D. K. Bhattacharya

A high dimensional delay selection for the reconstruction of proper phase space with cross auto-correlation

For the purpose of phase space reconstruction from nonlinear time series, delay selection is one of the most vital criteria. This is normally done by using a general measure viz., mutual information (MI). However, in that case, the delay selection is limited to the estimation of a single delay using MI between two variables only. The corresponding reconstructed phase space is also not satisfactory. To overcome the situation, a high-dimensional estimator of the MI is used; it selects more than one delay between more than two variables. The quality of the reconstructed phase space is tested by shape distortion parameter (SD), it is found that even this multi-dimensional MI sometimes fails to produce a less distorted phase space. In this paper, an alternative nonlinear measure-cross auto-correlation (CAC) is introduced. A comparative study is made between the reconstructed phase spaces of a known three dimensional Neuro-dynamical model, Lorenz dynamical model and a three dimensional food-web model under MI for two and higher dimensions and also under cross auto-correlation separately. It is found that the least distorted phase space is obtained only under the notion of cross auto-correlation.

New types of nonlinear auto-correlations of bivariate data and their applications

Abstract The paper introduces new types of nonlinear correlations between bivariate data sets and derives nonlinear auto-correlations on the same data set. These auto-correlations are of different types to match signals with different types of nonlinearities. Examples are cited in all cases to make the definitions meaningful. Next correlogram diagrams are drawn separately in all cases; from these diagrams proper time lags/delays are determined. These give rise to independent coordinates of the attractors. Finally three dimensional attractors are reconstructed in each case separately with the help of these independent coordinates. Moreover for the purpose of making proper distinction between the signals, the attractors so reconstructed are quantified by a new technique called ‘ellipsoid fit’.

Is one dimensional Poincaré map sufficient to describe the chaotic dynamics of a three dimensional system

Study of continuous dynamical system through Poincare map is one of the most popular topics in nonlinear analysis. This is done by taking intersections of the orbit of flow by a hyper-plane parallel to one of the coordinate hyper-planes of co-dimension one. Naturally for a 3D-attractor, the Poincare map gives rise to 2D points, which can describe the dynamics of the attractor properly. In a very special case, sometimes these 2D points are considered as their 1D-projections to obtain a 1D map. However, this is an artificial way of reducing the 2D map by dropping one of the variables. Sometimes it is found that the two coordinates of the points on the Poincare section are functionally related. This also reduces the 2D Poincare map to a 1D map. This reduction is natural, and not artificial as mentioned above. In the present study, this issue is being highlighted. In fact, we find out some examples, which show that even this natural reduction of the 2D Poincare map is not always justified, because the resultant 1D map may fail to generate the original dynamics properly. This proves that to describe the dynamics of the 3D chaotic attractor, the minimum dimension of the Poincare map must be two, in general.

A Comparative Study on Three Different Types of Music Based on Same Indian Raga and Their Effects on Human Autonomic Nervous Systems

Complex heart dynamics reflects activities of human non-autonomous system through Heart rate variability (HRV). Poincare plot is one of the fascinating geometrical tools, which can properly describe the complex heart dynamics. In this chapter, the effect of music on HRV is studied by observing the geometric pattern of Poincare plot. In this concern, Indian classical music based on Raga ‘Malkaunsh’ is selected in different forms, and HRV signals are collected from different persons. Then, we have identified the differences (if any) in the pattern of music in the three cases, where by pattern we understand dynamics, timber, rhythm and tonality. Next, by using Poincare plot it is investigated whether the different types of music have different types of effects on HRV. The whole study has been carried out for both of Indian Raga music initiated and non-initiated (IRM and NIRM) persons.

Phase synchronization of instrumental music signals

Abstract Signal analysis is one of the finest scientific techniques in communication theory. Some quantitative and qualitative measures describe the pattern of a music signal, vary from one to another. Same musical recital, when played by different instrumentalists, generates different types of music patterns. The reason behind various patterns is the psycho-acoustic measures – Dynamics, Timber, Tonality and Rhythm, varies in each time. However, the psycho-acoustic study of the music signals does not reveal any idea about the similarity between the signals. For such cases, study of synchronization of long-term nonlinear dynamics may provide effective results. In this context, phase synchronization (PS) is one of the measures to show synchronization between two non-identical signals. In fact, it is very critical to investigate any other kind of synchronization for experimental condition, because those are completely non identical signals. Also, there exists equivalence between the phases and the distances of the diagonal line in Recurrence plot (RP) of the signals, which is quantifiable by the recurrence quantification measure τ-recurrence rate. This paper considers two nonlinear music signals based on same raga played by two eminent sitar instrumentalists as two non-identical sources. The psycho-acoustic study shows how the Dynamics, Timber, Tonality and Rhythm vary for the two music signals. Then, long term analysis in the form of phase space reconstruction is performed, which reveals the chaotic phase spaces for both the signals. From the RP of both the phase spaces, τ-recurrence rate is calculated. Finally by the correlation of normalized tau-recurrence rate of their 3D phase spaces and the PS of the two music signals has been established. The numerical results well support the analysis.

