Multiplicative-cascade dynamics supports whole-body coordination for perception via effortful touch

Human Movement Science 70 (2020) 102595

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Multiplicative-cascade dynamics supports whole-body coordination for perception via effortful touch

T

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Madhur Mangalama, , Damian G. Kelty-Stephenb a b

Department of Physical Therapy, Movement and Rehabilitation Sciences, Northeastern University, Boston 02115, MA, USA Department of Psychology, Grinnell College, Grinnell 50112, IA, USA

A R T IC LE I N F O

ABS TRA CT

Keywords: Biotensegrity Center of pressure Dynamic touch Effortful touch Haptic perception Multifractality Proprioception Psychophysics Tensegrity

Effortful touch by the hand is essential to engaging with and perceiving properties of objects. The temporal structure of whole-body coordination must reflect the prospective control that provides for both the engagement with and perception of properties of the hefted objects. In the present study, we found signatures of multifractality in the time series of fluctuations in Euclidean displacement in the participants' center of pressure (CoP) as they hefted weighted objects to perceive their heaviness and length. Comparisons of widths of the multifractal spectrums of CoP series with 32 Iterative Amplitude Adjusted Fourier Transform (IAAFT) surrogates provided evidence for multiplicative-cascade dynamics and interactivity across scales, through the continuous t-statistic comparing the original and surrogate widths (tMF). After controlling for the linear properties of CoP series and their interactions with the informational variable (i.e., the moment of inertia of the hefted objects), regression modeling of unsigned error in judgments of heaviness and length revealed that the multifractal evidence for nonlinearity (tMF) significantly influenced unsigned error. The two indicators showed opposite, task-specific effects on accuracy: accuracy in judgments of heaviness and length decreased and increased, respectively, with greater tMF. These findings suggest that multiplicative-cascade dynamics in posture play a role in prospective coordination during the engagement with objects and perception of their properties via effortful touch by the hand. Future work may elucidate how constraint(s) on exploratory kinematics influence the multifractal behavior in such suprapostural perceptual tasks as effortful touch.

1. Introduction 1.1. Fractal fluctuations in the human sensory and motor system The human sensory and movement system operates over theoretically infinite degrees of freedom (DoFs), but coordinated behavior appears to follow relatively few functional parameters (Bernstein, 1967). Consequently, much consistency observed in the patterning of coordinated movement is accompanied by a significant amount of fluctuations. A prevalent notion is that low-dimensional control strategies recruit individual DoFs as needed, to suit the environment and task constraints and the accompanying fluctuations reflect the selective deployment of low-dimensional movement patterns (Berniker, Jarc, Bizzi, & Tresch, 2009). What principles govern this weaving of consistency with fluctuations? A recent proposal is that fluctuations provide a window into the

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Corresponding author. E-mail address: [email protected] (M. Mangalam).

https://doi.org/10.1016/j.humov.2020.102595 Received 19 November 2019; Received in revised form 3 January 2020; Accepted 14 February 2020 Available online 22 February 2020 0167-9457/ © 2020 Elsevier B.V. All rights reserved.

Human Movement Science 70 (2020) 102595

M. Mangalam and D.G. Kelty-Stephen

cascade-like dynamics that govern the relationship between organism and environment and underlies the observed consistency. These cascades entail nonlinear interactions: larger fluctuations engender smaller fluctuations and smaller fluctuations support and constrain larger fluctuations across scales, yielding fractality in fluctuations. Healthy fluctuations exogenous to heartbeat (Ivanov et al., 1999, 2001), inter-breath (Suki, 2002), and inter-spike intervals (Stratimirović, Milošević, Blesić, & Ljubisavljević, 2001; Zheng, Gao, Sanchez, Principe, & Okun, 2005), all show fractality, reflecting nonlinear interactions across scales consistent with cascade-like dynamics—the loss of which due to the gradual weakening of across-scale interactions appears to be a universal signature of the aging process (Goldberger et al., 2002).

