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First published online June 15, 2006
Journal of Experimental Biology 209, 2535-2553 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02276
Storage and recovery of elastic potential energy powers ballistic prey capture in toads

Department of Biological Sciences, Northern Arizona University, Flagstaff, AZ 86011-5640, USA
Author for correspondence (e-mail:
Kiisa.Nishikawa{at}nau.edu)
Accepted 18 April 2006
| Summary |
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Key words: contractile properties, depressor mandibulae, elastic properties, elastic recoil model, load clamp, load dependence, parallel elastic component, series elastic component, power output, toad
| Introduction |
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Ballistic tongue projection is a feeding mechanism in which the tongue is
propelled from the oral cavity. Among vertebrates, only some anurans, some
salamanders, and chameleons use ballistic tongue projection to capture prey
(Nishikawa, 2000
). Whereas
power augmentation during jumping has been investigated in a wide variety of
animals (e.g. Alexander, 1968
;
Lutz and Rome, 1994
;
Peplowski and Marsh, 1997
;
Aerts, 1998
), fewer studies
have addressed the question of power augmentation during ballistic tongue
projection in chameleons (Wainwright and
Bennett, 1992a
; Wainwright and
Bennett, 1992b
; Meyers and
Nishikawa, 2000
; de Groot and
van Leeuwen, 2004
) and salamanders
(Deban et al., 1997
;
Deban and Dicke, 1999
). Yet,
these ballistic tongue projection systems may provide ideal models for
studying mechanisms of power augmentation because their anatomical
arrangements are relatively simple compared to those of the limbs (e.g. the
jaws and hyolingual apparatus tend to possess fewer muscles and fewer joints
than limbs). Also, in general, the origins and insertions of cranial and
hyolingual muscles in vertebrates tend to be fleshy or possess very short
aponeuroses and/or tendons compared to distal limb muscles. Thus, it is
expected that cranial and hyolingual muscles would contribute more to the
total strain and strain energy than muscles that possess longer tendons
(Alexander, 1988
).
Recent comparative studies have shown that, among anurans, toads (genus
Bufo) appear to be particularly adapted for ballistic tongue
projection (Nishikawa, 1999
;
Nishikawa, 2000
).
Forward-dynamic biomechanical models have shown that the paired depressor
mandibulae muscles produce >90% of the force needed for ballistic tongue
projection in anurans (Mallett et al.,
2001
). When these muscles open the mouth rapidly, momentum is
transferred from the lower jaw to the tongue pad. Although previous studies
have addressed the mechanism of tongue projection and have described the
kinematics of ballistic jaw and tongue movements
(Nishikawa and Gans, 1992
;
Nishikawa and Gans, 1996
), no
previous study has investigated mechanisms of power augmentation during
ballistic mouth opening in toads.
The goals of the present study were to: (1) quantify the power output and
patterns of activation of the depressor mandibulae muscles; (2) quantify the
factors that contribute to high in vivo power output by estimating
the elastic properties (i.e. displacement and stiffness) of the depressor
mandibulae muscles and their series connective tissues; and (3) develop and
test an elastic recoil model, based on in situ elastic properties of
the depressor mandibulae muscles and series connective tissues, that accounts
for the observed displacement, velocity, acceleration and power output during
ballistic mouth opening in vivo. Whereas many studies have compared
the elastic potential energy needed to power ballistic movements to that
stored in elastic structures prior to movement (e.g.
Alexander, 1968
;
Alexander and Bennet-Clark,
1977
), the present study seeks not only to quantify the storage of
elastic potential energy prior to movement, but also the contribution of
elastic properties to the rate of energy recovery and power output.
| Materials and methods |
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Kinematic analysis of ballistic mouth opening
Adult Colorado River toads, Bufo alvarius Girard (N=4),
were imaged digitally at 1000 Hz while feeding using a RedlakeTM
Motionscope high-speed digital imaging system with synchronized, stroboscopic,
infrared illumination. A grid of 1 cm squares was used to calculate the
scaling factor and aspect ratio. Toads were placed on a flat stage, oriented
perpendicular to the camera, and allowed to feed unrestrained on adult
crickets (Acheta domesticus, length=2.5 cm). The crickets were placed
at varying distances from the toads (111 cm) to elicit variability in
effort during prey capture behavior (see below). Three feeding sequences were
recorded for each toad. Electrodes were implanted into the depressor
mandibulae and levator mandibulae posterior longus muscles on the right and
left sides, the toads were again imaged at 1000 Hz while feeding, and
1122 sequences were obtained after implantation (total of 71 sequences
from 4 individuals). After a series of feeding sequences synchronized with
electromyograms had been recorded, the toads were prepared for in
situ force-lever studies (see below). Feeding sequences were digitized
using Didge 2.2 motion analysis software (A. Cullum, Creighton University,
Omaha, NE, USA). For each frame, the digitized points included the upper jaw
tip, jaw joint, lower jaw tip and prey item. Only the ballistic phase of mouth
opening (i.e. first 20 ms of the prey capture sequence) was analyzed.
