Emergence of dynamical complexity related to human heart rate variability

We apply the refined composite multiscale entropy (MSE) method to a one-dimensional directed small-world network composed of nodes whose states are binary and whose dynamics obey the majority rule. We find that the resulting fluctuating signal becomes dynamically complex. This dynamical complexity is caused (i) by the presence of both short-range connections and long-range shortcuts and (ii) by how well the system can adapt to the noisy environment. By tuning the adaptability of the environment and the long-range shortcuts we can increase or decrease the dynamical complexity, thereby modeling trends found in the MSE of a healthy human heart rate in different physiological states. When the shortcut and adaptability values increase, the complexity in the system dynamics becomes uncorrelated.

Figure 1

(a) An illustration of the structure of a one-dimensional directed SW network, where the system size $N=100$ and the mean excess degree $f_e=0.2$. (b) The temporal state fluctuation of nodes $i=1000,...,2000$ after waiting a transient 8192 time steps, where $N=4096$, $f_e=0.2$, and ${\Pi }_n=0.2$. White and black regions represent ${\sigma }_i\left(t\right)=-1$ and $+1$, respectively. (c) The system dynamical behavior $S\left(t\right)={\sum }_{i=1}^{4096}{\sigma }_i\left(t\right)$, where $f_e=0.2$ and ${\Pi }_n=0.2.$

Figure 2

(a) The sample entropy (SampEn) of revised ${\mathrm{RR}}^{\prime }$ time series of 14 healthy young people during waking period, 7 men and 7 women, aged $30.5\pm 4.4$ years (mean $\pm$ SD), range 20–35 years. Symbols represent the average values of sample entropy, and the bars are the standard error. The time series length is $3\times {10}^4$ beats. (b) The black circles showing the sample entropy of fluctuating $S^{\prime }\left(t\right)$ by RCMSE from scale factor $\tau =10$–400, where $f_e=0.2$, ${\Pi }_n=0.1$, and the rescaled ${\tau }^{\prime }$ is given by dividing $\tau$ by 20. The red squares depict the entropy results from the shuffled $S^{\prime }\left(t\right)$. Each symbol is averaged by ten rounds. The bars are the standard error.

Figure 3

(a) The sample entropy (SampEn) of revised ${\mathrm{RR}}^{\prime }$ series of 14 healthy young subjects (blue circles) and the 16 healthy elderly subjects (wine squares) during waking period. The 16 healthy elderly subjects are 9 men and 7 women, aged $70.0\pm 3.1$ years (mean $\pm$ SD), range 68–76 years (t-test, $p<0.05$). Symbols represent the average values of sample entropy, and the bars are the standard error. The time series length is $3\times {10}^4$ beats. The entropy of elderly subjects is lower than that of young subjects over all scale factors. (b) and (c) The sample entropy of $S^{\prime }\left(t\right)$ for ${\Pi }_n=0.05–0.10$, ${\Pi }_n=0.10–0.22$, and ${\Pi }_n=1.0$, where $f_e=0.2$, $\tau =10$–400, and the rescaled ${\tau }^{\prime }$ is given by dividing $\tau$ by 20. Each symbol is averaged by ten rounds. The bars are the standard error.

Figure 4

The complexity index obtained by summing the area of sample entropy in Figs. 3 and 3. Squares and circles represent the range for ${\tau }^{\prime }=1$–10 and 11–20, respectively. The bars are the standard error.

Figure 5

(a) The panels of how to develop a SW network by adding $\Delta f_{e}N$ long-range links to make $f_e=0.08$ (left), 0.12 (middle), and 0.16 (right), where $N=100$ and $\Delta f_e=0.04$. (b) The corresponding sample entropy (SampEn) of $S^{\prime }\left(t\right)$ for $f_e=0.08$, 0.12, 0.16, 0.20, and 1.00, where ${\Pi }_n=0.10$, $\tau =10$–400, and the rescaled ${\tau }^{\prime }$ is given by dividing $\tau$ by 20. Each symbol is averaged by ten rounds. The bars are the standard error.