1.2. Fractal fluctuations in whole-body coordination support engagement with perception of object properties via dynamic touch The haptic [sub]modality of “effortful touch” supports perceptions of such properties of a wielded object as heaviness, length, width, shape, and orientation in the hand (Carello & Turvey, 2000; Mangalam, Barton, Wagman, Fragaszy, Newell, 2017; Mangalam, Wagman, Newell 2018; Mangalam, Pacheco, Fragaszy, & Newell, 2019; Michaels, Weier, & Harrison, 2007; Turvey & Carello, 2011; Wagman & Carello, 2001). The applied muscular effort to passively support or actively wield an object shapes the medium by which an action-relevant perceptual judgment is derived (Mangalam, Conners, Kelty-Stephen, & Singh, 2019; Mangalam, Conners, & Singh, 2019; Waddell & Amazeen, 2017, 2018; Waddell, Fine, Likens, Amazeen, & Amazeen, 2016). It is being increasingly recognized that the biophysical substrate of the body-wide haptic perceptual system engaged during effortful touch is a neurally-tunable multifractal tensegrity (MFT) system that consists of the skin, connective-tissue net, muscles, tendons, bones, ligaments, and nerve fibers, all held together by tensional and compressional forces (Cabe, 2018; Schleip, Mechsner, Zorn, & Klingler, 2014; Turvey & Fonseca, 2014). In this body-wide MFT, the global realignment of forces engenders local perturbations, and local perturbations support and constrain the global realignment of forces, yielding fractality in fluctuations (Mangalam, Chen, McHugh, Singh, & Kelty-Stephen, 2020; Mangalam, Conners, Kelty-Stephen, & Singh, 2019). Recent investigations into fractality in fluctuations that appear throughout the whole-body movement have provided evidence that fractality in exploratory movement supports individuals' engagement with and perception of properties of manually wielded objects. Investigations on the role of fractality in effortful touch have found that the time series of Euclidean displacement in manual wielding trajectories exhibit temporal correlations across scales associated with fractality, and trial-by-trial variation in fractality helps to predict the effect of objects' inertial properties—the physical variable that provides the informational support for perception via dynamic touch (Carello & Turvey, 2000; Turvey & Carello, 2011)—on trial-by-trial variation in judgment of length (Stephen, Arzamarski, & Michaels, 2010). This effect extends to wielding across disparate anatomical parts (e.g., from the hand and foot to the foot and hand, respectively) (Stephen & Hajnal, 2011). Fractality in manual exploration predicts differences in effortful touch capabilities across populations, such as between children with Attention-Deficit Hyperactivity Disorder (ADHD) and typical development (Avelar et al., 2019). Other studies have tapped into the complex relationship between manual activity and posture, entailing perception of the intended properties of the wielded objects through the redistribution of forces throughout the body. Earlier work in this domain investigated the role of postural sway in exploring properties of objects resting on the back of the shoulders (Kelty-Stephen & Dixon, 2014; Palatinus, Dixon, & Kelty-Stephen, 2013; Palatinus, Kelty-Stephen, Kinsella-Shaw, Carello, & Turvey, 2014). However, this passive support has none of the dynamical properties of active manipulation by the hand. Without the finely tuned orchestrated activity of joints and muscles supporting manual exploration, we might easily suspect that the unwitting and unintended fluctuations in the postural center of pressure (CoP) had sooner to do with stabilizing posture than with exploring objects resting across the shoulder. Nonetheless, the findings of these studies suggest that fractality in postural sway predicts judgments of objects resting across the shoulder just as fractality in hand movements predicts judgments of objects wielded by the hand. The fluctuations necessary for perceiving the length of a passively supported object are available in the smaller scales of postural sway that reflect the perceiver's intention (Palatinus et al., 2013). The degree of trial-by-trial variations in fractality in the time series of planar Euclidean displacements in individuals' COP significantly improved the prediction of trial-by-trial variability in judgment of length, above and beyond prediction by traditional predictors of effortful touch (i.e., the moment of inertia) and conventional measures of COP variability (Palatinus et al., 2014). Variation in fractality of body sway also supports the use of visual feedback during effortful touch (Kelty-Stephen & Dixon, 2014). The latest evidence in this line of research has shown body-wide sharing of fractal fluctuations between postural sway and hand movements supporting perception of properties of objects hefted by the hand (Mangalam et al., 2020). Specifically, the findings indicate that fractality of COP can actually predict judgments of heaviness and length following manual wielding. This predictive role of variation in fractality encourages the alignment of these findings with the multifractal formalism, and the body-wide availability of these predictive variations in fractality likewise suggests that these results support the MFT hypothesis. However, the prior work showing that COP variations in fractality predict judgments of manually wielded objects have only estimated multifractality as a temporally local variation in monofractal scaling. It remains to test whether the multifractal spectrum entailed by these predictive variations reflect the nonlinear correlations across time that we expect in a tensegrity framework with nonlinear interactions across many nested time scales. So, we present a reanalysis of this recent work to further diagnose and estimate the role of multiplicativecascade dynamics and cross-scale interactivity that supposedly underlie all the above multifractal expressions in perception and action.

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Human Movement Science 70 (2020) 102595

