EEG functional connectivity as a Riemannian mediator: An application to malnutrition and cognition.

Mediation analysis assesses whether an exposure directly produces changes in cognitive behavior or is influenced by intermediate "mediators". Electroencephalographic (EEG) spectral measurements have been previously used as effective mediators representing diverse aspects of brain function. However, it has been necessary to collapse EEG measures onto a single scalar using standard mediation methods. In this article, we overcome this limitation and examine EEG frequency-resolved functional connectivity measures as a mediator using the full EEG cross-spectral tensor (CST). Since CST samples do not exist in Euclidean space but in the Riemannian manifold of positive-definite tensors, we transform the problem, allowing for the use of classic multivariate statistics. Toward this end, we map the data from the original manifold space to the Euclidean tangent space, eliminating redundant information to conform to a "compressed CST." The resulting object is a matrix with rows corresponding to frequencies and columns to cross spectra between channels. We have developed a novel matrix mediation approach that leverages a nuclear norm regularization to determine the matrix-valued regression parameters. Furthermore, we introduced a global test for the overall CST mediation and a test to determine specific channels and frequencies driving the mediation. We validated the method through simulations and applied it to our well-studied 50+-year Barbados Nutrition Study dataset by comparing EEGs collected in school-age children (5-11 years) who were malnourished in the first year of life with those of healthy classmate controls. We hypothesized that the CST mediates the effect of malnutrition on cognitive performance. We can now explicitly pinpoint the frequencies (delta, theta, alpha, and beta bands) and regions (frontal, central, and occipital) in which functional connectivity was altered in previously malnourished children, an improvement to prior studies. Understanding the specific networks impacted by a history of postnatal malnutrition could pave the way for developing more targeted and personalized therapeutic interventions. Our methods offer a versatile framework applicable to mediation studies encompassing matrix and Hermitian 3D tensor mediators alongside scalar exposures and outcomes, facilitating comprehensive analyses across diverse research domains.

Accurate and Efficient Simulation of Very High-Dimensional Neural Mass Models with Distributed-Delay Connectome Tensors.

This paper introduces methods and a novel toolbox that efficiently integrates high-dimensional Neural Mass Models (NMMs) specified by two essential components. The first is the set of nonlinear Random Differential Equations (RDEs) of the dynamics of each neural mass. The second is the highly sparse three-dimensional Connectome Tensor (CT) that encodes the strength of the connections and the delays of information transfer along the axons of each connection. To date, simplistic assumptions prevail about delays in the CT, often assumed to be Dirac-delta functions. In reality, delays are distributed due to heterogeneous conduction velocities of the axons connecting neural masses. These distributed-delay CTs are challenging to model. Our approach implements these models by leveraging several innovations. Semi-analytical integration of RDEs is done with the Local Linearization (LL) scheme for each neural mass, ensuring dynamical fidelity to the original continuous-time nonlinear dynamic. This semi-analytic LL integration is highly computationally-efficient. In addition, a tensor representation of the CT facilitates parallel computation. It also seamlessly allows modeling distributed delays CT with any level of complexity or realism. This ease of implementation includes models with distributed-delay CTs. Consequently, our algorithm scales linearly with the number of neural masses and the number of equations they are represented with, contrasting with more traditional methods that scale quadratically at best. To illustrate the toolbox's usefulness, we simulate a single Zetterberg-Jansen and Rit (ZJR) cortical column, a single thalmo-cortical unit, and a toy example comprising 1000 interconnected ZJR columns. These simulations demonstrate the consequences of modifying the CT, especially by introducing distributed delays. The examples illustrate the complexity of explaining EEG oscillations, e.g., split alpha peaks, since they only appear for distinct neural masses. We provide an open-source Script for the toolbox.

Optimal placement of sensor and actuator for controlling low-dimensional chaotic systems based on global modeling.

Controlling chaos is fundamental in many applications, and for this reason, many techniques have been proposed to address this problem. Here, we propose a strategy based on an optimal placement of the sensor and actuator providing global observability of the state space and global controllability to any desired state. The first of these two conditions enables the derivation of a model of the system by using a global modeling technique. In turn, this permits the use of feedback linearization for designing the control law based on the equations of the obtained model and providing a zero-flat system. The procedure is applied to three case studies, including two piecewise linear circuits, namely, the Carroll circuit and the Chua circuit whose governing equations are approximated by a continuous global model. The sensitivity of the procedure to the time constant of the dynamics is also discussed.

