Context-dependent changes in strength and efficacy of leg coordination mechanisms
Abteilung für Biologische Kybernetik und Theoretische Biologie, Fakultät für Biologie, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany
e-mail: volker.duerr{at}uni-bielefeld.de
Accepted 12 April 2005
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Summary |
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The present study analyses stepping patterns of the stick insect Carausius morosus during straight and curve walking sequences. To infer strength and efficacy of coupling between pairs of sender and receiver legs, the likelihood of the receiver leg being in swing is determined, given a certain delay relative to the time of a swingstance (or stanceswing) transition in the sender leg. This is compared to a corresponding measure for independent, hence uncoupled, step sequences. The difference is defined as coupling strength. The ratio of coupling strength and its theoretical maximum is defined as efficacy.
Irrespective of the coordination rule, coupling strength between ipsilateral leg pairs is at least twice that of contralateral leg pairs, being strongest between ipsilateral hind and middle legs and weakest between contralateral middle legs. Efficacy is highest for inhibitory rule 1, reaching 8495% for ipsilateral and 2965% for contralateral leg pairs. Efficacy of excitatory rules 2 and 3 ranges between 3556% for ipsilateral and 821% for contralateral leg pairs. The behavioural transition from straight to curve walking is associated with context-dependent changes in coupling strength, increasing in both outer leg pairs and decreasing between inner hind and middle leg. Thus, the coordination rules that are thought to underlie many adaptive properties of the walking system, themselves adapt in a context-dependent manner.
Key words: leg coupling, coupling strength, context dependence, stick insect, Carausius morosus
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Introduction |
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The step cycle sequence of each walking leg consists of alternating swing
and stance modes. As the body weight is supported during stance only,
transitions between the support and non-support states immediately affect
static stability. There is a qualitative difference between the two possible
types of state transition: whereas the swingstance transition, i.e.
touch-down of the leg, adds support to the body, the stanceswing
transition, i.e. lift-off of the leg, reduces support and, therefore,
potentially destabilises the animal. Thus, appropriate timing of lift-off is
particularly important to stability. Indeed, behavioural experiments on
different insect species have gathered evidence for three coordination
mechanisms that affect timing of the stanceswing transition during
walking (reviewed by Cruse,
1990). Throughout this study, they will be referred to as rules 1
to 3, sensu Cruse et al.
(1995
), where rule 1
postulates an inhibitory mechanism acting on anterior and contralateral
neighbour legs, rule 2 postulates an excitatory mechanism acting on anterior
and contralateral neighbour legs, and rule 3 postulates an excitatory
mechanism acting on posterior and contralateral neighbour legs.
On the basis of a number of kinematic simulations (Dean,
1991b,
1992b
;
Müller-Wilm et al., 1992
;
Cruse et al., 1995
,
1998
) and robot hardware tests
(Espenschied et al., 1993
;
Pfeiffer et al., 1995
) it is
well established that behaviourally derived coordination rules are sufficient
to generate natural hexapod gaits that are robust against perturbation
(Dean, 1992a
;
Kindermann, 2002
). Moreover,
theoretical analyses (Dean,
1991a
,b
;
Calvitti and Beer, 2000
) and
sensitivity analyses in software (Dean,
1992b
) and hardware tests
(Espenschied et al., 1993
)
have highlighted the importance of the three coordination mechanisms that
affect the timing of stanceswing transitions. In contrast to the
parameter ranges derived from theoretical and engineering studies, coupling
strengths associated with coordination rules 1 to 3 have never been determined
experimentally. Rather, coupling between adjacent legs is typically quantified
by measures of relative coordination
(Holst, 1943
), expressing the
phase lag of the step cycle in a receiver leg within the step cycle of the
sender leg. The resulting phase histograms and corresponding circular
statistics measures quantify the coherence of two rhythmic patterns (eg.
Clarac and Chasserat, 1986
;
Cruse and Knauth, 1989
) but
they cannot be related to any of the proposed leg coordination rules in
particular.