M. Mangalam and D.G. Kelty-Stephen

1.3. Can multiplicative-cascade dynamics and interactivity support engagement with perception of object properties via dynamic touch? An increasing number of feedback loops at multiple levels of organizational hierarchy may contribute towards the prospective control of whole-body coordination as the manual act of perceiving via effortful touch becomes progressively dynamic. Specifically, both internal and external factors that influence these feedback loops may cause changes in the heterogeneity of interactions, which may be reflected in the whole-body coordination, such as in the temporal relations in fluctuations in CoP (Avelar et al., 2019; Palatinus et al., 2013, 2014; Stephen et al., 2010; Stephen & Hajnal, 2011). Manipulating the constraints on exploratory movement and manipulating the informational support for perception via effortful touch provides a robust test for whether multiplicativecascade dynamics and interactivity support engagement with and perception of properties of manually hefted objects. We can also investigate how various internal and external factors jointly influence the multiplicative-cascade dynamics and interactivity. In the present study, we sought signatures of multifractality in time series of fluctuations in the Euclidean displacement in individuals' CoP as they (blindfoldedly) manually hefted weighted objects to perceive their heaviness and length. We and our colleagues previously reported that fractality in CoP displacement supports judgments of heaviness and length of manually hefted objects and that this support increases with exposure to the task, indicating that individuals progressively implicate the fractal scaling of CoP fluctuations in the perception of heaviness and length (Mangalam et al., 2020). The present study extends this finding beyond fractality to the realm of multifractality by specifically investigating the multiplicativity of processes through which posture may support manual exploration, entailing perception of the intended properties of hefted objects (heaviness or length) through the redistribution of forces throughout the body. The goal of the present study is to elevate the discourse about the role of temporal correlations in the perceptuomotor exploration of task constraints during effortful touch. Specifically, we hope to show that multifractal evidence of nonlinear correlations in perceptuomotor exploration is much clearly related to the accuracy of perceptual judgments than monofractal scaling. An ongoing point of contention in theorizing on links between perception and action has been whether monofractal scaling in the perceptuomotor exploration of task constraints leads to greater accuracy. The evidence for this view has been strongly contingent upon task constraints and the experience of a participant (Kelty-Stephen & Dixon, 2014; Stephen et al., 2010; Stephen & Hajnal, 2011), if not strongly against this view (Kuznetsov & Wallot, 2011). What the expectation for monofractal scaling to support accuracy misses is the point that the human body is not a single point-mass approximable with a single value, but instead, humans embody a diversity of fractal scalings—a multifractality across space and across time—that better explains how humans can accomplish such a variety of perceptuomotor exploration tasks under a variety of constraints (Kelty-Stephen, 2017; Kelty-Stephen & Dixon, 2014; Stephen et al., 2010; Stephen & Hajnal, 2011). Recent work has begun to show that perceptual accuracy has less to do with monofractal evidence of linear correlations but much specifically to do with the multifractal evidence of nonlinearity (Bell, Carver, Zbaracki, & Kelty-Stephen, 2019; Carver, Bojovic, & Kelty-Stephen, 2017). In the present study, we seek to show that multifractal evidence of nonlinear correlations in postural fluctuations is related to perceptual accuracy through interactions with the informational variable for perception via effortful touch (i.e., the moment of inertia (Palatinus, Carello, & Turvey, 2011; Turvey & Carello, 2011)). 2. Materials and methods 2.1. Participants Eight adult men and seven adult women (M = 23.4, SD = 3.4 years, self-identified right-handed) voluntarily participated in the present study. Each participant provided written consent by signing a form with information about the purposes of the study, the procedures, and the potential risks and benefits of participation. The Institutional Review Board (IRB) at the University of Georgia (Athens, GA) approved the present study. 2.2. Experimental objects Each participant manually hefted six experimental objects, each object consisting of a dowel (oak, hollow aluminum, or solid aluminum; diameter = 1.2 cm, length = 75.0 cm) weighted by stacked steel rings (4 or 12 rings attached to the dowel at 20.0 or 60.0 cm, respectively, opposite to the grasping location; each ring's inner diameter = 1.4 cm, outer diameter = 3.4 cm, thickness = 0.2 cm; mass = 14 g) (Table 1). The dowels were weighted such that the resulting six objects systematically differed in the moment of inertia, Ilongitudinal (Object 1 < Object 2, Object 3 < Object 4, Object 5 < Object 6), reflecting the resistance of the object to rotation about the participant's wrist about the longitudinal axis. A cotton tape of negligible mass was enfolded on each dowel to prevent the cutaneous perception of its composition. 2.3. Experimental setup and procedure Each participant stood on a pair of force plates (600 × 400 mm each; Bertec Inc., Columbus, OH), hefted each object, and reported judgments of heaviness and length. We imposed external constraints on the participant's wrist movement. In a “static” condition, we asked the participant to lift and hold each object static for 5 s, and in the two “dynamic” conditions, to lift and wield each object for 5 s synchronously with metronome beats at 2 Hz or 3 Hz. The participant lifted a “reference” object of an arbitrary mass of 100 (no units) before the first and after every six trials thereafter, and assigned heaviness values proportionally greater than 100 to objects perceived heavier than the reference object (e.g., 3

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M. Mangalam and D.G. Kelty-Stephen

Table 1 Experimental objects (n = 6). Object

1 2 3 4 5 6

Dowel

Attached weights

Object parameters

Composition

Length (cm)

Mass (g)

Mass (g)

Location (cm)

Mass, m (g)

Moment of inertia, Ilongitudinala (g·cm2)

Oak wood Oak wood Hollow aluminum Hollow aluminum Solid aluminum Solid aluminum

75 75 75 75 75 75

68 68 109 109 266 266

168 56 168 56 168 56

20 60 20 60 20 60

236 124 277 165 434 332

153,500 278,850 194,720 321,770 459,850 586,720

a We calculated the values of a 3 × 3 inertia tensor matrix for each object, each value corresponding to rotations about the wrist, assuming 5-cm distance between the location of grasp and the object's proximal end. Diagonalizing the 3 × 3 inertia tensor matrix using MATLAB function “eig (A)” yielded the eigenvalues of the tensor.

