Is there chaos in the brain? II. Experimental evidence and related models

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.

Linear response in neuronal networks: From neurons dynamics to collective response

We review two examples where the linear response of a neuronal network submitted to an external stimulus can be derived explicitly, including network parameters dependence. This is done in a statistical physicslike approach where one associates, to the spontaneous dynamics of the model, a natural notion of Gibbs distribution inherited from ergodic theory or stochastic processes. These two examples are the Amari-Wilson-Cowan model [S. Amari, Syst. Man Cybernet. SMC-2, 643-657 (1972); H. R. Wilson and J. D. Cowan, Biophys. J. 12, 1-24 (1972)] and a conductance based Integrate and Fire model [M. Rudolph and A. Destexhe, Neural Comput. 18, 2146-2210 (2006); M. Rudolph and A. Destexhe, Neurocomputing 70(10-12), 1966-1969 (2007)].

Diagnosis of attention deficit hyperactivity disorder using non-linear analysis of the EEG signal

Attention deficit hyperactivity disorder (ADHD) is a common behavioural disorder that may be found in 5%-8% of the children. Early diagnosis of ADHD is crucial for treating the disease and reducing its harmful effects on education, employment, relationships, and life quality. On the other hand, non-linear analysis methods are widely applied in processing the electroencephalogram (EEG) signals. It has been proved that the brain neuronal activity and its related EEG signals have chaotic behaviour. Hence, chaotic indices can be employed to classify the EEG signals. In this study, a new approach is proposed based on the combination of some non-linear features to distinguish ADHD from normal children. Lyapunov exponent, fractal dimension, correlation dimension and sample, fuzzy and approximate entropies are the non-linear extracted features. For computing, the chaotic time series of obtained EEG in the brain frontal lobe (FP1, FP2, F3, F4, and Fz) need to be analysed. Experiments on a set of EEG signal obtained from 50 ADHD and 26 normal cases yielded a sensitivity, specificity, and accuracy of 98, 92.31, and 96.05%, respectively. The obtained accuracy provides a significant improvement in comparison to the other similar studies in identifying and classifying children with ADHD.

Complex temporal patterns processing by a neural mass model of a cortical column

It is well known that neuronal networks are capable of transmitting complex spatiotemporal information in the form of precise sequences of neuronal discharges characterized by recurrent patterns. At the same time, the synchronized activity of large ensembles produces local field potentials that propagate through highly dynamic oscillatory waves, such that, at the whole brain scale, complex spatiotemporal dynamics of electroencephalographic (EEG) signals may be associated to sensorimotor decision making processes. Despite these experimental evidences, the link between highly temporally organized input patterns and EEG waves has not been studied in detail. Here, we use a neural mass model to investigate to what extent precise temporal information, carried by deterministic nonlinear attractor mappings, is filtered and transformed into fluctuations in phase, frequency and amplitude of oscillatory brain activity. The phase shift that we observe, when we drive the neural mass model with specific chaotic inputs, shows that the local field potential amplitude peak appears in less than one full cycle, thus allowing traveling waves to encode temporal information. After converting phase and amplitude changes obtained into point processes, we quantify input-output similarity following a threshold-filtering algorithm onto the amplitude wave peaks. Our analysis shows that the neural mass model has the capacity for gating the input signal and propagate selected temporal features of that signal. Finally, we discuss the effect of local excitatory/inhibitory balance on these results and how excitability in cortical columns, controlled by neuromodulatory innervation of the cerebral cortex, may contribute to set a fine tuning and gating of the information fed to the cortex.

Hold Your Methods! How Multineuronal Firing Ensembles Can Be Studied Using Classical Spike-Train Analysis Techniques

Responses of neuronal populations play an important role in the encoding of stimulus related information. However, the inherent multidimensionality required to describe population activity has imposed significant challenges and has limited the applicability of classical spike train analysis techniques. Here, we show that these limitations can be overcome. We first quantify the collective activity of neurons as multidimensional vectors (patterns). Then we characterize the behavior of these patterns by applying classical spike train analysis techniques: peri-stimulus time histograms, tuning curves and auto- and cross-correlation histograms. We find that patterns can exhibit a broad spectrum of properties, some resembling and others substantially differing from those of their component neurons. We show that in some cases pattern behavior cannot be intuitively inferred from the activity of component neurons. Importantly, silent neurons play a critical role in shaping pattern expression. By correlating pattern timing with local-field potentials, we show that the method can reveal fine temporal coordination of cortical circuits at the mesoscale. Because of its simplicity and reliance on well understood classical analysis methods the proposed approach is valuable for the study of neuronal population dynamics.

