Journal of Experimental Psychology: Human Perception and Performance 2009, Vol. 35, No. 6, 1811–1832

© 2009 American Psychological Association 0096-1523/09/$12.00 DOI: 10.1037/a0014510

Dynamics of Representational Change: Entropy, Action, and Cognition Damian G. Stephen

James A. Dixon

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

Center for the Ecological Study of Perception and Action, University of Connecticut, and Haskins Laboratories

Robert W. Isenhower Center for the Ecological Study of Perception and Action, University of Connecticut Explaining how the cognitive system can create new structures has been a major challenge for cognitive science. Self-organization from the theory of nonlinear dynamics offers an account of this remarkable phenomenon. Two studies provide an initial test of the hypothesis that the emergence of new cognitive structure follows the same universal principles as emergence in other domains (e.g., fluids, lasers). In both studies, participants initially solved gear-system problems by manually tracing the force across a system of gears. Subsequently, they discovered that the gears form an alternating sequence, thereby demonstrating a new cognitive structure. In both studies, dynamical analyses of action during problem solving predicted the spontaneous emergence of the new cognitive structure. Study 1 showed that a peak in entropy, followed by negentropy, key indicators of self-organization, predicted discovery of alternation. Study 2 replicated these effects, and showed that increasing environmental entropy accelerated discovery, a classic prediction from dynamics. Additional analyses based on the relationship between phase transitions and power-law behavior provide converging evidence. The studies provide an initial demonstration of the emergence of cognitive structure through self-organization. Keywords: representation, cognition, dynamic systems, embodiment, entropy

& Yarlett, 2007), cognitive development (e.g., Karmiloff-Smith, 1992; Piaget, 1952, 1954; Smith, 2005), and category induction (e.g., Hummel & Holyoak, 2003; Kalish, Lewandowski, & Davies, 2005). Addressing the problem of new structure will have deep implications for our understanding of cognition. Not only will it ground a theory of how cognition changes, it will constrain theories of cognitive architecture, because the nature of cognitive structure must be intimately related to its ability to spontaneously reorganize. The theory of nonlinear dynamics has offered self-organization as an account of the emergence of new structure (Beck & Schogl, 1993; Hilborn, 1994; Nayfeh & Balachandran, 1995). Self-organization, a natural property of complex systems, explains the emergence of new structure in fluid (Lorenz, 1963), lasers (Haken, 1983), single-cell organisms (Webster & Goodwin, 1996), and molecular networks (Sardanyes & Sole´, 2006), to name just a few areas. Across these quite-disparate domains, self-organization is characterized by a set of higher-order principles; these principles form part of the larger theory of nonlinear dynamics. Within cognitive science, self-organization has been shown to explain new structures in locomotion (Kugler & Turvey, 1987; Swenson & Turvey, 1991), reaching (Mottet & Bootsma, 1999), and timing behavior (Kelso, 1995). A number of theorists have proposed that these principles should extend to higherorder cognition as well (Thelen & Smith, 1994; Van Orden, Holden, & Turvey, 2003, 2005). However, it has been unclear how one might tap into the dynamical processes that drive self-organization for phenomena in cognition. A major impediment, we suggest, has been the fundamental difficulty of accessing cognitive dynamics. A central goal of the manuscript is to show how the gap between high-level cognitive structures and dynamic processes can be bridged. Doing so,

The problem of new structure has been a long-standing issue for psychological theory. How can a system produce new structure without prior knowledge of that new structure’s impending form? New structures in cognition (e.g., concepts, categories, memories) must emerge from the activity of the current structures, without the guiding influence of an external or internal intelligent agent. This issue has shaped theories of perception-action (e.g., Adolph & Berger, 2006; Kugler & Turvey, 1987), learning (e.g., Jacobs & Michaels, 2007; Lattal & Bernardi, 2007; Shrager & Johnson, 1996), language acquisition (e.g., Bates & Goodman, 1997; MacWhinney, 1999; Ramscar

Damian G. Stephen, Department of Psychology and Center for the Ecological Study of Perception and Action, University of Connecticut; James A. Dixon, Department of Psychology and Center for the Ecological Study of Perception and Action, University of Connecticut and Haskins Laboratories; Robert W. Isenhower, Department of Psychology and Center for the Ecological Study of Perception and Action, University of Connecticut. This research was partially funded by the National Science Foundation (BCS-0643271). We extend heartfelt thanks to Rebecca Boncoddo, Alen Hajnal, Steven Harrison, Anne Olmstead, and Navin Viswanathan for their patience, encouragement, and criticism, all of which was essential to the work. We thank Claudia Carello, Bruce Kay, and Michael Turvey for technical and theoretical support. We thank Heidi Kloos, Claire Michaels, Guy Van Orden, Iris van Rooij, Peter Vishton, Eric-Jan Wagenmakers, and a number of anonymous reviewers for their helpful insights on the long path from laboratory to publication. We thank Rick Dale, Geoff Hollis, and others for their faith, in the meantime, that the work would in fact be published. Correspondence concerning this article should be addressed to Damian G. Stephen, Department of Psychology, 406 Babbidge Road, Unit 1020, University of Connecticut, Storrs, CT 06269-1020. E-mail: damian [email protected] 1811

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we suggest, has deep theoretical implications for understanding cognitive structure and how it might change. Two recent developments, one in cognitive science, the other in nonlinear dynamics, allow for substantial progress on the problem of capturing cognitive dynamics. First, a growing body of work shows that action and cognition are tightly intertwined in an interactive system. For example, research on embodied cognition has demonstrated that sentence comprehension is affected by the compatibility of concurrently performed actions (see Glenberg & Kaschak, 2002; Zwaan & Taylor, 2006). In one such experiment, Zwaan and Taylor had participants listen to sentences that implied rotation in one direction (e.g., “Turn up the volume”), while judging the sensibility of the sentence by turning a knob. When the direction of the motor response corresponded to the direction of the motion implied by the sentence, sensibility judgments were significantly faster. Similarly, sensibility judgments were significantly slower when the turning direction of the motor response conflicted with the direction implied by the sentence (see also, Dale, Kehoe, & Spivey, 2007; Solomon & Barsalou, 2001). Behavioral work showing interactivity converges with findings from neuroscience that demonstrate surprising relationships among brain areas subserving perception, action, and cognition. For example, individual neurons within the F5c-PF circuit fire both during specific motor actions (e.g., grasping) and observing a conspecific performing that same action (Gallese & Lakoff, 2005; Wilson & Knoblich, 2005). Similarly, Buccino et al. (2005) demonstrated that motor-evoked potentials in the hand and foot muscles were differentially affected by reading sentences about hand and foot action, respectively. Multiple levels of brain and body appear to continuously interact in cognition. The second advance comes from nonlinear dynamics and the study of complex systems. A primary concern in dynamics is to quantify a system’s trajectory in phase space. Phase space is defined as the set of potential states a system may take; different states correspond to different points in phase space (Hilborn, 1994). Each dimension in phase space reflects a variable in the system. The trajectory through phase space specifies the system’s behavior. Many methods in nonlinear dynamics require some means of inferring phase space from a set of observations. Takens (1981) demonstrated that phase space can be reconstructed from a single, densely sampled, univariate time series. His method has since been refined and become a cornerstone for nonlinear dynamics (see, Abarbanel, 1996; Marwan, Romano, Thiel, & Kurths, 2007; Sauer, Yorke, & Casdagli, 1991). Because the properties of self-organizing, complex systems may be unfamiliar to some readers, we next briefly outline the process of self-organization, and discuss a foundational principle of selforganization, the relationship between entropy and the emergence of new structure. We then present a paradigm in which cognitive structure has been shown to emerge spontaneously and explain how changes in entropy should predict cognitive emergence. Finally, we provide a concrete example of how entropy can be assessed to predict the emergence of new structure using recurrence quantification analysis, a recently developed method grounded in Takens’ theorem.

A Brief Account of Self-Organization Self-organization occurs in systems composed of a huge number of constituent parts. Self-organized structure emerges from the

spontaneous breaking and reforming of constraints binding these parts together. This breaking and reforming of constraints is often understood in terms of entropy, an index of disorder or instability. The existing structure (i.e., configuration of constituent parts) of a system will remain unchanged until entropy (i.e., disorder) overwhelms its internal constraints. The overwhelming influence of entropy manifests itself as a critical instability. As the system approaches a critical instability, constraints dissolve thereby freeing previously cohering parts and causing them to interact. These interactions may allow the system to reach a new configuration, better suited to disperse entropy. Therefore, in self-organization, new structures are predicted by an increase in entropy (indicative of critical instability) followed by a decrease in entropy (indicative of the onset of new constraints). In nonlinear dynamics, this sequence of changes is a phase transition.1 (For a more complete discussion of this account of self-organization, see Prigogine & Stengers, 1984, or Schneider & Sagan, 2005). Phase transitions are changes in the macroscopic, qualitative behavior of the system. One of the foundational tenets of dynamic theory is that changes in behavior on the microscopic scale drive the changes at the macroscopic scale. From the perspective of psychology, this implies that the emergence of a new macroscopic cognitive structure may be explained by microscopic fluctuations in behavior that occur within the current macroscopic organization.