200 to an object perceived twice as heavy), and proportionally less than 100 to objects perceived lighter than the reference object (e.g., 50 to an object perceived half as heavy). In each trial, a ‘lift’ signal indicated the participant to lift the object by about 5 cm and hold it static or wield it at 2 Hz or 3 Hz. After 5 s, a ‘stop’ signal indicated the participant to place the object back. The participant then reported judgments of heaviness (no units) and length (by manually adjusting the position of a marker by pulling a string on a string-pulley assembly). The experimenter noted the judgment of length (cm) on a meter-scale attached to the string-pulley assembly. We instructed each participant to minimize the torso and upper hand motion, as well as the amplitude of wielding movements. Each participant completed 108 trials (3 conditions of exploratory kinematics × 6 objects × 2 trials) in 90–105-min session. We blocked trials for each condition of exploratory kinematics but randomized the order of the conditions of exploratory kinematics. Additionally, we randomized the order of the six objects and the two trials within each condition of exploratory kinematics. 2.4. Obtaining CoP Planar Euclidean Displacement (PED) series The ground reaction forces recorded using force plates yielded a two-dimensional CoP time series for each trial, with each dimension describing the position of CoP along the medial: lateral and anterior: posterior axes. Each trial yielded a two-dimensional CoP time series of 10,000 samples (5-s recording at 2000 Hz), and the corresponding time series of CoP displacement consisted of 9999 samples. We combined the two dimensions to finally obtain a one-dimensional planar Euclidean displacement (PED) series describing the displacement of CoP in two-dimensions. 2.5. Assessing multifractality and interactivity 2.5.1. Direct estimation of the multifractal spectrum We estimated the multifractal spectrums of each CoP PED series using the direct method first described by Chhabra and Jensen (1989) and Chhabra, Meneveau, Jensen, and Sreenivasan (1989). This method begins by sampling a time series u(t)at progressively larger scales such that the proportion of signal Pi(L)falling within the ithbin of scale L is iL

Pi (L) =

∑k = (i − 1) L + 1 u (k ) ∑ u (t )

(1)

As L increases, Pi(L) represents progressively larger proportion of u(t), such that

P (L) ∝ Lα

(2)

If P(L) shows monofractal dynamics, then it grows homogeneously across time scales according to a single, potentially fractional (noninteger) “singularity” strength α(Mandelbrot, 1982). If P(L) shows multifractal dynamics, then it grows heterogeneously across time scales L according to a range of singularity strengths, such that

Pi (L) ∝ Lαi

(3)

whereby each i bin may show a distinct relationship of P(L) with L. The width of this singularity spectrum, αmax − αmin, indicates the heterogeneity of relationships between P(L) and L(Halsey, Jensen, Kadanoff, Procaccia, & Shraiman, 1986; Mandelbrot, 1997). Chhabra and Jensen (1989) and Chhabra et al. (1989) method involves first estimating P(L) for NL nonoverlapping bins of L-sizes and then transforming them with a parameter q, emphasizing higher or lower P(L) for q > 1 and q < 1, respectively. Thus, for each ith bin of size L, the transformed proportions, th

μi (q, L) =

[Pi (L)]q NL ∑i = 1 [Pi (L)]q

(4)

reflecting different q-weighted distributions of fluctuations in the magnitude of the signal at scale L. For each q, each estimated value of α(q) belongs to the singularity spectrum only when the Shannon entropy of μ(q, l) scales with L according to the Hausdorff 4

Human Movement Science 70 (2020) 102595

M. Mangalam and D.G. Kelty-Stephen

dimension f(q), where

f (q) = − lim L →∞

= lim L→0

1 lnL

1 lnL

N

∑i =1 N

∑i =1

μi (q, L) ln μi (q, L)

μi (q, L) ln μi (q, L)

(5)

Finally, α(q) is estimated as;

α (q) = − lim L →∞

1 = lim L → 0 lnL

1 lnL

N

∑i =1 N

∑i =1

μi (q, L) ln Pi (L)

μi (q, L) ln Pi (L)

(6)

