Exploring MEG brain fingerprints: Evaluation, pitfalls, and interpretations

The effects of spatial leakage correction on the reliability of EEG-based functional connectivity networks

Electroencephalography (EEG) functional connectivity (FC) estimates are confounded by the volume conduction problem. This effect can be greatly reduced by applying FC measures insensitive to instantaneous, zero-lag dependencies (corrected measures). However, numerous studies showed that FC measures sensitive to volume conduction (uncorrected measures) exhibit higher reliability and higher subject-level identifiability. We tested how source reconstruction contributed to the reliability difference of EEG FC measures on a large (n = 201) resting-state data set testing eight FC measures (including corrected and uncorrected measures). We showed that the high reliability of uncorrected FC measures in resting state partly stems from source reconstruction: idiosyncratic noise patterns define a baseline resting-state functional network that explains a significant portion of the reliability of uncorrected FC measures. This effect remained valid for template head model-based, as well as individual head model-based source reconstruction. Based on our findings we made suggestions how to best use spatial leakage corrected and uncorrected FC measures depending on the main goals of the study.

Bayesian reduced rank regression models generalizable neural fingerprints that differentiate between individuals in magnetoencephalography data

Recent magnetoencephalography (MEG) studies have reported that functional connectivity (FC) and power spectra can be used as neural fingerprints in differentiating individuals. Such studies have mainly used correlations between measurement sessions to distinguish individuals from each other. However, it has remained unclear whether such correlations might reflect a more generalizable principle of individually distinctive brain patterns. Here, we evaluated a machine-learning based approach, termed latent-noise Bayesian reduced rank regression (BRRR) as a means of modelling individual differences in the resting-state MEG data of the Human Connectome Project (HCP), using FC and power spectra as neural features. First, we verified that BRRR could model and reproduce the differences between metrics that correlation-based fingerprinting yields. We trained BRRR models to distinguish individuals based on data from one measurement and used the models to identify subsequent measurement sessions of those same individuals. The best performing BRRR models, using only 20 spatiospectral components, were able to identify subjects across measurement sessions with over 90% accuracy, approaching the highest correlation-based accuracies. Using cross-validation, we then determined whether that BRRR model could generalize to unseen subjects, successfully classifying the measurement sessions of novel individuals with over 80% accuracy. The results demonstrate that individual neurofunctional differences can be reliably extracted from MEG data with a low-dimensional predictive model and that the model is able to classify novel subjects.

The Effect of Sleep Deprivation on Brain Fingerprint Stability: A Magnetoencephalography Validation Study

This study examined the stability of the functional connectome (FC) over time using fingerprint analysis in healthy subjects. Additionally, it investigated how a specific stressor, namely sleep deprivation, affects individuals' differentiation. To this aim, 23 healthy young adults underwent magnetoencephalography (MEG) recording at three equally spaced time points within 24 h: 9 a.m., 9 p.m., and 9 a.m. of the following day after a night of sleep deprivation. The findings indicate that the differentiation was stable from morning to evening in all frequency bands, except in the delta band. However, after a night of sleep deprivation, the stability of the FCs was reduced. Consistent with this observation, the reduced differentiation following sleep deprivation was found to be negatively correlated with the effort perceived by participants in completing the cognitive task during sleep deprivation. This correlation suggests that individuals with less stable connectomes following sleep deprivation experienced greater difficulty in performing cognitive tasks, reflecting increased effort.

Rapid dynamics of electrophysiological connectome states are heritable

Time-varying changes in whole-brain connectivity patterns, or connectome state dynamics, are a prominent feature of brain activity with broad functional implications. While infra-slow (<0.1Hz) connectome dynamics have been extensively studied with fMRI, rapid dynamics highly relevant for cognition are poorly understood. Here, we asked whether rapid electrophysiological connectome dynamics constitute subject-specific brain traits and to what extent they are under genetic influence. Using source-localized EEG connectomes during resting-state (N=928, 473 females), we quantified heritability of multivariate (multi-state) features describing temporal or spatial characteristics of connectome dynamics. States switched rapidly every ~60-500ms. Temporal features were heritable, particularly, Fractional Occupancy (in theta, alpha, beta, and gamma bands) and Transition Probability (in theta, alpha, and gamma bands), representing the duration spent in each state and the frequency of state switches, respectively. Genetic effects explained a substantial proportion of phenotypic variance of these features: Fractional Occupancy in beta (44.3%) and gamma (39.8%) bands and Transition Probability in theta (38.4%), alpha (63.3%), beta (22.6%), and gamma (40%) bands. However, we found no evidence for heritability of spatial features, specifically states' Modularity and connectivity pattern. We conclude that genetic effects strongly shape individuals' connectome dynamics at rapid timescales, specifically states' overall occurrence and sequencing.

