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INTRODUCTION |
A fundamental goal of
neuroscience is to understand the neural basis of behavior. A central
component of this goal is to understand the genesis and
control of motor activity. During the last 50 years, tremendous strides
have been made, most notably in particularly advantageous, primarily
invertebrate, model systems, in describing the neural mechanisms that
generate the motorneuron firing patterns that induce movement. During
the same period, tremendous strides have also been made in describing
muscle excitation:contraction coupling and muscle input:output
characteristics. This work has shown that muscles are highly nonlinear
and that in general, movement cannot be predicted by linear
extrapolation from motorneuron activity. Unfortunately the work on the
neural mechanisms that generate motorneuron patterned activity, and the
work on muscle contraction characteristics, have generally occurred in
different preparations, and hence in no system do we have a complete
understanding of the neural genesis of motor behavior. This is even
more unfortunate because, since neural networks evolved to create
behavior, it is possible that certain aspects of these networks exist
to compensate for, or take advantage of, muscle nonlinearities.
We have been attempting to remedy this situation in one of
the best understood of all neural networks, the pyloric neural network
of the lobster (Panulirus interruptus) stomatogastric system, by studying the input:output characteristics of the muscles the
network innervates (Ellis et al. 1996
; Koehnle et
al. 1997
; Morris and Hooper 1997
, 1998a
,b
;
Morris et al. 2000
). This work has shown that these
muscles are very slow (relaxation time constants in the 3- to 5-s
range). The pyloric network produces a neural output pattern in which
the motorneurons fire 100-250 ms bursts every 0.5-2 s, and
consequently the pyloric muscle contractions each motorneuron burst
induces cannot fully relax during the subsequent interburst interval.
The contraction induced by each motorneuron burst therefore
"builds" on the relaxation phase of the contraction induced by the
preceding motorneuron burst. The response to repeated motorneuron
bursts is thus a staircasing in which the contraction each burst
induces (the phasic component of the contraction) is added
to a temporally summated background contraction (the tonic component of the contraction) that is a function of system history.
Temporal summation occurs whenever a repetitive input activates a
response whose relaxation is slow in comparison with the input's
repeat period. Temporal summation is found not only in a variety of
rhythmically driven slow muscles (Atwood 1973
;
Carrier 1989
; Ellis et al. 1996
;
Hall and Lloyd 1990
; Hetherington and Lombard
1983
; Koehnle et al. 1997
; Mason and
Kristan 1982
; McPherson and Blankenship 1992
;
Morris and Hooper 1998a
; Morris et al.
2000
), but also in a variety of neurobiological processes
including changes in intracellular calcium concentration and enzyme
activity in response to synaptic input (Blinks et al.
1978
; De Koninck and Schulman 1998
;
Hanson et al. 1994
; Putney 1998
;
Rome et al. 1996
), summation of postsynaptic potentials
in neurons and muscle fibers, and fusion of muscle twitches into tetani.
An important aspect of many temporally summating systems is that system
response often reaches stable state at amplitudes less than the maximal
(saturated) response of the system
e.g., in response to rhythmic
neural activity, at steady state most slow muscles reach maximum
contraction amplitudes that are less than the largest contraction the
muscle can produce. To achieve steady state at sub-maximal levels, some
characteristics of the phasic contractions must change during the
rhythm's initial cycles so that the increase in phasic contraction
amplitude that each motorneuron burst induces and the decrease in
phasic contraction amplitude that occurs during each interburst
interval become equal, and thus the temporally summated tonic
contraction ceases to change. Prediction of motor output from neural
input critically depends on these changes in phasic contraction
characteristics, and these changes have seldom been examined in detail.
We have therefore investigated how contraction characteristics change
as isotonic contractions of the slow, nontwitch pyloric dorsal dilator
muscle of the lobster stabilize in response to rhythmic neural input.
Both the amplitude and the temporal characteristics of the phasic
contraction change to allow steady state to be attained. We have
developed a method to quantitatively compare these changes, and report
that at different cycle periods and duty cycles the muscle uses
different mechanisms to achieve steady state. A preliminary report of
these data has appeared in abstract form (Morris and Hooper
1998b
).
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METHODS |
Spiny lobsters (500-1,000 g) were obtained from Don Tomlinson
Commercial Fishing (San Diego, CA) and maintained in aquaria with
chilled (12°C) circulating artificial seawater. Stomachs were
dissected from the animals in the standard manner (Selverston et
al. 1976
) except that the hypodermal origin of the dorsal
dilator (cpv1b) muscle pair was preserved. Care was taken to ensure
that digestive juices never contacted the muscles and that the muscles were never stretched. Preparations were continuously superfused with
chilled (12-15°C), oxygenated Panulirus saline with 40 mM glucose. The data shown here are from four experiments.
The dorsal dilator (cpv1b) muscles are a bilaterally symmetric muscle
pair inserting on ossicles XXXI on the medial dorsal surface of the
pylorus and originating on the dorsal carapace (Fig.
1). They are innervated by the two
pyloric dilator (PD) neurons (Maynard and Dando 1974
),
which travel to the muscles through the dorsal ventricular (dvn),
lateral ventricular (lvn), and dorsal lateral ventricular (dlvn) and/or
gastropyloric (gpn) nerves. Contractions were induced by stimulation of
the lvn or the pyloric dilator (pdn) nerve (which also contains PD
neuron axons) after the dvn was cut to prevent spontaneous pyloric
network activity from reaching the muscle. Each experiment lasted
between 24 and 36 h. Muscle stability during this time was
assessed by occasionally delivering a standard pulse train to the
muscle and observing whether the muscle response was the same as at
earlier times; in all cases reported here, muscle response properties remained constant throughout the experiment.

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Fig. 1.
Experimental preparation. The dorsal dilator muscles insert on ossicles
XXXI of the pyloric stomach and originate on the dorsal carapace. They
are innervated by the pyloric dilator (PD) neurons of the
stomatogastric ganglion (STG). The PD neuron axons travel to the
muscles via the dorsal ventricular (dvn), lateral ventricular (lvn),
and dorsal lateral ventricular (dlvn) and/or gastropyloric (gpn) nerves
and are also present in the pyloric dilator (pdn) nerves. Muscle
contractions were induced by stimulation of the lvn or pdn after the
dvn had been cut to prevent spontaneous PD neuron activity from
reaching the muscle.