The present study supplies the first empirical values of coupling strengths associated with individual mechanisms. Taking a stochastic view of leg coordination, the strength of each one of three competing coordination mechanisms is assessed by its likelihood of enforcing a stanceswing transition. The likelihood of a receiver leg being in swing mode depending on the time relative to a state transition in the sender leg is determined for each coupled pair of legs, and in a specific way for each one of the three major coordination rules.
The rules under investigation are associated with putative neural
mechanisms, one inhibitory and two excitatory. Wilson
(1966) deduced the existence
of an inhibitory mechanism from observations on insects. The associated rule 1
states that lift-off in a receiver leg is suppressed while the sender leg is
in swing. This rule was also proposed for walking scorpions
(Bowerman, 1975
), suggesting
that it is widespread among arthropods. Careful analysis of irregular stepping
patterns in grasshoppers (Graham,
1978b
) and direct evidence from perturbation experiments on stick
insects (Cruse and Epstein,
1982
) unequivocally demonstrated the existence of rule 1 in
insects. Rules 2 and 3 have been derived from studies on walking stick
insects, based on perturbation experiments
(Cruse and Schwarze, 1988
) and
observations on mechanically uncoupled steps of contralateral neighbours
(Cruse and Knauth, 1989
). Rule
2 states that an excitatory mechanism favours stanceswing transition in
a receiver leg shortly after the sender leg commences stance. Rule 3 refers to
a position-dependent excitatory mechanism that favours swingstance
transition in a receiver leg with increasingly posterior tarsus position of
the sender leg. Similar rules have been described for walking crayfish
(Cruse and Müller, 1985
)
where they mainly affect swingstance transition.
Based on the experimental data of the accompanying paper
(Dürr and Ebeling, 2005),
the present study measures coupling strength and efficacy associated with
coordination rules 13 in all four ipsilateral and three contralateral
leg pairs. Whereas coupling strength supposedly indicates the strength of the
underlying neural mechanism, coupling efficacy expresses the relative effect
of this mechanism on the prevalent gait. Context-dependent modulation of
coupling strength is shown to occur during the behavioural transition from
straight to curve walking. Due to the specificity of the analysis to
individual coupling mechanisms, the results require postulating adaptive
modulation of leg coordination mechanisms and demand changes to current
hexapod walking models. In addition, the relative differences of coupling
strength between leg pairs and coordination rules provide a behavioural
reference for further neurophysiological investigations. Preliminary results
have been published by Dürr
(2002
).
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Material and methods |
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Experimental animals and set-up
Experiments were carried out on eight adult female stick insects of the
species Carausius morosus Brunner. Stick insects walked on the
surface of a hollow Styrofoam sphere that floated on an air cushion. The
sphere was covered by evenly spaced patches of reflective foil, the movement
of which was recorded by an optic tracking system. Animals were tethered to a
balsa wood support above the sphere, restricting any horizontal displacement
relative to the set-up. A balancing system ensured that animals needed to
carry their own weight and adjust their body height. Walking stick insects
rotated the sphere underneath them. The resulting walking path was
reconstructed from the displacement readings of the tracking system. All three
degrees of freedom of horizontal walking could thus be measured: forward and
sideward translation and yaw rotation around the vertical axis. Additionally
to the tracking system, a monochrome CCD video camera was mounted 1.4 m above
the set-up. All walking sequences were videotaped using standard PAL video
equipment.
The tracking system was centred in a motor-driven paper drum that was lined with a black-and-white grating of wavelength 24° (Fig. 1A). The pattern was diffusely illuminated from the outside (contrast of 90%, luminance 60 cd m2) and rotated around the tracking system at 38.1 deg. s1.
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Experimental procedure
Single trials lasted 22.5 s and were divided into three periods of equal
duration. Following a pre-stimulus period of 7.5 s, during which the drum
stood still, the drum was rotated for 15 s. This stimulus period was divided
into an early transition phase and a late steady-state phase. The large-field
visual motion stimulus reliably elicited an optomotor turning response in the
direction of stimulus motion. All analyses presented in this study compare leg
coordination during straight walking in the pre-stimulus period, to that
during curve walking in the late stimulus period.
