Is there chaos in the brain? I. Concepts of nonlinear dynamics and methods of investigation

Examining electroencephalogram signatures of people with multiple sclerosis using a nonlinear dynamics approach: a systematic review and bibliographic analysis

Background

Considering that brain activity involves communication between millions of neurons in a complex network, nonlinear analysis is a viable tool for studying electroencephalography (EEG). The main objective of this review was to collate studies that utilized chaotic measures and nonlinear dynamical analysis in EEG of multiple sclerosis (MS) patients and to discuss the contributions of chaos theory techniques to understanding, diagnosing, and treating MS.

Methods

Using the preferred reporting items for systematic reviews and meta-analysis (PRISMA), the databases EbscoHost, IEEE, ProQuest, PubMed, Science Direct, Web of Science, and Google Scholar were searched for publications that applied chaos theory in EEG analysis of MS patients.

Results

A bibliographic analysis was performed using VOSviewer software keyword co-occurrence analysis indicated that MS was the focus of the research and that research on MS diagnosis has shifted from conventional methods, such as magnetic resonance imaging, to EEG techniques in recent years. A total of 17 studies were included in this review. Among the included articles, nine studies examined resting-state, and eight examined task-based conditions.

Conclusion

Although nonlinear EEG analysis of MS is a relatively novel area of research, the findings have been demonstrated to be informative and effective. The most frequently used nonlinear dynamics analyses were fractal dimension, recurrence quantification analysis, mutual information, and coherence. Each analysis selected provided a unique assessment to fulfill the objective of this review. While considering the limitations discussed, there is a promising path forward using nonlinear analyses with MS data.

How can artificial neural networks approximate the brain?

The article reviews the history development of artificial neural networks (ANNs), then compares the differences between ANNs and brain networks in their constituent unit, network architecture, and dynamic principle. The authors offer five points of suggestion for ANNs development and ten questions to be investigated further for the interdisciplinary field of brain simulation. Even though brain is a super-complex system with 1011 neurons, its intelligence does depend rather on the neuronal type and their energy supply mode than the number of neurons. It might be possible for ANN development to follow a new direction that is a combination of multiple modules with different architecture principle and multiple computation, rather than very large scale of neural networks with much more uniformed units and hidden layers.

Cupolets in a chaotic neuron model

This paper reports the first finding of cupolets in a chaotic Hindmarsh-Rose neural model. Cupolets (chaotic, unstable, periodic, orbit-lets) are unstable periodic orbits that have been stabilized through a particular control scheme by applying a binary control sequence. We demonstrate different neural dynamics (periodic or chaotic) of the Hindmarsh-Rose model through a bifurcation diagram where the external input current, I, is the bifurcation parameter. We select a region in the chaotic parameter space and provide the results of numerical simulations. In this chosen parameter space, a control scheme is applied when the trajectory intersects with either of the two control planes. The type of the control is determined by a bit in a binary control sequence. The control is either a small microcontrol (0) or a large macrocontrol (1) that adjusts the future dynamics of the trajectory by a perturbation determined by the coding function _r ( x ). We report the discovery of many cupolets with corresponding control sequences and comment on the differences with previously reported cupolets in the double scroll system. We provide some examples of the generated cupolets and conclude by discussing potential implications for biological neurons.

Brain entropy, fractal dimensions and predictability: A review of complexity measures for EEG in healthy and neuropsychiatric populations

There has been an increasing trend towards the use of complexity analysis in quantifying neural activity measured by electroencephalography (EEG) signals. On top of revealing complex neuronal processes of the brain that may not be possible with linear approaches, EEG complexity measures have also demonstrated their potential as biomarkers of psychopathology such as depression and schizophrenia. Unfortunately, the opacity of algorithms and descriptions originating from mathematical concepts have made it difficult to understand what complexity is and how to draw consistent conclusions when applied within psychology and neuropsychiatry research. In this review, we provide an overview and entry-level explanation of existing EEG complexity measures, which can be broadly categorized as measures of predictability and regularity. We then synthesize complexity findings across different areas of psychological science, namely, in consciousness research, mood and anxiety disorders, schizophrenia, neurodevelopmental and neurodegenerative disorders, as well as changes across the lifespan, while addressing some theoretical and methodological issues underlying the discrepancies in the data. Finally, we present important considerations when choosing and interpreting these metrics.

Auditory attention decoding from EEG-based Mandarin speech envelope reconstruction

In the cocktail party circumstance, the human auditory system extracts the information from a specific speaker of interest and ignores others. Many studies have focused on auditory attention decoding (AAD), but the stimulation materials were mainly non-tonal languages. We used a tonal language (Mandarin) as the speech stimulus and constructed a Long Short-Term Memory (LSTM) architecture for speech envelope reconstruction based on electroencephalogram (EEG) data. The correlation coefficient between the reconstructed and candidate envelopes was calculated to determine the subject's auditory attention. The proposed LSTM architecture outperformed the linear models. The average decoding accuracy in cross-subject and inter-subject cases varies from 63.02 to 74.29%, with the highest accuracy rate of 89.1% in a decision window of 0.15 s. In addition, the beta-band rhythm was found to play an essential role in identifying the attention and the non-attention state. These results provide a new AAD architecture to help develop neuro-steered hearing devices, especially for tonal languages.

NoLiTiA: An Open-Source Toolbox for Non-linear Time Series Analysis

In many scientific fields including neuroscience, climatology or physics, complex relationships can be described most parsimoniously by non-linear mechanics. Despite their relevance, many neuroscientists still apply linear estimates in order to evaluate complex interactions. This is partially due to the lack of a comprehensive compilation of non-linear methods. Available packages mostly specialize in only one aspect of non-linear time-series analysis and most often require some coding proficiency to use. Here, we introduce NoLiTiA, a free open-source MATLAB toolbox for non-linear time series analysis. In comparison to other currently available non-linear packages, NoLiTiA offers (1) an implementation of a broad range of classic and recently developed methods, (2) an implementation of newly proposed spatially and time-resolved recurrence amplitude analysis and (3) an intuitive environment accessible even to users with little coding experience due to a graphical user interface and batch-editor. The core methodology derives from three distinct fields of complex systems theory, including dynamical systems theory, recurrence quantification analysis and information theory. Besides established methodology including estimation of dynamic invariants like Lyapunov exponents and entropy-based measures, such as active information storage, we include recent developments of quantifying time-resolved aperiodic oscillations. In general, the toolbox will make non-linear methods accessible to the broad neuroscientific community engaged in time series processing.

