Physica A 390 (2011) 1539–1545

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The scaling behavior of hand motions reveals self-organization during an executive function task Jason R. Anastas a,b,∗ , Damian G. Stephen c,d , James A. Dixon a,b,e a

Department of Psychology, University of Connecticut, United States

b

Center for the Ecological Study of Perception and Action, University of Connecticut, United States

c

Wyss Institute for Biologically Inspired Engineering, Harvard University, United States

d

Department of Psychiatry, Children’s Hospital Boston, Harvard Medical School, United States

e

Haskins Laboratories, United States

article

info

Article history: Received 2 August 2010 Received in revised form 14 November 2010 Available online 8 December 2010 Keywords: Executive function Detrended fluctuation analysis Interaction-dominance Power-laws Self-organization

abstract Recent approaches to cognition explain cognitive phenomena in terms of interactiondominant dynamics. In the current experiment, we extend this approach to executive function, a construct used to describe flexible, goal-oriented behavior. Participants were asked to perform a widely used executive function task, card sorting, under two conditions. In one condition, participants were given a rule with which to sort the cards. In the other condition, participants had to induce the rule from experimenter feedback. The motion of each participant’s hand was tracked during the sorting task. Detrended fluctuation analysis was performed on the inter-point time series using a windowing strategy to capture changes over each trial. For participants in the induction condition, the Hurst exponent sharply increased and then decreased. The Hurst exponents for the explicit condition did not show this pattern. Our results suggest that executive function may be understood in terms of changes in stability that arise from interaction-dominant dynamics. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The cognitive system is a complex aggregate of many mutually dependent subsystems, such as those that support working memory [1] and goal-directed behavior [2]. Despite the heterogeneity of the processes and structures involved in cognition, the cognitive system is capable of sustained, directed action. This interesting phenomenon is part of a larger group of functional attributes called executive function [3]. While traditional theories of cognition tend to implicitly treat executive function as if it were an intelligent internal agent that runs the cognitive system [4], recent developments in cognitive science have begun to discard these notions. Current theory emphasizes the relationship between stability and instability to explain the maintenance of current structures and the transition to new structures [5–7]. Under this description of cognition, structures do not rise and fall from the workings of a central executive; instead, they are self-organized from the interaction of components in the cognitive system [8,9]. The phenomena that define executive function would likewise emerge from the self-organization of the system. Systems that exhibit self-organization have interaction-dominant dynamics. Interaction-dominant dynamics entails that the activity of a system cannot be separated into the activity of each component alone [7]. Activity of this kind will result in power-law distributions, where the magnitude of an event response is inversely related to its frequency. Power-laws have

∗ Corresponding address: Department of Psychology, 406 Babbidge Road, Unit 1020, University of Connecticut, Storrs, CT 06269-1020, United States. Tel.: +1 603 809 7467. E-mail address: [email protected] (J.R. Anastas). 0378-4371/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2010.11.038

