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In this chapter we investigate a range of dynamic behaviors accessible to a continuum model of the cerebral cortex placed close to the anesthetic phase transition. If the anesthetic transition from the high-firing (conscious) to the low-firing (comatose) state can be modeled as a jump between two equilibrium states of the cortex, then we can draw an analogy with the vapor-to-liquid phase transition of the van der Waals gas of classical thermodynamics. In this analogy, specific volume (inverse density) of the gas maps to cortical activity, with pressure and temperature being the analogs of anesthetic concentration and subcortical excitation. It is well known that at the thermodynamic critical point, large fluctuations in specific volume are observed; we find analogous critically-slowed fluctuations in cortical activity at its critical point. Unlike the van der Waals system, the cortical model can also exhibit nonequilibrium phase transitions in which the homogeneous equilibrium can destabilize in favor of slow global oscillations (Hopf temporal instability), stationary structures (Turing spatial instability), and chaotic spatiotemporal activity patterns (Hopf–Turing interactions). We comment on possible physiological and pathological interpretations for these dynamics. In particular, the turbulent state may correspond to the cortical slow oscillation between “up” and “down” states observed in nonREM sleep and clinical anesthesia.

The proliferation of electric vehicles (EV) creates a need for fast battery chargers. In order for systems of supercapacitors (SC) to reach adequate energy density levels to replace battery packs, fast SC chargers will also be required. However, when a fully discharged supercapacitor pack is charged from an ideal voltage source, the five-time-constant charging bottleneck comes into play. This issue can be addressed by using the high-voltage charging topology proposed here. The new approach partitions the SC bank into two parts. One part is charged by a high voltage charging source, and the other is replenished by the energy stored in a coupled inductor within the charging path used for the first half of the bank. Early experimental results for a 30 V, 15 F SC bank are presented.

Supercapacitor circulation techniques can be used to improve the end-to-end efficiency of linear regulators based on commercial low drop-out regulators (LDO). In this approach a single or multiple supercapacitors are used in series and parallel to the input of a LDO IC and circulated at a very low frequency, to increase the end-to-end efficiency by a multiplication factor compared with the efficiency of the linear regulator circuit. The paper presents the essential theory of the supercapacitor circulation, together with MATLAB simulation results compared with experimental results of a 12V to 5V implementation.

We present a model for the dynamics of a cerebral cortex in which inputs to neuronal assemblies are treated as random Gaussian fluctuations about a mean value. We incorporate the effect of general anesthetic agents on the cortex as a modulation of the inhibitory neurotransmitter rate constant. Stochastic differential equations are derived for the state variable ${h}_{e},$ the average excitatory soma potential, coherent fluctuations of which are believed to be the source of scalp-measured electroencephalogram (EEG) signals. Using this stochastic approach we derive a stationary (long-time limit) fluctuation spectrum for ${h}_{e}.$ The model predicts that there will be three distinct stationary (equilibrium) regimes for cortical activity. In region I (``coma''), corresponding to a strong inhibitory anesthetic effect, ${h}_{e}$ is single valued, large, and negative, so that neuronal firing rates are suppressed. In region II for a zero or small anesthetic effect, ${h}_{e}$ can take on three values, two of which are stable; we label the stable solutions as ``active'' (enhanced firing) and ``quiescent'' (suppressed firing). For region III, corresponding to negative anesthetic (i.e., analeptic) effect, ${h}_{e}$ again becomes single valued, but is now small and negative, resulting in strongly elevated firing rates (``seizure''). If we identify region II as associated with the conscious state of the cortex, then the model predicts that there will be a rapid transit between the active-conscious and comatose unconscious states at a critical value of anesthetic concentration, suggesting the existence of phase transitions in the cortex. The low-frequency spectral power in the ${h}_{e}$ signal should increase strongly during the initial stage of anesthesia induction, before collapsing to much lower values after the transition into comatose-unconsciousness. These qualitative predictions are consistent with clinical measurements by B\"uhrer et al. [Anaesthesiology 77, 226 (1992)], MacIver et al. [ibid. 84, 1411 (1996)], and Kuizenga et al. [Br. J. Anaesthesia 80, 725 (1998)]. This strong increase in EEG spectral power in the vicinity of the critical point is similar to the divergences observed during thermodynamic phase transitions. We show that the divergence in low-frequency power in our model is a natural consequence of the existence of turning points in the trajectory of stationary states for the cortex.

