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First published online June 6, 2005
Journal of Experimental Biology 208, 2253-2267 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01638

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Context-dependent changes in strength and efficacy of leg coordination mechanisms

Volker Dürr

Abteilung für Biologische Kybernetik und Theoretische Biologie, Fakultät für Biologie, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany

View larger version (47K): [in a new window]   Fig. 1. Optomotor-induced curve walking behaviour. (A) Stick insects walked on the apex of a light-weight sphere that floated on an air cushion. Animals were tethered to a support that required them to carry their own weight by adjustment of body height, but restricted horizontal translation and rotation. A vertical stripe pattern was rotated around the animal, reliably eliciting an optomotor turning response in the direction of stimulus motion. (B) Walking legs are denoted as outer legs (O1 to O3) and inner legs (I1 to I3), depending on turning direction (broken arrow). For example, in clockwise rotations left legs are outer legs and right legs are inner legs. (Ci–Cviii) Examples of walking paths (left) and corresponding gait patterns (right) of each one of the eight animals used in this study. Walking paths: Head position (circles) and orientation of the body axis (line segments) are indicated for every 20th record of the tracking system. Open and filled symbols indicate the pre-stimulus period (first 7.5 s) and stimulus period (last 15 s), respectively. Width of the scale grid, 10 cm. Path curvature is always such that a loop in the path remains within one field of the grid. Gait patterns: Stance phases (power strokes) of each leg are indicated by a row of black bars. Rows from top to bottom show left front, middle and hind leg, followed by right front, middle and hind legs, respectively. Time runs from left to right, with vertical lines spaced by 7.5 s, indicating the pre-stimulus, transition and curve-walk period of each trial. The bold vertical line marks stimulus onset. Gait patterns are sometimes time-varying within single stimulus periods and vary strongly between animals, even if the walked path is fairly similar (e.g. compare Ci, Cvi and Cvii). Red diagonal lines highlight back-to-front waves of step cycles during tetrapod-like gait, blue vertical lines highlight in-phase step cycles of ipsilateral front and hind legs during tripod-like gait.

View larger version (21K): [in a new window]   Fig. 2. Leg coordination rules 1 and 2. (A) Leg coordination rules 1 and 2, sensu Cruse et al. (1995), act ipsilaterally in an anterior direction and contralaterally between intrasegmental leg pairs (arrows point from sender to receiver leg). Rule 1 supposedly does not act between middle and front legs (broken arrows). Legs are labelled according to the standardised clockwise turning direction. (B) Quantification of coupling strength associated with rules 1 and 2. Step cycle timing of a leg is considered a sequence of alternating, mutually exclusive states S over time t, assuming value 0 for stance and value 1 for swing. In an ipsilateral leg pair, rule 1 inhibits stance–swing transition in the anterior leg (dotted line, receiver leg) whenever the posterior leg (broken line, sender leg) is in state 1 (large grey arrows). In the same leg pair, rule 2 excites stance–swing transition in the anterior leg soon after swing–stance transition of the posterior leg. Thus, if rules 1 and 2 were effective, the receiver leg should undergo a stance–swing transition at the time of a swing–stance transition of the sender leg (trans10). Coupling strength is calculated by summing the state of the receiver leg for a given time bin (t–ttrans10) within a time window (horizontal arrows between stops) for each one of N steps belonging to the same stimulus period. Division by N gives the likelihood of the receiver leg to be in state 1, given a particular time delay relative to the swing–stance transition in the sender leg. If coupling according to rules 1 and 2 is strong, values are expected to be close to zero before ttrans10 (rule 1) and close to unity after ttrans10 (rule 2).

View larger version (25K): [in a new window]   Fig. 3. Coupling strength of rules 1 and 2 between ipsilateral legs. Likelihood of protraction in anterior legs (receiver legs) relative to ttrans10 (see Fig. 2), i.e. time of touch-down in posterior legs (sender legs). Coupling strength is plotted for the straight walking sequence of the pre-stimulus period (solid lines) and the curve walking sequence of the second half of the stimulus period (dotted lines). Error bars indicate 95% confidence intervals. Inserts indicate the leg pair concerned (see Fig. 1B). Horizontal line segments marked with arrowheads indicate the baseline, i.e. the level expected without any coordinating influences (solid arrowheads: straight walking; open arrowheads: curve walking). The latter was evaluated from step sequence pairs taken from independent trials. Rule 1 is always acting strongly, as values prior to touch-down are close to zero and significantly below the baseline. Values are similar for straight and curve walking. Coupling strength associated with rule 2 changes with the behavioural context of the animal. During curve walking it increases in outer legs and decreases or remains similar in inner legs.