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been found in a number of cognitive tasks [10], such as word naming [7], problem solving [11], memory retrieval [12], and lexical decision making [13]. On such an account, the stability and change in executive control would be driven by the activity of the system. Under some conditions, the activity of a complex system may increase the stability of a structure by creating additional constraints among its elements. Alternatively, when constraints in the system become loosened, allowing interactions among the elements, a new structure may emerge. The interactions drive the system into a new configuration. Once the reorganization has occurred, the constraints tighten, and the system settles into a new kind of order [14]. These changes in stability can be indexed using the Hurst exponent [15]. The Hurst exponent is an estimate of the long-range correlations in a time series of activity, such as hand or eye motions. The Hurst exponent has been found to increase as the system becomes less stable and decreases as the system settles into a new regime [16]. 1.1. Card-sorting tasks and executive function In the current manuscript, we investigate a primary phenomenon used to assess executive function: the ability to flexibly switch amongst rules. A large body of work has demonstrated that the ability to switch rules during card-sorting tasks provides a measure of executive function. For example, Zelazo et al. [17] asked children, aged 3 to 5 years, to sort cards into two piles along a given dimension (e.g., color). After sorting 10 cards, the rule was explicitly changed (e.g., children were now asked to sort by shape). They showed that young children had difficulty sorting by the new rule, and that the degree to which they could easily switch rules predicted performance on other cognitive-control tasks. In other card-sorting tasks, the rule is changed without the participants’ knowledge [18]. In these tasks, participants must induce the new rule. As mentioned above, performance on these tasks predict other cognitive-control phenomena [17]. Thus, card-sorting tasks provide a widely accepted instance of the cognitive system imposing new constraints on itself. When a new rule is given explicitly, the constraints change on a fast-time scale. When the rule must be induced over multiple trials, the constraints change more slowly. The current constraints are loosened as the participant encounters errors and then become stronger as the new rule coalesces. In the present experiment, we tracked the hand motions of adult participants during a card-sorting task. Detrended fluctuation analysis [19] was performed on the inter-point time series to compute the Hurst exponents, as explained below. We investigated the dynamics underlying the ability to switch rules under two conditions. In one condition, the rule was explicitly stated. In the other condition, participants had to induce the rule. In both conditions, participants sorted cards until they consecutively placed 10 cards successfully, according to the current rule. We predicted that in the explicitrule condition participants would rapidly organize to the stated rule (i.e., even before sorting had begun), and that their cognitive organization would become more stable as they continued the task. In the induction condition, we predicted that participants’ cognitive organization would become increasingly unconstrained as they approached discovery of the new rule, and then more constrained as the new rule emerged. We therefore predicted that the Hurst exponent would rise as the participants in the induction condition became less constrained in response to the task, and that it would then peak, and fall as the participants reorganized to the rule at hand. 2. Method Twenty-six University of Connecticut undergraduate and graduate students participated. The participants were assigned to either the ‘‘explicit’’ or ‘‘induction’’ condition, with thirteen students in each group. All participants gave consent in accordance with University of Connecticut informed consent procedures. Undergraduate participants received credit toward course completion; graduate students were not compensated for participation. Participants were required to take cards from a facedown deck and place them into one of four piles, based on one of three dimensions. Each dimension contained four levels, with one level of each dimension on a card. For example, one card had a picture of a red wolf wearing a hat; ‘‘wolf’’, ‘‘red’’, and ‘‘hat’’ each represented a level of one dimension (‘‘animal’’, ‘‘color’’, and ‘‘clothing item’’, respectively). The deck that each participant used contained one card for every unique combination of the three dimensions, making for a deck with sixty-four cards. Decks were randomly shuffled between trials. During each trial, participants were to sort along one dimension or ‘‘rule’’. Each participant completed 5 trials. Participants placed each card into one of four guide piles. Each pile contained a guide card, with each level of each rule represented once across the guide cards. Participants placed cards based on the rule for that trial. For example, one guide card contained a picture of a red wolf wearing a hat. This was the only guide card to contain these three characteristics; participants would then place into this pile red cards, hat cards, or wolf cards, depending on which rule was active (see Fig. 1). The experimenter told participants after each card placement whether or not they had put the card into the correct pile based on the current rule. Participants sorted cards into piles until they correctly placed ten cards in succession. Each trial had one unchanging rule. Participants in the explicit condition were given the rule in advance of the trial. Participants in the induction condition were not told in advance what the rule would be; instead, they had to determine the rule based on experimenter feedback (i.e., whether each card placement was correct or not). Participants were asked to wear a motion tracker on their sorting hand. Motion tracking data was collected during sorting using a magnetic motion-capture device (Polhemus Fastrak, Polhemus

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Fig. 1. An example of the card-sorting task is shown above. Participants were presented with four guide cards. The guide cards each have a unique value of each of the three dimensions (animal, color, and clothing item). Participants were required to draw a card and place it into the correct guide pile, as determined by the rule for that trial. In the example above, the blue cow with glasses would be placed into the second pile if the rule was ‘‘color’’, the third pile if the rule was ‘‘clothing item’’, or the fourth pile if the rule was ‘‘animal’’.

