Temporal complexity measure of reaction time series: Operational versus event time

Abstract

Introduction: Detrended fluctuation analysis (DFA) is a well-established method to evaluate scaling indices of time series, which categorize the dynamics of complex systems. In the literature, DFA has been used to study the fluctuations of reaction time Y(n) time series, where n is the trial number.

Methods: Herein we propose treating each reaction time as a duration time that changes the representation from operational (trial number) time n to event (temporal) time t, or X(t). The DFA algorithm was then applied to the X(t) time series to evaluate scaling indices. The dataset analyzed is based on a Go-NoGo shooting task that was performed by 30 participants under low and high time-stress conditions in each of six repeated sessions over a 3-week period.

Results: This new perspective leads to quantitatively better results in (1) differentiating scaling indices between low versus high time-stress conditions and (2) predicting task performance outcomes.

Conclusion: We show that by changing from operational time to event time, the DFA allows discrimination of time-stress conditions and predicts performance outcomes.

Keywords: detrended fluctuation analysis; reaction time series; temporal complexity; time-stress.

FIGURE 1

Left panel: Virtual reality Go–NoGo shooting task via participant's first‐person perspective in HTC Vive; right panel: example participant's reaction time trial series in low (top) and high (bottom) time‐stress conditions. Red circles indicate errors of commission; gaps in time series indicate lack of trigger responses on those trials (i.e., correct omissions to friendly targets or errors of omission to enemy targets).

FIGURE 2

The schematics of changing data from operational time Y(n) (a) to event time X(t) (c): (Panel a) the initial Y(n) data; (Panel b) each data point in operational time Y(n) is considered a duration time; (Panel c) to create X(t), each duration is filled with +1 or −1, assigned randomly (in this example, each latency interval is filled with +1, −1, +1, +1, and −1, respectively); (Panel d) the S(t) trajectory is the cumulative sum of X(t), notice that because of the random assignment of signs, there are many possible trajectories in event time X(t)’s. Thus, we use an ensemble average for our analysis to consider this variety, especially for short time series. In panels (b) and (c), t represents accumulated durations of reaction times. Note that the trajectory X(t) hosts both CEs and non‐CEs from the empirical trials. The trajectory is then analyzed using detrended fluctuation analysis (DFA) to determine the scaling index α, which is not the same index as that obtained by applying DFA to Y(n). This difference is discussed in the following section.

FIGURE 3

The left panels show the graph for detrended fluctuation analysis (DFA) on the Y(n) time series of one participant during an individual session (top–left) and during all sessions appended (bottom–left). The right panels show the graph for DFA on the corresponding X(t) time series (created according to the description in Figure 2) of the same participant during an individual session (top–right) and during all sessions appended (bottom–right).

FIGURE 4

The detrended fluctuation analysis (DFA) of simulated reaction time series. Each Y(n) is a simulated duration time generated by the idealized Manneville map (Buiatti et al., 1999) with corresponding temporal complexity index μ. The simulated Y(n) time series used to generate the corresponding simulated X(t) time series using the description of Figure 2. The length of the simulated trajectories (X(t) time series) were chosen to be similar to the X(t) dataset in the session level (blue line) and the appended sessions (black line). The dotted red line is the linear fit (Equation 4). The dotted green lines show an example of estimating the temporal complexity index μ from measured α using DFA applied to the X(t) time series.

FIGURE 5

Left panel: scaling indices measured by detrended fluctuation analysis (DFA) analysis of the Y(n) time series of participants in high (red dots) and low time‐stress (black squares) conditions; right panel: scaling indices measured by DFA analysis of the X(t) time series of each participant in high (red dots) and low time‐stress (black squares) conditions. Paired data are sorted by indices in low time‐stress condition in both panels. Each data point is averaged over 100 DFA analyses of the X(t) time series. Note the dispersion of red dots around black squares in the left panel (Y(n)) versus the consistently lower red dots around black squares in the right panel (X(t)), highlighting the superiority of the latter in differentiating scaling indices in the low versus high time‐stress conditions.

FIGURE 6

The left panel shows the results of applying detrended fluctuation analysis (DFA) to the Y(n) (black squares) and to shuffled Y(n) (red dots). The right panel depicts the results of applying DFA to X(t) created from Y(n) (black squares) and to X(t) created form the shuffled Y(n) (red dots).

FIGURE 7

The figures show the probability of errors of commission versus scaling indices of detrended fluctuation analysis (DFA) on Y(n)’s of all sessions of all participants (top figures) and appended sessions of all the participants (bottom figures) for two cases of low (left figures) and high time‐stress conditions (right figures).

FIGURE 8

The figures show the trend between the probability of errors of commission and scaling indices of detrended fluctuation analysis (DFA) analysis on the X(t) time series of all sessions of all participants (top figures) and appended sessions of all the participants (bottom figures) for two cases of low (left figures) and high time‐stress conditions (right figures). Each data point is averaged over 100 DFA on X(t)s.