From conformons to human brains: an informal overview of nonlinear dynamics and its applications in biomedicine <s> Higuchi's fractal dimension in time domain <s> We present a technique to measure the fractal dimension of the set of points (t, f(t)) forming the graph of a function f defined on the unit interval. First we apply it to a fractional Brownian function [1] which has a property of self-similarity for all scales, and we can get the stable and precise fractal dimension. This technique is also applied to the observational data of natural phenomena. It does not show self-similarity all over the scale but has a different self-similarity across the characteristic time scale. The present method gives us a stable characteristic time scale as well as the fractal dimension. <s> BIB001 </s> From conformons to human brains: an informal overview of nonlinear dynamics and its applications in biomedicine <s> Higuchi's fractal dimension in time domain <s> Human brain is a highly complex nonlinear system. So it is not surprising that in analysis of EEG‐signal, which represents overall activity of the brain, the methods of Nonlinear Dynamics (or Chaos Theory as it is commonly called) can be used. Even if the signal is not chaotic these methods are a motivating tool to explore changes in brain activity due to different functional activation states, e.g. different sleep stages, or to applied therapy, e.g. exposure to chemical agents (drugs) and physical factors (light, magnetic field). The methods supplied by Nonlinear Dynamics reveal signal characteristics that are not revealed by linear methods like FFT. Better understanding of principles that govern dynamics and complexity of EEG‐signal can help to find ‘the signatures’ of different physiological and pathological states of human brain, quantitative characteristics that may find applications in medical diagnostics. <s> BIB002
As an example of nonlinear methods we take Higuchi's fractal dimension in time domain BIB001 . It is, in fact, fractal dimension of the curve representing amplitude of the signal under consideration on a plane as a function of time. So, it is always between 1 and 2, since as everybody knows a simple curve has dimension equal 1 and a plane has dimension equal 2. Imagine a classical registration device that records a signal like EEG by drawing with a pen a curve on a moving paper tape. Such a curve may in some degree fill out the plane (here the tape surface); if the device registers pure noise the plane will be filled out curve. Higuchi's fractal dimension should not be confused with fractal dimension of an attractor in the phase space. D f may be calculated in a time window containing so few as ca. 100 data points and the window can be moved along the signal, so giving substantial compression of data. Using moving window one obtains running fractal dimension D f (t) that shows changes of the signal's complexity in time. Higher values of D f correspond to presence of higher frequencies in the signal's Fourier spectrum (cf. BIB002 ). While running fractal dimension may be used to characterize short-lasting phenomena like eye blinking, the mean fractal dimension value averaged over longer period of time also serves as a meaningful characteristic of studied biosignals. Moreover, while linear methods, like FFT or WT, work properly only for stationary signals, Higuchi's methods may be used to characterize nonstationary signals too (Fig. 3) -signals shown at the upper part look slightly different and their mean fractal dimensions are also only slightly different but their power spectra (the lower part) are immensely different; it is so because the second signal is nonstationary. Signal presented in Fig. 3a upper part was 'synthesized' by adding together 5 simple sinusoidal signals of different frequencies; when this artificial signal is analyzed using FFT we can clearly see in its spectrum those 5 components (Fig. 3a bottom) . Signal presented in Fig. 3b upper part was obtained from this shown on Fig. 3a by repeating several times the following procedure: 1. choosing randomly a short signal epoch; 2. removing the chosen epoch; 3. 'stitching together' two remaining parts of the signal. In the place of such a stitching 'jump' of signal's amplitude usually arises. This procedure is in fact very similar to artifacting of EEG-signal. One can observe that after such a procedure it is not possible to see in the signal's spectrum