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

Unified theory of alpha, mu, and tau rhythms via eigenmodes of brain activity

A compact description of the frequency structure and topography of human alpha-band rhythms is obtained by use of the first four brain activity eigenmodes previously derived from corticothalamic neural field theory. Just two eigenmodes that overlap in frequency are found to reproduce the observed topography of the classical alpha rhythm for subjects with a single, occipitally concentrated alpha peak in their electroencephalograms. Alpha frequency splitting and relative amplitudes of double alpha peaks are explored analytically and numerically within this four-mode framework using eigenfunction expansion and perturbation methods. These effects are found to result primarily from the different eigenvalues and corticothalamic gains corresponding to the eigenmodes. Three modes with two non-overlapping frequencies suffice to reproduce the observed topography for subjects with a double alpha peak, where the appearance of a distinct second alpha peak requires an increase of the corticothalamic gain of higher eigenmodes relative to the first. Conversely, alpha blocking is inferred to be linked to a relatively small attention-dependent reduction of the gain of the relevant eigenmodes, whose effect is enhanced by the near-critical state of the brain and whose sign is consistent with inferences from neural field theory. The topographies and blocking of the mu and tau rhythms within the alpha-band are explained analogously via eigenmodes. Moreover, the observation of three rhythms in the alpha band is due to there being exactly three members of the first family of spatially nonuniform modes. These results thus provide a simple, unified description of alpha band rhythms and enable experimental observations of spectral structure and topography to be linked directly to theory and underlying physiology.

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.

Neural Field Theory of Evoked Response Sequences and Mismatch Negativity With Adaptation

Physiologically based neural field theory of the corticothalamic system is used to calculate the responses evoked by trains of auditory stimuli that correspond to different cortical locations via the tonotopic map. The results are shown to account for standard and deviant evoked responses to frequent and rare stimuli, respectively, in the auditory oddball paradigms widely used in human cognitive studies, and the so-called mismatch negativity between them. It also reproduces a wide range of other effects and variants, including the mechanism by which a change in standard responses relative to deviants can develop through adaptation, different responses when two deviants are presented in a row or a standard is presented after two deviants, relaxation of standard responses back to deviant form after a stimulus-free period, and more complex sequences. Some cases are identified in which adaptation does not account for the whole difference between standard and deviant responses. The results thus provide a systematic means to determine how much of the response is due to adaptation in the system comprising the primary auditory cortex and medial geniculate nucleus, and how much requires involvement of higher-level processing.

Determination of Dynamic Brain Connectivity via Spectral Analysis

Spectral analysis based on neural field theory is used to analyze dynamic connectivity via methods based on the physical eigenmodes that are the building blocks of brain dynamics. These approaches integrate over space instead of averaging over time and thereby greatly reduce or remove the temporal averaging effects, windowing artifacts, and noise at fine spatial scales that have bedeviled the analysis of dynamical functional connectivity (FC). The dependences of FC on dynamics at various timescales, and on windowing, are clarified and the results are demonstrated on simple test cases, demonstrating how modes provide directly interpretable insights that can be related to brain structure and function. It is shown that FC is dynamic even when the brain structure and effective connectivity are fixed, and that the observed patterns of FC are dominated by relatively few eigenmodes. Common artifacts introduced by statistical analyses that do not incorporate the physical nature of the brain are discussed and it is shown that these are avoided by spectral analysis using eigenmodes. Unlike most published artificially discretized “resting state networks” and other statistically-derived patterns, eigenmodes overlap, with every mode extending across the whole brain and every region participating in every mode—just like the vibrations that give rise to notes of a musical instrument. Despite this, modes are independent and do not interact in the linear limit. It is argued that for many purposes the intrinsic limitations of covariance-based FC instead favor the alternative of tracking eigenmode coefficients vs. time, which provide a compact representation that is directly related to biophysical brain dynamics.

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.

