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

A comprehensive investigation of intracortical and corticothalamic models of the alpha rhythm

The electroencephalographic alpha rhythm is one of the most robustly observed and widely studied empirical phenomena in all of neuroscience. However, despite its extensive implication in a wide range of cognitive processes and clinical pathologies, the mechanisms underlying alpha generation in neural circuits remain poorly understood. In this paper we offer a renewed foundation for research on this question, by undertaking a systematic comparison and synthesis of the most prominent theoretical models of alpha rhythmogenesis in the published literature. We focus on four models, each studied intensively by multiple authors over the past three decades: (i) Jansen-Rit, (ii) Moran-David-Friston, (iii) Robinson-Rennie-Wright, and (iv) Liley-Wright. Several common elements are identified, such as the use of second-order differential equations and sigmoidal potential-to-rate operators to represent population-level neural activity. Major differences are seen in other features such as wiring topologies and conduction delays. Through a series of mathematical analyses and numerical simulations, we nevertheless demonstrate that the selected models can be meaningfully compared, by associating parameters and circuit motifs of analogous biological significance. With this established, we conduct explorations of rate constant and synaptic connectivity parameter spaces, with the aim of identifying common patterns in key behaviours, such as the role of excitatory-inhibitory interactions in the generation of oscillations. Finally, using linear stability analysis we identify two qualitatively different alpha-generating dynamical regimes across the models: (i) noise-driven fluctuations and (ii) self-sustained limit-cycle oscillations, emerging due to an Andronov-Hopf bifurcation. The comprehensive survey and synthesis developed here can, we suggest, be used to help guide future theoretical and experimental work aimed at disambiguating these and other candidate theories of alpha rhythmogenesis.

Method for quantifying arousal and consciousness in healthy states and severe brain injury via EEG-based measures of corticothalamic physiology

BACKGROUND

Characterization of normal arousal states has been achieved by fitting predictions of corticothalamic neural field theory (NFT) to electroencephalographic (EEG) spectra to yield relevant physiological parameters.

NEW METHOD

A prior fitting method is extended to distinguish conscious and unconscious states in healthy and brain injured subjects by identifying additional parameters and clusters in parameter space.

RESULTS

Fits of NFT predictions to EEG spectra are used to estimate neurophysiological parameters in healthy and brain injured subjects. Spectra are used from healthy subjects in wake and sleep and from patients with unresponsive wakefulness syndrome, in a minimally conscious state (MCS), and emerged from MCS. Subjects cluster into three groups in parameter space: conscious healthy (wake and REM), sleep, and brain injured. These are distinguished by the difference X-Y between corticocortical (X) and corticothalamic (Y) feedbacks, and by mean neural response rates α and β to incoming spikes. X-Y tracks consciousness in healthy individuals, with smaller values in wake/REM than sleep, but cannot distinguish between brain injuries. Parameters α and β differentiate deep sleep from wake/REM and brain injury.

COMPARISON WITH EXISTING METHODS

Other methods typically rely on laborious clinical assessment, manual EEG scoring, or evaluation of measures like Φ from integrated information theory, for which no efficient method exists. In contrast, the present method can be automated on a personal computer.

CONCLUSION

The method provides a means to quantify consciousness and arousal in healthy and brain injured subjects, but does not distinguish subtypes of brain injury.

Preservation of thalamocortical circuitry is essential for good recovery after cardiac arrest

Continuous electroencephalographam (EEG) monitoring contributes to prediction of neurological outcome in comatose cardiac arrest survivors. While the phenomenology of EEG abnormalities in postanoxic encephalopathy is well known, the pathophysiology, especially the presumed role of selective synaptic failure, is less understood. To further this understanding, we estimate biophysical model parameters from the EEG power spectra from individual patients with a good or poor recovery from a postanoxic encephalopathy. This biophysical model includes intracortical, intrathalamic, and corticothalamic synaptic strengths, as well as synaptic time constants and axonal conduction delays. We used continuous EEG measurements from hundred comatose patients recorded during the first 48 h postcardiac arrest, 50 with a poor neurological outcome [cerebral performance category ( CPC = 5 ) ] and 50 with a good neurological outcome ( CPC = 1 ). We only included patients that developed (dis-)continuous EEG activity within 48 h postcardiac arrest. For patients with a good outcome, we observed an initial relative excitation in the corticothalamic loop and corticothalamic propagation that subsequently evolved towards values observed in healthy controls. For patients with a poor outcome, we observed an initial increase in the cortical excitation-inhibition ratio, increased relative inhibition in the corticothalamic loop, delayed corticothalamic propagation of neuronal activity, and severely prolonged synaptic time constants that did not return to physiological values. We conclude that the abnormal EEG evolution in patients with a poor neurological recovery after cardiac arrest may result from persistent and selective synaptic failure that includes corticothalamic circuitry and also delayed corticothalamic propagation.