Corporation, Colchester, VT and 6-D Research System software, Skill Technologies, Inc., Phoenix, AZ). The position of the participant’s hand was sampled at 60 Hz. For each trial, we created a time series of the inter-point distances that was submitted to detrended fluctuation analysis (DFA) [19]. DFA is occasionally susceptible to sinusoidal trends in time series data. In these cases, the scaling parameter produced by DFA is affected by low-frequency trends [20]. To protect against this, each time series was filtered prior to analysis using Fourier DFA, as described by Chianca et al. [21]. High-amplitude, low-frequency oscillations were removed on a trial-by-trial basis. Fig. 2 shows examples of the time series before and after filtering. 2.1. Detrended fluctuation analysis Detrended fluctuation analysis assesses long-range correlations in non-stationary time series x(t ) of length N. First, the time series is integrated to produce a trajectory y(t ): y(t ) =

N −

x(i) − x(t ),

i=1

where x(i) is the ith interpoint distance and x(t ) is the average interpoint distance. Next, the integrated time series is segmented into non-overlapping bins of length n, such that 4 ≥ n ≥ N /4. DFA proceeds with a least-squares regression within each bin. The residuals of these regressions provide an estimate of root mean square (RMS) error: F (n) =



(1/N )

− [y(t ) − yn (t )]2 ,

where yn (t ) is the y coordinate of the local trend within each bin. DFA treats the average RMS error as the fluctuation F (n) for bin size n. The relationship between F (n) and n is the fluctuation function increasing as: F (n) ∼ nH . When the fluctuation function is plotted on double-logarithmic axes, the relationship between log F (n) and log n may be linear. The slope of this linear relationship is an estimate of the Hurst exponent (see Fig. 3). 2.2. Epoching In order to track changes in the Hurst exponent over time, we used an epoching approach [22]. Epoching is a useful method for examining a non-stationary time series. In the epoch approach a sliding window is moved across the time series; each window is an epoch. The window was 800 cycles wide; it was shifted 300 cycles on each step, creating an overlap of 500 cycles per shift. DFA is performed on each epoch. This allows us to quantify changes in the Hurst exponent as the participant sorts cards over the course of a trial.

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Fig. 2. The figure shows examples of the interpoint-distance time series for the explicit condition (left panels) and the induction condition (right panels). The upper panels show the unfiltered time series, the lower panels show the series after filtering.

3. Results Participants completed each trial fairly quickly (median = 55 s), with those in the induction condition taking slightly longer to reach the required 10 correct sorts. Relatively few incorrect card placements were committed across trials. Participants in the explicit condition almost never committed an error in sorting, while participants in the induction condition typically did not commit more than one or two errors before inducing the proper sorting rule. Given that the task was self-paced, the trials varied in duration. Because participants sorted to a criterion on each trial (i.e., 10 correct card placements), it seems reasonable to consider their performance as roughly equivalent at that point in time (i.e., the end of the trial). In order to avoid the undue influence of very long trials, we consider only the last 65 s of each trial. 80% of the trials were shorter than this 65 s time frame (and therefore were not trimmed at all). The remaining 20% of trials were trimmed such that only the last 65 s were included in the analysis. We analyzed changes in the Hurst exponent across the 65 s prior to the end of each trial. Fig. 4 shows the Hurst exponent as a function of epochs for individual trials within each condition. In the figure, and for the analyses below, the epochs are aligned at the end of the trial. Growth curve modeling (GCM) was used to predict changes in the Hurst exponent over the length of the trial. GCM is a maximum-likelihood, hierarchical linear regression technique suited to modeling over time data. In GCM, a time-varying dependent measure is modeled as the weighted sum of main effects and interactions. Whereas OLS regression usually assumes equal variance across measurements, GCM includes random effects to capture heteroscedastic and auto-correlated error terms [23]. Where OLS uses increases in R-squared to evaluate contributions to the model, GCM evaluates added predictors via reductions in −2 log likelihood (−2LL). Change in −2LL may be tested against a chi-square distribution with as many degrees of freedom as added predictors [23]. GCM has proven effective for modeling general trajectories in Hurst exponents while controlling for individual differences during cognitive and perceptual tasks [16,24]. Our primary prediction concerns change in the Hurst exponent over time. For the induction condition, changes in the Hurst exponent should be an inverted-u shaped curve. For the explicit condition, the Hurst exponent should decrease as the rule becomes more stable.

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Fig. 3. Sample fluctuation functions for the explicit (upper panels) and induction (lower panels) conditions. The log of the residual (RMSE) is plotted as a function of the log of the bin size. The solid curve shows the best-fitting (OLS) line. The slope is an estimate of the Hurst exponent. Table 1 Estimated growth curve parameters.