Scale-resolved analysis of brain functional connectivity networks with spectral entropy.

Functional connectivity is derived from inter-regional correlations in spontaneous fluctuations of brain activity, and can be represented in terms of complete graphs with continuous (real-valued) edges. The structure of functional connectivity networks is strongly affected by signal processing procedures to remove the effects of motion, physiological noise and other sources of experimental error. However, in the absence of an established ground truth, it is difficult to determine the optimal procedure, and no consensus has been reached on the most effective approach to remove nuisance signals without unduly affecting the network intrinsic structural features. Here, we use a novel information-theoretic approach, based on von Neumann entropy, which provides a measure of information encoded in the networks at different scales. We also define a measure of distance between networks, based on information divergence, and optimal null models appropriate for the description of functional connectivity networks, to test for the presence of nontrivial structural patterns that are not the result of simple local constraints. This formalism enables a scale-resolved analysis of the distance between a functional connectivity network and its maximally random counterpart, thus providing a means to assess the effects of noise and image processing on network structure. We apply this novel approach to address a few open questions in the analysis of brain functional connectivity networks. Specifically, we demonstrate a strongly beneficial effect of network sparsification by removal of the weakest links, and the existence of an optimal threshold that maximizes the ability to extract information on large-scale network structures. Additionally, we investigate the effects of different degrees of motion at different scales, and compare the most popular processing pipelines designed to mitigate its deleterious effect on functional connectivity networks. We show that network sparsification, in combination with motion correction algorithms, dramatically improves detection of large scale network structure.

Visual behaviors in disorders of consciousness: Disentangling conscious visual processing by a multimodal approach.

One of the major challenges for clinicians who treat patients with Disorders of Consciousness (DoCs) concerns the detection of signs of consciousness that distinguish patients in Vegetative State from those in Minimally Conscious State. Recent studies showed how visual responses to tailored stimuli are one of the first evidence revealing that one patient is changing from one state to another. This study aimed to explore the integrity of the neural structures being part of the visual system in patients with DoCs manifesting a reflexive behavior (visual blink) and in those manifesting a cognitively and cortically mediated behavior (visual pursuit). We collected instrumental data using specialized equipment (EEG following the rules of the International 10-20 system, 3T Magnetic Resonance, and Positron Emission Tomography) in 54 DoC patients. Our results indicated that visual pursuit group showed a better fVEPs response than the visual blink group, because of a greater area under the N2/P2 component of fVEPs (AUC could be seen as an indicator of the residual activity of visual areas). Considering neuroimaging data, the main structural differences between groups were found in the retrochiasmatic areas, specifically in the right optic radiation and visual cortex (V1), areas statistically less impaired in patients able to perform a visual pursuit. FDG-PET analysis confirmed difference between groups at the level of the right calcarine cortex and neighboring right lingual gyrus. In conclusion, although there are methodological and theoretical limitations that should be considered, our study suggests a new perspective to consider for a future diagnostic protocol.

Generic Rotating-Frame-Based Approach to Chaos Generation in Nonlinear Micro- and Nanoelectromechanical System Resonators.

This Letter provides a low-power method for chaos generation that is generally applicable to nonlinear micro- and nanoelectromechanical systems (MNEMS) resonators. The approach taken is independent of the material, scale, design, and actuation of the device in question; it simply assumes a good quality factor and a Duffing type nonlinearity, features that are commonplace to MNEMS resonators. The approach models the rotating-frame dynamics to analytically constrain the parameter space required for chaos generation. By leveraging these common properties of MNEMS devices, a period-doubling route to chaos is generated using smaller forcing than typically reported in the literature.

SpiSeMe: A multi-language package for spike train surrogate generation.

Many studies in nonlinear science heavily rely on surrogate-based hypothesis testing to provide significance estimations of analysis results. Among the complex data produced by nonlinear systems, spike trains are a class of sequences requiring algorithms for surrogate generation that are typically more sophisticated and computationally demanding than methods developed for continuous signals. Although algorithms to specifically generate surrogate spike trains exist, the availability of open-source, portable implementations is still incomplete. In this paper, we introduce the SpiSeMe (Spike Sequence Mime) software package that implements four algorithms for the generation of surrogate data out of spike trains and more generally out of any sequence of discrete events. The purpose of the package is to provide a unified and portable toolbox to carry out surrogate generation on point-process data. Code is provided in three languages, namely, C++, Matlab, and Python, thus allowing straightforward integration of package functions into most analysis pipelines.