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All electronics were standard. Nerves were stimulated using
polyethylene suction electrodes. Stimulation voltages were increased until maximum muscle contraction amplitudes were achieved, and hence
presumably the axons of both PD neurons were being stimulated. Contractions were measured by attaching a Harvard Apparatus 60-3000 isotonic transducer to a wire hooked through the hypodermis between the
cpv1b muscle pair. Rest muscle length was maintained at approximately physiological levels. Muscle loading was determined by observing contractions elicited by nerve stimulation with a single burst of
shocks using physiologically relevant parameters. We consistently found
that loads that produced the largest muscle contractions were often
insufficient to return the muscle to its rest length. Muscle load was
therefore adjusted to achieve the maximum contraction amplitude
consistent with full muscle relaxation subsequent to the stimulation. A
support bar was then placed under the end of the transducer arm to
prevent muscle overstretching between contraction trains. Transducer
output was amplified 20- to 50-fold by a Tektronix AM502 differential
amplifier. Contraction characteristics were measured using Spike II
(Cambridge Electronics Design) and Kaleidagraph (Synergy Software)
after transfer (Cambridge Electronics Design 1401 laboratory interface)
to a Gateway 2000 P5. Statistics were performed with Kaleidagraph or
SPSS software. Although the full range of stimulation paradigms were
attempted with all muscles, in some cases, data from only three of the
muscles were usable. We were therefore unable to use paired sample
t-tests in all cases. In Fig. 6 [comparison of single (1st)
and stable contractions], all data were paired, and paired sample
t-tests were used. Alternatively, in the comparisons
between different data points in Fig. 9, the data are not necessarily
paired, and for these data, an independent sample t-test,
with equal variances not assumed, was used. A P < 0.05 significance level was used throughout. Figure 3 was made using a model
developed with Stella II (High Performance Systems) software.
Much of the analysis reported here required linear approximation of the
rhythmic and single contraction profiles (Fig.
2). As a first step in this process, we
observed the muscle relaxations after the rhythmic stimulations to
determine the relaxation dynamics from the contraction amplitude
induced by each stimulation paradigm. Figure 2A shows 10 examples of 1.5 s of these relaxations drawn from the data set so
as to span the entire contraction amplitude range. In all cases the
relaxation phases during this period would be well captured by a linear
fit. Since in all our stimulation paradigms intercontraction interval
is <1.5 s, we therefore approximated contraction relaxation with a
straight line.

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Fig. 2.
Linearization of dorsal dilator muscle contractions. A:
1.5 s of relaxation of temporally summated muscle contractions
drawn from the data set so as to span entire range of observed
contraction amplitudes. In all cases a linear fit would well capture
the data. B: explanation of linearization procedure for
phasic (top) and single (bottom)
contractions. See text for details. Filled rectangles under traces
indicate the durations of the motor neuron bursts used to induce the
contractions.
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The majority of the rise phase is also well fit with a straight line
(Fig. 2B). How to define the beginning and end of the broad,
relatively flat region at the peak of the contraction (the plateau
phase) was more difficult. Trial and error showed that the following
procedure (Fig. 2B, top) was robust, appeared to well capture the changes in contraction shape that occurred in the
various trains, and did not induce gross distortions in contraction amplitude. Lines were drawn to approximate the rise and relaxation phases of the contractions (solid sloping lines). The slopes of the
rise and relaxation lines were determined, and new lines with slopes
one-third of these lines were drawn (dashed sloping lines marked
"relax 1/3" and "rise 1/3"). These lines were moved to the part
of the contraction with a matching slope, and horizontal lines were
drawn from the middle of the region of overlap between the contraction
and the one third slope lines on both the rise and relaxation sides
(horizontal dashed lines marked "relax horiz" and "rise
horiz"). A new horizontal line (top solid horizontal line) was drawn
and centered vertically between the rise and relaxation horizontal
lines. This line and the original rise and relaxation lines were
extended to meet, and the length of the resulting horizontal line was
identified as plateau duration. A line whose length equaled cycle
period was then drawn, and it was moved, and the original rise and
relaxation lines extended, until all three lines touched (bottom
horizontal line). This linearized contraction was then used to measure
rise, plateau, and relaxation durations, rise and relaxation slope, and
contraction amplitude.
This procedure had to be modified for contractions induced by a single
burst because in these contractions the relaxation phase is not cut
short by a subsequent contraction (Fig. 2B,
bottom), and hence it is unclear where on the relaxation
phase to place the original line. For these contractions the rise line,
the one-third slope rise line, and the rise horizontal line were all
determined as in the preceding text. The intersection of the rise
horizontal line and the relaxation phase of the contraction was then
identified as the approximate time of the plateau end, and the slope of
the contraction between this point and the end of the relaxation
duration present in the corresponding rhythmic contractions (vertical
lines with arrows) was used to draw the relaxation line in the bottom trace. The one-third relaxation slope and relaxation horizontal lines,
and the plateau line, were then determined as usual. Contraction duration was determined by drawing a vertical line at the beginning of
the contraction, extending the rise line to meet it, and then extending
a horizontal line from this intersection to meet the relaxation line;
contraction characteristics were again measured from the resulting trapezoid.
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RESULTS |
The top traces with rectangles in Fig.
3, A and B, show
the motor nerve stimulations that were used to induce the dorsal
dilator muscle contractions shown in the graphs below the traces. In
Fig. 3A a 2-s cycle period, 60-Hz intraburst spike
frequency, 0.25 duty cycle stimulation paradigm was used. Duty cycle
("DC" on figure) equals burst duration ("BD") divided by period
("Per"), and so in this example each rectangle in the upper line
represents a 500-ms duration, 31 spike shock burst. In all data
reported here, a 60-Hz intraburst spike frequency, which is
physiological for the motor neurons that innervate the dorsal dilator
muscles, was used. Figure 3B shows the muscle contractions
induced by a stimulation paradigm with the same intraburst spike
frequency and duty cycle, but a cycle period of 0.5 s.

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Fig. 3.
Comparison of contraction stabilization at 2- and 0.5-s cycle periods
and constant duty cycle (0.25) and interburst spike frequency (60 Hz)
of the dorsal dilator muscle (A and B)
and a simple muscle model (C and D).
A and B, top: schematic of
nerve stimulation; in A, each rectangle represents a
500-ms, 31-spike shock burst; in B, each rectangle
represents a 125-ms, 8-spike shock burst. Arrows, labels, and equation
define duty cycle (see text). A: muscle stabilization at
a 2-s cycle period. In response to rhythmic nerve stimulation, the
contractions induced by each burst temporally summate so that the
muscle contraction consists of phasic and tonic (dashed line)
components. Eventually the tonic component becomes largely unchanging,
which we refer to as stabilization. Arrows, labels, and asterisk on
contraction trace define various terms used to describe temporally
summated contractions; see text. B: muscle stabilization
at a 0.5-s cycle period. C: model muscle contraction at
a 2-s cycle period. D: model muscle contraction at a
0.5-s period. Insets: phasic contraction amplitude
decreases, and relaxation slope increases, as the tonic component of
the contraction stabilizes. In C and D,
all model parameters are constant; contraction stabilization occurs
solely as a result of the increased relaxation slope that occurs as
contraction amplitude increases.
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In each case, the muscle contracts with each burst and relaxes between
each burst. However, at the beginning of the stimulation train the
muscle relaxes too slowly to fully relax during the interburst
intervals present in Fig. 3, A or B. Subsequent
contractions therefore "build" on the preceding relaxation phase,
and the muscle develops a sustained tonic contraction (dashed lines).