Introducing chaotic codes for the modulation of code modulated visual evoked potentials (c-VEP) in normal adults for visual fatigue reduction

Code modulated Visual Evoked Potentials (c-VEP) based BCI studies usually employ m-sequences as a modulating codes for their broadband spectrum and correlation property. However, subjective fatigue of the presented codes has been a problem. In this study, we introduce chaotic codes containing broadband spectrum and similar correlation property. We examined whether the introduced chaotic codes could be decoded from EEG signals and also compared the subjective fatigue level with m-sequence codes in normal subjects. We generated chaotic code from one-dimensional logistic map and used it with conventional 31-bit m-sequence code. In a c-VEP based study in normal subjects (n = 44, 21 females) we presented these codes visually and recorded EEG signals from the corresponding codes for their four lagged versions. Canonical correlation analysis (CCA) and spatiotemporal beamforming (STB) methods were used for target identification and comparison of responses. Additionally, we compared the subjective self-declared fatigue using VAS caused by presented m-sequence and chaotic codes. The introduced chaotic code was decoded from EEG responses with CCA and STB methods. The maximum total accuracy values of 93.6 ± 11.9% and 94 ± 14.4% were achieved with STB method for chaotic and m-sequence codes for all subjects respectively. The achieved accuracies in all subjects were not significantly different in m-sequence and chaotic codes. There was significant reduction in subjective fatigue caused by chaotic codes compared to the m-sequence codes. Both m-sequence and chaotic codes were similar in their accuracies as evaluated by CCA and STB methods. The chaotic codes significantly reduced subjective fatigue compared to the m-sequence codes.

Embodied neuromechanical chaos through homeostatic regulation

In this paper, we present detailed analyses of the dynamics of a number of embodied neuromechanical systems of a class that has been shown to efficiently exploit chaos in the development and learning of motor behaviors for bodies of arbitrary morphology. This class of systems has been successfully used in robotics, as well as to model biological systems. At the heart of these systems are neural central pattern generating (CPG) units connected to actuators which return proprioceptive information via an adaptive homeostatic mechanism. Detailed dynamical analyses of example systems, using high resolution largest Lyapunov exponent maps, demonstrate the existence of chaotic regimes within a particular region of parameter space, as well as the striking similarity of the maps for systems of varying size. Thanks to the homeostatic sensory mechanisms, any single CPG "views" the whole of the rest of the system as if it was another CPG in a two coupled system, allowing a scale invariant conceptualization of such embodied neuromechanical systems. The analysis reveals chaos at all levels of the systems; the entire brain-body-environment system exhibits chaotic dynamics which can be exploited to power an exploration of possible motor behaviors. The crucial influence of the adaptive homeostatic mechanisms on the system dynamics is examined in detail, revealing chaotic behavior characterized by mixed mode oscillations (MMOs). An analysis of the mechanism of the MMO concludes that they stems from dynamic Hopf bifurcation, where a number of slow variables act as "moving" bifurcation parameters for the remaining part of the system.

Clarifying cognitive control and the controllable connectome

Cognitive control researchers aim to describe the processes that support adaptive cognition to achieve specific goals. Control theorists consider how to influence the state of systems to reach certain user-defined goals. In brain networks, some conceptual and lexical similarities between cognitive control and control theory offer appealing avenues for scientific discovery. However, these opportunities also come with the risk of conceptual confusion. Here, I suggest that each field of inquiry continues to produce novel and distinct insights. Then, I describe opportunities for synergistic research at the intersection of these subdisciplines with a critical stance that reduces the risk of conceptual confusion. Through this exercise, we can observe that both cognitive neuroscience and systems engineering have much to contribute to cognitive control research in human brain networks. This article is categorized under: Neuroscience > Cognition Computer Science > Neural Networks Neuroscience > Clinical Neuroscience.

The Anatomy of Inference: Generative Models and Brain Structure

To infer the causes of its sensations, the brain must call on a generative (predictive) model. This necessitates passing local messages between populations of neurons to update beliefs about hidden variables in the world beyond its sensory samples. It also entails inferences about how we will act. Active inference is a principled framework that frames perception and action as approximate Bayesian inference. This has been successful in accounting for a wide range of physiological and behavioral phenomena. Recently, a process theory has emerged that attempts to relate inferences to their neurobiological substrates. In this paper, we review and develop the anatomical aspects of this process theory. We argue that the form of the generative models required for inference constrains the way in which brain regions connect to one another. Specifically, neuronal populations representing beliefs about a variable must receive input from populations representing the Markov blanket of that variable. We illustrate this idea in four different domains: perception, planning, attention, and movement. In doing so, we attempt to show how appealing to generative models enables us to account for anatomical brain architectures. Ultimately, committing to an anatomical theory of inference ensures we can form empirical hypotheses that can be tested using neuroimaging, neuropsychological, and electrophysiological experiments.