For each frame, the distance to the prey item (i.e. shortest distance from
the lower jaw tip to prey) was calculated. Gape distance (i.e. distance
between the upper and lower jaw tips) and gape angle (i.e. angle subtended by
the upper and lower jaw tips with the jaw joint at the vertex) were computed
using the Euclidean distance and the law of cosines, respectively. The total
distance shortened by the depressor mandibulae muscles during the ballistic
phase was calculated by dividing the gape distance by the in-lever/out-lever
ratio of the muscle. The position data were smoothed using a heptic spline
function (QuickSAND) (Walker,
1998
). Instantaneous velocity (v) and acceleration
(a) were estimated as the first and second derivatives of the
smoothed position data for each 1 ms time step. For each feeding sequence,
peak instantaneous power was calculated as:
![]() | (1) |
The general equation of motion for a damped massspring system is:
![]() | (2) |
![]() | (3) |
n is the natural frequency of vibration,
is the
damping ratio, and
d is the damped frequency of vibration:
![]() | (4) |
![]() | (5) |
![]() | (6) |
For each feeding trial, Berkeley Madonna (version 8.0.1) was used to find the values of the non-linear spring constant (kt) and effective damping coefficient (beff) that minimized the squared difference between the position predicted from the time solution of the general equation of motion for a damped mass-spring system (Eqn. 2) and the observed kinematic data [initial displacement x0 and the total displacement over time xt(t)], given the meff estimated for each toad from anatomical measurements and initial velocity (v0)=0.
Electromyography
Electromyograms were recorded from the depressor mandibulae muscles and the
largest of the six jaw adductor muscles, the levator mandibulae posterior
longus, on the right and left sides of four toads (Ba 14). Bipolar hook
electrodes were constructed from 0.05 mm diameter formvar-coated stainless
steel wire (California Fine Wire Co., Grover Beach, CA, USA) with bared tips
(0.5 mm). The dipole spacing of the electrodes was
1 mm. Toads were
anesthetized in a solution of tricaine methanesulfonate (MS-222) (0.1 g
l1 water) buffered to a pH of 7.0 using sodium bicarbonate.
Incisions (
0.5 cm) were made to expose the jaw muscles. Using a 23-gauge
hypodermic needle, the tips of the electrodes were implanted into the belly of
each muscle. Electrode leads were threaded under the skin to the back of the
toad, where they were sutured to the skin using 6.0 surgical silk. The leads
were connected to a differential AC amplifier (A-M Systems, model 1700,
Everett, WA, USA).
Analog signals were band-passed (105000 Hz), notch-filtered at 60 Hz, amplified (1000x), and digitized at 4000 Hz (Peak Performance Technologies, Inc., Centennial, CO, USA). The signals were later high-pass filtered at 300 Hz to remove low-frequency motion artifacts. The onset of EMG activity was defined as the time at which the EMG amplitude rose above twice the mean baseline voltage. Likewise, the muscle was considered to be electrically inactive when the EMG amplitude fell below twice the mean baseline voltage. EMG recordings were synchronized with digital images (1000 Hz). For each trial, the peak amplitude (mV), duration (ms), and full-wave rectified, integrated area (mVs) of EMG activity that occurred prior to the onset of mouth opening were calculated. For each toad, the relationships between EMG activity (i.e., duration, peak amplitude, rectified integrated area), prey distance (cm), and total displacement (xt) were examined (Pearson productmoment correlation coefficients, P=0.05). For two toads (Ba 1 and 3), the amplitude of the EMG signal declined significantly over time (P<0.05), so only the duration data were analyzed.
In situ force-lever experiments
A dual servo-motor force lever (Aurora Scientific, Inc., Series 305B,
Aurora, ON, Canada) was used to examine the contractile and elastic properties
of the depressor mandibulae muscles. For each muscle, a maximum isometric
tetanic contraction was performed, followed by an after-loaded
forcevelocity experiment (N=3), a lengthtension
experiment (N=3), or a load-clamp experiment (N=3; Ba
13).