The muscle contraction thus consists of phasic contractions and
relaxations in time with the bursts of rhythmic nerve stimulation and
an underlying tonic contraction.
The presence of two components in summated muscle contractions can
potentially lead to confusion when referring to various aspects of the
contraction. Throughout this article contraction amplitude refers to
the muscle's present contraction amplitude; for instance, at the point
at ~9 s marked "a" on the contraction curve in Fig.
3A, the muscle's contraction amplitude is ~0.4 mm. Tonic
amplitude refers to the total amplitude of the tonic component (double
headed arrow on Fig. 3A marked "c"). Phasic contraction amplitude refers to the total amplitude of the phasic component alone
(double headed arrow on Fig. 3A marked "b"); when the
amplitudes of the rise and relaxation of the phasic component differ
(e.g., the first phasic contraction of the train, asterisk), the terms phasic rise amplitude and phasic relaxation amplitude are used. In this
article we are concerned with changes in the characteristics of the
phasic contractions, and throughout, all references to contraction
characteristics (rise, plateau, and relaxation duration; rise and
relaxation slope, see METHODS and Fig. 2) refer to the phasic contractions. A final convention that is used throughout concerns relaxation slope. Relaxation slope is a negative number. When
relaxation becomes more rapid, this slope becomes more negative, and
hence, although the absolute magnitude of the slope increases, the
slope itself, strictly speaking, decreases. However, in our experience
stating that relaxation slope is decreasing inevitably leads to the
mistaken perception that relaxation is slowing instead of accelerating.
Therefore throughout this article when relaxation becomes more rapid
(relaxation slope becomes more negative), we state that relaxation
slope increases.
As is apparent in Fig. 3B, the tonic amplitude increases
rapidly early in the train and then increases only very slowly. If the
train is continued, this late slow increase also ceases, but this can
take as long as 1-2 min, which, given the number of conditions tested,
was too long a stimulation time to use in these experiments. Control
experiments showed that in all stimulation paradigms the tonic
component achieved 95% of its final amplitude within the first 10 s of the stimulation. We therefore defined tonic amplitude as having
reached steady state after 10 s of stimulation (although steady-state measurements were always taken well after this time; generally 25-30 s after the beginning of the train), and we refer to
the process by which tonic amplitude ceases changing as stabilization.
The purpose of the work reported here was to investigate the mechanisms
underlying tonic amplitude stabilization. One possible mechanism is
simply that the muscle is reaching the maximum contraction it can
produce. However, in Fig. 3, A and B, the muscle
stabilized at amplitudes well below the amplitude (~1.75 mm) that
would be achieved with a 60-Hz tetanic stimulation; this suggests that other mechanisms underlie the tonic stabilization process.
Insight into these mechanisms can be gained by considering the results
of a very simple model of a slow muscle. In this model, each motor
neuron spike induces a constant, 0.03 mm, increase in muscle
contraction amplitude, and the muscle relaxes toward zero with a single
exponential that is sufficiently slow (relaxation time constant 5 s) that at the beginning of the stimulation the contractions cannot
fully relax during the subsequent interburst intervals. Figure
3C shows the output of the model muscle in response to a
slow cycle period stimulation paradigm (2-s cycle period, 0.3 duty
cycle, 60-Hz intraburst spike frequency), and is thus comparable to
Fig. 3A; Fig. 3D shows the output in response to a fast cycle period stimulation paradigm (0.5-s cycle period, 0.3 duty
cycle, 60-Hz intraburst spike frequency) and is thus comparable to Fig.
3B. In each case tonic amplitude stabilized 5-6 s after the
train began, and in each case the stable-state contraction amplitude
was well below the amplitude (~2 mm) that would be achieved with a
60-Hz tetanic stimulation.
The model contractions stabilize because the model muscle's
relaxation is exponential, and thus as contraction amplitude increases, relaxation slope increases. [During the relaxation phase, contraction amplitude (Amp) equals the amplitude at the beginning of the
relaxation (Amp0) times the
exponential decline, or
and so
and so as contraction amplitude increases so does relaxation
slope.] This increase in relaxation slope has two effects (Fig. 3D, insets). First, as contraction amplitude increases the
rise amplitude of the phasic contraction induced by each motorneuron burst decreases [compare rise amplitudes of the first (left
inset) and final (right inset) contractions] because,
although each motorneuron spike induces the same increase in muscle
contraction, the relaxation amplitude between each spike in the burst
increases as contraction amplitude increases. Second, it increases the
amplitude of the relaxation that occurs between bursts [compare
relaxation amplitudes of the first (left inset) and final
(right inset) contractions]. Eventually phasic rise
amplitude decreases sufficiently, and phasic relaxation amplitude
increases sufficiently, to become equal, at which point tonic
contraction amplitude stabilizes.
The model thus suggests that in the muscle tonic amplitude stabilizes
because phasic relaxation slope increases as contraction amplitude
increases. However, close comparison of the muscle and model reveals
that at least one aspect of the stabilization process in the two cases
differs. In the model, the relative decrease in phasic contraction
amplitude that occurs as tonic amplitude stabilizes is the same in both
stimulation paradigms (in Fig. 3, C and D, phasic
contraction amplitude decreases by 23% from the 1st to the final
contraction). However, in the muscle, the relative change in phasic
contraction amplitude between the first phasic contraction and those
where tonic amplitude is stable differs considerably in the two cycle
period stimulations. In the 2-s cycle period stimulation (Fig.
3A), phasic contraction amplitude decreases ~23%, whereas
in the 0.5-s cycle period stimulation (Fig. 3B), phasic
contraction amplitude decreases ~70%.
These data suggested that dorsal dilator tonic amplitude stabilization
might not be a simple function of relaxation slope increasing as
contraction amplitude increases with phasic contraction temporal
summation. In a system with multiple characteristics such as that shown
in Fig. 2B, many different mechanisms can result in tonic
amplitude stabilization. Figure 4 shows
one such mechanism. The left panel shows the muscle's
response if no characteristic of the phasic contractions changed; the
phasic contractions would continually temporally summate until the
muscle achieved its maximum contraction amplitude (not shown). The
right panel shows the muscle response if phasic contraction
rise and plateau duration decreased without changing rise or relaxation
slope. Decreasing rise and plateau duration increases relaxation
duration. The increased relaxation duration increases phasic relaxation
amplitude (since phasic relaxation amplitude equals relaxation duration
times relaxation slope). Decreasing rise duration also decreases phasic
rise amplitude (since phasic rise amplitude equals rise duration times
rise slope). The decrease in phasic rise amplitude and increase in
phasic relaxation amplitude eventually result in phasic rise and
relaxation amplitudes equalizing, and tonic amplitude therefore
stabilizes.

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Fig. 4.
In multi-component contractions, contraction stabilization need not
occur as a result of relaxation slope increases. Left:
temporal summation if no phasic contraction component changes.
