Muscle produces two kinds of force, active and passive, which sum to compose a muscle’s total force. A muscle’s contractile elements provide its active force through the actin and myosin “ratcheting” mechanism. Noncontractile elements contribute its passive force. Technically, a muscle’s passive element has properties most correctly termed elastic, but it can be modeled more simply as a spring. Because this spring-like element attaches in series with the contractile element, you can think of the force that the contractile element produces as an active force transmitted to the skeleton via a series elastic element. Muscles, however, have another elastic element, as well, called a parallel elastic element, that also contributes to its passive force.
1922, A. V. Hill
(see Hill, 1970)
first noted that activated muscles produce more force when held
isometrically (i.e., at a length fixed) than when they shorten.
When muscles shorten, they appear to waste some of their active force in
overcoming an inherent resistance. This
resistance could not result from the series elastic element because it resists
lengthening not shortening. So Hill
thought of this resistance as another kind of passive force in the muscle.
He found that the faster a muscle shortens, the less total force it
produces. Assuming a constant
active force, Hill concluded that the faster shortening leads to a larger
Hill drew an analogy between the resistive force a shock absorber.
A piston in a viscous fluid exemplifies a simple shock absorber, also
known as a damper (Figure
1). If you push on its
piston, a shock absorber will resist by a tension T (equivalent to a
force) that depends on the viscosity b of the fluid in its cavity.
The faster you try to push the piston, the stronger the fluid resists.
For a given speed
, the force that you need to move the piston is . To account for the fact that
muscle produces less force when it shortens, Hill proposed that this viscous
element lies in parallel with the contractile element.
Accordingly, this component can be called a "parallel elastic element".
To investigate the properties of this viscous element, Hill and his colleagues performed a simple experiment. They attached a muscle to a bar that pivoted around a point (Figure 2A). One end of the bar had a catch mechanism that they could release at any time. A basket held a weight on the other end of the bar. When Hill released the catch, this weight would pull on the muscle by a force T. The experiment began with the catch in place and the muscle stimulated maximally. The stimulation resulted in the production of force To in the muscle. Because the muscle pulled on a bar that could not move, the force that the muscle produced did not change the muscle’s length.
At this point, the experimenters released the catch. Note, in Figure 2C, how the length of the muscle suddenly shortened and, in Figure 2B, how the force dropped to T from To . After this rapid phase of shortening, Figure 2C shows how the muscle continued to shorten, but now gradually. The fact that the muscle immediately shortened by amount Dx1 and reduced its force from To to T suggests that something in the muscle acted like a spring. If you put tension on a spring by pulling it, then suddenly release it, the spring will rapidly shorten. This spring is the series elastic (SE) element referred to above and its stiffness is KSE in Figure 2E. Recall that stiffness relates changes in force (or tension) to changes in length:
After the immediate change in muscle length and force, a slow, gradual
change in length developed (Figure
2C), without any change in force (Figure
2B). Whereas a part of the
muscle’s mechanism changed length rapidly in response to the force change,
another part did not change as quickly—as if a “shock absorber” acted on
the “spring,” slowing its response to the force change. The parallel elastic (PE) element, referred to above,
represents this second passive element in the muscle, and its stiffness is KPE
The muscle’s viscosity, the parallel elastic element and the series
elastic element compose the passive components of an elementary model muscle.
2, the length of the series elastic element is x1 and
the length of the parallel elastic element is x2.
Note that this is a model of a muscle: it does not imply that the
various components have this physical arrangement within a muscle.
Further, the viscous component of muscle tension results from mechanisms
very different from the mechanical damper depicted in Figure
1. Mathematically, however,
the characteristics of the muscle accord reasonably well with this depiction.
You might think of the model as a simile.
In the model, like in a muscle, when the tension in the system
suddenly decreases, the series elastic element responds immediately, but the
parallel elastic element responds gradually because of its viscous component.
