Prediction and verification of nonlinear sleep spindle harmonic oscillations

The Relationship Between the Rhythmic Components of the Brain Electrical Activity During the Development of Status Epilepticus: An Operational Model of Brain Rhythms Generation

Background/Introduction: Despite the fact that brain rhythms are widely studied and officially classified, there is no consensus on their relationship, which can shed light on the genesis of rhythmic activity, its synchronization, functional role, and the formation of pathological reactions. Using the experimental status epilepticus (SE) as a model of brain in a hypersynchronized state with well-defined rhythms, we aimed to study the relationship between the rhythmic components of the brain electrical activity. Materials and Methods: Local field potentials (LFPs) were recorded simultaneously from the hippocampus, entorhinal cortex, medial septum, and amygdala during normal conditions and after kainic acid (KA) administration in waking guinea pigs. The dynamical spectral LFP properties were analyzed with the aid of Fast Fourier transform. Results: KA induces prominent SE with periodic combination of epileptiform discharge complexes and relatively quiet interdischarge intervals in the electrical activity of the brain. We have shown that new components appeared in the LFP spectra during the development of SE, representing a sequential doubling of the frequency, which had initially been dominating in the background records. Discussion: The phenomenon of frequency doubling can be interpreted as the octave principle of the LFP spectrum rhythmic carcass structure. The spectra of discharge complexes represent an alternation of harmonic spectra, where fundamental frequency coincides with one of the doubled frequencies dominating in the interdischarge activity. Using a nonlinear recurrent operation of rhythm multiplication and the obtained data we propose an operational model of the generation of rhythms and pathological discharges in the brain based on the octave principle. Impact statement In this study, we examined the relationship between the rhythmic components of the electrical activity of the limbic structures during the experimental status epilepticus and propose an operational model of brain rhythms generation based on the octave principle. Our study demonstrates that using fundamental principles (nonlinearity and the presence of recurrence), it is possible to explain the genesis and phenomenology of the electrical activity of brain structures in normal and pathological conditions.

Discrete spectral eigenmode-resonance network of brain dynamics and connectivity

The problem of finding a compact natural representation of brain dynamics and connectivity is addressed using an expansion in terms of physical spatial eigenmodes and their frequency resonances. It is demonstrated that this discrete expansion via the system transfer function enables linear and nonlinear dynamics to be analyzed in compact form in terms of natural dynamic "atoms," each of which is a frequency resonance of an eigenmode. Because these modal resonances are determined by the system dynamics, not the investigator, they are privileged over widely used phenomenological patterns, and obviate the need for artificial discretizations and thresholding in coordinate space. It is shown that modal resonances participate as nodes of a discrete spectral network, are noninteracting in the linear regime, but are linked nonlinearly by wave-wave coalescence and decay processes. The modal resonance formulation is shown to be capable of speeding numerical calculations of strongly nonlinear interactions. Recent work in brain dynamics, especially based on neural field theory (NFT) approaches, allows eigenmodes and their resonances to be estimated from data without assuming a specific brain model. This means that dynamic equations can be inferred using system identification methods from control theory, rather than being assumed, and resonances can be interpreted as control-systems data filters. The results link brain activity and connectivity with control-systems functions such as prediction and attention via gain control and can also be linked to specific NFT predictions if desired, thereby providing a convenient bridge between physiologically based theories and experiment. Amplitudes of modes and resonances can also be tracked to provide a more direct and temporally localized representation of the dynamics than correlations and covariances, which are widely used in the field. By synthesizing many different lines of research, this work provides a way to link quantitative electrophysiological and imaging measurements, connectivity, brain dynamics, and function. This underlines the need to move between coordinate and spectral representations as required. Moreover, standard theoretical-physics approaches and mathematical methods can be used in place of ad hoc statistical measures such as those based on graph theory of artificially discretized and decimated networks, which are highly prone to selection effects and artifacts.

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.