Direction of stimulus motion was randomised and leftward and rightward turns were pooled, assuming bilaterally symmetric animals. According to the location of the legs relative to the turning direction, they are denoted as inner legs (I1 to I3) and outer legs (O1 to O3) throughout this study (Fig. 1B). In total, 33 out of 321 trials from eight animals were selected according to steadiness of forward walking and maximum diameter of the walked curve (see Fig. 1C).
Step sequences of each leg were recorded manually from digitised video files, using a custom-written program. Only the timing of touch-down and lift-off was considered here, as these transition times are equivalent to the beginning of stance movement, i. e. power stroke, and swing movement, i.e. return stroke, respectively. Representative step sequences are shown in Fig. 1C.
Data analysis and terminology
Coupling strength and efficacy was determined for pairs of step sequences
of sender and receiver legs, where the leg that exerts the effect is called
the `sender' and the leg being influenced is called the `receiver'.
Mathematically, step sequences were treated as functions of a discrete state
variable over time, assuming value 0 during stance and value 1 during swing
movement. State transitions were considered as triggering events, and the
effect of these events upon adjacent legs was quantified as the likelihood of
a receiver leg to be in state 1, given a particular time delay relative to the
transition event in a sender leg, e.g.
P(S=1|tttrans10|),
where S is a state variable and P is probability. According
to their sign, transitions are called `trans01' for the stanceswing
transition and `trans10' for the swingstance transition. The peak
likelihood minus its baseline value is a measure of `coupling strength', where
the `baseline' is calculated by shuffling step sequences from independent,
hence uncoupled, pairs of trials. As expected, the resulting baseline
likelihood is time-invariant. It can be interpreted as the average probability
of the receiver leg to be in swing or, equivalently, as the average ratio of
swing duration over step period. The fraction of step cycles consistent with a
given coordination rule is defined as `coupling efficacy'. Accordingly,
efficacy is equal to (peakbaseline)/(1baseline) for rules 2 and
3 and (baselineminimum)/baseline for rule 1. The rationale underlying
these formulae will be explained in more detail in relation to Figs
2 and
5. Note that if a coordination
rule is violated in a fraction of steps, this does not imply that the
associated neural coordination mechanism is not present. Rather, it suggests
that it is being overridden by other competing mechanisms or noise. Thus,
coupling efficacy indicates how strongly a mechanism affects the gait of the
animal.
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Results |
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Accordingly, gaits may not be a helpful concept for describing leg coordination in all walking arthropods, because often the prevalent gait is discernable for a short period of time only. This situation may be called a free gait, a term frequently used by engineers that design step controllers for legged machines. It expresses the fact that the step cycle of each leg depends on a number of internal and context variables, and that a stable repetitive pattern is not always produced. Fig. 1 shows that a free gait was observed for each one of eight walking stick insects.
The lack of a stable repetitive pattern, however, poses severe problems for
the quantification of leg coordination. As the cycle periods change
continuously, relating the step cycle of a given leg to the step cycle of its
neighbour would be appropriate to quantify the coherence of two step rhythms,
but coherence is a result of several interacting physiological coordination
mechanisms and does not indicate the efficacy of any known mechanism in
particular. Although phase analyses are a powerful descriptive tool, they do
not relate the timing of a step to an observable event or a measurable state
variable and, therefore, provide only indirect information about causal
coordination mechanisms. In order to quantify the coupling strength of a
coordination mechanism, it is desirable to determine the significance of a
given event or state variable for entrainment or reset of a given pattern. For
this reason, the following analysis introduces measures of coupling strength
and efficacy that are directly related to coordination rules derived from
behavioural experiments on walking arthropods (reviewed by
Cruse, 1990).
The role of touch-down as a coordinating event
From behavioural experiments on stick insects, three coordination rules
have been proposed to act between ipsilateral legs. In two of these, rules 1
and 2 sensu Cruse et al.