Toads were cooled in a refrigerator at 4°C for 1 h anddouble-pithed
prior to each force-lever experiment. On the temporal aspect of the cranium,
the depressor mandibulae muscles lie under the parotid glands, which in toads
secrete a moderately potent toxin. To avoid contact between toxic parotid
gland secretions and the muscle tissue, secretions were squeezed from the
parotid glands with paper towels prior to removal of the skin and connective
tissue overlying the depressor mandibulae muscles. We used an in situ
preparation that maintained the muscle's natural origin, insertion, and line
of action (Fig. 1A). This
procedure also avoided trauma to the muscle's blood supply, which remained
intact. The mandible was transected
5 mm anterior of the jaw joint and
freed of its attached hyolingual musculature. All muscles that resist joint
rotation due to shortening of the depressor mandibulae muscle (i.e. jaw
adductors) were removed (Fig.
1B), and it was assumed that resistance to joint rotation was
negligible. During experiments, the muscle was irrigated frequently with
amphibian Ringer's solution (6 g NaCl, 0.075 g KCl, 0.1 g CaCl2,
and 0.1 g NaHCO3 l1 H2O).
|
The viscoelastic properties of Spider WireTM were investigated by performing load-clamp tests on 2.5 cm lengths of Spider Wire alone. The Spider WireTM was loaded with forces of 4.05.1 N, which were reduced rapidly to 2.81.0 N. When the force was reduced, there was a small step decrease in the length of the Spider Wire with no observable oscillations. Over the range of loads used in this study, the decrease in length was independent of clamp load (N=5 trials, r2=0.06), and averaged 0.06 mm. The contribution of the Spider Wire, including its connections to muscle and lever, to the total displacement during load-clamp experiments was 0.350.37% of muscle resting length.
Force-lever experiments were performed with the depressor mandibulae muscle
at its in vivo resting length
(Fig. 2), which corresponds to
the length at which its active force production is greatest
(L0). The resting length is the greatest length that the
depressor mandibulae muscles can attain in vivo because, at this
length, the mouth is fully closed. When the muscle insertions were isolated
initially, the muscles assumed a length that was shorter than their in
vivo resting length by
1 mm (6% ML). Using the lever, the
length of the muscles was increased slowly until the passive tension was 0.1 N
(711 mN mm2). At this passive tension, the length of
the muscles corresponded to the in vivo resting length, as measured
to the nearest 0.1 mm with digital calipers prior to isolation. The
contribution of the passive muscle strain (i.e. at resting length and a
passive tension of 0.1 N) to the total displacement during the load-clamp
experiments was calculated from the passive lengthtension curve as the
strain at a passive force of 0.1 N minus the strain at the in situ
load (0.030.08 N).
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For stimulation, two electrodes (diameter 0.076 mm) were implanted into the denervated depressor mandibulae muscle. To achieve maximum activation, 5 mm of each electrode tip was bared of insulation, and the electrodes were inserted into the muscle belly so that the distance between them was half the total muscle length (e.g. a 20 mm muscle would have electrodes implanted 5 mm from the origin and 5 mm from the insertion, spaced 10 mm apart). The electrodes were attached to a Grass SIU5 Isolation Unit in series with a Grass S48 Stimulator. The voltage needed to produce maximum activation was determined using twitch contractions (1 mspulse duration), and this voltage was increased by 20% in subsequent tests (i.e. supramaximal voltage). Supramaximal voltage ranged from 80120 V. Tetanic stimuli were delivered in single trains of 400 ms duration at a frequency of 80 pulses s1. To avoid fatigue, the muscles were rested for 15 min between trains. In these experiments, maximum isometric force never fell below 22 N cm2, which indicates that muscle recruitment was maximal and fatigue was minimal.
External load was controlled and length changes were measured (in after-load and load-clamp experiments) using the series 305B lever, which is capable of recording forces and excursions up to 5 N and 10 mm, respectively. The lever was tuned to factory specifications for minimal damping and step response time (2 ms) and was calibrated using weights of known mass. For data capture, the servo-motor output was sent to a GW Instruments InstruNet Model 100B A/D I/O System connected to a Macintosh 7200 computer running GW Instruments SuperScope II software sampling at 4000 Hz.