Modal-Polar Representation of Evoked Response Potentials in Multiple Arousal States

An expansion of the corticothalamic transfer function into eigenmodes and resonant poles is used to derive a simple formula for evoked response potentials (ERPs) in various states of arousal. The transfer function corresponds to the cortical response to an external stimulus, which encodes all the information and properties of the linear system. This approach links experimental observations of resonances and characteristic timescales in brain activity with physically based neural field theory (NFT). The present work greatly simplifies the formula of the analytical ERP, and separates its spatial part (eigenmodes) from the temporal part (poles). Within this framework, calculations involve contour integrations that yield an explicit expression for ERPs. The dominant global mode is considered explicitly in more detail to study how the ERP varies with time in this mode and to illustrate the method. For each arousal state in sleep and wake, the resonances of the system are determined and it is found that five poles are sufficient to study the main dynamics of the system in waking eyes-open and eyes-closed states. Similarly, it is shown that six poles suffice to reproduce ERPs in rapid-eye movement sleep, sleep state 1, and sleep state 2 states, whereas just four poles suffice to reproduce the dynamics in slow wave sleep. Thus, six poles are sufficient to preserve the main global ERP dynamics of the system for all states of arousal. These six poles correspond to the dominant resonances of the system at slow-wave, alpha, and beta frequencies. These results provide the basis for simplified analytic treatment of brain dynamics and link observations more closely to theory.

Brain dynamics and structure-function relationships via spectral factorization and the transfer function

It is shown how the brain's linear transfer function provides a means to systematically analyze brain connectivity and dynamics, and to infer connectivity, eigenmodes, and activity measures such as spectra, evoked responses, coherence, and causality, all of which are widely used in brain monitoring. In particular, the Wilson spectral factorization algorithm is outlined and used to efficiently obtain linear transfer functions from experimental two-point correlation functions. The algorithm is tested on a series of brain-like structures of increasing complexity which include time delays, asymmetry, two-dimensionality, and complex network connectivity. These tests are used to verify the algorithm is suitable for application to brain dynamics, specify sampling requirements for experimental time series, and to verify that its runtime is short enough to obtain accurate results for systems of similar size to current experiments. The results can equally well be applied to inference of the transfer function in complex linear systems other than brains.

Coupling of gamma band activity to sleep spindle oscillations - a combined EEG/MEG study

Sleep spindles are crucial to memory consolidation. Cortical gamma oscillations (30-100 Hz) are considered to reflect processing of memory in local cortical networks. The temporal and regulatory relationship between spindles and gamma activity might therefore provide clues into how sleep strengthens cortical memory representations. Here, combining EEG with MEG recordings during sleep in healthy humans (n = 12), we investigated the temporal relationships of cortical gamma band activity, always measured by MEG, during fast (12-16 Hz) and slow (8-12 Hz) sleep spindles detected in the EEG or MEG. Time-frequency distributions did not show a consistent coupling of gamma to the spindle oscillation, although activity in the low gamma (30-40 Hz) and neighboring beta range (<30 Hz) was generally increased during spindles. However, more fine-grained analyses of cross-frequency interactions revealed that both low and high gamma power (30-100 Hz) was coupled to the phase of slow and fast EEG spindles, importantly, with this coupling at a fixed phase only for the oscillations within an individual spindle, but with variable phase across spindles. We did not observe any coupling of gamma activity for spindles detected solely in the MEG and not in parallel EEG recordings, raising the possibility that these are more local spindles of different quality. Similar to fast spindle activity, low gamma band power followed a ~0.025 Hz infraslow rhythm during sleep whose frequency, however, was significantly faster than that of spindle activity. Our findings suggest a general function of fast and slow spindles that by spanning larger cortical networks might serve to synchronize gamma band activity occurring in more local but distributed networks. Thereby, spindles might help linking local memory processing between distributed networks.