Finding synaptic couplings from a biophysical model of motor evoked potentials after theta-burst transcranial magnetic stimulation

OBJECTIVE

We aimed to use measured input-output (IO) data to identify the best fitting model for motor evoked potentials.

METHODS

We analyzed existing IO data before and after intermittent and continuous theta-burst stimulation (iTBS & cTBS) from a small group of subjects (18 for each). We fitted individual synaptic couplings and sensitivity parameters using variations of a biophysical model. A best performing model was selected and analyzed.

RESULTS

cTBS gives a broad reduction in MEPs for amplitudes larger than resting motor threshold (RMT). Close to threshold, iTBS gives strong potentiation. The model captures individual IO curves. There is no change to the population average synaptic weights post TBS but the change in excitatory-to-excitatory synaptic coupling is strongly correlated with the experimental post-TBS response relative to baseline.

CONCLUSIONS

The model describes population-averaged and individual IO curves, and their post-TBS change. Variation among individuals is accounted for with variation in synaptic couplings, and variation in sensitivity of neural response to stimulation.

SIGNIFICANCE

The best fitting model could be applied more broadly and validation studies could elucidate underlying biophysical meaning of parameters.

Selection of stimulus parameters for enhancing slow wave sleep events with a neural-field theory thalamocortical model

Slow-wave sleep cortical brain activity, conformed by slow-oscillations and sleep spindles, plays a key role in memory consolidation. The increase of the power of the slow-wave events, obtained by auditory sensory stimulation, positively correlates with memory consolidation performance. However, little is known about the experimental protocol maximizing this effect, which could be induced by the power of slow-oscillation, the number of sleep spindles, or the timing of both events' co-occurrence. Using a mean-field model of thalamocortical activity, we studied the effect of several stimulation protocols, varying the pulse shape, duration, amplitude, and frequency, as well as a target-phase using a closed-loop approach. We evaluated the effect of these parameters on slow-oscillations (SO) and sleep-spindles (SP), considering: (i) the power at the frequency bands of interest, (ii) the number of SO and SP, (iii) co-occurrences between SO and SP, and (iv) synchronization of SP with the up-peak of the SO. The first three targets are maximized using a decreasing ramp pulse with a pulse duration of 50 ms. Also, we observed a reduction in the number of SO when increasing the stimulus energy by rising its amplitude. To assess the target-phase parameter, we applied closed-loop stimulation at 0°, 45°, and 90° of the phase of the narrow-band filtered ongoing activity, at 0.85 Hz as central frequency. The 0° stimulation produces better results in the power and number of SO and SP than the rhythmic or random stimulation. On the other hand, stimulating at 45° or 90° change the timing distribution of spindles centers but with fewer co-occurrences than rhythmic and 0° phase. Finally, we propose the application of closed-loop stimulation at the rising zero-cross point using pulses with a decreasing ramp shape and 50 ms of duration for future experimental work.

Computational Models in Electroencephalography

Computational models lie at the intersection of basic neuroscience and healthcare applications because they allow researchers to test hypotheses in silico and predict the outcome of experiments and interactions that are very hard to test in reality. Yet, what is meant by “computational model” is understood in many different ways by researchers in different fields of neuroscience and psychology, hindering communication and collaboration. In this review, we point out the state of the art of computational modeling in Electroencephalography (EEG) and outline how these models can be used to integrate findings from electrophysiology, network-level models, and behavior. On the one hand, computational models serve to investigate the mechanisms that generate brain activity, for example measured with EEG, such as the transient emergence of oscillations at different frequency bands and/or with different spatial topographies. On the other hand, computational models serve to design experiments and test hypotheses in silico. The final purpose of computational models of EEG is to obtain a comprehensive understanding of the mechanisms that underlie the EEG signal. This is crucial for an accurate interpretation of EEG measurements that may ultimately serve in the development of novel clinical applications.

Modeling motor-evoked potentials from neural field simulations of transcranial magnetic stimulation

OBJECTIVE

To develop a population-based biophysical model of motor-evoked potentials (MEPs) following transcranial magnetic stimulation (TMS).

