Sayan Mukherjee

Can complexity decrease in congestive heart failure

The complexity of a signal can be measured by the Recurrence period density entropy (RPDE) from the reconstructed phase space. We have chosen a window based RPDE method for the classification of signals, as RPDE is an average entropic measure of the whole phase space. We have observed the changes in the complexity in cardiac signals of normal healthy person (NHP) and congestive heart failure patients (CHFP). The results show that the cardiac dynamics of a healthy subject is more complex and random compare to the same for a heart failure patient, whose dynamics is more deterministic. We have constructed a general threshold to distinguish the border line between a healthy and a congestive heart failure dynamics. The results may be useful for wide range for physiological and biomedical analysis.

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’.

Complexity in congestive heart failure: A time-frequency approach

Reconstruction of phase space is an effective method to quantify the dynamics of a signal or a time series. Various phase space reconstruction techniques have been investigated. However, there are some issues on the optimal reconstructions and the best possible choice of the reconstruction parameters. This research introduces the idea of gradient cross recurrence (GCR) and mean gradient cross recurrence density which shows that reconstructions in time frequency domain preserve more information about the dynamics than the optimal reconstructions in time domain. This analysis is further extended to ECG signals of normal and congestive heart failure patients. By using another newly introduced measure-gradient cross recurrence period density entropy, two classes of aforesaid ECG signals can be classified with a proper threshold. This analysis can be applied to quantifying and distinguishing biomedical and other nonlinear signals.

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.

Computing two dimensional Poincar maps for hyperchaotic dynamics

Poincar map (PM) is one of the felicitous discrete approximation of the continuous dynamics. To compute PM, the discrete relation(s) between the successive point of interactions of the trajectories on the suitable Poincar section (PS) are found out. These discrete relations act as an amanuensis of the nature of the continuous dynamics. In this article, we propose a computational scheme to find a hyperchaotic PM (HPM) from an equivalent three dimensional (3D) subsystem of a 4D (or higher) hyperchaotic model. For the experimental purpose, a standard four dimensional (4D) hyperchaotic Lorenz-Stenflo system (HLSS) and a five dimensional (5D) hyperchaotic laser model (HLM) is considered. Equivalent 3D subsystem is obtained by comparing the movements of the trajectories of the original hyperchaotic systems with all of their 3D subsystems. The quantitative measurement of this comparison is made promising by recurrence quantification analysis (RQA). Various two dimensional (2D) Poincar mas are computed for several suitable Poincar sections for both the systems. But, only some of them are hyperchaotic in nature. The hyperchaotic behavior is verified by positive values of both one dimensional (1D) Lyapunov Exponent (LE-I) and 2D Lyapunov Exponent (LE-II). At the end, similarity of the dynamics between the hyperchaotic systems and their 2D hyperchaotic Poincar maps (HPM) has been established through mean recurrence time (MRT) statistics for both of 4D HLSS and 5D HLM and the best approximated discrete dynamics for both the hyperchaotic systems are found out.

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.

Synchronization between two discrete chaotic systems for secure communications

In this article, we have investigated synchronization phenomenon between two discrete chaotic systems. A general scheme for synchronization between two discrete maps with adaptive coupling has been studied analytically. The scheme can be successfully implemented for generalized synchronization between two chaotic maps. Conditional Lyapunov exponents (CLE) and Transverse Lyapunov exponents (TLE) can quantifies the robustness of synchronization. A secure communication scheme based on synchronization between two Logistic maps is also demonstrated. Numerical results show the effectiveness of our proposed scheme.