Bifurcation structure determines different phase-amplitude coupling patterns in the activity of biologically plausible neural networks

How to design optimal brain stimulation to modulate phase-amplitude coupling?

Objective.Phase-amplitude coupling (PAC), the coupling of the amplitude of a faster brain rhythm to the phase of a slower brain rhythm, plays a significant role in brain activity and has been implicated in various neurological disorders. For example, in Parkinson's disease, PAC between the beta (13-30 Hz) and gamma (30-100 Hz) rhythms in the motor cortex is exaggerated, while in Alzheimer's disease, PAC between the theta (4-8 Hz) and gamma rhythms is diminished. Modulating PAC (i.e. reducing or enhancing PAC) using brain stimulation could therefore open new therapeutic avenues. However, while it has been previously reported that phase-locked stimulation can increase PAC, it is unclear what the optimal stimulation strategy to modulate PAC might be. Here, we provide a theoretical framework to narrow down the experimental optimisation of stimulation aimed at modulating PAC, which would otherwise rely on trial and error.Approach.We make analytical predictions using a Stuart-Landau model, and confirm these predictions in a more realistic model of coupled neural populations.Main results.Our framework specifies the critical Fourier coefficients of the stimulation waveform which should be tuned to optimally modulate PAC. Depending on the characteristics of the amplitude response curve of the fast population, these components may include the slow frequency, the fast frequency, combinations of these, as well as their harmonics. We also show that the optimal balance of energy between these Fourier components depends on the relative strength of the endogenous slow and fast rhythms, and that the alignment of fast components with the fast rhythm should change throughout the slow cycle. Furthermore, we identify the conditions requiring to phase-lock stimulation to the fast and/or slow rhythms.Significance.Together, our theoretical framework lays the foundation for guiding the development of innovative and more effective brain stimulation aimed at modulating PAC for therapeutic benefit.

Irregular optogenetic stimulation waveforms can induce naturalistic patterns of hippocampal spectral activity

Objective. Therapeutic brain stimulation is conventionally delivered using constant-frequency stimulation pulses. Several recent clinical studies have explored how unconventional and irregular temporal stimulation patterns could enable better therapy. However, it is challenging to understand which irregular patterns are most effective for different therapeutic applications given the massively high-dimensional parameter space.Approach. Here we applied many irregular stimulation patterns in a single neural circuit to demonstrate how they can enable new dimensions of neural control compared to conventional stimulation, to guide future exploration of novel stimulation patterns in translational settings. We optogenetically excited the septohippocampal circuit with constant-frequency, nested pulse, sinusoidal, and randomized stimulation waveforms, systematically varying their amplitude and frequency parameters.Main results.We first found equal entrainment of hippocampal oscillations: all waveforms provided similar gamma-power increase, whereas no parameters increased theta-band power above baseline (despite the mechanistic role of the medial septum in driving hippocampal theta oscillations). We then compared each of the effects of each waveform on high-dimensional multi-band activity states using dimensionality reduction methods. Strikingly, we found that conventional stimulation drove predominantly 'artificial' (different from behavioral activity) effects, whereas all irregular waveforms induced activity patterns that more closely resembled behavioral activity.Significance. Our findings suggest that irregular stimulation patterns are not useful when the desired mechanism is to suppress or enhance a single frequency band. However, novel stimulation patterns may provide the greatest benefit for neural control applications where entraining a particular mixture of bands (e.g. if they are associated with different symptoms) or behaviorally-relevant activity is desired.

A phase-shifting anterior-posterior network organizes global phase relations

Prior research has identified a variety of task-dependent networks that form through inter-regional phase-locking of oscillatory activity that are neural correlates of specific behaviors. Despite ample knowledge of task-specific functional networks, general rules governing global phase relations have not been investigated. To discover such general rules, we focused on phase modularity, measured as the degree to which global phase relations in EEG comprised distinct synchronized clusters interacting with one another at large phase lags. Synchronized clusters were detected with a standard community-detection algorithm, and the degree of phase modularity was quantified by the index q. Notably, we found that the mechanism controlling phase modularity is remarkably simple. A network comprising anterior-posterior long-distance connectivity coherently shifted phase relations from low-angles (|Δθ| < π/4) in low-modularity states (bottom 5% in q) to high-angles (|Δθ| > 3π/4) in high-modularity states (top 5% in q), accounting for fluctuations in phase modularity. This anterior-posterior network may play a fundamental functional role as (1) it controls phase modularity across a broad range of frequencies (3-50 Hz examined) in different behavioral conditions (resting with the eyes closed or watching a silent nature video) and (2) neural interactions (measured as power correlations) in beta-to-gamma bands were consistently elevated in high-modularity states. These results may motivate future investigations into the functional roles of phase modularity as well as the anterior-posterior network that controls it.