Contraction amplitude continually increases and would continue to do so
until the maximum contraction the muscle could produce was achieved
(not shown). Right: as the train continues, phasic rise
and plateau duration progressively decrease while keeping rise and
relaxation slope constant. The changes in rise and plateau duration
both decrease phasic contraction amplitude and increase relaxation
duration, and hence the contraction can stabilize at an amplitude less
than the maximum the muscle can produce.
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The possibility that changes in phasic contraction characteristics
other than relaxation slope are involved in tonic amplitude stabilization was supported by experiments in which contractions were
induced by constant cycle period stimulation with a variety of duty
cycles (Fig. 5; cycle period, 2 s;
the filled rectangles underneath the plot are schematic representations
of the stimulation pattern used in the rhythmic muscle stimulations).
These data show that tonic contraction amplitude at stable state
increases as duty cycle increases. If tonic amplitude stabilization was due to relaxation slope increasing with tonic amplitude, the relaxation slope of the higher duty cycle contractions should be greater than
those of the lower duty cycle contractions. However, comparison of the
stable contractions in the 0.4, 0.6, and 0.8 duty cycle stimulations
shows that, despite the large increase in tonic amplitude, very little,
if any, increase in relaxation slope occurs as duty cycle and tonic
contraction amplitude increases. Similarly, if an increase in
relaxation slope was responsible for stabilization, one would also
expect to see, in each individual stimulation train, a progressive
increase in relaxation slope as the train progressed. This comparison
is difficult, particularly in the high duty cycle trains, because the
relaxation phase of the early phasic contractions is cut off by the
next contraction in the train. However, comparison of the relaxation
phases of the first, second, and third (at which point stabilization is
largely complete) contractions in the 0.1, 0.25, and 0.4 duty cycle
trains shows no apparent increase in relaxation slope.

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Fig. 5.
Dorsal dilator muscle stabilization different duty cycles does not
appear to depend solely on an increase in relaxation slope. Despite
large increases in contraction amplitude as duty cycle increases from
0.4 to 0.8, relaxation slope increases only slightly. Similarly, in
those conditions in which the comparison is possible (the 0.1, 0.25, and 0.4 duty cycle trains), relaxation slope does not appear to
markedly increase within a train as it progresses (compare the
relaxation phases of the first, second, and third phasic contractions
in each train). Numbers on right of plot indicate the
duty cycle of the various trains. All trains had a 2-s cycle period.
The filled rectangles under the plot are schematic representations of
the stimulation pattern used in the rhythmic muscle stimulations. Burst
duration equals duty cycle times 2 s; intraburst spike frequency
was 60 Hz in all stimulation paradigms.
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Although these data suggest that changes in phasic contraction
characteristics other than an increase in relaxation slope are involved
in tonic amplitude stabilization, observation of rhythmic trains alone
cannot be used to address the question of what changes occur as a train
progresses because, particularly at more rapid cycle periods, the
relaxation and plateau phases of the early contractions are cut off by
the next contraction in the train (see, for instance, Fig.
3B and the 0.6 and 0.8 duty cycle stimulations in Fig. 5).
We therefore modified our stimulation protocol so that we also
delivered a single burst of shocks with parameters that matched the
bursts given in the rhythmic stimulation trains (single contractions).
These contractions should be identical to the first contractions in a
train, but since they are not followed by subsequent contractions,
their plateau durations and relaxation slopes can be measured. We then
linearized the single contractions, and the rhythmic contractions at
stable state (METHODS, Fig. 2B) and used these
data to compare the rise, plateau, and relaxation durations; rise and
relaxation slopes; and phasic contraction amplitudes of the first
(single) and stable state contractions.
Figure 6 shows summary data for rise
duration (row 1), plateau duration (row 2),
relaxation duration (row 3), phasic amplitude (row
4), rise slope (row 5), and relaxation slope (row
6) for single (1st) contractions (
) and stabilized contractions
(
) for 0.1, 0.2, 0.3, 0.4, 0.6, and 0.8 duty cycles at cycle periods of 0.5, 1, and 2 s and for 0.2, 0.4, and 0.8 duty cycles at a 1.5-s cycle period (all stimulations had a 60-Hz intraburst spike frequency). These cycle periods span the physiological range, and in
each plot,
shows the physiological duty cycle range of the input
this muscle receives (J. B. Thuma, L. G. Morris, and S. L. Hooper, unpublished data); muscle response across the entire duty
cycle range was measured to obtain a general understanding of the
muscle's response profile.

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Fig. 6.
Variation in phasic contraction components as tonic contraction
amplitude stabilizes. Rise duration, plateau duration, relaxation
duration, phasic amplitude, and rise slope (rows 1-5)
all decrease as tonic amplitude stabilizes, although the magnitude of
these decreases is cycle period and duty cycle dependent. Relaxation
slope (row 6) increases becomes more negative at low
duty cycles but remains unchanged or even decreases (which would oppose
contraction stabilization) at higher duty cycles. Shading represents
physiological duty cycle range. *, cases in which stable and single
(1st) contractions were significantly different. For single
contractions, a single shock burst with a burst length matching that
applied in the various rhythmic conditions was applied to the motor
nerve (for example, in the 1-s cycle period, 0.4 duty cycle case, the
single contraction was elicited with a 400-ms duration shock burst).
All stimulations were with a 60-Hz intraburst spike frequency.
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Rise duration decreases (which would promote tonic contraction
stabilization) between the first and the stable contraction at all
cycle periods and duty cycles. These decreases are significant (*) for
almost all data points in the 0.5-, 1-, and 1.5-s period regimes but
are significant in the 2-s regime only for the 0.3 duty cycle
stimulation. Plateau duration decreases (which would promote tonic
contraction stabilization) between the first and stable contraction at
all cycle periods and duty cycles. However, the magnitude of this
decrease decreases as cycle period increases. In the 2-s cycle period
regime, the decrease in plateau duration also decreases with duty
cycle, and in the 0.8 duty cycle case, the decrease is no longer
significant. Relaxation duration decreases (which would promote tonic
contraction stabilization) between the first and stable contraction at
all cycle periods and duty cycles, and there is no apparent dependence
of this decrease on cycle period or duty cycle. Phasic amplitude
decreases (which would promote tonic contraction stabilization) between
the first and stable contractions at all cycle periods and duty cycles, and in each cycle period regime this decrease increases with duty cycle. Rise slope decreases (which would promote tonic contraction stabilization) between the first and stable contractions at all cycle
periods and duty cycles, and at all cycle periods this decrease increases with duty cycle. Relaxation slope changes relatively little,
and these changes are seldom significant except in the 2-s cycle period
regime. At all cycle periods at low duty cycles relaxation slope
increases (becomes more negative) between the first and stable
contractions; this would promote tonic contraction stabilization.
However, this decrease decreases as duty cycle increases, and at all
cycle periods at high duty cycles relaxation, slope is less (more
positive) than the single (first) contraction; this would oppose
contraction stabilization.
