Symmetry and order parameter dynamics of the human odometer

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.

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:

$$\begin{aligned} \gamma (\textit{estim})&= \frac{1-a_1 (\textit{estim})}{\Delta t_e}\nonumber \\ Q(\textit{estim})&= \frac{\textit{VAR}}{2\Delta t_e} \end{aligned}$$

(6)

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

$$\begin{aligned} \delta (\textit{estim})=v_m \cdot \gamma (\textit{estim}) \end{aligned}$$

(7)

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.