Hierarchical network structure as the source of hierarchical dynamics (power-law frequency spectra) in living and non-living systems: How state-trait continua (body plans, personalities) emerge from first principles in biophysics

Living systems are hierarchical control systems that display a small world network structure. In such structures, many smaller clusters are nested within fewer larger ones, producing a fractal-like structure with a 'power-law' cluster size distribution (a mereology). Just like their structure, the dynamics of living systems shows fractal-like qualities: the timeseries of inner message passing and overt behavior contain high frequencies or 'states' (treble) that are nested within lower frequencies or 'traits' (bass), producing a power-law frequency spectrum that is known as a 'state-trait continuum' in the behavioral sciences. Here, we argue that the power-law dynamics of living systems results from their power-law network structure: organisms 'vertically encode' the deep spatiotemporal structure of their (anticipated) environments, to the effect that many small clusters near the base of the hierarchy produce high frequency signal changes and fewer larger clusters at its top produce ultra-low frequencies. Such ultra-low frequencies exert a tonic regulatory pressure that produces morphological as well as behavioral traits (i.e., body plans and personalities). Nested-modular structure causes higher frequencies to be embedded within lower frequencies, producing a power-law state-trait continuum. At the heart of such dynamics lies the need for efficient energy dissipation through networks of coupled oscillators, which also governs the dynamics of non-living systems (e.q., earthquakes, stock market fluctuations). Since hierarchical structure produces hierarchical dynamics, the development and collapse of hierarchical structure (e.g., during maturation and disease) should leave specific traces in system dynamics (shifts in lower frequencies, i.e. morphological and behavioral traits) that may serve as early warning signs to system failure. The applications of this idea range from (bio)physics and phylogenesis to ontogenesis and clinical medicine.

Theta-gamma phase amplitude coupling in a hippocampal CA1 microcircuit

Phase amplitude coupling (PAC) between slow and fast oscillations is found throughout the brain and plays important functional roles. Its neural origin remains unclear. Experimental findings are often puzzling and sometimes contradictory. Most computational models rely on pairs of pacemaker neurons or neural populations tuned at different frequencies to produce PAC. Here, using a data-driven model of a hippocampal microcircuit, we demonstrate that PAC can naturally emerge from a single feedback mechanism involving an inhibitory and excitatory neuron population, which interplay to generate theta frequency periodic bursts of higher frequency gamma. The model suggests the conditions under which a CA1 microcircuit can operate to elicit theta-gamma PAC, and highlights the modulatory role of OLM and PVBC cells, recurrent connectivity, and short term synaptic plasticity. Surprisingly, the results suggest the experimentally testable prediction that the generation of the slow population oscillation requires the fast one and cannot occur without it.

Dynamic theta-modulated high frequency oscillations in rat medial prefrontal cortex during spatial working memory task

Interaction of oscillatory rhythms at different frequencies is considered to provide a neuronal mechanism for information processing and transmission. These interactions have been suggested to have a vital role in cognitive functions such as working memory and decision-making. Here, we investigated the medial prefrontal cortex (mPFC), which is known to have a critical role in successful execution of spatial working memory tasks. We recorded local field potential oscillations from mPFC while rats performed a delayed-non-match-to-place (DNMTP) task. In the DNMTP task, the rat needed to decide actively about the pathway based on the information remembered in the first phase of each trial. Our analysis revealed a dynamic phase-amplitude coupling (PAC) between theta and high frequency oscillations (HFOs). This dynamic coupling emerged near the turning point and diminished afterward. Further, theta activity during the delay period, which is thought of as the maintenance phase, in the absence of the coupling, can predict task completion time. We previously reported diminished rat performance in the DNMTP task in response to electromagnetic radiation. Here, we report an increase in the theta rhythm during delay activity besides diminishing the coupling after electromagnetic radiation. These findings suggest that the different roles of the mPFC in working memory could be supported by separate mechanisms: Theta activity during the delay period for information maintenance and theta-HFOs phase-amplitude coupling relating to the decision-making procedure.