Methods used to infer sea surface temperatures (SSTs) from satellite have traditionally been based on regression-tuned split-window fixed-coefficient algorithms. These can give inaccurate SST results when local atmospheric conditions are significantly different from those encapsulated by the regression averaging. The new generation of SST algorithms attempts to correct for atmospheric variability. These approaches include the R54 transmittance-ratio methods of other workers, and the dynamic water vapour (DWV) correction method of the authors. The relative performances of the various methods are compared by applying each to an ocean and satellite dataset obtained off the west coast of Tasmania, Australia in 1987. We also investigate the performance of the NESDIS operational multi-channel, cross-product, and nonlinear formulas for NOAA-9, -11, -12, and-14 when applied to the same dataset. We find the DWV method gives SST retrievals which have significantly smaller bias errors than those returned by the thre...

We present a method which permits retrievals of land surface temperatures (LSTs) from AVHRR (advanced very high resolution radiometer) radiances sensed through atmospheres which may contain a large and strongly varying water vapor content. This new method is an extension of the dynamic water vapor (DWV) algorithm which was designed to retrieve sea surface temperatures. The generalization to LST retrievals recognizes that in general, land emissivities are unknown, may be spectrally dependent, and are less than unity. Because the LST retrieval problem is inherently underconstrained (there are more unknowns than radiative transfer equations), some knowledge of surface emissivity is required in order to establish upper and lower bounds on surface temperature. We demonstrate our method by comparing DWV-LST retrievals with point surface measurements made by a cluster of eight infrared thermometers (IRTs) deployed over a grasslands prairie site in eastern Kansas in July and August 1989; this deployment was part of the First International Satellite Land Surface Climatology Project (ISLSCP) Field Experiment (FIFE). We find that several of the AVHRR images supplied on FIFE CD-ROM contain navigation errors of ∼30 km, consistent with a misidentification of the Tuttle Reservoir ground control point. After correcting the navigation, we identified the IRT pixels and computed the bias and rms errors for a DWV-IRT comparison. For night passes we obtained agreement to.+0.39±1.11 K, while for day passes the comparison yielded +4.09±3.10 K. The large daytime bias is probably the result of the IRT readings not being representative of the ∼1 km2-scale areas sensed by AVHRR (the IRT views vegetation; the AVHRR field of view includes warmer, less well vegetated surfaces). Our results show that while a fixed-coefficient, global split-window algorithm does not perform well in the relatively moist FIFE atmosphere, it is quite feasible to use the DWV-LST to derive a local split-window algorithm whose coefficient is tuned on a per-pass basis.

Electrical recordings of brain activity during the transition from wake to anesthetic coma show temporal and spectral alterations that are correlated with gross changes in the underlying brain state. Entry into anesthetic unconsciousness is signposted by the emergence of large, slow oscillations of electrical activity (≲1 Hz) similar to the slow waves observed in natural sleep. Here we present a two-dimensional mean-field model of the cortex in which slow spatiotemporal oscillations arise spontaneously through a Turing (spatial) symmetry-breaking bifurcation that is modulated by a Hopf (temporal) instability. In our model, populations of neurons are densely interlinked by chemical synapses, and by interneuronal gap junctions represented as an inhibitory diffusive coupling. To demonstrate cortical behavior over a wide range of distinct brain states, we explore model dynamics in the vicinity of a general-anesthetic-induced transition from “wake” to “coma.” In this region, the system is poised at a codimension-2 point where competing Turing and Hopf instabilities coexist. We model anesthesia as a moderate reduction in inhibitory diffusion, paired with an increase in inhibitory postsynaptic response, producing a coma state that is characterized by emergent low-frequency oscillations whose dynamics is chaotic in time and space. The effect of long-range axonal white-matter connectivity is probed with the inclusion of a single idealized point-to-point connection. We find that the additional excitation from the long-range connection can provoke seizurelike bursts of cortical activity when inhibitory diffusion is weak, but has little impact on an active cortex. Our proposed dynamic mechanism for the origin of anesthetic slow waves complements—and contrasts with—conventional explanations that require cyclic modulation of ion-channel conductances. We postulate that a similar bifurcation mechanism might underpin the slow waves of natural sleep and comment on the possible consequences of chaotic dynamics for memory processing and learning.

