior of a mean-field cortical model that includes chemical and electrical (gap-junction) synapses, focusing on two limiting cases: the "slow-soma" limit with slow voltage feedback from soma to dendrite, and the "fast-soma" limit in which the feedback action of soma voltage onto dendrite reversal potentials is instantaneous. For slow soma-dendrite feedback, we find a low-frequency (approximately 1 Hz) dynamic Hopf instability, and a stationary Turing instability that catalyzes formation of patterned distributions of cortical firing-rate activity with pattern wavelength approximately 2 cm. Turing instability can only be triggered when gap-junction diffusion between inhibitory neurons is strong, but patterning is destroyed if the tonic level of subcortical excitation is raised sufficiently. Interaction between the Hopf and Turing instabilities may describe the non-cognitive background or "default" state of the brain, as observed by BOLD imaging. In the fast-soma limit, the model predicts a high-frequency Hopf (approximately 35 Hz) instability, and a traveling-wave gamma-band instability that manifests as a 2-D standing-wave pattern oscillating in place at approximately 30 Hz. Small levels of inhibitory diffusion enhance and broaden the definition of the gamma antinodal regions by suppressing higher-frequency spatial modes, but gamma emergence is not contingent on the presence of inhibitory gap junctions; higher levels of diffusion suppress gamma activity. Fast-soma instabilities are enhanced by increased subcortical stimulation. Prompt soma-dendrite feedback may be an essential component of the genesis and large-scale cortical synchrony of gamma activity observed at the point of cognition.

Population based models of cortical drug response: insights from anaesthesia

A great explanatory gap lies between the molecular pharmacology of psychoactive agents and the neurophysiological changes they induce, as recorded by neuroimaging modalities. Causally relating the cellular actions of psychoactive compounds to their influence on population activity is experimentally challenging. Recent developments in the dynamical modelling of neural tissue have attempted to span this explanatory gap between microscopic targets and their macroscopic neurophysiological effects via a range of biologically plausible dynamical models of cortical tissue. Such theoretical models allow exploration of neural dynamics, in particular their modification by drug action. The ability to theoretically bridge scales is due to a biologically plausible averaging of cortical tissue properties. In the resulting macroscopic neural field, individual neurons need not be explicitly represented (as in neural networks). The following paper aims to provide a non-technical introduction to the mean field population modelling of drug action and its recent successes in modelling anaesthesia.

Characteristics of temporal fluctuations in the hyperpolarized state of the cortical slow oscillation

We present evidence for the hypothesis that transitions between the low- and high-firing states of the cortical slow oscillation correspond to neuronal phase transitions. By analyzing intracellular recordings of the membrane potential during the cortical slow oscillation in rats, we quantify the temporal fluctuations in power and the frequency centroid of the power spectrum in the period of time before "down" to "up" transitions. By taking appropriate averages over such events, we present these statistics as a function of time before transition. The results demonstrate an increase in fluctuation power and time scale broadly consistent with the slowing of systems close to phase transitions. The analysis is complicated and limited by the difficulty in identifying when transitions begin, and removing dc trends in membrane potential.

Anesthetic-induced transitions by propofol modeled by nonlocal neural populations involving two neuron types

The present work derives the spatiotemporal field equation of neural populations considering two types of neurons. The model considers pyramidal cells, which may excite or inhibit other neurons, and GABAergic interneurons inhibiting terminal neurons. Additionally, taking into account excitatory and inhibitory synapses, the neural population obeys a vector-field equation involving nonlocal spatial interactions. The work studies the effect of the anesthetic agent propofol, which increases the decay time of inhibitory synapses. In addition, it explains the bifurcation mechanism in some detail and finds a saddle-node bifurcation subject to the propofol concentration. This bifurcation may model the transition between consciousness and nonconsciousness and vice versa during the administration of general anesthetics in medicine.