New Structure in the Gear-System Paradigm Previous research has demonstrated that participants spontaneously discover a new representation or cognitive structure as they solve gear-system problems (Dixon & Bangert, 2002, 2004; Dixon & Dohn, 2003; Schwartz & Black, 1996). In this paradigm, participants are asked to predict the movement of a target gear, given the turning direction of a driving gear. The gear systems are presented as static images on a computer display (see Figure 1). 1 Entropy is the amount of disorder in a system. The second law of thermodynamics states that the entropy of an isolated system increases to a maximum level. This maximum level of entropy constitutes equilibrium, a state in which structure is absent and concentrations of matter and energy are equal throughout. Under the second law, order decays and yields disorder as time progresses, leading towards a final state of “heat death” (Helmholtz, 1854/1995). However, in open systems, this trend does not hold. Open systems are in constant interaction with the environment, exchanging both energy and matter. This exchange allows them to dissipate entropy into the environment. This dissipation maintains the system’s ordered state. Interaction with the environment leaves an open system susceptible to perturbations. These perturbations result in increased entropy at the microscopic scale. If the system takes on too much entropy, it runs the risk of becoming fatally disordered. The continued survival of the system’s structure depends on its ability to dissipate entropy. Here, the story gets very interesting. If the perturbation brings entropy at the microscopic scale beyond the limit of what the system’s current structure can dissipate, new structure may emerge (Prigogine, 1980). That is, to dissipate entropy, an open system self-organizes into a dissipative structure, a state of increased order on the macroscopic scale. Whereas processes leading a system towards equilibrium are entropic, self-organization is negentropic. Thus, in open systems, the influx of entropy drives the emergence of new structure (Prigogine, 1976). Self-organization is the process through which order (i.e., new structure) arises from disorder (i.e., entropy).

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Figure 1. Examples of the different types of gear system problems are shown in the figure. Gear systems varied along three dimensions: size, number of pathways, and whether an extraneous gear was present. The gear system on the upper left is labeled to identify the driving gear and target gear. The three response options in the display are also labeled: chutes to indicate clockwise or counterclockwise rotation (e.g., the fuel would slide down the chute on the right, given clockwise rotation of the target gear) and the “jams” button to indicate that the target gear would not turn.

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Prior work showed that participants’ understanding of the gearsystem task changed dramatically over the course of the experiment, as evidenced by the strategies they employed. Initially, the majority of participants traced the force from the driving gear to the target gear. That is, they manually simulated the turning of each gear, the pushing of that gear’s teeth on the next gear in the series, ultimately determining the turning direction of the target gear. This approach to the problem, which we call force tracing (Dixon & Kelley, 2006, 2007), arises from a meshing of affordances available in the display (i.e., body-scaled information about the potential movement of the gears) with the participants’ current understanding of the simple physics of gear systems (e.g., gears turn and teeth push) (see Glenberg, 1997, for a discussion of “meshing,” and Turvey, 1992, or Warren, 1984, for a discussion of affordances). Force tracing creates new information about a higher-order property of the gear system: the gears form an alternating sequence. Specifically, because the gears are configured on a two-dimensional plane, participants’ tracing motions alternate from clockwise to counterclockwise rotation, as they move from gear to gear. In this way, participants’ actions, based on the available affordances and the macroscopic state of the cognitive system, yield higher-order relational information. Consistent with the hypothesis that participants’ actions create new information, use of the force-tracing strategy predicted participants’ spontaneous discovery of alternation. Participants discovered that the gears alternate turning direction, as evidenced by their sudden shift to classifying gears as clockwise turning and counterclockwise turning, completely dropping any references to the physics of the system. That is, the representation of the problem based on simple physical interactions changed to one based on a higher-order property, alternation. Importantly, the discovery of alternation occurs without the participants having ever seen the interlocking gears move. Dixon and Bangert (2002) showed that the discovery of alternation was predicted by (a) having a history of correctly using the force-tracing strategy on previous trials, and (b) concentrated use of force tracing on recent trials. We hypothesize that alternation, a new cognitive structure, emerges through the self-organization of the cognitive system as the person interacts with the task. According to the theory of nonlinear dynamics, the emergence of new structure should be predicted by an increase in entropy as the current structure begins to break down. Eventually, entropy overwhelms the organization and the system reorganizes into a new structure. This reorganization is marked by a drop in entropy as the new structure is formed. Therefore, the discovery of alternation will be predicted by two signature changes in entropy: a peak in entropy in recent history (i.e., at moderate lag) and a decrease in entropy immediately before discovery (i.e., at a very short lag). In the next section, we illustrate these predictions more explicitly, by showing how these relationships appear in a known system.

responds to an equation in the model. When the system continually returns to a specific region of phase space, the system is considered to be in an attractor. An attractor, the repeated convergence of the trajectory toward a set of points in phase space, corresponds to a mode of macroscopic organization or behavior (see Hilborn, 1994, for a more formal definition). A phase transition can be modeled as a shift from one attractor to another. Figure 2 shows the trajectory of the Lorenz model as temperature is increased logistically. At low temperatures, the Lorenz system converges to a disk attractor, but at greater temperatures, it abruptly shifts to a new attractor in which it orbits two points (see Hilborn, 1994). This radical shift from one attractor to another illustrates the emergence of new structure in an idealized model. Of course, it would be very rare in empirical research to have measures of all the dimensions of a system. Fortunately, nonlinear dynamics has provided tools for reconstructing phase space and assessing the dynamic organization (e.g., entropy) of complex systems based only on a single univariate time series. To illustrate the approach in a known system, we apply it to the phase transition in the Lorenz model. More specifically, we assess entropy over the course of the phase transition working only with a single dimension from the Lorenz model. First, we show how Takens’s theorem allows phase space to be reconstructed. Then, we show how the dynamical organization of that phase space can be assessed.

Investigating Self-Organization in Complex Systems Reconstructing Phase Space Takens (1981) showed that phase space could be reconstructed by plotting a time series against successively time-lagged copies of itself. Phase space reconstruction projects a one-dimensional time series into a higher-dimensional space that preserves the geometry (i.e., topology) of the original phase space. In Figure 3, a sample time series has been lagged by two time steps. Each lagged copy of the time series constitutes another dimension, often called an embedding dimension (see Abarbanel [1996] for a discussion of embedding dimensions). The reconstructed dimensions are topological mappings (also called homeomorphisms) of the original dimensions of the system (see Aleksandrov, 1998, for a discussion of topological mapping). Although these dimensions are not individually meaningful, together they allow the extraction of global variation of the complex system from the initial time series.2 The result is a multidimensional trajectory that recreates key geometric properties of the system’s trajectory through phase space. Recent work recommends determining the amount of lag by using the time step of the earliest minimum in a nonlinear autocorrelation function of the time series (i.e., the first time step, t, with a lower value than those at both t - 1 and t ⫹ 1). Using this time lag minimizes the collinearity among the copies (Webber & Zbilut, 2005). As an example of reconstructed phase space, compare the two panels of Figure 4. The panel on the left shows a detailed view of the

An Example of a Phase Transition 2

To provide a concrete example of phase transition, we draw on the classic Lorenz (1963) model. Lorenz simulated fluid convection in a system with three differential equations. These equations determine the evolution of the system’s trajectory over time. Thus, the trajectory occurs over three dimensions; each dimension cor-

Rarely do we know the exact dimensionality of a complex system. For biological systems, Webber and Zbilut (2005) recommended simply using 10 embedding dimensions so as to introduce adequate variability in reconstructed phase space whereas also minimizing the effect of noise. Alternatively, dimensionality can be estimated by such nonlinear dynamics analyses as global false nearest neighbors (see Abarbanel, 1999).

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Figure 2. An example of an attractor shift in the classic Lorenz model. As heat increases, the system departs from the preshift, disk attractor and enters a new regime. Ultimately, the system settles into the postshift, butterfly attractor. Each of the three dimensions, X, Y, and Z, specify a property of the system’s behavior. The X dimension indexes the intensity of the flow. Y indexes the difference between the ascending and descending convection currents, and Z scales with the deviation of the vertical temperature profile from linearity.

disk attractor from the Lorenz model (from Figure 2). The panel on the right side of Figure 4 shows the same attractor reconstructed using Takens’ approach. Specifically, we plotted the X variable against two time-lagged copies of itself. Note that the topological characteristics of the two attractors are the same; the ordinal relations (Stevens, 1951) among the orbits are preserved. In other words, except for deformations involving stretching or bending, the topology of the original attractor is preserved in the reconstructed attractor. Thus, the figure illustrates graphically how phase space reconstruction reproduces geometric properties of an attractor. To make the analogy to psychological theory more explicit, consider that the X variable (which is the only measured variable in hand) indexes a particular property of the system, the intensity of the flow. The Y dimension, however, indexes a very different property, the temperature difference between the rising and falling currents (and Z measures yet another property of the system). Despite the very different meaning of these properties, a measure of any one variable allows us to reconstruct the behavior of the system across all three variables, because the properties are mutually interdependent (i.e., interactive). Thus, to the degree that a system is interactive, we can recover key aspects of its global dynamics from a single time series. This point is central to the present work. Reconstructing the phase space from a single behavioral time series should likewise recover key aspects of the

global dynamics across the cognitive properties that are currently in play. In the example above, reconstructing phase space allows us to visualize the organization of the attractor. Because the Lorenz model has three dimensions, its phase space can be easily plotted. The phase space of more complex systems with higher dimensionality, such as those within biology and psychology, would, of course, be difficult to visualize. Fortunately, powerful methods for quantifying the organizational properties of phase space have been developed; these methods do not require a visual representation of phase space. In the next section, we outline one such approach for analyzing phase space, recurrence quantification analysis (RQA).