When DC loads are powered by a DC-microgrid (DCμG) based on a renewable source such as a PV system, energy storage becomes mandatory due to fluctuating nature of the source. Localized DC-energy storage within DC-DC converters could address this requirement. A variation of supercapacitor-assisted low-dropout regulators (SCALDO) could provide localized energy storage with low-noise and fast transient response. For a generalized SCALDO configuration, end-to-end efficiency is given by (1 + k)(V reg /V p ) where k equals n or 1/n, and n is the number of supercapacitors required to step-down an input voltage of V p , to an output regulated voltage of V reg . For SCALDO-based converters, (1 + k) is the efficiency improvement factor of a V p -to-V reg standard linear regulator.

A low frequency supercapacitor circulation technique coupled with a commercial low dropout regulators(LDO), namely the supercapacitor assisted LDO (SCALDO), can achieve significantly high end-to-end efficiency(ETEE) for linear regulators. The ETEE could be closer to the efficiencies of practical switching regulators, but without having the negative aspects of switching regulators such as RFI/EMI issues and the use of bulky inductors. In these supercapacitor assisted linear regulator topologies, the efficiency improvement compared to linear regulators is given by a special figure of merit, efficiency improvement ratio, which can be in the range of 1.33 to 3 depending on the SCALDO configuration. Compared to the six different possible loss elements in a switching regulator, in the SCALDO technique losses are mainly contributed by equivalent series resistance of the supercapacitor, R DS (on) of the switches, parasitics in the PCB traces and losses due to paralleling of the supercapacitor with a small buffer capacitor in addition to the losses of the LDO stage. This paper presents a Laplace transform-based analytical solution to estimate the losses during the four phases of SCALDO technique with an example for a 12V-to-5V SCALDO converter.

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Supercapacitor-based energy recovery techniques can be coupled with low-dropout regulators to enhance the end-to-end efficiency (ETEE) of a linear regulator by multiplication factors such as 1.33, 2 or 3. For example, a 12-to-5 V supercapacitor assisted low-dropout regulator (SCALDO) theoretically achieves 84% ETEE compared to the 42% efficiency of a simple linear regulator. The experimental prototype achieved an approximate ETEE of 80%. The supercapacitor performs as a lossless voltage-dropper in this SCALDO design, cyclically absorbing energy and releasing to the low-dropout (LDO) regulator at frequencies in the order of few millihertz to a few hundreds of hertz. The reduced-switch SCALDO regulator (RS-SCALDO) is an extension of the basic SCALDO topology that improves current handling capabilities and reducing the number of power switches while combining multiple LDOs with a common output rail. VRMs can be designed with digital potentiometer-assisted voltage identification (VID) feature using the RS-SCALDO. The main focus of this paper is to provide implementation details of an RS-SCALDO circuit to design a low-noise VRM that cycles at extremely low frequencies (e.g. 1 to 500 mHz) and to present the experimental results of a 1.5 V, 5 A proof of concept prototype

