The authors of this chapter measured plethysmography and calculated the Largest Lyapunov Expornent (LLE) using non-linear analysis. They found that the value of LLE was significantly related to the severity of dementia and the communication skill in the ADL index for 144 elderly individuals. The authors developed a mathematical model to analyze the results by studying the information extracted from the plethysmogram data. Furthermore, data were collected when the central nerve was blocked by general anesthesia to evaluate the mathematical model. The pulse wave data indicated that the authors included information from the nucleus of the brain origin. In other words, they obtained conclusive evidence of dementia using the LLE and communication skills. The authors measured pulse waves while elderly individuals had a conversation. They calculated the activation of the sympathetic nerve and the parasympathetic (LF/HF, HF) response simultaneously. LLE that was activated by communication had a low HF, and the HF was high in individuals who were not activated. In other words, an effect of communication was observed in conscious elderly individuals. Communication scientifically indicated the mental activity of elderly individuals.

Top## Introduction

Biogenic information, such as heartbeat, blood pressure, and blood flow, is coded in complex systems. The dynamic rhythm of the biogenic information is neither constant, such as that of a metronome, nor random. Most of the natural world has chaos in the dynamic rhythm.

The blood flow in capillaries of the finger was examined by infrared rays. Figure 1 shows the method of measurement of finger plethysmography using a sampling rate of 200 Hz and a 12-bit resolution.

*Figure 1. *Method of recording finger wave pulses

In a given time series (*x*(*i*), with *i*=1…*N*), the phase space is reconstructed using a method of delays. Assuming that we create a d-dimensional phase space using a τ constant delay lag, the vectors in the space are formed by d-tuples from the time series and are given by**x**(*i*)=(*x*(*i*),…,*x*(*i*-(*d*-1)τ))={*x*_{k}(*i*)} *(1)* where *x*_{k}(*i*)=*x*(*i*-(*k*-1)τ), with *k*=1,..., *d*. To correctly reconstruct the phase space, the parameters of delay lag τand embedding dimension d should be carefully chosen. For the reconstructed phase space, one of the important complexity measures is the LLE. The LLE characterizes how a set of infinite orthonormal small distances evolves according to its dynamics. For a chaotic system, there is at least one positive Lyapunov exponent (λ_{1}> 0 is the largest exponent). The defining property of chaos is dependent on the initial conditions. Given an initial infinitely small distance ∆*x*(0), its evolution obeys

*(2)*For an M-dimensional dynamical system, there are M Lyapunov exponents. We estimated only λ_{1} using the algorithm of Sano and Sawada (1985; paraneters: d-dimensional phase space d= 4. τ: time delay 50 ms).

Top## Mathematical Model

To understand changes in chaos in the finger plethysmograms, a mathematical model was proposed. Figure 2 shows a schematic description of the model used in this paper. The model consists of a feedback loop and physiological factors (Miao et al 2006). The pressure receptors sense and transmit neural afferent signals to the cardio-vascular center. Neural efferent signals are created and then sent to effectors. Influences come from respiratory centers and from higher cerebral regions.

*Figure 2. *Schematic representation of the model

Pulsation of the blood in the ear was represented as a response function to pulsation in the radial artery, and a proportional relationship between finger plethysmogram and artery blood pressure was approximated. Thus, for the sake of simplifying unimportant details, our model concentrated on the dynamics of blood pressure in finger plethysmograms without a loss of generalization.

Top## Verification From Anesthetizing Experiment

We used anesthetized subjects to verify the hypothetical mathematical model. The patient who participated in the experiment was a 71-year-old male. He was put into deep anesthesia during cancer surgery.

The surgery took place at Rakuwakai Otowa Hospital, Kyoto on December 12, 2008. The participant provided informed consent.