Journal of Experimental Psychology: Human Perception and Performance 2014, Vol. 40, No. 6, 2289 –2309
© 2014 American Psychological Association 0096-1523/14/$12.00 http://dx.doi.org/10.1037/a0038159
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Interwoven Fluctuations During Intermodal Perception: Fractality in Head Sway Supports the Use of Visual Feedback in Haptic Perceptual Judgments by Manual Wielding Damian G. Kelty-Stephen
James A. Dixon
Grinnell College
University of Connecticut
Intermodal integration required for perceptual learning tasks is rife with individual differences. Participants vary in how they use perceptual information to one modality. One participant alone might change her own response over time. Participants vary further in their use of feedback through one modality to inform another modality. Two experiments test the general hypothesis that perceptual-motor fluctuations reveal both information use within modality and coordination among modalities. Experiment 1 focuses on perceptual learning in dynamic touch, in which participants use exploratory hand-wielding of unseen objects to make visually guided length judgments and use visual feedback to rescale their judgments of the same mechanical information. Previous research found that the degree of fractal temporal scaling (i.e., “fractality”) in hand-wielding moderates the use of mechanical information. Experiment 1 shows that head-sway fractality moderates the use of visual information. Further, experience with feedback increases head-sway fractality and prolongs its effect on later hand-wielding fractality. Experiment 2 replicates effects of head-sway fractality moderating use of visual information in a purely visualjudgment task. Together, these findings suggest that fractal fluctuations may provide a modal-general window onto not just how participants use perceptual information but also how well they may integrate information among different modalities. Keywords: intermodal perception, haptics, vision, feedback, dynamic touch, fractal
What is not so plainly obvious is how or why participants might arrive at different intermodal coordinations of their perceptual faculties. The intermodal integration required for perceptual-learning tasks is rife with individual differences (Withagen & van Wermeskerken, 2009; Spence & Squire, 2003). The simple experimental manipulation of providing feedback compounds the challenges found in perception within a single modality: Not only may participants differ (or, themselves, vary over time) in how they use perceptual information to either modality alone, but then these individual differences compound as feedback to one modality informs another. Experimentally, we try to provide enough redundant information between feedback and response that the leaps from response modality to feedback modality and back again are as short and direct as possible. Nevertheless, participants bring into our laboratories a wealth of free-ranging parameters— endogenous to their own physiology, history of experiences and preferences— that can upset even the most elegant perceptual-learning design. We aim in this manuscript to present an account of individual differences in intermodal perception that rests on an analysis of perceptual-motor fluctuations at two different locations across the body. This approach to the individual differences in intermodal perception departs from a more conventional class of approaches that explains individual differences in terms of explicitly cognitive, central nervous processes. We build from previous research into perceptual learning that has only examined fluctuations at single locations at a time. In this previous research, we had used timeseries methods to estimate trial-by-trial temporal correlations of
Perceptual-learning paradigms frequently make use of the observation that one perceptual modality can support another. A popular understanding of perceptual learning involves integrating information from at least two sensory modalities (Bahrick & Lickliter, 2002; Lickliter, Bahrick, & Markham, 2006). Essentially, one modality must prop up another where the other falls short. Feedback to one modality may change responses based on another. These notions are commonplace facts of the literature.
This article was published Online First October 20, 2014. Damian G. Kelty-Stephen, Psychology Department, Grinnell College; James A. Dixon, Psychology Department, University of Connecticut, and Center for Ecological Study of Perception and Action, University of Connecticut, Haskins Laboratory. We thank two anonymous reviewers and Jay Holden and Brett Fajen for their helpful guidance through important revisions supporting the hopeful readability of the remarks. Damian G. Kelty-Stephen would particularly like to thank coauthor James A. Dixon as well as Claire F. Michaels and Michael T. Turvey, all three of whom sat patiently on the dissertation committee that suffered through an earlier, gustier version of this work and provided insights invaluable for this final version. Harvard’s Wyss Institute supported important revisions and reanalyses contributing this work as well. Zsolt Palatinus and Emma Kelty-Stephen provided further feedback crucial for the completion (and recompletion) of this work through its many iterations over five years of work. Correspondence concerning this article should be addressed to Damian G. Kelty-Stephen, 1115 8th Avenue, Grinnell College, Grinnell, IA 50112. E-mail:
[email protected] 2289
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perceptual-motor fluctuations leading up to a response, and then we demonstrated that these trial-by-trial measures of temporal correlations significantly improved regression models of trial-bytrial perceptual judgments. In the present research, we elaborate our earlier approach to include measures of temporal correlations at two locations, one whose fluctuations served the exploration of mechanical information for touch and another whose fluctuations served the exploration of visual information. We will incorporate both locations’ trial-by-trial measures of temporal correlations into a regression model of trial-by-trial perceptual response. Intermodal perception is much more than the concurrent use of two subsystems; however, it reflects the sharing of information between modalities or the influence of one on another. We will use a vector time-series method to demonstrate specifically that visual feedback for touchbased judgments causes temporal correlations of fluctuations in visual exploration to have a more sustained effect on the temporal correlations serving touch-based exploration.
Two Approaches to Individual Differences in Perceptual Learning One strategy for confronting the individual differences in perceptual learning is to use the variability to infer the internal, mediating mechanisms that might enrich or distort incoming information (Jonassen & Grabowski, 1993; Goldstone, Schyns, & Medin, 1997). Certainly, neurological evidence confirms the intuition that intermodal perception involves activity in the central nervous system (Mapstone, Logan, & Duffy, 2006; Meeren, van Heijnsbergen, & de Gelder, 2005). Further neurological evidence points to specific brain regions that may contribute to individual differences in perceptual learning (Eisner, McGettigan, Faulkner, Rosen, & Scott, 2010; Martin, Barnes, & Stevens, 2012; Mukai et al., 2007; Ong, Russell, & Helton, 2013; Bridwell, Hecker, Serences, & Srinivasan, 2013). In many of these cases, the neurological evidence aligns with and supports the implication of specifically cognitive mechanisms such as attention and motivation. Another line of research has sought to understand perceptual learning by probing the temporal structure of perceptual-motor variability during the task. Without denying a role for more central, neural underpinnings, this approach has instead focused on the fluctuations incident to the exploration at relatively more peripheral regions of the perceiving-acting system. Often this approach seeks to examine fluctuations near more immediate points of contact between perceptual apparatus and task stimulus, such as the fingertip tracing an image (Stephen, Dixon, & Isenhower, 2009; Stephen & Dixon, 2009) or point of gaze on a computer screen (Stephen, Boncoddo, Magnuson, & Dixon, 2009). Fluctuations in the periphery play a subtle but important role in shaping an organism’s experience of the energy distributions serving as perceptual information. Perhaps across the periphery of the perception-action system we may see only “downstream” echoes of the central-nervous, cognitive factors exerting their “top-down” effects on perceptual learning, but perhaps we may see also novel heterogeneities that allow “bottom-up” seeming perceptual-motor “noise” to percolate into major changes in cognitive structure (Stephen, Boncoddo, et al., 2009; Stephen & Dixon, 2009; Stephen, Dixon, & Isenhower, 2009).
A central premise of this latter approach is that no measurable fluctuation occurs without an energy flow to set it into motion. Perception through any single modality depends on a perceptual system sampling an energy distribution. Fluctuations in perceptual-motor behavior provide a negative image of energy flows: they are not energy themselves, but they reflect the impact of any number of energy distributions sampled for perceptual impressions. For instance, fluctuations like that of a flapping flag provide empirical information for inferring the kinetic energy of the wind and the inertial properties of the flag. The flag is not itself energy, but its fluctuations reflect the confluence of energy flows giving rise to flag flapping. We can infer the statistics of energy flow from the statistics of fluctuations. Thus, the spreading of fluctuations from one perceptual subsystem to another should thus provide insights into the coordination underlying intermodal integration.
Fluctuations in Perceptual-Motor Exploration Are “Fractal,” and Fractality Predicts the Use of (Unimodal) Perceptual Information Fluctuations in perceptual-motor behavior often bear what is known as “fractal” temporal structure. Temporally fractal fluctuations are prevalent throughout the physiology and motor variability of the perceiving-acting system. Temporally fractal fluctuations are random sequences that can exhibit a patchy, intermittent clustering over time, for example, clustering of big fluctuations with other big fluctuations, or clustering of small fluctuations with other small fluctuations. They exhibit long-range structure wherein the similarity of fluctuations from one time point to another lingers over a wide range of time scales, falling off with longer time scales but only very slowly. They strike a statistical compromise between two extremes of temporal correlation: “uncorrelated” sequences that have no dependence or predictability between time points across any time scales (e.g., “white noise”) and “nonstationary” or “persistent” sequences that show strong, relatively uninterrupted trends. The strength of temporal correlations within stationary bounds serves as a measure of the degree of fractal structure (i.e., “fractality”; Eke, Herman, Kocsis, & Kozak, 2002). The fractality of perceptual-motor fluctuations has predicted how well participants use perceptual information in a number of domains and task varieties (Stephen, Boncoddo, et al., 2009; Stephen & Dixon, 2009; Stephen & Hajnal, 2011; Stephen, Arzamarski, & Michaels, 2010; Palatinus, Dixon, & Kelty-Stephen, 2013; Nonaka & Bril, 2014; Palatinus, Kelty-Stephen, KinsellaShaw, Carello, & Turvey, 2014). We should underscore that fractal fluctuations are not explicitly or specific “for” perception, and their origins are hotly contested (Torre & Wagenmakers, 2009). Their prevalence through biological structure—and even throughout many physical, nonbiological systems—suggests that they do not arise by a conscious will to perceive (Kelty-Stephen, 2014). Some scholars have taken fractality to reflect a self-organizing process in which many nested processes interact over many nested time scales in cascade-like fashion (Van Orden, Holden, & Turvey, 2003; Holden, Van Orden, & Turvey, 2009). Others have been more skeptical that it means anything straightforward or meaningful at all (e.g., Uttal, 2003). Whatever its origins, we only note that fractal fluctuations in perceptual-motor contact with task stimuli predict how participants make use of the contacted stimulus energy
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distributions. Stronger evidence of temporally fractal structure has so far been associated with better use of feedback and more accurate judgments, suggesting that perception benefits from temporal correlations balanced more evenly between the uncorrelated and nonstationary extremes.