Electrical recordings of brain activity show that entry into anesthetic unconsciousness is signposted by the emergence of large, slow oscillations of electrical activity (~1 Hz) that appear very similar to the slow waves observed in natural sleep. In this phase, populations of cortical neurons periodically switch between hyperpolarized inactivity (“down” state), and wake-like depolarized activation (“up” state) [1]. The origin of the slow oscillation has not yet been unambiguously established, and remains an area of intense research and debate [2,3]. Here we suggest a novel mechanism in which the up- and down-states are generated spontaneously by emergent chaotic waves of spatiotemporal activity that sweep the cortex. We present a mean-field model of the cortex in which populations of neurons are densely interlinked by both chemical synapses—including idealized long-range spatially heterogeneous connections—and by direct electrical connections forming a continuous network of interneuronal gap junctions. Anesthetic effect is modeled as a moderate reduction in inhibitory diffusion, paired with an increase in inhibitory postsynaptic potential. We explore model dynamics in the vicinity of a general-anesthetic induced transition from wake to coma. In this region the system is poised at a codimension-2 point where competing Turing (spatial) and Hopf (temporal) instabilities co-exist. We argue that normal functioning of the resting “default-wake” brain requires a delicate balance between these instabilities. Reduction of gap-junction diffusivity disturbs the balance in favor of the Hopf instability, eventually predicting global seizure in the limit of severe imbalance. Our cortical model predicts that introduction of anesthetic to the awake brain will force a subtle rebalancing of dynamic pressures resulting in a coma state that is characterized by emergent low-frequency oscillations whose dynamics is chaotic in time and space: see Fig. ​Fig.1.1. We quantify cortical dynamics in terms of a phase coherence measure and demonstrate that the model-predicted turbulent slow-wave state is characterized by low phase coherence. This prediction is supported by clinical studies of phase synchronization changes in EEG during induction of propofol anesthesia [4]. Figure 1 Spontaneous slow-wave oscillations in cortical firing-rate patterns during 20 s of simulated anesthesia. Traces were recorded from five equally-spaced points lying along the midline of the 25-×25-cm simulated cortical grid. Time-series are chaotic ...

Background. Analysis of the bispectrum of EEG waveforms is a component of the proprietary BIS index—a commonly used commercial monitor of depth of anaesthesia. Does the use of the bispectrum give more information about depth of anaesthesia than the power spectrum? Methods. We collected and analysed EEG waveforms during induction of general anaesthesia in 39 patients, comparing the changes in bispectral parameter (SynchFastSlow), with an analogous power spectrum-based parameter (PowerFastSlow). Both compare the logarithmic ratio of high frequency components (40–47 Hz) with the total (1–47 Hz). Because the changes in bispectrum are affected by signal amplitude, we also calculated a third parameter (SFSbicoh) from the bicoherence, which is an amplitude-independent statistic. Results. The SynchFastSlow and PowerFastSlow were correlated (r=0.84) and neither was superior in predicting the awake or anaesthetized state (area under receiver operating characteristic curves=0.85 vs 0.93). There was no change in the SFSbicoh over the induction period, and it did not correlate with SynchFastSlow (r=0.07). Conclusions. We could not show that bispectral analysis gave more information than power spectral-based analysis. Most of the changes in the bispectral values result from decreases in the relative high frequency content of the EEG caused by anaesthesia.

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.

A diverse range of modelling approaches have been applied to try and understand some of the neural mechanisms that underlie transitions between wake-sleep (and rapid-eye movement-to-slow-wave sleep) states. There is a strong evolutionary argument that general anaesthesia exists because it is a form of drug-induced harnessing of natural sleep mechanisms. The theoretical models tend to either describe specific interactions between various brain-stem nuclei; or at the other extreme, assume that sleep is a universal property of all neural assemblies, and therefore follow a thalamo-cortico-centric approach Using a general cortex-based mean field model we propose that: (1) Unconsciousness during natural slow wave sleep is caused by blockade of cortical connectivity; which is induced by increased gamma-amino-butyric acid(GABA)-ergic activity and diminished excitatory neuromodulators—and hence relative cortical hyperpolarization. (2) The sleeping subject can be woken because the normal homeostatic effects of arousal neuromodulators are able to depolarize the cortex, and switch off the GABAergic systems. (3) Sedative doses of GABAergic general anaesthetic drugs augment GABAergic systems which then inhibit excitatory neuromodulators and trigger a sleep-like state. However excessive nociceptive activation of the brainstem arousal systems is still able to depolarize the cortex and switch off the GABAergic systems. (4) Larger doses of GABAergic general anaesthetics cause an irreversible global increase in the total charge carried by the inhibitory post synaptic potential. This causes an increased negative feedback loop in the cortex, which is not able to be overcome by intrinsic neuronal currents, and hence the patient cannot be woken up even by the most extreme nociceptive stimuli—the definition of general anaesthesia.