For values of q yielding a strong relationship between Eqs. (5) & (6)—in this study, exhibited a correlation coefficient, r > 0.999, the parametric curve(α(q), f(q)) constitutes the singularity spectrum. 2.5.2. Surrogate testing Nonzero singularity spectrum can reflect multiplicative interactions across timescales or linear autocorrelation/heavy-tailed distributions (Kelty-Stephen, Palatinus, Saltzman, & Dixon, 2013; Veneziano, Moglen, & Bras, 1995). The two possibilities can be distinguished by comparing the original spectrum to those of surrogate time series in which cross-scale interactivity is eliminated (Ihlen & Vereijken, 2010). The most commonly used surrogate for testing for nonlinear origins of the multifractal spectrum of a time series is the Iterated Amplitude Adjusted Fourier Transformation (IAAFT). IAAFT has been extensively studied and used in a wide range of applications and all its pitfalls have been shown to be milder in comparison to other methods (Lancaster, Iatsenko, Pidde, Ticcinelli, & Stefanovska, 2018). IAAFT algorithm sorts the original values in a time-symmetric fashion around the overall autoregressive structure, generating surrogates that randomize the phase ordering of the spectral amplitudes of the original series without affecting the probability density and autocorrelation function (Schreiber & Schmitz, 1996). If the width of the original singularity spectrum is outside 95% confidence interval (CI) of mean width for a given sample of IAAFT spectums (according to a p-value of less than 0.05), then the original time series is considered to have nonlinear origins. The t-statistic (henceforth, tMF) indexes the contribution of multiplicative interactions across timescales to the fractality of time series. 2.6. Statistical analysis We modeled the accuracy of perception as unsigned error in judgments of heaviness (|Hperceived – Hactual|) and length (|Lperceived − Lactual|). Because judgments of heaviness followed a proportion relative to a 109-g reference object, we calculated the percentage that this judgment represented of what would have been the accurate percentage value based on each object's actual mass. For instance, if a participant judged Object 2 to have a length of 62.5 cm and heaviness 120 relative to 100 of the reference object, then they would have signed error in judgments of length, Lerror = 62.5–75.0 = −12.5 and signed heaviness error, Herror = 100 × ((120 × 109)/100)/236 = 55.42. For calculating signed error in judgments of length, we subtracted the actual length of objects (75 cm) from all judgments of length. Finally, for calculating unsigned error, we calculated the absolute value of 100 less than the percentage value corresponding to Hperceived, and the absolute value of error in judgments of length. Accordingly, for the same judgment of the same object, the unsigned error in judgment of length would be 12.5, and the unsigned error in judgment of heaviness would be |55.34–100| = 44.66. We rounded the unsigned error for judgments of heaviness. Perceived heaviness was a nonlinear dependent measure, given the instruction to report heaviness in terms of ratios to the reference object (e.g., 50 to indicate half and 200 to indicate twice the reference 100). Thus, we modeled signed error in judgments of heaviness using the generalized linear model (GLM) of Poisson regression. In contrast, perceived length was explicitly interval based on how we defined it to the participants. We thus used the GLM of Poisson regression using “lme4” package for Rstudio (Bates, Sarkar, Bates, & Matrix, 2007), to examine variation in signed error in judgments of heaviness, and linear mixed-effect (LME) model (Singer & Willett, 2003), using the “nlme” package for Rstudio (Pinheiro, Bates, DebRoy, Sarkar, & Team, 2018), to examine variation in unsigned error in judgments of length, as a function of three classes of predictors described below. Given that our goal was to test whether the multifractal evidence of nonlinearity supports accuracy in perceptual judgments, it was crucial for our modeling to give a full chance for the linear structure of exploratory behaviors to predict this accuracy as well. Conveniently, a general linear model (GLM) is compactly described through no more than three parameters: the mean, the standard deviation (or its square, i.e., variance), and the linear autocorrelation (Mandic, Chen, Gautama, Van Hulle, & Constantinides, 2008). Hence, we include the purely linear descriptors of the CoP PED series, namely, the mean (M), the standard deviation (SD), and the monofractal HfGn describing the power-law decay in the linear autocorrelation, serving as the first class of predictors. It is common for monofractal scaling to be mistaken for a nonlinear descriptor, despite that it refers to power-law divergence of the Fourier amplitude spectrum or, equivalently, the linear autocorrelation. Thus, although monofractal analysis of nonlinearity-driven time series often returns HfGn estimates in the pink noise range, this HfGn is a linear-model parameter that fails to speak conclusively about nonlinearity (e.g., Kelty-Stephen & Wallot, 2017). The linear autocorrelation is a set of N–1 coefficients for lagged values of a given time series x(t), but the power-law behavior estimable through monofractal analysis entails that this linear autocorrelation is compactly describable by the single monofractal exponent HfGn that defines the scale-invariant form of the autocorrelation. Previous work has repeatedly found that the effect of fractality on perceptual accuracy depends on the informational variable (i.e., the moment 5

Human Movement Science 70 (2020) 102595

M. Mangalam and D.G. Kelty-Stephen

Fig. 1. Multifractal analysis. A: representative original CoP PED series. B: A representative IAAFT surrogate series of the original CoP PED series. C: Multifractal spectrums (α(q), f(q)) of the original CoP PED series (solid line) in (a) and that of its five representative IAAFT surrogates (dashed lines).

