Computational Topology Techniques for Characterizing Time-Series Data

Topological data analysis (TDA), while abstract, allows a characterization of time-series data obtained from nonlinear and complex dynamical systems. Though it is surprising that such an abstract measure of structure - counting pieces and holes - could be useful for real-world data, TDA lets us compare different systems, and even do membership testing or change-point detection. However, TDA is computationally expensive and involves a number of free parameters. This complexity can be obviated by coarse-graining, using a construct called the witness complex. The parametric dependence gives rise to the concept of persistent homology: how shape changes with scale. Its results allow us to distinguish time-series data from different systems - e.g., the same note played on different musical instruments.Comment: 12 pages, 6 Figures, 1 Table, The Sixteenth International Symposium on Intelligent Data Analysis (IDA 2017

Time-Frequency Representations as Phase Space Reconstruction in Symbolic Recurrence Structure Analysis

To appear as a book chapter in the Springer series “Contributions to Statistics”Recurrence structures in univariate time series are challenging to detect. We propose a combination of symbolic and recurrence analysis in order to identify recurrence domains in the signal. This method allows to obtain a symbolic representation of the data. Recurrence analysis produces valid results for multidimensional data, however, in the case of univariate time series one should perform phase space reconstruction first. In this chapter, we propose a new method of phase space reconstruction based on signal's time-frequency representation and compare it to the delay embedding method. We argue that the proposed method outperforms the delay embedding reconstruction in the case of oscillatory signals. We also propose to use recurrence complexity as a quantitative feature of a signal. We evaluate our method on synthetic data and show its application to experimental EEG signals

Consequences of converting graded to action potentials upon neural information coding and energy efficiency

Information is encoded in neural circuits using both graded and action potentials, converting between them within single neurons and successive processing layers. This conversion is accompanied by information loss and a drop in energy efficiency. We investigate the biophysical causes of this loss of information and efficiency by comparing spiking neuron models, containing stochastic voltage-gated Na+ and K+ channels, with generator potential and graded potential models lacking voltage-gated Na+ channels. We identify three causes of information loss in the generator potential that are the by-product of action potential generation: (1) the voltage-gated Na+ channels necessary for action potential generation increase intrinsic noise and (2) introduce non-linearities, and (3) the finite duration of the action potential creates a ‘footprint’ in the generator potential that obscures incoming signals. These three processes reduce information rates by ~50% in generator potentials, to ~3 times that of spike trains. Both generator potentials and graded potentials consume almost an order of magnitude less energy per second than spike trains. Because of the lower information rates of generator potentials they are substantially less energy efficient than graded potentials. However, both are an order of magnitude more efficient than spike trains due to the higher energy costs and low information content of spikes, emphasizing that there is a two-fold cost of converting analogue to digital; information loss and cost inflation

Sensitivity of Chaos Measures in Detecting Stress in the Focusing Control Mechanism of the Short-Sighted Eye

yesWhen fixating on a stationary object, the power of the eye’s lens fluctuates. Studies have suggested that changes in these so-called microfluctuations in accommodation may be a factor in the onset and progression of short-sightedness. Like many physiological signals, the fluctuations in the power of the lens exhibit chaotic behaviour. A breakdown or reduction in chaos in physiological systems indicates stress to the system or pathology. The purpose of this study was to determine whether the chaos in fluctuations of the power of the lens changes with refractive error, i.e. how short-sighted a subject is, and/or accommodative demand, i.e. the effective distance of the object that is being viewed. Six emmetropes (EMMs, non-short-sighted), six early-onset myopes (EOMs, onset of short-sightedness before the age of 15), and six late-onset myopes (LOMs, onset of short-sightedness after the age of 15) took part in the study. Accommodative microfluctuations were measured at 22 Hz using an SRW-5000 autorefractor at accommodative demands of 1 D (dioptres), 2 D, and 3 D. Chaos theory analysis was used to determine the embedding lag, embedding dimension, limit of predictability, and Lyapunov exponent. Topological transitivity was also tested for. For comparison, the power spectrum and standard deviation were calculated for each time record. The EMMs had a statistically significant higher Lyapunov exponent than the LOMs ( 0.64±0.330.64±0.33 vs. 0.39±0.20 D/s0.39±0.20 D/s ) and a lower embedding dimension than the LOMs ( 3.28±0.463.28±0.46 vs. 3.67±0.493.67±0.49 ). There was insufficient evidence (non-significant p value) of a difference between EOMs and EMMs or EOMs and LOMs. The majority of time records were topologically transitive. There was insufficient evidence of accommodative demand having an effect. Power spectrum analysis and assessment of the standard deviation of the fluctuations failed to discern differences based on refractive error. Chaos differences in accommodation microfluctuations indicate that the control system for LOMs is under stress in comparison to EMMs. Chaos theory analysis is a more sensitive marker of changes in accommodation microfluctuations than traditional analysis methods

Nonlinear Time Series Analysis of Nodulation Factor Induced Calcium Oscillations: Evidence for Deterministic Chaos?