(1995), the sender leg is
posterior to the receiver leg, i. e. hind legs affect middle legs and middle
legs affect front legs (Fig.
2A). Rule 1 states that stanceswing transition in the
receiver leg is inhibited during swing movement of the sender leg. Rule 2
states that stanceswing transition in the receiver leg is excited for a
brief period after the sender leg begins or resumes stance movement (the
typical case after touch-down). Note that inhibition does not imply prevention
of lift-off, and excitation does not imply that lift-off is invoked. Rather,
both of these rules affect the likelihood of a stanceswing transition
in the receiver leg, depending on the time relative to touch-down of the
sender leg (Fig. 2B). Moreover,
by affecting the likelihood of a stanceswing transition, the rules have
a sustained effect on the likelihood of a leg to be in swing mode. This is
because swing phases typically last for at least 200 ms, and stance phases
last even longer. Accordingly, the likelihood to be in swing mode is zero for
some time before a given stanceswing transition, and one for some time
after the transition.
A useful measure of coupling strength can be obtained by calculating how
reliable the effect of a coordination rule is. For example, coupling strength
of rules 1 and 2 can be determined by looking up each touch-down event in a
sender leg and calculating the fraction of events in which the receiver leg
was in swing mode within some time window before and after the event. The
corresponding mathematical equation is:
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The likelihood minima in Fig. 3, indicating coupling according to rule 1, are very similar during straight and curve walking. As the baseline is elevated during curve walking, coupling strength according to rule 1 increases too. In contrast, there is a clear context-dependent increase of peak likelihood associated with rule 2 in outer leg pairs. Larger peak values are accompanied by a reduced delay relative to touch-down and a moderate rise in baseline. Inner leg pairs undergo different changes during curve walking. There is a significant reduction of the peak likelihood for the rear pair of inner legs, but an increase for the front pair. However, as the baseline of the front pair rises by the same amount as the peak value, there is no change in coupling strength.
In summary, as the stick insects change from straight to curve walking, coupling strength according to rule 1 remains the same (from 0.214 to 0.220) between inner hind and middle leg, but increases in all other ipsilateral leg pairs (straight: 0.146 to 0.231; curve: 0.253 to 0.275). This increase is due to a raised baseline, accompanied by nearly constant likelihood minima. Coupling strength of rule 2 increases in outer legs (0.424 to 0.518 between O3/02, 0. 280 to 0.444 between O2/O1), remains stable between inner middle and front legs (0.308 and 0.279) and decreases between inner hind and middle leg (from 0.391 to 0.232).
Contralateral leg pairs
Rule 2 has been suggested to be active symmetrically between contralateral
leg pairs as well (Cruse and Knauth,
1989), whereas rule 1 is typically claimed to be absent (but see
Dean, 1991b
). Naturally, the
above analysis can be applied to pairs of intrasegmental front, middle and
hind legs (Fig. 4), using each
leg once as a sender leg and once as a receiver leg. Compared to ipsilateral
leg pairs, contralateral coupling is much weaker. The resulting plots of
coupling strength do not always show a pronounced trough related to rule 1 or
peak related to rule 2. Between middle leg pairs, the lack of either peak or
trough suggests the absence of significant contralateral coupling.
For hind leg and front leg pairs, likelihood minima range between 0.06 and 0.152, occurring 0.04 to 0.16 s prior to touch-down of the contralateral leg. Peak likelihood lies between 0.247 and 0.452, occurring 0.2 to 0.44 s after to touch-down of the contralateral leg (Table 1). As baselines range between 0.178 and 0.291 for these leg pairs, coupling strength is in the range of 0.101 to 0.157 for rule 1 and 0.069 to 0.169 for rule 2. This is approximately half the strength of ipsilateral coupling.
When changing from straight to curve walking, peak likelihood undergoes a positive shift in front leg pairs and a negative shift in the hind leg pair O3/I3. In most contralateral leg pairs, the corresponding shift in baseline is of similar magnitude, indicating that contralateral coupling strength of rules 1 and 2 changes little or not at all in a context-dependent manner.