of inertia), implicating specifically the interaction of fractality with the inertial values (Kelty-Stephen & Dixon, 2014; Mangalam et al., 2020; Stephen et al., 2010). The interactions of these linear descriptors with the informational variable, ILogitudinal (i.e., the logarithmically scaled first moment of inertia of the hefted object): M × LogILogitudinal, SD × LogILogitudinal, and HfGn × LogILogitudinal, served as the second class of predictors. These effects serve only as control effects to ensure that our multifractal-based predictors address specifically nonlinear contributions to error. 3. Results 3.1. Widths of multifractal spectrums Each participant (n = 15) completed 108 trials (3 conditions of exploratory kinematics × 6 objects/condition of exploratory kinematics × 6 trials/object). So, we obtained a total of 1620 CoP series. We estimated widths of the multifractal spectrums for each original CoP PED series and 32 IAAFT surrogates for each original CoP PED series. Fig. 1A & 1B depicts a representative original CoP PED series and a representative IAAFT surrogate series of the original CoP PED series, respectively. Fig. 1C depicts the resulting multifractal spectrums of the original CoP PED series, as well as that of its five representative IAAFT surrogates. We found evidence that multiplicative-cascade dynamics and interactivity supports perception of object properties via effortful touch by the hand in the form of non-zero multifractality-spectrum widths (i.e., αmax – αmin > 0) for all 1620 original CoP PED series (M = 1.37 × 10−2, SD = 5.17 × 10−3). 3.2. Narrower multifractal spectrum of surrogate CoP PED series We used one-sample t-tests to individually contrast the width of the multifractal spectrum of each original CoP PED series with those of its 32 surrogates. The widths of the multifractal spectrums (M = 1.37 × 10−2, SD = 5.17 × 10−3) differed significantly from those of the mean widths of their IAAFT surrogates (M = 8.61 × 10−3, SD = 6.12 × 10−4), t(1619) = 40.23, p < .001. Fig. 2 depicts a frequency distribution of t-statistics comparing widths of the multifractal spectrum of the original CoP PED series and that of its 32 IAAFT surrogates. The widths of 1362 of 1620 spectrums differed significantly from that of their surrogates (p < .05), (Table 2). A significant proportion of the original CoP PED series (1291 out of 1620) exhibited wider multifractal spectrums than did their 32 surrogates, confirming the role of multiplicative-cascade dynamics and interactivity governing the suprapostural perceptual task of effortful touch. 3.3. Variation in multifractality and interactivity across conditions of exploratory kinematics and experimental objects Table 3 provides the coefficients of the GLM model of Poisson regression examining the influence of various linear and nonlinear measures of variation in CoP displacement and the object's moment of inertia on the unsigned error in judgments of heaviness. Crucial to our prediction that multifractality in postural fluctuations support perceptual judgments of heaviness via effortful touch by 6

Human Movement Science 70 (2020) 102595

M. Mangalam and D.G. Kelty-Stephen

Fig. 2. Frequency distribution of tMF comparing widths of multifractal spectrum of the original CoP PED series and that of its 32 IAAFT surrogates. tTM > 0 indicates that the original spectrum was wider than the surrogate spectrums and vice versa. The dashed vertical lines indicate cutoffs for statistical significance at the two-tailed alpha level of 0.05 for 31 DoFs. Table 2 tTM comparing widths of multifractal spectrum of the original CoP PED series and that of its 32 IAAFT surrogates.† Original narrower

Not significant

Original wider

71

258

1291

†

Significance was evaluated at the alpha level of 0.05.

Table 3 Coefficients of the generalized linear model (GLM) of Poisson regression examining the influence of various linear and nonlinear measures of variation in CoP displacement and the object's moment of inertia, Ilongitudinal, on the unsigned error in judgments of heaviness, Herror. Effectsa,b

b

SE

z

pc

(Intercept) tTM M SD HfGn LogIlongitudinal M × LogIlongitudinal SD × LogIlongitudinal HfGn × LogIlongitudinal

9.69 0.00098 1804.00 1365.00 −3134.00 −1.02 −323.00 −560.00 0.55

3.02 0.00045 30.68 14.32 29.84 0.55 5.71 56.22 0.98

3.21 2.17 58.79 95.35 −105.03 −1.86 −56.55 99.61 0.56

0.001 0.030 < 0.001 0.001 0.490 0.064 < 0.001 < 0.001 0.575

a b c

Fitted model: absolute(Herror) ~ tDM + (M + SD + HfGn) * (LogILongitudinal) + (1|Participant). 95% confidence intervals are calculable as b ± 1.96 SE. Boldface values indicate statistical significance at the two-tailed alpha level of 0.05.

the hand, we found a significant positive effect of tMF (b = 0.00098, SE = 0.00045) on unsigned error in judgments of heaviness. This result suggests that the extent of nonlinearity (tMF) in CoP fluctuations has a magnifying effect on unsigned error or higher perceptual accuracy for judgments of heaviness. Table 4 provides the coefficients of the LME model examining the influence of various linear and nonlinear measures of variation in CoP displacement and the object's moment of inertia on the unsigned error in judgments of length. Crucial to our prediction that multifractality in postural fluctuations support perceptual judgments of length via effortful touch by the hand, we found a significant negative effect of tMF (b = −0.05, SE = 0.02) on unsigned error in judgments of length. This result suggests that the extent of nonlinearity (tMF) in CoP fluctuations has a diminishing effect on unsigned error or higher perceptual accuracy for judgments of length. In both models, we found significant interactions of the three linear descriptors of exploratory behaviors (M, SD, and HfGn) with inertial values. These effects all show the opposite sign of the main effect of the corresponding linear descriptor, suggesting that the contingency on the moment of inertia reduces the effect of each linear descriptor on error. Because these effects only serve as control effects to ensure that our multifractal-based predictors address specifically nonlinear contributions to error, we do not comment 7

