Alternate Entropy Computations by Applying Recurrence Matrix Masking.

In practicality, recurrence analyses of dynamical systems can only process short sections of signals that may be infinitely long. By necessity, the recurrence plot and its quantifications are constrained within a truncated triangle that clips the signals at its borders. Recurrence variables defined within these confining borders can be influenced more or less by truncation effects depending upon the system under evaluation. In this study, the question being asked is what if the boundary borders were tilted, what would be the effect on all recurrence variables? This question was prompted by the observation that line entropy values are maximized for highly periodic systems in which the infinitely long line elements are truncated to different unique lengths. However, by redefining the recurrence plot area to a 45-degree tilted box within the triangular area, the diagonal lines would consequently be truncated to identical lengths. Such masking would minimize the line entropy to 0.000 bits/bin. However, what new truncation influences would be imposed on the other recurrence variables? This question is examined by comparing recurrence variables computed with the triangular recurrence area versus boxed recurrence area. Examples include the logistic equation (mathematical series), the Dow Jones Industrial Average over a decade (real-word data), and a square wave pulse (toy series). Good agreement among the variables in terms of timing and amplitude was found for most, but not all variables. These important results are discussed.

Recurrence quantification analysis of complex-fractionated electrograms differentiates active and passive sites during atrial fibrillation.

OBJECTIVES: To differentiate electrograms representing sites of active atrial fibrillation (AF) drivers from passive ones. BACKGROUND: Ablation of complex-fractionated atrial electrograms (CFAEs) is controversial due to difficulty in distinguishing CFAEs representing sites of active AF drivers from passive mechanisms. We hypothesized that active CFAE sites exhibit repetitive wavefront directionality, thereby inscribing an electrogram conformation (Egm-C) that is more recurrent compared with passive CFAE sites; and that can be differentiated from passive CFAEs using nonlinear recurrence quantification analysis (RQA). METHODS: We developed multiple computer models of active CFAE mechanisms (ie, rotors) and passive CFAE mechanisms (ie, wavebreak, slow conduction, and double potentials). CFAE signals were converted into discrete time-series representing Egm-C. The RQA algorithm was used to compare signals derived from active CFAE sites to those from passive CFAEs sites. The RQA algorithm was then applied to human CFAE signals collected during AF ablation (n = 17 patients). RESULTS: RQA was performed in silico on simulated bipolar CFAEs within active (n = 45) and passive (n = 60) areas. Recurrence of Egm-C was significantly higher in active compared with passive CFAE sites (31.8% ± 19.6% vs 0.3% ± 0.5%, respectively, P < .0001) despite no difference in mean cycle length (CL). Similarly, for human AF (n = 39 signals), Egm-C recurrence was higher in active vs passive CFAE areas despite similar CLs (%recurrence 13.6% ± 15.5% vs 0.1% ± 0.3%, P < .002; mean CL 102.5 ± 14.3 vs 106.6 ± 14.4, P = NS). CONCLUSION: Active CFAEs critical to AF maintenance exhibit higher Egm-C recurrence and can be differentiated from passive bystander CFAE sites using RQA.

New skip parameter to facilitate recurrence quantification of signals comprised of multiple components.