Population dynamics: variance and the sigmoid activation function

This paper demonstrates how the sigmoid activation function of neural-mass models can be understood in terms of the variance or dispersion of neuronal states. We use this relationship to estimate the probability density on hidden neuronal states, using non-invasive electrophysiological (EEG) measures and dynamic casual modelling. The importance of implicit variance in neuronal states for neural-mass models of cortical dynamics is illustrated using both synthetic data and real EEG measurements of sensory evoked responses.

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.

Monitoring of global cerebral ischemia using instantaneous phase variation plots

In this paper, the derivative of the instantaneous phase of electroencephalographic (EEG) signals is used as a basis for monitoring of global cerebral ischemia. Visual and quantitative results were obtained from six rodents that were subject to 3, 5 and 7 minutes of global ischemic brain injury by asphyxic cardiac arrest. Results show that the variations in the instantaneous phase are capable of amplifying the variations during the various stages of the recovery process and may serve as a novel analytical approach to grade and classify brain rhythms during global ischemic brain injury and recovery.

Modeling electrocortical activity through improved local approximations of integral neural field equations

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular, we are able to treat "patchy" connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a "lattice-directed" traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs.

Neural rate equations for bursting dynamics derived from conductance-based equations

A method of obtaining rate equations from conductance-based equations is developed and applied to fast-spiking and bursting neocortical neurons. It involves splitting systems of conductance-based equations into fast and slow subsystems, and averaging the effects of fast terms that drive the slowly varying quantities by showing that their average is closely proportional to the firing rate. The dependence of the firing rate on the injected current is then approximated in the analysis. The resulting behavior of the slow variables is then substituted back into the fast equations, with the further approximation of replacing the fast voltages in these terms by effective values. For bursting neurons the method yields two coupled limit-cycle oscillators: a self-exciting oscillator for the slow variables that commences limit-cycle oscillations at a critical current and modulates a fast spike-generating oscillator, thereby leading to slowly modulated bursts with a group of spikes in each burst. The dynamics of these coupled oscillators are then verified against those of the conductance-based equations. Finally, it is shown how to place the results in a form suitable for use in mean-field equations for neural population dynamics.

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.

Gap junctions mediate large-scale Turing structures in a mean-field cortex driven by subcortical noise

One of the grand puzzles in neuroscience is establishing the link between cognition and the disparate patterns of spontaneous and task-induced brain activity that can be measured clinically using a wide range of detection modalities such as scalp electrodes and imaging tomography. High-level brain function is not a single-neuron property, yet emerges as a cooperative phenomenon of multiply-interacting populations of neurons. Therefore a fruitful modeling approach is to picture the cerebral cortex as a continuum characterized by parameters that have been averaged over a small volume of cortical tissue. Such mean-field cortical models have been used to investigate gross patterns of brain behavior such as anesthesia, the cycles of natural sleep, memory and erasure in slow-wave sleep, and epilepsy. There is persuasive and accumulating evidence that direct gap-junction connections between inhibitory neurons promote synchronous oscillatory behavior both locally and across distances of some centimeters, but, to date, continuum models have ignored gap-junction connectivity. In this paper we employ simple mean-field arguments to derive an expression for D2, the diffusive coupling strength arising from gap-junction connections between inhibitory neurons. Using recent neurophysiological measurements reported by Fukuda [J. Neurosci. 26, 3434 (2006)], we estimate an upper limit of D2 approximately 0.6cm2. We apply a linear stability analysis to a standard mean-field cortical model, augmented with gap-junction diffusion, and find this value for the diffusive coupling strength to be close to the critical value required to destabilize the homogeneous steady state. Computer simulations demonstrate that larger values of D2 cause the noise-driven model cortex to spontaneously crystalize into random mazelike Turing structures: centimeter-scale spatial patterns in which regions of high-firing activity are intermixed with regions of low-firing activity. These structures are consistent with the spatial variations in brain activity patterns detected with the BOLD (blood oxygen-level-dependent) signal detected with magnetic resonance imaging, and may provide a natural substrate for synchronous gamma-band rhythms observed across separated EEG (electroencephalogram) electrodes.