METHODS

We combined an existing MEP model with population-based cortical modeling. Layer 2/3 excitatory and inhibitory neural populations, modeled with neural-field theory, are stimulated with TMS and feed layer 5 corticospinal neurons, which also couple directly but weakly to the TMS pulse. The layer 5 output controls mean motoneuron responses, which generate a series of single motor-unit action potentials that are summed to estimate a MEP.

RESULTS

A MEP waveform was generated comparable to those observed experimentally. The model captured TMS phenomena including a sigmoidal input-output curve, common paired pulse effects (short interval intracortical inhibition, intracortical facilitation, long interval intracortical inhibition) including responses to pharmacological interventions, and a cortical silent period. Changes in MEP amplitude following theta burst paradigms were observed including variability in outcome direction.

CONCLUSIONS

The model reproduces effects seen in common TMS paradigms.

SIGNIFICANCE

The model allows population-based modeling of changes in cortical dynamics due to TMS protocols to be assessed in terms of changes in MEPs, thus allowing a clear comparison between population-based modeling predictions and typical experimental outcome measures.

COALIA: A Computational Model of Human EEG for Consciousness Research

Understanding the origin of the main physiological processes involved in consciousness is a major challenge of contemporary neuroscience, with crucial implications for the study of Disorders of Consciousness (DOC). The difficulties in achieving this task include the considerable quantity of experimental data in this field, along with the non-intuitive, nonlinear nature of neuronal dynamics. One possibility of integrating the main results from the experimental literature into a cohesive framework, while accounting for nonlinear brain dynamics, is the use of physiologically-inspired computational models. In this study, we present a physiologically-grounded computational model, attempting to account for the main micro-circuits identified in the human cortex, while including the specificities of each neuronal type. More specifically, the model accounts for thalamo-cortical (vertical) regulation of cortico-cortical (horizontal) connectivity, which is a central mechanism for brain information integration and processing. The distinct neuronal assemblies communicate through feedforward and feedback excitatory and inhibitory synaptic connections implemented in a template brain accounting for long-range connectome. The EEG generated by this physiologically-based simulated brain is validated through comparison with brain rhythms recorded in humans in two states of consciousness (wakefulness, sleep). Using the model, it is possible to reproduce the local disynaptic disinhibition of basket cells (fast GABAergic inhibition) and glutamatergic pyramidal neurons through long-range activation of vasoactive intestinal-peptide (VIP) interneurons that induced inhibition of somatostatin positive (SST) interneurons. The model (COALIA) predicts that the strength and dynamics of the thalamic output on the cortex control the local and long-range cortical processing of information. Furthermore, the model reproduces and explains clinical results regarding the complexity of transcranial magnetic stimulation TMS-evoked EEG responses in DOC patients and healthy volunteers, through a modulation of thalamo-cortical connectivity that governs the level of cortico-cortical communication. This new model provides a quantitative framework to accelerate the study of the physiological mechanisms involved in the emergence, maintenance and disruption (sleep, anesthesia, DOC) of consciousness.

Neural field theory of evoked response potentials in a spherical brain geometry

Evoked response potentials (ERPs) are calculated in spherical and planar geometries using neural field theory of the corticothalamic system. The ERP is modeled as an impulse response and the resulting modal effects of spherical corticothalamic dynamics are explored, showing that results for spherical and planar geometries converge in the limit of large brain size. Cortical modal effects can lead to a double-peak structure in the ERP time series. It is found that the main difference between infinite planar geometry and spherical geometry is that the ERP peak is sharper and stronger in the spherical geometry. It is also found that the magnitude of the response decreases with increasing spatial width of the stimulus at the cortex. The peak is slightly delayed at large angles from the stimulus point, corresponding to group velocities of 6-10 m s^{-1}. Strong modal effects are found in the spherical geometry, with the lowest few modes sufficing to describe the main features of ERPs, except very near to spatially narrow stimuli.

Necessity of the sleep-wake cycle for synaptic homeostasis: system-level analysis of plasticity in the corticothalamic system

Neural field theory is used to study the system-level effects of plasticity in the corticothalamic system, where arousal states are represented parametrically by the connection strengths of the system, among other physiologically based parameters. It is found that the plasticity dynamics have no fixed points or closed cycles in the parameter space of the connection strengths, but parameter subregions exist where flows have opposite signs. Remarkably, these subregions coincide with previously identified regions that correspond to wake and slow-wave sleep, thus demonstrating state dependence of the sign of synaptic modification. We then show that a closed cycle in the parameter space is possible when the plasticity dynamics are driven by the ascending arousal system, which cycles the brain between sleep and wake to complete a closed loop that includes arcs through the opposite-flow subregions. Thus, it is concluded that both wake and sleep are necessary, and together are able to stabilize connection weights in the brain over the daily cycle, thereby providing quantitative realization of the synaptic homeostasis hypothesis.