The muscle’s active component contributes the final piece of the
mathematical muscle model. This
active force acts against the passive components of the model (and muscles) to
produce the final force that acts on the bar in Figure
2D. Function A
indicates the active component in Figure
The model can now describe how the total force produced by the muscle depends on its passive and active components.
Assume that the series elastic element—its most spring-like
component—has a resting length
and the parallel elastic spring has
a resting length
. The same force T develops
in both of these elements because the muscle can have only one force at any
given time. Thus,
the total length of the muscle must be the sum of the lengths of the series
elastic and parallel elastic elements:
that in this equation, the active force A, muscle length
rate of change
, and force T are all function of time.
How well does this model capture the behavior of a real muscle? Figure 3A shows the tension of a frog gastrocnemius muscle under various conditions. The gastrocnemius is the calf muscle, and somehow the frog’s gastrocnemius became a prototype for skeletal muscles in the early days of motor physiology. Other muscles may differ in certain details, but the basic principles of striated muscle are highly conserved in evolution. In the experiment described here, the muscle was stimulated electrically. As Figure 3A shows, the first excitatory input caused an initial isometric twitch. Later, during maximal activation, tetanus occurred, which represents the maximal sustained output that the muscle can produce. During maximal activation, the muscle was stretched (Figure 3B) and held in a fixed, stretched state for about one second, then released. Shortly after release, the stimulation stopped.
after the stretch, the muscle’s force level increased rapidly.
According to the model, the series elastic element must have caused this
increase because the active component had reached its maximal output prior to
the stretch—recall that the muscle was in tetany.
3 shows that this force increase amounted to 150 g, which, for a change
in length of 1.1 cm, means that stiffness:
the stretch, the force in the muscle slowly declined to a steady-state value
(arrow in Figure
3A). You can compare the
steady-state values of force before and after the stretch and use the change to
solve for parameter KPE in the model.
Using Eq. (2),
Thus, the stiffness of the series elastic element exceeds that of the
parallel elastic element. However,
stiffness does not completely describe the passive component of the parallel
elasticity because it also has viscosity. Accordingly,
your final step is to estimate the viscosity of the parallel elastic element.
Consider the conditions after the sudden stretch: the length of the
muscle, as shown in Figure
3B, did not change (i.e.,
). Nevertheless, the force
gradually dropped, as shown in Figure
3A (arrow). If you
arbitrarily call the time of the sudden muscle stretch t = 0, then the
force in the muscle was:
statement is a differential equation of the form
, which has a general solution of the form
, where c is a constant that depends on the initial conditions, e.g.,
force at time zero. Solving the
differential equation produces
that in an exponential function of the form
, the time constant
specifies the time when the
exponential has declined by about 63%. From the muscle force trace of Figure
3A, you might estimate that
sec for the frog’s calf muscle. From this you have
that you have the parameters of the muscle model, you can simulate its behavior
and compare the results of that simulation to empirical data.
shows the results of
the simulation. Assume that a
muscle is at rest at t = 0 and
its length is 1.0 cm. You simulate
pulling on the muscle for 0.05 sec so that its length increases linearly by 1.0
cm over this period. Hold the muscle at this new length for an additional 0.95
seconds, at which this point you release the muscle and its length returns to
the original 1.0 cm at time 1.05 sec. Using
you can simulate the force produced in this muscle (
If you increase viscosity, the trace declines slower when the muscle is
stretched, and rises slower when the muscle is released compared to the trace
shown in the figure.
This simple muscle model illustrates that the total force in a muscle
depends not only on the actively produced force, but also on the passive
elements of the muscle. Because the
active force has to work against a passive viscosity, muscles produce the most
force in isometric conditions. Passive
forces also provide a virtually immediate influence that promotes limb
stability. Because muscles act like
springs and have viscosity—that is, because they have viscoelastic
properties—forces imposed on the limb meet resistance immediately, long before
signals could reach a central controller and return.