Recurrence analyses are typically performed on discretized time series after applying proper embeddings, delays, and thresholds. In our study of atrial electrograms, we found limitations to this approach when sequential bipolar complexes were defined as the timings of the first two zero crosses following the initiation of each event. The reason for this is that each bipolar component consists of two points in odd-even pairings. Since recurrence analysis starts vectors on each sequential point, incorrect even-odd pairings occur for every other vector. To overcome this limitation, a new parameter SKIP is introduced such that recurrence vectors can skip 1 (or 2) points for signals with defined multiple components. To demonstrate the utility of parameter SKIP, we used the Courtemanche model to simulate the electrical activity in the human atrium on a square, two-dimensional plane with 800 × 800 nodes. Over this plane, a grid of 39 × 39 virtual unipoles was created. Neighboring unipoles formed 39 × 38 bipolar pairs, which were recorded as 1482 continuous and synchronous time series. At each unipolar site, the actual wavefront direction was determined by comparing the relative activation timings of the local intracellular potentials. Parameters were set such that the "tissue" exhibited both spiral waves (organized activity) and wave breakups (chaotic activity). For each bipolar complex in the continuous electrogram, discretized electrogram conformation was defined as the timing delays from the start of the complex to the first two zero-crosses. Long sequences of paired zero-cross timings were subjected to recurrence analysis using SKIP values of 0 (no skipping) and 1 (single skipping). Recurrence variables were computed and correlated with the absolute wavefront directions. The results showed that the introduction of the skipping window improved the correlations of some recurrence variables with absolute wavefront directions. This is critically important because such variables may be better markers for wavefront directions in human recordings when the absolute wavefront directions cannot be calculated directly.

A method for quantifying recurrent patterns of local wavefront direction during atrial fibrillation.

INTRODUCTION: Spiral wave reentry is a potential mechanism of atrial fibrillation (AF), but is difficult to differentiate clinically from multiple wavelet breakup using standard bipolar recordings. We developed a new methodology using bipolar recordings to estimate the direction of local activation wavefronts during AF by calculating the electrogram conformation (Egm-C). We subsequently used recurrence quantification analysis (RQA) of Egm-C to differentiate regions of spiral wave reentry from wavelet breakup. METHODS: A 2D computer simulation was created with regions containing a stable spiral wave and also regions of wavebreak. A grid of 40 × 40 unipolar electrodes was superimposed. At each site, the actual wavefront direction (WD) was determined by comparing relative activation timings of the local intracellular recordings, and the estimated wavefront direction (Egm-C) was determined from the morphology of the local bipolar electrogram. RQA of Egm-C was compared to RQA of actual WD in order to differentiate AF mechanisms. RESULTS: RQA of actual WD and Egm-C both distinguished regions of spiral wave reentry from wavelet breakup with high correlation between the two methods (recurrence rate, r = 0.96; determinism, r = 0.61; line max, r = 0.95; entropy, r = 0.84; p < 0.001 for all). In areas of stable spiral wave reentry, the recurrence plots of both Egm-C and actual WD demonstrated stable, periodic dynamics, while regions of wavelet breakup demonstrated chaotic behavior largely devoid of repetitive activation patterns. CONCLUSION: Calculation of Egm-C allows RQA to be performed on bipolar electrograms during AF and differentiates regions of spiral wave reentry from multiple wavelet breakup.

Effect of growing media and fertilization on sugarcane flowering under artificial photoperiod.

The USDA-ARS Sugarcane Variety Development Program in Houma, LA aims to maximize the number of panicles available for crossing through artificial manipulation of the environment. In a three-year study, the effect of growing media, fertilizer treatment, and their interaction on sugarcane flowering (% of panicles emerged), and number of days to flowering (DTF) under an artificial photoperiod treatment were assessed. The commercially-available sugarcane cultivar, 'HoCP 96-540' was planted in 2.8-L pots and subjected to the standard local photoperiod treatment. The cultivar was planted in four growing media (RediEarth Seedling and Germination Mix, Fafard, Metro-Mix®902, and Metro-Mix®900) and subjected to three different fertilizer applications. In the control treatment, fertilizer application was stopped prior to the commencement of the photoperiod treatment as practiced in some sugarcane breeding programs. The continuous treatment consisted of an application of a 10 ml solution of a NPK three times a week between June and October. The partial treatment consisted of applications of the same NPK solution applied post-initiation between September and October. Nitrogen starvation prior to the commencement of the photoperiod treatment is generally accepted to improve flower initiation; thus the standard practice is to cease nitrogen application two weeks prior to beginning a photoperiod regime. The growing media used in this study did not have a significant effect on days to flowering or percent panicle emergence. In our study, the control fertilizer treatment showed a flowering percentage across all growing media types of 21.2% less than a continuous fertilization regime. Furthermore, a significant trend was observed between fertilization treatments and days to flowering, with the continuous treatment producing panicles, on average across growing media, four days earlier than the control treatment, and six days earlier than the partial treatment. Evidence across this three-year experiment indicates that we should consider modifying plant nutrition management as soil fertility was found to be inadequate.