Competition of noise and collectivity in global cryptocurrency trading: Route to a self-contained market.

Cross correlations in fluctuations of the daily exchange rates within the basket of the 100 highest-capitalization cryptocurrencies over the period October 1, 2015-March 31, 2019 are studied. The corresponding dynamics predominantly involve one leading eigenvalue of the correlation matrix, while the others largely coincide with those of Wishart random matrices. However, the magnitude of the principal eigenvalue, and thus the degree of collectivity, strongly depends on which cryptocurrency is used as a base. It is largest when the base is the most peripheral cryptocurrency; when more significant ones are taken into consideration, its magnitude systematically decreases, nevertheless preserving a sizable gap with respect to the random bulk, which in turn indicates that the organization of correlations becomes more heterogeneous. This finding provides a criterion for recognizing which currencies or cryptocurrencies play a dominant role in the global cryptomarket. The present study shows that over the period under consideration, the Bitcoin (BTC) predominates, hallmarking exchange rate dynamics at least as influential as the U.S. dollar (USD). Even more, the BTC started dominating around the year 2017, while other cryptocurrencies, such as the Ethereum and even Ripple, assumed similar trends. At the same time, the USD, an original value determinant for the cryptocurrency market, became increasingly disconnected, and its related characteristics eventually started approaching those of a fictitious currency. These results are strong indicators of incipient independence of the global cryptocurrency market, delineating a self-contained trade resembling the Forex.

Generation of diverse insect-like gait patterns using networks of coupled Rössler systems.

The generation of walking patterns is central to bio-inspired robotics and has been attained using methods encompassing diverse numerical as well as analog implementations. Here, we demonstrate the possibility of synthesizing viable gaits using a paradigmatic low-dimensional non-linear entity, namely, the Rössler system, as a dynamical unit. Through a minimalistic network wherein each instance is univocally associated with one leg, it is possible to readily reproduce the canonical gaits as well as generate new ones via changing the coupling scheme and the associated delays. Varying levels of irregularity can be introduced by rendering individual systems or the entire network chaotic. Moreover, through tailored mapping of the state variables to physical angles, adequate leg trajectories can be accessed directly from the coupled systems. The functionality of the resulting generator was confirmed in laboratory experiments by means of an instrumented six-legged ant-like robot. Owing to their simple form, the 18 coupled equations could be rapidly integrated on a bare-metal microcontroller, leading to the demonstration of real-time robot control navigating an arena using a brain-machine interface.

Generation of surrogate event sequences via joint distribution of successive inter-event intervals.

The study of many dynamical systems relies on the analysis of experimentally-recorded sequences of events for which information is encoded in the sequence of interevent intervals. A correct interpretation of the results of the application of analytical techniques to these sequences requires the assessment of statistical significance. In most cases, the corresponding null-hypothesis distribution is unknown, thus forbidding an evaluation of the significance. An alternative solution, which is efficient in the case of continuous signals, is provided by the generation of surrogate data that share statistical and spectral properties with the original dataset. However, in the case of event sequences, the available algorithms for the generation of surrogate data can become cumbersome and computationally demanding. In this work, we present a new method for the generation of surrogate event sequences that relies on the joint distribution of successive interevent intervals. Our method, which was tested on both synthetic and experimental sequences, performs equally well or even better than conventional methods in terms of interevent interval distribution and autocorrelation while abating the computational time by at least one order of magnitude.

Warped phase coherence: An empirical synchronization measure combining phase and amplitude information.

The entrainment between weakly coupled nonlinear oscillators, as well as between complex signals such as those representing physiological activity, is frequently assessed in terms of whether a stable relationship is detectable between the instantaneous phases extracted from the measured or simulated time-series via the analytic signal. Here, we demonstrate that adding a possibly complex constant value to this normally null-mean signal has a non-trivial warping effect. Among other consequences, this introduces a level of sensitivity to the amplitude fluctuations and average relative phase. By means of simulations of Rössler systems and experiments on single-transistor oscillator networks, it is shown that the resulting coherence measure may have an empirical value in improving the inference of the structural couplings from the dynamics. When tentatively applied to the electroencephalogram recorded while performing imaginary and real movements, this straightforward modification of the phase locking value substantially improved the classification accuracy. Hence, its possible practical relevance in brain-computer and brain-machine interfaces deserves consideration.