It is a well established clinical observation that, at low concentrations, most anesthetic agents produce a surge in brain activity that occurs around the time of loss of consciousness. At higher concentrations, brain activity slows, and eventually tends towards electrical silence. A second surge in EEG power occurs during the return to consciousness. These induction and recovery biphasic power surges were first explained in terms of a first-order switching transition between distinct active and quiescent neural states, but subsequent modeling by other researchers has demonstrated that biphasic surges can also be generated by a smooth, graduated transition between normal and suppressed levels of cortical activity. In this chapter we examine the contrasting predictions of the switching model versus the continuous model for anesthetic induction. If the continuous non-switching picture is correct, then the return path to recovery will retrace the trajectory for induction, so the biphasic peaks should occur at identical drug concentrations. In contrast, the switching model predicts that there must be a hysteresis separation between the entry and recovery EEG power maxima, and that the patient will awaken at a lower drug concentration than that required to put her to sleep.

Epilepsy affects nearly 3 million people in the United States alone. Given the fact that many people suffer from seizures that are intractable to pharmacological intervention, research groups are investigating the use of electrical stimulation to interact with and ameliorate symptoms of epileptic seizures. In mouse cortical slices made seizuregenic through chemical means, we applied precision controlled current pulses and measured local field potentials through a four point probe system to investigate the response of seizing tissue to electrical stimulation. We have determined that the frequency of the spontaneous seizure-like events may be modified by low amplitude, current controlled stimulation (0.5 microA). Differently from previously thought, this change in frequency is however not accompanied by any alteration of the tissue permittivity or conductivity during the inter-seizure interval.

Objective : Large-scale synchronous firing of neurons during seizures is modulated by electrotonic coupling between neurons via gap junctions. To explore roles for connexin36 (Cx36) gap junctions in seizures, we examined the seizure threshold of connexin36 knockout (Cx36KO) mice using a pentylenetetrazol (PTZ) model. Methods : Mice (2–3 months old) with Cx36 wildtype (WT) or Cx36KO genotype were treated with vehicle or 10–40 mg/kg of the convulsant PTZ by intraperitoneal injection. Seizure and seizure-like behaviors were scored by examination of video collected for 20 min. Quantitative real-time PCR (QPCR) was performed to measure potential compensatory neuronal connexin (Cx30.2, Cx37, Cx43 and Cx45), pannexin (PANX1 and PANX2) and gamma-aminobutyric acid type A (GABA A ) receptor α1 subunit gene expression. Results : Cx36KO animals exhibited considerably more severe seizures; 40 mg/kg of PTZ caused severe generalized (≥ grade III) seizures in 78% of KO, but just 5% of WT mice. A lower dose of PTZ (20 mg/kg) induced grade II seizure-like behaviors in 40% KO vs. 0% of WT animals. There was no significant difference in either connexin, pannexin or GABA A α1 gene expression between WT and KO animals. Conclusion : Increased sensitivity of Cx36KO animals to PTZ-induced seizure suggests that Cx36 gap junctional communication functions as a physiological anti-convulsant mechanism, and identifies the Cx36 gap junction as a potential therapeutic target in epilepsy.