A neural mass model of spectral responses in electrophysiology

We present a neural mass model of steady-state membrane potentials measured with local field potentials or electroencephalography in the frequency domain. This model is an extended version of previous dynamic causal models for investigating event-related potentials in the time-domain. In this paper, we augment the previous formulation with parameters that mediate spike-rate adaptation and recurrent intrinsic inhibitory connections. We then use linear systems analysis to show how the model's spectral response changes with its neurophysiological parameters. We demonstrate that much of the interesting behaviour depends on the non-linearity which couples mean membrane potential to mean spiking rate. This non-linearity is analogous, at the population level, to the firing rate–input curves often used to characterize single-cell responses. This function depends on the model's gain and adaptation currents which, neurobiologically, are influenced by the activity of modulatory neurotransmitters. The key contribution of this paper is to show how neuromodulatory effects can be modelled by adding adaptation currents to a simple phenomenological model of EEG. Critically, we show that these effects are expressed in a systematic way in the spectral density of EEG recordings. Inversion of the model, given such non-invasive recordings, should allow one to quantify pharmacologically induced changes in adaptation currents. In short, this work establishes a forward or generative model of electrophysiological recordings for psychopharmacological studies.

Comparison of spectral analysis methods for characterizing brain oscillations

Spectral analysis methods are now routinely used in electrophysiological studies of human and animal cognition. Although a wide variety of spectral methods has been used, the ways in which these methods differ are not generally understood. Here we use simulation methods to characterize the similarities and differences between three spectral analysis methods: wavelets, multitapers and P(episode). P(episode) is a novel method that quantifies the fraction of time that oscillations exceed amplitude and duration thresholds. We show that wavelets and P(episode) used side-by-side helps to disentangle length and amplitude of a signal. P(episode) is especially sensitive to fluctuations around its thresholds, puts frequencies on a more equal footing, and is sensitive to long but low-amplitude signals. In contrast, multitaper methods are less sensitive to weak signals, but are very frequency-specific. If frequency specificity is not essential, then wavelets and P(episode) are recommended.

White-noise susceptibility and critical slowing in neurons near spiking threshold

We present mathematical and simulation analyses of the below-threshold noisy response of two biophysically motivated models for excitable membrane due to H. R. Wilson: a squid axon ("resonator") and a human cortical neuron ("integrator"). When stimulated with a low-intensity white noise superimposed on a dc control current, both membrane types generate voltage fluctuations that exhibit critical slowing down--that is, the voltage responsiveness to noisy input currents grows in amplitude while slowing in frequency--as the membrane approaches spiking threshold from below. We define threshold unambiguously as that dc current that renders a zero real eigenvalue for the Jacobian matrix for the integrator neuron, and, for the resonator neuron, as the dc current that gives a complex eigenvalue pair whose real part is zero. Using a linear Ornstein-Uhlenbeck analysis, we give exact small-noise expressions for the variance, power spectrum, and correlation function of the voltage fluctuations, and we derive the scaling laws for the divergence of susceptibility and correlation times for approach to threshold. We compare these predictions with numerical simulations of the nonlinear stochastic equations, and demonstrate that, provided the white-noise perturbations are kept sufficiently small, the linearized theory works well. These predictions should be testable in the laboratory using a current-clamped cell configuration. If confirmed, then the proximity of a neuron to its spike-transition point can be judged by measuring its subthreshold susceptibility to white-noise stimulation. We postulate that such temporally correlated fluctuations could provide a means of subthreshold signaling via gap-junction connections with neighboring neurons.