Recurrence Quantification Analysis RQA measures important properties of attractors in reconstructed phase space (see Webber & Zbilut, 1994, 2005). The fundamental insight for understanding RQA is simple: attractors are defined by the system revisiting regions in the multidimensional space. RQA assesses the degree to which regions recur and computes measures of the system’s organization based on that recurrence. The first step in RQA is to construct a distance matrix of the points constituting phase space. As a simplified example, consider the top left panel of Figure 5 that shows portions of a system’s hypothetical

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Figure 3. The method of embedding dimensions for reconstructing the phase space of a system from a univariate time series is illustrated schematically. The initial time series is lagged two time points to make the second embedding dimension, and another two time points to make the third embedding dimension.

trajectory through a reconstructed phase space. Assume that we have sampled six points, A-F, of this phase space, where A is sampled at time t, B at t ⫹ 1, etc. To simplify matters, we assume the space has only two dimensions, X and Y, but the method extends without complication to multiple dimensions. The sampled points, labeled A-F in the figure, constitute a very short time series. (In practice, sampled phase space would usually consist of hundreds or thousands

of points.) The top right panel of Figure 5 shows the 6 ⫻ 6 matrix containing the Euclidean distances between all sampled points in phase space. The positions of each sampled point on the dimensions X and Y are shown next to the matrix. The next step is to divide the distances by their maximum (henceforth, max-scaling). In our example, the maximum distance is 87.92, so the values in the distance matrix were divided by that value. The bottom left panel of Figure 5

Figure 4. Panel (a) shows a detailed view of the disk attractor from Figure 2. Panel (b) shows the same phase space reconstructed using the method of embedding dimensions outlined in the text.

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Figure 5. The top left panel shows a hypothetical trajectory through phase space. Six points, A-F, have been sequentially sampled by measuring their locations on dimensions X and Y. The top right panel presents the distance matrix for six hypothetical points in phase space. The values of X and Y for each of the six points is shown along the side of the distance matrix, and again along the bottom of the matrix. For example, point A has X ⫽ 234, Y ⫽ 153. The entries in the matrix give the Euclidean distance between pairs of points (e.g., A and B) indexed on the row and column. The bottom left panel shows the same matrix rescaled by its maximum value. The bottom right panel presents a discretization of the max-scaled matrix; values below .3 are labeled as recurrences (REC), values greater than .3 as nonrecurrences (NON). Recurrences that are adjacent in time create lines; in the current example A-D, B-E, C-F, create a line, indicating that the system has returned to its prior trajectory.

shows the max-scaled distance matrix. This max-scaled distance matrix is then used to construct the recurrence matrix. Whereas the distance matrix contains continuous measures of interpoint distance in phase space, the recurrence matrix is a discretization of the distances into recurrences and nonrecurrences. Interpoint distances that are within a defined radius qualify as recurrences; interpoint distances that exceed the criterion radius are nonrecurrences.3 Assuming a criterion radius of 0.30 for this example, the bottom right panel of Figure 5 illustrates the discretization of the distance matrix. Given a recurrence matrix like the one in Figure 5, RQA extracts a number of parameters that describe different aspects of dynam-

ical organization. Consecutive recurrences form diagonal lines in the recurrence matrix; these lines indicate instances of convergence in phase space, and thus contain information about the properties of attractors in phase space. (The center diagonal is

3 Percent-recurrence, the proportion of recurrent points to all points in phase space, is a diagnostic of the criterion radius. Following recommendations in the literature, we kept recurrence reasonably sparse, around 2%, for both reported studies (Riley, Balasubramaniam, & Turvey, 1999; Webber & Zbilut, 2005).

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uninteresting because it reflects the proximity of each point to itself.) In the bottom right panel of Figure 5, for example, there is a line formed by the recurrences of points A with D, B with E, and C with F. Percent-determinism, calculated as the percentage of recurrent points on lines, provides a measure of the nonstochastic contribution to the trajectory. Each line is an instance of convergence; two additional measures, mean line and max line, are consequently indicative of attractor strength (see Shockley, 2005). Mean line, the average line length, provides a measure of the degree to which the system spends contiguous time in an attractor. Max line, the longest line length, assesses the longest instance of contiguous time in an attractor. RQA also calculates a measure of entropy based on the distribution of line lengths. Entropy is calculated according to Shannon’s (1948) formula,4 and is related to the variability of the trajectories through phase space.

Applying RQA to the Lorenz Model To demonstrate the ability of RQA to capture changes in phase space, based solely on a univariate time series, we analyzed the attractor shift evident in the Lorenz simulation using only the X dimension values to reconstruct phase space. We ran the RQA procedure on windows of 800 time steps, lagged every 300 time steps along the time course. This method of analyzing overlapping windows, called epoch analysis, gives a continuous portrayal of recurrence trends within a time series (Webber & Zbilut, 2005). Because the Lorenz model is deterministic, we anticipated high percent-determinism throughout. More importantly, the attractor shift, as an instance of self-organization, should be predicted by systematic changes in entropy. As can be seen in Figure 6, entropy is relatively constant for the first portion of the time series. Then, there is a sharp rise in entropy, followed by a decrease to a stable, lower level. The trajectory of the Lorenz model across the time series is also shown in Figure 6. The increase in entropy marks the system’s departure from the disk attractor. Negentropy, a decrease in entropy, occurs as the system reorganizes into the butterfly attractor. Thus, RQA captures these fundamental aspects of the attractor shift in the simulated Lorenz system: a peak in entropy followed by negentropy predicts the onset of a new organization. The changes in entropy over the course of the Lorenz phase transition provide a concrete example of the general principles of self-organization described earlier. The increase in entropy breaks the constraints within the system, creating a critical instability. The decrease in entropy indicates the formation of new constraints and the settling of the system into a new configuration. In the current study, we evaluate the hypothesis that new macroscopic cognitive structure may be predicted from microscopic behavior as specified by the principles of self-organization. Specifically, we expect that a critical instability, as indexed by a peak in entropy, followed by a subsequent drop in entropy will precede the emergence of new cognitive structure. In the first study, we asked adult participants to solve 36 gear-system problems in any way they wished. On each trial, we recorded the motion of their dominant hand via an infrared motion tracker. We reconstructed the phase space of the cognitive system with a time series extracted from the motion data (i.e., angular velocity). Using RQA, we measured the entropy of system behavior, as well as other aspects of phase space, for each trial. We used

these measures of dynamic organization to predict the discovery of alternation on subsequent trials.

Study 1 Method Participants Thirty-eight college students participated as one option to fulfill a course requirement.

Materials and Procedure Each gear system consisted of a driving gear with a clockwisepointing arrow, a variable number of intermediate gears, and a target gear. Participants were asked to predict whether the target gear would turn clockwise, counterclockwise, or jam (if two intermediate gears acted on the target gear in opposing directions, the target gear would not move). The gear systems varied along three dimensions: size (small: 4 or 5 gears; or large: 7 or 8 gears), number of pathways (one or two), and whether an extraneous gear was present. Extraneous gears were not involved in the causal pathway from driving to target gear (see Figure 1 for examples). The gear problems were presented on a Macintosh PowerPC. Each problem appeared in a 500 (horizontal) ⫻ 480 (vertical) pixel window centered on the computer screen. Presentation order was randomized for each participant. With the exception of the target gear, which moved only after all the other gears were covered, the gear-system displays were completely static. Participants were asked to solve the gear systems in any way that they wished and told to take as long as they needed to generate a solution. They were encouraged to think aloud as they solved the problems, thereby allowing us to code their strategies. Motion of the forefinger of the participant’s dominant hand was tracked through three-dimensional space at 100 Hz using a Northern Digital Optotrak Certus. Based on previous work, we knew that the vast majority of participants move their hand while solving the problem (Dixon & Bangert, 2002). The task was presented in the context of a train race in which the participant’s train was competing with a train controlled by the computer. Selecting a response option (clockwise, counterclockwise, or jams) positioned the participant’s train on the screen. If the participant correctly predicted the turning direction of the target gear, his or her train would be positioned to catch a load of fuel that would make the train go faster. If jamming was correctly predicted, the train was positioned to leave the station thus saving time in the race. After the participant made his or her prediction, a virtual screen covered all the gears except the target gear. The target gear then turned appropriately or jammed. Two subsequent scenes showed the participant’s train moving relative to the com4 Shannon entropy is calculated as follows: Entropy ⫽ ⫺⌺ (Pbin)log2(Pbin), where Pbin is the probability of a diagonal line of a particular length. The original purpose of Shannon entropy was to quantify the amount of information in bits needed to communicate an electrical signal. However, Shannon entropy is calculated using the same formula that Boltzmann (1886/1974) used to define thermodynamic entropy and covaries with thermodynamic entropy (Jaynes, 1957; Pierce, 1980; Trambouze, 2006; Waterman, 1968).

