Abstract
Bipedal gaits have been classified on the basis of the group symmetry of the minimal network of identical differential equations (alias cells) required to model them. Primary bipedal gaits (e.g., walk, run) are characterized by dihedral symmetry, whereas secondary bipedal gaits (e.g., gallop-walk, gallop- run) are characterized by a lower, cyclic symmetry. This fact has been used in tests of human odometry (e.g., Turvey et al. in P Roy Soc Lond B Biol 276:4309–4314, 2009, J Exp Psychol Hum Percept Perform 38:1014–1025, 2012). Results suggest that when distance is measured and reported by gaits from the same symmetry class, primary and secondary gaits are comparable. Switching symmetry classes at report compresses (primary to secondary) or inflates (secondary to primary) measured distance, with the compression and inflation equal in magnitude. The present research (a) extends these findings from overground locomotion to treadmill locomotion and (b) assesses a dynamics of sequentially coupled measure and report phases, with relative velocity as an order parameter, or equilibrium state, and difference in symmetry class as an imperfection parameter, or detuning, of those dynamics. The results suggest that the symmetries and dynamics of distance measurement by the human odometer are the same whether the odometer is in motion relative to a stationary ground or stationary relative to a moving ground.
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Notes
A similar axiomatic approach has been taken previously to derive the relevant dynamical evolution equations known to govern bi-manual rhythmic coordination (Frank et al. 2012b).
Another primary motivation was the difficulty of scheduling a university gymnasium facility during weekdays.
For physical systems (e.g., lasers, convection cells), order parameters have been identified by means of bottom-up approaches, that is, by means of mechanistic modeling (Haken 1977). In other instances, typically biological, order parameters have been suggested based on a top-down modeling approach, one based on experimental observations (e.g., Frank et al. 2009, 2012a; Haken 1996; Kelso 1995).
For a detailed discussion of the challenges posed for a principled derivation of imperfection parameters (detuning) see Appendix B in Turvey et al. (2012).
These were the target distances. Actual distances differed slightly from these target distances (see Table 2) due to variation in the experimenter’s signaling of when to stop and variation in the participant’s ability to stop on cue.
Qualitatively, Eq. (5) reproduces the \(M -R\) distance results of Turvey et al. (2012) summarized in the final two paragraphs of the present Introduction. This can be seen by noting that the stationary mean relative velocity \(v_{m}\) is given by the fixed point: \(v_{m}= \delta /\gamma \) of Eq. (4), i.e., the deterministic part of Eq. (5). Consequently, for \(\delta =0,\delta > 0\), and \(\delta < 0\), the model exhibits mean relative velocities of \(v_{m} = 0,\,v_{m} > 0\), and \(v_{m}< 0\), and predicts the distance relations of \({R} = {M, R} >{M}\), and \(R <M\), respectively.
Such might well be the case for overground locomotion.
For a theoretical overview see Frank et al. (2012a).
The Berthoz et al. experiment could be conducted with an \(M\) robot and an \(R\) robot, where \(M\) robot = \(R\) robot, or \(M\) robot \(<R\) robot, or \(M\) robot \(>R\) robot. The inequalities could be introduced by manipulating an extensive quantity, such as robot weight or robot width. According to Eq. (5), the two inequalities should have opposite effects. If one robot inequality yielded \(M\) distance \(>R\) distance, the other robot inequality should yield \(M\) distance \(<R\) distance, with robot equality yielding \(M\) distance = \(R\) distance.