Dual Mechanism for the Emergence of Synchronization in Inhibitory Neural Networks

During cognitive tasks cortical microcircuits synchronize to bind stimuli into unified perception. The emergence of coherent rhythmic activity is thought to be inhibition-driven and stimulation-dependent. However, the exact mechanisms of synchronization remain unknown. Recent optogenetic experiments have identified two neuron sub-types as the likely inhibitory vectors of synchronization. Here, we show that local networks mimicking the soma-targeting properties observed in fast-spiking interneurons and the dendrite-projecting properties observed in somatostatin interneurons synchronize through different mechanisms which may provide adaptive advantages by combining flexibility and robustness. We probed the synchronization phase diagrams of small all-to-all inhibitory networks in-silico as a function of inhibition delay, neurotransmitter kinetics, timings and intensity of stimulation. Inhibition delay is found to induce coherent oscillations over a broader range of experimental conditions than high-frequency entrainment. Inhibition delay boosts network capacity (ln2)^−N-fold by stabilizing locally coherent oscillations. This work may inform novel therapeutic strategies for moderating pathological cortical oscillations.

Synchronization transition in neuronal networks composed of chaotic or non-chaotic oscillators

Chaotic dynamics has been shown in the dynamics of neurons and neural networks, in experimental data and numerical simulations. Theoretical studies have proposed an underlying role of chaos in neural systems. Nevertheless, whether chaotic neural oscillators make a significant contribution to network behaviour and whether the dynamical richness of neural networks is sensitive to the dynamics of isolated neurons, still remain open questions. We investigated synchronization transitions in heterogeneous neural networks of neurons connected by electrical coupling in a small world topology. The nodes in our model are oscillatory neurons that – when isolated – can exhibit either chaotic or non-chaotic behaviour, depending on conductance parameters. We found that the heterogeneity of firing rates and firing patterns make a greater contribution than chaos to the steepness of the synchronization transition curve. We also show that chaotic dynamics of the isolated neurons do not always make a visible difference in the transition to full synchrony. Moreover, macroscopic chaos is observed regardless of the dynamics nature of the neurons. However, performing a Functional Connectivity Dynamics analysis, we show that chaotic nodes can promote what is known as multi-stable behaviour, where the network dynamically switches between a number of different semi-synchronized, metastable states.

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.

Nonlinear Dynamic Complexity and Sources of Resting-state EEG in Abstinent Heroin Addicts

It has been reported that chronic heroin intake induces both structural and functional changes in human brain; however, few studies have investigated the carry-over adverse effects on brain after heroin withdrawal. In this paper, we examined the neurophysiological differences between the abstinent heroin addicts (AHAs) and healthy controls (HCs) using nonlinear dynamic analysis and source localization analysis in resting-state electroencephalogram (EEG) data; 5 min resting EEG data from 20 AHAs and twenty age-, education-, and gender-matched HCs were recorded using 64 electrodes. The results of nonlinear characteristics (e.g., the correlation dimension, Kolmogorov entropy, and Lempel-Ziv complexity) showed that the EEG signals in alpha band from AHAs were significantly more irregular. Moreover, the source localization results confirmed the neuronal activities in alpha band in AHAs were significantly weaker in parietal lobe (BA3 and BA7), frontal lobe (BA4 and BA6), and limbic lobe (BA24). Together, our analysis at both the sensor level and source level suggested the functional abnormalities in the brain during heroin abstinence, in particular for the neuronal oscillations in alpha band.

Nonlinear Recurrent Dynamics and Long-Term Nonstationarities in EEG Alpha Cortical Activity: Implications for Choosing Adequate Segment Length in Nonlinear EEG Analyses

Nonlinear analysis of EEG recordings allows detection of characteristics that would probably be neglected by linear methods. This study aimed to determine a suitable epoch length for nonlinear analysis of EEG data based on its recurrence rate in EEG alpha activity (electrodes Fz, Oz, and Pz) from 28 healthy and 64 major depressive disorder subjects. Two nonlinear metrics, Lempel-Ziv complexity and scaling index, were applied in sliding windows of 20 seconds shifted every 1 second and in nonoverlapping windows of 1 minute. In addition, linear spectral analysis was carried out for comparison with the nonlinear results. The analysis with sliding windows showed that the cortical dynamics underlying alpha activity had a recurrence period of around 40 seconds in both groups. In the analysis with nonoverlapping windows, long-term nonstationarities entailed changes over time in the nonlinear dynamics that became significantly different between epochs across time, which was not detected with the linear spectral analysis. Findings suggest that epoch lengths shorter than 40 seconds neglect information in EEG nonlinear studies. In turn, linear analysis did not detect characteristics from long-term nonstationarities in EEG alpha waves of control subjects and patients with major depressive disorder patients. We recommend that application of nonlinear metrics in EEG time series, particularly of alpha activity, should be carried out with epochs around 60 seconds. In addition, this study aimed to demonstrate that long-term nonlinearities are inherent to the cortical brain dynamics regardless of the presence or absence of a mental disorder.