Human Movement Science 70 (2020) 102595

M. Mangalam and D.G. Kelty-Stephen

Table 4 Coefficients of the linear mixed-effects (LME) model examining the influence of various linear and nonlinear measures of variation in CoP displacement and the object's moment of inertia, Ilongitudinal on the unsigned error in judgments of length, Lerror. Effectsa,b

b

SE

t(1597)

pc

(Intercept) tTM M SD HfGn LogIlongitudinal M × LogIlongitudinal SD × LogIlongitudinal HfGn × LogIlongitudinal

−178.14 −0.05 −28,567.30 46,302.42 550.00 36.47 5159.82 −8372.82 −99.75

146.65 0.02 12,895.93 20,899.53 256.54 26.72 2326.38 3773.38 46.78

−1.21 −2.28 −2.22 2.22 2.14 1.36 2.22 −2.22 −2.13

0.224 0.023 0.027 0.032 0.032 0.173 0.027 0.032 0.033

a b c

Fitted model: absolute(Lerror) ~ tMF + (M + SD + HfGn) * (LogILongitudinal) + (1|Participant). 95% confidence intervals are calculable as b ± 1.96SE. Boldface values indicate statistical significance at the two-tailed alpha level of 0.05.

further than to note that these contributions were significant, indicating that the effect of nonlinearity on accuracy does not preclude the effect of linear aspects of exploratory behaviors. 4. Discussion Effortful touch by the hand is essential to engaging with and perceiving properties of objects. In the present study, we sought signatures of multifractality in the time series of fluctuations in Euclidean displacement in individuals' CoP as they (blindfoldedly) manually hefted weighted objects to perceive their heaviness and length. We found signatures of multifractality in CoP PED series as they hefted weighted objects to perceive their heaviness and length. Comparisons of widths of the multifractal spectrums of CoP series with 32 IAAFT surrogates provided evidence for multiplicative-cascade dynamics and interactivity across scales, through the continuous t-statistic comparing the original and surrogate widths (tMF). After controlling for the linear properties of CoP series and their interactions with the informational variable (i.e., the moment of inertia of the hefted objects), regression modeling of unsigned error in judgments of heaviness and length revealed that the multifractal evidence for nonlinearity (tMF) significantly influenced unsigned error. The two indicators showed opposite, task-specific effects on accuracy: accuracy in judgments of heaviness and length decreased and increased, respectively, with greater tMF. The task-specificity of the relationship between multifractal nonlinearity and perceptual accuracy is puzzling, but the resolution of this puzzle will involve the prior understanding that nonlinearity across time is bidirectional and not a one-way attribute. A significant deviation of original spectrum width from surrogate spectrum width remains evidence of nonlinear interactions across time, a failure of only linear contributions to variability. Linearity is a presumption of symmetry across time, a statistical expectation of roughly the same variability across time given the mean and variance. The variation of t-statistics above and below the range of null-hypothesis reflects different ways in which time-variability can exceed or fall short of the linear expectation. Specifically, significantly positive values of t-statistic reflect the propagation of more variability across time than linearity would predict, and significantly negative values of t-statistic sooner reflect the constriction of variability across time than linearity would predict (Lee & Kelty-Stephen, 2017). Hence, we can find narrower-than-surrogate spectra for postural sway when sway is constricted (e.g., KeltyStephen, 2018), suggesting that judgments of heaviness may be most accurate when nonlinear interactions serve to constrict postural sway. Collectively, the present findings suggest that multiplicative-cascade dynamics in posture play a role in prospective coordination during the engagement with objects and perception of their properties via effortful touch by the hand. Future work may elucidate how constraint(s) on exploratory kinematics influence the multifractal behavior in such suprapostural perceptual tasks as effortful touch. Specifically, the present work replicates prior findings that linear aspects of exploratory behavior (e.g., M, SD, and HfGn) contribute to perceptual accuracy and that they do so contingently on the informational variable, that is, the moment of inertia (e.g., M × LogIlongitudinal, SD × LogIlongitudinal, and HfGn × LogIlongitudinal). The novel progress in the present study lies in showing that the multifractal evidence of nonlinearity in exploratory behavior promotes perceptual accuracy. The major significance of the present findings is that they extend and refine the relative utility of multifractal vs. monofractal perceptuomotor exploration in supporting perception via effortful touch. Monofractal descriptors of perceptuomotor exploration are certainly useful, but their effects appear to be firmly rooted in the specifics of task constraints. On the contrary, multifractal descriptors of perceptuomotor exploration do not necessarily interact with the informational variables to exert their effects on accuracy, and accordingly, we found that nonlinearity does support accuracy. That is, when portrayed through a single point of measurement (i.e., CoP), the support for accuracy from nonlinearity ends up differing by the type of perceptual task, that is, that the benefit of nonlinearity for promoting perceptual accuracy can depend either on the dichotomous difference from surrogates or on the continuous extent of difference from surrogates. In short, the novelty of the present findings lies in demonstrating that multifractality is a key property of fluctuations in CoP, and may reflect the very many [multiplicative] processes underlying the suprapostural perceptual task of effortful touch. Previous work has shown a tight relationship between fractality in postural control and fractality in manual activities, such as aimed throwing (Bell et al., 2019; Carver, Bojovic, & Kelty-Stephen, 2017). Extending the role of fractality to the perceptual realm, the work of others 8