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Figure 6. The results from the epoch RQA on the Lorenz-model attractor shift are presented. Time is shown along the horizontal axis. The vertical axes show % Determinism and Entropy on the left and right, respectively. The activity of the Lorenz system is shown schematically above the horizontal axis for each of the model’s four phases: occupying the initial attractor, departing from the attractor, settling into the second attractor, and occupying the second attractor. The darkened areas of the trajectory indicate the model’s global status during that time period.

puter’s train. These events provided feedback about the participant’s prediction. Participants’ strategy use on each trial was coded by the experimenter and videotaped. Force tracing is indicated by the participant simulating the turning motion of each gear with his or her finger. These motions are typically accompanied by verbalizations that describe the “turning” of the gears and “pushing” of the interlocking teeth. Alternation is indicated by the participant classifying each gear as “clockwise” or “counterclockwise” in rapid sequence without any reference to the physics of the system. The first use of alternation, which we refer to as the discovery of alternation, is often marked by participants explicitly mentioning that they have noticed something new (i.e., a “pattern” or “rule”). An additional observer independently coded 180 randomly selected trials to assess reliability of the strategy coding. The observers showed agreement on 90% of the trials; analyses are based on the primary observer’s coding.

Results Overview Strategy data. Consistent with previous work, the majority of participants (84%) first used force tracing to solve the problems.5 The remaining participants (16%) initially moved their heads and eyes to trace the force through the system. Many participants

discovered alternation fairly quickly; 50% discovered within the first nine trials, 68% (26 participants) discovered alternation at some point during the study. Participants accurately predicted the movement of the target gear on 81% of trials. However, they were significantly more accurate with the alternation strategy (90%) than force-tracing (77%), F(1, 26) ⫽ 6.40. Motion data. As a participant simulated the turning and pushing of the gears, his or her fingertip trajectory approximated a series of circles traced predominantly upon a single plane in space. We used principal component analysis to reduce the threedimensional coordinates sampled by the Optotrak to twodimensional trajectories on the plane of tracing. The two dimensions accounted for an average of 96% (SD ⫽ 2%) of the variance. This result is consistent with Mitra and Turvey (2004) who showed 5

Placing the motion tracking marker on the participants’ finger may have encouraged the use of their hand to solve the task. However, the percentage of initial force tracing reported here is only moderately higher than that found in previous work in which motion tracking was not used. Dixon and Bangert (2002) reported that 70% of college participants first used force tracing; Dixon and Dohn (2003) reported 78%. We note that other effects (e.g., accuracy of the different strategies, timing of the transition to alternation) also square quite well with previous work. Motion tracking does not appear to substantially affect how participants operate in the task.

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that trajectories of rotation of the hand occur in two dimensions. Participants moved their finger sufficiently to generate an adequate time series for analysis on 93% of prediscovery trials. Participants used force-tracing on the vast majority of these trials, 83%. We constructed the predictors described below from the force-tracing trials, because previous work demonstrated a link between force tracing and the discovery of alternation (Dixon & Bangert, 2002; Dixon & Dohn, 2003; Trudeau & Dixon, 2007). However, the same pattern of results obtain if the predictors are computed on all trials with adequate motion data. Because we predicted that the key actions involved circular motion, we created a time series of the angular velocities for each trial. Angular velocity captures important aspects of the force-tracing motion, including direction and speed of movement. Consider any three noncolinear points, A, B, and C, as shown in Figure 7. These points define a circle with center D. For each set of three consecutive points, we determined the proportion of the arc length ABC to the circumference of their defined circle. This proportion is equivalent to the proportion of the angle ADC subtended by the arc ABC to the 360º of the entire circle (see Figure 7). When treated as an arc, each set of three consecutive points described the angular sweep of the fingertip over 0.03 s. The change quantified by this angle over time is angular velocity (Angeles, 2003; Synge & Griffith, 2007). To evaluate the necessity of a nonlinear method such as RQA, we performed surrogate data analysis to test for nonlinearity in the time series (Schreiber & Schmitz, 1996, 2000; Theiler, Eubank, Longtin, Galdrikian, & Farmer, 1992). This analysis, reported in the Appendix, confirmed the nonlinearity of the time series. We performed RQA on the angular-velocity time series from each trial separately, yielding measures of dynamic organization. Table 1 presents the average values for percent-recurrence, percent-determinism, mean line, max line, and entropy for prediscovery trials (i.e., trials before discovering alternation) on which force-tracing was used. We use parameters from these trials as predictors of discovery in the subsequent analyses.

Figure 7. The figure shows how the angles used for angular velocity are determined from three consecutive points (Panel a). Panels (b and c) shows how three points imply a center point, and thus determine a circle (Panel d), Angle ADC, ␪, is then easily determined, Panel (e).

Table 1 Recurrence Quantification Analysis Parameters on Prediscovery Trials Parameter

M

(SD)

% Recurrence % Determinism Mean line Maximum line Entropy

2.11 24.40 2.92 27.79 1.23

(2.49) (18.71) (1.77) (31.14) (.95)

Modeling the Discovery of Alternation We modeled the probability of discovering alternation using event history analysis (Allison, 1984; Singer & Willett, 2003). In brief, event history analysis is a statistical technique developed for longitudinal data in which the probability of a discrete event occurring over time is modeled as a function of a set of predictors. Conceptually, event history is closely analogous to multiple regression in that continuous and categorical predictors can be used to model the dependent measure. However, the dependent measure in event history is a discrete event that may occur at any point during the study. The predictors in event history can be betweensubject, time-invariant differences (e.g., between-subject condition) or within-subject, time-varying differences (e.g., proportion correct on previous trials). Event history naturally integrates both time-invariant and time-varying predictors into a familiar, regression-type format. Contribution to model fit is quantified not in terms of an ordinary least-squares (OLS) r-squared but, instead, in terms of reductions in the maximum likelihood (ML) deviance statistic, ⫺2 multiplied by the log-likelihood (henceforth, ⫺2 LL). Two concepts are central to understanding the analyses reported below. The first is the risk set. The risk set is comprised of participants who have not yet experienced the event. These participants are “at risk” for its occurrence, regardless of whether they ever actually experience it. After a participant experiences the event (e.g., discovering alternation), he or she is dropped from the risk set (and, hence, the analysis). Therefore, the risk set can change from trial to trial. Only participants in the risk set contribute to the analyses on a given trial. We refer to the trial currently under consideration as trial j. The second concept, the hazard, refers to the probability of the event occurring on trial j. The hazard is estimated as the proportion of participants who experience the event on that trial relative to the number of participants at risk. It is worth noting that event history appropriately handles participants who do not experience the event (right-censored cases); these participants remain in the risk set and, therefore, in the denominator of the hazard. Thus, event history requires that the predictors simultaneously explain both the timing of the observed events and the failure to observe the event (for right-censored individuals) in a single model. The upper panel of Figure 8 shows the cumulative hazard function for discovering alternation. The cumulative hazard is a simple and convenient way to display the probability of an event occurring over time. It contains information about the hazard on each trial and also shows how the hazards accumulate over trials. The hazard for each trial (estimated as the proportion of participants currently in the risk set who experience the event on that

DYNAMICS AND REPRESENTATION

Figure 8. The upper panel shows the cumulative hazard (i.e., sum of per-trial probabilities up to and including the current trial) over trials. The hazard rate for any particular trial is shown by the change in the cumulative hazard from the previous trial. The lower panel shows the cumulative hazard for different levels of the two major, time-varying predictors: peak entropy and prior entropy. To illustrate the relationship between these predictors and discovering alternation, the risk set of each trial was divided into four groups, the factorial combination of high and low levels of peak and prior entropy. The hazard for each group was computed. Note that, for an individual participant, group membership can change from trial to trial, thus capturing the time-varying nature of the predictors. The cumulative hazard functions show how the probability of discovery changes dramatically when participants’ peak and prior entropy take high and low values. Trial 3 is the first trial under consideration, because the lagged predictors require a minimum of two previous trials for their computation.