Physiology-based ERPs in normal and abnormal states

Evoked response potentials (ERPs) and other transients are modeled as impulse responses using physiology-based neural field theory (NFT) of the corticothalamic system of neural activity in the human brain that incorporates synaptic and dendritic dynamics, firing response, axonal propagation, and corticocortical and corticothalamic pathways. The properties of model-predicted ERPs are explored throughout the stability zone of the corticothalamic system, and predicted time series and wavelet spectra are also analyzed. This provides a unified treatment of predicted ERPs for both normal and abnormal states within the brain's stability zone, including likely parameters to represent abnormal states of reduced arousal.

Single-molecule studies of flavivirus envelope dynamics: Experiment and computation

Flaviviruses are simple enveloped viruses exhibiting complex structural and functional heterogeneities. Decades of research have provided crucial basic insights, antiviral medication and moderately successful gene therapy trials. The most infectious particle is, however, not always the most abundant one in a population, questioning the utility of classic ensemble-averaging virology approaches. Indeed, viral replication is often not particularly efficient, prone to errors or containing parallel routes. Here, we review different single-molecule sensitive fluorescence methods that are employed to investigate flaviviruses. In particular, we review how (i) time-resolved Förster resonance energy transfer (trFRET) was applied to probe dengue envelope conformations; (ii) FRET-fluorescence correlation spectroscopy to investigate dengue envelope intrinsic dynamics and (iii) single particle tracking to follow the path of dengue viruses in cells. We also discuss how such methods may be supported by molecular dynamics (MD) simulations over a range of spatio-temporal scales, to provide complementary data on the structure and dynamics of flaviviral systems. We describe recent improvements in multiscale MD approaches that allowed the simulation of dengue particle envelopes in near-atomic resolution. We hope this review is an incentive for setting up and applying similar single-molecule studies and combine them with MD simulations to investigate structural dynamics of entire flavivirus particles over the nanosecond-to-millisecond time-scale and follow viruses during infection in cells over milliseconds to minutes.

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

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

Multistability in the corticothalamic system

Neural field theory of the corticothalamic system is used to analyze the properties of its steady-state solutions, including their linear stability, in the parameter space of synaptic couplings for physiological parameter ranges representing normal arousal waking states in adult humans. The independent connections of the corticothalamic model define an eight-dimensional parameter space, while specific combinations of these connections parameterize intracortical, corticothalamic, and intrathalamic loops. Multistable regions are systematically identified and the existence of up to five steady-state solutions is confirmed, up to three of which are linearly stable. A key determinant for the existence of five steady states is found to be the number of nonzero connections. This finding had not been previously proposed as the determining factor of high multiplicities of multistability in mesoscopic models of the brain. In the corticothalamic model presented here, multistability occurs when the intrathalamic loop is present (i.e., the reticular nucleus inhibits the relay nuclei), and when the net synaptic effect of the intracortical loop is inhibitory. The signature of these additional waking states is an overall increased level of thalamic activity. It is argued that the additional steady states found may represent hyperarousal states which occur when the corticothalamic projections do not attenuate the activity of the cortex.

Unified neural field theory of brain dynamics underlying oscillations in Parkinson's disease and generalized epilepsies