Chaotic and stochastic dynamics of epileptiform-like activities in sclerotic hippocampus resected from patients with pharmacoresistant epilepsy

The types of epileptiform activity occurring in the sclerotic hippocampus with highest incidence are interictal-like events (II) and periodic ictal spiking (PIS). These activities are classified according to their event rates, but it is still unclear if these rate differences are consequences of underlying physiological mechanisms. Identifying new and more specific information related to these two activities may bring insights to a better understanding about the epileptogenic process and new diagnosis. We applied Poincaré map analysis and Recurrence Quantification Analysis (RQA) onto 35 in vitro electrophysiological signals recorded from slices of 12 hippocampal tissues surgically resected from patients with pharmacoresistant temporal lobe epilepsy. These analyzes showed that the II activity is related to chaotic dynamics, whereas the PIS activity is related to deterministic periodic dynamics. Additionally, it indicates that their different rates are consequence of different endogenous dynamics. Finally, by using two computational models we were able to simulate the transition between II and PIS activities. The RQA was applied to different periods of these simulations to compare the recurrences between artificial and real signals, showing that different ranges of regularity-chaoticity can be directly associated with the generation of PIS and II activities.

Temporal lobe epilepsy (TLE) is the most prevalent type of epilepsy in adults and hippocampal sclerosis is the major pathophysiological substrate of pharmaco-refractory TLE. Different patterns of epileptiform-like activity have been described in human hippocampal sclerosis, but the standard analysis applied to characterize the activities usually do not consider the nonlinear features that epileptiform patterns exhibit. Here, using Poincaré map and Recurrence Quantitative Analysis we characterized the most prevalent type of epileptiform-like activities—interictal-like events (II) and periodic ictal spiking (PIS), recorded in vitro from resected hippocampi of pharmacoresistant patients with TLE—according to their levels of stochasticity, chaoticity and determinism. The II activities showed to be more chaotic with complex rhythmicity than PIS activities. The nonlinear dynamic differences between II and PIS leads us to conjecture that they are expressions of different seizure susceptibility. We also identified that each hippocampal subfield expresses II and PIS activities in a specific and different way. Finally, from the modulation of internal parameters of two computational models, we show the conversion of one type of activity into the other, showing how specific neuron networks synchronize over time, leading to II and PIS activities and then into a generalized seizure.

STAnet: A Spatiotemporal Attention Network for Decoding Auditory Spatial Attention from EEG

OBJECTIVE

Humans are able to localize the source of a sound. This enables them to direct attention to a particular speaker in a cocktail party. Psycho-acoustic studies show that the sensory cortices of the human brain respond to the location of sound sources differently, and the auditory attention itself is a dynamic and temporally based brain activity. In this work, we seek to build a computational model which uses both spatial and temporal information manifested in EEG signals for auditory spatial attention detection (ASAD).

METHODS

We propose an end-to-end spatiotemporal attention network, denoted as STAnet, to detect auditory spatial attention from EEG. The STAnet is designed to assign differentiated weights dynamically to EEG channels through a spatial attention mechanism, and to temporal patterns in EEG signals through a temporal attention mechanism.

RESULTS

We report the ASAD experiments on two publicly available datasets. The STAnet outperforms other competitive models by a large margin under various experimental conditions. Its attention decision for 1-second decision window outperforms that of the state-of-the-art techniques for 10-second decision window. Experimental results also demonstrate that the STAnet achieves competitive performance on EEG signals ranging from 64 to as few as 16 channels.

CONCLUSION

This study provides evidence suggesting that efficient low-density EEG online decoding is within reach.

SIGNIFICANCE

This study also marks an important step towards the practical implementation of ASAD in real life applications.

Auditory Attention Detection with EEG Channel Attention

Auditory attention detection (AAD) seeks to detect the attended speech from EEG signals in a multi-talker scenario, i.e. cocktail party. As the EEG channels reflect the activities of different brain areas, a task-oriented channel selection technique improves the performance of brain-computer interface applications. In this study, we propose a soft channel attention mechanism, instead of hard channel selection, that derives an EEG channel mask by optimizing the auditory attention detection task. The neural AAD system consists of a neural channel attention mechanism and a convolutional neural network (CNN) classifier. We evaluate the proposed framework on a publicly available database. We achieve 88.3% and 77.2% for 2-second and 0.1-second decision windows with 64-channel EEG; and 86.1% and 83.9% for 2-second decision windows with 32-channel and 16-channel EEG, respectively. The proposed framework outperforms other competitive models by a large margin across all test cases.

Radius selection using kernel density estimation for the computation of nonlinear measures

When nonlinear measures are estimated from sampled temporal signals with finite-length, a radius parameter must be carefully selected to avoid a poor estimation. These measures are generally derived from the correlation integral, which quantifies the probability of finding neighbors, i.e., pair of points spaced by less than the radius parameter. While each nonlinear measure comes with several specific empirical rules to select a radius value, we provide a systematic selection method. We show that the optimal radius for nonlinear measures can be approximated by the optimal bandwidth of a Kernel Density Estimator (KDE) related to the correlation sum. The KDE framework provides non-parametric tools to approximate a density function from finite samples (e.g., histograms) and optimal methods to select a smoothing parameter, the bandwidth (e.g., bin width in histograms). We use results from KDE to derive a closed-form expression for the optimal radius. The latter is used to compute the correlation dimension and to construct recurrence plots yielding an estimate of Kolmogorov-Sinai entropy. We assess our method through numerical experiments on signals generated by nonlinear systems and experimental electroencephalographic time series.