Legume plants form beneficial symbiotic interactions with nitrogen fixing bacteria (called rhizobia), with the rhizobia being accommodated in unique structures on the roots of the host plant. The legume/rhizobial symbiosis is responsible for a significant proportion of the global biologically available nitrogen. The initiation of this symbiosis is governed by a characteristic calcium oscillation within the plant root hair cells and this signal is activated by the rhizobia. Recent analyses on calcium time series data have suggested that stochastic effects have a large role to play in defining the nature of the oscillations. The use of multiple nonlinear time series techniques, however, suggests an alternative interpretation, namely deterministic chaos. We provide an extensive, nonlinear time series analysis on the nature of this calcium oscillation response. We build up evidence through a series of techniques that test for determinism, quantify linear and nonlinear components, and measure the local divergence of the system. Chaos is common in nature and it seems plausible that properties of chaotic dynamics might be exploited by biological systems to control processes within the cell. Systems possessing chaotic control mechanisms are more robust in the sense that the enhanced flexibility allows more rapid response to environmental changes with less energetic costs. The desired behaviour could be most efficiently targeted in this manner, supporting some intriguing speculations about nonlinear mechanisms in biological signaling

Kinematic variability, fractal dynamics and local dynamic stability of treadmill walking

<p>Abstract</p> <p>Background</p> <p>Motorized treadmills are widely used in research or in clinical therapy. Small kinematics, kinetics and energetics changes induced by Treadmill Walking (TW) as compared to Overground Walking (OW) have been reported in literature. The purpose of the present study was to characterize the differences between OW and TW in terms of stride-to-stride variability. Classical (Standard Deviation, SD) and non-linear (fractal dynamics, local dynamic stability) methods were used. In addition, the correlations between the different variability indexes were analyzed.</p> <p>Methods</p> <p>Twenty healthy subjects performed 10 min TW and OW in a random sequence. A triaxial accelerometer recorded trunk accelerations. Kinematic variability was computed as the average SD (MeanSD) of acceleration patterns among standardized strides. Fractal dynamics (scaling exponent α) was assessed by Detrended Fluctuation Analysis (DFA) of stride intervals. Short-term and long-term dynamic stability were estimated by computing the maximal Lyapunov exponents of acceleration signals.</p> <p>Results</p> <p>TW did not modify kinematic gait variability as compared to OW (multivariate T², p = 0.87). Conversely, TW significantly modified fractal dynamics (t-test, p = 0.01), and both short and long term local dynamic stability (T²p = 0.0002). No relationship was observed between variability indexes with the exception of significant negative correlation between MeanSD and dynamic stability in TW (3 × 6 canonical correlation, r = 0.94).</p> <p>Conclusions</p> <p>Treadmill induced a less correlated pattern in the stride intervals and increased gait stability, but did not modify kinematic variability in healthy subjects. This could be due to changes in perceptual information induced by treadmill walking that would affect locomotor control of the gait and hence specifically alter non-linear dependencies among consecutive strides. Consequently, the type of walking (i.e. treadmill or overground) is important to consider in each protocol design.</p

Information Transmission in Cercal Giant Interneurons Is Unaffected by Axonal Conduction Noise

What are the fundamental constraints on the precision and accuracy with which nervous systems can process information? One constraint must reflect the intrinsic “noisiness” of the mechanisms that transmit information between nerve cells. Most neurons transmit information through the probabilistic generation and propagation of spikes along axons, and recent modeling studies suggest that noise from spike propagation might pose a significant constraint on the rate at which information could be transmitted between neurons. However, the magnitude and functional significance of this noise source in actual cells remains poorly understood. We measured variability in conduction time along the axons of identified neurons in the cercal sensory system of the cricket Acheta domesticus, and used information theory to calculate the effects of this variability on sensory coding. We found that the variability in spike propagation speed is not large enough to constrain the accuracy of neural encoding in this system

HERMES: Towards an Integrated Toolbox to Characterize Functional and Effective Brain Connectivity