Contractile properties
To generate a lengthtension curve for the depressor mandibulae
muscles, each muscle (N=3) was stimulated isometrically over a series
of 68 lengths at 0.5 mm increments. Stimulation parameters were as
described above. From these experiments, we measured the passive, active and
total tension at each length. To determine where on its lengthtension
curve the depressor mandibulae muscles operate in vivo, muscle length
was measured to the nearest 0.1 mm with digital calipers with the mouth in the
fully closed position, prior to isolation of the muscles.
To generate a forcevelocity curve for the depressor mandibulae
muscles, we performed a series of isotonic after-loaded contractions at
68 loads per muscle (N=3). From these data,
forcevelocity curves were generated and the maximum velocity of
shortening, Vmax, was estimated using Hill's equation
(Hill, 1938
):
![]() | (7) |
Elastic properties
It is widely believed (e.g. Alexander
and Bennet-Clark, 1977
) that muscles themselves contribute little
to the power of ballistic movements because muscles shorten slowly, relative
to elastic recoil in tendon or cuticle, even under very low loads. This idea
follows directly from Hill's forcevelocity curve
(Hill, 1938
): the velocity of
muscle shortening increases hyperbolically with decreasing load. Because power
is the product of force and velocity, the forcevelocity property of
muscle necessarily limits power output. When the force is large, the velocity
is small, and when the velocity is large, the force is small.
The apparent trade-off between force and velocity is, at least in part, an
artifact of Hill's after-loaded isotonic paradigm for generating the
forcevelocity curve (Hill,
1938
). In this paradigm, not only does the external load vary, but
the duration of muscle stimulation prior to the onset of shortening also
varies with the load. This means that the muscle is stimulated for very short
durations (as little as 1015 ms) at the smallest loads and for much
longer durations (>250 ms) at the largest loads. Hill designed the
after-loaded paradigm specifically to remove any contribution of muscle series
elasticity to the observed shortening, in order to study the properties of the
contractile elements (Hill,
1938
). However, if muscles are activated for relatively long
durations prior to shortening, as often occurs during natural movements, then
recoil of series elastic elements within the muscle itself will contribute to
power output. The question is, for a given duration of pre-movement
activation, how much power can muscles produce as the load decreases?
For these reasons, we chose to quantify the elastic properties of the depressor mandibulae muscles using the load-clamp technique. The load-clamp experiments were designed to mimic the in vivo behavior of the depressor mandibulae muscles. In vivo, the depressor mandibulae muscles contract prior to mouth opening, producing strain in series elastic elements of the cranium and jaws, as well as within the muscle itself. When the mouth opens, the load decreases rapidly. In load-clamp tests, the load is reduced following isometric stimulation, which results in biphasic shortening of the muscle. An initial rapid change in length (due to recoil of series elastic elements within the muscle) is followed by a slower change in length (due to cyclic interactions of contractile proteins within the muscle). Damped oscillations appear during the transition (Fig. 3).
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Estimating strain in series elastic elements during depressor mandibulae stimulation
The goal of these experiments was to estimate the degree to which the
depressor mandibulae muscles impose strains on series elastic structures of
the cranium and lower jaw prior to mouth opening. As for the force-lever
experiments, toads (N=3) were cooled and double-pithed, and all
jaw-adductor muscles were removed. The intact lower jaw was sutured to the
maxilla to prevent mouth opening. The cranium was clamped to a metal frame on
a vibration-resistant table, and stimulating electrodes were implanted in the
depressor mandibulae muscle as described above. The depressor mandibulae
muscle was stimulated supramaximally, and movements of the cranium and lower
jaw were recorded digitally at 500 Hz. Points at the origin of the depressor
mandibulae on the cranium, the insertion of the depressor mandibulae on the
retroarticular process, and a fixed point on the clamp were digitized at 20 ms
intervals (every tenth frame) beginning with the frame preceding the first
perceptible movement. Total shortening of the depressor mandibulae muscle, as
well as strain at the cranial insertion and retroarticular process, were
measured as a function of time from the onset of movement. We assumed that the
total shortening of the depressor mandibulae muscle is equal to the total
strain in all extra-muscular structures combined (i.e. the sum of the strain
at the muscle origin and insertion), and that strains in directions other than
the direction of muscle shortening were negligible. Immediately after the
in situ strain data had been recorded, each specimen was prepared as
described above for the force-lever experiments. The muscle insertion was
attached to the arm of the force lever, the muscle was stimulated tetanically
until it reached its maximum isometric force, and the force was measured as a
function of time from the onset of stimulation. For each depressor mandibulae
muscle, the strain vs time and force vs time plots were
aligned at t=0 to obtain the relationship between force and
strain.