Taken together, these data suggest that in different cycle period and
duty cycle regimes, changes in rise duration, plateau duration,
relaxation duration, phasic amplitude, rise slope, and relaxation slope
may have different relative importance in tonic contraction
stabilization. For instance, the change in plateau duration is large in
the 0.5-s cycle period condition at all duty cycles but continuously
decreases as duty cycle increases in the 2-s cycle period condition.
Similarly, changes in relaxation slope in the 0.2, 0.3, and 0.4 duty
cycles in the 2-s cycle period regime are much larger than the changes
at these duty cycles in the 0.5-s cycle period regime. However, as we
show later, determining relative importance of changes in these
characteristics is not straightforward and cannot be determined by
simple comparison among these data. [As an aside, also note the
striking nonproportionality of rise duration on motorneuron burst
length. In the 0.5-, 1-, and 2-s cycle period cases, as duty cycle
increases from 0.1 to 0.8, burst length increases eightfold, but in no
case does rise duration increase more than 3.4-fold (2-s cycle period
plot, stable contraction line), and for the stable contraction line in
the 0.5-s period regime, the increase in rise duration over the 0.1 to
0.8 duty cycle range is only 1.25-fold. Although not directly relevant to the issue of tonic amplitude stabilization, this observation does
highlight the nonlinearity of the input:output relationship in this
muscle and the need for detailed, quantitative analysis to relate
neural activity to behavioral output.]
The data presented in Fig. 6 summarize the quantitative changes that
occur in all phasic contraction characteristics as tonic amplitude
stabilizes, but this presentation does not provide an intuitive sense
for the changes in the phasic contractions that occur as stabilization
proceeds in the different stimulation regimes. We therefore used these
values to construct average phasic contraction profiles (Fig.
7) for the 0.2, 0.4, and 0.8 duty cycle
cases at all cycle periods for both single (1st) contractions (the
large amplitude, long duration contractions in the plots) and phasic contractions when tonic amplitude had stabilized (the small amplitude, short-duration contractions in the plots). The rectangles under each
plot are schematic representations of the stimulation pattern used in
that plot; the first rectangle is black to emphasize that the single
(1st) contractions were induced with a single stimulus burst. Note
that, as required for tonic contraction stabilization, in all cases the
total contraction duration of the phasic contractions equaled
stimulation cycle period. These data again suggest that the relative
importance for stabilization of changes in the various phasic
contraction characteristics vary under different stimulation conditions. For instance, in the 0.2 duty cycle regime
(top), decreases in rise slope appear to play a relatively
small role in contraction stabilization, whereas in the 0.8 duty cycle
regime (bottom), decreases in rise slope appear to play a
major role. Similarly, at low duty cycles relaxation slope increases as
the trains stabilize, whereas at 0.8 duty cycle relaxation slope either remains constant (long cycle periods) or even decreases (short cycle
periods).

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Fig. 7.
Averaged profiles of the 1st and stable phasic contractions at 0.2, 0.4, and 0.8 duty cycle (rows) and 0.5-, 1-, 1.5-, and 2-s periods.
Again, different mechanisms appear to be more important in different
stimulation regimes. For instance, in the 0.2 duty cycle regime, rise
slope changes relatively little, whereas in the 0.8 duty cycle regime
rise slope changes are large. Rectangles under plots are schematics
indicating the burst duration and cycle period of the stimulation used
to produce the muscle contractions; the 1st rectangle is filled to
emphasize that the first contractions (the large, long duration
contractions) were induced with a single burst stimulus.
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Although these plots provide a qualitative understanding of the
changes that cause tonic amplitude stabilization, neither they nor Fig.
6 quantify the relative importance that changes in each characteristic
contribute to stabilization. That is, they do not show that, for
instance, in the 0.8 duty cycle, 0.5-s cycle period stimulation changes
in plateau duration are responsible for 25% of stabilization, whereas
in the 0.8 duty cycle, 2-s cycle period changes in plateau duration are
responsible for only 5% of stabilization. How to perform this
quantification is complicated. Contraction duration (Tcon)
equals rise duration (Rt) plus plateau duration
(Pt) plus relaxation duration (Relt).
Relt equals contraction phasic amplitude divided by the
negative of the relaxation slope (Relm; the negative is
required because Relm is negative), and phasic amplitude
equals rise slope (Rm) times Rt. Putting these together gives Tcon = Rt + Pt
Rt * Rm/Relm.
Tonic contraction amplitude stabilizes because as the stimulation train
proceeds Rt, Pt, Rm, and
Relm change so that Tcon becomes equal to cycle period (since temporal summation between phasic contractions ceases when Tcon equals cycle period). Thus one's initial approach
to quantifying the relative importance in the tonic contraction
stabilization process of changes in Rt, Pt,
Rm, and Relm would be to change only one of these
terms, calculate the associated change in Tcon with the
preceding equation, and compare this change to the total change in
Tcon that occurs in the real stabilization process. However,
the presence of the multiplication and division terms involving
Rt, Rm, and Relm in the expression for
Tcon means that one obtains different answers for their
relative importance depending on how the calculation is performed.
To make this difficulty clear, consider a theoretical example in which
changes in only Rt and Rm are responsible for
tonic contraction stabilization (Fig.
8A). The first contraction
(large, long profile) has a 1-s Rt, 1-mm/s Rm,
0.5-s Pt, and
1-mm/s Relm; use of the preceding
relations shows that this contraction has a 1-s Relt, 1-mm
phasic amplitude, and 2.5-s Tcon. The stable contraction
(small, short profile) has a 0.75-s Rt and 0.33-mm/s Rm. Pt and Relm are the same as in the
first contraction, and so the stable contraction has a 0.25-s
Relt, 0.25-mm phasic amplitude, and 1.5-s Tcon.

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Fig. 8.
Relative importance to tonic amplitude stabilization of changes in
phasic contraction components depends on the order in which changes in
the phasic contraction components are made. A: initial
(large, long contraction) and stable (small, short contraction) phasic
contraction profiles. The change between the initial and the stable
contractions are solely due to changes in rise slope (Rm)
and rise duration (Rt). B: if first
Rm is changed (which results in gray contraction profile),
the change in total contraction duration so induced is 67% of the
change in total contraction duration between the initial and stable
contraction, and hence when analyzed in this manner, the change in
Rm is 67% responsible, and the change in Rt 33%
responsible, for stabilization. C: if first
Rt is changed (which results in gray contraction profile),
the change in total contraction duration so induced is 50% of the
change in total contraction duration between the initial and stable
contraction, and hence when analyzed in this manner, the change in
Rt and Rm are each 50% responsible for
stabilization.
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In this example, obviously only changes in Rt and
Rm contribute to tonic stabilization, but what is the
relative importance of the changes in each? Fig. 8B shows
the result if we first change Rm (which results in the gray
profile) and then change Rt. In this case, changing
Rm results in a 667-ms change in total contraction duration.