Experiment 1
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Toward a Fractal Model of Intermodal Perception in Dynamic Touch A severe limitation of the previous work on the fractality of perceptual-motor fluctuations has been that each demonstration has only documented an effect of fractality on a single modality at once. To illustrate what this point means, we now review the paradigm of dynamic touch that has provided the testbed for most of the previous work in this vein. First, we will review the traditional understanding of dynamic touch as a specifically haptic, nonvisual process. Second, we will review how dynamic touch lends itself to intermodal perception, particularly with the inclusion of visual feedback. Third we will outline the predictions for elaborating the fractal model of unimodal perceptual responses to the intermodal case. Dynamic touch is a variety of haptic perception that involves grasping an unseen object and exploring its geometric properties by wielding it. Also called “effortful touch” or the “muscle sense,” dynamic touch is distinct from cutaneous haptic perception because it does not simply depend on skin sensitivity but on the participation of neuromuscular factors (Carello & Turvey, 2004; Turvey & Carello, 2011). Experiments in the dynamic-touch paradigm often require participants to sit with their wielding hand hidden from their own view; in the present experiment, participants sit with an opaque curtain flush against the right shoulder, leaving the right hand free to wield without being visible (Figure 1; see also Hajnal, Fonseca, Harrison, Kinsella-Shaw, & Carello [2007]; Hajnal, Fonseca, Kinsella-Shaw et al. [2007] for extensions of dynamic touch to the foot and Palatinus, Carello, & Turvey, [2011] for extension to the torso). The dynamic-touch paradigm thus involves a rather straightforward decoupling of haptic information from visual information. Dynamic touch also shows how perceptual information can spread from one part of the body to another, and sometimes, from one modality into another. For instance, in providing a sense of mechanical loads under a variety of neuromuscular orientations, dynamic touch is a fundamental aspect of posture in and locomotion through a cluttered, heterogeneous environment (Carello & Turvey, 2004; Turvey & Carello, 2011). However, in order to support posture and locomotion, information detected through dynamic touch must often be coordinated with information detected through other modalities, whether vision (Gibson, 1979; Lee & Lishman, 1975; Kugler & Turvey, 1987) or, in the case of blind locomotion, sound (Easton, Greene, DiZio, & Lackner, 1998; Strelow & Brabyn, 1982; Dufour, Després, & Candas, 2005). That is, the use of dynamic touch for tailoring the behavior of a perceptual system to the perceptual system’s environment, may quickly become a deeply complex matter of reconciling two allegedly distinct modalities. Though often unrecognized in the dynamic-touch literature, measurement of performance in dynamic
Figure 1. The experimental apparatus of Experiment 1. The screen occludes the participant’s view of the wielded object. Participants pull on the rope to make the response marker move forward or backward to make length judgments. The length of the bracket on the right indicates the distance from the table’s edge near the participant to the response marker, that is, the length judgment. The dashed, contorted line above the unseen object is intended here to schematize the wielding trajectory whose detrended fluctuations were used for previous fractal analysis. Text boxes summarize the findings of these previous analyses that the present work seeks to replicate and elaborate to include fluctuations in head sway.
touch usually depends on the participant being able to see the response system. So, lurking within the phenomenon of dynamic touch is a case of visually guided distance perception even without any visual feedback. Perception of distance to a specific target will depend on the visual perception of the surfaces surrounding the target (e.g., Meng & Sedgwick, 2002). Providing new visual information in the form of feedback about previous haptic judgments can change subsequent haptic judgments (Wagman, McBride, & Trefzger, 2008). Hence, despite experimental controls to the contrary, important perceptual information still bleeds through from one modality to another. Besides suggesting the role of potentially invariant amodal information (e.g., Stoffregen & Bardy, 2001), the dynamic-touch literature has not yet offered an explanation of how visual feedback operates on haptic judgments. In light of the visual guidance of judgments in dynamic touch, it is a very natural expectation that vision should play a role in dynamic touch. The key question relevant to intermodal perception is how visual feedback can jump the experimentally controlled divide to influence haptic perception as a result of feedback. We hope to close this divide by implicating the fluctuations of head sway as a predictor of how well participants use visual feedback. We will augment our existing unimodal fractal model by incorporating a measure of trial-by-trial fractality for head sway. Previous research has suggested that head sway is an important aspect of visual exploration (Paulus, Straube, & Brandt, 1984; Bronstein & Buckwell, 1997; Ehrenfried, Guerraz, Thilo, Yardley, & Gresty, 2003; Mitra & Fraizer, 2004; Vuillerme, Burdet, Is-
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ableu, & Demetz, 2006). Fluctuations in head sway also appear to be temporally fractal (Ashenfelter, Boker, Waddell, & Vitanov, 2009). Just in the same way that grasping an object by the fingers allows wielding by the hand, inspection of a scene with eyes may involve comparable wielding by the head. We begin from the premise that stronger signatures of fractality at the hand are associated with better performance over the course of the perceptual learning task. Hence, the perceptual-learning task will ideally engender stronger signatures of fractality, and our current understanding suggests that visual feedback accomplishes this change. In fractal terms, we might enrich this account to suggest that head-sway fractality moderates the use of visual information, including the use of visual feedback. Further, the experimental manipulation of providing visual feedback may itself promote greater fractality in head-sway fluctuations. Subsequently, increases in head-sway fractality may contribute to later and, more importantly, sustained increases in fractality in handwielding fluctuations. To test these ideas, we now outline our hypotheses. Our first substantial hypothesis was that effects of visual feedback on trialby-trial judgments would depend on head-sway fractality and not hand-wielding fractality (Hypothesis 1a; Figure 2). If head-sway fractality were an important factor in incorporating visual feedback during training, a natural corollary hypothesis was that it would also significantly improve the regression model of trial-by-trial response in the block immediately following training after feed-
back has ceased (Hypothesis 1b). Our second substantial hypothesis bore upon our proposed relationship among visual feedback, head-sway fractality, and hand-wielding fractality. Here, we expected that visual feedback would increase head-sway fractality (Hypothesis 2a) and subsequently, that increases in head-sway fractality would have significantly longer-lasting positive effects on hand-wielding fractality (Hypothesis 2b; Figure 3).1 As the results will show, there is no significant effect of adding visual feedback to the regression model of trial-by-trial judgments before either fractal predictor enters into the model. Entering head-sway fractality into the model will make the subsequent inclusion of visual feedback a significant improvement, but entering hand-wielding fractality will not. A regression model comparable to that for testing Hypotheses 1a and 1b will indicate positive effects of visual feedback on head-sway fractality. A vector autogressive model will allow forecasting the unique effects of head-sway fractality on later hand-wielding fractality, and these forecasts will reveal that positive effects of head-sway fractality on later hand-wielding fractality last significantly longer over subsequent trials for the group receiving visual feedback than for the group not receiving feedback.
Method Participants Fourteen right-handed male students participated in the study for partial credit in an introductory psychology course. Participants ranged in age from 18 to 22 years and provided informed consent according to University of Connecticut’s Institutional Review Board.
Apparatus Participants sat in a chair with an opaque screen flush with the participant’s right shoulder, blocking view of the right arm, and
Figure 2. Schematic of experimenter feedback. Following each judgment during the two training blocks, participants randomly assigned to the visual-feedback condition received visual feedback regarding the veridical length of the wielded object. The speech bubble provides the verbal input from the experimenter (not included in the picture but intended to be just offscreen to the right). The arrow pointing to the rope indicates where an experimenter might point as a means of gesturing to the participant where, in terms of the response system, the response marker might have been for a veridical response. The dashed, contorted line above the participant’s head is intended here to schematize head sway whose detrended fluctuations are used in the present work to test the novel hypothesis that fractality of head sway moderates use of visual information.
1 Trial-by-trial series lengths were participant-controlled. The time series ended when the participant completed registering her judgment. A GCM tested the effects of condition, currently receiving feedback, having once received feedback, as well as block and trials within block on time-series length. All predictors’ effects were significant and confirmed a number of intuitive expectations about how time-series length changed. As both tedium and expertise grow over the course of the participant’s experience, length of time series leading up to each judgment completion decreased over the entire task, by 55.30 points (SE ⫽ 15.75) with each block; above and beyond these block effects, a negative effect of trials and a positive interaction between trial and block (SEs ⫽ 1.64 and .60) together indicated that length decreased by 4.20 points per trial in the first block and by 1.78 points per trial in the second block, but then increased by .22 points per trial in the third block and by 2.22 points per trial in the fourth block. Participants in the condition receiving feedback yielded, on average, series 69.79 points shorter than participants in the condition without feedback. We hesitate to interpret this (69.79-points/94-Hz ⫽) .74-s margin too deeply, but it is possible that the promise of feedback catches the participants’ competitive spirit and makes them more eager. Feedback in the middle two blocks increased time-series length by 106.98 points, indicating that participants took roughly 1.14 seconds longer than average to complete judgments that would receive feedback. Once feedback had ended, the experience of having been trained left participants more cautious and thoughtful in the last block, yielding time-series lengths 125.16 points longer than average, indicating that participants took roughly 1.33 seconds longer than average to complete their judgment.
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participant to judge “length” as the object’s extent beyond the handle. Objects were presented in randomized order. On each trial, the experimenter placed an object in the participant’s right hand. Wielding and motion capture began on each trial when the participants took hold of the object’s handle. Participants were free to wield as long as needed to make their length judgment. Participants moved the response marker by manipulating the pulley system with their left hand, and motion capture ended on each trial when they were satisfied with the marker position. The marker was reset to zero cm before each new trial. The experimenter provided visual feedback to participants in the experimental condition on each trial during Blocks 2 and 3. Once the participant had registered a judgment, the experimenter provided visual feedback by adjusting the marker to the object’s actual length. The entire experiment lasted 70 –90 minutes, including breaks whenever the participant requested and a mandatory 5-min break taken after the third block.