Anatomical measurements
The effective mass (meff) moved by the depressor
mandibulae muscles in vivo was estimated as the mass of the lower jaw
plus tongue, multiplied by the moment of this mass (i.e. distance from center
of mass to center of rotation divided by distance from muscle insertion to
center of rotation), plus the mass of the depressor mandibulae muscles and
retroarticular processes. The effective mass ranged from 9.1212.94 g in
the four individuals for which in vivo kinematic data were collected
(Ba 14). The in vivo inertial load
(meffg) was estimated as the effective
mass multiplied by the acceleration due to gravity (g).
Because the load is distributed over the right and left depressor mandibulae
muscles arranged in parallel, the in vivo loads (mean=0.1 N)
represent only 1.31.8% of the maximum isometric force that the two
depressor mandibulae muscles can produce (
58 N).
To estimate total shortening of the depressor mandibulae muscles and associated connective tissues during ballistic mouth opening, gape distances measured from digital images were converted to total shortening distances using the in-lever/out-lever ratio (i.e. distance from muscle insertion to center of rotation divided by distance from center of rotation to tip of the lower jaw). The lower jaw was dissected from the specimen and cleaned to reveal the anatomy of the jaw joint and retroarticular process. Measurements used for in-lever/out-lever calculations were taken from digital images of the dorsal surface of the isolated lower jaw as straight line distances parallel to the longitudinal axis of the head. We assumed that the center of rotation of the lower jaw is at the center of the jaw joint itself. The in-lever/out-lever ratio ranged from 1:7.71:8.6 in the four individuals used in this part of the study. The shortest (deepest) and longest (most superficial) distances along the muscle surface from origin to insertion were measured with digital calipers, and the mean of these distances was used to calculate strain (ML) and strain rate (ML s1).
Sarcomere length
Sarcomere lengths of the depressor mandibulae muscle at resting length were
measured from specimens (N=3) prepared for transmission electron
microscopy using standard techniques (e.g.
Nishikawa et al., 1999
). To
ensure that the muscles maintained their natural position during fixation, the
entire posterior half of the head and lower jaw (from the eye to the jaw
joint) was submerged in fixative (6.25% glutaraldehyde in sodium cacodylate
buffer, pH=7.4). The mouth was sutured shut to ensure that the muscle
maintained its resting length during fixation. Sarcomere lengths were measured
from digital images taken from four regions within the muscle: superficial
belly, deep belly, origin and insertion. For each region in each muscle, ten
sarcomeres were measured from at least two different micrographs in three
toads (N=120). A two-way factorial ANOVA was used to test the effects
of region within the muscle (fixed) and individual (random) on sarcomere
length.
Elastic recoil model of ballistic mouth opening in toads
In order to account for the power output of the depressor mandibulae
muscles during ballistic mouth opening, we developed an elastic recoil model
of the toad cranium and jaws (Fig.
4). In the model, each depressor mandibulae muscle
(Fig. 4, red symbols) is
represented as a force generator (i.e. cross bridges) in series with a spring
(i.e. series elastic component of the muscle). Extra-muscular connective
tissues at the origin and insertion are represented together as a single
spring in series with the muscle (Fig.
4, blue symbols). The springs originate on the cranium and suspend
an effective mass (meff) equal to the product of the mass
of the lower jaw plus tongue and its moment plus the mass of the depressor
mandibulae muscles and retroarticular processes. Dashpots
(Fig. 4, purple symbols)
represent the total effective damping (beff) of the
feeding apparatus. When the depressor mandibulae muscles become active prior
to mouth opening, they store elastic potential energy in both intramuscular
(red) and extra-muscular (blue) springs. When the mouth opens rapidly, the
springs return to their equilibrium position (xt) at a
rate that depends on the total stiffness (kt), effective
mass (meff), and effective damping coefficient
(beff). Given the anatomical arrangement of the various
spring elements (Fig. 4), the
expressions for the total displacement (xt) and total
stiffness (kt) are as follows:
![]() | (8) |
![]() | (9) |
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To implement the model, the non-linear, load-dependent displacements (xm) and spring constants (km) of the depressor mandibulae muscles at in vivo loads were estimated from load-clamp data. The displacement (xe) and spring constant (ke) of the extra-muscular connective tissues were estimated from in situ muscle stimulation experiments. Eqn 8 and 9 were used to calculate the total spring constant (kt) at the in vivo load that corresponds to a given total displacement (xt) and total force (Ft). For a given total force (Ft), the total spring constants (kt) predicted by the model were compared to the total spring constants (kt) estimated from kinematics as described above.