Shared functional connectome fingerprints following ritualistic ayahuasca intake

The knowledge that brain functional connectomes are unique and reliable has enabled behaviourally relevant inferences at a subject level. However, whether such "fingerprints" persist under altered states of consciousness is unknown. Ayahuasca is a potent serotonergic psychedelic which produces a widespread dysregulation of functional connectivity. Used communally in religious ceremonies, its shared use may highlight relevant novel interactions between mental state and functional connectome (FC) idiosyncrasy. Using 7T fMRI, we assessed resting-state static and dynamic FCs for 21 Santo Daime members after collective ayahuasca intake in an acute, within-subject study. Here, connectome fingerprinting revealed FCs showed reduced idiosyncrasy, accompanied by a spatiotemporal reallocation of keypoint edges. Importantly, we show that interindividual differences in higher-order FC motifs are relevant to experiential phenotypes, given that they can predict perceptual drug effects. Collectively, our findings offer an example of how individualised connectivity markers can be used to trace a subject's FC across altered states of consciousness.

Multiplexity of human brain oscillations as a personal brain signature

Human individuality is likely underpinned by the constitution of functional brain networks that ensure consistency of each person's cognitive and behavioral profile. These functional networks should, in principle, be detectable by noninvasive neurophysiology. We use a method that enables the detection of dominant frequencies of the interaction between every pair of brain areas at every temporal segment of the recording period, the dominant coupling modes (DoCM). We apply this method to brain oscillations, measured with magnetoencephalography (MEG) at rest in two independent datasets, and show that the spatiotemporal evolution of DoCMs constitutes an individualized brain fingerprint. Based on this successful fingerprinting we suggest that DoCMs are important targets for the investigation of neural correlates of individual psychological parameters and can provide mechanistic insight into the underlying neurophysiological processes, as well as their disturbance in brain diseases.

The neurophysiological brain-fingerprint of Parkinson's disease

In this study, we investigate the clinical potential of brain-fingerprints derived from electrophysiological brain activity for diagnostics and progression monitoring of Parkinson's disease (PD). We obtained brain-fingerprints from PD patients and age-matched healthy controls using short, task-free magnetoencephalographic recordings. The rhythmic components of the individual brain-fingerprint distinguished between patients and healthy participants with approximately 90% accuracy. The most prominent cortical features of the Parkinson's brain-fingerprint mapped to polyrhythmic activity in unimodal sensorimotor regions. Leveraging these features, we also show that Parkinson's disease stages can be decoded directly from cortical neurophysiological activity. Additionally, our study reveals that the cortical topography of the Parkinson's brain-fingerprint aligns with that of neurotransmitter systems affected by the disease's pathophysiology. We further demonstrate that the arrhythmic components of cortical activity are more variable over short periods of time in patients with Parkinson's disease than in healthy controls, making individual differentiation between patients based on these features more challenging and explaining previous negative published results. Overall, we outline patient-specific rhythmic brain signaling features that provide insights into both the neurophysiological signature and clinical staging of Parkinson's disease. For this reason, the proposed definition of a rhythmic brain-fingerprint of Parkinson's disease may contribute to novel, refined approaches to patient stratification and to the improved identification and testing of therapeutic neurostimulation targets.

Neural fingerprinting on MEG time series using MiniRocket

Neural fingerprinting is the identification of individuals in a cohort based on neuroimaging recordings of brain activity. In magneto- and electroencephalography (M/EEG), it is common practice to use second-order statistical measures, such as correlation or connectivity matrices, when neural fingerprinting is performed. These measures or features typically require coupling between signal channels and often ignore the individual temporal dynamics. In this study, we show that, following recent advances in multivariate time series classification, such as the development of the RandOm Convolutional KErnel Transformation (ROCKET) classifier, it is possible to perform classification directly on short time segments from MEG resting-state recordings with remarkably high classification accuracies. In a cohort of 124 subjects, it was possible to assign windows of time series of 1 s in duration to the correct subject with above 99% accuracy. The achieved accuracies are vastly superior to those of previous methods while simultaneously requiring considerably shorter time segments.

