We propose a definition of complexity based on the observation that self-organization and synchronization of individual constituents (neurons) generate non-ergodic and non-Poisson renewal events. Thus, we define a system as complex if it produces non-ergodic and non-Poisson sequences of events. According to this definition, a complex system is characterized by a complexity index ranging from 1 to 2.
We illustrate also a technique of statistical analysis of real data to assess if a system is complex and, if it is, which is its complexity index. We show how this technique works with applications to EEG data and to music composition. We prove that complex systems do not respond to harmonic perturbations.
We argue that the popular phenomenon of stochastic resonance can be interpreted as a form of rate matching, thereby explaining why systems without a fixed time scale cannot respond to perturbations producing a finite number of events per unit of time. With the same arguments we reach the conclusion that the transfer of information from a complex perturbation, with index º(P), to a complex system, with index º(S), reaches the efficiency maximum at the matching point º(P) = º(S).