Evoked response activity eigenmode analysis in a convoluted cortex via neural field theory

Neural field theory of the corticothalamic system is used to explore evoked response potentials (ERPs) caused by spatially localized impulse stimuli on the convoluted cortex and on a spherical cortex. Eigenfunctions are calculated analytically on the spherical cortex and numerically on the convoluted cortex via eigenfunction expansions. Eigenmodes on a convoluted cortex are similar to those of the spherical cortex, and a few such modes are found to be sufficient to reproduce the main ERP features. It is found that the ERP peak is stronger in spherical cortex than convoluted cortex, but in both cases the peak decreases monotonically with increasing distance from the stimulus point. In the convoluted case, cortical folding causes ERPs to differ between locations at the same distance from the stimulus point and spherical symmetries are only approximately preserved.

Spherical-harmonics mode decomposition of neural field equations

Large-scale neural networks can be described in the spatial continuous limit by neural field equations. For large-scale brain networks, the connectivity is typically translationally variant and imposes a large computational burden upon simulations. To reduce this burden, we take a semiquantitative approach and study the dynamics of neural fields described by a delayed integrodifferential equation. We decompose the connectivity into spatially variant and invariant contributions, which typically comprise the short- and long-range fiber systems, respectively. The neural fields are mapped on the two-dimensional spherical surface, which is choice consistent with routine mappings of cortical surfaces. Then, we perform mathematically a mode decomposition of the neural field equation into spherical harmonic basis functions. A spatial truncation of the leading orders at low wave number is consistent with the spatially coherent pattern formation of large-scale patterns observed in simulations and empirical brain imaging data and leads to a low-dimensional representation of the dynamics of the neural fields, bearing promise for an acceleration of the numerical simulations by orders of magnitude.

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.

Neural field theory of effects of brain modifications and lesions on functional connectivity: Acute effects, short-term homeostasis, and long-term plasticity

Neural field theory is used to predict the functional connectivity effects of lesions or other modifications to effective connectivity. Widespread initial changes are predicted after localized or diffuse changes to white or gray matter, consistent with observations, and enabling lesion severity indexes to be defined. It is shown how short-term homeostasis and longer-term plasticity can reduce perturbations while maintaining brain criticality under conditions where some connections remain fixed because of damage in the lesion core. The extent to which such effects can compensate for initial connectivity changes is then explored, showing that the strongest corrective changes are concentrated toward the edges of the perturbation if it is localized and its core is fixed. The results are applicable to inferring underlying connectivity changes and to interpreting and monitoring functional connectivity modifications after lesions, injury, surgery, drugs, or brain stimulation.

Physical brain connectomics

Brain connectivity and structure-function relationships are analyzed from a physical perspective in place of common graph-theoretic and statistical approaches that overwhelmingly ignore the brain's physical structure and geometry. Field theory is used to define connectivity tensors in terms of bare and dressed propagators, and discretized representations are implemented that respect the physical nature and dimensionality of the quantities involved, retain the correct continuum limit, and enable diagrammatic analysis. Eigenfunction analysis is used to simultaneously characterize and probe patterns of brain connectivity and activity, in place of statistical or phenomenological patterns. Physically based measures that characterize the connectivity are then developed in coordinate and spectral domains; some of which generalize or rectify graph-theoretic measures to implement correct dimensionality and continuum limits, and some replace graph-theoretic quantities. Traditional graph-based measures are shown to be highly prone to artifacts introduced by discretization and threshold, often because essential physical constraints have not been imposed, dimensionality has not been included, and/or distinctions between scalar, vector, and tensor quantities have not been considered. The results can replace them in ways that converge correctly and measure properties of brain structure, rather than of its discretization, and thus potentially enable physical interpretation of the many phenomenological results in the literature. Geometric effects are shown to dominate in determining many brain properties and care must be taken not to interpret geometric differences as differences in intrinsic neural connectivity. The results demonstrate the need to use systematic physical methods to analyze the brain and the potential of such methods to obtain new insights from data, make new predictions for experimental test, and go beyond phenomenological classification to dynamics and mechanisms.

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.