The mechanisms underlying pathologically synchronized neural oscillations in Parkinson's disease (PD) and generalized epilepsies are explored in parallel via a physiologically-based neural field model of the corticothalamic-basal ganglia (CTBG) system. The basal ganglia (BG) are approximated as a single effective population and their roles in the modulation of oscillatory dynamics of the corticothalamic (CT) system and vice versa are analyzed. In addition to normal EEG rhythms, enhanced activity around 4 Hz and 20 Hz exists in the model, consistent with the characteristic frequencies observed in PD. These rhythms result from resonances in loops formed between the BG and CT populations, analogous to those that underlie epileptic oscillations in a previous CT model, and which are still present in the combined CTBG system. Dopamine depletion is argued to weaken the dampening of these loop resonances in PD, and network connections then explain the significant coherence observed between BG, thalamic, and cortical population activity around 4-8 Hz and 20 Hz. Parallels between the afferent and efferent connection sites of the thalamic reticular nucleus (TRN) and BG predict low dopamine to correspond to a reduced likelihood of tonic-clonic (grand mal) seizures, which agrees with experimental findings. Furthermore, the model predicts an increased likelihood of absence (petit mal) seizure resulting from pathologically low dopamine levels in accordance with experimental observations. Suppression of absence seizure activity is demonstrated when afferent and efferent BG connections to the CT system are strengthened, which is consistent with other CTBG modeling studies. The BG are demonstrated to have a suppressive effect on activity of the CTBG system near tonic-clonic seizure states, which provides insight into the reported efficacy of current treatments in BG circuits. Sleep states of the TRN are also found to suppress pathological PD activity in accordance with observations. Overall, the findings demonstrate strong parallels between coherent oscillations in generalized epilepsies and PD, and provide insights into possible comorbidities.

K-complexes, spindles, and ERPs as impulse responses: unification via neural field theory

To interrelate K-complexes, spindles, evoked response potentials (ERPs), and spontaneous electroencephalography (EEG) using neural field theory (NFT), physiology-based NFT of the corticothalamic system is used to model cortical excitatory and inhibitory populations and thalamic relay and reticular nuclei. The impulse response function of the model is used to predict the responses to impulses, which are compared with transient waveforms in sleep studies. Fits to empirical data then allow underlying brain physiology to be inferred and compared with other waves. Spontaneous K-complexes, spindles, and other transient waveforms can be reproduced using NFT by treating them as evoked responses to impulsive stimuli with brain parameters appropriate to spontaneous EEG in sleep stage 2. Using this approach, spontaneous K-complexes and sleep spindles can be analyzed using the same single theory as previously been used to account for waking ERPs and other EEG phenomena. As a result, NFT can explain a wide variety of transient waveforms that have only been phenomenologically classified to date. This enables noninvasive fitting to be used to infer underlying physiological parameters. This physiology-based model reproduces the time series of different transient EEG waveforms; it has previously reproduced experimental EEG spectra, and waking ERPs, and many other observations, thereby unifying transient sleep waveforms with these phenomena.

Real-time automated EEG tracking of brain states using neural field theory

A real-time fitting system is developed and used to fit the predictions of an established physiologically-based neural field model to electroencephalographic spectra, yielding a trajectory in a physiological parameter space that parametrizes intracortical, intrathalamic, and corticothalamic feedbacks as the arousal state evolves continuously over time. This avoids traditional sleep/wake staging (e.g., using Rechtschaffen-Kales stages), which is fundamentally limited because it forces classification of continuous dynamics into a few discrete categories that are neither physiologically informative nor individualized. The classification is also subject to substantial interobserver disagreement because traditional staging relies in part on subjective evaluations. The fitting routine objectively and robustly tracks arousal parameters over the course of a full night of sleep, and runs in real-time on a desktop computer. The system developed here supersedes discrete staging systems by representing arousal states in terms of physiology, and provides an objective measure of arousal state which solves the problem of interobserver disagreement. Discrete stages from traditional schemes can be expressed in terms of model parameters for backward compatibility with prior studies.

Generalized seizures in a neural field model with bursting dynamics

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

Physiologically based arousal state estimation and dynamics

A neural field model of the brain is used to represent brain states using physiologically based parameters rather than arbitrary, discrete sleep stages. Each brain state is represented as a point in a physiologically parametrized space. Over time, changes in brain state cause these points to trace continuous trajectories, unlike the artificial discrete jumps in sleep stage that occur with traditional sleep staging. The discrete Rechtschaffen and Kales sleep stages are associated with regions in the physiological parameter space based on their electroencephalographic features, which enables interpretation of traditional sleep stages in terms of physiological trajectories. Wake states are found to be associated with strong positive corticothalamic feedback compared to sleep. The existence of physiologically valid trajectories between brain states in the model is demonstrated. Actual trajectories for an individual can be determined by fitting the model using EEG alone, and enable analysis of the physiological differences between subjects.