Pathological pattern formation and cortical propagation of epileptic seizures

The stochastic partial differential equations (SPDEs) stated by Steyn-Ross and co-workers constitute a model of mesoscopic electrical activity of the human cortex. A simplification in which spatial variation and stochastic input are neglected yields ordinary differential equations (ODEs), which are amenable to analysis by techniques of dynamical systems theory. Bifurcation diagrams are developed for the ODEs with increased subcortical excitation, showing that the model predicts oscillatory electrical activity in a large range of parameters. The full SPDEs with increased subcortical excitation produce travelling waves of electrical activity. These model results are compared with electrocortical data recorded at two subdural electrodes from a human subject undergoing a seizure. The model and observational results agree in two important respects during seizure: (i) the average frequency of maximum power, and (ii) the speed of spatial propagation of voltage peaks. This suggests that seizing activity on the human cortex may be understood as an example of pathological pattern formation. Included is a discussion of the applications and limitations of these results.

Bifurcation control of a seizing human cortex

We consider as a mathematical model of human cortical electrical activity a system of fourteen ordinary differential equations. With appropriate parameters, the model produces activity characteristic of a seizure. To prevent such seizures, we incorporate feedback controllers into the model dynamics. We show that three controllers--a linear feedback controller, a differential controller, and a filter controller--can be used to eliminate seizing activity in the model system. We show how bifurcations induced by the linear controller alter those present in the original dynamics.

The sleep cycle modelled as a cortical phase transition

We present a mean-field model of the cortex that attempts to describe the gross changes in brain electrical activity for the cycles of natural sleep. We incorporate within the model two major sleep modulatory effects: slow changes in both synaptic efficiency and in neuron resting voltage caused by the ∼90-min cycling in acetylcholine, together with even slower changes in resting voltage caused by gradual elimination during sleep of somnogens (fatigue agents) such as adenosine. We argue that the change from slow-wave sleep (SWS) to rapid-eye-movement (REM) sleep can be understood as a first-order phase transition from a low-firing, coherent state to a high-firing, desychronized cortical state. We show that the model predictions for changes in EEG power, spectral distribution, and correlation time at the SWS-to-REM transition are consistent not only with those observed in clinical recordings of a sleeping human subject, but also with the on-cortex EEG patterns recently reported by Destexhe et al. [J. Neurosci.19(11), (1999) 4595-4608] for the sleeping cat.

Proposed mechanism for learning and memory erasure in a white-noise-driven sleeping cortex

Understanding the structure and purpose of sleep remains one of the grand challenges of neurobiology. Here we use a mean-field linearized theory of the sleeping cortex to derive statistics for synaptic learning and memory erasure. The growth in correlated low-frequency high-amplitude voltage fluctuations during slow-wave sleep (SWS) is characterized by a probability density function that becomes broader and shallower as the transition into rapid-eye-movement (REM) sleep is approached. At transition, the Shannon information entropy of the fluctuations is maximized. If we assume Hebbian-learning rules apply to the cortex, then its correlated response to white-noise stimulation during SWS provides a natural mechanism for a synaptic weight change that will tend to shut down reverberant neural activity. In contrast, during REM sleep the weights will evolve in a direction that encourages excitatory activity. These entropy and weight-change predictions lead us to identify the final portion of deep SWS that occurs immediately prior to transition into REM sleep as a time of enhanced erasure of labile memory. We draw a link between the sleeping cortex and Landauer's dissipation theorem for irreversible computing [R. Landauer, IBM J. Res. Devel. 5, 183 (1961)], arguing that because information erasure is an irreversible computation, there is an inherent entropy cost as the cortex transits from SWS into REM sleep.

Predictions and simulations of cortical dynamics during natural sleep using a continuum approach

In this paper we use a continuum model in two spatial dimensions to study the dynamics of the cortex during natural sleep, including explicitly the effects of two key neuromodulators. The model predicts that a number of states could be available to the cortex. We identify two of these with slow-wave sleep and rapid eye movement (REM) sleep, and focus on the transition between the two. Eigenvalue analysis of the linearized model, together with simulations on a two-dimensional grid, show that a number of oscillatory states exist; the occurrence of these is particularly dependent upon the duration in time of the inhibitory postsynaptic potential. These oscillatory states are similar to the cortical slow oscillation and certain types of seizure. Power spectra are evaluated for different parameter sets and compare favorably with experiment. Grid simulations show that transitions between cortical states (e.g., slow-wave to REM) can be seeded at any point in space by random fluctuations in subcortical input.