Recurrence Quantification Analysis of F-Waves and the Evaluation of Neuropathies.

Electrodiagnostic (EDX) patterns of neuropathic dysfunction have been based on axonal/demyelinating criteria requiring prior assumptions. This has not produced classifications of desired sensitivity or specificity. Furthermore, standard nerve conduction studies have limited reproducibility. New methodologies in EDX seem important. Recurrent Quantification Analysis (RQA) is a nonlinear method for examining patterns of recurrence. RQA might provide a unique method for the EDX evaluation of neuropathies. RQA was used to analyze F-wave recordings from the abductor hallucis muscle in 61 patients with neuropathies. Twenty-nine of these patients had diabetes as the sole cause of their neuropathies. In the other 32 patients, the etiologies of the neuropathies were diverse. Commonly used EDX variables were also recorded. RQA data could separate the 29 patients with diabetic neuropathies from the other 32 patients (P < 0.009). Statistically significant differences in two EDX variables were also present: compound muscle action potential amplitudes (P < 0.007) and F-wave persistence (P < 0.001). RQA analysis of F-waves seemed able to distinguish diabetic neuropathies from the other neuropathies studied, and this separation was associated with specific physiological abnormalities. This study would therefore support the idea that RQA of F-waves can distinguish between types of neuropathic dysfunction based on EDX data alone without prior assumptions.

Pyraclostrobin wash-off from sugarcane leaves and aerobic dissipation in agricultural soil.

To mitigate damage from the fungal sugarcane pathogen brown rust (Puccinia melanocephala), a Section 18 Emergency Use Label was put in place by the United States Environmental Protection Agency (U.S. EPA) for the application of pyraclostrobin (trade name Headline SC, produced by BASF, Research Triangle Park, NC) on sugarcane in 2008. To assess the dynamics of this fungicide in Louisiana soil, samples (n = 24) from a non-treated field were spiked with pyraclostrobin (3.1 µg g(-1)) and analyzed in laboratory conditions over the course of 63 days using quick, easy, cheap, effective, rugged, and safe (QuEChERS) dispersive solid-phase extraction/high-performance liquid chromatography with ultraviolet-visible detection (dSPE/HPLC-UV). Modeling was performed using Microsoft Excel to predict DTx values. Pyraclostrobin was found to follow biphasic kinetics with DT50 and DT90 values of 60 and 282 days, suggesting that it is moderately persistent to persistent in soils. Wash-off studies on sugarcane indicate that very little fungicide is in the wash-off after 48 h. If applied to sugarcane according to label recommendations, the fungicide should have minimal dissipation from rainfall events.

Recurrence quantification of fractal structures.

By definition, fractal structures possess recurrent patterns. At different levels repeating patterns can be visualized at higher magnifications. The purpose of this chapter is threefold. First, general characteristics of dynamical systems are addressed from a theoretical mathematical perspective. Second, qualitative and quantitative recurrence analyses are reviewed in brief, but the reader is directed to other sources for explicit details. Third, example mathematical systems that generate strange attractors are explicitly defined, giving the reader the ability to reproduce the rich dynamics of continuous chaotic flows or discrete chaotic iterations. The challenge is then posited for the reader to study for themselves the recurrent structuring of these different dynamics. With a firm appreciation of the power of recurrence analysis, the reader will be prepared to turn their sights on real-world systems (physiological, psychological, mechanical, etc.).

Magnetosensory evoked potentials: consistent nonlinear phenomena.