Measuring Phase-Amplitude Coupling Based on the Jensen-Shannon Divergence and Correlation Matrix

Phase-amplitude coupling (PAC) measures the relationship between the phase of low-frequency oscillations (LFO) and the amplitude of high-frequency oscillations (HFO). It plays an important functional role in neural information processing and cognition. Thus, we propose a novel method based on the Jensen-Shannon (JS) divergence and correlation matrix. The method takes the amplitude distributions of the HFO located in the corresponding phase bins of the LFO as multichannel inputs to construct a correlation matrix, where the elements are calculated based on the JS divergence between pairs of amplitude distributions. Then, the omega complexity extracted from the correlation matrix is used to estimate the PAC strength. The simulation results demonstrate that the proposed method can effectively reflect the PAC strength and slightly vary with the data length. Moreover, it outperforms five frequently used algorithms in the performance against additive white Gaussian noise and spike noise and the ability of detecting PAC in wide frequency ranges. To validate our proposed method with real data, it was applied to analyze the local field potential recorded from the dorsomedial striatum in a male Sprague-Dawley rat. It can replicate previous results obtained with other PAC metrics. In conclusion, these results suggest that our proposed method is a powerful tool for measuring the PAC between neural oscillations.

Spatiotemporal dynamics of maximal and minimal EEG spectral power

Oscillatory neural activities are prevalent in the brain with their phase realignment contributing to the coordination of neural communication. Phase realignments may have especially strong (or weak) impact when neural activities are strongly synchronized (or desynchronized) within the interacting populations. We report that the spatiotemporal dynamics of strong regional synchronization measured as maximal EEG spectral power-referred to as activation-and strong regional desynchronization measured as minimal EEG spectral power-referred to as suppression-are characterized by the spatial segregation of small-scale and large-scale networks. Specifically, small-scale spectral-power activations and suppressions involving only 2-7% (1-4 of 60) of EEG scalp sites were prolonged (relative to stochastic dynamics) and consistently co-localized in a frequency specific manner. For example, the small-scale networks for θ, α, β1, and β2 bands (4-30 Hz) consistently included frontal sites when the eyes were closed, whereas the small-scale network for γ band (31-55 Hz) consistently clustered in medial-central-posterior sites whether the eyes were open or closed. Large-scale activations and suppressions involving over 17-30% (10-18 of 60) of EEG sites were also prolonged and generally clustered in regions complementary to where small-scale activations and suppressions clustered. In contrast, intermediate-scale activations and suppressions (involving 7-17% of EEG sites) tended to follow stochastic dynamics and were less consistently localized. These results suggest that strong synchronizations and desynchronizations tend to occur in small-scale and large-scale networks that are spatially segregated and frequency specific. These synchronization networks may broadly segregate the relatively independent and highly cooperative oscillatory processes while phase realignments fine-tune the network configurations based on behavioral demands.

Probabilistic, entropy-maximizing control of large-scale neural synchronization

Oscillatory neural activity is dynamically controlled to coordinate perceptual, attentional and cognitive processes. On the macroscopic scale, this control is reflected in the U-shaped deviations of EEG spectral-power dynamics from stochastic dynamics, characterized by disproportionately elevated occurrences of the lowest and highest ranges of power. To understand the mechanisms that generate these low- and high-power states, we fit a simple mathematical model of synchronization of oscillatory activity to human EEG data. The results consistently indicated that the majority (~95%) of synchronization dynamics is controlled by slowly adjusting the probability of synchronization while maintaining maximum entropy within the timescale of a few seconds. This strategy appears to be universal as the results generalized across oscillation frequencies, EEG current sources, and participants (N = 52) whether they rested with their eyes closed, rested with their eyes open in a darkened room, or viewed a silent nature video. Given that precisely coordinated behavior requires tightly controlled oscillatory dynamics, the current results suggest that the large-scale spatial synchronization of oscillatory activity is controlled by the relatively slow, entropy-maximizing adjustments of synchronization probability (demonstrated here) in combination with temporally precise phase adjustments (e.g., phase resetting generated by sensorimotor interactions). Interestingly, we observed a modest but consistent spatial pattern of deviations from the maximum-entropy rule, potentially suggesting that the mid-central-posterior region serves as an "entropy dump" to facilitate the temporally precise control of spectral-power dynamics in the surrounding regions.