Data Analysis
Figure 3. Schematic of hypothesized effects of head-sway fractality on hand-wielding fluctuations during visual feedback in dynamic touch. As in Figures 2 and 3, the dashed, contorted lines above the unseen object and above the participant’s head are intended here to schematize hand-wielding and head sway, respectively. Hypothesis 2b is the expectation that visual feedback will promote the effect of head-sway fractality on hand-wielding fractality during later trials.
with the table’s right edge (see Figure 1). Participants registered judgments by using a string-and-pulley system to position a marker along the length of the table. The object set consisted of 24 wooden parallelepipeds (see Table 1). The 1.8-cm diameter wooden dowels extended 12.5 cm beyond each object’s base, providing a handle with roughly 5 cm between the participant’s hand and parallelepiped’s base. Wireless magnetic markers recorded 94-Hz position data of manual wielding and head sway (Polhemus Liberty Latus, Polhemus Corporation, Colchester, VT). Velcro secured each of two markers to a neoprene glove and a headband on wielding hand and head, respectively.
Design In a 4-block, pretest-training1-training2-posttest design with 24 trials in each block, participants manually wielded an unseen parallelepiped on each trial to judge its length. Participants were randomly assigned into either a feedback condition or a nofeedback condition. In the feedback condition, participants received visual feedback after each judgment in Blocks 2 and 3 only. Participants in the no-feedback condition never received feedback.
Analysis included three analytical methods: detrended fluctuation analysis (DFA), growth curve modeling (GCM), and vector autoregression (VAR). DFA assesses fractal temporal correlation in terms of scaling exponent H. GCM is a longitudinal regression technique used for testing change over time, ideal for perceptuallearning tasks (e.g., Blau, Stephen, Carello, & Turvey, 2009). VAR estimates unique effects for mutually influencing variables. DFA estimated scaling exponents H for head sway (HHead) and for hand wielding (HHand) during the exploratory period before participants completed their judgments on the response pulley. Both scaling exponents HHead and HHand appeared as predictors in GCM testing their effects on subsequent length judgments for Hypotheses 1a and 1b. GCM tested the effect of visual feedback on scaling exponents HHead for Hypothesis 2a, and for Hypothesis 2b, VAR tested for unique effects of HHead on HHand in the condition receiving feedback as compared with the condition not receiving feedback. Detrended fluctuation analysis. Given time series x(t) of length N, DFA begins by integrating, computing successive cumulative sums to produce a new series y(t): y(t) ⫽
N
, x(i) ⫺ x(t) 兺 i⫽1
is the time-series mean. Next, DFA quantifies the where x共t兲 average root mean square (RMS) of residuals for linear regressions yn(t) for nonoverlapping n-length windows of y(t), producing a fluctuation function F(n): F(n) ⫽
冑
N
(1 ⁄ N)
[y(t) ⫺ yn(t)]2 兺 i⫽1
Experimenters showed the participant an example object (not part of the set in Table 1) in plain view and informed her that all objects were attached to a uniform handle. They instructed the
(2)
for n ⬍ N/4, where F(n) forms a power law F(n) ~ nH ,
Procedure
(1)
(3)
where H is the scaling exponent (e.g., Kantelhardt et al., 2002). For uncorrelated or anticorrelated series, H ⱕ .5, but for fractally temporal-correlated series, .5 ⬍ H ⬍ 1.5, with stronger fractal temporal correlations yielding a scaling exponent closer to H ⫽ 1
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Table 1 Details of Objects in Experiments 1 and 2 Stimulus
Length (cm)
Width (cm)
Mass (g)
I1 (gⴱcm2)
I3 (gⴱcm2)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
3.7 7.5 7.6 7.7 7.9 9.6 10.1 10.2 10.7 11.4 11.7 12.3 12.7 12.9 15.2 16.5 17.8 18.1 19 21.7 23.2 24.9 27.9 28.5
22.7 14 18.2 17.3 22.8 11.1 10.1 21 8.7 13.9 12.6 3.8 13.8 2.6 3.8 16.5 2.6 7.6 11.5 8.7 3.8 2.6 12.7 8.7
1184 747 1267 880 2400 496 410 1746 434 1162 710 94 959 95 118 1730 118 401 1728 889 174 156 1649 1186
241831 160783 291053 200634 606830 117768 99739 483439 109160 315594 193885 26450 280454 27618 38879 637924 44679 153944 705948 412488 86181 83586 1034139 754597
86715 22864 64114 40473 188158 10057 7054 119677 5846 36612 18660 512 30168 405 627 78517 504 4766 41212 13591 1002 729 49426 19168
(Eke et al., 2002; West, Geneston, & Grigolini, 2008). As H increases beyond 1, temporal correlations converge closer to the integration of white noise (i.e., integrated Brownian motion). Fractal analyses often estimate H by logarithmically scaling Eq. 3 as log F(n) ~ log nH
(4)
log F(n) ~ H log n.
(5)
and
The slope of the F(n) in double-log plots is an estimate for H (Peng et al., 1994). Alternate fractal analyses. Relative dispersion analysis and autoregressive fractional-integrated moving-average (ARFIMA) modeling served as additional checks confirming fractal temporal correlations. Dispersion analysis first estimates sample means of x(t) for nonoverlapping windows of length n and then evaluates standard deviation D over the same range for n as in DFA. The power law exponent relating D to n is estimated as log D(n) ~ ⫺c log n, and a fractal dimension (FD) is calculated as FD ⫽ 1 ⫹ c, with 1.5 ⬎ FD ⱖ 1 for fractal series (Bassingthwaighte & Raymond, 1995). ARFIMA modeling involves fitting series with all combinations of 0, 1 or 2 autoregressive (AR) components and 0, 1 or 2 moving-average (MA) components and subsequently testing whether fractional integration term d above and beyond the ARMA model significantly improves model fit. This improvement is estimated both in terms of AIC and BIC comparisons as well as the significant difference of d from zero (Torre, Delignières, & Lemoine, [2007]; but see also Wijnants [2014] for criticism of ARFIMA).
Despite enlisting alternate analyses to confirm fractal results, we used DFA scaling exponents H for all subsequent modeling because they are known to be the most reliable for empirical time series (Bashan, Bartsch, Kantelhardt, & Havlin, 2008). Growth curve modeling. GCM is a maximum-likelihood (ML) hierarchical multiple linear regression technique modeling a single time-varying dependent measure as a weighted sum of main effects and interactions. The effect of adding m predictors can be tested by treating the change in a deviance measure (i.e., ⫺2 log likelihood [⫺2LL]) as a chi-square statistic with m degrees of freedom (Singer & Willett, 2003). Vector autoregression. Vector autoregressive models allow modeling the interacting variables endogenous to a multivariate time series (Lütkepohl, 2007; Sims, 1980). In the simplest case, m ⫽ 2, with time series f(t) and g(t) each defined at each value of time t from t ⫽ 1 to t ⫽ N, where N is the length of the time series. VAR can produce a system of m regression equations predicting each variable as a function of lagged values of themselves and of each other. In the simplest case where m ⫽ 2, a VAR model for f(t) and g(t) would have the following structure: f(t) ⫽ A1 f t⫺1 ⫹ B2gt⫺1 ⫹ C f h ⫹ ε f , g(t) ⫽ B1gt⫺1 ⫹ A2 f t⫺1 ⫹ Cgh ⫹ εg ,
(6)
where Aj and Bj are the coefficients quantifying the effects of previous values of f and g, respectively, with j indexing the variable to which these previous values contribute. VAR also fits an error term to the regression for each individual variable, such as ε f and εg in Eq. 6 (Kennedy, 2008). VAR models can include exogenous variables, for example, experimental-design factors standing outside the mutual relationship among the variables within the system. For instance, the time series h(t) can induce
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changes in either f(t) or g(t), but changes in neither f(t) nor g(t) can induce changes in h(t). Endogenous variables are simply those variables internal to the system, that is, f(t) and g(t), that may respond to changes in other variables and that may induce changes in other variables, and h is the exogenous variable from our example and where Cf and Cg are the coefficients indicating the effect of h(t) on f(t) and g(t), respectively (Kennedy, 2008). VAR models provide forecasts of unique effects of endogenous variable into the future through impulse-response functions (IRFs). Whereas standard regression evaluates the relationship between f(t) and g(t), IRFs can evaluate relationships between f(t) and g共t ⫹ 兲 or between g(t) and f共t ⫹ 兲, where is a whole number. First, orthogonalizing the regression equations and, second, inducing an “impulse” to the system of regression equations by adding 1 standard error to any single variable will propagate responses across the other variables. The plot of an IRF presents the change in predicted later values of one time series because of the impulse from another time series (Elder & Serletis, 2009).
Results and Discussion Experiment 1 tested two major hypotheses, each with two subparts. Hypothesis 1 was the twofold prediction that the effect of visual feedback on haptic judgments of length is contingent upon head-sway fractality both during the training blocks (Hypothesis 1a) and in the posttest block after feedback is no longer available (Hypothesis 1b). Hypothesis 2 was the twofold prediction that visual feedback would increase head-sway fractality (Hypothesis 2a) and that experience with visual feedback promotes the effects of head-sway fractality on subsequent hand-wielding fractality (Hypothesis 2b). Results of Experiment 1 were consistent with these hypotheses. Below, we first outline preliminary descriptives (using DFA and GCM) confirming necessary premises to these hypotheses, and then we review the GCMs and VARs testing the substantive questions raised by these hypotheses.
Preliminary Descriptives and Findings Overall, length judgments exhibited moderately positive correlation with actual object lengths, r ⫽ .44, p ⬍ .0001 (e.g., Figure 4). Motion capture of the hand and head in three-dimensional space produced 1344 trajectories for hand and head (i.e., 14 (participants) ⫻ 4 (blocks) ⫻ 24 (trials in each block) ⫽ 1344) trajectories for hand and head. Sampling at 94 Hz produced time series with a mean length of 608 data points (SE ⫽ 3.66).1 Displacement time series were produced by taking the euclidean distance between each pair of consecutive points. DFA for each original 1344 displacement time series returned scaling exponents H estimates consistent with fractal fluctuations for original series at the hand and the head (Ms ⫽ .88 and .85, SEs ⫽ .01 and .01, respectively), exceeded scaling exponents H for shuffled displacement series for hand and for head (Ms ⫽ .47, SEs ⫽ .01 for both), paired-samples ts(1343) ⫽ 112.12 and 79.45, ps ⬍ .0001, respectively (e.g., Figures 5 and 6). Dispersion and ARFIMA modeling provided converging evidence of fractal structure, yielding FDs for hand and head-sway fluctuations (Ms ⫽ 1.22 and 1.31, SEs ⫽ .01 and .01, respectively), as well as significant ARFIMA ds for over 90% of series, and over
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Figure 4. Plots of perceived length against actual length for an example participant from the length-judgment condition during an example block.