Nonlinear wave-wave interactions in the brain

Neural field theory of the corticothalamic system is used to analyze nonlinear wave-wave interactions in steady state visual evoked potential responses. The nonlinear power spectrum is analytically calculated by convolving the linear power spectrum with itself and other factors. Periodic sine and square wave stimuli are used to generate steady state visual evoked potential responses and to study stimulus-driven nonlinear corticothalamic dynamic interactions. Moreover, we use dual sine drives to analyze the driven dynamics. Numerical analysis shows that the nonlinear power spectrum embodies key nonlinear features, including harmonic and subharmonic generation, entrainment of the alpha rhythm to periodic stimuli at the drive frequency, sum and difference frequencies due to wave-wave coalescence and decay. Further, the scaling properties of the key phenomena observed in nonlinear interactions are studied, verifying some of the theoretical predictions for these being generated by three-wave processes.

Effects of physiological parameter evolution on the dynamics of tonic-clonic seizures

The temporal and spectral characteristics of tonic-clonic seizures are investigated using a neural field model of the corticothalamic system in the presence of a temporally varying connection strength between the cerebral cortex and thalamus. Increasing connection strength drives the system into ∼ 10 Hz seizure oscillations once a threshold is passed and a subcritical Hopf bifurcation occurs. In this study, the spectral and temporal characteristics of tonic-clonic seizures are explored as functions of the relevant properties of physiological connection strengths, such as maximum strength, time above threshold, and the ramp rate at which the strength increases or decreases. Analysis shows that the seizure onset time decreases with the maximum connection strength and time above threshold, but increases with the ramp rate. Seizure duration and offset time increase with maximum connection strength, time above threshold, and rate of change. Spectral analysis reveals that the power of nonlinear harmonics and the duration of the oscillations increase as the maximum connection strength and the time above threshold increase. A secondary limit cycle at ∼ 18 Hz, termed a saddle-cycle, is also seen during seizure onset and becomes more prominent and robust with increasing ramp rate. If the time above the threshold is too small, the system does not reach the 10 Hz limit cycle, and only exhibits 18 Hz saddle-cycle oscillations. It is also seen that the time to reach the saturated large amplitude limit-cycle seizure oscillation from both the instability threshold and from the end of the saddle-cycle oscillations is inversely proportional to the square root of the ramp rate.

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.

Sleep Modelling across Physiological Levels

Sleep and circadian rhythms are regulated across multiple functional, spatial and temporal levels: from genes to networks of coupled neurons and glial cells, to large scale brain dynamics and behaviour. The dynamics at each of these levels are complex and the interaction between the levels is even more so, so research have mostly focused on interactions within the levels to understand the underlying mechanisms—the so-called reductionist approach. Mathematical models were developed to test theories of sleep regulation and guide new experiments at each of these levels and have become an integral part of the field. The advantage of modelling, however, is that it allows us to simulate and test the dynamics of complex biological systems and thus provides a tool to investigate the connections between the different levels and study the system as a whole. In this paper I review key models of sleep developed at different physiological levels and discuss the potential for an integrated systems biology approach for sleep regulation across these levels. I also highlight the necessity of building mechanistic connections between models of sleep and circadian rhythms across these levels.

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.

Spectrum of connectivity fluctuations including the effect of activity-dependent feedback

The spatiotemporal spectrum of feedback-driven fluctuations of brain connectivity is investigated using nonlinear neural field theory of the corticothalamic system. Weakly nonlinear dynamics of neural feedbacks are expanded in terms of first order perturbations of neural activity relative to a fixed point. Susceptibilities are used to quantify the change in connectivity per unit change in presynaptic or postsynaptic activity caused by nonlinear feedbacks such as facilitation, depression, sensitization, potentiation, and the effects of discrete eigenmode structure are included for a spherical brain geometry. Spectral signatures such as resonances are identified that allow the presence of particular presynaptic and postsynaptic feedback effects to be inferred. These include additional resonances at high frequencies and shifts of existing spectral peaks, mostly visible in the lowest spatial modes of the response.