Differential metabolic network construction for personalized medicine: Study of type 2 diabetes mellitus patients' response to gliclazide-modified-release-treated

Individual variation in genetic and environmental factors can cause the differences in metabolic phenotypes, which may have an effect on drug responses of patients. Deep exploration of patients' responses to therapeutic agents is a crucial and urgent event in the personalized treatment study. Using machine learning methods for the discovery of suitability evaluation biomarkers can provide deep insight into the mechanism of disease therapy and facilitate the development of personalized medicine. To find important metabolic network signals for the prediction of patients' drug responses, a novel method referred to as differential metabolic network construction (DMNC) was proposed. In DMNC, concentration changes in metabolite ratios between different pathological states are measured to construct differential metabolic networks, which can be used to advance clinical decision-making. In this study, DMNC was applied to characterize type 2 diabetes mellitus (T2DM) patients' responses against gliclazide modified-release (MR) therapy. Two T2DM metabolomics datasets from different batches of subjects treated by gliclazide MR were analyzed in depth. A network biomarker was defined to assess the patients' suitability for gliclazide MR. It can be effective in the prediction of significant responders from nonsignificant responders, achieving area under the curve values of 0.893 and 1.000 for the discovery and validation sets, respectively. Compared with the metabolites selected by the other methods, the network biomarker selected by DMNC was more stable and precise to reflect the metabolic responses in patients to gliclazide MR therapy, thereby contributing for the personalized medicine of T2DM patients. The better performance of DMNC validated its potential for the identification of network biomarkers to characterize the responses against therapeutic treatments and provide valuable information for personalized medicine.

The metastable brain associated with autistic-like traits of typically developing individuals

Metastability in the brain is thought to be a mechanism involved in the dynamic organization of cognitive and behavioral functions across multiple spatiotemporal scales. However, it is not clear how such organization is realized in underlying neural oscillations in a high-dimensional state space. It was shown that macroscopic oscillations often form phase-phase coupling (PPC) and phase-amplitude coupling (PAC), which result in synchronization and amplitude modulation, respectively, even without external stimuli. These oscillations can also make spontaneous transitions across synchronous states at rest. Using resting-state electroencephalographic signals and the autism-spectrum quotient scores acquired from healthy humans, we show experimental evidence that the PAC combined with PPC allows amplitude modulation to be transient, and that the metastable dynamics with this transient modulation is associated with autistic-like traits. In individuals with a longer attention span, such dynamics tended to show fewer transitions between states by forming delta-alpha PAC. We identified these states as two-dimensional metastable states that could share consistent patterns across individuals. Our findings suggest that the human brain dynamically organizes inter-individual differences in a hierarchy of macroscopic oscillations with multiple timescales by utilizing metastability.

The human brain organizes cognitive and behavioral functions dynamically. For decades, the dynamic organization of underlying neural oscillations has been a fundamental topic in neuroscience research. Even without external stimuli, macroscopic oscillations often form phase-phase coupling and phase-amplitude coupling (PAC) that result in synchronization and amplitude modulation, respectively, and can make spontaneous transitions across synchronous states at rest. Using resting-state electroencephalography signals acquired from healthy humans, we show evidence that these two neural couplings enable amplitude modulation to be transient, and that this transient modulation can be viewed as the transition among oscillatory states with different PAC strengths. We also demonstrate that such transition dynamics are associated with the ability to maintain attention to detail and to switch attention, as measured by autism-spectrum quotient scores. These individual dynamics were visualized as a trajectory among states with attracting tendencies, and involved consistent brain states across individuals. Our findings have significant implications for unraveling the variability in the individual brains showing typical and atypical development.