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trial) is shown by the changes in the function. Increases in the cumulative hazard indicate the proportion of (risk-set) participants who discovered alternation on that trial. For example, in the upper panel of Figure 8 the difference between Trials 6 and 7 is .13, indicating that 13% of participants in the risk set on Trial 7 discovered alternation. The cumulative nature of the function shows how the relatively small, trial-level hazards can jointly create large overall hazard. We note that the cumulative hazard can (and often does) exceed 1.0. Although this may initially seem counterintuitive, consider a series of three coin flips where the event of interest is “heads.” If all three flips are “tails,” the cumulative hazard is 1.5. Thus, cumulative hazard, as the sum of per-trial probabilities, can exceed 1.0. Predictors. To predict the discovery of alternation, we constructed three predictors from the measures of dynamic organization computed in RQA. One predictor uses mean line length, an index of attractor strength. Higher values of mean line indicate that the system has longer spells of recurrence. Put another way, mean line measures the degree to which the system repeatedly spends contiguous time in an attractor. Thus, this predictor captures the requirement that the system should become organized before the shift. We used the mean line length computed from the motion data on the immediately prior trial (j-1) to predict discovery on trial j. We call this predictor, mean line from the immediately prior trial, prior mean line. The other two predictors were based on system entropy, the central indicator of self-organization. The first of these predictors, which we call peak entropy, was the maximum value of entropy on previous trials j-2 through j-5. This variable is intended to capture the increase in entropy that triggers reorganization. (We selected this range of previous trials based on prior work which demonstrated that concentrated use of force tracing across this range predicted discovery [Dixon & Bangert, 2002, see also Dixon & Bangert, 2005]. Other, similar ranges yield very similar effects.) The second of these predictors, which we call prior entropy, was simply entropy on the immediately prior trial, j-1. This variable allows us to capture the predicted decrease in entropy (i.e., negentropy) that marks the emergence of the new organization. We emphasize that all three substantive predictors are lagged variables. One important implication of using lagged predictors is that, functionally, the model is always predicting future behavior (i.e., the discovery of alternation on the next trial), based on prior dynamics. A second implication is that the measures of dynamic organization that we use to predict discovery of alternation are always computed from force-tracing trials. Put differently, the predictors only reflect changes in dynamical organization that precede the shift to alternation; they do not notice changes in actions that are because of using the new strategy (e.g., the rhythmic pointing to each gear that typically accompanies alternation). We modeled the probability of discovering alternation on trial j, using prior mean line, peak entropy, and prior entropy, as well as trial number, B ⫽ ⫺.09. All three substantive predictors contributed significantly (i.e., at p ⬍ .05) to the model, change in ⫺2 LL ␹2(1) ⫽ 4.80, 6.20, 5.28, B’s ⫽ .42, .66, ⫺.97, for prior mean line, peak entropy and prior entropy, respectively. The probability of discovery was greater for those trials, j, with higher values of prior mean line (computed from trial j-1) and peak entropy (computed from trials j-2 through j-5), and lower values of prior entropy

(computed from trial j-1). Increases in attractor strength predict discovery of alternation. More importantly, increases in peak entropy and decreases in prior entropy (i.e., negentropy) predicted the discovery of alternation. The lower panel of Figure 8 illustrates the relationships between the entropy measures and discovering alternation. Four cumulative hazard functions are displayed that show the factorial combination of high-versus-low peak entropy and high-versus-low prior entropy (split, for purposes of illustration, on the sample mean). Because these predictors are time varying, the combination of levels is created separately for each trial; the hazard reflects the probability of discovery for participants who had that combination of levels on that trial. Thus, the functions provide an estimate of the probability of discovery for participants whose predictors take high and low values for peak and prior entropy for each trial. Conceptually, the functions show how the probability of discovery changes as participants’ peak and prior entropy levels change across trials. For example, an individual participant might initially have low peak entropy and high prior entropy, and therefore have a low probability of discovery. If, on a subsequent trial, the participant had high peak entropy and low prior entropy, his or her probability of discovery would increase dramatically. As can be seen in the figure, the two curves with low values of prior entropy have much higher cumulative hazard. The two curves with high values of peak entropy are also, on average, somewhat higher than their counterparts with low values. Although the figure gives a sense of the relationships between the entropy measures and discovering alternation, we note that the event history analysis captures these relationships more comprehensively, in part because it treats the predictors as continuous, rather than discretizing them. Finally, we note that standard measures, such as response time, mean and standard deviation of angular velocity, explain very little variance in the measures from RQA, entropy and mean line. Response time and mean angular velocity explain less than 1% of the variance in entropy and mean line. The standard deviation of angular velocity explains ⬃3% of the variance in entropy, r ⫽ ⫺.18, and less than 2% in mean line, r ⫽ ⫺.13.

Discussion Consistent with the principles of self-organization, the emergence of a new representation of the gear domain was predicted by a peak in entropy on recent trials and subsequent decrease in entropy immediately before discovery. We suggest that the new representation emerges as the ability of the current structure to offload entropy is exceeded by the entropy entering the system from the environment. The current structure is capable of dispersing input entropy, as long as input entropy remains within a limited range. When input entropy begins to exceed that range, system entropy starts to increase. Eventually the system becomes so entropic that it exceeds its critical threshold and the system reorganizes into a new representation. In the gear-system paradigm, an important source of input entropy is the new relation (i.e., alternation) generated by performing force-tracing. The introduction of new relations into a selfmodifying system initially increases entropy. For example, de Pinho, Mazza, Piqueira, and Roque (2002) presented a sophisticated model of tonotopic organization in the auditory system. They showed that the entropy of simulated layers in auditory cortex

DYNAMICS AND REPRESENTATION

increased when a simple relation was introduced (i.e., conditioned stimulus-unconditioned stimulus pairing). In the current scenario, input entropy increases because the participant’s actions embody the novel relation (Trudeau & Dixon, 2007; Zwaan & Taylor, 2006). That is, a participant’s force-tracing actions initially arise from an organizational state that does not specify alternation. However, because repeated force tracing embodies the alternation relation, it introduces disorder into the original organization. In this way, the interface between the organism and the environment (i.e., action) can increase the degree of entropy entering the system, ultimately driving the system to reorganize. Although our primary focus has been on the predictive power of microscopic fluctuations in behavior (i.e., entropy), we wish to emphasize that the cognitive structures that give rise, in part, to force tracing (e.g., conceptual understanding of the simple physics of gear systems) play a central role here as well. These higher-order cognitive structures support the organization of action. Because action is an exchange between the cognitive system and environment, it allows microscopic fluctuations (i.e., entropy) to enter the system. These microscopic fluctuations are the impetus for a phase transition in the macroscopic structure of the cognitive system. Higher-order cognitive structures support action, and the entropy introduced by action in turn drives the emergence of novel cognitive structure. Given the central role of entropy in the discussion above, a logical next step would be to manipulate the amount of entropy entering the system. Interestingly, research in nonlinear dynamics has shown that introducing fluctuations (i.e., input entropy) can induce a phase transition (see Shinbrot & Muzzio, 2001, or Wio, 2004, for reviews). When entropy is added to a nonlinear dynamic system, the system reaches a critical instability more quickly than it would otherwise. A large body of work has addressed this counterintuitive phenomenon in which increasing the disorder entering a system creates new order. In the second study, we test whether increasing the input entropy accelerates the emergence of new cognitive structure. Because actions are an interface between cognitive structures and the environment, and hence a source of entropy, we expect that randomly perturbing participants’ actions should increase entropy. Random perturbations, by definition, are not specified by the current organization and should, therefore, increase the disorder or entropy entering the system. In the second study, we introduced a random component into participants’ actions by manipulating the task environment. We predict that this additional source of input entropy should accelerate the self-organization of a new cognitive structure.

Study 2 We introduced random perturbations into the task environment by having the computer-screen window that contained the gear system shift unpredictably in two of three conditions; in a third condition the window was stationary. When the window shifts position, participants must reorient their motions to the new location to continue tracing the force across the system. We predict that these small perturbations in the participants’ actions will provide an additional source of input entropy and, therefore, lead to earlier discovery of alternation. Although it may initially seem counterintuitive that the addition of noise into the task environment could result in earlier discovery of a new relation, as discussed above, noise is a classic catalyst for new structure in nonlinear dynamical systems.

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Participants Thirty-five college students participated as one option to fulfill a course requirement.

Materials and Procedure We employed the same materials and procedures used in Study 1 in Study 2. Participants were randomly assigned to one of three conditions: low window shifting, high window shifting, or no shift. In the first two conditions, the computer-screen window containing the gear system shifted both vertically and horizontally at randomly sequenced intervals of 1 or 2 s. The direction of vertical and horizontal movement was also randomized for each shift. The magnitude of vertical and horizontal shifts in the low windowshifting condition was 30 pixels; in the high window-shifting condition it was 60 pixels. The vertical and horizontal shifts occurred simultaneously. Thus, the window appeared to jump to a random location every 1 to 2 s in these conditions. In the no-shift condition, the window remained stationary at the center of the screen.