Stochasticity, Nonlinear Value Functions, and Update Rules in Learning Aesthetic Biases

A theoretical framework for the reinforcement learning of aesthetic biases was recently proposed based on brain circuitries revealed by neuroimaging. A model grounded on that framework accounted for interesting features of human aesthetic biases. These features included individuality, cultural predispositions, stochastic dynamics of learning and aesthetic biases, and the peak-shift effect. However, despite the success in explaining these features, a potential weakness was the linearity of the value function used to predict reward. This linearity meant that the learning process employed a value function that assumed a linear relationship between reward and sensory stimuli. Linearity is common in reinforcement learning in neuroscience. However, linearity can be problematic because neural mechanisms and the dependence of reward on sensory stimuli were typically nonlinear. Here, we analyze the learning performance with models including optimal nonlinear value functions. We also compare updating the free parameters of the value functions with the delta rule, which neuroscience models use frequently, vs. updating with a new Phi rule that considers the structure of the nonlinearities. Our computer simulations showed that optimal nonlinear value functions resulted in improvements of learning errors when the reward models were nonlinear. Similarly, the new Phi rule led to improvements in these errors. These improvements were accompanied by the straightening of the trajectories of the vector of free parameters in its phase space. This straightening meant that the process became more efficient in learning the prediction of reward. Surprisingly, however, this improved efficiency had a complex relationship with the rate of learning. Finally, the stochasticity arising from the probabilistic sampling of sensory stimuli, rewards, and motivations helped the learning process narrow the range of free parameters to nearly optimal outcomes. Therefore, we suggest that value functions and update rules optimized for social and ecological constraints are ideal for learning aesthetic biases.

EEG-based detection of the locus of auditory attention with convolutional neural networks

In a multi-speaker scenario, the human auditory system is able to attend to one particular speaker of interest and ignore the others. It has been demonstrated that it is possible to use electroencephalography (EEG) signals to infer to which speaker someone is attending by relating the neural activity to the speech signals. However, classifying auditory attention within a short time interval remains the main challenge. We present a convolutional neural network-based approach to extract the locus of auditory attention (left/right) without knowledge of the speech envelopes. Our results show that it is possible to decode the locus of attention within 1-2 s, with a median accuracy of around 81%. These results are promising for neuro-steered noise suppression in hearing aids, in particular in scenarios where per-speaker envelopes are unavailable.

EEG-based detection of the locus of auditory attention with convolutional neural networks

In a multi-speaker scenario, the human auditory system is able to attend to one particular speaker of interest and ignore the others. It has been demonstrated that it is possible to use electroencephalography (EEG) signals to infer to which speaker someone is attending by relating the neural activity to the speech signals. However, classifying auditory attention within a short time interval remains the main challenge. We present a convolutional neural network-based approach to extract the locus of auditory attention (left/right) without knowledge of the speech envelopes. Our results show that it is possible to decode the locus of attention within 1-2 s, with a median accuracy of around 81%. These results are promising for neuro-steered noise suppression in hearing aids, in particular in scenarios where per-speaker envelopes are unavailable.

Noise-driven multistability vs deterministic chaos in phenomenological semi-empirical models of whole-brain activity

An outstanding open problem in neuroscience is to understand how neural systems are capable of producing and sustaining complex spatiotemporal dynamics. Computational models that combine local dynamics with in vivo measurements of anatomical and functional connectivity can be used to test potential mechanisms underlying this complexity. We compared two conceptually different mechanisms: noise-driven switching between equilibrium solutions (modeled by coupled Stuart-Landau oscillators) and deterministic chaos (modeled by coupled Rossler oscillators). We found that both models struggled to simultaneously reproduce multiple observables computed from the empirical data. This issue was especially manifested in the case of noise-driven dynamics close to a bifurcation, which imposed overly strong constraints on the optimal model parameters. In contrast, the chaotic model could produce complex behavior over a range of parameters, thus being capable of capturing multiple observables at the same time with good performance. Our observations support the view of the brain as a non-equilibrium system able to produce endogenous variability. We presented a simple model capable of jointly reproducing functional connectivity computed at different temporal scales. Besides adding to our conceptual understanding of brain complexity, our results inform and constrain the future development of biophysically realistic large-scale models.

Brain network dynamics fingerprints are resilient to data heterogeneity

CONTEXT

Large multi-site neuroimaging datasets have significantly advanced our quest to understand brain-behavior relationships and to develop biomarkers of psychiatric and neurodegenerative disorders. Yet, such data collections come at a cost, as the inevitable differences across samples may lead to biased or erroneous conclusions.

OBJECTIVE

We aim to validate the estimation of individual brain network dynamics fingerprints and appraise sources of variability in large resting-state functional magnetic resonance imaging (rs-fMRI) datasets by providing a novel point of view based on data-driven dynamical models.

APPROACH

Previous work has investigated this critical issue in terms of effects on static measures, such as functional connectivity and brain parcellations. Here, we utilize dynamical models (Hidden Markov models - HMM) to examine how diverse scanning factors in multi-site fMRI recordings affect our ability to infer the brain's spatiotemporal wandering between large-scale networks of activity. Specifically, we leverage a stable HMM trained on the Human Connectome Project (homogeneous) dataset, which we then apply to an heterogeneous dataset of traveling subjects scanned under a multitude of conditions.

MAIN RESULTS

Building upon this premise, we first replicate previous work on the emergence of non-random sequences of brain states. We next highlight how these time-varying brain activity patterns are robust subject-specific fingerprints. Finally, we suggest these fingerprints may be used to assess which scanning factors induce high variability in the data.

SIGNIFICANCE

These results demonstrate that we can i) use large scale dataset to train models that can be then used to interrogate subject-specific data, ii) recover the unique trajectories of brain activity changes in each individual, but also iii) urge caution as our ability to infer such patterns is affected by how, where and when we do so.

ApEn for assessing hypoxemia severity in obstructive sleep apnea hypopnea syndrome patients

OBJECTIVE

A new index, approximate entropy (ApEn) of oxygen saturation, was used to assess the severity of hypoxemia in patients with obstructive sleep apnea-hypopnea syndrome (OSAHS), determine the correlation with other parameters, and explore its clinical value.