Method for retrospective estimation of natural head movement during structural MRI.

BACKGROUND: Head motion during brain structural MRI scans biases brain morphometry measurements but quantitative retrospective methods estimating head motion from structural MRI have not been evaluated. PURPOSE: To verify the hypothesis that two metrics retrospectively computed from MR images: 1) average edge strength (AES, reduced with image blurring) and 2) entropy (ENT, increased with blurring and ringing artifacts) could be sensitive to in-scanner head motion during acquisition of T1 -weighted MR images. STUDY TYPE: Retrospective. POPULATION/SUBJECTS/PHANTOM/SPECIMEN/ANIMAL MODEL: In all, 83 healthy control (HC) and 120 Parkinson's disease (PD) patients. FIELD STRENGTH/SEQUENCE: 3D magnetization-prepared rapid gradient-echo (MPRAGE) images at 3T. ASSESSMENT: We 1) compared AES and ENT distribution between HC and PD; 2) evaluated the correlation between tremor score (TS) and AES (or ENT) in PD; and 3) investigated cortical regions showing an association between AES (or ENT) and local and network-level covariance measures of cortical thickness (CT), gray to white matter contrast (GWC) and gray matter density maps (GMx). STATISTICAL TESTS: 1) Student's t-test. 2) Spearman's rank correlation. 3) General linear model and partial least square analysis. RESULTS: AES, but not ENT, differentiated HC and PD (P = 0.02, HC median AES = 39.8, interquartile range = 9.8, PD median AES = 37.6, interquartile range = 8.1). In PD, AES correlated negatively with TS (ρ = -0.21, P = 0.02) and showed a significant relationship (|Z| >3, P < 0.001) with structural covariance of CT and GWC in 54 out of 68 cortical regions. DATA CONCLUSION: In clinical populations prone to head motion, AES can provide a reliable retrospective index of motion during structural scans, identifying brain areas whose morphometric measures covary with motion. LEVEL OF EVIDENCE: 3 Technical Efficacy: Stage 2 J. Magn. Reson. Imaging 2018;48:927-937.

Apparent remote synchronization of amplitudes: A demodulation and interference effect.

A form of "remote synchronization" was recently described, wherein amplitude fluctuations across a ring of non-identical, non-linear electronic oscillators become entrained into spatially-structured patterns. According to linear models and mutual information, synchronization and causality dip at a certain distance, then recover before eventually fading. Here, the underlying mechanism is finally elucidated through novel experiments and simulations. The system non-linearity is found to have a dual role: it supports chaotic dynamics, and it enables the energy exchange between the lower and higher sidebands of a predominant frequency. This frequency acts as carrier signal in an arrangement resembling standard amplitude modulation, wherein the lower sideband and the demodulated baseband signals spectrally overlap. Due to a spatially-dependent phase relationship, at a certain distance near-complete destructive interference occurs between them, causing the observed dip. Methods suitable for detecting non-trivial entrainment, such as transfer entropy and the auxiliary system approach, nevertheless, reveal that synchronization and causality actually decrease with distance monotonically. Remoteness is, therefore, arguably only apparent, as also reflected in the propagation of external perturbations. These results demonstrate a complex mechanism of dynamical interdependence, and exemplify how it can lead to incorrectly inferring synchronization and causality.

High-dimensional dynamics in a single-transistor oscillator containing Feynman-Sierpinski resonators: Effect of fractal depth and irregularity.

Fractal structures pervade nature and are receiving increasing engineering attention towards the realization of broadband resonators and antennas. We show that fractal resonators can support the emergence of high-dimensional chaotic dynamics even in the context of an elementary, single-transistor oscillator circuit. Sierpinski gaskets of variable depth are constructed using discrete capacitors and inductors, whose values are scaled according to a simple sequence. It is found that in regular fractals of this kind, each iteration effectively adds a conjugate pole/zero pair, yielding gradually more complex and broader frequency responses, which can also be implemented as much smaller Foster equivalent networks. The resonators are instanced in the circuit as one-port devices, replacing the inductors found in the initial version of the oscillator. By means of a highly simplified numerical model, it is shown that increasing the fractal depth elevates the dimension of the chaotic dynamics, leading to high-order hyperchaos. This result is overall confirmed by SPICE simulations and experiments, which however also reveal that the non-ideal behavior of physical components hinders obtaining high-dimensional dynamics. The issue could be practically mitigated by building the Foster equivalent networks rather than the verbatim fractals. Furthermore, it is shown that considerably more complex resonances, and consequently richer dynamics, can be obtained by rendering the fractal resonators irregular through reshuffling the inductors, or even by inserting a limited number of focal imperfections. The present results draw attention to the potential usefulness of fractal resonators for generating high-dimensional chaotic dynamics, and underline the importance of irregularities and component non-idealities.