Ongoing spontaneous activity controls access to consciousness: a neuronal model for inattentional blindness

Even in the absence of sensory inputs, cortical and thalamic neurons can show structured patterns of ongoing spontaneous activity, whose origins and functional significance are not well understood. We use computer simulations to explore the conditions under which spontaneous activity emerges from a simplified model of multiple interconnected thalamocortical columns linked by long-range, top-down excitatory axons, and to examine its interactions with stimulus-induced activation. Simulations help characterize two main states of activity. First, spontaneous gamma-band oscillations emerge at a precise threshold controlled by ascending neuromodulator systems. Second, within a spontaneously active network, we observe the sudden “ignition” of one out of many possible coherent states of high-level activity amidst cortical neurons with long-distance projections. During such an ignited state, spontaneous activity can block external sensory processing. We relate those properties to experimental observations on the neural bases of endogenous states of consciousness, and particularly the blocking of access to consciousness that occurs in the psychophysical phenomenon of “inattentional blindness,” in which normal subjects intensely engaged in mental activity fail to notice salient but irrelevant sensory stimuli. Although highly simplified, the generic properties of a minimal network may help clarify some of the basic cerebral phenomena underlying the autonomy of consciousness.

Computer simulations of the circuits of activity in the cerebral cortex and underlying thalamus suggest that precisely controlled oscillatory states can control the central neural processing of sensory information

Modeling the effects of anesthesia on the electroencephalogram

Changes to the electroencephalogram (EEG) observed during general anesthesia are modeled with a physiological mean field theory of electrocortical activity. To this end a parametrization of the postsynaptic impulse response is introduced which takes into account pharmacological effects of anesthetic agents on neuronal ligand-gated ionic channels. Parameter sets for this improved theory are then identified which respect known anatomical constraints and predict mean firing rates and power spectra typically encountered in human subjects. Through parallelized simulations of the eight nonlinear, two-dimensional partial differential equations on a grid representing an entire human cortex, it is demonstrated that linear approximations are sufficient for the prediction of a range of quantitative EEG variables. More than 70,000 plausible parameter sets are finally selected and subjected to a simulated induction with the stereotypical inhaled general anesthetic isoflurane. Thereby 86 parameter sets are identified that exhibit a strong "biphasic" rise in total power, a feature often observed in experiments. A sensitivity study suggests that this "biphasic" behavior is distinguishable even at low agent concentrations. Finally, our results are briefly compared with previous work by other groups and an outlook on future fits to experimental data is provided.

Cortical entropy changes with general anaesthesia: theory and experiment

Commonly used general anaesthetics cause a decrease in the spectral entropy of the electroencephalogram as the patient transits from the conscious to the unconscious state. Although the spectral entropy is a configurational entropy, it is plausible that the spectral entropy may be acting as a reliable indicator of real changes in cortical neuronal interactions. Using a mean field theory, the activity of the cerebral cortex may be modelled as fluctuations in mean soma potential around equilibrium states. In the adiabatic limit, the stochastic differential equations take the form of an Ornstein-Uhlenbeck process. It can be shown that spectral entropy is a logarithmic measure of the rate of synaptic interaction. This model predicts that the spectral entropy should decrease abruptly from values approximately 1.0 to values of approximately 0.7 as the patient becomes unconscious during induction of general anaesthesia, and then not decrease significantly on further deepening of anaesthesia. These predictions were compared with experimental results in which electrocorticograms and brain concentrations of propofol were recorded in seven sheep during induction of anaesthesia with intravenous propofol. The observed changes in spectral entropy agreed with the theoretical predictions. We conclude that spectral entropy may be a sensitive monitor of the consciousness-unconsciousness transition, rather than a progressive indicator of anaesthetic drug effect.