The change in total contraction duration between the initial and stable
contractions in Fig. 8A is 1 s. Hence by this analysis,
the change in Rm is 67% responsible for changing the initial contraction into the stable contraction, and thus the change in
Rt is responsible for the remaining 33%. Figure
8C shows the result if we first change Rt (which
results in the gray profile) and then change Rm. In this
case, changing Rt results in a 500-ms change in total
contraction duration, and hence by this analysis, the changes in
Rt and in Rm are each 50% responsible for
changing the initial contraction into the stable contraction.
Which of these analyses is correct? In fact, depending on the physical
reality of the stabilization process, both are. If tonic amplitude
stabilizes because Rm first decreases to 0.33 mm/s
while Rt remains constant, and only after this process is complete, does Rt decrease to 0.75 s, then the analysis
shown in Fig. 8B is correct. Alternatively, if stabilization
occurs because first Rt decreases to 0.75 s while
Rm remains constant, and only then does Rm
decrease to 0.33 mm/s, then the analysis shown in Fig. 8C is
correct. As such, the relative importance that changes in phasic
contraction characteristics have in tonic amplitude stabilization
depends on the temporal evolution of these characteristics relative to
each other as the stabilization process proceeds.
This is not a problem specific to temporally summating muscle
contractions but will occur in any system in which independent variables interact nonlinearly in determining a dependent variable and
is thus one that arises frequently in biological and other systems. If
the temporal evolution of the independent variables is known, calculus
can be used to resolve this problem unambiguously by integrating along
the path the independent variables follow from their initial to final
state. In brief, this procedure consists of the following steps.
First, differentiate the equation y = f(x1,
x2,
x3, ...) that defines the
relationship between the dependent variable y and the
independent variables xi to
obtain dy = f'1(x1,
x2,
x3, ...) × dx1 + f'2(x1,
x2,
x3,
...)dx2 + f'3(x1,
x2,
x3,
...)dx3 + ... .
Second, note that the contribution to dy made by each
independent variable xi is
Third, use the experimentally determined temporal evolution
equations, dx2 = g2(x1)dx1,
dx3 = g3(x1)dx1
... to express the equations in the second step in terms of only one
of independent variables (in this case,
x1).
Fourth, integrate the equations derived in the third step between
x1F and
x10 and divide by the total
change in y to calculate the relative contribution that the
change in each independent variable makes to the total change in
y.
We provide a detailed explanation (both for a simple illustrative case
and for the data presented here) of this procedure in the
APPENDIX.
As noted earlier, tonic contraction amplitude stabilization depends
only on changes in rise slope, rise duration, plateau duration, and
relaxation slope (since phasic amplitude and relaxation time are
derived consequences of these characteristics). We performed the
analysis outlined above on these four characteristics to determine their relative importance in tonic contraction stabilization as a
function of cycle period and duty cycle (Fig.
9). The * indicate whether the point in
question was different from zero (P < 0.05); * that
are part of line segments connecting different points indicate whether
the two points were different from each other (P < 0.05). For instance, in the 0.5-s panel for rise duration (row
1), the points at duty cycles of 0.2, 0.3, 0.4, 0.6, and 0.8 differ from zero and the 0.1 and 0.8 duty cycle points differ from each
other.

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Fig. 9.
Relative contribution of changes in rise and plateau duration and rise
and relaxation slope to tonic amplitude stabilization at different duty
cycles and cycle periods. For rise and plateau duration and relaxation
slope, it appears that the contributions of changes in each contraction
component depend in many cases on both duty cycle and cycle period. The
contribution of changes in rise slope, alternatively, appear to depend
only on duty cycle. See text for details. Shading represents
physiological duty cycle range.
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For 1-, 1.5-, and 2-s cycle periods, the change in rise duration
(row 1) made an approximately constant relative contribution of ~35% to tonic amplitude stabilization. However, in the 0.5-s cycle period regime, the contribution of change in rise duration appeared to be duty cycle dependent, rising from a low of 25% contribution at 0.1 duty cycle to a high of 50% at 0.8 duty cycle.
Change in plateau duration (row 2) made an approximately
constant relative contribution of ~25% in the 0.5-s cycle period regime. As cycle period increased, a negative duty cycle dependence (relative contribution declined as duty cycle increased) of the contribution of change in plateau duration became increasingly apparent. This dependence reached significance in the 1.5- and 2-s
cycle period regimes with the relative contribution of change in
plateau duration decreasing from 30 to 15% in the 1.5-s cycle period
regime, and from 25 to 5% in the 2-s cycle period regime, as duty
cycle increased from 0.1 to 0.8. Although this dependence was
sufficient to be significant within the 1.5-and 2-s cycle period
regimes, it was not large enough to separate equal duty cycles in
different cycle period regimes except in the 0.3 duty cycle case in the
0.5- and 2-s cycle period regimes (open data points on plots; the
P value comparing the 0.6 duty cycle cases in the 0.5- and
2-s period regimes was 0.054, the P value comparing the 0.8 duty cycle cases in the 0.5- and 2-s period regimes was 0.052).
In all cycle period regimes, the relative contribution of change in
rise slope (row 3) to tonic contraction amplitude
stabilization was strongly positively dependent on duty cycle (relative
contribution increased as duty cycle increased), rising from
contributions of near 0% at a duty cycle of 0.1 to contributions
ranging from 60 to 90% at a duty cycle of 0.8. No dependence on cycle
period was apparent when equal duty cycle points in different cycle
period regimes were compared.
In all cycle periods, the relative contribution of change in relaxation
slope (row 4) to tonic contraction amplitude stabilization declined as duty cycle increased, decreasing from contributions ranging
from 50% (0.5-s cycle period) to 20% (1-, 1.5-s cycle periods) at a
duty cycle of 0.1 to 0% (1.5-, 2-s cycle periods) to
40% (0.5-, 1-s
cycle periods) at a duty cycle of 0.8, although only in the 0.5- and
1-s cycle period regimes was this decrease significant. Thus, in
keeping with the data presented in Figs. 6 and 7, at high duty cycles
in the short cycle period regimes the change in relaxation slope
actually opposed tonic contraction amplitude stabilization. The
suggestion that the decline in contribution with duty cycle is stronger
at shorter cycle periods is supported by comparison of the 0.1, 0.2, and 0.3 duty cycle cases between the 0.5- and 2-s cycle regimes (
).
At each of these duty cycles, the contribution of change in relaxation
slope was significantly less in the 0.5-s cycle period regime than in
the 2-s cycle period regime. Thus, the decline in contribution was not
only larger but also occurred at lower duty cycles in the 0.5- than in
the 2-s cycle period regime.