Bitcoin market route to maturity? Evidence from return fluctuations, temporal correlations and multiscaling effects.

Based on 1-min price changes recorded since year 2012, the fluctuation properties of the rapidly emerging Bitcoin market are assessed over chosen sub-periods, in terms of return distributions, volatility autocorrelation, Hurst exponents, and multiscaling effects. The findings are compared to the stylized facts of mature world markets. While early trading was affected by system-specific irregularities, it is found that over the months preceding April 2018 all these statistical indicators approach the features hallmarking maturity. This can be taken as an indication that the Bitcoin market, and possibly other cryptocurrencies, carry concrete potential of imminently becoming a regular market, alternative to the foreign exchange. Since high-frequency price data are available since the beginning of trading, the Bitcoin offers a unique window into the statistical characteristics of a market maturation trajectory.

Evidence of altered pressure pain thresholds in persons with disorders of consciousness as measured by the Nociception Coma Scale-Italian version.

Pain assessment in patients with disorders of consciousness (DoC) is a controversial issue for clinicians, who require tools and standardised procedures for testing nociception in non-communicative patients. The aims of the present study were, first, to analyse the psychometric properties of the Italian version of the Nociception Coma Scale and, second, to evaluate pressure pain thresholds in a group of patients with DoC. The authors conducted a multi-centre study on 40 healthy participants and 60 DoC patients enrolled from six hospitals in Italy. For each group an electronic algometer was used to apply all nociceptive pressure stimuli. Our results show that the Italian version of the NCS retains the good psychometric properties of the original version and is therefore suitable for standardised pain assessment in clinical practice. In our study, pressure pain thresholds measured in a group of patients in vegetative and minimally conscious state were relatively lower than pain threshold values found in a group of healthy participants. Such findings motivate additional investigation on possible pain sensitisation in patients with severe brain injury and multiple co-morbidities, and on application of tailored therapeutic approaches useful for pain management in patients unable verbally to communicate their feelings.

Free water elimination improves test-retest reproducibility of diffusion tensor imaging indices in the brain: A longitudinal multisite study of healthy elderly subjects.

Free water elimination (FWE) in brain diffusion MRI has been shown to improve tissue specificity in human white matter characterization both in health and in disease. Relative to the classical diffusion tensor imaging (DTI) model, FWE is also expected to increase sensitivity to microstructural changes in longitudinal studies. However, it is not clear if these two models differ in their test-retest reproducibility. This study compares a bi-tensor model for FWE with DTI by extending a previous longitudinal-reproducibility 3T multisite study (10 sites, 7 different scanner models) of 50 healthy elderly participants (55-80 years old) scanned in two sessions at least 1 week apart. We computed the reproducibility of commonly used DTI metrics (FA: fractional anisotropy, MD: mean diffusivity, RD: radial diffusivity, and AXD: axial diffusivity), derived either using a DTI model or a FWE model. The DTI metrics were evaluated over 48 white-matter regions of the JHU-ICBM-DTI-81 white-matter labels atlas, and reproducibility errors were assessed. We found that relative to the DTI model, FWE significantly reduced reproducibility errors in most areas tested. In particular, for the FA and MD metrics, there was an average reduction of approximately 1% in the reproducibility error. The reproducibility scores did not significantly differ across sites. This study shows that FWE improves sensitivity and is thus promising for clinical applications, with the potential to identify more subtle changes. The increased reproducibility allows for smaller sample size or shorter trials in studies evaluating biomarkers of disease progression or treatment effects. Hum Brain Mapp 38:12-26, 2017. © 2016 Wiley Periodicals, Inc.

Multimodal study of default-mode network integrity in disorders of consciousness.