Results As in Study 1, the majority of participants, 79% initially used force tracing to solve the problems. The remaining participants traced the motion with their eyes and head, but did not use their hands. Participants’ performance with force tracing was quite accurate in terms of predicting the turning direction of the target gear, 75% correct (similar to the performance of participants in Study 1, 77% correct). Further, there was no effect of the windowshifting conditions on accuracy with force tracing, F ⬍ 1, where accuracy is again defined as making the correct final prediction. The random perturbations did not interfere with successfully reaching the correct solution via the force-tracing strategy. Figure 9 shows the cumulative hazard function for each condition separately. As can be seen in the figure, the probability of discovery increased systematically as a function of shift condition. We again modeled the probability of discovering alternation using event history analysis. First, we examined whether the effects of prior mean line, peak and prior entropy on discovery, were replicated in Study 2. Recall that prior mean line and prior entropy were the RQA parameters from the immediately prior trial, j-1. Peak entropy was computed as the maximum value of entropy from trials j-2 through j-5. Prior mean line, peak and prior entropy all contributed significantly (i.e., at p ⬍ .05) to the model, change in ⫺2 LL ␹2(1) ⫽ 5.82, 7.45, 6.04, Bs ⫽ .71, .69, ⫺1.24. All three effects from the first study were replicated in Study 2. The major manipulation, shift condition, was also included in the model. Shift condition was coded 0, 1, and 2, for no, low, and high window shifting, respectively. As predicted, shift condition affected the probability of discovering alternation, change in ⫺2 LL ␹2(1) ⫽ 6.19, B ⫽ .68. Dummy-coding the conditions such that low-shift is compared separately to the no-shift and high-shift conditions shows that it is significantly different from both, Wald ␹2(1) ⫽ 4.56, 5.57, Bs ⫽ ⫺.63, .76. The random movement of the window increased the likelihood that participants would discover alternation.

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Figure 9. The cumulative hazard over trials is shown for each of the three conditions (i.e., High, Low, and No Window-Shifting) in Study 2.

The sample in Study 1 discovered alternation somewhat more quickly than the control-condition (i.e., no-shift) participants in Study 2, despite their nearly identical total cumulative hazard. We note that all participants in Study 2 were sampled and randomly assigned during the same time of the semester, after Study 1 was completed.

Discussion The second study replicated the major results obtained in Study 1: measures of dynamic organization, prior mean line, peak and prior entropy, predicted the discovery of a new representation in real time (i.e., on a particular trial). In addition, the second study also showed that the introduction of random perturbations into the task environment increased the probability of discovering alternation. Each time the display shifted randomly, participants’ actions had to incorporate that random motion to remain on task. Because actions impact the cognitive system, these random motions should provide another source of input entropy and, therefore, increase the probability of discovery. Recall that input entropy within a given range is offloaded by the current structure, allowing system entropy to remain relatively stable. However, when input entropy begins to exceed that range, system entropy increases, leading to a reorganization. Input entropy comes from a variety of external sources, including random variations in the environment (e.g., changes in sound, lighting, etc.), as well as the sources already emphasized: the random shifting of the window and the introduction of a new relation. Because the current structure offloads input entropy across a given range, system entropy fluctuates modestly until that range is exceeded. Put

differently, system entropy is a very nonlinear function of input entropy. One consequence of this nonlinear relationship is that system entropy will not, on average, be greater for the window shifting conditions. Rather, system entropy will remain relatively stable until input entropy exceeds the dissipative capacity of the structure. Introducing an additional source of input entropy (e.g., window shifting) makes it more likely that the total amount of input entropy will exceed the structure’s dissipative limit. The nonlinear nature of the function creates dependencies among these additive effects of input entropy; that is, the effect of window shifting on system entropy will depend on the values of the other sources of input entropy. Therefore, an additional consequence of the nonlinear relationship between input and system entropy is that system entropy will not mediate the relationship between input entropy and the probability of discovery. These relationships can be demonstrated concretely with any thresholding function.6 Note that the introduction of random motion into the task environment did not appear to negatively impact participants’ force tracing. Participants in Study 2 showed approximately the same level of accuracy as those in Study 1. Further, there was no effect of window-shifting condition on force tracing accuracy 6 Because lack of mediation, although consistent with the nonlinear functioning of the system, is not a compelling finding, we simply note that the effect of window-shifting condition on discovery is not mediated by measures of system entropy. Details regarding this analysis, as a well as a model demonstrating the prediction, are available from the second author.

DYNAMICS AND REPRESENTATION

within Study 2. Therefore, force tracing should create information about the alternation relation, and thus provide a source of input entropy.

Power-Law Behavior We have presented preliminary evidence that the discovery of the alternation relation is a phase transition in the cognitive system. Admittedly, this is an extraordinary claim in some respects, particularly because it requires that our measures have captured cognitive dynamics to a substantial degree. Thus far, the dynamic measures we have employed were all computed on reconstructed phase space via RQA. Although phase space reconstruction is a well-understood methodology in physics and other fields, we recognize that evidence from unfamiliar, complex methods may not be wholly convincing. Therefore, in this section, we use a different, complementary approach to address the hypothesis that the discovery of alternation constitutes a phase transition. Specifically, we test two interrelated predictions regarding changes in powerlaw behavior as a system approaches a phase transition. First, we briefly introduce power-law behavior as a general phenomenon, and then we describe how it changes around phase transitions, using the Lorenz system as an example. Then, we test these predictions on the time-series data from both studies. Power-law behavior is a widespread, general phenomenon (Newman, 2005; Schroeder, 1992). It is observed in a variety of fields and has stimulated considerable empirical and theoretical work. The basic power-law phenomenon is simply stated: a systematic relationship exists between the magnitude of system behavior and its frequency. The relationship, however, is nonlinear; it follows a power-law function. The left panel of Figure 10 shows an example of one such relationship, taken from a single trial of a single participant in Study 1. The frequency of each value of angular velocity is plotted as a function of its magnitude. The relationship is clearly nonlinear when plotted on the original scales. The log-log plot, shown in right panel of Figure 10, is another way to represent this relationship. Not only does it make the relationship somewhat easier to see by making it linear, it turns out that the slope of the linear portion of the curve contains important information about the nature of the system under consideration, because it quantifies the exponent of the power-law function. Crucially, as a system approaches a phase transition, the (unsigned) value of the power-law exponent increases; that is, the slope of the linear portion of the log-log plot becomes increasingly steep (Jensen, 1998; Schroeder, 1992). (For purposes of exposition, we treat ␣ as unsigned; the slopes of all the log-log plots discussed below are negative.) Power-law behavior has, thus, been taken as index of the degree to which the system is “poised” for a phase transition.7 To illustrate the relationship between power-law behavior and phase transitions concretely, we again briefly consider the Lorenz model. Figure 11 shows the slope (␣) of the log-log frequency histogram of the X-variable in the Lorenz system computed for each epoch. (Recall that the epochs are partially overlapping-time windows.) As previously, for purposes of exposition, we show the portion of the Lorenz trajectory under consideration in each of four different periods. Note that power-law slopes increase dramatically as the system approaches the phase transition. As systems become increasingly “poised” for a phase transition, ␣ becomes larger.

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However, as soon as the system enters the phase transition itself, ␣ begins to decrease. Thus, increases in ␣ indicate that the system is approaching the phase transition, and subsequent decreases indicate that it is reorganizing into the new structure (Grebogi, Ott, Romeiras, & Yorke, 1987). These relationships between power-law exponents and phase transitions allow us to make two predictions about the time-series data obtained in our studies. First, if participants’ force-tracing activity is driving them towards a phase transition, then they should become increasingly poised for a phase shift, the longer they stay in the risk set. Therefore, over prediscovery trials, the average ␣ for participants in the risk set should increase. However, we should also see evidence of the second effect, a decrease in ␣ as the system enters the phase transition and settles into the new regime. During the period just before discovery, ␣ should decrease. To test these predictions, we calculated ␣ for each trial for each subject as follows: ␣ ⫽ 1 ⫹ n (⌺ ln(xi/xmin)),⫺1 where xi is the absolute value of angular velocity for time sample i, xmin is the smallest value of x in the time series for this trial, and n is the number of points in the time series.8 This method of computing ␣ is functionally equivalent to taking the slope of the log-log plot of the frequency histogram via regression, but avoids some well-known biases introduced by that method (Newman, 2005). Because the predictions are the same for both studies, we treat them jointly in a single set of analyses. This gives us sufficient data to apply an appropriate over-time analytic method, growth-curve analysis (including study as a predictor does not significantly improve the fit of the model presented below). The means for participants at risk for discovering alternation is show in the top panel of Figure 12. Consistent with the prediction that participants will become increasingly poised for a phase transition, ␣ increases over trials. Of course, the discoveries of individual participants are embedded within this averaged curve. To examine graphically whether the ␣ values decreased as predicted before discovery, we can align participants on their discovery trial; the mean ␣ is plotted as a function of prediscovery trial in the lower panel of Figure 12 (because nondiscoverer’s have no discovery trial, they are not represented in the lower panel). Note that, as predicted, the ␣ values decrease before discovery. Both these effects can be captured simultaneously in a simple growth-curve model. The dependent measure, ␣, is modeled as function of trial number and prediscovery trial. The latter variable 7 We note that the relationship between power-law behavior and phasetransitions is noncontroversial. Indeed, straightforward mathematical proofs are readily available (cf., Jensen, 1998). Controversy has arisen, however, over the claim that some systems continually exhibit power-law behavior because they reset themselves to “critical” points. That is, the system is set up such that it resides in the “poised” state (i.e., the state just before a phase transition), and returns to that poised state after the phase transition (Bak, 1996). This very interesting claim has far-reaching implications, but it is not germane to the current argument. Rather, the current analyses rest on the more mundane relationship between power-law behavior and a system approaching a phase transition. 8 To take advantage of this convenient computational method, we added a constant, 1, to the time series, thereby avoiding taking the natural log of values between 0 and 1. Other methods of obtaining ␣, reviewed in Newman (2005), yield the same substantive results as those reported here. All the angular velocity time series were greater than 300 points in length.