The decline in the contribution of relaxation slope as duty cycle
increased and cycle period decreased is particularly interesting because under these conditions average muscle contraction amplitude increased (Figs. 3, A and B, and 5), and hence
relaxation slope should increase for an exponential process with an
unchanging time constant. Alternatively, if the time constant of the
exponential increased with contraction amplitude, then the
amplitude-induced increase in relaxation slope would lessen, or even
reverse (as is seen in the short cycle period, high duty cycle
conditions in Figs. 6 and 7, in which relaxation slope
decreased
became "shallower"
as the contractions stabilized). To
determine if this was occurring, for three of our experiments we made
exponential fits to the relaxation at the end of every stimulation
train. Figure 10 shows the relaxation time constants of these fits as a function of contraction total amplitude (the amplitude at the top of the phasic contractions; b plus
c in Fig. 3A); relaxation exponential time constant can increase as much as a second as total contraction amplitude increases from 0.2 to 2 mm.

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Fig. 10.
The dorsal dilator muscle relaxes more slowly as contraction amplitude
increases. Exponential fits to the relaxations following the rhythmic
stimulations were made in 3 experiments, and the time constants of the
exponential were plotted against contraction total amplitude (b plus c
in Fig. 3). Tau can increase by as much as a second as total
contraction amplitude increases from 0.2 to 2 mm.
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DISCUSSION |
We have described the changes in the phasic contractions of the
dorsal dilator muscle that allow the tonic component of this muscle's
temporally summating contractions to stabilize in response to
stimulation with rhythmic burst trains. We have also presented a
generally applicable method for quantifying the relative importance of
each change in this process and shown that the relative importance of
each change varies as a function of stimulation cycle period and duty cycle.
Generality of the path-dependent relative contribution analysis
Processes in which one variable depends on the product, power, or
similarly nonadditive interaction of two or more other variables are
common in biological systems, and we are unaware of a widely used
method in biology for apportioning changes in the dependent variable to
changes in the independent variables in such cases. The method we
describe here is a straightforward application of calculus, its path
dependence means that it reflects many of the properties of the system
being analyzed, and its ability to express the relative importance of
changes in complex systems as single numbers is both a useful, compact
description of the system and conceptually satisfying. The use of such
integrals is common in engineering and other fields but does not seem
to be widespread in biology. These integrals can determine the relative
contribution changes in independent variables make to changes of the
system as a whole in any biological system in which the temporal
evolution of the independent variables can be determined and thus would seem to have widespread utility in biological studies.
Implications for the pyloric neuromuscular system
The pyloric neural network has been intensively studied for almost
25 yr and is one of the best understood of all neural networks (Harris-Warrick et al. 1992
). However, because the
pylorus is an internal organ and hence not directly observable, almost
no data exist as to what motor pattern the network's activity induces. It has often been assumed that the pyloric muscles would be fast enough
to accurately follow their neural input and that the motor pattern
would thus be a rapid (1 Hz) series of repeated movements (Selverston et al. 1976
; Turrigiano and Heinzel
1992
). The data presented here and elsewhere showing that many
pyloric muscles are extremely slow, that these muscles respond to
pyloric-timed rhythmic input with temporally summated contractions with
large tonic components, and that these muscles can express the activity of other, longer cycle period, stomatogastric neural networks that
modulate pyloric network neural output (Ellis et al.
1996
; Koehnle et al. 1997
; Morris and
Hooper 1997
, 1998a
,b
; Morris et al. 2000
) throw
this assumption into question. Unfortunately, due to the internal
location and small size of the pylorus, direct measurement of pyloric
movements will likely continue to be difficult. An alternative approach
is to predict pyloric movement patterns using computer simulation. The
data shown here cannot be used directly for this purpose because muscle
movement, not tension, was measured. However, the insights into the
temporal summation process gained from this work and the analytical
techniques developed here should materially advance efforts in
developing such models and linking pyloric neural network activity to
pyloric behavior.
Relevance to fundamental mechanisms underlying muscle response
The rising phases of the muscle contractions shown here are in
response to bursts of action potentials, and as such are a combined
response of the muscle to unitary contraction events associated with
each neuron spike. The changes in rise slope, rise duration, and
plateau duration shown in Figs. 6 and 7 presumably occur as a result of
changes in these unitary contractions as a result of stimulation
history and the muscle's increased contraction amplitude as the
stimulation continues. Unfortunately these unitary contractions are too
small to be observed, and it is thus impossible to associate the
changes in the characteristics of burst-induced contractions with
changes in unitary contraction characteristics.
Nonetheless these data do suggest some tentative conclusions. As was
noted in the model contractions shown in Fig. 3, changes in relaxation
slope can alter rise slope and amplitude by increasing the amount of
relaxation that occurs between each unitary contraction, and it is thus
possible that the changes in other contraction components described
here could also arise indirectly from changes in relaxation slope
alone. However, inspection of Fig. 6 suggests that this is not the
case. For instance, in several stimulation regimes relaxation slope
does not change as the contractions stabilize (e.g., the low duty cycle
cases at 0.5-s cycle period), and yet stabilization still proceeds.
Similarly, in several cases relaxation slope decreases, which would,
all other things being equal, increase rise slope and phasic amplitude,
but in all cases both these components decrease as contractions
stabilize. Another mechanism likely to play a role in decreasing rise
slope is the amplitude of the unitary contractions decreasing as the
muscle enters the declining portion of its length-tension curve as
muscle contraction amplitude increases. However, neither of these
mechanisms explains the large decreases in rise duration and plateau
duration that are also observed during tonic amplitude stabilization,
as it is unclear why decreasing unitary contraction amplitude would
decrease unitary contraction rise duration or plateau duration, and the
decreases in rise duration and plateau duration appear to be
independent of changes in relaxation slope (compare 1st and
2nd rows to bottom row, Fig. 6). Similarly unexplained is the large increase in relaxation time constant seen as
contraction amplitude increases (Fig. 10). Resolution of these issues
will require a much deeper understanding of the physical properties of,
and excitation-contraction coupling in, this muscle, but at present it
appears that tonic amplitude stabilization cannot be explained solely
by increasing relaxation slope or decreasing unitary contraction amplitude.
Implications for other motor systems
Considerable evidence from a variety of preparations suggests that
temporal summation of muscle contractions could occur in several
preparations (Atwood 1973
; Carrier 1989
;
Ellis et al. 1996
; Hall and Lloyd 1990
;
Hetherington and Lombard 1983
; Koehnle et al.
1997
; Mason and Kristan 1982
; McPherson
and Blankenship 1992
; Morris and Hooper 1998a
;
Morris et al. 2000
; Weiss et al. 1992
).
Most researchers have implicitly or explicitly considered the tonic
contraction associated with temporal summation to be contrary to
function (e.g., Weiss et al. 1992
), but tonic
contraction could be behaviorally beneficial by providing fine motor
control by allowing antagonistic muscle co-contraction (Bizzi
and Abend 1983
), by promoting muscle stiffness and thus
allowing muscles to serve as structural elements (Altringham et
al. 1993
), or by allowing muscles to express the motor patterns
of neural networks whose neurons do not innervate them (Morris
et al. 2000
). In cases in which temporal summation of muscle
contractions is behaviorally relevant, prediction of muscle response to
varying neural input will critically depend on how phasic contraction
characteristics change as a function of muscle contraction history and
attained amplitude. Given the exponential nature of muscle relaxation, and the fact that relaxation slope increases with amplitude in exponential processes, it is natural to assume that this increase in
relaxation slope would play a, or possibly the, major role in
contraction response. However, our data showing that changes in
relaxation slope can play a negative (opposing) role in muscle stabilization in certain duty cycle and cycle period regimes and that
the relative contributions of other contraction components play large
duty cycle and cycle period dependent roles in stabilization, suggest
that full understanding of muscle response in such systems may require
a detailed analysis of all muscle contraction components. As such,
these data underscore the central necessity of a detailed understanding
of the neural input to muscle output transfer function for describing
the neural genesis of motor behavior.