The role of lift-off as a coordinating event
The third leg coordination rule that affects the timing of
stanceswing transitions has been described to couple ipsilateral leg
pairs in a posterior direction (Cruse and
Schwarze, 1988). This coordination rule will be called rule 3
sensu Cruse et al.
(1995
). Like rule 2, it has
been described to act contralaterally as well
(Cruse and Knauth, 1989
). Apart
from its direction of action (Fig.
5A), the major difference to rules 1 and 2 is that it is active
during late stance. Thus, the part of the step cycle in which rule 3 is active
is related to the stanceswing transition, i.e. lift-off, of the sender
leg. Accordingly, coupling strength due to rule 3 must be calculated as the
likelihood of a receiver leg to be in swing mode, depending on the delay
relative to a stanceswing transition
(tttrans01) in a sender leg.
Fig. 5B shows the corresponding
scheme, where an excitatory influence, that occurs during late stance of the
sender leg, raises the likelihood of a stanceswing transition in a
receiver leg. The corresponding equation follows the same rationale as
explained in relation to Fig.
2B. If rule 3 was in effect in each single step cycle, the
likelihood of receiver leg to be in swing would equal one prior to time of
lift-off of the sender leg (ttrans01).
Ipsilateral leg pairs
Fig. 6 shows that coupling
strength associated with rule 3 reaches a peak prior to lift-off, followed by
a significant trough soon after lift-off. Peak values lie within 0.341 and
0.729 and these are reached 0.36 to 0.16 s prior to lift-off
(Table 1).Differences are
mainly due to behavioural context as peak values during straight walking vary
only between 0.514 and 0.594, which is only about half the range of rule 2.
Baselines range between 0.180 and 0.299. Coupling strength minima are
similarly pronounced as those related to rule 1 (compare with
Fig. 3), which could indicate
the presence of an inhibitory effect after lift-off. Note that this would be
equivalent to a posterior action of rule 1, i.e. suppressed lift-off in a
posterior leg during swing mode of an anterior leg.
During curve walking, peak likelihood increases in outer leg pairs (O2/O3, 0.688; O1/O2, 0.729) and decreases in inner leg pairs (I1/I2, 0.504; I2/I3, 0.341). Also, in three of four leg pairs, peaks shift closer to time of lift-off in the sender leg. As none of these changes can be explained by equivalent shifts in baseline, they reflect changes in coupling strength. Whereas coupling strength of rule 3 is between 0.264 and 0.342 during straight walking, it decreases during curve walking to 0.161 or 0.247 in inner leg pairs, and increases to 0.430 in outer leg pairs. Thus, coupling due to rule 3 is context-dependent.
Contralateral leg pairs
When applied to contralateral leg pairs, the analysis reveals similar
results for rule 3 as have been described previously for rule 2. As can be
seen in Fig. 7, contralateral
coupling strength is marked by relatively shallow peaks in hind and front
legs. They range from 0.269 to 0.492 and occur 0.48 to 0.2 s prior to lift-off
of the sender leg. No clear peak is discernable for middle legs, indicating
that they are not coupled according to rule 3. Finally, during curve walking
the sizes of the observed changes in peak likelihood are approximately the
same as the shift of the baseline, indicating little or no context
dependence.
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Discussion |
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Measuring coupling strength and efficacy
The present study applies a stochastic measure to quantify the strength of
a given leg coordination mechanism. Moreover, it uses transitions between
swing and stance mode as reference events, rendering the analysis
time-dependent. Both of these properties, its stochastic nature and its time
dependence, are somewhat different from the properties of measures previously
applied. Typically, coordination of adjacent legs is described by phase
histograms (e.g. Holst, 1943),
or circular statistics measures thereof (e.g.
Cruse and Knauth, 1989
;
Clarac and Chasserat, 1986
).