Approximate discrete dynamics of EMG signal

Approximation of a continuous dynamics by discrete dynamics in the form of Poincare map is one of the fascinating mathematical tool, which can describe the approximate behaviour of the dynamics of the dynamical system in lesser dimension than the embedding dimension. The present article considers a very rare biomedical signal like Electromyography (EMG) signal. It determines suitable time delay and reconstruct the attractor with embedding dimension three. By measuring its Lyapunov exponent, the attractor so reconstructed is found to be chaotic. Naturally the Poincare map obtained by corresponding Poincare section has to be chaotic too. This may be verified by calculation of Lyapunov exponent of the map. The main objective of this article is to show that Poincare map exists in this case as a 2D map for a suitable Poincare section only. In fact, the article considers two Poincare sections of the attractor for construction of the Poincare map. It is seen that one such map is chaotic but the other one is not so - both are verified by calculation of Lyapunov exponent of the map.

Some Time-Delay Finding Measures and Attractor Reconstruction

Topologically equivalent attractor reconstruction is one of the major issues in nonlinear analysis. This is because of the fact that the underlying dynamical model of some nonlinear phenomena may not be known and thus it is necessary to retrieve the dynamics from the data it generates. One way to achieve this is the reconstruction of the attractor. The basis of such reconstruction is the famous Taken’s embedding theorem, which asserts that an equivalent phase space trajectory,preserving the topological structures of the original phase space trajectory, can be reconstructed by using only one observation of the time series. However, in some cases topologically equivalent attractor reconstructions can also be done by using multiple observations. All these things involve the choice of suitable time-delay(s) and embedding dimension. Various measures are available to find out the suitable time-delay(s). Among them, linear auto-correlation, Average mutual information, higher dimensional mutual information are mostly used measures for the reconstruction of the attractors. Every measures have certain limitations in the sense that they are not always useful in finding suitable time-delay(s). Thus it is necessary to introduce few more nonlinear measures, which may be useful if the aforesaid measures fail to produce suitable time-delay/time-delays. In this chapter, some comparatively new nonlinear measures namely generalized auto-correlation, Cross auto-correlation and a new type of nonlinear auto-correlation of bivariate data for finding suitable time-delay(s) have been discussed. To establish their usefulness, attractors of some known dynamical systems have been reconstructed from their solution components with suitable time-delay(s) obtained by each of the measures. These attractors are then compared with their corresponding original attractor by a shape distortion parameter Sd. This shape distortion parameter actually checks how much distorted the reconstructed attractor is from its corresponding original attractor. The main objective of this chapter is to address the problem of reconstruction of a least distorted topologically equivalent attractor. The reason is that if the reconstructed attractor is least distorted from its original one, the dynamics of the system can be retrieved more accurately from it. This would help in identifying the dynamics of the corresponding system, even when the dynamical model is not known. Out of the three measures discussed in this chapter, the generalized and cross auto-correlation measures produce least distorted topologically equivalent attractor only by consideration of multiple solution components of the dynamical system. On the other hand, by using the measure—new type of nonlinear auto-correlation of bivariate data, one can reconstruct a least distorted topologically attractor from single solution component of the dynamical system. Various numerical results on Lorenz system, Neuro-dynamical system and also on two real life signals are presented to prove the effectiveness of the aforesaid three comparatively new nonlinear time-delay finding measures. Finding of suitable embedding dimension is another important issue for attractor reconstruction. However, this issue has not been highlighted in this chapter because we have restricted this discussion only to three dimensional attractor reconstruction.

Does Music Affect HRV Impulse? A Time Domain Study

The two-dimensional (2D) Poincare plot of HRV signal reflects the effect of different external stimuli on ANS. The classification is generally done by fitting an ‘ellipse’ on the dense region of the constructed Poincare plot of HRV signal. However, 2D Poincare plot sometimes fails to describe the proper behaviour of the system. One such example is 2D Poincare plot of HRV signal in pre-music and on-music condition. In fact, 2D Poincare plots in pre and on-music condition look almost similar for few subjects. So a proper classification tool is sought for. In this article, an improved technique called ‘3D Poincare plot with proper delay’ has been applied to properly distinguish the pre-music and on-music state of some normal healthy subjects.