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.

Searching for Chaos Evidence in Eye Movement Signals

Most naturally-occurring physical phenomena are examples of nonlinear dynamic systems, the functioning of which attracts many researchers seeking to unveil their nature. The research presented in this paper is aimed at exploring eye movement dynamic features in terms of the existence of chaotic nature. Nonlinear time series analysis methods were used for this purpose. Two time series features were studied: fractal dimension and entropy, by utilising the embedding theory. The methods were applied to the data collected during the experiment with “jumping point” stimulus. Eye movements were registered by means of the Jazz-novo eye tracker. One thousand three hundred and ninety two (1392) time series were defined, based on the horizontal velocity of eye movements registered during imposed, prolonged fixations. In order to conduct detailed analysis of the signal and identify differences contributing to the observed patterns of behaviour in time scale, fractal dimension and entropy were evaluated in various time series intervals. The influence of the noise contained in the data and the impact of the utilized filter on the obtained results were also studied. The low pass filter was used for the purpose of noise reduction with a 50 Hz cut-off frequency, estimated by means of the Fourier transform and all concerned methods were applied to time series before and after noise reduction. These studies provided some premises, which allow perceiving eye movements as observed chaotic data: characteristic of a space-time separation plot, low and non-integer time series dimension, and the time series entropy characteristic for chaotic systems.

Sensory Stream Adaptation in Chaotic Networks

Implicit expectations induced by predictable stimuli sequences affect neuronal response to upcoming stimuli at both single cell and neural population levels. Temporally regular sensory streams also phase entrain ongoing low frequency brain oscillations but how and why this happens is unknown. Here we investigate how random recurrent neural networks without plasticity respond to stimuli streams containing oddballs. We found the neuronal correlates of sensory stream adaptation emerge if networks generate chaotic oscillations which can be phase entrained by stimulus streams. The resultant activity patterns are close to critical and support history dependent response on long timescales. Because critical network entrainment is a slow process stimulus response adapts gradually over multiple repetitions. Repeated stimuli generate suppressed responses but oddball responses are large and distinct. Oscillatory mismatch responses persist in population activity for long periods after stimulus offset while individual cell mismatch responses are strongly phasic. These effects are weakened in temporally irregular sensory streams. Thus we show that network phase entrainment provides a biologically plausible mechanism for neural oddball detection. Our results do not depend on specific network characteristics, are consistent with experimental studies and may be relevant for multiple pathologies demonstrating altered mismatch processing such as schizophrenia and depression.

The influence of filtering and downsampling on the estimation of transfer entropy

Transfer entropy (TE) provides a generalized and model-free framework to study Wiener-Granger causality between brain regions. Because of its nonparametric character, TE can infer directed information flow also from nonlinear systems. Despite its increasing number of applications in neuroscience, not much is known regarding the influence of common electrophysiological preprocessing on its estimation. We test the influence of filtering and downsampling on a recently proposed nearest neighborhood based TE estimator. Different filter settings and downsampling factors were tested in a simulation framework using a model with a linear coupling function and two nonlinear models with sigmoid and logistic coupling functions. For nonlinear coupling and progressively lower low-pass filter cut-off frequencies up to 72% false negative direct connections and up to 26% false positive connections were identified. In contrast, for the linear model, a monotonic increase was only observed for missed indirect connections (up to 86%). High-pass filtering (1 Hz, 2 Hz) had no impact on TE estimation. After low-pass filtering interaction delays were significantly underestimated. Downsampling the data by a factor greater than the assumed interaction delay erased most of the transmitted information and thus led to a very high percentage (67–100%) of false negative direct connections. Low-pass filtering increases the number of missed connections depending on the filters cut-off frequency. Downsampling should only be done if the sampling factor is smaller than the smallest assumed interaction delay of the analyzed network.

A new description of epileptic seizures based on dynamic analysis of a thalamocortical model

Increasing evidence suggests that the brain dynamics can be interpreted from the viewpoint of nonlinear dynamical systems. The aim of this paper is to investigate the behavior of a thalamocortical model from this perspective. The model includes both cortical and sensory inputs that can affect the dynamic nature of the model. Driving response of the model subjected to various harmonic stimulations is considered to identify the effects of stimulus parameters on the cortical output. Detailed numerical studies including phase portraits, Poincare maps and bifurcation diagrams reveal a wide range of complex dynamics including period doubling and chaos in the output. Transition between different states can occur as the stimulation parameters are changed. In addition, the amplitude jump phenomena and hysteresis are shown to be possible as a result of the bending in the frequency response curve. These results suggest that the jump phenomenon due to the brain nonlinear resonance can be responsible for the transitions between ictal and interictal states.

