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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In tasks with two categorically distinct behavioral possibilities a person beginning with one option will typically switch to the other at a higher value of a control parameter in an ascending (increasing) sequence than in a descending (decreasing) sequence. For example, the switch from walking to running on an accelerating treadmill occurs at a higher speed than the switch from running to walking on a decelerating treadmill. The reported research posed the question of whether this variant of behavioral hysteresis was affected by concurrent cognitive activity. Participants walked or ran on a treadmill with a constant acceleration or deceleration while counting backwards by sevens or ones, or not counting. The degree of hysteresis, the difference between walk-to-run and run-to-walk transition speeds, increased with cognitive difficulty. Specifically, the increased hysteresis was shown to be due to lower run-to-walk transition speeds for the more difficult concurrent cognitive tasks. These results support the hypothesis that cognitive load occupies attentional resources that contribute to triggering human gait transitions.

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