Healthy ageing and cognitive impairment alter EEG functional connectivity in distinct frequency bands

Functional connectivity (FC) indicates the interdependencies between brain signals recorded from spatially distinct locations in different frequency bands, which is modulated by cognitive tasks and is known to change with ageing and cognitive disorders. Recently, the power of narrow-band gamma oscillations induced by visual gratings have been shown to reduce with both healthy ageing and in subjects with mild cognitive impairment (MCI). However, the impact of ageing/MCI on stimulus-induced gamma FC has not been well studied. We recorded electroencephalogram (EEG) from a large cohort (N = 229) of elderly subjects (>49 years) while they viewed large cartesian gratings to induce gamma oscillations and studied changes in alpha and gamma FC with healthy ageing (N = 218) and MCI (N = 11). Surprisingly, we found distinct differences across age and MCI groups in power and FC. With healthy ageing, alpha power did not change but FC decreased significantly. MCI reduced gamma but not alpha FC significantly compared with age and gender matched controls, even when power was matched between the two groups. Overall, our results suggest distinct effects of ageing and disease on EEG power and FC, suggesting different mechanisms underlying ageing and cognitive disorders.

Reduced clinical connectome fingerprinting in multiple sclerosis predicts fatigue severity

BACKGROUND

Brain connectome fingerprinting is progressively gaining ground in the field of brain network analysis. It represents a valid approach in assessing the subject-specific connectivity and, according to recent studies, in predicting clinical impairment in some neurodegenerative diseases. Nevertheless, its performance, and clinical utility, in the Multiple Sclerosis (MS) field has not yet been investigated.

METHODS

We conducted the Clinical Connectome Fingerprint (CCF) analysis on source-reconstructed magnetoencephalography signals in a cohort of 50 subjects: twenty-five MS patients and twenty-five healthy controls.

RESULTS

All the parameters of identifiability, in the alpha band, were reduced in patients as compared to controls. These results implied a lower similarity between functional connectomes (FCs) of the same patient and a reduced homogeneity among FCs in the MS group. We also demonstrated that in MS patients, reduced identifiability was able to predict, fatigue level (assessed by the Fatigue Severity Scale).

CONCLUSION

These results confirm the clinical usefulness of the CCF in both identifying MS patients and predicting clinical impairment. We hope that the present study provides future prospects for treatment personalization on the basis of individual brain connectome.

Unsupervised representation learning of spontaneous MEG data with Nonlinear ICA

Resting-state magnetoencephalography (MEG) data show complex but structured spatiotemporal patterns. However, the neurophysiological basis of these signal patterns is not fully known and the underlying signal sources are mixed in MEG measurements. Here, we developed a method based on the nonlinear independent component analysis (ICA), a generative model trainable with unsupervised learning, to learn representations from resting-state MEG data. After being trained with a large dataset from the Cam-CAN repository, the model has learned to represent and generate patterns of spontaneous cortical activity using latent nonlinear components, which reflects principal cortical patterns with specific spectral modes. When applied to the downstream classification task of audio-visual MEG, the nonlinear ICA model achieves competitive performance with deep neural networks despite limited access to labels. We further validate the generalizability of the model across different datasets by applying it to an independent neurofeedback dataset for decoding the subject's attentional states, providing a real-time feature extraction and decoding mindfulness and thought-inducing tasks with an accuracy of around 70% at the individual level, which is much higher than obtained by linear ICA or other baseline methods. Our results demonstrate that nonlinear ICA is a valuable addition to existing tools, particularly suited for unsupervised representation learning of spontaneous MEG activity which can then be applied to specific goals or tasks when labelled data are scarce.