Addressing Pitfalls in Phase-Amplitude Coupling Analysis with an Extended Modulation Index Toolbox

Phase-amplitude coupling (PAC) is proposed to play an essential role in coordinating the processing of information on local and global scales. In recent years, the methods able to reveal trustworthy PAC has gained considerable interest. However, the intrinsic features of some signals can lead to the identification of spurious or waveform-dependent coupling. This prompted us to develop an easily accessible tool that could be used to differentiate spurious from authentic PAC. Here, we propose a new tool for more reliable detection of PAC named the Extended Modulation Index (eMI) based on the classical Modulation Index measure of coupling. eMI is suitable both for continuous and epoched data and allows estimation of the statistical significance of each pair of frequencies for phase and for amplitude in the whole comodulogram in the framework of extreme value statistics. We compared eMI with the reference PAC measures-direct PAC estimator (a modification of Mean Vector Length) and standard Modulation Index. All three methods were tested using computer-simulated data and actual local field potential recordings from freely moving rats. All methods exhibited similar properties in terms of sensitivity and specificity of PAC detection. eMI proved to be more selective in the dimension of frequency for phase. One of the novelty's offered by eMI is a heuristic algorithm for classification of PAC as Reliable or Ambiguous. It relies on analysis of the relation between the spectral properties of the signal and the detected coupling. Moreover, eMI generates visualizations that support further evaluation of the coupling properties. It also introduces the concept of the polar phase-histogram to study phase relations of coupled slow and fast oscillations. We discuss the extent to which eMI addresses the known problems of interpreting PAC. The Matlab® toolbox implementing eMI framework, and the two reference PAC estimators is freely available as EEGLAB plugin at https://github.com/GabrielaJurkiewicz/ePAC .

Dissociating harmonic and non-harmonic phase-amplitude coupling in the human brain

Phase-amplitude coupling (PAC) has been hypothesized to coordinate cross-frequency interactions of neuronal activity in the brain. However, little is known about the distribution of PAC across the human brain and the frequencies involved. Furthermore, it remains unclear to what extent PAC may reflect spurious cross-frequency coupling induced by physiological artifacts or rhythmic non-sinusoidal signals with higher harmonics. Here, we combined MEG, source-reconstruction and different measures of cross-frequency coupling to systematically characterize local PAC across the resting human brain. We show that cross-frequency measures of phase-amplitude, phase-phase, and amplitude-amplitude coupling are all sensitive to signals with higher harmonics. In conjunction, these measures allow to distinguish harmonic and non-harmonic PAC. Based on these insights, we found no evidence for non-harmonic local PAC in resting-state MEG. Instead, we found cortically and spectrally wide-spread PAC driven by harmonic signals. Furthermore, we show how physiological artifacts and spectral leakage cause spurious PAC across wide frequency ranges. Our results clarify how different measures of cross-frequency interactions can be combined to characterize PAC, and cast doubt on the presence of prominent non-harmonic phase-amplitude coupling in human resting-state MEG.

Complex interplay between spectral harmonicity and different types of cross-frequency couplings in nonlinear oscillators and biologically plausible neural network models

Cross-frequency coupling (CFC) refers to the nonlinear interaction between oscillations in different frequency bands, and it is a rather ubiquitous phenomenon that has been observed in a variety of physical and biophysical systems. In particular, the coupling between the phase of slow oscillations and the amplitude of fast oscillations, referred as phase-amplitude coupling (PAC), has been intensively explored in the brain activity recorded from animals and humans. However, the interpretation of these CFC patterns remains challenging since harmonic spectral correlations characterizing nonsinusoidal oscillatory dynamics can act as a confounding factor. Specialized signal processing techniques are proposed to address the complex interplay between spectral harmonicity and different types of CFC, not restricted only to PAC. For this, we provide an in-depth characterization of the time locked index (TLI) as a tool aimed to efficiently quantify the harmonic content of noisy time series. It is shown that the proposed TLI measure is more robust and outperforms traditional phase coherence metrics (e.g., phase locking value, pairwise phase consistency) in several aspects. We found that a nonlinear oscillator under the effect of additive noise can produce spurious CFC with low spectral harmonic content. On the other hand, two coupled oscillatory dynamics with independent fundamental frequencies can produce true CFC with high spectral harmonic content via a rectification mechanism or other post-interaction nonlinear processing mechanisms. These results reveal a complex interplay between CFC and harmonicity emerging in the dynamics of biologically plausible neural network models and more generic nonlinear and parametric oscillators. We show that, contrary to what is usually assumed in the literature, the high harmonic content observed in nonsinusoidal oscillatory dynamics is neither a sufficient nor necessary condition to interpret the associated CFC patterns as epiphenomenal. There is mounting evidence suggesting that the combination of multimodal recordings, specialized signal processing techniques, and theoretical modeling is becoming a required step to completely understand CFC patterns observed in oscillatory rich dynamics of physical and biophysical systems.