Now return to the force trace of the frog calf muscle (Figure 3). At the beginning of the trace, the muscle was held in an isometric condition and electrical stimulation produced a single twitch. In the model of Figure 2E, activation of the muscle engages the contractile element that produced an active force A(t). This active force interacts with a viscosity and the two elastic elements to result in a muscle force of T(t). Using Eq. (3), under isometric conditions you have
function T(t) simply describes the force trace during the twitch.
You can sample this function at a few points and approximate it by
fitting it to the sum of two exponential functions (Figure
5A). Using the values for
the elastic and viscous elements derived in the previous section and Eq. (4),
you can now calculate the shape of the active force trace (Figure 5B).
active force function
that was computed from a single
twitch in Eq. (4)
is the impulse response of the contractile element:
If you assume that the contractile element is approximately a linear system, then the impulse response is all that you need to estimate the active force for any sequence of impulses. You can ask how your muscle responds when a train of action potentials, each an impulse, acts on the muscle. The resulting active force will, in turn, interact with the passive properties of the muscle to produce a total force. Figure 6 shows the resulting total forces for various stimulation rates. As the impulses arrive at a faster rate, the total force in the muscle builds, according to the following relations:
is, the total force of the muscle is a function of the force produced by a
current excitatory input and the residual forces from previous inputs that have
not yet decayed completely. Of
course, the faster the inputs arrive, the less decay occurs, and the total force
of the muscle begins to add up. At
its limit, the muscle reaches tetanus, the state in which the
experimenters placed the muscle in Figure
The final component of the muscle model follows from the observation that
active force not only depends on the time course of action potentials, but also
on muscle length. Muscle length
determines the amount of overlap between the thick and thin filaments of the
muscle fibers, affecting the number of myosin heads eligible for binding with
actin. To capture this dependence,
you can scale the active force produced in Eq. (5)
by a function s(x),
increases value from 0 to
1 as muscle length increases and then declines as
muscle length increases further. When
the value of s(x) is zero, tetanus produces no force, which occurs when the muscle
is much shorter than its resting length. In
essence, in this state the myosin “ratchets” have no mechanical advantage
and cannot generate any force, regardless of the amount of excitatory input.
When s(x) is unity, which
occurs one at some length in a given muscle, the muscle produces its maximal
force. At greater lengths, s(x)
gradually decreases as the ratcheting mechanisms falters because of a mechanical
disadvantage at excessive lengths.
the model, two elastic elements and one viscous element represent the passive
properties of the muscle. The
physical properties of muscle fibers readily account for the elastic elements,
which result from the spring-like properties of the connectins within the muscle
fibers, as well as the connective tissue that surrounds them.
The viscous element, however, does not result from any actual fluid that
resists length change in the muscle. Recall
that the viscous element in the model accounts for the observation that an
activated muscle produces less force when allowed to shorten than in isometric
conditions. The model assumes that active force is more or less the same
when a muscle is held in an isometric state as when it shortens.
However, this is not the case: in a shortening muscle the myosin heads
detach and need to be recocked. During
this time, they cannot contribute to the contractile element’s active force.
Accordingly, the fact that active force is smaller during shortening is
not due to an actual, physical viscosity but rather to the characteristics of
the actin–myosin contractile mechanism, which you can model as a viscosity.
The model portrayed in Figure
2E should thus be viewed as a mathematical description of the mechanical
properties of the muscle, not as a representation of how muscles actually
generate force. The Figure
2E model depicts a mechanical analogy to the muscle in that it captures
the dependence of force on muscle length and the rate of length change.
Knowledge of these properties and the ability to model them will aid in
understanding how the CNS controls muscles and how their passive properties
provide stability in controlling a limb.
mathematical muscle model developed in the previous section for extrafusal
muscle fibers can be extended to intrafusal fibers, as well.
7 shows schematic of the muscle-spindle system, analogous to the model
presented in Figure
2. The contractile element
of the spindle lies at the pole region, which receives synaptic inputs from g-motor
neurons and sensory innervation from a secondary (Group II) muscle-spindle
afferents. The central
region—called the nuclear bag region—lacks contractile properties and
receives sensory innervation from primary (Group Ia) muscle-spindle afferents. Forces that stretch the muscle spindle result in length
changes in the nuclear bag and pole regions, and the muscle-spindle afferents
transduce this length change into firing rate.