Analysis of Similarity Between Protein Sequences Through the Study of Symbolic Dynamics

Protein sequence analysis is an important tool to decode the logic of life. The rapid growth of protein sequences is constantly throwing several challenges to the bioscientists. So several methods are being improvised to assign mathematical descriptors to protein sequences, in order to quantitatively compare the sequences and determine similarities and dissimilarities between them. In this paper, for the analysis of protein sequences their 4-bit ranks and entropies are studied and compared with each other which yields satisfactorily convenient results.

Symbolic representation of protein sequences on the basis of fuzzy representation of amino acids

The source and sink balance of fuel on earth, has been time and again challenged by the growing consumerism of fuel by people. To meet these demands, alternative fuel technology is currently a growing industry of research, especially the use of organisms like algae to harvest fuel, has been one of the recent developments in plant biotechnology. Algae being photosynthetic in nature, utilizes the sugar they produce, to harvest fuel in a scientifically driven manner. However there is a limitation to the variety of oil produced from the same algae decreasing the yield efficiency. But thanks to science, we have come up with new skills to harvest different types of oil products from the same algae using different sugar inputs in a dark acclimatized metabolic pathway. A few algae like Chlorella protothecoids and Botryococcus braunii are fed with rich carbohydrate sources (grass and woodchips) in dark, where it gets converted into oil using underlying principles of fermentation. The heterotrophic growth mode of algae favors an oil yield 3.4 times higher than autotrophic one, with a cetane rate of the fuel obtained as 74, which is ideal for any fuel. The Oil can further be upgraded using nano-sized catalysts like commercially available calcium oxide nano-crystals which can increase the reaction rate up to 99%.Watermelon (Citrullus lanatus) is commonly grown in traditional agrosystems throughout the drought-prone Saudi Arabia. There has been little work on investigating the relationships between wild and cultivated forms, and to study amount and partitioning of genetic variation, to allow for better conservation strategies. Previous studies have reported relatively low levels of genetic diversity in cultivated watermelon, but these have been based mainly on US plant introductions and modern watermelon cultivars linked to breeding programmes for disease resistance. The genetic relationships among six cultivars from different countries of origin and with different horticultural characteristics and one related wild-type species (Citrullus colocynthis) were assessed using inter-simple sequence repeats (ISSR) markers. Also, the presence of the resistance stress gene (DREB) as a marker of drought tolerance in watermelon cultivars and wild-type species was investigated. The cluster analysis results demonstrated low genetic diversity among commercial cultivars and high genetic diversity between wild species and commercial cultivars. Dendrograms produced two major clusters; one with all the watermelon cultivars and the other with the wild-type species. The low level of marker polymorphism among the adapted cultivars implies that a severe bottleneck in genetic diversity existed in watermelon during the initial breeding practices. There were no significant differences in chlorophyll content and plant height of wild-type species either in three-day intervals or watered daily. All watermelon cultivars gave lower chlorophyll content and plant height at three- and five-day intervals compared to watered daily. Specific PCR assays using AP2 primers (designed on the basis of the AP2/EREBP sequence of gene DREB) represent a sensitive tool for screening watermelon genotypes for the DREB gene. The DREB gene was present only in wild-type species. The amplifying DREB gene could be valuable in watermelon breeding programs for selection of desirable alleles under drought stress.H immunodeficiency virus (HIV) is a lenti virus that causes acquired immunodeficiency syndrome (AIDS), a condition in humans in which the immune system begins to fail, leading to life-threatening opportunistic infections. The RNA genome of HIV consists of nine genes (gag, pol, and env, tat, rev, nef, vif, vpr, vpu, and tev) encoding 19 proteins. The HIV-1 protease enzyme is responsible for the post-translational processing of the viral Gag and Gag-Pol polyproteins to yield the structural proteins and enzymes of the virus. With its integral role in HIV replication, HIV-PR has been a prime target for drug therapy. However, mutations associated with protease-inhibitor’s resistance have been reported at more than 20 different residues of the enzyme. Methods: In this experiment, HIV-1 DNA in 12 infants (of HIV-positive mother’s) dried blood spot (DBS) or whole blood sample (WBS) was first detected and the DNA containing the provirus from the positive samples was extracted and nested PCR was done to extract the first 400bp of pol region (p16) of HIV-1 genome which encodes for protease. The samples were then purified and sequenced by the Sanger’s method. Bioinformatics tools were used to analyze the sequenced samples. Results: BLAST search revealed that our isolates belonged to HIV-1 pol gene (protease region). There existed a close resemblance among the viral strains. Of the 12 sequences, 13 established drug resistance and 13 additional mutations were found. Only a few non-polymorphic mutations were found and most mutations were polymorphic. Few important drug resistant mutations included K 20 R/M/I, M 36 I/V, L 63 P/A/Q/S/H/C/T/I and I 93 L. Nidhi Raina, J Biotechnol Biomater 2013, 3:3 http://dx.doi.org/10.4172/2155-952X.S1.025