Coexisting Behaviors of Asymmetric Attractors in Hyperbolic-Type Memristor based Hopfield Neural Network

A new hyperbolic-type memristor emulator is presented and its frequency-dependent pinched hysteresis loops are analyzed by numerical simulations and confirmed by hardware experiments. Based on the emulator, a novel hyperbolic-type memristor based 3-neuron Hopfield neural network (HNN) is proposed, which is achieved through substituting one coupling-connection weight with a memristive synaptic weight. It is numerically shown that the memristive HNN has a dynamical transition from chaotic, to periodic, and further to stable point behaviors with the variations of the memristor inner parameter, implying the stabilization effect of the hyperbolic-type memristor on the chaotic HNN. Of particular interest, it should be highly stressed that for different memristor inner parameters, different coexisting behaviors of asymmetric attractors are emerged under different initial conditions, leading to the existence of multistable oscillation states in the memristive HNN. Furthermore, by using commercial discrete components, a nonlinear circuit is designed and PSPICE circuit simulations and hardware experiments are performed. The results simulated and captured from the realization circuit are consistent with numerical simulations, which well verify the facticity of coexisting asymmetric attractors' behaviors.

Structure Function Revisited: A Simple Tool for Complex Analysis of Neuronal Activity

Neural systems are characterized by their complex dynamics, reflected on signals produced by neurons and neuronal ensembles. This complexity exhibits specific features in health, disease and in different states of consciousness, and can be considered a hallmark of certain neurologic and neuropsychiatric conditions. To measure complexity from neurophysiologic signals, a number of different nonlinear tools of analysis are available. However, not all of these tools are easy to implement, or able to handle clinical data, often obtained in less than ideal conditions in comparison to laboratory or simulated data. Recently, the temporal structure function emerged as a powerful tool for the analysis of complex properties of neuronal activity. The temporal structure function is efficient computationally and it can be robustly estimated from short signals. However, the application of this tool to neuronal data is relatively new, making the interpretation of results difficult. In this methods paper we describe a step by step algorithm for the calculation and characterization of the structure function. We apply this algorithm to oscillatory, random and complex toy signals, and test the effect of added noise. We show that: (1) the mean slope of the structure function is zero in the case of random signals; (2) oscillations are reflected on the shape of the structure function, but they don't modify the mean slope if complex correlations are absent; (3) nonlinear systems produce structure functions with nonzero slope up to a critical point, where the function turns into a plateau. Two characteristic numbers can be extracted to quantify the behavior of the structure function in the case of nonlinear systems: (1). the point where the plateau starts (the inflection point, where the slope change occurs), and (2). the height of the plateau. While the inflection point is related to the scale where correlations weaken, the height of the plateau is related to the noise present in the signal. To exemplify our method we calculate structure functions of neuronal recordings from the basal ganglia of parkinsonian and healthy rats, and draw guidelines for their interpretation in light of the results obtained from our toy signals.

EEG analysis of seizure patterns using visibility graphs for detection of generalized seizures

BACKGROUND

The traditional EEG features in the time and frequency domain show limited seizure detection performance in the epileptic population with intellectual disability (ID). In addition, the influence of EEG seizure patterns on detection performance was less studied.

NEW METHOD

A single-channel EEG signal can be mapped into visibility graphs (VGS), including basic visibility graph (VG), horizontal VG (HVG), and difference VG (DVG). These graphs were used to characterize different EEG seizure patterns. To demonstrate its effectiveness in identifying EEG seizure patterns and detecting generalized seizures, EEG recordings of 615h on one EEG channel from 29 epileptic patients with ID were analyzed.

RESULTS

A novel feature set with discriminative power for seizure detection was obtained by using the VGS method. The degree distributions (DDs) of DVG can clearly distinguish EEG of each seizure pattern. The degree entropy and power-law degree power in DVG were proposed here for the first time, and they show significant difference between seizure and non-seizure EEG. The connecting structure measured by HVG can better distinguish seizure EEG from background than those by VG and DVG.

COMPARISON WITH EXISTING METHOD

A traditional EEG feature set based on frequency analysis was used here as a benchmark feature set. With a support vector machine (SVM) classifier, the seizure detection performance of the benchmark feature set (sensitivity of 24%, FD_t/h of 1.8s) can be improved by combining our proposed VGS features extracted from one EEG channel (sensitivity of 38%, FD_t/h of 1.4s).

CONCLUSIONS

The proposed VGS-based features can help improve seizure detection for ID patients.