90% proportion of summed Akaike weights, for both AIC and BIC criteria.
Predictors to Be Used for Building Growth Curve Models Table 2 lists all GCM predictors along with their definitions. Exogenous predictors reflecting experimental design factors included trial number (Trial), block number (Block), and the first and third inertial moments for object presented on each trial (I1 and I3). “FB” encoded availability of visual feedback, equaling 0 throughout except during Blocks 2 and 3 for participants in the feedback condition when FB equaled 1. The predictor PostFB encoded effects of having had feedback previously, equaling 0 throughout except during the posttest (Block 4) for participants in the feedback condition when PostFB equaled 1. Endogenous predictors represented various trial-by-trial features of displacement series for hand and head, including mean and standard deviation for the hand and head displacement series (MeanHand, StDevHand, MeanHead, StDevHead, respectively) and fractal scaling exponents H (HHand and HHead, respectively) so as to distinguish effects of fractal temporal correlations from effects of simply “moving more” or “moving more variably” (Stephen et al., 2010; Stephen & Hajnal, 2011). Collinearity among MeanHand, StDevHand, MeanHead and StDevHead led to convergence errors in regression modeling, especially when including scaling exponents HHand and HHead. We reduced these four variables to their first two orthogonal principal components (PC1 and PC2) for each participant. PC1 and PC2 captured 94.32% of the four statistics’ variability and served as predictors in place of the means and standard deviations.
Replication of Earlier Work: Hand-Wielding Fractality Moderates the Effect of Inertial Moments on Haptic Perceptual Judgments of Length We begin building a model of trial-by-trial length judgments with a series of growth curve models (GCMs) of length judgments that replicate the earlier findings that fractal temporal correlations at the wielding point moderated the effect of inertial moments on length judgment (Stephen et al., 2010; Stephen & Hajnal, 2011). All models appear in terms of the highest available interactions tested, and all lower-order interactions were included implicitly.
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Figure 5. Fluctuation functions from detrended fluctuation analysis (DFA) on a sample displacement time series for the hand during a single trial of wielding by a single participant. Points are overlaid with trend lines, and axes are logarithmically scaled. The upper fluctuation function (circles with solid trend line) describes the fluctuation function for the original displacement time series, and its slope indicates a scaling exponent HHand ⫽ .88. The lower fluctuation function (triangles with dashed trend line) describes the fluctuation function for a shuffled version of the original displacement time series, and its slope indicates a scaling exponent HHand ⫽ .53.
The base model for perceptual judgments is I1 ⫹ I3, respecting the observation that these two moments of inertia predict independent aspects of perceptual length (Fitzpatrick, Carello, & Turvey, 1994). Our model building involves adding new effects as well as their full-factorial interactions with all previous terms in the model. Table 3 outlines the details of this sequence. Model 5 replicated earlier findings that fluctuations in wielding predict how participants make use of inertial moments: including HHand, and its interactions with all terms in Model 4 leads to a significant improvement in prediction of perceptual judgments in Model 5, 2(48) ⫽ 84.24, p ⬍ .01. This point of replication is crucial for our present purposes, as we have based our proposals regarding intermodal integration on the premise that hand-wielding fractality moderates haptic use of mechanical information. The fact that we were able to replicate this finding warranted further inquiry as to head-sway fractality’s contribution to intermodal perception. As noted above, dynamic touch is classically cast as haptic perception, but all judgments are visually guided, and we now show that head-sway fractality may thus contribute significantly to length judgments. To accomplish this test, we removed the effect of HHand that we had entered in Model 5 and replaced it in Model 6 with HHead. Model 6 includes HHead and its interactions with all terms in Model 4 and provides a significant improvement of prediction from Model 4, 2(48) ⫽ 100.22, p ⬍ .0001. This test suggests that something about head-sway fluctuations matters for dynamic touch judgments, and we suspect that it might specifically support visual feedback in a way that hand-wielding fluctuations do not. We test and confirm this suspicion in the follow section.
Testing Hypothesis 1a: Effect of Visual Feedback During Training Blocks (FB) Depends on Head-Sway Fractality—and Not on Hand-Wielding Fractality Having adequately laid preliminary groundwork, we now test our first substantive hypothesis regarding explicitly intermodal perception. The finding above comparing Models 6 and 4 impli-
cated head-sway fractality in the dynamic-touch judgments, but this comparison did not yet entail that head-sway fractality is important for the use of visual feedback for dynamic-touch judgments. In this section, we will first encounter the potentially surprising fact that including visual feedback into the model does not necessarily improve predictions of trial-by-trial judgments. What this failure to improve prediction essentially means is that changes in perceptual judgments are not attributable to visual feedback. This observation may seem either absurd in light of evidence that feedback can improve judgments (Wagman et al., 2008), or this observation may fall neatly into the general understanding that perceptual learning is prone to individual differences (Withagen & van Wermeskerken, 2009). Taking the latter view, we tested whether including either of the trial-by-trial fractal measures in the GCM might bring this more textured effect of feedback into relief. That is, entering feedback into the model might become significant if the regression model has the benefit of “knowing” about the participant’s fractality. It will emerge that visual feedback will produce a significant improvement in the model following inclusion of only head-sway fractality— but not following the inclusion of only hand-wielding fractality. Hence, as Figure 2 had schematized, head-sway fractality appears to moderate the use of visual feedback in dynamic touch. GCMs building on the nested models from Models 1 through 5 fail to replicate previous findings that including FB and its interactions with the predictors from Model 4 significantly improves prediction of length judgments (Stephen et al., 2010; Stephen & Hajnal, 2011). In the present case, adding FB and its interactions with all previous terms (described in Table 3) does not significantly improve prediction of length judgments for any of the Models 1 through 5, all chi-square statistics being nonsignificantly small, p ⬎ .05. The failure of FB to improve the prediction of Model 5 suggests that hand-wielding fractality alone may not help predict changes in the use of visual feedback during training. Instead, the effect of visual feedback depends specifically on
Figure 6. Fluctuation functions from detrended fluctuation analysis (DFA) on a sample displacement time series for the head during a single trial of wielding by a single participant. Points are overlaid with trend lines, and axes are logarithmically scaled. The upper fluctuation function (circles with solid trend line) describes the fluctuation function for the original displacement time series, and its slope indicates a scaling exponent HHand ⫽ .77. The lower fluctuation function (triangles with dashed trend line) describes the fluctuation function for a shuffled version of the original displacement time series, and its slope indicates a scaling exponent HHand ⫽ .46.
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Table 2 Definition of Predictors for Growth Curve Models in Experiment 1 Definition Exogenous predictors Trial Block I1 I3 FB PostFB
Current trial number Current block number Current object’s first moment of inertia Current object’s third moment of inertia Concurrent feedback, equals 0 except during Blocks 2 and 3 in feedback condition Previously having had feedback, equals 0 except during Block 4 in feedback condition
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Endogenous predictors First principal component of means and standard deviations of hand and head displacement series Second principal component of means and standard deviations of hand and head displacement series Fractal scaling exponent indicating temporal correlations in hand displacement series Fractal scaling exponent indicating temporal correlations in head displacement series
PC1 PC2 HHand HHead
head-sway fractality and not on hand-wielding fractality. Table 4 describes models including HHead. As Model 7 shows, Adding FB and its interactions with all terms in Model 6 significant improves prediction of length judgments, 2(96) ⫽ 123.60, p ⬍ .05. Hence, significant effects of FB on length judgment during Blocks 2 and 3 appear to depend on the fractality of head sway displacements. However, the effect of head sway in moderating the use of visual feedback on dynamic-touch judgments does not preclude our ability to confirm, once again, that hand-wielding fractality contributes to the use of mechanical information (as noted earlier in comparing Models 4 and 5). Adding HHand and its interactions with all terms in Model 7 improves model fit 2(192) ⫽ 289.00, p ⬍ .0001, confirming both that fractality of hand-wielding displacements moderates effects of inertial moments on length judgments and that it also moderates the effect of visual feedback during training blocks on length judgments (as in Stephen et al., 2010; Stephen & Hajnal, 2011). Hand-wielding fractality and head-sway fractality both cooperate together in the same regression model to support the coordination of haptic and visual information in dynamic touch.