METHODS

A retrospective analysis was performed on 1200 patients with OSAHS and snorers (normal control). All subjects underwent sleep apnea monitoring for 6 h. Subjects were divided into four subgroups by apnea-hypopnea index (AHI): normal control (AHI < 5), mild OSAHS (5 ≤ AHI < 15), moderate OSAHS (15 ≤ AHI < 30), and severe OSAHS 104 (AHI ≥ 30). ApEn was initially compared among the subgroups. Then a correlation analysis of AHI with ApEn and a correlation analysis of ApEn with oxygen desaturation index (ODI), lowest oxygen saturation (LO2), and T < 90% were performed. (2) The AHI was used as the gold standard, and an attempt was performed to determine the value of ApEn to assess the severity of hypoxemia in OSAHS.

RESULTS

Among the 1200 subjects, 822 subjects were men (72%) and mean age was 53.2 ± 15.2 years (range 24-95 years). The ApEn in each group was significantly different (P <0.001), and the ApEn synchronously increased with AHI. Furthermore, a significant difference in ApEn was found among the groups (P <0.001). In addition, ApEn had a good correlation with ODI, LO2, and T <90%. According to the ROC analysis results, the boundary value of ApEn to judge OSAHS patients with mild, moderate, and severe hypoxia was 16.72, 17.84, and 20.06, respectively.

CONCLUSION

ApEn synchronously increased with the AHI and had a good correlation with AHI, ODI, LO2, and T <90%. These findings suggest that ApEn may have clinical value for assessing hypoxia severity in OSAHS patients.

ChaosNet: A chaos based artificial neural network architecture for classification

Inspired by chaotic firing of neurons in the brain, we propose ChaosNet-a novel chaos based artificial neural network architecture for classification tasks. ChaosNet is built using layers of neurons, each of which is a 1D chaotic map known as the Generalized Luröth Series (GLS) that has been shown in earlier works to possess very useful properties for compression, cryptography, and for computing XOR and other logical operations. In this work, we design a novel learning algorithm on ChaosNet that exploits the topological transitivity property of the chaotic GLS neurons. The proposed learning algorithm gives consistently good performance accuracy in a number of classification tasks on well known publicly available datasets with very limited training samples. Even with as low as seven (or fewer) training samples/class (which accounts for less than 0.05% of the total available data), ChaosNet yields performance accuracies in the range of 73.89%-98.33%. We demonstrate the robustness of ChaosNet to additive parameter noise and also provide an example implementation of a two layer ChaosNet for enhancing classification accuracy. We envisage the development of several other novel learning algorithms on ChaosNet in the near future.

Decoding Selective Attention in Normal Hearing Listeners and Bilateral Cochlear Implant Users With Concealed Ear EEG

Electroencephalography (EEG) data can be used to decode an attended speech source in normal-hearing (NH) listeners using high-density EEG caps, as well as around-the-ear EEG devices. The technology may find application in identifying the target speaker in a cocktail party like scenario and steer speech enhancement algorithms in cochlear implants (CIs). However, the worse spectral resolution and the electrical artifacts introduced by a CI may limit the applicability of this approach to CI users. The goal of this study was to investigate whether selective attention can be decoded in CI users using an around-the-ear EEG system (cEEGrid). The performances of high-density cap EEG recordings and cEEGrid EEG recordings were compared in a selective attention paradigm using an envelope tracking algorithm. Speech from two audio books was presented through insert earphones to NH listeners and via direct audio cable to the CI users. 10 NH listeners and 10 bilateral CI users participated in the study. Participants were instructed to attend to one out of the two concurrent speech streams while data were recorded by a 96-channel scalp EEG and an 18-channel cEEGrid setup simultaneously. Reconstruction performance was evaluated by means of parametric correlations between the reconstructed speech and both, the envelope of the attended and the unattended speech stream. Results confirm the feasibility to decode selective attention by means of single-trial EEG data in NH and CI users using a high-density EEG. All NH listeners and 9 out of 10 CI achieved high decoding accuracies. The cEEGrid was successful in decoding selective attention in 5 out of 10 NH listeners. The same result was obtained for CI users.

General Principles of Neurorobotic Models Employing Entrainment and Chaos Control

Neurorobotic models are perfect candidates for proving the validity of embodied/dynamical approaches to cognition. In order to control bodies of arbitrary complexity in complex environments, it is necessary to coordinate vast numbers of sensory and motor components. In this review, we took at several studies of neurorobotics and generalize common principles of how they achieve this massive feat, relying on the key concepts of entrainment and chaos control. We discuss current limitations and ways that these techniques could be expanded to cover more wide ranges of behavior, for example by taking inspiration from ecological psychology.

Interactive Sonification Exploring Emergent Behavior Applying Models for Biological Information and Listening

Sonification is an open-ended design task to construct sound informing a listener of data. Understanding application context is critical for shaping design requirements for data translation into sound. Sonification requires methodology to maintain reproducibility when data sources exhibit non-linear properties of self-organization and emergent behavior. This research formalizes interactive sonification in an extensible model to support reproducibility when data exhibits emergent behavior. In the absence of sonification theory, extensibility demonstrates relevant methods across case studies. The interactive sonification framework foregrounds three factors: reproducible system implementation for generating sonification; interactive mechanisms enhancing a listener's multisensory observations; and reproducible data from models that characterize emergent behavior. Supramodal attention research suggests interactive exploration with auditory feedback can generate context for recognizing irregular patterns and transient dynamics. The sonification framework provides circular causality as a signal pathway for modeling a listener interacting with emergent behavior. The extensible sonification model adopts a data acquisition pathway to formalize functional symmetry across three subsystems: Experimental Data Source, Sound Generation, and Guided Exploration. To differentiate time criticality and dimensionality of emerging dynamics, tuning functions are applied between subsystems to maintain scale and symmetry of concurrent processes and temporal dynamics. Tuning functions accommodate sonification design strategies that yield order parameter values to render emerging patterns discoverable as well as rehearsable, to reproduce desired instances for clinical listeners. Case studies are implemented with two computational models, Chua's circuit and Swarm Chemistry social agent simulation, generating data in real-time that exhibits emergent behavior. Heuristic Listening is introduced as an informal model of a listener's clinical attention to data sonification through multisensory interaction in a context of structured inquiry. Three methods are introduced to assess the proposed sonification framework: Listening Scenario classification, data flow Attunement, and Sonification Design Patterns to classify sound control. Case study implementations are assessed against these methods comparing levels of abstraction between experimental data and sound generation. Outcomes demonstrate the framework performance as a reference model for representing experimental implementations, also for identifying common sonification structures having different experimental implementations, identifying common functions implemented in different subsystems, and comparing impact of affordances across multiple implementations of listening scenarios.