Detection of nonstationary transition to synchronized states of a neural network using recurrence analyses

We study the stability of asymptotic states displayed by a complex neural network. We focus on the loss of stability of a stationary state of networks using recurrence quantifiers as tools to diagnose local and global stabilities as well as the multistability of a coupled neural network. Numerical simulations of a neural network composed of 1024 neurons in a small-world connection scheme are performed using the model of Braun et al. [Int. J. Bifurcation Chaos 08, 881 (1998)IJBEE40218-127410.1142/S0218127498000681], which is a modified model from the Hodgkin-Huxley model [J. Phys. 117, 500 (1952)]. To validate the analyses, the results are compared with those produced by Kuramoto's order parameter [Chemical Oscillations, Waves, and Turbulence (Springer-Verlag, Berlin Heidelberg, 1984)]. We show that recurrence tools making use of just integrated signals provided by the networks, such as local field potential (LFP) (LFP signals) or mean field values bring new results on the understanding of neural behavior occurring before the synchronization states. In particular we show the occurrence of different stationary and nonstationarity asymptotic states.

The stochastic nature of action potential backpropagation in apical tuft dendrites

In cortical pyramidal neurons, backpropagating action potentials (bAPs) supply Ca²⁺ to synaptic contacts on dendrites. To determine whether the efficacy of AP backpropagation into apical tuft dendrites is stable over time, we performed dendritic Ca²⁺ and voltage imaging in rat brain slices. We found that the amplitude of bAP-Ca²⁺ in apical tuft branches was unstable, given that it varied from trial to trial (termed "bAP-Ca²⁺ flickering"). Small perturbations in dendritic physiology, such as spontaneous synaptic inputs, channel inactivation, or temperature-induced changes in channel kinetics, can cause bAP flickering. In the tuft branches, the density of Na⁺ and K⁺ channels was sufficient to support local initiation of fast spikelets by glutamate iontophoresis. We quantified the time delay between the somatic AP burst and the peak of dendritic Ca²⁺ transient in the apical tuft, because this delay is important for induction of spike-timing dependent plasticity. Depending on the frequency of the somatic AP triplets, Ca²⁺ signals peaked in the apical tuft 20-50 ms after the 1st AP in the soma. Interestingly, at low frequency (<20 Hz), the Ca²⁺ peaked sooner than at high frequency, because only the 1st AP invaded tuft. Activation of dendritic voltage-gated Ca²⁺ channels is sensitive to the duration of the dendritic voltage transient. In apical tuft branches, small changes in the duration of bAP voltage waveforms cause disproportionately large increases in dendritic Ca²⁺ influx (bAP-Ca²⁺ flickering). The stochastic nature of bAP-Ca²⁺ adds a new perspective on the mechanisms by which pyramidal neurons combine inputs arriving at different cortical layers.NEW & NOTEWORTHY The bAP-Ca²⁺ signal amplitudes in some apical tuft branches randomly vary from moment to moment. In repetitive measurements, successful AP invasions are followed by complete failures. Passive spread of voltage from the apical trunk into the tuft occasionally reaches the threshold for local Na⁺ spike, resulting in stronger Ca²⁺ influx. During a burst of three somatic APs, the peak of dendritic Ca²⁺ in the apical tuft occurs with a delay of 20-50 ms depending on AP frequency.

Emergence of coupling-induced oscillations and broken symmetries in heterogeneously driven nonlinear reaction networks

Many natural systems including the brain comprise coupled elements that are stimulated non-uniformly. In this paper we show that heterogeneously driven networks of excitatory-inhibitory units exhibit a diverse range of collective phenomena, including the appearance of spontaneous oscillations upon coupling quiescent elements. On varying the coupling strength a previously unreported transition is seen wherein the symmetries of the synchronization patterns in the stimulated and unstimulated groups undergo mutual exchange. The system also exhibits coexisting chaotic and non-chaotic attractors - a result that may be of interest in connection to earlier reports of varying degrees of chaoticity in the brain.

Hyperpolarization-Activated Current Induces Period-Doubling Cascades and Chaos in a Cold Thermoreceptor Model

In this article, we describe and analyze the chaotic behavior of a conductance-based neuronal bursting model. This is a model with a reduced number of variables, yet it retains biophysical plausibility. Inspired by the activity of cold thermoreceptors, the model contains a persistent Sodium current, a Calcium-activated Potassium current and a hyperpolarization-activated current (I_h) that drive a slow subthreshold oscillation. Driven by this oscillation, a fast subsystem (fast Sodium and Potassium currents) fires action potentials in a periodic fashion. Depending on the parameters, this model can generate a variety of firing patterns that includes bursting, regular tonic and polymodal firing. Here we show that the transitions between different firing patterns are often accompanied by a range of chaotic firing, as suggested by an irregular, non-periodic firing pattern. To confirm this, we measure the maximum Lyapunov exponent of the voltage trajectories, and the Lyapunov exponent and Lempel-Ziv's complexity of the ISI time series. The four-variable slow system (without spiking) also generates chaotic behavior, and bifurcation analysis shows that this is often originated by period doubling cascades. Either with or without spikes, chaos is no longer generated when the I_h is removed from the system. As the model is biologically plausible with biophysically meaningful parameters, we propose it as a useful tool to understand chaotic dynamics in neurons.