References
Alton F, Baldey L, Caplan S, Morrissey MC (1998) A kinematic comparison of overground and treadmill walking. Clin Biomech 13: 434–440
Amazeen P, Amazeen EL, Turvey MT (1998) Dynamics of human intersegmental coordination: theory and research. In: Collyer C, Rosenbaum D (eds) Timing of behavior: neural, psychological and computational perspectives. MIT Press, Cambridge, pp 237–260
Barth FG (2004) Spider mechanoreceptors. Curr Opin Neurobiol 14:415–422
Beek PJ, Peper CE, Stegeman DF (1995) Dynamical models of movement coordination. Hum Mov Sci 14:573–608
Berthoz A, Israël I, Georges-François P, Grasso R, Tsuzuku T (1995) Spatial memory of body linear displacement: What is being stored? Science 269:95–98
Berthoz A, Viaud-Delmon I (1999) Multisensory integration in spatial orientation. Curr Opin Biol 9:708–712
Cruse H, Wehner R (2011) No need for a cognitive map: decentralized memory for insect navigation. PLoS Comput Biol 7:e1002009. doi:10.1371/journal.pcbi.1002009
Diggle PJ (1990) Time series: a biostatistical introduction. Clarendon Press, Oxford
Dotov DG, Frank TD, Turvey MT (2013) Balance affects prism adaptation: evidence from the latent after effect. Exp Brain Res 231:425–432
Etienne A, Berlie J, Georgakopolous J, Maurer R (1998) Role of dead reckoning in navigation. In: Healy S (ed) Spatial representation in animals. Oxford University Press, Oxford, pp 54–68
Frank TD (2005) Nonlinear Fokker-Planck equations: fundamentals and applications. Springer, Berlin
Frank TD, Blau J, Turvey MT (2009) Nonlinear attractor dynamics in the fundamental and extended prism adaptation paradigm. Phys Lett A 373:1022–1030
Frank TD, Blau J, Turvey MT (2012a) Symmetry breaking analysis of prism adaptation’s latent aftereffect. Cogn Sci 36:674–697
Frank TD, Silva PL, Turvey MT (2012b) Symmetry axiom of Haken-Kelso-Bunz coordination dynamics revisited in the context of cognitive activity. J Math Psychol 56:149–165
Gibson JJ (1966) The senses considered as perceptual systems. Houghton Mifflin, Boston
Golubitsky M, Stewart I (1999) Symmetry and pattern formation in coupled cell networks. Pattern Form Contin Coupled Syst 115:65–82
Golubitsky M, Stewart I, Buono P, Collins JJ (1998) A modular network for legged locomotion. Physica D 115:56–72
Golubitsky M, Stewart I, Buono P, Collins JJ (1999) Symmetry in locomotor central pattern generators and animal gaits. Nature 401:693–695
Haken H (1977) Synergetics. An introduction. Springer, Berlin
Haken H (1983) Advanced synergetics. Springer, Berlin
Haken H (1996) Principles of brain functioning. Springer, Berlin
Isenhower RW, Kant V, Frank TD, Pinto CMA, Carello C, Turvey MT (2012) Equivalence of human odometry by walk and run is indifferent to self-selected speed. J Mot Behav 44:47–52
Kelso JAS (1995) Dynamic patterns. Bradford/MIT Press, Cambridge
Klatzky RL, Loomis JM, Golledge RG, Cicinelli JG, Doherty S, Pellegrino JW (1990) Acquisition of route and survey knowledge in the absence of vision. J Mot Behav 22:19–43
Lee SJ, Hidler J (2008) Biomechanics of overground vs. treadmill walking in healthy individuals. J Appl Physiol 104:747–755
Mittelstaedt ML, Mittelstaedt H (2001) Idiothetic navigation in humans: estimation of path length. Exp Brain Res 139:318–332
Neath I, Surprenant A (2003) Human memory. Thompson/Wadsworth, Belmont
Park H, Turvey MT (2008) Imperfect symmetry and the elementary coordination law. In: Fuchs A, Jirsa VK (eds) Coordination: neural, behavioral and social dynamics. Springer, Berlin, pp 3–25
Pellecchia GL, Shockley K, Turvey MT (2005) Concurrent cognitive task modulates coordination dynamics. Cogn Sci 29:531–557