Testing Hypothesis 1b: Head-Sway Fractality Also Contributes to the Use of Visual Feedback After It Is No Longer Available This section deals with the fourth block of the dynamic touch perceptual-learning paradigm. This block tests how well-trained participants can use the feedback that they had previously received to now produce judgments in the absence of feedback. Presum-
ably, the visual feedback addressing each judgment in the second and third block finds expression in the fourth block through the visual guidance of dynamic-touch responses. A crucial indicator of head-sway fractality’s involvement in intermodal perception will be whether it contributes a significant effect on length judgments specifically in the fourth block. As above, we leave the individual steps of model building for Table 5. Adding PostFB and its interactions (Model 9) and later adding HHead and its interactions (Model 11) both improved prediction of length judgment, 2(6) ⫽ 25.69, p ⬍ .001 and 2(24) ⫽ 44.24, p ⬍ .01, respectively. Significant improvement because of PostFB terms indicates that having once had visual feedback significantly impacted length judgments in the posttest block (Block 4; e.g., Wagman et al., 2008). Significant improvement because of HHead and its interactions with PostFB terms indicates that fractal temporal correlations in head-sway displacements moderate effects of having once had visual feedback on length judgments, even when visual feedback is not available. Finally, adding HHand and its interactions with all previously added terms (Model 12), significantly improving prediction of length judgment, 2(48) ⫽ 92.46, p ⬍ .001. This last effect indicates that, above and beyond the effect of fractal temporal correlations in head-sway displacements, fractal temporal correlations in hand-wielding also moderate the effect of having once had feedback on length judgments. Overall, predictions from Model 12 correlate with actual length judgments, r(1342) ⫽ .86, p ⬍ .0001 (see Figures 7 and
Table 3 Preliminary Model Building in Terms of Highest-Order Interactions and Chi-Square (2) Tests for Improvements in Model Fit Model 1 2 3 4 5
Terms I1 I1 I1 I1 I1
⫹ ⫻ ⫻ ⫻ ⫻
I3 Trial Trial Trial Trial
⫻ ⫻ ⫻ ⫻
Block Block Block Block
⫹ ⫻ ⫻ ⫻
I3 ⫻ Trial ⫻ Block PC1 ⫹ I3 ⫻ Trial ⫻ Block ⫻ PC1 PC1 ⫻ PC2 ⫹ I3 ⫻ Trial ⫻ Block ⫻ PC1 ⫻ PC2 PC1 ⫻ PC2 ⫻ HHand ⫹ I3 ⫻ Trial ⫻ Block ⫻ PC1 ⫻ PC2 ⫻ HHand
2
df
p
669.66 13.91 21.29 84.24
8 12 24 48
⬍.0001 .31 .62 ⬍.01
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Table 4 Model Building Leading Up to Test of Hypothesis 1a in Terms of Highest-Order Interactions and Chi-Square (2) Tests for Improvements in Model Fit Model 6 7
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8
Terms I1 ⫻ Trial ⫻ Block I1 ⫻ Trial ⫻ Block ⫻ HHead ⫻ FB I1 ⫻ Trial ⫻ Block ⫻ PC1 ⫻ PC2 ⫻
⫻ PC1 ⫻ PC2 ⫻ HHead ⫹ I3 ⫻ Trial ⫻ Block ⫻ PC1 ⫻ PC2 ⫻ HHead ⫻ PC1 ⫻ PC2 ⫻ HHead ⫻ FB ⫹ I3 ⫻ Trial ⫻ Block ⫻ PC1 ⫻ PC2 ⫻ PC1 ⫻ PC2 ⫻ HHead ⫻ FB ⫻ HHand ⫹ I3 ⫻ Trial ⫻ Block HHead ⫻ FB ⫻ HHand
8 and Table 6 for individual participant comparisons to the model predictions).2
Testing Hypothesis 2a: Visual Feedback Increases Head-Sway Fractality This section presents modeling of scaling exponents themselves as dependent measures, rather than of length judgments, to determine whether visual feedback leads to changes in hand-wielding fractality or in head fractality. For this purpose, we use the same terms as in Model 2 (FBⴱI1ⴱTrialⴱBlock and FBⴱI3ⴱTrialⴱBlock) to predict changes in the scaling exponents HHead and HHand. We model both scaling exponents in the same model: the dependent measure is most accurately described as fractality in general. A new dummy-code predictor Hand distinguished effects on handwielding fractality (Hand ⫽ 1) from those on head fractality (Hand ⫽ 0). In keeping with earlier modeling practices, Model 13 Tests HandⴱFBⴱI1ⴱTrialⴱBlock and HandⴱFBⴱI3ⴱTrialⴱBlock (see Table 7). The significant main effect of FB indicates that visual feedback increases fractality for both head and hand. However, the foregoing GCMs are univariate models blind to any potential effects that HHead might exert on HHand. For instance, one possible reason for these results is that head-sway fractality may induce changes in hand-wielding fractality, and, as predicted by Hypothesis 2b, experience with visual feedback might accentuate such effects of head sway on hand-wielding.
Testing Hypothesis 2b: Experience With Visual Feedback Significantly Extends the Effect of HeadSway Fractality on Later Hand-Wielding Fractality The goal in this section is to determine whether experience with visual feedback leads head-sway fractality to propagate through the perceptual system and so to promote later hand-wielding fractality. VAR modeling of scaling exponents will test for relationships between changes in HHead and changes in HHand across multiple trials in each condition (see Figure 3). The general structure of the proposed VAR model (schematized in Figure 9) is written as follows: HHead(t) ⫽ A1HHead(t ⫺ 1) ⫹ B2HHand(t ⫺ 1) HHand(t) ⫽ B1HHand(t ⫺ 1) ⫹ A2HHead(t ⫺ 1)
(7)
Equation 7 shows that VAR modeling allowed us to treat HHead and HHand as endogeneous variables composing a single system, both having effects on each other. This framework allowed us to further extrapolate effects of current HHead and on subsequent HHand into the future in the form of IRFs based on each partici-
2
df
p
100.22 (from Model 4) 123.60
48 96
⬍.0001 ⬍.05
289.00
192
⬍.0001
pant’s 96 experiment-wide series of trial-by-trial HHead and HHand values. For each participant we used a VAR model with 13 predictors: two predictors HHead and HHand were treated as endogenous (e.g., Eq. 7), and 11 exogenous predictors (i.e., interactions from Model 2: I1ⴱTrialⴱBlock, I3ⴱTrialⴱBlock and all constituent lower-order terms) accounted for effects of experimental design on HHead and HHand. Adjusted R-squared indicated that VAR modeling accounted for, on average, 97.57% and 97.93% (SEs ⫽ .01) of the variability in HHead and HHand, respectively. Figure 10 shows the average IRF across participants in the visual-feedback condition (left panel) and the no-feedback condition (right panel) illustrating the effects of current HHead on later HHand. As can be seen, the response of hand-wielding fractality to an impulse from head-sway fractality decays rapidly in the no-feedback condition, but the response of hand-wielding fractality to an impulse from head-sway fractality is much more prolonged. A new GCM Model 14 revealed that the difference in IRFs by condition was significant. Model 14 enlists three predictors and their interactions to fit the curvilinear decay of the IRFs: Trial, Trial2 and Condition. Trial is the number of trials following the current trial (1, 2, 3 . . . , 10), Trial2 is the quadratic term of Trial (1, 4, 9 . . . , 100), and Condition is a dichotomous variable indicating participant membership in the visual-feedback or nofeedback condition (Condition ⫽ 1 or 0, respectively). The significant interactions of Condition with Trial and with Trial2 (Bs ⫽ .0057 and ⫺.0004, ps ⬍ .001 and .01, respectively) indicate a more gradual decrease of effects from head-sway fractality on hand-wielding fractality, more positively linear and following a shallower quadratic pattern (see Table 8). This pattern of significant results held for alternate modeling using orthogonal polynomial terms in place of Trial and Trial.2
2 Alternative modeling has entered time-series length as a predictor in comparable position as the scaling exponent terms in the regression (i.e., replacing H terms with the time-series length instead). Subsequently adding each H term to the regression model after thus controlling for trial-bytrial changes in time series length still leads to significant improvement in model fit. This alternative modeling thus confirms that effects of fractality are not reducible to effects of time series length. Furthermore, alternative modeling also tested the effects of downsampling the measured fluctuations series by only including every second, every third and every fourth on the estimated H, and in all cases the inclusion of H terms led to comparably significant improvements in prediction. The final model predictions in all downsampled-H cases correlated strongly with the model reported in the main text, rs above .95.
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Table 5 Model Building Leading Up to Test of Hypothesis 1b in Terms of Highest-Order Interactions and Chi-Square (2) Tests for Improvements in Model Fit Model
Terms
2
df
p
9
I1 ⫻ Trial ⫻ Block ⫻ PC1 ⫻ PC2 ⫻ HHead ⫻ FB ⫻ HHand ⫹ I3 ⫻ Trial ⫻ Block ⫻ PC1 ⫻ PC2 ⫻ HHead ⫻ FB ⫻ HHand ⫹ I1 ⫻ Trial ⫻ PostFB ⫹ I3 ⫻ Trial ⫻ PostFB I1 ⫻ Trial ⫻ Block ⫻ PC1 ⫻ PC2 ⫻ HHead ⫻ FB ⫻ HHand ⫹ I3 ⫻ Trial ⫻ Block ⫻ PC1 ⫻ PC2 ⫻ HHead ⫻ FB ⫻ HHand ⫹ I1 ⫻ Trial ⫻ PostFB ⫻ PC1 ⫻ PC2 ⫹ I3 ⫻ Trial ⫻ PostFB ⫻ PC1 ⫻ PC2 I1 ⫻ Trial ⫻ Block ⫻ PC1 ⫻ PC2 ⫻ HHead ⫻ FB ⫻ HHand ⫹ I3 ⫻ Trial ⫻ Block ⫻ PC1 ⫻ PC2 ⫻ HHead ⫻ FB ⫻ HHand ⫹ I1 ⫻ Trial ⫻ PostFB ⫻ PC1 ⫻ PC2 ⫻ HHead ⫹ I3 ⫻ Trial ⫻ PostFB ⫻ PC1 ⫻ PC2 ⫻ HHead I1 ⫻ Trial ⫻ Block ⫻ PC1 ⫻ PC2 ⫻ HHead ⫻ FB ⫻ HHand ⫹ I3 ⫻ Trial ⫻ Block ⫻ PC1 ⫻ PC2 ⫻ HHead ⫻ FB ⫻ HHand ⫹ I1 ⫻ Trial ⫻ PostFB ⫻ PC1 ⫻ PC2 ⫻ HHead ⫻ HHand ⫹ I3 ⫻ Trial ⫻ PostFB ⫻ PC1 ⫻ PC2 ⫻ HHead ⫻ HHand
25.69
6
⬍.001
71.03
18
⬍.0001
44.24
24
⬍.01
92.46
48
⬍.001
10
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11
12
Increased Fractality of Both Head Sway and HandWielding Together Predict Best Receptivity to Feedback in Perceptual Learning A straightforward question to ask at this point is whether stronger temporal correlations in the fractal range (i.e., scaling exponent H closer to H ⫽ 1) are any good for perceptual learning. Previous work had used a final regression model to project the likely average discrepancies between length or width judgment and actual object length or width for wielding characterized by ideally fractal fluctuations (H ⫽ 1) and for wielding characterized by ideally nonfractal fluctuations (H ⫽ .5), and this projection revealed that wielding characterized by ideally fractal fluctuations would have less discrepancy be-
tween judgment and actual dimension. Here, we attempt to replicate this question, only now elaborating it to include all four possible pairings of ideally fractal or nonfractal hand fluctuations (HHand ⫽ 1 or .5) and ideally fractal or nonfractal head-sway fluctuations (HHead ⫽ 1 or .5). Figure 11 shows the outcome of these projections from Model 12. What these projections suggest is that ideally fractal head-sway and handwielding fluctuations would predict the best use of the available information to produce the nearest-to-accurate length judgments. We do not want this potentially intriguing figure to be misleading. That time-varying scaling exponents HHand and HHead significantly improved models of length judgments is indication that the fractality of head-sway and hand-wielding fluctuations varied ap-
Figure 7. Comparison of judgments by trial with predictions from Model 12 for four individual participants in the visual-feedback condition, the two with the strongest correlation between model predictions and judgments and the two with the weakest correlations between model predictions and judgments. The black curve represents the judgments, and the gray curve represents the model predictions.