Using the terminology of coupled sender and receiver legs, phase histograms
relate the occurrence of a step transition in a receiver leg to the normalised
step cycle period of the sender leg. Due to the variability of both the step
cycle period of the sender leg and the relative timing of transitions in the
receiver leg, the width of the histogram peak cannot be related to an event in
the step cycle of either leg. However, since coupling mechanisms trigger
discrete transitions between antagonistic actions, they must either depend on
a discrete triggering event, e.g. an entraining reference clock signal, or on
thresholds of one or more physiological parameters, e.g. the spike thresholds
in a motoneuron pool. Phase histograms reveal dependence on discrete events or
thresholds only if either the rhythm of the sender leg is very regular, or if
the peak of the histogram is very narrow, i.e. shows very little variability.
In the first case, the result indicates a clear temporal relation to the state
transitions in the sender leg. In the second case, the result indicates the
presence of a threshold that is reached at a certain phase of the sender's
step cycle.
In contrast to phase histograms, the stochastic measure applied here is always time-locked to a behavioural event: lift-off or touch-down of a leg. Thus, coupling strength can be interpreted as the fraction of transition events that is attributable to a given coordination rule. Since leg mechanosensors encode changes in load and ground contact, information about transition events is supplied to the nervous system. Assuming that three distinct neural mechanisms give rise to coordination rules 1 to 3, coupling strength indicates the strength by which these mechanisms act to shift the likelihood of a transition event away from baseline. Therefore, this measure can serve as a quantitative behavioural reference value for electrophysiological studies on reduced preparations.
A further advantage of this approach is that each rule makes clear-cut predictions and, therefore, allows calculation of its efficacy. Efficacy, indicated by the arrow widths in Fig. 8, is defined as the fraction of step cycles that is consistent with a given rule. The more consistent a step pattern is with a given coordination rule, the more important must be the associated mechanism in generating the stepping pattern. Irrespective of the nature of the underlying neural mechanism, efficacy can be compared between types of rules, contralateral and ipsilateral leg pairs, and behavioural contexts.
Evidence for the three coordination rules is based on statistics of
stepping patterns (Wilson,
1966; Graham,
1978a
), disturbance of single step cycles (e.g.
Cruse and Epstein, 1982
;
Cruse and Schwarze, 1988
),
mechanical uncoupling of left and right legs (e.g.
Cruse and Knauth, 1989
) or
comparison of model simulations with natural step sequences (e.g. Graham,
1978a
,b
).
Rules 1 and 2 are related in the sense that both have been postulated to be
time-dependent with respect to the swingstance transition of the sender
leg. Rule 1 acts whenever the sender leg is in swing mode
(Wilson, 1966
;
Graham, 1978a
;
Cruse and Epstein, 1982
). Rule
2 acts for a brief period after the sender leg begins or resumes stance
(Cruse and Schwarze, 1988
;
Cruse and Knauth, 1989
).
Although the excitatory effect of rule 2 immediately follows the inhibition
mediated by rule 2, there is evidence that rule 2 is not just due to a
post-inhibitory rebound effect: the onset of a stance can raise the
probability of lift-off in the receiver leg if stance movement is resumed
after an experimental interruption (Cruse
and Schwarze, 1988
). Other than rules 1 and 2, rule 3 is not
time-dependent but position-dependent
(Dean and Wendler, 1983
;
Cruse and Schwarze, 1988
).
Moreover, it is thought to increase in strength with increasingly caudal
tarsus position of the sender leg. Indeed, Ludwar et al.
(2005
) have recently found in
a stick insect preparation with a single walking front leg that the spike rate
of a mesothoracic levator neuron increases during the stance phase of the
ipsilateral front leg. This may be indicative of a posterior directed,
position-dependent synaptic drive that increasingly facilitates lift-off in a
middle leg. By measuring coupling strength associated with rule 3 as depending
on time rather than on position of lift-off, the magnitude obtained is a
weighted average within the range of lift-off positions. Assuming that the
strength of the underlying mechanism linearly increases with position (as done
by Müller-Wilm et al.,
1992
; Espenschied et al.,
1993
; Calvitti and Beer,
2000
; but see Dean,
1991b
), coupling strength as determined here equals the strength
at the average lift-off position. An additional velocity dependence of rule 3,
as suggested by Dean (1991b
),
would also have some effect on the coupling strength measure used here.