View larger version (29K): [in a new window]   Fig. 4. Coupling strength of rules 1 and 2 between contralateral legs. Likelihood of protraction in receiver legs of one side relative to time to of touch-down (t–ttrans10) in sender legs of the other side. Same plot details as in Fig. 3. (Top) Outer legs are sender legs, inner legs are receiver legs; (bottom) inner legs are sender legs, outer legs are receiver legs. For middle legs, likelihood values deviate little from the baseline, indicating complete lack of contralateral coupling. Coupling strength of rules 1 and 2 between intrasegmental front and hind legs is a lot weaker than for ipsilateral leg pairs. During curve walking the observed changes are in the same range as the shift of the baseline, indicating little or no context dependence.

View larger version (21K): [in a new window]   Fig. 5. Leg coordination rule 3. (A) Leg coordination rule 3, sensu Cruse et al. (1995), acts ipsilaterally in a posterior direction and contralaterally between intrasegmental leg pairs (arrows point from sender to receiver leg). Legs are labelled according to the standardised clockwise turning direction. (B) Quantification of coupling strength associated with rule 3. Step cycle sequences of state S over time t for an ipsilateral leg pair, as in Fig. 2. Rule 3 excites stance–swing transition in the posterior leg (broken line, receiver leg) if the anterior leg (broken line, sender leg) is close to a stance–swing transition. Thus, if rule 3 is in effect, the receiver leg should undergo a stance–swing transition prior to a stance–swing transition of the sender leg (trans01). The coupling strength of this rule is calculated by summing the state of the receiver leg for a given time bin (t–ttrans01) within a time window (horizontal arrows between stops) for each one of N steps belonging to the same stimulus period. Division by N gives the likelihood of the receiver leg to be in state 1, given a particular time delay relative to the stance–swing transition in the sender leg. If coupling according to rule 3 was strong, values would be expected to be close to unity before ttrans01.

View larger version (26K): [in a new window]   Fig. 6. Coupling strength of rule 3 between ipsilateral legs. Likelihood of protraction in posterior legs (receiver legs) relative to ttrans01, i.e. time to of lift-off, in anterior legs (sender legs, see Fig. 5). Coupling strength is evaluated for the straight walking sequence of the pre-stimulus period (solid lines, baseline marked by solid arrowhead) and the curve walking sequence of the late stimulus period (dotted lines, baseline marked by open arrowhead). Error bars indicate 95% confidence intervals. Inserts indicate the leg pair concerned (see Fig. 5B). Coupling strength associated with rule 3 is indicated by the peak likelihood of a receiver leg to be in swing mode prior to lift-off of the sender leg. Coupling strength changes with the behavioural context of the animal. During curve walking it increases in outer legs and decreases or remains similar in inner legs.

View larger version (29K): [in a new window]   Fig. 7. Coupling strength of rule 3 between contralateral legs. Likelihood of protraction in receiver legs of one side relative to time of lift-off (ttrans10) in sender legs of the other side. Same plot details as in Fig. 6. (Top) Outer legs are sender legs, inner legs are receiver legs; (bottom) inner legs are sender legs, outer legs are receiver legs. Peak likelihood was close to the expected baseline for the middle leg pair, indicating complete lack of contralateral coupling. In front and hind legs, coupling strength associated with rule 3 is much weaker than for ipsilateral leg pairs. During curve walking, the observed changes are in the same range as the shift of the baseline, indicating little or no context dependence.

View larger version (38K): [in a new window]   Fig. 8. Local and context-dependent differences in leg coupling strength. Coupling strength (numbers) and efficacy (percentages) of the three main coordination rules (sensu Cruse et al., 1995) known to be present in the walking stick insect. (A–C) Rules 1–3, respectively, for straight walking. L, left; R, right leg. (D–F) Rules 1–3, respectively, for curve walking. O, outer; I, inner leg. Arrows point from sender to receiver legs. Arrow size is scaled to the efficacy of the rule. Coupling strength is the difference between the likelihood maxima (rules 2 and 3) or minima (rule 1) and the baseline. Efficacy indicates the percentage of the maximum coupling strength possible, i.e. the situation if a rule held in each single step cycle. This is equivalent to the fraction of step cycles by which a given coordination rule increased (rules 2 and 3) or decreased (rule 1) the likelihood of protraction. Coloured arrows mark coordination rules for which the coupling strength changed significantly (red, increase in strength; blue, decrease in strength). Coordination strength and efficacy vary strongly between rules and between leg pairs. During curve walking, coordination rules undergo a context-dependent change in strength and efficacy. Rule 1 becomes more effective between ipsilateral front/middle leg pairs. Efficacy of rules 2 and 3 increases between outer leg pairs and decreases between inner hind and middle leg.