| Results |
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The total spring constant (kt) and effective damping coefficient (beff) were estimated by fitting the raw kinematic data to the time solution (Eqn 3) of the general equation of motion for a single degree-of-freedom damped mass-spring system. Mean values of kt for each toad ranged from 508 to 712 N m1 (Table 1). Mean values of beff ranged from 4.1 to 5.6 Ns m1 (Table 1), which correspond to damping ratios of 85.7108.5% (mean=94.5%). Eqn 2 produced good fits to the experimental data. For 71 trials from four toads, r2 ranged from 0.9810.999 (Table 1, Fig. 5A).
Electromyography
The depressor mandibulae and levator mandibulae muscles exhibited
bilaterally symmetrical activity when crickets were captured at distances
ranging from 111 cm. The depressor mandibulae muscles consistently
exhibited a biphasic activity pattern, with a first large burst that began
49247 ms before the onset of mouth opening
(Fig. 6A,
Table 3). Ballistic mouth
opening begins at the end of the first burst, and is followed by a second,
shorter burst of activity associated with prey transport. Prior to mouth
opening, the levator mandibulae posterior longus showed low amplitude activity
(
20%) relative to its activity during mouth closing
(Fig. 6A).
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Two toads showed a significant positive association between total displacement (xt) and total integrated area of depressor mandibulae activity preceding mouth opening (Fig. 6B,C, all P<0.05, r2=0.220.27). These results are consistent with the hypothesis that the total displacement (xt) of the depressor mandibulae muscles and series connective tissues increases with the force (Ft) that develops in the depressor mandibulae muscles prior to movement. All four toads showed a significant positive association between the duration of depressor mandibulae activity preceding mouth opening and the distance to prey (Fig. 6D, all P<0.05, r2=0.260.66), suggesting that the muscles are activated earlier, relative to the onset of mouth opening, when prey are farther away.
Contractile properties of the depressor mandibulae muscles
At resting length, the depressor mandibulae muscles of toads (N=3)
are also at their optimum length, L0
(Fig. 7A). The slack length of
the depressor mandibulae muscles is
1 mm less than their resting length.
Stretching the muscles to their resting length produced a passive tension of
0.1 N (711 mN mm2,
2.5% P0).
Below L0, both active and passive tension decline rapidly.
Above L0, active tension decreases rapidly, but total
tension increases rapidly due to the increase in passive tension.
|
4) than the shortening velocity of frog limb
muscles (Vmax
12 ML s1)
(Lutz and Rome, 1994
Elastic properties of the depressor mandibulae muscles
Load-clamp experiments were used to investigate the elastic properties of
the depressor mandibulae muscles during elastic recoil. Original records for
the load-clamp experiments in which the muscle was activated isometrically for
200 ms prior to load reduction are shown in
Fig. 8A,B. From these records,
the distance shortened during the fast phase was calculated for each
load-clamp trial as described above (see
Fig. 3). The maximum
displacements (xm) obtained in situ using the
force lever were 2.6 mm (15.3% ML), 3.0 mm (17.5% ML), and
2.5 mm (15.3% ML), corresponding to changes in force
(
F) of 3.27 N, 3.34 N and 2.45 N, respectively
(Fig. 8B,
Table 4).
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The load-clamp experiments demonstrate that the depressor mandibulae
muscles behave as non-linear springs in which muscle displacement
(xm) during elastic recoil increases exponentially with
the change in load
F=FbeforeFafter.
The shape of the exponential functions:
![]() | (10) |
r2
0.997; Fig. 9A).
|
F=c1+c2log(xm)]:
![]() | (11) |
F is
small. During elastic recoil, the stiffness (km) of the
depressor mandibulae muscles decreases rapidly and non-linearly as the change
in force during unloading (
F) increases
(Fig. 9B). The stiffness
(km) decreases as
F increases because the
displacement (xm) increases exponentially with
F.
During load-clamp experiments,
F can be controlled by
varying either the duration of muscle stimulation
(Fbefore) or the clamp load (Fafter).
However, in vivo, Fafter is constant for a given toad at a
given time, but toads can vary Fbefore (and therefore
F) by modulating the duration and/or amplitude of muscle
activation prior to movement. In order to model ballistic mouth opening in
vivo, the constants c1 and c2
from all six series of load-clamp experiments were regressed with
Fbefore (Fig.
