Universal Scaling Law in Human Behavioral Organization

We describe the nature of human behavioral organization, specifically how resting and active periods are interwoven throughout daily life. Active period durations with physical activity count successively above a predefined threshold, when rescaled with individual means, follow a universal stretched exponential (gamma-type) cumulative distribution with characteristic time, both in healthy individuals and in patients with major depressive disorder. On the other hand, resting period durations below the threshold for both groups obey a scale-free power-law cumulative distribution over two decades, with significantly lower scaling exponents in the patients. We thus find universal distribution laws governing human behavioral organization, with a parameter altered in depression.

Figure 1

Illustrative examples of locomotor activity data for a healthy control subject (a) and a depression patient (b) over five consecutive days (upper) and 4-hour periods during the 4th day (lower). The overall average of nonzero activity counts is used for a threshold, and the period in which the counts are successively below or above the threshold is coded as a resting (open bar in bottom panels) or active (closed bar) period, respectively.

Figure 2

Cumulative distributions of resting and active period durations in human locomotor activity. (a) Double logarithmic plots of cumulative distributions $P(x\ge a)$ of resting period durations $a$ for control subjects (open circles) and for depression patients (filled squares). The error bars indicate the standard error of the mean. The straight lines are eye-guides with the mean scaling exponents from 2 min to 200 min for each group as the slopes. (b) The same as (a) but for active period durations. The curves are stretched exponential functions $P(x\ge a)=\exp (-\alpha a^{\beta })$ with the mean $\alpha$ and $\beta$ values for each group. (c) Rescaled cumulative distributions of resting period durations with the rescaled time $a/\overline{a}$, where $\overline{a}$ is the individual mean of the resting period durations, for both healthy subjects (dotted lines) and patients with major depression (solid lines). (d) The same as (c) but for active period durations. (e) Group average rescaled cumulative distributions of resting period durations for controls (open circles) and depression (filled squares). The straight lines are eye-guides with the mean $\overline{\gamma }$ for each group as the slopes. (f) The same as (e) but for active period durations. The curve is a stretched exponential function with the overall mean $\overline{\alpha }$ and $\overline{\beta }$ values for both groups.

Figure 3

Dependency of cumulative distribution of locomotor activity on the threshold to determine resting and active periods. (a) Double logarithmic plots of rescaled (by the mean resting duration $\overline{a}$) cumulative distributions $P(x\ge a/\overline{a})$ for control subjects with different threshold values of 0.6, 0.8, 1.0, 1.2, 1.4, and 1.6 times the overall average for nonzero activity counts. The straight line is an eye-guide with the mean $\overline{\gamma }$ for the control subjects as the slope. (b)–(d) The same as (a), but of active period durations (b), of resting periods for depression patients (c), and of active periods for depression patients.

Figure 4

Cumulative distributions of the burst size ($S$) for active periods. The threshold is the same as that used in Fig. 2. (a) Double logarithmic plots of cumulative distributions $P(x\ge S)$ for control subjects, with the solid regression line for [200, 2000] counts. The error bars indicate the standard error of the mean. The dotted line is an eye-guide with the slope of $-0.5$. (b) The same as (a), but for depression patients.