In nature we find many instances of complex behavior for example the dynamics of stock markets, power grids, internet networks, highway traffic, social networks, heartbeat dynamics, neural dynamics, dynamics of living organisms, etc. The study of these complex systems involves the use of tools of non-linear dynamics and non-equilibrium statistical physics. This dissertation is devoted to understanding two different sources of complex behavior – non-poissonian renewal events also called crucial events and infinite memory of fractional Brownian motion. They both generate 1/f noise frequency spectrum. Thus, we studied examples of both processes and also their joint action. We also tried to establish the role of crucial events in biological and physiological processes like biophoton emission during the germination of seeds, the dynamics of heartbeat and neural dynamics. Using a statistical method of analyzing the time series of bio signals we were able to quantify the complexity associated with the underlying dynamics of these processes. Finally, we adopted a model that unifies both crucial events and memory fluctuations to study the rhythmic behavior observed in heart rate variability of people during meditation. We were able to also quantify the level of stress reduction during meditation. The work presented in this dissertation may help us understand the communication and transfer of information in complex systems.

This dissertation is an attempt at establishing a bridge between biology and physics leading naturally from the field of phase transitions in physics to the cooperative nature of living systems. We show that this aim can be realized by supplementing the current field of evolutionary game theory with a new form of self-organized temporal criticality. In the case of ordinary criticality, the units of a system choosing either cooperation or defection under the influence of the choices done by their nearest neighbors, undergo a significant change of behavior when the intensity of social influence has a critical value. At criticality, the behavior of the individual units is correlated with that of all other units, in addition to the behavior of the nearest neighbors. The spontaneous transition to criticality of this work is realized as follows: the units change their behavior (defection or cooperation) under the social influence of their nearest neighbors and update the intensity of their social influence spontaneously by the feedback they get from the payoffs of the game (environment). If units, which are selfish, get higher benefit with respect to their previous play, they increase their interest to interact with other units and vice versa. Doing this, the behavior of single units and the whole system spontaneously evolve towards criticality, thereby realizing a global behavior favoring cooperation. In the case when the interacting units are oscillators with their own periodicity, homeodynamics concerns, the individual payoff is the synchronization with the nearest neighbors (i.e., lowering the energy of the system), the spontaneous transition to criticality generates fluctuations characterized by the joint action of periodicity and crucial events of the same kind as those revealed by the current analysis of the dynamics of the brain. This result is expected to explain the efficiency of enzyme catalyzers, on the basis of a new non-equilibrium statistical physics. We argue that the results obtained apply to sociological and psychological systems as well as to elementary biological systems.

This dissertation is devoted to study of complex systems in human physiology particularly heartbeats and brain dynamics. We have studied the dynamics of heartbeats that has been a subject of investigation of two independent groups. The first group emphasized the multifractal nature of the heartbeat dynamics of healthy subjects, whereas the second group had established a close connection between healthy subjects and the occurrence of crucial events. We have analyzed the same set of data and established that in fact the heartbeats are characterized by the occurrence of crucial and Poisson events. An increase in the percentage of crucial events makes the multifractal spectrum broader, thereby bridging the results of the former group with the results of the latter group. The crucial events are characterized by a power index that signals the occurrence of 1/f noise for complex systems in the best physiological condition. These results led us to focus our analysis on the statistical properties of crucial events. We have adopted the same statistical analysis to study the statistical properties of the heartbeat dynamics of subjects practicing meditation. The heartbeats of people doing meditation are known to produce coherent fluctuations. In addition to this effect, we made the surprising discovery that meditation makes the heartbeat depart from the ideal condition of 1/f noise. We also discussed how to combine the wave-like nature of the dynamics of the brain with the existence of crucial events that are responsible for the 1/f noise. We showed that the anomalous scaling generated by the crucial events could be established by means of a direct analysis of raw data. The efficiency of the direct analysis procedure is made possible by the fact that periodicity and crucial events is the product of a spontaneous process of self-organization. We argue that the results of this study can be used to shed light into the nature of this process of self-organization.