Levels of Complexity in Scale-Invariant Neural Signals
  Many physical and physiological signals exhibit complex scale-invariant features characterized by $1/f$ scaling and long-range power-law correlations, suggesting a possibly common control mechanism. Specifically, it has been suggested that dynamical processes influenced by inputs and feedback on multiple time scales may be sufficient to give rise to $1/f$ scaling and scale invariance. Two examples of physiologic signals that are the output of hierarchical, multi-scale physiologic systems under neural control are the human heartbeat and human gait. Here we show that while both cardiac interbeat interval and gait interstride interval time series under healthy conditions have comparable $1/f$ scaling, they still may belong to different complexity classes. Our analysis of the magnitude series correlations and multifractal scaling exponents of the fluctuations in these two signals demonstrates that in contrast with the nonlinear multifractal behavior found in healthy heartbeat dynamics, gait time series exhibit less complex, close to monofractal behavior and a low degree of nonlinearity. These findings are of interest because they underscore the limitations of traditional two-point correlation methods in fully characterizing physiologic and physical dynamics. In addition, these results suggest that different mechanisms of control may be responsible for varying levels of complexity observed in physiological systems under neural regulation and in physical systems that possess similar $1/f$ scaling. 
Plamen Ch. Ivanov, Jeffrey M. Hausdorff, S. Havlin, Lu\'is A. Nunes   Amaral, Kuniharu Arai, Verena Schulte-Frohlinde, Mitsu...