However, because the likelihood of the receiver leg being in swing mode
changes for the duration of an entire swing movement, the overlap would
compensate for much of the velocity-dependence of the shift, changing location
but not magnitude of the peak.
Differences between leg pairs
In the analyses shown in Figs
2,
3,
4,
5,
6,
7, a leg pair is considered
coupled according to a given coordination rule, if coupling-related likelihood
is sufficiently different from the value expected for uncoupled step patterns.
With the threshold chosen in this study, all adjacent leg pairs are coupled
according to each one of the three coordination rules, the only exception
being the contralateral pair of middle legs (broken arrows in
Fig. 8). Coupling strength of
ipsilateral leg pairs is about twice that of contralateral leg pairs. Coupling
due to rule 2 also differs considerably between ipsilateral leg pairs, being
stronger in the rear pair.
Comparing the three rules, coupling strength due to rule 2 is slightly
larger than that due to rule 3, and yet again larger than that due to rule 1.
This ranking is of functional importance where a receiver leg is influenced by
several mechanisms and/or by several sender legs. This is particularly so if
all mechanisms act on the same variable, which is a feature common to all leg
coordination schemes suggested for insects later than that proposed by Dean
(1991b). However, ranking of
coordination rules looks different when considering the efficacy of each rule
(arrow width in Fig. 8) rather
than coupling strength. In terms of efficacy, rule 1 is about twice as strong
as rules 2 and 3. In other words, 100% efficacy of rule 1 in suppressing
swing-movements requires less coupling strength than does 100% efficacy of
rule 2 in inducing a swing movement. This is because any leg is less likely to
be in swing mode than in stance mode, as is reflected by the baselines in Figs
2,
3,
4,
5,
6,
7, which are always asymmetric
with respect to the two extremes. Whereas the behavioural relevance of a
coordination rule is best reflected by its efficacy, the strength of an
underlying physiological mechanism is best reflected by coupling strength. For
instance, a neural signal from a sender leg may directly affect the likelihood
of lift-off in a receiver leg, but the same signal will result in different
efficacy, depending on the ratio of protraction and retraction time.
It is important to note that coupling strength as measured in this study
refers to regularities in behaviour rather than to an identified neural
mechanism. Because several neural mechanisms may cause inter-leg coupling
either directly or indirectly (see below), the values given in
Fig. 8 must be considered
relative measures that can be interpreted as a summed effect of various neural
mechanisms contributing to a given coordination rule. An example where an
indirect effect of local sensory feedback mechanism contributes to coupling
strength is rule 1. Contralateral coupling due to rule 1 has often been ruled
out because contralateral legs may swing simultaneously at times, although in
general they step alternately. The low coupling strength reported in Figs
5 and
8 reflects this ratio. To date,
it cannot be excluded that neural inhibitory coupling of contralateral legs
exists, since bilateral swing movements could occur if weak contralateral
coupling was overridden by a stronger ipsilateral mechanism, e.g. if rule 3
tended to enforce lift-off in a hind leg while rule 1 tended to suppress it.
On the other hand, there are local sensory feedback signals that affect the
likelihood of stanceswing transitions in addition to coordinating
signals from adjacent legs. For instance, increased load
(Cruse, 1985) and increasingly
caudal tarsus position (Bässler,
1977
) have competing effects on stanceswing transition. In
particular, a load-dependent inhibition of lift-off can create the same effect
as a contralateral rule 1: during swing movement of a leg, the contralateral
leg in stance takes on an additional load. If this load increase was
sufficient to suppress lift-off, the effect of rule 1 would be caused by
mechanical dependencies rather than neural connections. This effect has been
documented in curve-walking stick insects
(Jander, 1985
). Similarly, the
strong contralateral coupling according to rule 1 between hind legs in walking
cats (Cruse and Warnecke,
1992
) is probably mostly due to unilateral load-dependent feedback
from Golgi tendon organs that prevent flexion and, therefore, lift-off
(reviewed by Pearson, 1995
).