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(Kelty-Stephen & Dixon, 2014; Palatinus et al., 2013, 2014) and our own (Mangalam et al., 2020; Mangalam, Conners, Kelty-Stephen, & Singh, 2019) show that fluctuations in whole-body movement show fractality and that fractality supports perceptual activities involving manual exploration. It should be noted that individuals in the present study held each experimental object static or wielded it at 2 Hz or 3 Hz synchronously with metronome beats. The requirement of this behavior is different than either passively supporting an object (Palatinus et al., 2013, 2014) or freely wielding it (Kelty-Stephen & Dixon, 2014). The extent to which multiplicative-cascade dynamics and interactivity is present under highly constrained behavior remain to be explored. A reasonable expectation is that reducing the available DoFs by imposing constraints on movement would decrease the potential interactions between disparate processes, thereby resulting in weaker signatures of multifractality. We thus feel that the widths of spectrums we obtained in the present study may at worst suffer underestimation but definitely not an overestimation. In summary, the present findings provide evidence that multiplicative-cascade dynamics and interactivity governing the temporal structure of fluctuation in the whole-body coordination reflects the prospective control that provides for both the engagement with and perception of properties of the hefted objects. Understanding the relationship between constraints arising out of the organism–environment interactions and multifractality is essential for understanding the role of posture in coordination in suprapostural perceptual activities. Previous work has demonstrated that the interdependency of postural control and manual behavior develops with experience. The ability to either suppress or allow fluctuations in posture as per the constraint(s) of a suprapostural task begins to develop at around 10 years of age and is essential for learning specific segmental movements required to complete a task within an environmental context (Haddad et al., 2012). Such interdependency is best explained using tensegrity principles (Cabe, 2018; Schleip et al., 2014; Turvey & Fonseca, 2014) and multifractality offers the best way to identify the cross-scale interactions that the bodywide MFT systems can use to support suprapostural perceptual activities. Tensegrity principles currently remain at the extreme peripheries of mainstream behavioral neuroscience. To make tensegrity principles part of mainstream behavioral neurscience, future work should take up the challenge of articulating the coordination of tensegrity structures in conjunction with neural dynamics and not simply before neural dynamics. The questions of whether or exactly how a tensegrity-like system supports effortful perception require much further work. The present findings are in no way conclusive, but they are promising and open the door to further inquiry towards developing the MFT hypothesis of the haptic perceptual system. We should clarify that we never intended to glean the full meaning of nonlinearity from one and only one measured point on the body. As noted above, the human body is not a single point-mass approximable with a single value, and accordingly, prior work shows that accuracy likely rests on a blend of nonlinearity spanning across the body (Bell et al., 2019; Carver et al., 2017). We must thus acknowledge that perceiving and acting organisms embody continuous fields of nonlinear temporal correlations distributed throughout the extent of their bodies. The goal of each task may recruit nonlinearity through the body differently, and the different goals of judging heaviness and length may lean differently on different aspects of nonlinearity (i.e., either its simple presence or extent or both). The present findings are strongly suggestive that network analysis of how fractality in fluctuations change and flow across the body should be an essential next step. Author contributions M.M. conceived and designed research; M.M. performed experiments; M.M. and D.G.K-S. analyzed data; M.M. and D.G.K-S. interpreted results of experiments; M.M. prepared figures; M.M. and D.G.K-S. drafted manuscript; M.M. and D.G.K-S. edited and revised manuscript; M.M. and D.G.K-S. approved final version of manuscript. Open practices statement The data and materials for the experiment are available upon request, and the experiment was not preregistered. Declaration of Competing Interest The authors declare that no competing interests exist. Acknowledgments We thank Jeffrey W. Wagman for calculating the eigenvalues of the inertia tensor of the experimental objects. We thank Ryan Chen and Terrence R. McHugh for help with testing the participants. References Avelar, B. S., Mancini, M. C., Fonseca, S. T., Kelty-Stephen, D. G., de Miranda, D. M., Romano-Silva, M. A., ... Silva, P. L. (2019). Fractal fluctuations in exploratory movements predict differences in dynamic touch capabilities between children with Attention-Deficit Hyperactivity Disorder and typical development. PLoS One, 14(5), https://doi.org/10.1371/journal.pone.0217200 e0217200. Bates, D., Sarkar, D., Bates, M., & Matrix, L. (2007). The lme4 package. Bell, C., Carver, N., Zbaracki, J., & Kelty-Stephen, D. (2019). Nonlinear amplification of variability through interaction across scales supports greater accuracy in manual aiming: Evidence from a multifractal analysis with comparisons to linear surrogates in the Fitts task. 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