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.

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.

Neural field theory of perceptual echo and implications for estimating brain connectivity

Neural field theory is used to predict and analyze the phenomenon of perceptual echo in which random input stimuli at one location are correlated with electroencephalographic responses at other locations. It is shown that this echo correlation (EC) yields an estimate of the transfer function from the stimulated point to other locations. Modal analysis then explains the observed spatiotemporal structure of visually driven EC and the dominance of the alpha frequency; two eigenmodes of similar amplitude dominate the response, leading to temporal beating and a line of low correlation that runs from the crown of the head toward the ears. These effects result from mode splitting and symmetry breaking caused by interhemispheric coupling and cortical folding. It is shown how eigenmodes obtained from functional magnetic resonance imaging experiments can be combined with temporal dynamics from EC or other evoked responses to estimate the spatiotemporal transfer function between any two points and hence their effective connectivity.

The balanced and introspective brain

Transfers of large-scale neural activity into, within and between corticothalamic neural populations and brain hemispheres are analysed using time-integrated transfer functions and state parameters obtained from neural field theory for a variety of arousal states. It is shown that the great majority of activity results from feedbacks within the corticothalamic system, including significant transfer between hemispheres, but only a small minority arises via net input from the external world, with the brain thus in a near-critical, highly introspective state. Notably, the total excitatory and inhibitory influences on cortical neurons are balanced to within a few per cent across arousal states. Strong negative intrahemispheric feedforward exists to the cortex, and even larger interhemispheric positive feedforward, but these are modified by feedback loops to yield near-critical positive overall gain. The results underline the utility of transfer functions for the analysis of brain activity.

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.

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.

Shock-like haemodynamic responses induced in the primary visual cortex by moving visual stimuli

It is shown that recently discovered haemodynamic waves can form shock-like fronts when driven by stimuli that excite the cortex in a patch that moves faster than the haemodynamic wave velocity. If stimuli are chosen in order to induce shock-like behaviour, the resulting blood oxygen level-dependent (BOLD) response is enhanced, thereby improving the signal to noise ratio of measurements made with functional magnetic resonance imaging. A spatio-temporal haemodynamic model is extended to calculate the BOLD response and determine the main properties of waves induced by moving stimuli. From this, the optimal conditions for stimulating shock-like responses are determined, and ways of inducing these responses in experiments are demonstrated in a pilot study.

Response-mode decomposition of spatio-temporal haemodynamics

The blood oxygen-level dependent (BOLD) response to a neural stimulus is analysed using the transfer function derived from a physiologically based poroelastic model of cortical tissue. The transfer function is decomposed into components that correspond to distinct poles, each related to a response mode with a natural frequency and dispersion relation; together these yield the total BOLD response. The properties of the decomposed components provide a deeper understanding of the nature of the BOLD response, via the components' frequency dependences, spatial and temporal power spectra, and resonances. The transfer function components are then used to separate the BOLD response to a localized impulse stimulus, termed the Green function or spatio-temporal haemodynamic response function, into component responses that are explicitly related to underlying physiological quantities. The analytical results also provide a quantitative tool to calculate the linear BOLD response to an arbitrary neural drive, which is faster to implement than direct Fourier transform methods. The results of this study can be used to interpret functional magnetic resonance imaging data in new ways based on physiology, to enhance deconvolution methods and to design experimental protocols that can selectively enhance or suppress particular responses, to probe specific physiological phenomena.

Cortical geometry as a determinant of brain activity eigenmodes: Neural field analysis

Perturbation analysis of neural field theory is used to derive eigenmodes of neural activity on a cortical hemisphere, which have previously been calculated numerically and found to be close analogs of spherical harmonics, despite heavy cortical folding. The present perturbation method treats cortical folding as a first-order perturbation from a spherical geometry. The first nine spatial eigenmodes on a population-averaged cortical hemisphere are derived and compared with previous numerical solutions. These eigenmodes contribute most to brain activity patterns such as those seen in electroencephalography and functional magnetic resonance imaging. The eigenvalues of these eigenmodes are found to agree with the previous numerical solutions to within their uncertainties. Also in agreement with the previous numerics, all eigenmodes are found to closely resemble spherical harmonics. The first seven eigenmodes exhibit a one-to-one correspondence with their numerical counterparts, with overlaps that are close to unity. The next two eigenmodes overlap the corresponding pair of numerical eigenmodes, having been rotated within the subspace spanned by that pair, likely due to second-order effects. The spatial orientations of the eigenmodes are found to be fixed by gross cortical shape rather than finer-scale cortical properties, which is consistent with the observed intersubject consistency of functional connectivity patterns. However, the eigenvalues depend more sensitively on finer-scale cortical structure, implying that the eigenfrequencies and consequent dynamical properties of functional connectivity depend more strongly on details of individual cortical folding. Overall, these results imply that well-established tools from perturbation theory and spherical harmonic analysis can be used to calculate the main properties and dynamics of low-order brain eigenmodes.