Dissociation, Epileptiform Discharges and Chaos in the Brain: Toward a Neuroscientific Theory of Dissociation

Dissociated states represent pathological conditions when psychological trauma may emerge in a variety of forms such as psychic dissociative symptoms or, on the contrary, as paroxysms or other somatoform symptoms. There is evidence that epileptic activity plays an important role in the generation of dissociative states and it is able to generate various psychopathological processes as well as a wide spectrum of somatic symptoms or seizures. For the explanation of these connections between dissociative states and epileptic discharges the author proposes a neuroscientific model of dissociation based on the theory of competitive neural assemblies which can lead to chaotic self-organization in brain neural networks. This model is suggested as an integrative view interconnecting the various psychopathological and somatoform manifestations of dissociative states and suggests further possibilities for future research regarding common pathogenic mechanisms among epilepsy and mental disorders.

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.

Chaos and Hyperchaos in a Model of Ribosome Autocatalytic Synthesis

Any vital activities of the cell are based on the ribosomes, which not only provide the basic machinery for the synthesis of all proteins necessary for cell functioning during growth and division, but for biogenesis itself. From this point of view, ribosomes are self-replicating and autocatalytic structures. In current work we present an elementary model in which the autocatalytic synthesis of ribosomal RNA and proteins, as well as enzymes ensuring their degradation are described with two monotonically increasing functions. For certain parameter values, the model, consisting of one differential equation with delayed argument, demonstrates both stationary and oscillatory dynamics of the ribosomal protein synthesis, which can be chaotic and hyperchaotic dependent on the value of the delayed argument. The biological interpretation of the modeling results and parameter estimation suggest the feasibility of chaotic dynamics in molecular genetic systems of eukaryotes, which depends only on the internal characteristics of functioning of the translation system.

New insights into sucking, swallowing and breathing central generators: A complexity analysis of rhythmic motor behaviors

Sucking, swallowing and breathing are dynamic motor behaviors. Breathing displays features of chaos-like dynamics, in particular nonlinearity and complexity, which take their source in the automatic command of breathing. In contrast, buccal/gill ventilation in amphibians is one of the rare motor behaviors that do not display nonlinear complexity. This study aimed at assessing whether sucking and swallowing would also follow nonlinear complex dynamics in the newborn lamb. Breathing movements were recorded before, during and after bottle-feeding. Sucking pressure and the integrated EMG of the thyroartenoid muscle, as an index of swallowing, were recorded during bottle-feeding. Nonlinear complexity of the whole signals was assessed through the calculation of the noise limit value (NL). Breathing and swallowing always exhibited chaos-like dynamics. The NL of breathing did not change significantly before, during or after bottle-feeding. On the other hand, sucking inconsistently and significantly less frequently than breathing exhibited a chaos-like dynamics. Therefore, the central pattern generator (CPG) that drives sucking may be functionally different from the breathing CPG. Furthermore, the analogy between buccal/gill ventilation and sucking suggests that the latter may take its phylogenetic origin in the gill ventilation CPG of the common ancestor of extant amphibians and mammals.

Chaos and hyperchaos in simple gene network with negative feedback and time delays

Today there are examples that prove the existence of chaotic dynamics at all levels of organization of living systems, except intracellular, although such a possibility has been theoretically predicted. The lack of experimental evidence of chaos generation at the intracellular level in vivo may indicate that during evolution the cell got rid of chaos. This work allows the hypothesis that one of the possible mechanisms for avoiding chaos in gene networks can be a negative evolutionary selection, which prevents fixation or realization of regulatory circuits, creating too mild, from the biological point of view, conditions for the emergence of chaos. It has been shown that one of such circuits may be a combination of negative autoregulation of expression of transcription factors at the level of their synthesis and degradation. The presence of such a circuit results in formation of multiple branches of chaotic solutions as well as formation of hyperchaos with equal and sufficiently low values of the delayed argument that can be implemented not only in eukaryotic, but in prokaryotic cells.

A Linear Analysis of Coupled Wilson-Cowan Neuronal Populations

Let a neuronal population be composed of an excitatory group interconnected to an inhibitory group. In the Wilson-Cowan model, the activity of each group of neurons is described by a first-order nonlinear differential equation. The source of the nonlinearity is the interaction between these two groups, which is represented by a sigmoidal function. Such a nonlinearity makes difficult theoretical works. Here, we analytically investigate the dynamics of a pair of coupled populations described by the Wilson-Cowan model by using a linear approximation. The analytical results are compared to numerical simulations, which show that the trajectories of this fourth-order dynamical system can converge to an equilibrium point, a limit cycle, a two-dimensional torus, or a chaotic attractor. The relevance of this study is discussed from a biological perspective.