Patients with Chronic Obstructive Pulmonary Disease Walk with Altered Step Time and Step Width Variability as Compared with Healthy Control Subjects

RATIONALE

Compared with control subjects, patients with chronic obstructive pulmonary disease (COPD) have an increased incidence of falls and demonstrate balance deficits and alterations in mediolateral trunk acceleration while walking. Measures of gait variability have been implicated as indicators of fall risk, fear of falling, and future falls.

OBJECTIVES

To investigate whether alterations in gait variability are found in patients with COPD as compared with healthy control subjects.

METHODS

Twenty patients with COPD (16 males; mean age, 63.6 ± 9.7 yr; FEV₁/FVC, 0.52 ± 0.12) and 20 control subjects (9 males; mean age, 62.5 ± 8.2 yr) walked for 3 minutes on a treadmill while their gait was recorded. The amount (SD and coefficient of variation) and structure of variability (sample entropy, a measure of regularity) were quantified for step length, time, and width at three walking speeds (self-selected and ±20% of self-selected speed). Generalized linear mixed models were used to compare dependent variables.

RESULTS

Patients with COPD demonstrated increased mean and SD step time across all speed conditions as compared with control subjects. They also walked with a narrower step width that increased with increasing speed, whereas the healthy control subjects walked with a wider step width that decreased as speed increased. Further, patients with COPD demonstrated less variability in step width, with decreased SD, compared with control subjects at all three speed conditions. No differences in regularity of gait patterns were found between groups.

CONCLUSIONS

Patients with COPD walk with increased duration of time between steps, and this timing is more variable than that of control subjects. They also walk with a narrower step width in which the variability of the step widths from step to step is decreased. Changes in these parameters have been related to increased risk of falling in aging research. This provides a mechanism that could explain the increased prevalence of falls in patients with COPD.

Cocaine-Induced Changes in Low-Dimensional Attractors of Local Field Potentials in Optogenetic Mice

Optogenetically evoked local field potential (LFP) recorded from the medial prefrontal cortex (mPFC) of mice during basal conditions and following a systemic cocaine administration were analyzed. Blue light stimuli were delivered to mPFC through a fiber optic every 2 s and each trial was repeated 100 times. As in the previous study, we used a surrogate data method to check that nonlinearity was present in the experimental LFPs and only used the last 1.5 s of steady activity to measure the LFPs phase resetting induced by the brief 10 ms light stimulus. We found that the steady dynamics of the mPFC in response to light stimuli could be reconstructed in a three-dimensional phase space with topologically similar "8"-shaped attractors across different animals. Therefore, cocaine did not change the complexity of the recorded nonlinear data compared to the control case. The phase space of the reconstructed attractor is determined by the LFP time series and its temporally shifted versions by a multiple of some lag time. We also compared the change in the attractor shape between cocaine-injected and control using (1) dendrogram clustering and (2) Frechet distance. We found about 20% overlap between control and cocaine trials when classified using dendrogram method, which suggest that it may be possible to describe mathematically both data sets with the same model and slightly different model parameters. We also found that the lag times are about three times shorter for cocaine trials compared to control. As a result, although the phase space trajectories for control and cocaine may look similar, their dynamics is significantly different.

A New Chaotic System with a Self-Excited Attractor: Entropy Measurement, Signal Encryption, and Parameter Estimation

In this paper, we introduce a new chaotic system that is used for an engineering application of the signal encryption. It has some interesting features, and its successful implementation and manufacturing were performed via a real circuit as a random number generator. In addition, we provide a parameter estimation method to extract chaotic model parameters from the real data of the chaotic circuit. The parameter estimation method is based on the attractor distribution modeling in the state space, which is compatible with the chaotic system characteristics. Here, a Gaussian mixture model (GMM) is used as a main part of cost function computations in the parameter estimation method. To optimize the cost function, we also apply two recent efficient optimization methods: WOA (Whale Optimization Algorithm), and MVO (Multi-Verse Optimizer) algorithms. The results show the success of the parameter estimation procedure.

Chaos, Cognition and Disordered Brain

Important feature of chaotic transitions is a self-organization that presents a spontaneous order arising in a system when certain parameters of the system reach critical values. Recent findings suggest that these principles may implicate new concepts for understanding consciousness and cognition. Self-organization may produce random-like processes that could explain “randomness” in neural synchronization related to cognitive functions and consciousness, and also in mental disorganization related to psychopathological phenomena.

Toward a complex system understanding of bipolar disorder: A chaotic model of abnormal circadian activity rhythms in euthymic bipolar disorder

IMPORTANCE

In the absence of a comprehensive neural model to explain the underlying mechanisms of disturbed circadian function in bipolar disorder, mathematical modeling is a helpful tool. Here, circadian activity as a response to exogenous daily cycles is proposed to be the product of interactions between neuronal networks in cortical (cognitive processing) and subcortical (pacemaker) areas of the brain.

OBJECTIVE

To investigate the dynamical aspects of the link between disturbed circadian activity rhythms and abnormalities of neurotransmitter functioning in frontal areas of the brain, we developed a novel mathematical model of a chaotic system which represents fluctuations in circadian activity in bipolar disorder as changes in the model's parameters.