Brain-wide network analysis of resting-state neuromagnetic data

The present study performed a brain-wide network analysis of resting-state magnetoencephalograms recorded from 53 healthy participants to visualize elaborate brain maps of phase- and amplitude-derived graph-theory metrics at different frequencies. To achieve this, we conducted a vertex-wise computation of threshold-independent graph metrics by combining proportional thresholding and a conjunction analysis and applied them to a correlation analysis of age and brain networks. Source power showed a frequency-dependent cortical distribution. Threshold-independent graph metrics derived from phase- and amplitude-based connectivity showed similar or different distributions depending on frequency. Vertex-wise age-brain correlation maps revealed that source power at the beta band and the amplitude-based degree at the alpha band changed with age in local regions. The present results indicate that a brain-wide analysis of neuromagnetic data has the potential to reveal neurophysiological network features in the human brain in a resting state.

Task matters: Individual MEG signatures from naturalistic and neurophysiological brain states

The discovery that human brain connectivity data can be used as a "fingerprint" to identify a given individual from a population, has become a burgeoning research area in the neuroscience field. Recent studies have identified the possibility to extract these brain signatures from the temporal rich dynamics of resting-state magneto encephalography (MEG) recordings. Nevertheless, it is still uncertain to what extent MEG signatures can serve as an indicator of human identifiability during task-related conduct. Here, using MEG data from naturalistic and neurophysiological tasks, we show that identification improves in tasks relative to resting-state, providing compelling evidence for a task dependent axis of MEG signatures. Notably, improvements in identifiability were more prominent in strictly controlled tasks. Lastly, the brain regions contributing most towards individual identification were also modified when engaged in task activities. We hope that this investigation advances our understanding of the driving factors behind brain identification from MEG signals.

Uncovering individual differences in fine-scale dynamics of functional connectivity

Functional connectivity (FC) profiles contain subject-specific features that are conserved across time and have potential to capture brain-behavior relationships. Most prior work has focused on spatial features (nodes and systems) of these FC fingerprints, computed over entire imaging sessions. We propose a method for temporally filtering FC, which allows selecting specific moments in time while also maintaining the spatial pattern of node-based activity. To this end, we leverage a recently proposed decomposition of FC into edge time series (eTS). We systematically analyze functional magnetic resonance imaging frames to define features that enhance identifiability across multiple fingerprinting metrics, similarity metrics, and data sets. Results show that these metrics characteristically vary with eTS cofluctuation amplitude, similarity of frames within a run, transition velocity, and expression of functional systems. We further show that data-driven optimization of features that maximize fingerprinting metrics isolates multiple spatial patterns of system expression at specific moments in time. Selecting just 10% of the data can yield stronger fingerprints than are obtained from the full data set. Our findings support the idea that FC fingerprints are differentially expressed across time and suggest that multiple distinct fingerprints can be identified when spatial and temporal characteristics are considered simultaneously.

Fading of brain network fingerprint in Parkinson's disease predicts motor clinical impairment

The clinical connectome fingerprint (CCF) was recently introduced as a way to assess brain dynamics. It is an approach able to recognize individuals, based on the brain network. It showed its applicability providing network features used to predict the cognitive decline in preclinical Alzheimer's disease. In this article, we explore the performance of CCF in 47 Parkinson's disease (PD) patients and 47 healthy controls, under the hypothesis that patients would show reduced identifiability as compared to controls, and that such reduction could be used to predict motor impairment. We used source-reconstructed magnetoencephalography signals to build two functional connectomes for 47 patients with PD and 47 healthy controls. Then, exploiting the two connectomes per individual, we investigated the identifiability characteristics of each subject in each group. We observed reduced identifiability in patients compared to healthy individuals in the beta band. Furthermore, we found that the reduction in identifiability was proportional to the motor impairment, assessed through the Unified Parkinson's Disease Rating Scale, and, interestingly, able to predict it (at the subject level), through a cross-validated regression model. Along with previous evidence, this article shows that CCF captures disrupted dynamics in neurodegenerative diseases and is particularly effective in predicting motor clinical impairment in PD.

A 90-channel triaxial magnetoencephalography system using optically pumped magnetometers