NFTsim: Theory and Simulation of Multiscale Neural Field Dynamics

A user ready, portable, documented software package, NFTsim, is presented to facilitate numerical simulations of a wide range of brain systems using continuum neural field modeling. NFTsim enables users to simulate key aspects of brain activity at multiple scales. At the microscopic scale, it incorporates characteristics of local interactions between cells, neurotransmitter effects, synaptodendritic delays and feedbacks. At the mesoscopic scale, it incorporates information about medium to large scale axonal ranges of fibers, which are essential to model dissipative wave transmission and to produce synchronous oscillations and associated cross-correlation patterns as observed in local field potential recordings of active tissue. At the scale of the whole brain, NFTsim allows for the inclusion of long range pathways, such as thalamocortical projections, when generating macroscopic activity fields. The multiscale nature of the neural activity produced by NFTsim has the potential to enable the modeling of resulting quantities measurable via various neuroimaging techniques. In this work, we give a comprehensive description of the design and implementation of the software. Due to its modularity and flexibility, NFTsim enables the systematic study of an unlimited number of neural systems with multiple neural populations under a unified framework and allows for direct comparison with analytic and experimental predictions. The code is written in C++ and bundled with Matlab routines for a rapid quantitative analysis and visualization of the outputs. The output of NFTsim is stored in plain text file enabling users to select from a broad range of tools for offline analysis. This software enables a wide and convenient use of powerful physiologically-based neural field approaches to brain modeling. NFTsim is distributed under the Apache 2.0 license.

Dependence of absence seizure dynamics on physiological parameter evolution

A neural field model of the corticothalamic system is applied to investigate the temporal and spectral characteristics of absence seizures in the presence of a temporally varying connection strength between the cerebral cortex and thalamus. Increasing connection strength drives the system into an absence seizure-like state once a threshold is passed and a supercritical Hopf bifurcation occurs. The dynamics and spectral characteristics of the resulting model seizures are explored as functions of maximum connection strength, time above threshold, and the rate at which the connection strength increases (ramp rate). Our results enable spectral and temporal characteristics of seizures to be related to changes in the underlying physiological evolution of connections via nonlinear dynamics and neural field theory. Spectral analysis reveals that the power of the harmonics and the duration of the oscillations increase as the maximum connection strength and the time above threshold increase. It is also found that the time to reach the stable limit-cycle seizure oscillation from the instability threshold decreases with the square root of the ramp rate.

Explaining the heterogeneity of functional connectivity findings in multiple sclerosis: An empirically informed modeling study

To understand the heterogeneity of functional connectivity results reported in the literature, we analyzed the separate effects of grey and white matter damage on functional connectivity and networks in multiple sclerosis. For this, we employed a biophysical thalamo‐cortical model consisting of interconnected cortical and thalamic neuronal populations, informed and amended by empirical diffusion MRI tractography data, to simulate functional data that mimic neurophysiological signals. Grey matter degeneration was simulated by decreasing within population connections and white matter degeneration by lowering between population connections, based on lesion predilection sites in multiple sclerosis. For all simulations, functional connectivity and functional network organization are quantified by phase synchronization and network integration, respectively. Modeling results showed that both cortical and thalamic grey matter damage induced a global increase in functional connectivity, whereas white matter damage induced an initially increased connectivity followed by a global decrease. Both white and especially grey matter damage, however, induced a decrease in network integration. These empirically informed simulations show that specific topology and timing of structural damage are nontrivial aspects in explaining functional abnormalities in MS. Insufficient attention to these aspects likely explains contradictory findings in multiple sclerosis functional imaging studies so far.

Spectral signatures of activity-dependent neural feedback in the corticothalamic system

The modulation of neural quantities by presynaptic and postsynaptic activities via local feedback processes is investigated by incorporating nonlinear phenomena such as relative refractory period, synaptic enhancement, synaptic depression, and habituation. This is done by introducing susceptibilities, which quantify the response in either firing threshold or synaptic strength to unit change in either presynaptic or postsynaptic activity. Effects on the power spectra are then analyzed for a realistic corticothalamic model to determine the spectral signatures of various nonlinear processes and to what extent these are distinct. Depending on the feedback processes, there can be enhancements or reductions in low-frequency and/or alpha power, splitting of the alpha resonance, and/or appearance of new resonances at high frequencies. These features in the power spectra allow processes to be fully distinguished where they are unique, or partly distinguished if they are common to only a subset of feedbacks, and can potentially be used to constrain the types, strengths, and dynamics of feedbacks present.