DESIGN, SETTING AND PARTICIPANTS

A novel map-based chaotic system was developed to capture disturbances in circadian activity across the two extreme mood states of bipolar disorder. The model uses chaos theory to characterize interplay between neurotransmitter functions and rhythm generation; it aims to illuminate key activity phenomenology in bipolar disorder, including prolonged sleep intervals, decreased total activity and attenuated amplitude of the diurnal activity rhythm. To test our new cortical-circadian mathematical model of bipolar disorder, we utilized previously collected locomotor activity data recorded from normal subjects and bipolar patients by wrist-worn actigraphs.

RESULTS

All control parameters in the proposed model have an important role in replicating the different aspects of circadian activity rhythm generation in the brain. The model can successfully replicate deviations in sleep/wake time intervals corresponding to manic and depressive episodes of bipolar disorder, in which one of the excitatory or inhibitory pathways is abnormally dominant.

CONCLUSIONS AND RELEVANCE

Although neuroimaging research has strongly implicated a reciprocal interaction between cortical and subcortical regions as pathogenic in bipolar disorder, this is the first model to mathematically represent this multilevel explanation of the phenomena of bipolar disorder.

DYNAMICAL ANALYSIS OF EMOTIONAL STATES FROM ELECTROENCEPHALOGRAM SIGNALS

The current study evaluates the dynamics of the electroencephalogram signals during specific emotional states in order to obtain a detailed understanding of the affective EEG patterns. Employing recurrence analysis, the dynamical states of the emotional brain during visual stimuli is evaluated. Three channels of electroencephalogram time series (Fz, Cz, and Pz) available in eNTERFACE06_EMOBRAIN database are used in this study. Electroencephalogram signals are recorded from 5 subjects in three emotional categories: exciting negative (disgust), neutral and exciting positive (happy). Recurrence quantification analysis (RQA) was applied to study electroencephalogram morphological changes in different emotional states (happy, disgust, neutral). The ANOVA and [Formula: see text]-test are done to detect significant differences in RQA measures of the EEGs. It has shown that the phase space trajectory becomes more periodic during exciting negative. In addition, the results reveal that in comparison with negative emotion and neutral, the behavior of EEGs in positive emotion is highly chaotic. Performing statistical analysis, significant differences were observed in recurrence rate, determinism, average diagonal, line length, Shannon entropy, laminarity, trapping time among three emotional evoked groups. Although these changes occurred in all EEG channels, a better distinction between each emotional state can be observed in Pz location. It seems that recurrence analysis is a promising non-linear approach for detecting instantaneous changes in the emotional EEG induced by visual stimuli. RQA has the potential to discover differences of signal features in response to an emotional stimulus.

Can Lionel Messi's brain slow down time passing?

It seems that seeing others in slow-motion by heroes does not belong only to movies. When Lionel Messi plays football, you can hardly see anything from him that other players cannot do. Then why he is not stoppable really? It seems the answer may be that opponents do not have enough time to do what they want; because in Messi's neural system, time passes slower. In differential equations that model a single neuron, this speed can be generated by multiplying an equal term in all equations. Or maybe interactions between neurons and the structure of neural networks play this role.

Dissociative states in dreams and brain chaos: implications for creative awareness

This article reviews recent findings indicating some common brain processes during dissociative states and dreaming with the aim to outline a perspective that neural chaotic states during dreaming can be closely related to dissociative states that may manifest in dreams scenery. These data are in agreement with various clinical findings that dissociated states can be projected into the "dream scenery" in REM sleep periods and dreams may represent their specific interactions that may uncover unusual psychological potential of creativity in psychotherapy, art, and scientific discoveries.

[To err is human? Interests of chaotic models to study adult psychiatric disorders and developmental disorders]

BACKGROUND

Many clinical and biological parameters have nonlinear chaotic fluctuations. These variations result in unexpected pseudo-random transitions. In these models, few risk factors can lead to unexpected phenomena if oscillations and self-reinforcement patterns occur. Complex rhythms could ease the ability of a physiological system to adapt and react quickly to a constantly changing environment.

OBJECTIVES

It has been proposed that several psychiatric disorders and developmental disorders are characterized by a loss of complex rhythm in favor of a more organized pattern. We examine evidence to support these assumptions in literatures.

METHODS

We performed a literature review of the main computerized databases (Medline, PubMed) and manual searches of the literature concerning non dynamic rhythms in time series analysis, in adults with psychiatric disorder and children with developmental disorder. These results were interpreted through a developmental approach that highlights the role of the learning process in the emergence of abilities.

RESULTS

Analysis of clinical scores and electroencephalographic data have found that subjects with bipolar disorder or schizophrenia, tested over a time series, have lower chaotic rhythms compared with healthy subjects. Growing children share several properties of a complex system: the interdependence of developmental axes (motor, emotional, language, social skills), multiple hierarchical levels (i.e. genetic, biological, environmental, and cultural), the two-way transactions between the child and his environment, and the sensitivity to initial conditions. This could explain the difficulty to predict the emergence of abilities or the long-term prognosis of impairment in children. This limitation is not only due to errors in the explanatory model or the lack of explanatory variable. It is also caused by instability, which is a core characteristic of a chaotic system.

CONCLUSION

The study of chaotic rhythms in time-series clinical and nonclinical data (e.g. EEG, functional neuroimaging) could improve the prediction of an acute event, such as relapse of mood disorder. Moreover, the complex rhythms in children may play a major part in synchronicity during interactions with a caregiver, held as essential for later development of self-regulation skills, such as emotional stability.