KELTY-STEPHEN AND DIXON
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Figure 8. Comparison of judgments by trial with predictions from Model 12 for four individual participants in the no-feedback condition, the two with the strongest correlation between model predictions and judgments and the two with the weakest correlations between model predictions and judgments. The black curve represents the judgments, and the gray curve represents the model predictions.
preciably. In other words, Figure 11 does not describe any participant in particular in our sample, nor do we expect that we will ever directly observe a participant whose perceptual discrepancy mimics this trajectory. Figure 11 is essentially a cartoon derived from coefficients estimated from a regression model based on many participants each bringing multiple scaling exponents to bear on this perceptual task, and imagined for stereotyped kinds of fractality or nonfractality held constant over blocks. We offer these stereotyped instances of fractality (or nonfractality) only to illustrate that, based on the observation of several noisy participantgenerated fluctuations, we may produce a participant-independent notion of what head and hand fractality might be worth for the purpose of training perceptual faculties in dynamic touch. In fact,
Table 6 Pearson Correlation Coefficients R Comparing Actual Trial-byTrial Length Judgments With Trial-by-Trial Model Predictions in Terms of Model 12 Grouped by Condition Participant 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Condition FB FB FB FB FB FB FB No No No No No No No
FB FB FB FB FB FB FB
R .94 .91 .79 .81 .91 .87 .93 .80 .80 .86 .68 .76 .82 .82
our interest here was not in making our 14 participants do this task more accurately but in learning how the fractal fluctuations at the hand and the head individually contribute and coordinate to support what intermodal perception they shared with us. Actual discrepancy exhibited by individual participants would not be limited to the linear trajectories in Figure 11. Generally, if effects of head-sway fractality that support the use of visual feedback do in fact spill over to produce greater hand-wielding fractality, then we might find most of our actual participants’ data bobbling somewhere within the region bounded by the black-lined trajectory (“head fractal, hand fractal”) and the darkest-gray lined trajectory (“head fractal, hand nonfractal”). It is important to recall that a participant’s degrees of fractality and the hand and at the head will change over trials and so over blocks. Consequently, as Figure 12 illustrates, there are many ways we might envision a real trajectory of actual discrepancy. As shown in the left panel, there are a few likely trajectories that could lead participants to reduced discrepancy in Block 4 (Figure 12; left panel). They might begin
Table 7 Coefficients for Significant Effects in Model 13: Prediction Changes in Head and Hand Fractality Because of Feedback
Hand Trial Block FB Hand ⫻ Trial Hand ⫻ Block Trial ⫻ Block FB ⫻ Trial FB ⫻ Block ⫻ Trial
B
SE
p
.1597 .0094 .0552 .4263 ⫺.0058 ⫺.0425 ⫺.0028 ⫺.0239 .0071
.0532 .0019 .0135 .1562 .0027 .0191 .0007 .0080 .0031
⬍.01 ⬍.0001 ⬍.001 ⬍.01 ⬍.05 ⬍.05 ⬍.001 ⬍.01 ⬍.05
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Table 8 Coefficients for Significant Effects in Model 14: Testing for Condition Effects on the Decay of Head-Sway Fractality’s Effects on Hand-Wielding Fractality
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Intercept Trial F Trial F2 Condition Condition ⫻ Trial F Condition ⫻ Trial F2
Figure 9. Schematic illustrating the conceptual structure of the vector autoregressive (VAR) model proposed to model mutual contributions between head-sway fractality and hand-wielding fractality. Each kind of fractality is here represented as a time series of trial-by-trial scaling exponents. Arrows represent regression weights modeling the effects of previous scaling exponents on current scaling exponents.
the task with relatively strong head fractality and relatively weak hand fractality, and as suggested by the VAR model above, this strong head fractality might bolster subsequent hand fractality, thereby steering the participant from one of the dark gray linear trajectory to the black linear trajectory in Figure 11. Participants
B
SE
p
.0353 ⫺.0081 .0005 ⫺.0162 .0057 ⫺.0004
.0045 .0012 .0001 .0064 .0017 .0001
⬍.0001 ⬍.0001 ⬍.0001 ⬍.05 ⬍.001 ⬍.01
might begin with relatively weak fractality at both hand and head, but the increase in both types of fractality found in the mixed linear modeling and the head-to-hand relationship in the VAR might both steer the participant from the lightest gray linear trajectory to the black linear trajectory in Figure 11. If participants began with strong fractality at both hand and head and then if they retained this fractality-based profile, they might only experience a modest increase in discrepancy. There are many ways in which Figure 11 might seem like bad news for any attempt to relate fractal fluctuations to “improved” performance in tailoring judgments to the experimental feedback. For any hypothetical participants with invariantly strong signatures of fractality across all four blocks, fractality might seem to be good for nothing more than slowing the growth of discrepancy. However, all modeling so far has demonstrated effects of fractality on
Figure 10. Average impulse response functions (IRFs) for the no-feedback condition (left panel) and the visual-feedback condition (right panel). The y-axis denotes the response of an endogenous variable trials after a given current trial to an increase in (i.e., impulse from) another endogenous variable on a current trial. The x-axis denotes , the number of trials after a given current trial. VAR analysis for Experiment 2 solid lines indicates average response of hand-wielding fractality to (and trials after) an increase (i.e., “impulse”) from hand fractality. This average is across all participants’ individual IRF curves. Dashed lines 1 standard error above and below the condition average. Although the immediate positive effect of an increase in head-sway fractality appears to be greater in the no-feedback condition than in the visual-feedback condition (e.g., the values of the hand-wielding fractality response are greater over the next one to two trials), the effect of an increase in head-sway fractality drops off rather quickly across the next few trials in the no-feedback condition. However, the shallower decrease in the right panel suggests that the effect of head-sway fractality on hand-wielding fractality lasts longer in the visual-feedback condition.
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Figure 11. Projections over average discrepancy between length judgments and actual length by block for all four possible pairings of ideally fractal or nonfractal hand fluctuations (HHand ⫽ 1 or .5) and ideally fractal or nonfractal head-sway fluctuations (HHead ⫽ 1 or .5). A dashed, horizontal gray line indicates zero discrepancy (i.e., an exactly accurate length judgment). Each solid line segment represents a trajectory of the discrepancy across block for each of these four pairings, distinguished by color, from black to progressively lighter shades of gray. The black line reflects the by-block discrepancies for the pairing of ideally fractal head-sway fluctuations and ideally fractal hand-wielding fluctuations. The darkest gray line reflects the by-block discrepancies for the pairing of ideally fractal head-sway fluctuations and ideally nonfractal hand-wielding fluctuations. The lighter gray line reflects the by-block discrepancies for the pairing of ideally nonfractal head-sway fluctuations and ideally fractal hand-wielding fluctuations. The lightest gray line reflects the by-block discrepancies for the pairing of ideally nonfractal head-sway fluctuations and ideally nonfractal hand-wielding fluctuations.
judgments across time and further effect of one fractality on the other. Both observations suggest that participants who retained more or less the same fractal fluctuations across time would be something of a statistical oddity. Both regression methods require comparable patterns of variation among variables to return significant effects, suggesting that fractality is not a constant. Figure 12 illustrates many more possible ways for discrepancy to stray to greater absolute values (right panel). In short, any configuration of
hand or head fractality during Block 1 might lead to worse performance in the dynamic touch task provided that the relationships between hand and head fractality documented in modeling above do not steer participants toward relatively strong fractality for hand and head. Table 9 shows that simply evaluating average discrepancy will not reveal any effects of feedback, above and beyond any endogenous covariates: we might initially take comfort in the fact that the participants receiving visual feedback appear to have begun underestimating and to have ended up producing errors not distinguishable from zero in Block 4, but that observation does not seem as reassuring in demonstrating a lasting effect of visual feedback when we consider that the group never receiving visual feedback also exhibited average discrepancy not significantly different from zero in Block 4. Average discrepancy distinguished simply by random assignment to experimental group shows no effect of feedback, much in the same way that modeling of length judgment did not show an effect of including the variable FB without previous entry of HHead. The averages fail to reveal what a deeper accounting of the individual differences of these participants in terms of their endogenous fractal fluctuations as they explore the task environment. No matter the power of the experimenter to shape the environment of the unwitting perceiving-acting system with the broad strokes of an experimental manipulation, an important part of the “information” for the perceptual judgment and, more generally, for the intermodal coordination of perceptual information is bound up in each participant’s profile of fractal fluctuations. An important difference in understanding these results may be one’s preferred question. Some readers may prefer only to know whether visual feedback was effective for reducing discrepancy, and these readers may come away disappointed. Although it’s true that, as Table 9 shows, discrepancy began as statistically different from zero in both groups and vanished to values not significantly different from zero, these averages provide no indication that feedback is important for this reduced discrepancy. However, we suspect that these results are more encouraging to those interested in understanding more deeply how visual feedback works for some participants and not for others. At this point, we offer to any disappointed readers the possibility that knowing more about how
Figure 12. Hypothetical trajectories in discrepancy for individual participants overlaid on the trajectories shown in Figure 11. Left panel shows trajectories that would indicate intermodal perception leading to reduced discrepancy in Block 4, and right panel shows trajectories that would indicate intermodal perception leading to reduced discrepancy in Block 4.
