From conformons to human brains: an informal overview of nonlinear dynamics and its applications in biomedicine

Abstract

Methods of contemporary physics are increasingly important for biomedical research but, for a multitude of diverse reasons, most practitioners of biomedicine lack access to a comprehensive knowledge of these modern methodologies. This paper is an attempt to describe nonlinear dynamics and its methods in a way that could be read and understood by biomedical professionals who usually are not trained in advanced mathematics. After an overview of basic concepts and vocabulary of nonlinear dynamics, deterministic chaos, and fractals, application of nonlinear methods of biosignal analysis is discussed. In particular, five case studies are presented: 1. Monitoring the depth of anaesthesia and of sedation; 2. Bright Light Therapy and Seasonal Affective Disorder; 3. Analysis of posturographic signals; 4. Evoked EEG and photo-stimulation; 5. Influence of electromagnetic fields generated by cellular phones.

Figure 1

Example of hormesis – biological response to chemical and physical agents. Deficiency symptoms are caused by deficit of an agent (dose less than D); small doses (between D and T) are vital for good health (shaded area), doses higher than T cause toxic or other harmful effects. Dotted line represents linear no-threshold relationship, solid line represents hormetic dose-effect relationship (after [10])

Figure 2

Example of deterministic chaos – sensitivity of solution of Lorenz equations (3) illustrated by dependence of x on t; state variables y and z show similar chaotic behavior. a. sensitivity to system parameters for initial conditions (5a), solutions for parameters (4a) (crosses) and (4b) (squares), respectively; b. sensitivity to initial conditions for system parameters (4a), solutions for initial conditions (5a) (crosses), (5b) (squares), (5c) (diamonds) respectively.

Figure 3

FFT applied to similar stationary (upper a.) and nonstationary (upper b.) signals gives dramatically different results (bottom), while application of Higuchi's algorithm gives quite similar values of average fractal dimension, D_f, of both signals.

Figure 4

'Epileptic seizures' in 'economic organism'. a. EEG-epoch showing an epileptic seizure (top) and its fractal dimension (bottom); b. Dow Jones index from the period of 'big crash' (top) and its fractal dimension (bottom).

Figure 5

Measuring the depth of sedation (cf. [19]) – fractal and symbolic dynamics methods of EEG analysis give similar results as BIS method. In this case sedation has been employed to secure adequate comfort for 65-year-old male patient during long, unpleasant procedure (colonoscopy with polypectomy). Sedation has been controlled according to the BIS (target BIS value between 60 and 80) with intermittent boluses of propofol. At 16. and 27. minutes of the study sedation was lightened to level 3 in OAA/S score. Record showed that at those moments (D_f - 1)*100 rose rapidly to rich the highest value. BIS increased only once (at 27. minute) while at 16. minute remained unchanged. Awakening can be predicted by rise of the (D_f - 1)*100 towards its highest value. Exact dosage and timing of drugs administration are also shown.

Figure 6

Fractal dimension of 19 seconds EEG-epoch (P4-O2) with eyes-opening (in 4. s.) and eyes-closing (in 15. s.).

Figure 7

Fractal dimension of EEG-signal for 80 sec. epochs (channel P3-O1) for a patient with SAD. a. before BLT; b. after BLT.

Figure 8

Example of COM (a.) and COP (b.) signals recorded simultaneously in a subject 74 years old during quiet stance in a 2-minutes trial; displacements measured in millimeters.

Figure 9

Fractal dimensions (mean values with standard deviation bars) of COM (a.) and COP (b.) signals in the elderly subjects during quiet stance with eyes open (EO) and eyes closed (EC); AP – for anteroposterior displacement; ML – for mediolateral displacement.

Figure 10

Fractal dimension at T6-O2 channel during photo-stimulation of a healthy subject.

Figure 11

Higuchi's fractal dimension of 300 sec. long epochs of EEG-signals. a. for a person who does not show sensitivity to EMF; b. for a person (hyper)sensitive to EMF; basal – phone at place but not in use; sin – in use, without NPD; con – in use, with NPD; registered on 4 channels (respectively, in each column upside down): T6-O2; T4-T6; F8-T4; Fp2-F8 (from back to front of the head).