Fractality of sensations and the brain health: the theory linking neurodegenerative disorder with distortion of spatial and temporal scale-invariance and fractal complexity of the visible world

The theory that ties normal functioning and pathology of the brain and visual system with the spatial-temporal structure of the visual and other sensory stimuli is described for the first time in the present study. The deficit of fractal complexity of environmental influences can lead to the distortion of fractal complexity in the visual pathways of the brain and abnormalities of development or aging. The use of fractal light stimuli and fractal stimuli of other modalities can help to restore the functions of the brain, particularly in the elderly and in patients with neurodegenerative disorders or amblyopia. Non-linear dynamics of these physiological processes have a strong base of evidence, which is seen in the impaired fractal regulation of rhythmic activity in aged and diseased brains. From birth to old age, we live in a non-linear world, in which objects and processes with the properties of fractality and non-linearity surround us. Against this background, the evolution of man took place and all periods of life unfolded. Works of art created by man may also have fractal properties. The positive influence of music on cognitive functions is well-known. Insufficiency of sensory experience is believed to play a crucial role in the pathogenesis of amblyopia and age-dependent diseases. The brain is very plastic in its early development, and the plasticity decreases throughout life. However, several studies showed the possibility to reactivate the adult's neuroplasticity in a variety of ways. We propose that a non-linear structure of sensory information on many spatial and temporal scales is crucial to the brain health and fractal regulation of physiological rhythms. Theoretical substantiation of the author's theory is presented. Possible applications and the future research that can experimentally confirm or refute the theoretical concept are considered.

Nonlinear dynamics of mood regulation in bipolar disorder

OBJECTIVES

We sought to study the underlying dynamic processes involved in mood regulation in subjects with bipolar disorder and healthy control subjects using time-series analysis and to then analyze the relation between anxiety and mood using cross-correlation techniques.

METHODS

We recruited 30 healthy controls and 30 euthymic patients with bipolar disorder. Participants rated their mood, anxiety, and energy levels using a paper-based visual analog scale; and they also recorded their sleep and any life events. Information on these variables was provided over a three-month period on a daily basis, twice per day. We analyzed the data using Box-Jenkins time series analysis to obtain information on the autocorrelation of the series (for mood) and cross-correlation (mood and anxiety series).

RESULTS

Throughout the study, we analyzed 10,170 data points. Self-ratings for mood, anxiety, and energy were normally distributed in both groups. Autocorrelation functions for mood in both groups were governed by the autoregressive integrated moving average (ARIMA) (1,1,0) model, which means that current values in the series were related to one previous point only. We also found a negative cross-correlation between mood and anxiety.

CONCLUSIONS

Mood can be considered a memory stochastic process; it is a flexible, dynamic process that has a 'short memory' both in healthy controls and euthymic patients with bipolar disorder. This process may be quite different in untreated patients or in those acutely ill. Our results suggest that nonlinear measures can be applied to the study of mood disorders.

Differential regulation of observational fear and neural oscillations by serotonin and dopamine in the mouse anterior cingulate cortex

RATIONALE

The aberrant regulation of serotonin (5-HT) and dopamine (DA) in the brain has been implicated in neuropsychiatric disorders associated with marked impairments in empathy, such as schizophrenia and autism. Many psychiatric drugs bind to both types of receptors, and the anterior cingulate cortex (ACC) is known to be centrally involved with empathy. However, the relationship between the 5-HT/DA system in the ACC and empathic behavior is not yet well known.

OBJECTIVES

We investigated the role of 5-HT/DA in empathy-like behavior and in the regulation of ACC neural activity.

METHODS

An observational fear learning task was conducted following microinjections of 5-HT, DA, 5-HT and DA, methysergide (5-HT receptor antagonist), SCH-23390 (DA D1 receptor antagonist), or haloperidol (DA D2 receptor antagonist) into the mouse ACC. The ACC neural activity influenced by 5-HT and DA was electrophysiologically characterized in vitro and in vivo.

RESULTS

The microinjection of haloperidol, but not methysergide or SCH-23390, decreased the fear response of observing mice. The administration of 5-HT and 5-HT and DA together, but not DA alone, reduced the freezing response of observing mice. 5-HT enhanced delta-band activity and reduced alpha- and gamma-band activities in the ACC, whereas DA reduced only alpha-band activity. Based on entropy, reduced complexity of ACC neural activity was observed with 5-HT treatment.

CONCLUSIONS

The current results demonstrated that DA D2 receptors in the ACC are required for observational fear learning, whereas increased 5-HT levels disrupt observational fear and alter the regularity of ACC neural oscillations.

Spin-torque building blocks

The discovery of the spin-torque effect has made magnetic nanodevices realistic candidates for active elements of memory devices and applications. Magnetoresistive effects allow the read-out of increasingly small magnetic bits, and the spin torque provides an efficient tool to manipulate - precisely, rapidly and at low energy cost - the magnetic state, which is in turn the central information medium of spintronic devices. By keeping the same magnetic stack, but by tuning a device's shape and bias conditions, the spin torque can be engineered to build a variety of advanced magnetic nanodevices. Here we show that by assembling these nanodevices as building blocks with different functionalities, novel types of computing architecture can be envisaged. We focus in particular on recent concepts such as magnonics and spintronic neural networks.

Nonlinear measures and dynamics in psychophysiology of consciousness

According to recent findings nonlinear dynamic processes related to neural chaos and complexity likely play a crucial role in neural synchronization of distributed neural activities that enable information integration and conscious experience. Disturbances in these interactions produce patterns of temporal and spatial disorganization with decreased or increased functional connectivity and complexity that underlie specific changes of perceptual and cognitive states. These perceptual and cognitive changes may be characterized by neural chaos with significantly increased brain sensitivity that may underlie sensitization and kindling, and cognitive hypersensitivity in some mental disorders. Together these findings suggest that processes related to more irregular neural states with higher complexity that may lead to neural chaos, negatively affect information integration and processing in the brain, and may influence disintegrated conscious experience.

