David T. J. Liley

A spatially continuous mean field theory of electrocortical activity

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

Simulation of electrocortical waves

We report simulations of the electrocorticogram of the cat and human, based on estimates of fibre range, fibre density, axonal and dendritic delays, and cortical synaptic density. The long-range cortical connections of real cortex were simplified to couplings of symmetric density, decreasing in density with range, on a closed (toroidal) surface. Non-specific cortical activation was modelled as a diffuse global input and specific sensory input as a localised white noise input. Spectral properties of output included peak densities at the frequencies of the major cerebral rhythms, a ‘1/f’ spectral envelope and ‘shift to the right’ with increasing total power as non-specific activation increased. Steady-state travelling waves with a velocity of 5–7 m/s (human) and < 1 m/s (cat) were produced. Frequency/wavenumber analysis revealed an additional class of activity with wavenumbers independent of temporal frequency. All these findings accord qualitatively and quantitatively with existing physiological results. Global resonant modes were not prominent, but the simulations obey a restricted case of the analytical results of Nunez (1994). Wave/pulse relations resemble the findings of Freeman (1975).

A Continuum Theory of Electro-Cortical Activity

Abstract A set of non-linear continuum field equations are presented which describe the macroscopic dynamics of neural activity in cortex. Numerical solutions of the coupled non-linear system of partial differential equations show properties analogous to cortical evoked potentials, oscillations at the frequency of the mammalian alpha rhythm and non-stationary epileptic spikes.

Robust chaos in a model of the electroencephalogram: Implications for brain dynamics

Various techniques designed to extract nonlinear characteristics from experimental time series have provided no clear evidence as to whether the electroencephalogram (EEG) is chaotic. Compounding the lack of firm experimental evidence is the paucity of physiologically plausible theories of EEG that are capable of supporting nonlinear and chaotic dynamics. Here we provide evidence for the existence of chaotic dynamics in a neurophysiologically plausible continuum theory of electrocortical activity and show that the set of parameter values supporting chaos within parameter space has positive measure and exhibits fat fractal scaling. (c) 2001 American Institute of Physics.

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.

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.

Propofol and Remifentanil Differentially Modulate Frontal Electroencephalographic Activity

Background:The purpose of this study was to evaluate a new, physiologically inspired method for the analysis of the electroencephalogram during propofol–remifentanil anesthesia. Based on fixed-order autoregressive moving-average modeling, this method was hypothesized to be capable of dissociating the effects that hypnotic and analgesic agents have on brain electrical activity. Methods:Raw electroencephalographic waves from a previously published study were reanalyzed. In this study, 45 American Society of Anesthesiologists status I patients were randomly allocated to one of three groups according to a specific target effect-site remifentanil concentration (0, 2, and 4 ng/ml). All patients received stepwise-increased targeted effect-site concentrations of propofol (CePROP). At each step change in target CePROP, the Observers Assessment of Alertness/Sedation score was evaluated. Raw electroencephalograph was continuously acquired from frontal electrodes. Electroencephalography traces were analyzed using a fixed-order autoregressive moving average model to give derived measures of Cortical State and Cortical Input. Response surfaces were visualized and modeled using Hierarchical Linear Modeling. Results:Cortical State (a measure of cortical responsiveness) and Cortical Input (a measure of the magnitude of cortical input) were shown to respond differently to CePROP and effect-site remifentanil concentration. Cortical Input decreased significantly with increasing effect-site remifentanil concentration, whereas Cortical State remained unchanged with increasing effect-site remifentanil concentration but decreased with increasing CePROP. Conclusion:Because Cortical State responds principally to variations in CePROP, it is a potential measure of hypnosis, whereas the dependence of Cortical Input on effect-site remifentanil concentration suggests that it may be useful as a measure of analgesic efficacy and the nociceptive–antinociceptive balance.

The Mesoscopic Modeling of Burst Suppression during Anesthesia

The burst-suppression pattern is well recognized as a distinct feature of the mammalian electroencephalogram (EEG) waveform. Consisting of alternating periods of high amplitude oscillatory and isoelectric activity, it can be induced in health by deep anesthesia as well as being evoked by a range of pathophysiological processes that include coma and anoxia. While the electroencephalographic phenomenon and clinical implications of burst suppression have been studied extensively, the physiological mechanisms underlying its emergence remain unresolved and obscure. Because electroencephalographic bursting phenomenologically resembles the bursting observed in single neurons, it would be reasonable to assume that the theoretical insights developed to understand bursting at the cellular (“microscopic”) level would enable insights into the dynamical genesis of bursting at the level of the whole brain (“macroscopic”). In general action potential bursting is the result of the interplay of two time scales: a fast time scale responsible for spiking, and a slow time scale that modulates such activity. We therefore hypothesize that such fast-slow systems dynamically underpin electroencephalographic bursting. Here we show that a well-known mean field dynamical model of the electroencephalogram, the Liley model, while unable to produce burst suppression unmodified, is able to give rise to a wide variety of burst-like activity by the addition of one or more slow systems modulating model parameters speculated to be major “targets” for anesthetic action. The development of a physiologically plausible theoretical framework to account for burst suppression will lead to a more complete physiological understanding of the EEG and the mechanisms that serve to modify ongoing brain activity necessary for purposeful behavior and consciousness.

Self-organized 40Hz synchronization in a physiological theory of EEG

We present evidence that large-scale spatial coherence of 40Hz oscillations can emerge dynamically in a cortical mean field theory. The simulated synchronization time scale is about 150ms, which compares well with experimental data on large-scale integration during cognitive tasks. The same model has previously provided consistent descriptions of the human EEG at rest, with tranquilizers, under anesthesia, and during anesthetic-induced epileptic seizures. The emergence of coherent gamma band activity is brought about by changing just one physiological parameter until cortex becomes marginally unstable for a small range of wavelengths. This suggests for future study a model of dynamic computation at the edge of cortical stability.

Ketamine, propofol, and the EEG: a neural field analysis of HCN1-mediated interactions

Ketamine and propofol are two well-known, powerful anesthetic agents, yet at first sight this appears to be their only commonality. Ketamine is a dissociative anesthetic agent, whose main mechanism of action is considered to be N-methyl-d-aspartate (NMDA) antagonism; whereas propofol is a general anesthetic agent, which is assumed to primarily potentiate currents gated by γ-aminobutyric acid type A (GABAA) receptors. However, several experimental observations suggest a closer relationship. First, the effect of ketamine on the electroencephalogram (EEG) is markedly changed in the presence of propofol: on its own ketamine increases θ (4–8 Hz) and decreases α (8–13 Hz) oscillations, whereas ketamine induces a significant shift to beta band frequencies (13–30 Hz) in the presence of propofol. Second, both ketamine and propofol cause inhibition of the inward pacemaker current Ih, by binding to the corresponding hyperpolarization-activated cyclic nucleotide-gated potassium channel 1 (HCN1) subunit. The resulting effect is a hyperpolarization of the neuron’s resting membrane potential. Third, the ability of both ketamine and propofol to induce hypnosis is reduced in HCN1-knockout mice. Here we show that one can theoretically understand the observed spectral changes of the EEG based on HCN1-mediated hyperpolarizations alone, without involving the supposed main mechanisms of action of these drugs through NMDA and GABAA, respectively. On the basis of our successful EEG model we conclude that ketamine and propofol should be antagonistic to each other in their interaction at HCN1 subunits. Such a prediction is in accord with the results of clinical experiment in which it is found that ketamine and propofol interact in an infra-additive manner with respect to the endpoints of hypnosis and immobility.