Electromagnetic fields (EMFs) having strengths typically found in the general environment can alter brain activity, but the reported effects have been inconsistent. We theorized that the problem arose from the use of linear methods for analyzing what were actually nonlinear phenomena, and therefore studied whether the nonlinear signal-processing technique known as recurrence quantification analysis (RQA) could be employed as the basis of a reliable method for demonstrating consistent changes in brain activity. Our primary purpose was to develop such a method for observing the occurrence of evoked potentials in individual subjects exposed to magnetic fields (2G, 30 and 60 Hz). After all conditions that affected the analysis of the EEG were specified in advance, we detected magnetosensory evoked potentials (MEPs) in all 15 subjects (P<0.05 in each experiment). The MEPs, which occurred within the predicted latency interval of 109-504 ms, were independent of the frequency and the direction of the field, and were not detected using the traditional linear method of analysis, time averaging. When the results obtained within subjects were averaged across subjects, the evoked potentials could not be detected, indicating how real nonlinear phenomena can be averaged away when the incorrect method of analysis is used. Recurrence quantification analysis, but not linear analysis, permitted consistent demonstration of MEPs. The use of nonlinear analysis might also resolve apparent inconsistencies in other kinds of brain studies.

A topologically related singularity suggests a maximum preferred size for protein domains.

A variety of protein physicochemical as well as topological properties, demonstrate a scaling behavior relative to chain length. Many of the scalings can be modeled as a power law which is qualitatively similar across the examples. In this article, we suggest a rational explanation to these observations on the basis of both protein connectivity and hydrophobic constraints of residues compactness relative to surface volume. Unexpectedly, in an examination of these relationships, a singularity was shown to exist near 255-270 residues length, and may be associated with an upper limit for domain size. Evaluation of related G-factor data points to a wide range of conformational plasticity near this point. In addition to its theoretical importance, we show by an application of CASP experimental and predicted structures, that the scaling is a practical filter for protein structure prediction.

Detection of cardiac variability in the isolated rat heart.

Cardiac variability can be assessed from two perspectives: beat-to-beat performance and continuous performance during the cardiac cycle. Linear analysis techniques assess cardiac variability by measuring the physical attributes of a signal, whereas nonlinear techniques evaluate signal dynamics. This study sought to determine if recurrence quantification analysis (RQA), a nonlinear technique, could detect pharmacologically induced autonomic changes in the continuous left ventricular pressure (LVP) and electrographic (EC) signals from an isolated rat heart-a model that theoretically contains no inherent variability. LVP and EC signal data were acquired simultaneously during Langendorff perfusion of isolated rat hearts before and after the addition of acetylcholine (n = 11), norepinephrine (n = 12), or no drug (n = 12). Two-minute segments of the continuous LVP and EC signal data were analyzed by RQA. Findings showed that%recurrence,%determinism, entropy, maxline, and trend from the continuous LVP signal significantly increased in the presence of both acetylcholine and norepinephrine, although systolic LVP significantly increased only with norepinephrine. In the continuous EC signal, the RQA trend variable significantly increased in the presence of norepinephrine. These results suggest that when either the sympathetic or parasympathetic division of the autonomic nervous system overwhelms the other, the dynamics underlying cardiac variability become stationary. This study also shows that information concerning inherent variability in the isolated rat heart can be gained via RQA of the continuous cardiac signal. Although speculative, RQA may be a tool for detecting alterations in cardiac variability and evaluating signal dynamics as a nonlinear indicator of cardiac pathology.

Charge and hydrophobicity patterning along the sequence predicts the folding mechanism and aggregation of proteins: a computational approach.