Network connectivity modulates power spectrum scale invariance

Measures of complexity are sensitive in detecting disease, which has made them attractive candidates for diagnostic biomarkers; one complexity measure that has shown promise in fMRI is power spectrum scale invariance (PSSI). Even if scale-free features of neuroimaging turn out to be diagnostically useful, however, their underlying neurobiological basis is poorly understood. Using modeling and simulations of a schematic prefrontal-limbic meso-circuit, with excitatory and inhibitory networks of nodes, we present here a framework for how network density within a control system can affect the complexity of signal outputs. Our model demonstrates that scale-free behavior, similar to that observed in fMRI PSSI data, can be obtained for sufficiently large networks in a context as simple as a linear stochastic system of differential equations, although the scale-free range improves when introducing more realistic, nonlinear behavior in the system. PSSI values (reflective of complexity) vary as a function of both input type (excitatory, inhibitory) and input density (mean number of long-range connections, or strength), independent of their node-specific geometric distribution. Signals show pink noise (1/f) behavior when excitatory and inhibitory influences are balanced. As excitatory inputs are increased and decreased, signals shift towards white and brown noise, respectively. As inhibitory inputs are increased and decreased, signals shift towards brown and white noise, respectively. The results hold qualitatively at the hemodynamic scale, which we modeled by introducing a neurovascular component. Comparing hemodynamic simulation results to fMRI PSSI results from 96 individuals across a wide spectrum of anxiety-levels, we show how our model can generate concrete and testable hypotheses for understanding how connectivity affects regulation of meso-circuits in the brain.

Spontaneous slow oscillations and sequential patterns due to short-term plasticity in a model of the cortex

We study a realistic model of a cortical column taking into account short-term plasticity between pyramidal cells and interneurons. The simulation of leaky integrate-and-fire neurons shows that low-frequency oscillations emerge spontaneously as a result of intrinsic network properties. These oscillations are composed of prolonged phases of high and low activity reminiscent of cortical up and down states, respectively. We simplify the description of the network activity by using a mean field approximation and reduce the system to two slow variables exhibiting some relaxation oscillations. We identify two types of slow oscillations. When the combination of dynamic synapses between pyramidal cells and those between interneurons accounts for the generation of these slow oscillations, the end of the up phase is characterized by asynchronous fluctuations of the membrane potentials. When the slow oscillations are mainly driven by the dynamic synapses between interneurons, the network exhibits fluctuations of membrane potentials, which are more synchronous at the end than at the beginning of the up phase. Additionally, finite size effect and slow synaptic currents can modify the irregularity and frequency, respectively, of these oscillations. Finally, we consider possible roles of a slow oscillatory input modeling long-range interactions in the brain. Spontaneous slow oscillations of local networks are modulated by the oscillatory input, which induces, notably, synchronization, subharmonic synchronization, and chaotic relaxation oscillations in the mean field approximation. In the case of forced oscillations, the slow population-averaged activity of leaky integrate-and-fire neurons can have both deterministic and stochastic temporal features. We discuss the possibility that long-range connectivity controls the emergence of slow sequential patterns in local populations due to the tendency of a cortical column to oscillate at low frequency.

Chaotic Boltzmann machines

The chaotic Boltzmann machine proposed in this paper is a chaotic pseudo-billiard system that works as a Boltzmann machine. Chaotic Boltzmann machines are shown numerically to have computing abilities comparable to conventional (stochastic) Boltzmann machines. Since no randomness is required, efficient hardware implementation is expected. Moreover, the ferromagnetic phase transition of the Ising model is shown to be characterised by the largest Lyapunov exponent of the proposed system. In general, a method to relate probabilistic models to nonlinear dynamics by derandomising Gibbs sampling is presented.

The appropriate use of approximate entropy and sample entropy with short data sets

Approximate entropy (ApEn) and sample entropy (SampEn) are mathematical algorithms created to measure the repeatability or predictability within a time series. Both algorithms are extremely sensitive to their input parameters: m (length of the data segment being compared), r (similarity criterion), and N (length of data). There is no established consensus on parameter selection in short data sets, especially for biological data. Therefore, the purpose of this research was to examine the robustness of these two entropy algorithms by exploring the effect of changing parameter values on short data sets. Data with known theoretical entropy qualities as well as experimental data from both healthy young and older adults was utilized. Our results demonstrate that both ApEn and SampEn are extremely sensitive to parameter choices, especially for very short data sets, N ≤ 200. We suggest using N larger than 200, an m of 2 and examine several r values before selecting your parameters. Extreme caution should be used when choosing parameters for experimental studies with both algorithms. Based on our current findings, it appears that SampEn is more reliable for short data sets. SampEn was less sensitive to changes in data length and demonstrated fewer problems with relative consistency.

Human movement variability, nonlinear dynamics, and pathology: is there a connection?

Fields studying movement generation, including robotics, psychology, cognitive science, and neuroscience utilize concepts and tools related to the pervasiveness of variability in biological systems. The concept of variability and the measures for nonlinear dynamics used to evaluate this concept open new vistas for research in movement dysfunction of many types. This review describes innovations in the exploration of variability and their potential importance in understanding human movement. Far from being a source of error, evidence supports the presence of an optimal state of variability for healthy and functional movement. This variability has a particular organization and is characterized by a chaotic structure. Deviations from this state can lead to biological systems that are either overly rigid and robotic or noisy and unstable. Both situations result in systems that are less adaptable to perturbations, such as those associated with unhealthy pathological states or absence of skillfulness.

Spatial-temporal dynamics of chaotic behavior in cultured hippocampal networks

Using multiple nonlinear techniques, we revealed the existence of chaos in the spontaneous activity of neuronal networks in vitro. The spatial-temporal dynamics of these networks indicated that emergent transition between chaotic behavior and superburst occurred periodically in low-frequency oscillations. An analysis of network-wide activity indicated that chaos was synchronized among different sites. Moreover, we found that the degree of chaos increased as the number of active sites in the network increased during long-term development (over three months in vitro). The chaotic behavior of the dissociated networks had similar spatial-temporal characteristics (rapid transition, periodicity, and synchronization) as the intact brain; however, the degree of chaos depended on the number of active sites at the mesoscopic level. This work could provide insight into neural coding and neurocybernetics.

Psychosomatic symptoms as biomarkers: transcending the psyche-soma dichotomy

Following the advancement in understanding dynamical systems, the author presents a novel metaphor of psychosomatic symptoms as low-dimensional biomarkers. This metaphor, which transcends the old binary of psyche-soma, resonates with classical psychoanalytic concepts and with Matte-Blanco's idea of repetition as indicative of dimensionality reduction. The relevance of this metaphor for explanation, diagnosis, and treatment is illustrated through a case study of a male patient suffering from hyperprolactinemia.