Two types of ictal phase-amplitude couplings in epilepsy patients revealed by spectral harmonicity of intracerebral EEG recordings

OBJECTIVE

Spectral harmonicity of the ictal activity was analyzed regarding two clinically relevant aspects, (1) as a confounding factor producing 'spurious' phase-amplitude couplings (PAC) which may lead to wrong conclusions about the underlying ictal mechanisms, and (2) its role in how good PAC is in correspondence to the seizure onset zone (SOZ) classification performed by the epileptologists.

METHODS

PAC patterns observed in intracerebral electroencephalography (iEEG) recordings were retrospectively studied during seizures of seven patients with pharmacoresistant focal epilepsy. The time locked index (TLI) measure was introduced to quantify the degree of harmonicity between frequency bands associated to the emergence of PAC during epileptic seizures.

RESULTS

(1) Harmonic and non harmonic PAC patterns coexist during the seizure dynamics in iEEG recordings with macroelectrodes. (2) Harmonic PAC patterns are an emergent property of the periodic non sinusoidal waveform constituting the epileptiform activity. (3) The TLI metric allows to distinguish the non harmonic PAC pattern, which has been previously associated with the ictal core through the paroxysmal depolarizing shifts mechanism of seizure propagation.

CONCLUSIONS

Our results suggest that the spectral harmonicity of the ictal activity plays a relevant role in the visual analysis of the iEEG recordings performed by the epileptologists to define the SOZ, and that it should be considered for the proper interpretation of ictal mechanisms.

SIGNIFICANCE

The proposed harmonicity analysis can be used to improve the delineation of the SOZ by reliably identifying non harmonic PAC patterns emerging from fully recruited cortical and subcortical areas.

Spectral-power associations reflect amplitude modulation and within-frequency interactions on the sub-second timescale and cross-frequency interactions on the seconds timescale

We investigated the global structure of intrinsic cross-frequency dynamics by systematically examining power-based temporal associations among a broad range of oscillation frequencies both within and across EEG-based current sources (sites). We focused on power-based associations that could reveal unique timescale dependence independently of interacting frequencies. Large spectral-power fluctuations across all sites occurred at two characteristic timescales, sub-second and seconds, yielding distinct patterns of cross-frequency associations. On the fast sub-second timescale, within-site (local) associations were consistently between pairs of β—γ frequencies differing by a constant Δf (particularly Δf ~ 10 Hz at posterior sites and Δf ~ 16 Hz at lateral sites) suggesting that higher-frequency oscillations are organized into Δf amplitude-modulated packets, whereas cross-site (long-distance) associations were all within-frequency (particularly in the >30 Hz and 6–12 Hz ranges, suggestive of feedforward and feedback interactions). On the slower seconds timescale, within-site (local) associations were characterized by a broad range of frequencies selectively associated with ~10 Hz at posterior sites and associations among higher (>20 Hz) frequencies at lateral sites, whereas cross-site (long-distance) associations were characterized by a broad range of frequencies at posterior sites selectively associated with ~10 Hz at other sites, associations among higher (>20 Hz) frequencies among lateral and anterior sites, and prevalent associations at ~10 Hz. Regardless of timescale, within-site (local) cross-frequency associations were weak at anterior sites indicative of frequency-specific operations. Overall, these results suggest that the fast sub-second-timescale coordination of spectral power is limited to local amplitude modulation and insulated within-frequency long-distance interactions (likely feedforward and feedback interactions), while characteristic patterns of cross-frequency interactions emerge on the slower seconds timescale. The results also suggest that the occipital α oscillations play a role in organizing higher-frequency oscillations into ~10 Hz amplitude-modulated packets to communicate with other regions. Functional implications of these timescale-dependent cross-frequency associations await future investigations.