The procedure for determining relative contribution is somewhat
involved using the equations that define the phasic contractions (see
following text), and we have therefore chosen a simpler example with
which to demonstrate this analysis initially. Consider transforming one
line segment into another (Fig.
A1A, left). The
uppermost line segment runs from (0,0) to (5,35), and therefore has a
slope of 7 and an Xend of 5. The
bottom line segment runs from (0,0) to (3,6), and therefore has a slope
of 2 and an Xend of 3. In the transformation of the first line segment into the second line segment,
what is the relative importance of the change in slope versus the
change in Xend? Similar to the case
shown in Fig. 8, the answer to this question depends on which path is
used to make the transformation. For instance, first changing only
Xend from 5 to 3 (path 1) results in
arriving at the point marked "*1" on the plot. This point is
(3,21), and the change in Y due to changing Xend alone is thus 21
35 =14,
which is 48% of
29, the total amount Y changes in going
from line segment 1 to line segment 2. The remaining 52% of the change
along this path is due to the subsequent change in slope.
Alternatively, first changing only slope from 7 to 2 (path 2) results
in arriving at the point marked "*2" on the plot. This point is
(5,10), and along this path, the change in Y due to changing
slope is therefore 10
35 =
25. Thus along this path 86%
of the total change in Y is due to the change in slope and
only 14% is due to the change in
Xend.
Again, each of these values is correct for the different realities they
describe; path 1 represents a case in which first Xend and then slope changes; path 2 represents a case in which first slope and then
Xend changes. As such, to correctly
determine the relative importance of changes in each characteristic, it is again necessary to know how each changes as the others change. Our
muscle contractions are fully described by rise duration, rise slope,
plateau duration, and relaxation slope (see discussion of Fig. 8), and
how these characteristics change with time in a train is therefore the
issue at hand. Figure A1B, top, shows a 1.5-s cycle period,
0.4 duty cycle rhythmic stimulation, and its corresponding
linearizations for the 2nd (only rise duration and slope could be
measured for this contraction), 3rd-5th, 10th, and 25th contractions.
The rectangles under the trace are a schematic representation of the
stimulation used to generate the muscle contraction. Figure A1B,
bottom, shows the rise durations, rise slopes, plateau durations,
and, for ease of presentation, the absolute magnitude of the relaxation
slopes of these contractions. Also shown are the values of these
contraction characteristics for the corresponding single contraction
(the contraction 1 values on the graph). The dashed lines indicate that
for contraction 2 plateau duration and relaxation slope could not be
measured. It is apparent that all contraction characteristics change in approximate synchrony. This means that the relative change of each
characteristic is the same at all times (that is, if rise duration
changes by 10% of its total change, rise slope also changes by 10% of
its total change).
Knowledge of the path used to go from beginning to final state
allows the relative contributions of changes in each characteristic to
be unambiguously determined. To demonstrate this procedure, return
again to the example shown in Fig. A1A. The data shown in Fig. A1B, bottom, indicate that each independent variable is
changing in equal proportion of the total change in the variable at all times. Our goal is therefore to go from the original to the final line
segment in Fig. A1A by making sequential, constant
proportion step changes in line segment slope and
Xend. The gray and dashed lines in
Fig. A1A, right, show the two eight-step paths that result if these two paths are followed by sequentially changing slope and
Xend by 1/4 of their total value at
each step. The slope of the original line segment is 7, the slope of
the final line segment is 2, and thus each slope change will be 5/4;
the Xend of the original line segment
is 5, the Xend of the final line
segment is 3, and thus each Xend
change will be 2/4.
The reason there are two paths is because this procedure can be carried
out in two ways
first change slope, then
Xend, then slope, then
Xend, etc., or first change
Xend, then slope, then Xend, then slope, etc. In the dashed
path, first slope, then Xend, then
slope, then Xend, etc. are changed.
The first step (point marked "a" on plot) of the dashed path is
thus where the point would be for a slope of 7
(5/4) = 5.75 and an Xend of 5, the second step
(point marked "b" on plot) of this path is where the point would be
for a slope of 5.75 and an Xend of
5
(2/4) = 4.5, the third step (point marked "c" on
plot) of this path is where point would be for a slope of 5.75
(5/4) = 4.5 and an Xend of 4.5, etc. In the gray path, first Xend,
then slope, then Xend, then slope,
etc. are changed. The first step (point marked "d" on plot) of the
gray path is thus where the point would be for an
Xend of 5
(2/4) = 4.5 and
a slope of 7, the second step (point marked "b" on plot) of this
path is where the point would be for an
Xend of 4.5 and a slope of 7
(5/4) = 5.75, the third step (point marked "e" on plot) of
this path is where point would be for an
Xend of 4.5
(2/4) = 4 and
a slope of 5.75, etc. It appears that the gray and dashed paths are
converging on a single path that would be reached were the step sizes
infinitely reduced.
This line can be determined as follows. The equation for point
Y is y = m * x. We
are trying to determine the line connecting two such lines as
m and x change, and we therefore differentiate this equation to obtain dy = x * dm + m * dx. Solving this equation for x requires expressing dm and m in
terms of dx and x. The requirement that
dm and dx change by the same percentage of their
total change means that
Turning now to the data presented here, the percent contributions of
changes in rise duration (Rt), rise slope (Rm),
plateau duration (Pt), and relaxation slope
(Relm) can be determined as follows. Figure
A2 shows a first contraction (the
large, long duration contraction) and a stable contraction (the small,
short-duration contraction). Total contraction duration
(Tcon) equals Rt plus Pt plus the
relaxation duration. Relaxation duration equals contraction phasic amplitude divided by the negative of the relaxation slope (the
negative is necessary because relaxation slope is negative). Phasic
amplitude equals Rm * Rt, relaxation duration
is thus Rm * Rt/(
Relm) and so
Tcon = Rt + Pt - Rt * Rm/Relm. Differentiating gives
We thank J. S. Connor for useful discussion and advice on the
mathematical aspects of the work.
This work was supported by Ohio University, the National Science
Foundation, and the National Institute of Mental Health.
Address for reprint requests: S. L. Hooper, Neuroscience Program,
Dept. of Biological Sciences, Irvine Hall, Ohio University, Athens, OH
45701 (E-mail: Hooper{at}ohio.edu).