ANALYSIS OF RECURRENCE AND VARIABILITY OF NON LINEAR DYNAMICAL SYSTEMS: A review of methods for non linear analysis of interest in neurology and psycho-physiology

In the light of the results obtained during the last two decades in analysis of signals by time series, it has become evident that the tools of non linear dynamics have their elective role of application in biological, and, in particular, in neuro-physiological and psycho-physiological studies. A statement is repeatedly remarked in this review: the basic concept in non linear analysis of experimental time series is that one of recurrence whose conceptual counterpart is represented from variedness and variability that are the foundations of complexity in dynamic processes. As consequence of this statement, the recurrence plots and the Recurrence Quantification Analysis (RQA)are discussed . It is shown that RQA represents the most general and correct methodology in investigation of experimental time series. By it we arrive to inspect the inner structure of the time series connected to the signals under investigation. Linked to RQA we prospect also the method CZF, recently introduced by us. It is able to account for a true estimation of variability of signals in time as well as in frequency domain. and, consequently, it may be used in conjunction with classical Fourier analysis, accounting however that it is inappropriate in analysis of non linear and non stationary experimental time series. The use of CZF method in fractal analysis is also considered in addition to standard index as Hurst exponent. A large field of possible applications in neurological as well as in psycho-physiological studies is given. Also, there are given examples of other and (possibly linked) applications as example the analysis of beat –to-beat fluctuations of human heartbeat intervals that is sovereign in psycho-physiological studies. We give applications on some different planes to evidence the particular sensitivity of such methods. In particular, we consider the cases of young and old normal subjects and of old subjects about thirteen minutes before the advent of severe episodes of heart dynamics as the ventricular tachycardia and the ventricular fibrillation. We reach the objective to show that the previously exposed methods are also able to predict in advance the advent of ventricular tachycardia and /or of ventricular fibrillation. The RQA analysis gives good results. The CZF method gives the most excellent results showing that it is able to give very significant indexes of prediction. In analysis of variability of R-R signals in domain of frequency, CZF also reveals some feature of heart rate variability previously unknown as the presence or the absence of correlation of total variability with the VLF, LF, HF bands, usually taken in consideration in analysis of heart rate variability. Finally, we apply such methods in investigation of state anxiety, and proposing in detail also a quantum like model of such phenomenological status of the mind. 1. The Recurrences and the Variability of Signals in Nature Only few systems in Nature exhibit linearity. The greatest whole of natural systems, especially those who pertain to biological matter, to physiological (neuro-physiological and psychophysiological) and to psychological processes, possess a complexity that results in a great variedness and variability, linked to non linearity, to non stationarity, and to non predictability of their time dynamics. In the time domain, traditional methods were first used to describe the amplitude distribution of signals and later, methodologies used spectral analysis methods. However, they suffer of fundamental limits. They are applied assuming linearity and stationarity of signals that actually do not exist. The consequence is that such methods are unable to analyse in a proper way the irregularity present in most of signals. The results show that such irregularity is at the basis of the dynamics that we intend to explore. It reveals that complex behaviours of the system are very distant from previously accepted principles as it is the case, as example for biological signals, on the view of homeostatic equilibrium and of other similar mechanism of controls. The study of this very irregular behaviour requires the introduction of new basic principles. Therefore, nonlinear science is becoming an emerging methodological and theoretical framework that makes up what is called the science of the complexity, often called also chaos theory. 2. The Chaos Theory. The aim of non linear methodologies is a description of complexity and the exploration of the multidimensional interactions within and among components of given systems. An important concept here is that of chaotic behaviour. It will be defined chaotic if trajectories issuing from points of whatever degree of proximity in the space of phase, distance themselves from one another over time in an exponential way. In detail, the basic critical principles may be reassumed as it follows: 1) Non linear systems under certain conditions may exhibit chaotic behaviour; 2) The behaviour of a chaotic system can change drastically in response to small changes in the system’s initial conditions; 3) A chaotic system is deterministic; 4) In chaotic systems the output system is no more proportionate to system input. Chaos may be identified in systems also excluding the requirement of determinism. The standard approach to classical dynamics assumes the Laplace point of view that the time evolution of a system is uniquely determined by it’s initial conditions. Existence and uniqueness theorem of differential equations require that the equations of motion everywhere satisfy the Lipschitz condition. It has long been tacitly assumed that Nature is deterministic, and that correspondingly, the equations of motion describing physical systems are Lipschitz. However, there is no a priori reason to believe that Nature is unfailingly Lipschitzian. In very different conditions of interest, some systems exhibit physical solutions corresponding to equations of motion that violate the Lipschitz condition. The point is of particular interest. If a dynamical system is non-Lipschitz at a singular point, it is possible that several solutions will intersect at this point. This singularity is a common point among many trajectories, and the dynamics of the system, after the singular point is intersected, is not in any way determined by the dynamics before. Hence the term non-deterministic dynamics takes place. For a non-deterministic system, it is entirely possible (if not likely) that as the various solutions move away from the singularity, they will evolve very differently, and tend to diverge. Several solutions coincide at the non-Lipschitz singularity, and therefore whenever a phase space trajectory comes near this point, any arbitrarily small perturbation may push the trajectory on to a completely different solution. As noise is intrinsic to any physical system, the time evolution of a non-deterministic dynamical system will consist of a series of transient trajectories, with a new one being chosen randomly whenever the solution (in the presence of noise) nears the non-Lipschitz point. We term such behaviour non-deterministic chaos. This approach to chaos theory was initiated by Zak, Zbilut and Webber [1] and rather recently we have given several examples, theoretical and experimental verifications on this important chaotic behaviour [2]. 2.1Embedding time series in phase space. The notion of phase space is well known in physics. Let us consider a system, determined by the set of its variables. Since they are known, those values specify the state of the system at any time. We may represent one set of those values as a point in a space, with coordinates corresponding to those variables. This construction of space is called phase space. The set of states of the system is represented by the set of points in the phase space. The question of interest is that we perform an analysis of the topological properties of phase space but, as a counterpart, we obtain insights into the dynamic nature of the system. In experimental conditions, especially in experimental clinical studies, we are unable to measure all the variables of the system. In this case we may be able to reconstruct equally a phase space from experimental data where only one of the present variables (characterizing the whole system) is actually measured. The phase space is realized by a set of independent coordinates. Generally speaking, the attractor is the phase space set generated by a dynamical system represented by a set of difference or differential equations. In the actual case, let us take a non linear dynamical system represented by three independent variables ) ( ), ( ), ( t Z t Y t X , functions of time. The phase space set is given by the values of the variables at each time. The point ( ) , , z y x in phase space gives the values of the three variables and thus the state of the system at each time. Usually, in physics, for example, we plot one of the variables and its derivatives,