Phase transitions in single neurons and neural populations: Critical slowing, anesthesia, and sleep cycles.- Generalized state-space models for modeling nonstationary EEG time-series.- Spatiotemporal instabilities in neural fields and the effects of additive noise.- Spontaneous brain dynamics emerges at the edge of instability.- Limited spreading: How hierarchical networks prevent the transition to the epileptic state.- Bifurcations and state changes in the human alpha rhythm: Theory and experiment.- Inducing transitions in mesoscopic brain dynamics.- Phase transitions in physiologically-based multiscale mean-field brain models.- A continuum model for the dynamics of the phase transition from slow-wave sleep to REM sleep.- What can a mean-field model tell us about the dynamics of the cortex?.- Phase transitions, cortical gamma, and the selection and read-out of information stored in synapses.- Cortical patterns and gamma genesis are modulated by reversal potentials and gap-junction diffusion.

The firing of an action potential by a biological neuron represents a dramatic transition from small-scale linear stochastics (subthreshold voltage fluctuations) to gross-scale nonlinear dynamics (birth of a 1-ms voltage spike). In populations of neurons we see similar, but slower, switch-like there-and-back transitions between low-firing background states and high-firing activated states. These state transitions are controlled by varying levels of input current (single neuron), varying amounts of GABAergic drug (anesthesia), or varying concentrations of neuromodulators and neurotransmitters (natural sleep), and all occur within a milieu of unrelenting biological noise. By tracking the altering responsiveness of the excitable membrane to noisy stimulus, we can infer how close the neuronal system (single unit or entire population) is to switching threshold. We can quantify this “nearness to switching” in terms of the altering eigenvalue structure: the dominant eigenvalue approaches zero, leading to a growth in correlated, low-frequency power, with exaggerated responsiveness to small perturbations, the responses becoming larger and slower as the neural population approaches its critical point–-this is critical slowing. In this chapter we discuss phase-transition predictions for both single-neuron and neural-population models, comparing theory with laboratory and clinical measurement.

Previous studies have shown that activated cortical states (awake and rapid eye-movement (REM) sleep), are associated with increased cholinergic input into the cerebral cortex. However, the mechanisms that underlie the detailed dynamics of the cortical transition from slow-wave to REM sleep have not been quantitatively modeled. How does the sequence of abrupt changes in the cortical dynamics (as detected in the electrocorticogram) result from the more gradual change in subcortical cholinergic input? We compare the output from a continuum model of cortical neuronal dynamics with experimentally-derived rat electrocorticogram data. The output from the computer model was consistent with experimental observations. In slow-wave sleep, 0.5–2-Hz oscillations arise from the cortex jumping between “up” and “down” states on the stationary-state manifold. As cholinergic input increases, the upper state undergoes a bifurcation to an 8-Hz oscillation. The coexistence of both oscillations is similar to that found in the intermediate stage of sleep of the rat. Further cholinergic input moves the trajectory to a point where the lower part of the manifold in not available, and thus the slow oscillation abruptly ceases (REM sleep). The model provides a natural basis to explain neuromodulator-induced changes in cortical activity, and indicates that a cortical phase change, rather than a brainstem “flip-flop”, may describe the transition from slow-wave sleep to REM.

In this chapter we describe a continuum model for the cortex that includes both axon-to-dendrite chemical synapses and direct neuron-to-neuron gap-junction diffusive synapses. The effectiveness of chemical synapses is determined by the voltage of the receiving dendrite V relative to its Nernst reversal potential \(V^{\rm rev}{}\). Here we explore two alternative strategies for incorporating dendritic reversal potentials, and uncover surprising differences in their stability properties and model dynamics. In the “slow-soma” variant, the \((V^{\rm rev} - V)\) weighting is applied after the input flux has been integrated at the dendrite, while for “fast-soma”, the weighting is applied directly to the input flux, prior to dendritic integration. For the slow-soma case, we find that–-provided the inhibitory diffusion (via gap-junctions) is sufficiently strong–-the cortex generates stationary Turing patterns of cortical activity. In contrast, the fast-soma destabilizes in favor of standing-wave spatial structures that oscillate at low-gamma frequency (\(\sim\)30-Hz); these spatial patterns broaden and weaken as diffusive coupling increases, and disappear altogether at moderate levels of diffusion. We speculate that the slow- and fast-soma models might correspond respectively to the idling and active modes of the cortex, with slow-soma patterns providing the default background state, and emergence of gamma oscillations in the fast-soma case signaling the transition into the cognitive state.