Propofol-induced alpha rhythm

The electroencephalographic effects of two intravenous sedative/hypnotic drugs, propofol and thiopental, were studied at three stable blood concentrations in 52 normal healthy volunteers. The higher concentration resulted in unresponsiveness (lack of response to auditory/tactile stimuli) in all subjects. This report describes the strong frontal-central rhythms apparent in this state using a quantitative description of oscillatory systems underlying the rhythm. These rhythms occur when sedative drug concentrations are greater than those producing the well-described increase in broadband beta-power associated with many sedative drugs. Propofol induces rhythms in the alpha-range, while thiopental produces rhythms in the beta-range. Quasistationary for a period of about 1 h, these rhythms exceed the baseline alpha-rhythm in power. By their resonant nature, these propofol-induced rhythms are analogous to 'the classic alpha-rhythm', but quantitative characteristics of the underlying oscillatory systems are different. Baseline properties of the oscillatory system underlying the initial resting alpha-rhythm recover completely as drug concentration decays to negligible values.

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

Since 1997 we have been developing a theoretical foundation for general anaesthesia. We have been able to demonstrate that the abrupt change in brain state brought on by anaesthetic drugs can be characterized as a first-order phase transition in the population-average membrane voltage of the cortical neurons. The theory predicts that, as the critical point of phase change into unconsciousness is approached, the electrical fluctuations in cortical activity will grow strongly in amplitude while slowing in frequency, becoming more correlated both in time and in space. Thus the bio-electrical change of brain-state has deep similarities with thermodynamic phase changes of classical physics. The theory further predicts the existence of a second critical point, hysteretically separated from the first, corresponding to the return path from comatose unconsciousness back to normal responsiveness. There is a steadily accumulating body of clinical evidence in support of all of the phase-transition predictions: low-frequency power surge in EEG activity; increased correlation time and correlation length in EEG fluctuations; hysteresis separation, with respect to drug concentration, between the point of induction and the point of emergence.

Nonlinear analysis of EEG signals at different mental states

Background

The EEG (Electroencephalogram) is a representative signal containing information about the condition of the brain. The shape of the wave may contain useful information about the state of the brain. However, the human observer can not directly monitor these subtle details. Besides, since bio-signals are highly subjective, the symptoms may appear at random in the time scale. Therefore, the EEG signal parameters, extracted and analyzed using computers, are highly useful in diagnostics. This work discusses the effect on the EEG signal due to music and reflexological stimulation.

Methods

In this work, nonlinear parameters like Correlation Dimension (CD), Largest Lyapunov Exponent (LLE), Hurst Exponent (H) and Approximate Entropy (ApEn) are evaluated from the EEG signals under different mental states.

Results

The results obtained show that EEG to become less complex relative to the normal state with a confidence level of more than 85% due to stimulation.

Conclusions

It is found that the measures are significantly lower when the subjects are under sound or reflexologic stimulation as compared to the normal state. The dimension increases with the degree of the cognitive activity. This suggests that when the subjects are under sound or reflexologic stimuli, the number of parallel functional processes active in the brain is less and the brain goes to a more relaxed state

Drug-induced modification of the system properties associated with spontaneous human electroencephalographic activity