9C), so that the parameters xm and
km could be predicted for any value of
F
as follows:
![]() | (12) |
![]() | (13) |
Lever artifacts
Due to the oscillatory artifacts introduced by the interaction between the
muscle and the servo-controlled lever, previous workers (e.g.
Huxley and Simmons, 1971
;
Ford et al., 1977
) have
questioned the usefulness of the load-clamp technique for understanding the
viscoelastic properties of muscle. The oscillations in muscle length that
follow the rapid decrease in load are caused by several factors, including:
(1) conversion of elastic potential energy to kinetic energy in the muscle,
which causes it to overshoot its equilibrium position; (2) the inertia of the
force lever itself; and (3) the interaction between the muscle and the
servo-controlled force lever, which arises from the delay in the response of
the lever to changes in force. In the experiments reported here, the lever was
tuned to produce minimal damping (2 ms step response).
We used a modeling approach to estimate the contribution of lever artifacts
to the observed fast phase displacement. To estimate the expected
time-dependent shortening of the muscle itself (in the absence of lever
artifacts), we again used the time solution (Eqn 3) of the general equation of
motion for a damped mass-spring system. We modeled the expected shortening
behavior of the depressor mandibulae muscle using the load-clamp trial with
the greatest change in force (3.34 N, see
Fig. 8B). For this trial, Eqn 3
was used to predict the expected time-dependent shortening behavior of the
depressor mandibulae muscle (data not shown). The initial force, the change in
force (
F), and the total fast phase displacement
(xt) were used to calculate the muscle stiffness
(km) at a given clamp load using Eqn 11. Berkeley Madonna
(version 8.0.1) was used to find the value of the damping coefficient
(beff) that produced the best fit between the model and
the observed relationship between muscle length and time. This simple model,
which assumes no lever artifacts, explained 95.4% of the observed variance in
muscle length vs time (see Fig.
8B) in the load-clamp experiment with the largest oscillations and
the greatest change in force. It explained a higher proportion of the variance
in trials for which the change in force was smaller. This modeling approach
shows that, taken together, all lever artifacts combine to produce an increase
in the magnitude of the first oscillation, but they have no measurable effect
on the natural period of vibration.
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Estimating strain in series elastic elements during depressor mandibulae stimulation
During supramaximal depressor mandibulae stimulation in situ, the
total strain of extra-muscular series elastic structures of the cranium and
lower jaw ranged from 1.5 to 2.3 mm (8.914.1% ML) among the
three specimens (Fig. 10A,C).
Strain in the retroarticular process along the line of action of the depressor
mandibulae ranged from 8.1 to 12% ML, with a mean of 10.6% (data not
shown). Strain at the muscle's cranial origin ranged from 0.9 to 2.4%
ML (data not shown).
For each muscle, data on shortening of the depressor mandibulae vs time, collected using in situ muscle stimulation with synchronous digital imaging, were combined with force vs time data from the same muscle, collected using the force lever, to produce estimates of the total strain in extra-muscular connective tissues as a function of muscle force (Fig. 10B). The force vs strain relationship was linear in two of three cases (Fig. 10B, pink and blue symbols), and was linear when data from all three specimens were pooled. The spring constant for all extra-muscular structures combined (ke) was estimated as the slope of the linear relationship between total muscle shortening and force, constrained to pass through the origin. For any given force, the total strain in extra-muscular connective tissues (xe) was estimated as the product of muscle length (lm) and the linear function relating relative force (Frel) to relative strain (measured as a proportion of muscle length).
The spring constant for all extra-muscular structures combined
(ke), estimated as the slope of the linear relationship
between total muscle shortening and force
(Fig. 10B), was 1300 N
m1 (range=10401865 N m1). The total
strain (xe) in all extra-muscular structures combined was
estimated from the slope of the relationship between relative strain and
relative force (Fig. 10C) with
the intercept constrained to pass through the origin
(Fig. 10C):
![]() | (14) |
|
Elastic recoil model of ballistic mouth opening in toads
The purpose of the elastic recoil model was to account for the observed
strain, shortening velocity, acceleration and power output of the depressor
mandibulae muscles and series connective tissues during ballistic mouth
opening. The model is based on the elastic properties of the muscles (measured
in situ during load-clamp experiments, Eqn 10 and 11) and
extra-muscular connective tissues (measured in situ during muscle
stimulation experiments, Eqn 14), in the context of their anatomical
arrangement (Fig. 4, Eqn 8 and
9).
