The discovery of deterministic chaos in many fields of science, engineering and mathematics, especially biological rhythms have opened up a new avenue to explore the concepts that has been neglected over the years. The possibility of understanding and characterizing a complex system from the variation of a single state variable has gained immense significance in recent times.
To mention, careful analysis of the biological signals like ECG and EEG based on chaos theory tools can have a potential clinical use in distinguishing various physiological states of the system under consideration.
Many studies based on the metric properties such as the calculation of Correlation dimension and Lyapunov exponent have been applied with mixed success for ECG and EEG time series.
Since in the cases of ECG and EEG no clear distinction between normal and pathological cases has been made, based on the results of Correlation dimension and Lyapunov exponent, I had undertaken my doctoral work that revolves around the topological property, namely, the unstable periodic orbits (UPOs). We have for the first time used the UPOs to characterize the normal and different pathological conditions of cardiac system by using ECG and EEG as time series. From the number and the distribution of the dominant UPOs, normal and different pathological states were clearly distinguished.
In summary, the time series has been characterized mainly through the following:
We analyzed the dynamics of human cardiac system from the electrical activity of the heart namely Electrocardiogram (ECG) through the Unstable Periodic Orbits (UPOs) which is a topological property. UPOs represent the skeleton for the strange attractor of dynamical systems, and there by many quantities that characterize chaos such as fractal dimension, average Lyapunov exponent and entropy can be determined by knowing the properties of UPO. The UPOs are extracted from the attractor reconstructed by time-delay method proposed by Takens, followed by the method of close returns proposed by Kostelich et al with some modifications.
UPOs are extracted first for the standard systems such as Lorenz, Rossler and Rossler hyperchaos are subjected to the UPO analysis. The Lyapunov exponents of these systems are determined from the eigenvalues of the individual UPOs by weighted averages and are in good agreement with the values calculated directly from the entire attractor.
In this study, we extracted the unstable periodic orbits (UPOs) for subjects with ECGs showing normal or pathological conditions (such as premature ventricular contraction (PVC), atrio-ventricular (AV) block, ventricular tachyarrhythmia (VTA) and ventricular fibrillation (VF)). The UPOs were extracted from the state space (attractor) reconstructed from experimentally observed ECG.
We first demonstrated (3) the presence of deterministic chaotic character in the experimentally observed ECG using different quantities of nonlinear signal processing such as correlation dimension (4), Lyapunov exponent (19), predictability (10) and surrogate analysis (17) and then in this particular study, we used the recurrence method (9) to extract UPOs of the cardiac system.
On the extraction of UPOs it turned out that the human cardiac system behaved as a deterministic chaotic system with a limited number of dominant UPOs. The number of significant UPOs and their density distribution were found to be characteristic of the condition of normalcy and pathology of the heart. There were three to four dominant UPOs in the young healthy individual (Figure 1). The strongest UPO had a basic period of ~0.94–1.05 sec (in the order of normal heart rhythm). Healthy old subjects were also characterized by 3 to 4 UPOs. Here the basic periodicity was in the range of 0.66–0.88 sec which was lower than that observed in the young healthy subject (Figure 1). This may be a reflection of the increased heart rate that likely compensates for the decreased blood flow due to the thickening of blood vessels.
In sharp contrast to three to four UPOs for healthy cardiac system, in the case of PVC, significantly more number of UPOs of higher periods (approximately six UPOs) were observed (Figure 2). In the normal heart, the activity of the atria and ventricles are synchronized. In PVC, this synchronization was lost, resulted in irregular and higher R-R intervals. The higher number of UPOs probably reflected this larger number of possible dynamical states in PVC, occurring at different sites of the cardiac system. UPOs for the AV block were shifted to higher values of recurrence time (~ 1.63 sec) depending on the degree of the block, which was much higher than the normal case (Figure 2). This increase in recurrence might be due to delay in the conduction of electrical impulses from the atrium to the atrio-ventricular node.
For VTA, there were five to seven dominant UPOs (Figure 3) with the basic periodicity in the range of 0.50–0.72 sec. VTA results from a rapidly discharging focus developed in the ventricular myocardium usually from a single focus. Usually, it does not respond to the sino atrial (SA) node and both of them act independently of each other. Thus VTA is the result of a reentry mechanism within the ventricular myocardium. Consequently, the UPOs were shifted towards the lower values of recurrence time. The cases of VF were found to have more than ten UPOs (Figure 3) with a basic periodicity of 0.16–0.50 sec. The number of UPOs varied slightly, depending on the extent of fibrillation. In VF, rapidly discharging stimuli come from the single or multifoci within the ventricles. As a result, ventricles cannot effectively respond to each stimulus from the SA node and its rate is rapid and irregular. So many UPOs with significantly lower values of recurrence time were observed. From all these analyses, one can clearly note that the distribution of the UPOs could be utilized to distinguish between different states of the cardiac system (Figure 4) much more effectively than is possible from other types of analyses.
In addition to the above analyses, UPOs of some of the neurophysiological states of the brain such as eyes closed state, eyes opened state, chronic epilepsy and different stage of sleep are also extracted from Electroencephalogram (EEG) time series. All these states can be distinguished from one another through number and distribution of the dominant UPOs. There are about 16 dominant UPOs for the eyes closed cases, the recurrence time being less than 2 seconds. For the same individual, in the eyes open state, there are 12-14 UPOs, but more significantly they occur at much lower periods, i.e., with higher frequency. The maximum recurrence time is only ~ 1 sec. The number of dominant UPOs is limited to only 7 for the REM sleep stage, many of the UPOs of the eyes-closed stage merging to give fewer UPOs. There are also some UPOs occurring at much longer recurrence times in the REM stage. Distinction among the various sleep stages themselves is not however so clear-cut in terms of the UPOs. Generally sleep stage 2 has UPOs shifted to lower periodicity as compared to the REM stage.
The UPO analysis of pre-epilepsy (40 sec before epilepsy) and after-epilepsy (40 sec after epilepsy) are similar to results of the eyes-closed state where as during epileptic seizure, the UPOs are narrower than that of the REM stage based on the area under the peak. The most striking and distinguishing feature of epilepsy spectrum is that the low periodicity (high frequency) UPO dominates, accounting for about 40% of the total UPO intensities.
We have also calculated the eigenvalues of the UPOs to get the degree of unstableness and performed surrogate data analysis to test the statistical significance of the distribution of the UPOs.
We have also performed an electrical coupling between two Belousov-Zhabotinski (BZ) reactions in batch condition. One of the BZ oscillators is catalyzed by Cerium and the other by Manganese. The reaction catalyzed by Manganese oscillates at a higher frequency than the other. As far as our knowledge, this is the first instance that an electrical coupling of BZ reactions is carried out in batch conditions. The coupling strength is varied by using external resistance. We have found that the dynamics of the individual oscillators have moved from perfect limit cycle to chaotic behavior because of the coupling. The redox potential of the individual oscillators is used as time series for analysis. It is found from the studies such as Lyapunov exponent calculation, correlation dimension estimation, false-neighborhood method, predictability and surrogate analysis that the oscillator catalyzed by cerium undergoes more changes in its dynamics than the other oscillator i.e., the fast oscillating manganese system affects the dynamics of the slow oscillating cerium system.