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.

Critical dynamics of Hopf bifurcations in the corticothalamic system: Transitions from normal arousal states to epileptic seizures

A physiologically based corticothalamic model of large-scale brain activity is used to analyze critical dynamics of transitions from normal arousal states to epileptic seizures, which correspond to Hopf bifurcations. This relates an abstract normal form quantitatively to underlying physiology that includes neural dynamics, axonal propagation, and time delays. Thus, a bridge is constructed that enables normal forms to be used to interpret quantitative data. The normal form of the Hopf bifurcations with delays is derived using Hale's theory, the center manifold theorem, and normal form analysis, and it is found to be explicitly expressed in terms of transfer functions and the sensitivity matrix of a reduced open-loop system. It can be applied to understand the effect of each physiological parameter on the critical dynamics and determine whether the Hopf bifurcation is supercritical or subcritical in instabilities that lead to absence and tonic-clonic seizures. Furthermore, the effects of thalamic and cortical nonlinearities on the bifurcation type are investigated, with implications for the roles of underlying physiology. The theoretical predictions about the bifurcation type and the onset dynamics are confirmed by numerical simulations and provide physiologically based criteria for determining bifurcation types from first principles. The results are consistent with experimental data from previous studies, imply that new regimes of seizure transitions may exist in clinical settings, and provide a simplified basis for control-systems interventions. Using the normal form, and the full equations from which it is derived, more complex dynamics, such as quasiperiodic cycles and saddle cycles, are discovered near the critical points of the subcritical Hopf bifurcations.

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.

Inference of direct and multistep effective connectivities from functional connectivity of the brain and of relationships to cortical geometry

BACKGROUND

The problem of inferring effective brain connectivity from functional connectivity is under active investigation, and connectivity via multistep paths is poorly understood.

NEW METHOD

A method is presented to calculate the direct effective connection matrix (deCM), which embodies direct connection strengths between brain regions, from functional CMs (fCMs) by minimizing the difference between an experimental fCM and one calculated via neural field theory from an ansatz deCM based on an experimental anatomical CM.

RESULTS

The best match between fCMs occurs close to a critical point, consistent with independent published stability estimates. Residual mismatch between fCMs is identified to be largely due to interhemispheric connections that are poorly estimated in an initial ansatz deCM due to experimental limitations; improved ansatzes substantially reduce the mismatch and enable interhemispheric connections to be estimated. Various levels of significant multistep connections are then imaged via the neural field theory (NFT) result that these correspond to powers of the deCM; these are shown to be predictable from geometric distances between regions.

COMPARISON WITH EXISTING METHODS

This method gives insight into direct and multistep effective connectivity from fCMs and relating to physiology and brain geometry. This contrasts with other methods, which progressively adjust connections without an overarching physiologically based framework to deal with multistep or poorly estimated connections.

CONCLUSIONS

deCMs can be usefully estimated using this method and the results enable multistep connections to be investigated systematically.

Wake-sleep transition as a noisy bifurcation

A recent physiologically based model of the ascending arousal system is used to analyze the dynamics near the transition from wake to sleep, which corresponds to a saddle-node bifurcation at a critical point. A normal form is derived by approximating the dynamics by those of a particle in a parabolic potential well with dissipation. This mechanical analog is used to calculate the power spectrum of fluctuations in response to a white noise drive, and the scalings of fluctuation variance and spectral width are derived versus distance from the critical point. The predicted scalings are quantitatively confirmed by numerical simulations, which show that the variance increases and the spectrum undergoes critical slowing, both in accord with theory. These signals can thus serve as potential precursors to indicate imminent wake-sleep transition, with potential application to safety-critical occupations in transport, air-traffic control, medicine, and heavy industry.

Publisher's Note: Eigenvector dynamics under perturbation of modular networks [Phys. Rev. E 93, 062312 (2016)]

This corrects the article DOI: 10.1103/PhysRevE.93.062312.