Intercept Epoch Epoch2 Condition Trial: σ 2 Int Participant.: σ 2 Int Residual: σ 2 ∗Cond. × Epoch2

Estimate

SE

z

0.81 0.007 −0.0007 −0.0006 0.000001 0.002 0.005 −0.00029

0.012 0.002 0.0002 0.017

64.63 3.57 −3.65 −0.03

0.0001

−2.62

Note: Estimated coefficients and variances for the growth curve model. The base model included all terms except Condition × Epoch2 . This term was added to the model in the second step.

Fig. 5 shows the mean Hurst exponent, averaged over trials and participants, as a function of epochs for each condition. We modeled the Hurst exponent over epochs within each trial, within each participant. The base model included terms for condition, trial, epoch and Epoch2 ; the latter two terms allow for both a linear and quadratic effect of epoch. Random effects on the intercept were included for each trial within participant, and each participant within condition. The coefficients are shown in Table 1, along with the variances for the random effects. Including a random effect for epoch within trial (and its covariance with the intercept term) did not improve the fit of the model, σ 2 Epoch = 0.00, change in −2LL χ 2 (2) = 1.99 ns. The central prediction was tested by adding the interaction of Epoch2 with condition. The inclusion

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Fig. 4. The plots show the Hurst exponent as a function of epoch for ten individual trials, five from each condition. The epochs are aligned at the end of the trial.

Fig. 5. The plot shows the average Hurst exponent as a function of epoch with a separate curve for each condition. Hurst exponents were averaged across participants and trials by epoch. The magenta curve shows the results for the induction condition. The blue curve shows the results for the explicit condition. Error bars show the model-based, 95% confidence intervals. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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of this term significantly improved the fit of the model, change in −2LL χ 2 (1) = 6.56. The quadratic effect is significantly greater for the induction condition. This result is consistent with the pattern seen in Fig. 4; the induction condition shows a sharp increase in the Hurst exponent, followed by a prolonged decrease. A similar set of results can be obtained without trimming the front-end of the longer trials, but the model must be more complex to handle the larger range of durations. The analysis above considers the change in the Hurst exponent over epochs on the natural time scale, rather than normalizing each trial by its total duration (i.e., the maximum epoch in that trial). Normalizing each trial would be appropriate if all trials have roughly the same starting point, peak, and end point, but the trajectories are simply compressed or extended across the various trial durations [25]. More substantively, normalizing is consistent with the hypothesis that participants were starting each trial with a similar degree of constraint (as indexed by the Hurst exponent), but the constraints were changing at different rates, resulting in different trial completion times. Using the natural time scale would be appropriate if trials with different durations have different entry points onto the trajectory. For example, trials with shorter duration would have higher initial Hurst values, putting them further along the trajectory. This would be consistent with the hypothesis that participants sometimes had different degrees of constraint, and that these initial differences result in different trial completion times. To test between these two alternatives, we added trial length (i.e., the maximum epoch within the trial) as a predictor to the model above, thereby including a linear effect of trial length on the intercept. Consistent with the hypothesis that trials with different durations have different intercepts, there was a significant, negative effect of trial length, B = −0.004, change in −2LL χ 2 (1) = 40.03. Shorter trials have significantly higher intercepts than longer trials. 4. Conclusions We showed that the trajectory of the Hurst exponents for participants in the induction condition exhibited a sharp rise and fall as they induced the current rule. This result is consistent with an account of executive function in which the system breaks and then resets constraints as it self-organizes. The trajectory of the Hurst exponents for participants in the explicit condition showed a small decrease over epochs, consistent with an increase in stability. When participants were explicitly given the rule, sorting cards increases system stability. When participants must induce the rule, they initially become less stable, followed by an increase in stability as the new rule coalesces. These results provide new information on the dynamics underlying card-sorting tasks, an important measure of executive function. Traditional theories of executive function would suggest that certain scales of the system exert influence across all other scales; our results show that system activity is instead scale invariant. Rather than conceiving of executive control as coming from an internal agent, our results suggest that it may be better described in terms of stability-instability across the scales of the system. This experiment adds to the growing body of evidence that cognition arises from interaction-dominant dynamics. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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