In stick insects, the centre of gravity lies behind the hind leg coxae, so the
swing movements of a hind leg cause increased loading of the ipsilateral
middle leg, though to a lesser extent than of the contralateral hind leg.
Given that coupling strength due to rule 1 between ipsilateral leg pairs is
twice that between hind legs, this strong effect could only be caused by a
load reflex alone if the gain was much higher in middle legs than in hind
legs. Similar considerations lead to the conclusion that coupling strength
associated with rule 2 is partly due to unloading of a leg as a result of
touch-down and corresponding load-sharing by an adjacent leg.
To date, sensitivity studies on the relative contribution of the three
coordination rules to hexapod walking have been carried out in software
(Dean, 1992b;
Calvitti and Beer, 2000
) and
hardware models (Espenschied et al.,
1993
). All of these studies agree in that the three coordination
rules are redundant in that each one of them can be deleted without failure of
coordinated walking. The solutions preferred by these studies
(Table 2) differ mainly in
coupling strength according to rule 1, velocity dependence of rule 2 and an
inhibitory branch of the rule 3 characteristic as used by Dean
(1991b
). Comparing the
relative differences among rules between the present study and literature
values for straight walking (Table
2), the present results agree best with the settings used by
Espenschied et al. (1993
).
What is different according to the present results is the lower coupling
strength of ipsilateral rule 1 and contralateral rule 3. It is interesting to
note that the results of Espenschied et al.
(1993
) indicate that overall
stability is most sensitive to coupling strength associated with rule 3.
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Context dependence and functional significance
In contrast to all models of hexapod leg coordination, coupling strength
differs not only between the three rules but also between ipsilateral leg
pairs and between the two behavioural contexts. Curve walking naturally is an
asymmetrical locomotor behaviour, as legs on the inner side of the curve have
to produce different ground reaction forces than legs on the outer side of the
curve. Although this asymmetry in force need not necessarily be overt in
kinematic asymmetries of the leg movements, it is always accompanied by
changes in stance direction and other kinematic variables (see the
acccompanying paper, Dürr and
Ebeling, 2005, and references therein).
Fig. 8 illustrates the changes
in leg coupling strength and efficacy as determined for equal periods of
straight and curve walking. The four ipsilateral leg pairs stand out as being
subject to context-dependent changes. Associated with the transition from
straight to curve walking, rule 1 becomes stronger in both front/middle leg
pairs, rules 2 and 3 become stronger for outer leg pairs and weaker between
inner hind and middle leg (coloured arrows in
Fig. 8). Whether or not this
asymmetry in coupling alone is sufficient to give rise to a change in walking
direction will have to be tested in forthcoming simulation experiments.
Functionally, increased coupling strength between outer legs is likely to
enhance cooperative effects on yaw rotation by coherent movement along the
circular outer stance trajectory
(Dürr and Ebeling, 2005
).
Conversely, uncoupling of the inner hind leg allows its tarsus to remain in
stance for long periods and thus become the pivot around which the animal
turns (see also Jander, 1985
).
Nevertheless, modelling studies have shown that coupling strength does not
have to change to allow curve walking
(Cruse et al., 1998
;
Kindermann, 2002
) or to change
walking speed (Müller-Wilm et al.,
1992
), results that have contributed substantially to the view
that coordination rules 1 to 3 account for adaptive features of the locomotor
system. Here I report a change in coupling strength that is not accompanied by
a significant change in forward velocity, as shown in fig. 3 in the
acccompanying paper (Dürr and
Ebeling, 2005
). The present results show that coupling strength
and efficacy depend on behavioural context, so the underlying coordination
mechanisms must be subject to modulation. Therefore, the adaptive properties
of the locomotor system, which are due to these coordination mechanisms,
themselves adapt.
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