A Quantitative Model of Sleep-Wake Dynamics Based on the Physiology of the Brainstem Ascending Arousal System

A quantitative, physiology-based model of the ascending arousal system is developed, using continuum neuronal population modeling, which involves averaging properties such as firing rates across neurons in each population. The model includes the ventrolateral preoptic area (VLPO), where circadian and homeostatic drives enter the system, the monoaminergic and cholinergic nuclei of the ascending arousal system, and their interconnections. The human sleep-wake cycle is governed by the activities of these nuclei, which modulate the behavioral state of the brain via diffuse neuromodulatory projections. The model parameters are not free since they correspond to physiological observables. Approximate parameter bounds are obtained by requiring consistency with physiological and behavioral measures, and the model replicates the human sleep-wake cycle, with physiologically reasonable voltages and firing rates. Mutual inhibition between the wake-promoting monoaminergic group and sleep-promoting VLPO causes ``flip-flop'' behavior, with most time spent in 2 stable steady states corresponding to wake and sleep, with transitions between them on a timescale of a few minutes. The model predicts hysteresis in the sleep-wake cycle, with a region of bistability of the wake and sleep states. Reducing the monoaminergic-VLPO mutual inhibition results in a smaller hysteresis loop. This makes the model more prone to wake-sleep transitions in both directions and makes the states less distinguishable, as in narcolepsy. The model behavior is robust across the constrained parameter ranges, but with sufficient flexibility to describe a wide range of observed phenomena.

Eigenvector dynamics under perturbation of modular networks

Rotation dynamics of eigenvectors of modular network adjacency matrices under random perturbations are presented. In the presence of q communities, the number of eigenvectors corresponding to the q largest eigenvalues form a "community" eigenspace and rotate together, but separately from that of the "bulk" eigenspace spanned by all the other eigenvectors. Using this property, the number of modules or clusters in a network can be estimated in an algorithm-independent way. A general argument and derivation for the theoretical detectability limit for sparse modular networks with q communities is presented, beyond which modularity persists in the system but cannot be detected. It is shown that for detecting the clusters or modules using the adjacency matrix, there is a "band" in which it is hard to detect the clusters even before the theoretical detectability limit is reached, and for which the theoretically predicted detectability limit forms the sufficient upper bound. Analytic estimations of these bounds are presented and empirically demonstrated.

Eigenmodes of brain activity: Neural field theory predictions and comparison with experiment

Neural field theory of the corticothalamic system is applied to predict and analyze the activity eigenmodes of the bihemispheric brain, focusing particularly on their spatial structure. The eigenmodes of a single brain hemisphere are found to be close analogs of spherical harmonics, which are the natural modes of the sphere. Instead of multiple eigenvalues being equal, as in the spherical case, cortical folding splits them to have distinct values. Inclusion of interhemispheric connections between homologous regions via the corpus callosum leads to further splitting that depends on symmetry or antisymmetry of activity between brain hemispheres, and the strength and sign of the interhemispheric connections. Symmetry properties of the lowest observed eigenmodes strongly constrain the interhemispheric connectivity strengths and unihemispheric mode spectra, and it is predicted that most spontaneous brain activity will be symmetric between hemispheres, consistent with observations. Comparison with the eigenmodes of an experimental anatomical connectivity matrix confirms these results, permits the relative strengths of intrahemispheric and interhemispheric connectivities to be approximately inferred from their eigenvalues, and lays the foundation for further experimental tests. The results are consistent with brain activity being in corticothalamic eigenmodes, rather than discrete "networks" and open the way to new approaches to brain analysis.

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.

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.

Slow-wave oscillations in a corticothalamic model of sleep and wake

A physiologically-based corticothalamic neural field model is used to study slow wave oscillations including cortical UP and DOWN states in deep sleep by extending it to incorporate bursting dynamics of neurons in the thalamic reticular nucleus. The interplay of local bursting dynamics and network interactions produces the cortical UP and DOWN states of slow wave sleep while preserving previously verified model predictions in the wake state. Results show that EEG spectral features in wake and sleep are reproduced. The bursting is subthreshold but acts to intensify the amplitude of oscillations in slow wave sleep with deep UP/DOWN oscillations on the cortex emerging naturally. Furthermore, there is a continuous cycle between the two regimes, rather than a flip-flop between discrete states.