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.

Multiscale modeling of brain dynamics: from single neurons and networks to mathematical tools

The extreme complexity of the brain naturally requires mathematical modeling approaches on a large variety of scales; the spectrum ranges from single neuron dynamics over the behavior of groups of neurons to neuronal network activity. Thus, the connection between the microscopic scale (single neuron activity) to macroscopic behavior (emergent behavior of the collective dynamics) and vice versa is a key to understand the brain in its complexity. In this work, we attempt a review of a wide range of approaches, ranging from the modeling of single neuron dynamics to machine learning. The models include biophysical as well as data-driven phenomenological models. The discussed models include Hodgkin-Huxley, FitzHugh-Nagumo, coupled oscillators (Kuramoto oscillators, Rössler oscillators, and the Hindmarsh-Rose neuron), Integrate and Fire, networks of neurons, and neural field equations. In addition to the mathematical models, important mathematical methods in multiscale modeling and reconstruction of the causal connectivity are sketched. The methods include linear and nonlinear tools from statistics, data analysis, and time series analysis up to differential equations, dynamical systems, and bifurcation theory, including Granger causal connectivity analysis, phase synchronization connectivity analysis, principal component analysis (PCA), independent component analysis (ICA), and manifold learning algorithms such as ISOMAP, and diffusion maps and equation-free techniques. WIREs Syst Biol Med 2016, 8:438-458. doi: 10.1002/wsbm.1348 For further resources related to this article, please visit the WIREs website.

Big data to smart data in Alzheimer's disease: The brain health modeling initiative to foster actionable knowledge

Massive investment and technological advances in the collection of extensive and longitudinal information on thousands of Alzheimer patients results in large amounts of data. These "big-data" databases can potentially advance CNS research and drug development. However, although necessary, they are not sufficient, and we posit that they must be matched with analytical methods that go beyond retrospective data-driven associations with various clinical phenotypes. Although these empirically derived associations can generate novel and useful hypotheses, they need to be organically integrated in a quantitative understanding of the pathology that can be actionable for drug discovery and development. We argue that mechanism-based modeling and simulation approaches, where existing domain knowledge is formally integrated using complexity science and quantitative systems pharmacology can be combined with data-driven analytics to generate predictive actionable knowledge for drug discovery programs, target validation, and optimization of clinical development.

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.

Visually Evoked Spiking Evolves While Spontaneous Ongoing Dynamics Persist

Neurons in the primary visual cortex spontaneously spike even when there are no visual stimuli. It is unknown whether the spiking evoked by visual stimuli is just a modification of the spontaneous ongoing cortical spiking dynamics or whether the spontaneous spiking state disappears and is replaced by evoked spiking. This study of laminar recordings of spontaneous spiking and visually evoked spiking of neurons in the ferret primary visual cortex shows that the spiking dynamics does not change: the spontaneous spiking as well as evoked spiking is controlled by a stable and persisting fixed point attractor. Its existence guarantees that evoked spiking return to the spontaneous state. However, the spontaneous ongoing spiking state and the visual evoked spiking states are qualitatively different and are separated by a threshold (separatrix). The functional advantage of this organization is that it avoids the need for a system reorganization following visual stimulation, and impedes the transition of spontaneous spiking to evoked spiking and the propagation of spontaneous spiking from layer 4 to layers 2-3.

Analysis of Chaotic Resonance in Izhikevich Neuron Model

In stochastic resonance (SR), the presence of noise helps a nonlinear system amplify a weak (sub-threshold) signal. Chaotic resonance (CR) is a phenomenon similar to SR but without stochastic noise, which has been observed in neural systems. However, no study to date has investigated and compared the characteristics and performance of the signal responses of a spiking neural system in some chaotic states in CR. In this paper, we focus on the Izhikevich neuron model, which can reproduce major spike patterns that have been experimentally observed. We examine and classify the chaotic characteristics of this model by using Lyapunov exponents with a saltation matrix and Poincaré section methods in order to address the measurement challenge posed by the state-dependent jump in the resetting process. We found the existence of two distinctive states, a chaotic state involving primarily turbulent movement and an intermittent chaotic state. In order to assess the signal responses of CR in these classified states, we introduced an extended Izhikevich neuron model by considering weak periodic signals, and defined the cycle histogram of neuron spikes as well as the corresponding mutual correlation and information. Through computer simulations, we confirmed that both chaotic states in CR can sensitively respond to weak signals. Moreover, we found that the intermittent chaotic state exhibited a prompter response than the chaotic state with primarily turbulent movement.

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.

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.