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Figure 10. The left panel shows the frequency of angular velocity values as a function of their magnitude. The nonlinear relationship approximates a power-law. The right panel shows the same data plotted on (natural) log-log scales.

is coded as ⫺1, ⫺2, etc., (e.g., for one trial before discovery, two trials before discovery, respectively). For nondiscoverer’s, the prediscovery trial variable is always zero. Thus, the model simultaneously fits both an overall linear term for the effect of trials, and a separate linear term for the effect of discovery. (See Singer & Willett, 2003, for a discussion of this type of coding scheme for time-varying predictors.) Consistent with the effects evident in Figure 12, ␣ significantly increased as a function of trials, B ⫽ .01, change in ⫺2 LL ␹2(1) ⫽ 4.40, and decreased as participants approached discovery, B ⫽ ⫺.02, change in ⫺2 LL ␹2(1) ⫽7.40. The same set of predictions can be confirmed using spectral analysis (Thornton & Gilden, 2005; Van Orden et al., 2003). Because these analyses are somewhat more complex and offer little additional information, we do not report them here.9

General Discussion Previous work has demonstrated that force tracing and alternation are indicative of two distinct representations of the gear system. These two representations of the problem (a) are predicted by different prior behaviors, (b) are accompanied by different verbalizations, motions, and levels of accuracy, and (c) predict different subsequent behaviors, including generalization within a domain and transfer across domains (Dixon & Bangert, 2002; Dixon & Dohn, 2003; Dixon & Kelley, 2006; Lehrer & Schauble, 1998; Schwartz & Black, 1996; Trudeau & Dixon, 2007). Thus, the evidence suggests that force tracing and alternation reflect two different representations of the task. Both representations appear to systematically organize behavior at a number of levels (e.g., verbal, motor), and do so in demonstrably different ways. Therefore, although we have tapped the action level of the cognitive system to assess its dynamical organization, the phenomenon we are predicting is a change in representation. Clark (1997) argued that extending the reach of dynamics to representational phenomena is an important goal for dynamic systems approaches to cognition. In two experiments, we employed measures of dynamic organization to predict the real-time emergence of a new representation

of the gear-system task. In the first study, we showed that attractor strength (as measured by mean line) reliably increased before discovery. We also showed that peak entropy across recent trials, and negentropy on the immediately prior trial, predicted discovery of a new representation of the task. The peak in entropy triggers reorganization; subsequent negentropy marks the emergence of a new structure (see Dale, Roche, Snyder, & McCall [2008] for a parallel set of findings). In Study 2, we demonstrated a second, and related, classic effect from nonlinear dynamics noise can drive the shift to a new structure. By having the window containing the gear system shift randomly to different degrees, we manipulated the amount of noise entering the system. We predicted that the greater the degree of added input entropy or noise, the more rapidly the cognitive system would reorganize. The results showed that, as predicted, participants’ discovery of alternation was affected by the degree to which the window shifted randomly; increasing the noise injected into the system accelerated discovery. Study 2 also replicated the effects of prior mean line, peak, and prior entropy on discovery. We also demonstrated that participants‘ actions exhibit changing power-law behavior as they approach discovery of alternation. As predicted by the hypothesis that the discovery is a phase transition, power-law behavior increases as a function of time spent interacting with the task, and decreases just before the emergence of alternation. Participants who are at risk for discovery become increasingly poised for the transition. As they settle into 9 It is worth noting that entropy and ␣ respond to different aspects of the phase transition. Put metaphorically, ␣ signals an impending transition, entropy signals movement through the transition. More specifically, increases in ␣ precede the transition, during which entropy is relatively constant. Decreases in ␣ occur across the entire transition period (i.e., leaving one attractor and settling into another), but entropy both rises and falls during the transition. Thus, ␣ and entropy are not strongly correlated, even in the completely deterministic Lorenz model, r ⫽ .17; in our studies, the relationship between ␣ and entropy is similar, r ⫽ .22.

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Figure 11. Two curves show the changing values of ␣ and entropy as the Lorenz model undergoes a phase transition. Values of ␣ increase sharply before transition and then decrease as the trajectory leaves the initial attractor and settles into the new regime. The behavior of the system in four different periods is shown along the bottom axis.

the new mode of functioning, power-law behavior decreases. Note that the degree of power-law behavior is derived from straightforward measures computed on the frequency distribution of the time series. It does not involve phase-space reconstruction or other types of complex analysis. The power-law analyses serve as another window onto the dynamics of the cognitive system. Thus, the systematic changes in power-law behavior provide converging evidence with regard to our central hypothesis that the discovery of alternation, a new cognitive structure, is a product of selforganization.

Perception-Action and Entropy Our results with the gear-system paradigm suggest that perception-action may play a catalytic role in the emergence of new cognitive structure, rather than simply serve as the inputoutput device for an insular cognitive system. Because perceptionaction provides the interface between the system and environment, it is a major conduit for input entropy. We propose that perceptionaction can provide a source of entropy both when the cognitive system makes an error (e.g., moving one’s hand to the wrong location) and when it operates successfully. We expand on this second, more counterintuitive assertion below. The cognitive system organizes to anticipate events in the environment (Bickhard & Terveen, 1995; Glenberg, 1997; Rosen, 1985; Sommerville & Decety, 2006). These anticipations occur on multiple time scales simultaneously (Soodak & Iberall, 1987;

Stephen, Stepp, Dixon, & Turvey, 2008). For example, on a very short time scale, the cognitive system of a person reading this manuscript constantly anticipates the changes in physical support necessary to maintain posture. At a slightly longer time scale, the system organizes to anticipate the next word in the sentence, and at a yet longer scale, the reader’s system anticipates the semantic content of the argument being developed. In the gear task, participants’ force tracing arises, in part, from the organization on one time scale, at the temporal level of motion of the individual gears (i.e., pushing and turning affordances). Performing force tracing creates relational information at a longer time scale, at the level of the individual problem. The cognitive system is well organized at the temporal scale of the individual gears, as evidenced by successful force tracing, but it is poorly organized at the level of the individual problem. The relational information created by force tracing impacts the organization of the system at the longer temporal scale, and thus provides a source of entropy. Therefore, somewhat paradoxically, successful actions increase disorder at another level of organization, ultimately leading to reorganization. In this way, perception-action operating at one-time scale creates information about the relations in the environment on a longer time scale; these relations increase entropy because they are unanticipated by the current organization of the system. The account above emphasizes the effects of microscopic (i.e., short time scale) activity on macroscopic structure, but clearly

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Figure 12. The top panel shows the changing values of ␣ as a function of trials for all participants who remain in the risk set. The lower panel shows changing values of ␣ as a function of prediscovery trial (e.g., ⫺1 refers to one trial before discovery, ⫺2 refers to two trials prior) for those participants who discovered alternation (nondiscovers do not have a discovery trial).

macroscopic cognitive structure plays an indispensable role in the process (see Anderson, 2002, for a discussion of spanning scales in psychological research). Higher-order cognitive structure (e.g., the representation of the physical forces) and the properties of the gear display together organize the tracing actions generated by the participants. The actions create microscopic fluctuations that ultimately destabilize the macroscopic structure. However, these actions are predicated on the current macroscopic organization of the system. In this way, a particular set of existing higher-order structures allows the system to be driven through this phase transition. We suggest that action is both the product of the current organization in its environment and the catalyst for the transition.