Magnetoencephalography (MEG) measures the small magnetic fields generated by current flow in neural networks, providing a noninvasive metric of brain function. MEG is well established as a powerful neuroscientific and clinical tool. However, current instrumentation is hampered by cumbersome cryogenic field-sensing technologies. In contrast, MEG using optically pumped magnetometers (OPM-MEG) employs small, lightweight, noncryogenic sensors that provide data with higher sensitivity and spatial resolution, a natural scanning environment (including participant movement), and adaptability to any age. However, OPM-MEG is new and the optimum way to design a system is unknown. Here, we construct a novel, 90-channel triaxial OPM-MEG system and use it to map motor function during a naturalistic handwriting task. Results show that high-precision magnetic field control reduced background fields to ∼200 pT, enabling free participant movement. Our triaxial array offered twice the total measured signal and better interference rejection compared to a conventional (single-axis) design. We mapped neural oscillatory activity to the sensorimotor network, demonstrating significant differences in motor network activity and connectivity for left-handed versus right-handed handwriting. Repeatability across scans showed that we can map electrophysiological activity with an accuracy ∼4 mm. Overall, our study introduces a novel triaxial OPM-MEG design and confirms its potential for high-performance functional neuroimaging.

Brain fingerprints along the language hierarchy

Recent studies have shown that the brain functional connectome constitutes a unique fingerprint that allows the identification of individuals from a group. However, what information encoded in the brain that makes us unique remains elusive. Here, we addressed this issue by examining how individual identifiability changed along the language hierarchy. Subjects underwent fMRI scanning during rest and when listening to short stories played backward, scrambled at the sentence level, and played forward. Identification for individuals was performed between two scan sessions for each task as well as between the rest and task sessions. We found that individual identifiability tends to increase along the language hierarchy: the more complex the task is, the better subjects can be distinguished from each other based on their whole-brain functional connectivity profiles. A similar principle is found at the functional network level: compared to the low-order network (the auditory network), the high-order network is more individualized (the frontoparietal network). Moreover, in both cases, the increase in individual identifiability is accompanied by the increase in inter-subject variability of functional connectivities. These findings advance the understanding of the source of brain individualization and have potential implications for developing robust connectivity-based biomarkers.

State-dependent tDCS modulation of the somatomotor network: A MEG study

OBJECTIVE

Transcranial direct current stimulation (tDCS) is a non-invasive technique widely used to investigate brain excitability and activity. However, the variability in both brain and behavioral responses to tDCS limits its application for clinical purposes. This study aims to shed light on state-dependency, a phenomenon that contributes to the variability of tDCS.

METHODS

To this aim, we investigated changes in spectral activity and functional connectivity in somatomotor regions after Real and Sham tDCS using generalized additive mixed models (GAMMs), which allowed us to investigate how modulation depends on the initial state of the brain.

RESULTS

Results showed that changes in spectral activity, but not connectivity, in the somatomotor regions depend on the initial state of the brain, confirming state-dependent effects. Specifically, we found a non-linear interaction between stimulation conditions (Real vs Sham) and initial state: a reduction of alpha and beta power was observed only in participants that had higher alpha and beta power before Real tDCS.

CONCLUSIONS

This study highlights the importance of considering state-dependency to tDCS and shows how it can be taken into account with appropriate statistical models.

SIGNIFICANCE

Our findings bear insight into tDCS mechanisms, potentially leading to discriminate between tDCS responders and non-responders.

Human electromagnetic and haemodynamic networks systematically converge in unimodal cortex and diverge in transmodal cortex

Whole-brain neural communication is typically estimated from statistical associations among electromagnetic or haemodynamic time-series. The relationship between functional network architectures recovered from these 2 types of neural activity remains unknown. Here, we map electromagnetic networks (measured using magnetoencephalography (MEG)) to haemodynamic networks (measured using functional magnetic resonance imaging (fMRI)). We find that the relationship between the 2 modalities is regionally heterogeneous and systematically follows the cortical hierarchy, with close correspondence in unimodal cortex and poor correspondence in transmodal cortex. Comparison with the BigBrain histological atlas reveals that electromagnetic–haemodynamic coupling is driven by laminar differentiation and neuron density, suggesting that the mapping between the 2 modalities can be explained by cytoarchitectural variation. Importantly, haemodynamic connectivity cannot be explained by electromagnetic activity in a single frequency band, but rather arises from the mixing of multiple neurophysiological rhythms. Correspondence between the two is largely driven by MEG functional connectivity at the beta (15 to 29 Hz) frequency band. Collectively, these findings demonstrate highly organized but only partly overlapping patterns of connectivity in MEG and fMRI functional networks, opening fundamentally new avenues for studying the relationship between cortical microarchitecture and multimodal connectivity patterns.