The presence of partially folded intermediates along the folding funnel of proteins has been suggested to be a signature of potentially aggregating systems. Many studies have concluded that metastable, highly flexible intermediates are the basic elements of the aggregation process. In a previous paper, we demonstrated how the choice between aggregation and folding behavior was influenced by hydrophobicity distribution patterning along the sequence, as quantified by recurrence quantification analysis (RQA) of the Myiazawa-Jernigan coded primary structures. In the present paper, we tried to unify the "partially folded intermediate" and "hydrophobicity/charge" models of protein aggregation verifying the ability of an empirical relation, developed for rationalizing the effect of different mutations on aggregation propensity of acyl-phosphatase and based on the combination of hydrophobicity RQA and charge descriptors, to discriminate in a statistically significant way two different protein populations: (a) proteins that fold by a process passing by partially folded intermediates and (b) proteins that do not present partially folded intermediates.

EMG recurrence quantifications in dynamic exercise.

This study was designed to evaluate the suitability of nonlinear recurrence quantification analysis (RQA) in assessing electromyograph (EMG) signals during dynamic exercise. RQA has been proven to be effective in analyzing nonstationary signals. The subject group consisted of 19 male patients diagnosed with low back pain. EMG signals were recorded from left and right paraspinal muscles during isoinertial exercise both before and after 12 weeks of regimented physical therapy. Autorecurrence analysis was performed between the left and right EMG signals individually, and cross-recurrence analysis was performed on the left-right EMG pairs. Spectral analysis of the EMG signals was employed as an independent, objective measure of fatigue. Increase in the RQA variable % determinism during the 90-s dynamic tests was found to be a good marker for fatigue. Before physical therapy, this nonlinear marker revealed simultaneous increases in motor unit recruitment within each pool and between left and right pools. After physical therapy, the motor unit recruitment was less within and between pools, indicative of increased fatigue resistance. Finally, fatigue resistance (less increase in % determinism) correlated well with subjective scores of pain relief. Taken together, these latter results indicate that recurrence analysis may be useful in charting the efficacy of a specific exercise therapy program in reducing low back pain by elevating the fatigue threshold.

Protein aggregation/folding: the role of deterministic singularities of sequence hydrophobicity as determined by nonlinear signal analysis of acylphosphatase and Abeta(1-40).

The problem of protein folding vs. aggregation was investigated in acylphosphatase and the amyloid protein Abeta(1-40) by means of nonlinear signal analysis of their chain hydrophobicity. Numerical descriptors of recurrence patterns provided the basis for statistical evaluation of folding/aggregation distinctive features. Static and dynamic approaches were used to elucidate conditions coincident with folding vs. aggregation using comparisons with known protein secondary structure classifications, site-directed mutagenesis studies of acylphosphatase, and molecular dynamics simulations of amyloid protein, Abeta(1-40). The results suggest that a feature derived from principal component space characterized by the smoothness of singular, deterministic hydrophobicity patches plays a significant role in the conditions governing protein aggregation.

Review of nonlinear analysis of proteins through recurrence quantification.

This review considers the use of a nonlinear signal analysis tool, recurrence quantification analysis, as a method to study sequence/structure relationships of proteins. Four broad categories are discussed: (1) a point of view involving information contained in deterministic aspects of hydrophobicity; (2) the analysis of protein hydrophobicity "singularities"; (3) time-series analysis of protein dynamics simulations; and (4) prediction of protein secondary structure.

Recurrence quantification analysis as a tool for nonlinear exploration of nonstationary cardiac signals.

The complexity, nonlinearity and nonstationarity of the cardiovascular system typically defy comprehensive and deterministic mathematical modeling, except from a statistical perspective. Living systems are governed by numerous, continuously changing, interacting variables in the presence of noise. Cardiovascular signals can be shown to be discontinuous alternations between deterministic trajectories and stochastic pauses (terminal dynamics). One promising approach for assessing such nondeterministic complexity is recurrence quantification analysis (RQA). As reviewed in this paper, strategies implementing quantification of recurrences have been successful in diagnosing changes in nonstationary cardiac signals not easily detected by traditional methods. It is concluded that recurrence quantification analysis is a powerful discriminatory tool which, when properly applied to cardiac signals, can provide objectivity regarding the degree of determinism characterizing the system, state changes, as well as degrees of complexity and/or randomness.