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Table 9 Block-by-Block Average Discrepancy (Cm) Between Judged Length and Actual Length for Each Experimental Group (With Standard Errors in Parentheses) Block Group
1
2
3
4
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Visual Feedback ⫺1.49 (.56) ⫺1.27 (.50) ⫺.27 (.44) ⫺.32 (.42) No Visual Feedback .28 (.57) ⫺1.49 (.58) ⫺.85 (.57) ⫺.72 (.53)
the underlying fractal-fluctuation-based relationships moderating the effect of visual feedback might help us scientifically develop better forms of feedback and training that, when tailored to the endogenous noise processes of our participants, might allow for stronger differences to appear at the coarse grain of block means.
Is Fractal “Better?” We realize that a tantalizing conclusion to draw from Figure 11 might be “more fractal does better,” but we actually would like to discourage that short-hand conclusion in favor of the hopefully less reckless proposal that increased fractality is better associated with greater openness to novel information. For one thing, depending on the task, accurate performance in a task can be plainly at odds with fractal fluctuations. For instance, in the domain of “temporal estimation,” participants freely tap a button at what they perceive to be regular temporal intervals, usually 1-s intervals. Although the intertap-interval series that participants generate in this task exhibit fractally temporal correlations, experimentally providing feedback to participants during the temporal-estimation task will diminish the temporal correlations (Kuznetsov & Wallot, 2011). After all, greater accuracy in the temporal-estimation task requires the participant to settle into a period rhythm, assuming 1-s-long finger-tap oscillations to the exclusion of all others, and fractal temporal correlations are notoriously celebrated for their exhibiting fluctuations across all available time scales (Eke et al., 2002; Van Orden et al., 2003). Feedback in the temporalestimation task seeks to prod the participant’s endogenous variability into an explicitly periodic constraint. Fractal can thus lead to “poor” performance, and we expect that fractality has more to do with openness to novel information than “accurate” performance. The perceptual-learning case in the paradigm of dynamic touch exemplifies how openness to novel information might lead, for some tasks, to greater “accuracy.” Dynamic-touch tasks build in a degree of uncertainty, and each trial requires a novel search. Even if the participant has arrived at some heuristic or specifying variable that she will use for producing a judgment, she must stay flexibly poised to respond to randomized objects. In temporal estimation, the task constrains manual fluctuations so as to reproduce 1-s-long intervals and to decorrelate discrepancy between tapped interval and 1-s intervals: feedback pushes participants to extend intervals following too-short intervals and shorten intervals following too-long intervals. The dynamic-touch task does not require any particular temporal structure for manual fluctuations; it instead demands the participant to explore each new object mechanically and to incorporate new information about length from visual feedback.
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Effects of head-sway fractality in predicting use of optical information would be trivial if another perceptual task requiring vision did not show similar effects. We admit that dynamic touch strikes many—participants and readers alike—as a peculiar task, and perhaps the contrived instructions invite their own artifact into the measured data. Effects of head-sway fractality on visual perception might somehow be idiosyncratic to dynamic touch. To test this interpretation in the foregoing paragraph, we include this second experiment for comparison to the first. The present experiment seeks to export effects of head-sway fractality from dynamic touch to a different perceptual task, namely, visual perception of length. Using the same objects as in Experiments 1, we asked participants to judge object length visually. As in Experiment 1, we predict that head-sway fractality will moderate use of optical variables for visual perception. Unlike in Experiment 1, the measured head sway will not come from an organism whose hand is concurrently jostling an unseen object. Our hope is that the temporal correlations of head sway might support visual perception even without the peculiar instructions for dynamic touch. Previous research offers various visual cues for the dimensions of a visible object. Perceived object size may depend on distance and orientation (Todd & Norman, 2003), and research into the vertical-horizontal illusion suggests that length judgments will be sensitive to length and width (Robinson, 1998). Whatever the “best” informational variable may be, the crucial question for present purposes is whether head-sway fractality moderates use of likely optical variables. The foregoing experiment leads us to pose Hypothesis 3, namely, the prediction that head-sway fractality will predict use of optical information in visual perception of length.
Method Participants Six students (2 females, 4 males) participated for partial credit in an introductory psychology course. Participants ranged in age from 18 to 26 years and provided informed consent according to University of Connecticut’s Institutional Review Board.
Apparatus The participant sat with feet planted on the floor and next to the same table and pulley system from Experiment 1 (see Figure 13). A corridor 1.8 m wide and 13.5 m long extended immediately in front of the participant, with a passage immediately to the left at the end of the corridor. The object set and motion capture were the same as in Experiment 1, with the exception that motion capture recorded position at 188 Hz.
Design Participants visually judged the set of parallelepipeds over 4 blocks without feedback: blocks were identical except for randomized object presentation.
Procedure Presentation of an example object and definition of length were the same as in Experiment 1.
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83.06, ps ⬍ .0001 and t(574) ⫽ 83.06, p ⬍ .0001, respectively (e.g., Figures 15 and 16).
Definition of Predictors for Growth Curve Models
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Growth curve modeling of judgments enlisted nine predictors. Trial, Block, MeanHead, StDevHead and HHead were defined as in Experiment 1. Length and Width denoted trial-by-trial object dimensions. Distance and Orientation denoted number of meters from the participant (i.e., 6, 9 or 12) and angle between handle and participant’s line of sight (i.e., increments of 45 degrees between 0 degrees and 315 degrees), respectively.
Testing Hypothesis 3: Head-Sway Fractality Supports the Visual Judgment of Length Figure 13. The experimental apparatus of Experiment 3. Top panel shows a side view; bottom panel shows a view from above. Objects were placed on the floor in full view in front of the participants. Point N marks the tips of the participant’s toes. Point M marks the nearest face or edge of the object. The experimenter placed objects on the floor to manipulate the distance between M and N. The participant manipulated the same pulley system to register length judgments as in Experiments 1 and 2.
On each trial, an experimenter stepped into participant’s view from the left at the distal end of a corridor and placed an object on the floor 6, 9 or 12 m from the tips of the participant’s toes, centered in the corridor, in any of eight orientations, with handle pointing 0, 45, 90, 135, 180, 225, 270 or 315 degrees from the participant. Levels of distance and orientation of placement were counterbalanced evenly across 24 trials within each block. As in Experiment 1, motion capture began and ended on each trial with presentation of the object and completion of a length judgment using the pulley system, with marker reset to zero cm before each new trial. The entire experiment lasted 40 minutes, including breaks at participants’ request.
As in the Results section of Experiment 1, a number of successive GCMs of length judgments were tested. Also in the Results for Experiment 1, successive models are reported in terms of their highest-order interactions but include all lower-order constituent terms. The base model, that is, Model 15 tested the interaction TrialⴱBlock. Model 15 had a ⫺2LL deviance of 3742.62. Table 10 summarizes model building. Model 22 tested the interaction Trialⴱ BlockⴱLengthⴱWidthⴱDistanceⴱOrientationⴱMeanHeadⴱStDevHeadⴱ HHead. Including HHead significantly improved prediction of length judgments, 2(256) ⫽ 1078.14, p ⬍ .0001.2 This improvement in model fit suggests that head-sway fractality significantly contributes to visual judgments of object length, above and beyond effects of object length, object width, distance of object, angular orientation of object, magnitude of head-sway fluctuations and variability of head-sway fluctuations. We reduced the model for subsequent examination of fractality’s effect on perceptual discrepancy between length judgment and actual object length. Adding Distance and its interactions with all terms in Model 17 and adding MeanHead and its interactions with all terms in Model 19 did not improve model fit (see Table 10). When we remove these nonsignificant terms, we find that adding Orientation and its interactions with terms in Model 18 and that adding StDevHead and its interactions with terms in Model 20 no
Data Analysis Analysis involved DFA and GCM, as in Experiment 1.
Results and Discussion Descriptives on Visual Judgments and Scaling Exponents Length judgments correlated with object lengths, r ⫽ .82, p ⬍ .0001 (e.g., Figure 14). Three-dimensional motion capture at 188 Hz of head and hand for 6 participants produced 576 (i.e., 6 ⫻ 4 ⫻ 24) head-sway and hand-wielding trajectories with a mean length of 888 data points (SE ⫽ .08). DFA estimated scaling exponents HHead and HHand for each displacement time series consistent with fractal fluctuations (Ms ⫽ .68 and 1.03, SEs ⫽ both .01, respectively), significantly greater than for shuffled versions of the series for the head (Ms ⫽ both .48, SEs ⫽ both .01, respectively), paired-samples ts(574) ⫽ 21.60 and
Figure 14. Plot of perceived length against actual length for an example participant during an example block.
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Table 10 Model-Building Leading Up to Test of Hypothesis 3 in Terms of Highest-Order Interactions and Chi-Square (2) Tests for Improvements in Model Fit Model
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16 17 18 19 20 21 22
Figure 15. Fluctuation functions from detrended fluctuation analysis (DFA) on a sample displacement time series for the hand during a single trial of visual judgment by a single participant. Points are overlaid with trend lines, and axes are logarithmically scaled. The upper fluctuation function (circles with solid trend line) describes the fluctuation function for the original displacement time series, and its slope indicates a scaling exponent HHand ⫽ 1.04. The lower fluctuation function (triangles with dashed trend line) describes the fluctuation function for a shuffled version of the original displacement time series, and its slope indicates a scaling exponent HHand ⫽ .54.
longer produce significant improvements in model fit. However, without the intermediary terms added from Model 18 to Model 21, including HHead and its interactions with terms in Model 17 still improves model fit. This more compact model—which we may call Model 23—in the next section describes effects of fractality on accuracy in visual-perceptual judgments.