Pinto CMA, Golubitsky M (2006) Central pattern generators for bipedal locomotion. J Math Biol 53:474–489
Pinto CMA (2007) Numerical simulations in two CPG models for bipedal locomotion. J Vib Control 13:1487–1503
Riley PO, Dicharry J, Franz J, Croce UD, Wilder RP, Kerrigan DC (2008) A kinematics and kinetic comparison of overground and treadmill running. Med Sci Sport Exer 40:1093
Risken H (1989) The Fokker-Planck equation. Methods of solution and applications. Springer, Berlin
Schöner G, Haken H, Kelso JAS (1986) A stochastic theory of phase transitions in human hand movement. Biol Cybern 53:247–257
Schwartz M (1999) Haptic perception of the distance walked when blindfolded. J Exp Psychol Hum Percept Perform 25:852–865
Séguinot V, Cattet J, Benhamou S (1998) Path integration in dogs. Anim Behav 55:787–797
Seyfarth EA, Barth FG (1972) Compound slit sense organs on the spider leg: mechanoreceptors involved in kinesthetic orientation. J Comp Physiol 78:176–191
Silva PL, Turvey MT (2012) The role of haptic information in shaping coordination dynamics: inertial frame of reference hypothesis. Hum Mov Sci 31:1014–1036
Strogatz SH (1994) Nonlinear dynamics and chaos. Addison-Wesley, Reading
Turvey MT (2007) Action and perception at the level of synergies. Hum Mov Sci 26:657–697
Turvey MT, Romaniak-Gross C, Isenhower RW, Arzamarski R, Harrison S, Carello C (2009) Human odometer is gait-symmetry specific. P Roy Soc Lond B Bio 276:4309–4314
Turvey MT, Harrison SJ, Frank TD, Carello C (2012) Human odometry verifies the symmetry perspective on bipedal gaits. J Exp Psychol Hum Percept Perform 38:1014–1025
Walls ML, Layne JE (2009) Direct evidence for distance measurement via flexible stride integration in the fiddler crab. Curr Biol 19:25–29
Wittlinger M, Wehner R, Wolf H (2006) The ant odometer: stepping on stilts and stumps. Science 312:1965–1967
Wittlinger M, Wehner R, Wolf H (2007) The desert ant odometer: a stride integrator that accounts for stride length and walking speed. J Exp Biol 210:198–207
Wohlgemuth S, Ronacher B, Wehner R (2001) Ant odometry in the third dimension. Nature 411:795–798
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Appendix
Appendix
Details of parameter estimation method and self-consistency test for estimated parameter \(\gamma \)
Let \(x_{M}(t)\) and \(x_{R}(t)\) denote the participant positions from the perspective of the participant during \(M\) and \(R\) phases, that is, distances traveled up to time \(t.\,x_{M}(t)\) and \(x_{R}(t)\) are defined on the intervals [\(0,T_{M}\)] and [\(0,T_{R}\)], respectively, where \(T_{M}\) and \(T_{R}\) denote the durations of the \(M\) and \(R\) phases. This “Appendix” shows how the model parameters were estimated from these trajectories.
The position trajectories \(x_{k}(t)\) for \(k=M,\, R\) involved two components, an oscillatory component (related to the pendulum like rhythmic activity) and a directed, forward motion component. In a first approximation, it was assumed that both components added up linearly to the observed motion. The order parameter model in the text addressed the directed motion component. To remove the oscillatory component the trajectories \(x_{k}(t)\) were subjected to a low-pass filter with filter frequency determined by the number of steps (see e.g., Table 3). Explicitly, the filter frequency was determined as the step frequency (derived from the number of steps and \(T_{k})\) minus an offset of 2 Hz. Subsequently, the low-pass filtered trajectories were numerically differentiated with respect to \(t\) to obtain the velocities \(v_{k}(t)\).