Self-Organization of Cognitive Structure On this account, representations are stable, flexible cognitive structures (Smith, 2005; Thelen, Scho¨ner, Scheier, & Smith, 2001) that serve to maintain the system’s relationship with its environment. While the assertion that representations help maintain the system-environment relationship is perhaps noncontroversial, it has a deep and fundamental implication. At the most basic level, maintaining a system’s relationship to the environment is a matter of dispersing the entropy that enters it. If representations help maintain the relationship between the system and the environment, they must function to offload entropy. In systems that constantly maintain their distance from equilibrium, new structures form

DYNAMICS AND REPRESENTATION

when current structures are overwhelmed by entropy and begin to disintegrate. The breaking down of a current structure allows its smaller scale, constituent parts to interact. If the interactions do not create new structure, the system just moves closer to equilibrium. Of course, the more interesting case is when these interactions create a new structure (that better disperses entropy), thereby allowing the system to maintain its distance from equilibrium. The latter case is termed self-organization. Changes in power-law behavior tie naturally into this account and offer further insight into the nature of cognitive structure. In complex systems, the structures that become disintegrated (and reintegrated) are comprised of multiple, nested levels. That is, a structure is made up of other smaller structures that, in turn, are made up of yet smaller structures, etc. This type of structure is often referred to as fractal. Biological systems have been shown to have this type of structure (Schneider & Sagan, 2005; West, Brown, & Enquist, 1999). When such systems are stably organized into a structure, many of the nested levels are constrained (i.e., bonded together so they cannot move independently). When the structure begins to become disintegrated, the constraints break across the different scales, allowing more nested levels to become active. The degree of nesting among the levels that are currently unconstrained (i.e., free to interact) is indexed by the power-law exponent. The activity across such a nested structure produces power-law relationships because the number of active parts grows exponentially as one traverses the levels from larger to smaller. Our results suggest that the onset of new cognitive structure is preceded by the breaking of constraints across multiple scales, as evidenced by the power-law exponent increasing as a function of repeatedly using the lower-order representation. We also found that just before the emergence of a new cognitive structure the power-law exponent decreases, indicative of constraints across levels being reconstituted. Cognitive structures appear to be comprised of multiple, nested levels, and the degree of constraint among these levels changes as the system self-organizes into a new structure. The current findings complement previous work demonstrating a particular type of power-law relationship called 1/f scaling. Bak (1996) proposed that 1/f scaling is produced by systems that automatically and continually reset themselves to a critical threshold (i.e., a point just before a phase transition). The advantage of doing so is that the system is maximally poised to change when it is at a critical threshold. The 1/f scaling relation is often exhibited by systems that organize themselves into critical states. A fairly large body of work now suggests that 1/f scaling is found in a wide variety of standard cognitive tasks (Gilden, 2001; Thornton & Gilden, 2005). For example, Van Orden, Holden, and Turvey (2003) showed that the distribution of response times in a word-naming task exhibited the 1/f exponent (but see also Farrell, Wagenmakers, & Ratcliff, 2006; Wagenmakers, Farrell, & Ratcliff, 2004, 2005). Consistent with the theoretical work by Bak (1996), Van Orden et al. proposed that when repeatedly required to read aloud visually presented strings of letters, the cognitive system organizes itself to sit at the threshold of a phase transition. Sitting at this threshold involves keeping components in a loose but poised assembly capable of manifold organizations. This allows the system to rapidly recruit the appropriate articulatory degrees of freedom suited to verbalizing the visual stimulus.

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The common thread between the discovery of alternation and the case of word naming is that both are examples of a transition in the cognitive system. Van Orden et al. proposed a selforganization account of word naming, a relatively rapid transition from the visual perception of a letter string to the verbalization of the string. Their findings suggest a rapid resetting of the cognitive system, from trial to trial, to a critical state characterized by increased power-law exponents. On each trial, the cognitive system self-organizes to produce its response. We present a comparable account of cognitive transition; discovery of alternation is the self-organization of the cognitive system on a longer time scale. In the gear-system task, rather than examine the transitions underlying lexical verbalizations, we study the transition from one representation of the problem to another. We provide evidence of a gradual migration of cognitive system towards a critical state over the course of several trials. The changes in power-law behavior reported in our gear-system task reflect changes in the potential for a phase transition, and so stable configurations of the cognitive system exhibit weaker power-law behavior. When the cognitive system is in this regime, its constituent microelements are held relatively tightly in place, thereby creating stable (but not static) components. Initially, the relative dominance of these components yields lower power-law exponents. However, force tracing drives the system closer to criticality, as evidenced by increasing power-law exponents. The increasing power-law exponents index the gradual breaking of constraints. As power-law exponents increase, the stable configuration, that gave rise to the component behavior, begins to break apart, allowing interactions among the microelements to govern system activity. Thus, we propose that 1/f scaling sits within a broader framework in which cognitive structure is a self-organizing phenomenon. It is necessary to appreciate both the component dominance of relatively stable configurations and the interaction dominance that occurs near critical transitions. The self-organization of cognitive structure rests not only on rapid resettings to critical states but also a wider range of power-law behavior. We suggest that cognition runs on a mixture of componentdominant and interaction-dominant dynamics. Component-like dynamics tend to dominate when the system is functioning in a relatively stable configuration. That is, component dominance holds when higher-order cognitive structures are the crucial variables governing the behavior of the cognitive system. However, the emergence of new cognitive structure is based on interactions among the microelements that comprised those components. Interactions dominate when the system becomes unstable and approaches a transition. Once the system self-organizes from those interactions into a new stable mode of functioning, componentdominant dynamics re-emerge. In conclusion, the results presented here are consistent with the proposal that new structures in cognition emerge through selforganization. The theory of nonlinear dynamics has been extensively developed in domains that focus on the behavior generated by complex systems. Thus, it seems natural to extend the theory to the complex system that gives rise to cognition. We demonstrated that an increase in entropy followed by a drop in entropy predicts the emergence of a new cognitive structure; the relationship between entropy and new self-organized structure is a cornerstone of the theory of nonlinear dynamics. We also showed

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that increasing the entropy entering the system accelerated the transition to the new structure, a classic effect in self-organization. Finally, we demonstrated a third major effect from the theory of nonlinear dynamics: power-law behavior increases as the cognitive system approaches a phase transition, and then decreases as the constraints undergirding the new structure develop. Taken together, these results suggest that cognitive structure disassembles through the breaking of constraints and reassembles through the setting of new constraints.

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Appendix Recently, methods have been developed for detecting nonlinearity in time-series data. A number of researchers have suggested that establishing nonlinearity in a time series is an important prerequisite to performing nonlinear analysis (Schreiber & Schmitz, 1996, 2000; Theiler, Eubank, Longtin, Galdrikian, & Farmer, 1992). This appendix briefly reviews one of such method, surrogate data analysis, and reports the results from application of this method to time series obtained in both studies. In surrogate data analysis, the null hypothesis is that a linear process generated the observed times series. Theiler, Linsay, and Rubin (1993) define a Gaussian linear process as one in which all the properties of the time-series are encoded in the mean, variance, autocorrelation function. The core strategy in surrogate data analysis is to generate a set of artificial time-series data via a random process that reproduces these key aspects of the original data, most importantly the autocorrelation function. These artificial or surrogate time series function much like a sampling distribution to which the original time-series is then compared in two complementary ways. First, a nonlinear statistic capable of capturing nonlinear structure in the data is computed for all the series (original and surrogate). If the original time series has nonlinear structure, the nonlinear statistic should be significantly different from the surrogates. This demonstrates that there was additional structure that could not be captured by the linear process. Second, a linear statistic is likewise computed on all the time series. If the surrogate time series successfully captured the linear properties of the original series, then the original series should not be different from the surrogates. This latter test confirms that the process used to generate the surrogates successfully represented the linear structure in the original series (Palus, 1995). We randomly selected 34 participants from the two studies for this analysis, yielding 510 risk-set trials. We performed surrogate data analysis on the angular velocity time series for each trial separately. For each of these time series, we generated 100 surrogates via the Fourier method described by Theiler et al. (1993) (see also, Palus, 1995). The Fourier method produces surrogates with exactly the same power spectrum as the original sample and, therefore, the same autocorrelation function. We used a well-established measure of nonlinear autocorrelation, average mutual information (AMI), recommended by Palus

(1995). If the original time series is nonlinear, then its AMI should be stronger than that of the surrogates. A bootstrap t-test (see Nankervis & Savin, 1996) can be used to compare the AMI of the original time series against the AMI of the surrogates. A significant t-statistic from this comparison indicates nonlinearity of the original time series. Consistent with the hypothesis that a nonlinear process generated the time series obtained in our studies, ⬃95% of (486 of 510) AMI t tests were significant at the .05 level. The average value of t was 25.97, SD ⫽ 17.97. In nearly all cases, the original series had significantly greater AMI statistics. (The remaining nonsignificant tests, ⬃5%, are consistent with the Type 1 error rate.) Following Palus (1995), we used the linear redundancy of the time series as a measure of linear structure. In brief, linear redundancy aggregates information from the correlation matrix to give an overall measure of linear dependence. Again, if the surrogates successfully capture the linear structure of the series, the original and surrogates should not differ on this linear measure. A bootstrap t test can be used to compare the linear redundancy of the original time series against the linear redundancy of the surrogates. Ideally, the results of this t-test should indicate a null statistical difference. Of the 510 t-tests on the linear redundancy statistic, 485 (⬃95%) were nonsignificant. The average value of t was .63, SD ⫽ .72. The surrogates appear to have adequately represented the linear structure in the data. In summary, to justify the use of nonlinear analyses, we used surrogate data analysis to test for nonlinearity in the time-series data. Specifically, we compared the angular velocity time series described in the manuscript to surrogates generated under an assumption of linear autocorrelation. Each of 510 time series was compared to its respective surrogates using the average mutual information and linear redundancy measures recommended by Palus (1995). In the vast majority of cases, the AMI of the original time series was significantly greater than that of the surrogates. In addition, the linear redundancy of the original time series was not significantly different from the linear redundancy of the surrogates. Received April 27, 2008 Revision received October 18, 2008 Accepted October 21, 2008 䡲