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.

Causal Role of Thalamic Interneurons in Brain State Transitions: A Study Using a Neural Mass Model Implementing Synaptic Kinetics

Experimental studies on the Lateral Geniculate Nucleus (LGN) of mammals and rodents show that the inhibitory interneurons (IN) receive around 47.1% of their afferents from the retinal spiking neurons, and constitute around 20–25% of the LGN cell population. However, there is a definite gap in knowledge about the role and impact of IN on thalamocortical dynamics in both experimental and model-based research. We use a neural mass computational model of the LGN with three neural populations viz. IN, thalamocortical relay (TCR), thalamic reticular nucleus (TRN), to study the causality of IN on LGN oscillations and state-transitions. The synaptic information transmission in the model is implemented with kinetic modeling, facilitating the linking of low-level cellular attributes with high-level population dynamics. The model is parameterized and tuned to simulate alpha (8–13 Hz) rhythm that is dominant in both Local Field Potential (LFP) of LGN and electroencephalogram (EEG) of visual cortex in an awake resting state with eyes closed. The results show that: First, the response of the TRN is suppressed in the presence of IN in the circuit; disconnecting the IN from the circuit effects a dramatic change in the model output, displaying high amplitude synchronous oscillations within the alpha band in both TCR and TRN. These observations conform to experimental reports implicating the IN as the primary inhibitory modulator of LGN dynamics in a cognitive state, and that reduced cognition is achieved by suppressing the TRN response. Second, the model validates steady state visually evoked potential response in humans corresponding to periodic input stimuli; however, when the IN is disconnected from the circuit, the output power spectra do not reflect the input frequency. This agrees with experimental reports underpinning the role of IN in efficient retino-geniculate information transmission. Third, a smooth transition from alpha to theta band is observed by progressive decrease of neurotransmitter concentrations in the synaptic clefts; however, the transition is abrupt with removal of the IN circuitry in the model. The results imply a role of IN toward maintaining homeostasis in the LGN by suppressing any instability that may arise due to anomalous synaptic attributes.

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.

Neural field theory of nonlinear wave-wave and wave-neuron processes

Systematic expansion of neural field theory equations in terms of nonlinear response functions is carried out to enable a wide variety of nonlinear wave-wave and wave-neuron processes to be treated systematically in systems involving multiple neural populations. The results are illustrated by analyzing second-harmonic generation, and they can also be applied to wave-wave coalescence, multiharmonic generation, facilitation, depression, refractoriness, and other nonlinear processes.

Experimental observation of a theoretically predicted nonlinear sleep spindle harmonic in human EEG

OBJECTIVE

To investigate the properties of a sleep spindle harmonic oscillation previously predicted by a theoretical neural field model of the brain.

METHODS

Spindle oscillations were extracted from EEG data from nine subjects using an automated algorithm. The power and frequency of the spindle oscillation and the harmonic oscillation were compared across subjects. The bicoherence of the EEG was calculated to identify nonlinear coupling.

RESULTS

All subjects displayed a spindle harmonic at almost exactly twice the frequency of the spindle. The power of the harmonic scaled nonlinearly with that of the spindle peak, consistent with model predictions. Bicoherence was observed at the spindle frequency, confirming the nonlinear origin of the harmonic oscillation.

CONCLUSIONS

The properties of the sleep spindle harmonic were consistent with the theoretical modeling of the sleep spindle harmonic as a nonlinear phenomenon.

SIGNIFICANCE

Most models of sleep spindle generation are unable to produce a spindle harmonic oscillation, so the observation and theoretical explanation of the harmonic is a significant step in understanding the mechanisms of sleep spindle generation. Unlike seizures, sleep spindles produce nonlinear effects that can be observed in healthy controls, and unlike the alpha oscillation, there is no linearly generated harmonic that can obscure nonlinear effects. This makes the spindle harmonic a good candidate for future investigation of nonlinearity in the brain.