The paired depressor mandibulae muscles originate on the cranium and insert on the retroarticular processes of the lower jaw (Fig. 4). Each muscle was modeled as a force generator and a spring arranged in series (red symbols). On each side of the cranium, the depressor mandibulae muscle is arranged in series with a spring element that represents the sum of all extra-muscular structures that are strained by contraction of the depressor mandibulae prior to movement (i.e. cranium and retroarticular process). The springs on the left and right sides of the cranium are arranged in parallel and, from these, the effective mass is suspended (Fig. 4).
Given these anatomical arrangements, the total displacement
(xt) is equal to the sum of the displacements of the
muscle spring (xm) and connective tissue spring
(xe) arranged in series on each side of the cranium (Eqn
8). Substituting the experimentally derived expressions for
xm (Eqn 10) and xe (Eqn 14) gives the
expression for the predicted total displacement (xt):
![]() | (15) |
![]() | (16) |
In the model, connective tissue strain increases linearly with force, whereas muscle strain increases slightly non-linearly with force (Fig. 12A). At all but the lowest forces (<0.25 N), strain in the depressor mandibulae muscles is greater than strain in the extra-muscular connective tissues. Estimated maximum depressor mandibulae strain in vivo ranged from 1.3 to 2.2 mm, and estimated maximum connective tissue strain in vivo ranged from 1.2 to 1.6 mm (Table 5). In the in situ experiments (load-clamp and series connective tissue strain), the force that develops in the depressor mandibulae muscles (i.e. maximum isometric force) is higher than that achieved in vivo (i.e. voluntary force). Thus, the estimated in vivo displacement of the depressor mandibulae muscles and extra-muscular connective tissues is smaller than that observed in situ, as expected.
|
In the model, the total stiffness (kt) during elastic
recoil varies non-linearly as a function of both the force that develops prior
to movement and the external load during movement
(Fig. 12B). During elastic
recoil, the total stiffness is greater either when the force before movement
is smaller or when the external load is larger (i.e. whenever
F is smaller). The elastic properties of muscle during elastic
recoil thus differ substantially from the elastic properties of both inactive
muscle, in which stiffness increases non-linearly with stretch, and active
muscle under isometric conditions, in which stiffness increases linearly with
force.
The model contains three sources of random error, including the confidence intervals for the slopes of the coefficients c1 and c2 with respect to total force (Fig. 9C) and variation among individuals in the stiffness of the extra-muscular connective tissues (Fig. 10B). Because these sources of error are statistically independent, the probability that all three variables will exhibit values outside their 95% confidence intervals simultaneously is equal to (0.05)3, or 0.000125. Thus, the 95% confidence interval for the model as a whole was calculated from the 62.8% confidence intervals for each of the three variables [1(0.628)3=0.05; Fig. 12C, dotted lines]. The 95% confidence intervals of the model as a whole are relatively large because the estimated values of each variable are based upon relatively small samples (N=36).
The elastic recoil model was tested by comparing the total spring constant (kt) estimated from in vivo kinematics using Eqn 3 (Table 1) to the values of kt predicted by the elastic recoil model (Table 5). In general, there is a good match between the observed and predicted parameters (Fig. 12C). The observed values of kt ranged from 508712 N m1, with a mean of 581±47 N m1 (Table 1). Values of kt predicted by the model ranged from 550 to 714 N m1, with a mean of 608±37 N m1 (Table 5). For all four individuals, the values of kt estimated from kinematics were well within the 95% confidence interval for the kt predicted by the elastic recoil model and vice versa (Fig. 12C).
| Discussion |
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Electromyograms synchronized with digital images of freely behaving toads
demonstrate that the depressor mandibulae muscles are active for up to 250 ms
prior to the onset of ballistic mouth opening. The depressor mandibulae
muscles are deactivating, or electrically silent, during ballistic mouth
opening (Fig. 6). It appears
that de-activation during movement may increase the efficiency of recovery of
elastic strain energy. Lou et al. showed
(Lou et al., 1999
) that work
stored in the series elastic component of dogfish axial muscle could be
recovered completely as external work when the muscle shortened during
relaxation, whereas only 80% of the work was recovered if the muscle remained
active during shortening.
Strain of elastic elements in series with the depressor mandibulae muscles
Digital imaging demonstrates that both the cranium, at the muscle's origin,
and the retroarticular process, at the muscle's insertion, are strained d