Let \(y_{k}(t_{e})\) denote the position trajectories in event time with \(k =\hbox { M, R}\). Then, from \(t_{e }=t/T_{k}\) for \(k =\hbox { M, R}\) it follows that \(y_{k}(t_{e})=x_{k}(t_{e}\cdot T_{k})\). Let \(u_{k}(t_{e})\) denote the velocity in event time defined as the derivative of \(y_{k}\) with respect to \(t_{e}\). Then, using the chain rule of differentiation, we see that \(u_{k}(t_{e}) = T_{k}\cdot v_{k}(t_{e} \cdot T_{k})\). Consequently, the velocities of M and R phases in event time were calculated using \(u_{k}(t_{e})=T_{k} \cdot v_{k}(t_{e} \cdot T_{k})\) and the velocity trajectories \(v_{k}(t)\) mentioned above. Subsequently, the relative velocity in event time was calculated as the difference \(u(t_{e})=u_{R}(t_{e})-u_{M}(t_{e})\). In order to avoid an inflation of symbols, we replaced in the main text \(u(t_{e})\) by \(v(t_{e})\).
From the Ornstein–Ühlenbeck model defined by Eq. (5), it follows that in the stationary case the expectation value (ensemble average) of \(v(t_{e})\) equals the ratio \(\delta /\gamma \). Therefore, the ratio \(\delta /\gamma \) was estimated from the time-average \(v_{m}\) of \(v(t_{e})\) assuming that ensemble averaging can be approximated by time-averaging (ergodicity assumption). In addition, \(v_{m}\) was subtracted from the trajectory \(v(t_{e})\) and in doing so a centered trajectory with zero mean value was generated. The time-discrete version of the Ornstein–Ühlenbeck with zero mean value corresponds to an autoregressive (AR) model of order 1. Therefore, the model parameters of the Ornstein–Ühlenbeck model, \(\gamma \) and \(Q\), were estimated using the Yule-Walker method (Diggle 1990) for the \(AR-1\) model. Let \(a_{1}\) denote the first autoregressive parameter and VAR denote the variance of the noise term of the \(AR-1\) model. Then, a detailed calculation shows that \(\gamma \) and \(Q\) can be determined from \(a_{1}\) and VAR as follows:
Here \(\varDelta t_{e}\) denotes the single time step of the event time grid obtained from the laboratory time \(t\). To reiterate, the Yule-Walker method yielded the \(AR-1\) parameters \(a_{1}\) and VAR. Subsequently, \(\gamma \) and \(Q\) were calculated from Eq. (6). Having obtained \(\gamma \), the parameter \(\delta \) was calculated from the estimated ratio \(\delta /\gamma \), that is, from \(v_{m}\), according to
As a self-consistency test, the parameter \(\gamma \) was estimated in an alternative way, namely, from the power spectrum of \(v(t_{e})\). In a first step, the power spectrum of a given trajectory \(v(t_{e})\) was calculated. Subsequently, the analytical solution of the power spectrum (Diggle 1990) of an Ornstein–Ühlenbeck process was fitted to the observed spectrum using a nonlinear best-fit method (MATLAB function nonlfit). In doing so, a second estimate for \(\gamma \) was obtained.
In summary, for each pair of \(M\) and \(R\) phases two estimates for the parameter \(\gamma \) were obtained: one estimate from Eq. (6) involving the Yule-Walker method and another one via power spectral analysis. The two scores for \(\gamma \) were compared by a \(t\) test for dependent samples. The \(t\) test was not statistically significant indicating that the parameter estimation method involving the Yule-Walker method for the \(AR-1\) model produced consistent results with the power spectral analysis method tailored to an Ornstein–Ühlenbeck process.
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Abdolvahab, M., Carello, C., Pinto, C. et al. Symmetry and order parameter dynamics of the human odometer. Biol Cybern 109, 63–73 (2015). https://doi.org/10.1007/s00422-014-0627-1
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DOI: https://doi.org/10.1007/s00422-014-0627-1