The analysis of the interdependence between time series has become an important field of research in the last years, mainly as a result of advances in the characterization of dynamical systems from the signals they produce, the introduction of concepts such as generalized and phase synchronization and the application of information theory to time series analysis. In neurophysiology, different analytical tools stemming from these concepts have added to the ‘traditional’ set of linear methods, which includes the cross-correlation and the coherency function in the time and frequency domain, respectively, or more elaborated tools such as Granger Causality. This increase in the number of approaches to tackle the existence of functional (FC) or effective connectivity (EC) between two (or among many) neural networks, along with the mathematical complexity of the corresponding time series analysis tools, makes it desirable to arrange them into a unified-easy-to-use software package. The goal is to allow neuroscientists, neurophysiologists and researchers from related fields to easily access and make use of these analysis methods from a single integrated toolbox. Here we present HERMES (http://hermes.ctb.upm.es), a toolbox for the Matlab® environment (The Mathworks, Inc), which is designed to study functional and effective brain connectivity from neurophysiological data such as multivariate EEG and/or MEG records. It includes also visualization tools and statistical methods to address the problem of multiple comparisons. We believe that this toolbox will be very helpful to all the researchers working in the emerging field of brain connectivity analysis

Synchronous chaos and broad band gamma rhythm in a minimal multi-layer model of primary visual cortex

Visually induced neuronal activity in V1 displays a marked gamma-band component which is modulated by stimulus properties. It has been argued that synchronized oscillations contribute to these gamma-band activity [... however,] even when oscillations are observed, they undergo temporal decorrelation over very few cycles. This is not easily accounted for in previous network modeling of gamma oscillations. We argue here that interactions between cortical layers can be responsible for this fast decorrelation. We study a model of a V1 hypercolumn, embedding a simplified description of the multi-layered structure of the cortex. When the stimulus contrast is low, the induced activity is only weakly synchronous and the network resonates transiently without developing collective oscillations. When the contrast is high, on the other hand, the induced activity undergoes synchronous oscillations with an irregular spatiotemporal structure expressing a synchronous chaotic state. As a consequence the population activity undergoes fast temporal decorrelation, with concomitant rapid damping of the oscillations in LFPs autocorrelograms and peak broadening in LFPs power spectra. [...] Finally, we argue that the mechanism underlying the emergence of synchronous chaos in our model is in fact very general. It stems from the fact that gamma oscillations induced by local delayed inhibition tend to develop chaos when coupled by sufficiently strong excitation.Comment: 49 pages, 11 figures, 7 table

Recurrent, Robust and Scalable Patterns Underlie Human Approach and Avoidance

BACKGROUND. Approach and avoidance behavior provide a means for assessing the rewarding or aversive value of stimuli, and can be quantified by a keypress procedure whereby subjects work to increase (approach), decrease (avoid), or do nothing about time of exposure to a rewarding/aversive stimulus. To investigate whether approach/avoidance behavior might be governed by quantitative principles that meet engineering criteria for lawfulness and that encode known features of reward/aversion function, we evaluated whether keypress responses toward pictures with potential motivational value produced any regular patterns, such as a trade-off between approach and avoidance, or recurrent lawful patterns as observed with prospect theory. METHODOLOGY/PRINCIPAL FINDINGS. Three sets of experiments employed this task with beautiful face images, a standardized set of affective photographs, and pictures of food during controlled states of hunger and satiety. An iterative modeling approach to data identified multiple law-like patterns, based on variables grounded in the individual. These patterns were consistent across stimulus types, robust to noise, describable by a simple power law, and scalable between individuals and groups. Patterns included: (i) a preference trade-off counterbalancing approach and avoidance, (ii) a value function linking preference intensity to uncertainty about preference, and (iii) a saturation function linking preference intensity to its standard deviation, thereby setting limits to both. CONCLUSIONS/SIGNIFICANCE. These law-like patterns were compatible with critical features of prospect theory, the matching law, and alliesthesia. Furthermore, they appeared consistent with both mean-variance and expected utility approaches to the assessment of risk. Ordering of responses across categories of stimuli demonstrated three properties thought to be relevant for preference-based choice, suggesting these patterns might be grouped together as a relative preference theory. Since variables in these patterns have been associated with reward circuitry structure and function, they may provide a method for quantitative phenotyping of normative and pathological function (e.g., psychiatric illness).National Institute on Drug Abuse (14118, 026002, 026104, DABK39-03-0098, DABK39-03-C-0098); The MGH Phenotype Genotype Project in Addiction and Mood Disorder from the Office of National Drug Control Policy - Counterdrug Technology Assessment Center; MGH Department of Radiology; the National Center for Research Resources (P41RR14075); National Institute of Neurological Disorders and Stroke (34189, 05236