What is the relationship between functional network architectures inferred from electromagnetic and haemodynamic data? This study shows that superposition of electromagnetic networks at canonical frequency bands manifests as highly structured patterns of haemodynamic functional connectivity in the human brain, reflecting systematic variation in cytoarchitecture.

Identifying Individuals by fNIRS-Based Brain Functional Network Fingerprints

Individual identification based on brain functional network (BFN) has attracted a lot of research interest in recent years, since it provides a novel biometric for identity authentication, as well as a feasible way of exploring the brain at an individual level. Previous studies have shown that an individual can be identified by its BFN fingerprint estimated from functional magnetic resonance imaging, electroencephalogram, or magnetoencephalography data. Functional near-infrared spectroscopy (fNIRS) is an emerging imaging technique that, by measuring the changes in blood oxygen concentration, can respond to cerebral activities; in this paper, we investigate whether fNIRS-based BFN could be used as a “fingerprint” to identify individuals. In particular, Pearson's correlation is first used to calculate BFN based on the preprocessed fNIRS signals, and then the nearest neighbor scheme is used to match the estimated BFNs between different individuals. Through the experiments on an open-access fNIRS dataset, we have two main findings: (1) under the cases of cross-task (i.e., resting, right-handed, left-handed finger tapping, and foot tapping), the BFN fingerprints generally work well for the individual identification, and, more interestingly, (2) the accuracy under cross-task is well above the accuracy under cross-view (i.e., oxyhemoglobin and de-oxyhemoglobin). These findings indicate that fNIRS-based BFN fingerprint is a potential biometric for identifying individual.

Magnetoencephalography Brain Signatures Relate to Cognition and Cognitive Reserve in the Oldest-Old: The EMIF-AD 90 + Study

The oldest-old subjects represent the fastest growing segment of society and are at high risk for dementia with a prevalence of up to 40%. Lifestyle factors, such as lifelong participation in cognitive and leisure activities, may contribute to individual cognitive reserve and reduce the risk for cognitive impairments. However, the neural bases underlying cognitive functioning and cognitive reserve in this age range are still poorly understood. Here, we investigate spectral and functional connectivity features obtained from resting-state MEG recordings in a cohort of 35 cognitively normal (92.2 ± 1.8 years old, 19 women) and 11 cognitively impaired (90.9 ± 1.9 years old, 1 woman) oldest-old participants, in relation to cognitive traits and cognitive reserve. The latter was approximated with a self-reported scale on lifelong engagement in cognitively demanding activities. Cognitively impaired oldest-old participants had slower cortical rhythms in frontal, parietal and default mode network regions compared to the cognitively normal subjects. These alterations mainly concerned the theta and beta band and partially explained inter-subject variability of episodic memory scores. Moreover, a distinct spectral pattern characterized by higher relative power in the alpha band was specifically associated with higher cognitive reserve while taking into account the effect of age and education level. Finally, stronger functional connectivity in the alpha and beta band were weakly associated with better cognitive performances in the whole group of subjects, although functional connectivity effects were less prominent than the spectral ones. Our results shed new light on the neural underpinnings of cognitive functioning in the oldest-old population and indicate that cognitive performance and cognitive reserve may have distinct spectral electrophysiological substrates.

Brief segments of neurophysiological activity enable individual differentiation

Large, openly available datasets and current analytic tools promise the emergence of population neuroscience. The considerable diversity in personality traits and behaviour between individuals is reflected in the statistical variability of neural data collected in such repositories. Recent studies with functional magnetic resonance imaging (fMRI) have concluded that patterns of resting-state functional connectivity can both successfully distinguish individual participants within a cohort and predict some individual traits, yielding the notion of an individual's neural fingerprint. Here, we aim to clarify the neurophysiological foundations of individual differentiation from features of the rich and complex dynamics of resting-state brain activity using magnetoencephalography (MEG) in 158 participants. We show that akin to fMRI approaches, neurophysiological functional connectomes enable the differentiation of individuals, with rates similar to those seen with fMRI. We also show that individual differentiation is equally successful from simpler measures of the spatial distribution of neurophysiological spectral signal power. Our data further indicate that differentiation can be achieved from brain recordings as short as 30 seconds, and that it is robust over time: the neural fingerprint is present in recordings performed weeks after their baseline reference data was collected. This work, thus, extends the notion of a neural or brain fingerprint to fast and large-scale resting-state electrophysiological dynamics.