The benzodiazepine (BZ) class of minor tranquilizers are important modulators of the gamma-amino butyric acid (GABA(A))/BZ receptor complex that are well known to affect the spectral properties of spontaneous electroencephalographic activity. While it is experimentally well established that the BZs reduce total alpha band (8-13 Hz) power and increase total beta band (13-30 Hz) power, it is unclear what the physiological basis for this effect is. Based on a detailed theory of cortical electrorhythmogenesis it is conjectured that such an effect is explicable in terms of the modulation of GABAergic neurotransmission within locally connected populations of excitatory and inhibitory cortical neurons. Motivated by this theory, fixed order autoregressive moving average (ARMA) models were fitted to spontaneous eyes-closed electroencephalograms recorded from subjects before and approximately 2 h after the oral administration of a single 1 mg dose of the BZ alprazolam. Subsequent pole-zero analysis revealed that BZs significantly transform the dominant system pole such that its frequency and damping increase. Comparisons of ARMA derived power spectra with fast Fourier transform derived spectra indicate an enhanced ability to identify benzodiazepine induced electroencephalographic changes. This experimental result is in accord with the theoretical predictions implying that alprazolam enhances inhibition acting on inhibitory neurons more than inhibition acting on excitatory neurons. Further such a result is consistent with reported cortical neuronal distributions of the various GABA(A) receptor pharmacological subtypes. Therefore physiologically specified fixed order ARMA modeling is expected to become an important tool for the systematic investigation and modeling of a wide range of cortically acting compounds.

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.

Spatiotemporal forward solution of the EEG and MEG using network modeling

Dynamic systems have proven to be well suited to describe a broad spectrum of human coordination behavior such synchronization with auditory stimuli. Simultaneous measurements of the spatiotemporal dynamics of electroencephalographic (EEG) and magnetoencephalographic (MEG) data reveals that the dynamics of the brain signals is highly ordered and also accessible by dynamic systems theory. However, models of EEG and MEG dynamics have typically been formulated only in terms of phenomenological modeling such as fixed-current dipoles or spatial EEG and MEG patterns. In this paper, it is our goal to connect three levels of organization, that is the level of coordination behavior, the level of patterns observed in the EEG and MEG and the level of neuronal network dynamics. To do so, we develop a methodological framework, which defines the spatiotemporal dynamics of neural ensembles, the neural field, on a sphere in three dimensions. Using magnetic resonance imaging we map the neural field dynamics from the sphere onto the folded cortical surface of a hemisphere. The neural field represents the current flow perpendicular to the cortex and, thus, allows for the calculation of the electric potentials on the surface of the skull and the magnetic fields outside the skull to be measured by EEG and MEG, respectively. For demonstration of the dynamics, we present the propagation of activation at a single cortical site resulting from a transient input. Finally, a mapping between finger movement profile and EEG/MEG patterns is obtained using Volterra integrals.

Toward a theory of the general-anesthetic-induced phase transition of the cerebral cortex. I. A thermodynamics analogy

In a recent paper the authors developed a stochastic model for the response of the cerebral cortex to a general anesthetic agent. The model predicted that there would be an anesthetic-induced phase change at the point of transition into unconsciousness, manifested as a divergence in the electroencephalogram spectral power, and a change in spectral energy distribution from being relatively broadband in the conscious state to being strongly biased towards much lower frequencies in the unconscious state. Both predictions have been verified in recent clinical measurements. In the present paper we extend the model by calculating the equilibrium distribution function for the cortex, allowing us to establish a correspondence between the cortical phase transition and the more familiar thermodynamic phase transitions. This correspondence is achieved by first identifying a cortical free energy function, then by postulating that there exists an inverse relationship between an anesthetic effect and a quantity we define as cortical excitability, which plays a role analogous to temperature in thermodynamic phase transitions. We follow standard thermodynamic theory to compute a cortical entropy and a cortical "heat capacity," and we investigate how these will vary with anesthetic concentration. The significant result is the prediction that the entropy will decrease discontinuously at the moment of induction into unconsciousness, concomitant with a release of "latent heat" which should manifest as a divergence in the analogous heat capacity. There is clear clinical evidence of heat capacity divergence in historical anesthetic-effect measurements performed in 1977 by Stullken et al. [Anesthesiology 46, 28 (1977)]. The discontinuous step change in cortical entropy suggests that the cortical phase transition is analogous to a first-order thermodynamic transition in which the comatose-quiescent state is strongly ordered, while the active cortical state is relatively disordered.