OBJECTIVE: Understanding residual brain function in disorders of consciousness poses extraordinary challenges, and imaging examinations are needed to complement clinical assessment. The default-mode network (DMN) is known to be dysfunctional, although correlation with level of consciousness remains controversial. We investigated DMN activity with resting-state functional magnetic resonance imaging (rs-fMRI), alongside its structural and metabolic integrity, aiming to elucidate the corresponding associations with clinical assessment. METHODS: We enrolled 119 consecutive patients: 72 in a vegetative state/unresponsive wakefulness state (VS/UWS), 36 in a minimally conscious state (MCS), and 11 with severe disability. All underwent structural MRI and rs-fMRI, and a subset also underwent 18 F-fluorodeoxyglucose positron emission tomography (FDG-PET). Data were analyzed with manual and automatic approaches, in relation to diagnosis and clinical score. RESULTS: Excluding the quartile with largest head movement, DMN activity was decreased in VS/UWS compared to MCS, and correlated with clinical score. Independent-component and seed-based analyses provided similar results, although the latter and their combination were most informative. Structural MRI and FDG-PET were less sensitive to head movement and had better diagnostic accuracy than rs-fMRI only when all cases were included. rs-fMRI indicated relatively preserved DMN activity in a small subset of VS/UWS patients, 2 of whom evolved to MCS. The integrity of the left hemisphere appears to be predictive of a better clinical status. INTERPRETATION: rs-fMRI of the DMN is sensitive to clinical severity. The effect is consistent across data analysis approaches, but heavily dependent on head movement. rs-fMRI could be informative in detecting residual DMN activity for those patients who remain relatively still during scanning and whose diagnosis is uncertain. Ann Neurol 2016;79:841-853.

A comparative study on assessment procedures and metric properties of two scoring systems of the Coma Recovery Scale-Revised items: standard and modified scores.

OBJECTIVE: The study compared the metric characteristics (discriminant capacity and factorial structure) of two different methods for scoring the items of the Coma Recovery Scale-Revised and it analysed scale scores collected using the standard assessment procedure and a new proposed method. DESIGN: Cross sectional design/methodological study. SETTING: Inpatient, neurological unit. PARTICIPANTS: A total of 153 patients with disorders of consciousness were consecutively enrolled between 2011 and 2013. INTERVENTION: All patients were assessed with the Coma Recovery Scale-Revised using standard (rater 1) and inverted (rater 2) procedures. MAIN OUTCOME MEASURES: Coma Recovery Scale-Revised score, number of cognitive and reflex behaviours and diagnosis. RESULTS: Regarding patient assessment, rater 1 using standard and rater 2 using inverted procedures obtained the same best scores for each subscale of the Coma Recovery Scale-Revised for all patients, so no clinical (and statistical) difference was found between the two procedures. In 11 patients (7.7%), rater 2 noted that some Coma Recovery Scale-Revised codified behavioural responses were not found during assessment, although higher response categories were present. A total of 51 (36%) patients presented the same Coma Recovery Scale-Revised scores of 7 or 8 using a standard score, whereas no overlap was found using the modified score. Unidimensionality was confirmed for both score systems. CONCLUSION: The Coma Recovery Scale Modified Score showed a higher discriminant capacity than the standard score and a monofactorial structure was also supported. The inverted assessment procedure could be a useful evaluation method for the assessment of patients with disorder of consciousness diagnosis.

Atypical transistor-based chaotic oscillators: Design, realization, and diversity.

In this paper, we show that novel autonomous chaotic oscillators based on one or two bipolar junction transistors and a limited number of passive components can be obtained via random search with suitable heuristics. Chaos is a pervasive occurrence in these circuits, particularly after manual adjustment of a variable resistor placed in series with the supply voltage source. Following this approach, 49 unique circuits generating chaotic signals when physically realized were designed, representing the largest collection of circuits of this kind to date. These circuits are atypical as they do not trivially map onto known topologies or variations thereof. They feature diverse spectra and predominantly anti-persistent monofractal dynamics. Notably, we recurrently found a circuit comprising one resistor, one transistor, two inductors, and one capacitor, which generates a range of attractors depending on the parameter values. We also found a circuit yielding an irregular quantized spike-train resembling some aspects of neural discharge and another one generating a double-scroll attractor, which represent the smallest known transistor-based embodiments of these behaviors. Through three representative examples, we additionally show that diffusive coupling of heterogeneous oscillators of this kind may give rise to complex entrainment, such as lag synchronization with directed information transfer and generalized synchronization. The replicability and reproducibility of the experimental findings are good.