We study the dynamics of the transition between the low- and high-firing states of the cortical slow oscillation by using intracellular recordings of the membrane potential from cortical neurons of rats. We investigate the evidence for a bistability in assemblies of cortical neurons playing a major role in the maintenance of this oscillation. We show that the trajectory of a typical transition takes an approximately exponential form, equivalent to the response of a resistor–capacitor circuit to a step-change in input. The time constant for the transition is negatively correlated with the membrane potential of the low-firing state, and values are broadly equivalent to neural time constants measured elsewhere. Overall, the results do not strongly support the hypothesis of a bistability in cortical neurons; rather, they suggest the cortical manifestation of the oscillation is a result of a step-change in input to the cortical neurons. Since there is evidence from previous work that a phase transition exists, we speculate that the step-change may be a result of a bistability within other brain areas, such as the thalamus, or a bistability among only a small subset of cortical neurons, or as a result of more complicated brain dynamics.

It is a simple observation that the function of the central nervous system changes abruptly at certain critical brain concentrations of the anesthetic drug. This can be viewed as analogous to “state transitions” in systems of interacting particles, which have been extensively studied in the physical sciences. Theoretical models of the electroencephalogram (EEG) are in semi-quantitative agreement with experimental data and show some features suggestive of anesthetic-induced state transitions (critical slowing, the biphasic effect, increase in spatial correlations, and entropy changes). However, the EEG is only an indirect marker of the (unknown) networks of cortical connectivity that are required for the formation of consciousness. It is plausible that anesthetic-induced alteration in cortical and corticothalamic network topology prevents the formation of a giant component in the neuronal population network, and hence induces unconsciousness.

How does physiological sleep affect the Bispectral Index (BIS)? We collected electroencephalographic (EEG) data from five subjects during the early part of the night, comparing the changes in the BIS with the conventional EEG stages of sleep. We found that the BIS was a consistent marker of depth of sleep. Light sleep occurred at BIS values of 75‐90, slow-wave sleep occurred at BIS values of 20 ‐70, and rapid eye movement sleep occurred at BIS values of 75‐92. The effects of natural sleep on the BIS seem to be similar to the effects of general anesthesia on the BIS. The BIS may have a role in monitoring depth of sleep. Implications: Electroencephalographic data were collected from five subjects during sleep. We found that the Bispectral Index decreased during increasing depth of sleep in a fashion very similar to the decrease in Bispectral Index that occurs during general anesthesia. This study further highlights the electroencephalographic similarities of states of sleep and general anesthesia. (Anesth Analg 1999;88:659 ‐61)

Using a recently reported analysis of gap-junction density in the cat visual cortex, we have augmented a standard mean-field model of the cortex to include a dense network of diffusive connections between inhibitory neurons. Provided the diffusive connection strength exceeds a critical threshold, our augmented cortical model predicts the spontaneous emergence of Turing structures, patterned regions of high- and low-firing cortical activity distributed across the brain. In this paper we demonstrate that these patterns will become self-modulating if a second model parameter, the rate-constant for the inhibitory post-synaptic potential, is made sufficiently small, allowing a low-frequency Hopf temporal instability to emerge. This interaction between Turing and Hopf instabilities may explain the slow oscillations in coherent brain activity observed in BOLD (blood oxygen-level-dependent) imagery, and may have relevance to brain pathologies which are characterized by abnormally low (e.g., schizophrenia) or abnormally high (e.g., Parkinson’s disease) levels of cross-cortical neural synchrony.

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 ${D}_{2}$, the diffusive coupling strength arising from gap-junction connections between inhibitory neurons. Using recent neurophysiological measurements reported by Fukuda et al. [J. Neurosci. 26, 3434 (2006)], we estimate an upper limit of ${D}_{2}\ensuremath{\approx}0.6\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{2}$. 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 ${D}_{2}$ 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.