Terms Model Model Model Model Model Model Model
15 16 17 18 19 20 21
⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻
Length Width Distance Orientation MeanHead StDevHead HHead
2
df
p
975.49 48.88 18.54 46.86 74.55 234.09 1078.14
4 8 16 32 64 128 256
⬍.0001 ⬍.0001 .29 ⬍.05 .17 ⬍.0001 ⬍.0001
Increased Fractality of Both Head Sway Leads to Improved Visual Judgments, Contingent on Object Width We used Model 23 to predict the discrepancies between perceptual judgment of length and actual object length. Hence, the positive and negative coefficients for significant individual terms in this model of perceptual discrepancy will indicate overestimation and underestimation, respectively, of actual length. Intercept (B ⫽ 31.18, SE ⫽ 12.42, p ⬍ .05), HHead (B ⫽ ⫺36.14, SE ⫽ 16.47, p ⬍ .05) and the interaction Widthⴱ HHead (B ⫽ 2.22, SE ⫽ 1.13, p ⬍ .01) yielded the only significant effects. The positive Intercept term indicates average overestimation of length by 31 cm. Ideally fractal head sway would lead to more accurate judgments, with discrepancies indicating underestimation by just under 5 cm (31.18 – 36.14ⴱ(HHead ⫽ 1) ⫽ ⫺4.99). For ideally nonfractal head sway, length judgments will overestimate actual length by over 13 cm (i.e., 31.18 – 36.14ⴱ(HHead ⫽ .5) ⫽ 13.11), that is, almost three times more overestimation than there was underestimation with fractal head sway. So, yet again, it appears that fractal fluctuations support greater accuracy again. However, the interaction WidthⴱHHead indicates fractality of head sway may also lead the object width to contaminate length judgments, with a less-than-1-cm overestimation for the narrowest object (⫺4.99 ⫹ 2.22ⴱ2.6ⴱ(HHead ⫽ 1) ⫽ .78) and overestimation greater than 31 cm for the widest quarter of the stimulus set (⫺4.99 ⫹ 2.22ⴱ16.5ⴱ(HHead ⫽ 1) ⫽ 31.73). For ideally nonfractal head sway, width-sensitive overestimation was much greater (i.e., 15.89 cm) for the narrowest object (13.11 ⫹ 2.22ⴱ2.6ⴱ(HHead ⫽ .5) ⫽ 15.89) but grew much more slowly with greater width, only being overtaken by ideally fractal overestimation for the widest quarter of the set (13.11 ⫹ 2.22ⴱ16.5ⴱ(HHead ⫽ .5) ⫽ 31.45).
General Discussion Figure 16. Fluctuation functions from detrended fluctuation analysis (DFA) on a sample displacement time series for the head during a single trial of visual judgment by a single participant. Points are overlaid with trend lines, and axes are logarithmically scaled. The upper fluctuation function (circles with solid trend line) describes the fluctuation function for the original displacement time series, and its slope indicates a scaling exponent HHand ⫽ .69. The lower fluctuation function (triangles with dashed trend line) describes the fluctuation function for a shuffled version of the original displacement time series, and its slope indicates a scaling exponent HHand ⫽ .50.
The proposal that fractality promotes exploration might have foundations in its statistical relationship to diffusion. Compared with uncorrelated fluctuations, temporally correlated fluctuations reflect a more effective mechanism for searching the available space of possible variables that may inform a judgment. Fractal fluctuations are often known as “space filling” precisely because they represent a regime of relatively rapid diffusion, especially as indicated by the higher exponent H on time describing the growth of RMS fluctuation (e.g., Shlesinger, Zaslavsky, & Klafter, 1993). The .5 value of H would reflect the “ordinary” case of diffusion
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that hovers timidly around a mean position, and overly persistent fluctuations such as might arise from flinging an unseen object across the lab would exhibit the nonstationary fluctuations associated with H ⬎ 1.5. Fractal fluctuations represent a sequence of displacements that, in the observed order, accumulate to a trajectory that diverges farther (literally, in terms of mean-squared deviation of trajectory positions) from a mean position than would a trajectory built of the same displacements in a shuffled order (i.e., having the same mean and mean-squared deviation displacements). Fractal fluctuations reflect a compromise between overly constrained exploration (i.e., uncorrelated fluctuations) and overly ballistic (i.e., persistent fluctuations from flinging the object across the lab)—the latter of which simply reflects excessive impulse constrained within a narrow range of directions. Fractal fluctuations permit perceptual-motor exploration to spread themselves most widely through the constraints of the task and reflect a loosening of constraints within their own perceptual repertoire (Stephen & Hajnal, 2011), allowing novel configurations of motor degrees of freedom and sensory impressions to generate a perceptual response. The notion of fractal fluctuations promoting openness to novel information in the task is not just a rhetorical connection to statistical physics but has already been borne out in previous work in more cognitive domains. In many tasks, participants must induce a rule from limited information and feedback. In previous research we have studied the induction of rules in problem-solving behaviors with respect to gear system problems (Stephen, Boncoddo, et al., 2009; Stephen & Dixon, 2009) and in the executivecontrol task of the dimension card sort (Anastas, Stephen, & Dixon, 2011). In both domains, those participants who would go on to successfully induce the rules began interacting with the task environment using motor behaviors that exhibited progressively stronger signatures of fractality. Those who did not go on to induce the rules did not show any such strengthening of fractality. Fractality is not explicitly for perception, as noted above, but where it occurs during perception, it serves to put the perceiving-acting participant in a heightened state of poise in which they become sufficiently open to the novel structure to be discovered (Dixon, Holden, Mirman, & Stephen, 2012). This earlier work in rule induction had raised the question of how or why fractality might increase, and the present work offers the beginning of an explanation. We now see that fractality is not limited to one perceptual-motor point of contact between the organism and its task environment. Rather, the perception-action system exhibits fractal fluctuations at different locations, across different perceptual-motor points of contact with the task environment. These fractal fluctuations may differ across location on the body, and the differences among the fractal fluctuations may mediate the sharing of perceptual information under the anatomical constraints of motor connectivities. Even if the crucial mechanism for integrating across perceptual modalities reside solely within the central nervous system, it may be of use for subsequent research to know that the perception-action system wears indicators of intermodal integration at the periphery on its anatomical sleeves. Here, we see glimmers of an embodied sort of cognition that does not rely on the gross motor trajectory to be transparently gestural but opens the possibility that one key to assessing the flow of
information across the body is the waxing and waning of fractality in perceptual-motor fluctuations. Each individual stimulus might thus occasion a multifarious cascade of effects. The present experiment suggests that intermodal integration of perception might profit from such spreading fluctuation. An analogy to “spreading activation” implicated in parallel distributed processes (PDP; Rumelhart, McClelland, & the PDP Research Group, 1986) may be apt except for two points: it would not be limited to the brain or to neurons, and it would not necessarily be one-way. It might bear even closer resemblance to complex stochastic networks exhibiting continuous exchanges of flows (e.g., Ma, Holden, & Serota, 2013). When perturbed, physiological subsystems act to disperse perturbing forces and, in so doing, pass the buck onto neighboring subsystems (Gelfand & Tsetlin, 1971; Latash, Danion, Scholz, Zatsiorsky, & Schöner, 2003). Fractal fluctuations in the brain have already been implicated in the flexibility needed for the brain to anticipate novel structure in perceptual learning (Friston, Breakspear, & Deco, 2012). These fractal fluctuations extend well beyond the brain, across the perceptual-motor periphery. We used the paradigm of dynamic touch to determine whether fractality might provide new insights into intermodal perception. That is, we were curious whether previous work into a single modality might generalize to two modalities at once and, more than that, provide a window on the sharing of information between the two modalities. Specifically, we sought to test whether the effect of visual feedback on judgments might depend on head-sway fractality and whether the visual feedback elicited increases in head-sway fractality that might cascade downward to promote the fractal fluctuations of wielding by the hand. Experiment 1 provided evidence consistent with both expectations. Regression models found significant effects of visual feedback on dynamic-touchbased judgments of length only once the significant effect of head-sway fractality had been added. Head-sway fractality contributed also to the effect of visual feedback on length judgments after training had ceased. Further modeling revealed, first, that the group with visual feedback had significantly higher head-sway fractality and, second, that increases of headsway fractality on a current trial had a more sustained effect on later hand-wielding fractality in the group with visual feedback than in the no-feedback group. Experiment 2 found effects of fractality of head sway aligning with those of Experiment 1 in a unimodal case, suggesting that head-sway fractality does indeed moderate the use of optical information independently of the dynamic-touch task. Fractal fluctuations may provide a common currency for investigating coordination of perceptional information from one “mode” with another. In fact, the neural architecture supporting haptic perception draws heavily upon a vast network of connective tissues and extracellular media to facilitate its coordination of sensorimotor information (Marsden, Merton, & Morton, 1983; Ingber, 2006; Turvey & Fonseca, 2014). Clear distinctions between the role of neural dynamics and the role of bodily mechanics in exploration are thus unavailable, and we may not reach a clear understanding by setting nervous dynamics apart from the rest of the body. However, relinquishing such anatomical distinctions between bodily mechanics and the electrical precedents and consequents of these mechanics is not so
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much a loss of clarity as it may be an avenue for novel insight. Given the broadly distributed patterning of neural dynamics, the intermodal coordination appears to be legible in the fractal fluctuations of perceptual-motor fluctuations.
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Received May 6, 2014 Revision received August 29, 2014 Accepted September 10, 2014 䡲
Call for Nominations The Publications and Communications (P&C) Board of the American Psychological Association has opened nominations for the editorships of Developmental Psychology and the Journal of Consulting and Clinical Psychology for the years 2017–2022. Jacquelynne S. Eccles, PhD, and Arthur M. Nezu, PhD, respectively, are the incumbent editors. Candidates should be members of APA and should be available to start receiving manuscripts in early 2016 to prepare for issues published in 2017. Please note that the P&C Board encourages participation by members of underrepresented groups in the publication process and would particularly welcome such nominees. Self-nominations are also encouraged. Search chairs have been appointed as follows: ● Developmental Psychology, Suzanne Corkin, PhD, and Mark Sobell, PhD ● Journal of Consulting and Clinical Psychology, Neal Schmitt, PhD, and Annette LaGreca, PhD Candidates should be nominated by accessing APA’s EditorQuest site on the Web. Using your Web browser, go to http://editorquest.apa.org. On the Home menu on the left, find “Guests.” Next, click on the link “Submit a Nomination,” enter your nominee’s information, and click “Submit.” Prepared statements of one page or less in support of a nominee can also be submitted by e-mail to Sarah Wiederkehr, P&C Board Search Liaison, at
[email protected]. Deadline for accepting nominations is January 7, 2015, when reviews will begin.