A thalamo-cortical neural mass model for the simulation of brain rhythms during sleep

Cortico-thalamic interactions are known to play a pivotal role in many brain phenomena, including sleep, attention, memory consolidation and rhythm generation. Hence, simple mathematical models that can simulate the dialogue between the cortex and the thalamus, at a mesoscopic level, have a great cognitive value. In the present work we describe a neural mass model of a cortico-thalamic module, based on neurophysiological mechanisms. The model includes two thalamic populations (a thalamo-cortical relay cell population, TCR, and its related thalamic reticular nucleus, TRN), and a cortical column consisting of four connected populations (pyramidal neurons, excitatory interneurons, inhibitory interneurons with slow and fast kinetics). Moreover, thalamic neurons exhibit two firing modes: bursting and tonic. Finally, cortical synapses among pyramidal neurons incorporate a disfacilitation mechanism following prolonged activity. Simulations show that the model is able to mimic the different patterns of rhythmic activity in cortical and thalamic neurons (beta and alpha waves, spindles, delta waves, K-complexes, slow sleep waves) and their progressive changes from wakefulness to deep sleep, by just acting on modulatory inputs. Moreover, simulations performed by providing short sensory inputs to the TCR show that brain rhythms during sleep preserve the cortex from external perturbations, still allowing a high cortical activity necessary to drive synaptic plasticity and memory consolidation. In perspective, the present model may be used within larger cortico-thalamic networks, to gain a deeper understanding of mechanisms beneath synaptic changes during sleep, to investigate the specific role of brain rhythms, and to explore cortical synchronization achieved via thalamic influences.

Prediction and verification of nonlinear sleep spindle harmonic oscillations

This paper examines nonlinear effects in a neural field model of the corticothalamic system to predict the EEG power spectrum of sleep spindles. Nonlinearity in the thalamic relay nuclei gives rise to a spindle harmonic visible in the cortical EEG. By deriving an analytic expression for nonlinear spectrum, the power in the spindle harmonic is predicted to scale quadratically with the power in the spindle oscillation. By isolating sleep spindles from background sleep in experimental EEG data, the spindle harmonic is directly observed.

Modeling resting-state functional networks when the cortex falls asleep: local and global changes

The transition from wakefulness to sleep represents the most conspicuous change in behavior and the level of consciousness occurring in the healthy brain. It is accompanied by similarly conspicuous changes in neural dynamics, traditionally exemplified by the change from "desynchronized" electroencephalogram activity in wake to globally synchronized slow wave activity of early sleep. However, unit and local field recordings indicate that the transition is more gradual than it might appear: On one hand, local slow waves already appear during wake; on the other hand, slow sleep waves are only rarely global. Studies with functional magnetic resonance imaging also reveal changes in resting-state functional connectivity (FC) between wake and slow wave sleep. However, it remains unclear how resting-state networks may change during this transition period. Here, we employ large-scale modeling of the human cortico-cortical anatomical connectivity to evaluate changes in resting-state FC when the model "falls asleep" due to the progressive decrease in arousal-promoting neuromodulation. When cholinergic neuromodulation is parametrically decreased, local slow waves appear, while the overall organization of resting-state networks does not change. Furthermore, we show that these local slow waves are structured macroscopically in networks that resemble the resting-state networks. In contrast, when the neuromodulator decrease further to very low levels, slow waves become global and resting-state networks merge into a single undifferentiated, broadly synchronized network.

Critical fluctuations in cortical models near instability

Computational studies often proceed from the premise that cortical dynamics operate in a linearly stable domain, where fluctuations dissipate quickly and show only short memory. Studies of human electroencephalography (EEG), however, have shown significant autocorrelation at time lags on the scale of minutes, indicating the need to consider regimes where non-linearities influence the dynamics. Statistical properties such as increased autocorrelation length, increased variance, power law scaling, and bistable switching have been suggested as generic indicators of the approach to bifurcation in non-linear dynamical systems. We study temporal fluctuations in a widely-employed computational model (the Jansen–Rit model) of cortical activity, examining the statistical signatures that accompany bifurcations. Approaching supercritical Hopf bifurcations through tuning of the background excitatory input, we find a dramatic increase in the autocorrelation length that depends sensitively on the direction in phase space of the input fluctuations and hence on which neuronal subpopulation is stochastically perturbed. Similar dependence on the input direction is found in the distribution of fluctuation size and duration, which show power law scaling that extends over four orders of magnitude at the Hopf bifurcation. We conjecture that the alignment in phase space between the input noise vector and the center manifold of the Hopf bifurcation is directly linked to these changes. These results are consistent with the possibility of statistical indicators of linear instability being detectable in real EEG time series. However, even in a simple cortical model, we find that these indicators may not necessarily be visible even when bifurcations are present because their expression can depend sensitively on the neuronal pathway of incoming fluctuations.