7 represents the muscle spindle as a system with two elastic elements and
one viscous element, like the mathematical model of extrafusal muscles depicted
2. The length of the series
elastic (SE) element represents the length of the nuclear bag region.
The primary muscle-spindle afferent discharges at a rate proportional to
this length. Here x signifies how much the entire intrafusal muscle
extends beyond its resting length, and x2 to signify how much
the series elastic element extends beyond its resting length.
The contractile component lies in the pole region and it corresponds to
the length of the parallel elastic (PE) element: x1 signifies
how much it extends beyond its resting length.
If you assume that the primary muscle-spindle (Group Ia) afferent
responds linearly to length changes in the SE element, then you can write its
discharge rate as
. Similarly, the discharge at time t
of the secondary (Group II) afferent may be written as
. Although the responses of these
neurons does not correspond exactly to these assumptions of linearity, they
serve as useful approximations. Finally, when the intrafusal muscle contracts via stimulation
of the g-motor
neuron, the contractile component produces force g(t).
Remember the following relations:
Consider what the spindle afferents in the model should do if you
suddenly stretch the muscle-spindle and maintain it at a given increased length.
Remember that the parallel elastic element cannot change length
immediately, due to its viscous component, but the series elastic element can.
Accordingly, you should see a large initial increase in the discharge of
the primary muscle-spindle afferent. Gradually,
as the parallel elastic element overcomes the effect of its viscosity, it will
increase in length. Because the sum
of the lengths of the parallel and series spring-like elements equals the total
length of the muscle spindle, and the spindle length remains constant after the
stretch, length of the series elastic element should gradually decrease after
the stretching stops. As a result
of this relaxation in the polar region of the muscle-spindle, the nuclear-bad
regions should gradually shortening, which should result in a reduction in the
discharge of the primary muscle-spindle afferents. In fact, this predicted pattern of activity—a rapid
increase followed by a gradual decrease—resembles the discharge dynamics of
primary muscle-spindle afferents during a sudden stretch (Figure 7).
describe the dynamics of length changes in the muscle spindle, you take the
total tension on the muscle spindle T and relate the rate of change in
tension to the length x and the active forces g that g-motor
neurons generate. You can do this
with procedures analogous to those used for the mathematical model of the
extrafusal muscle. These
relationships are described as follows:
after a sudden stretch ends,
, which means that the muscle-spindle (and the muscle) have stopped changing
length and the tension on the spindle is the sum of the active force generated
by the parallel series element g and the passive, spring-like properties
of the series element. Solving the
above differential equation yields:
states that after the stretch, the tension in the spindle should gradually
decrease. Because the discharge in
the primary spindle afferent is proportional to the length of the series elastic
element, it is also proportional to this tension. This fact, which is essentially a restatement of the
length–tension relationship x = T/K, leads to the
implies that at the time of the stretch there should be a large response in the
primary muscle-spindle afferent, followed by a gradual decline to steady-state
the secondary muscle-spindle afferent, you have:
suggests that after the stretch, its discharge rate should gradually increase
Figure 9 provides a
simulation of the discharge patterns in muscle-spindle afferents. Reasonable parameter values for this simulation are: KSE
= 35 g/cm, KPE
= 5 g/cm, B = 10 g.s/cm, a = 220, g(t) = g(0) = 0.
The spindle is pulled from rest by a stretch that lasts 0.5 seconds and
lengthens the spindle by 0.5 cm. The
simulated primary muscle-spindle afferents show a great sensitivity to the
“dynamic” phase of the stretch, whereas the simulated secondary afferents
show a monotonic response that mainly reflects the “static” phase of the
Now consider how the velocity of stretch might affect the activity of the spindle afferents. Assume that you stretch a spindle 0.5 cm but at a very slow rate, 5 mm/s. In this case, the effect of the viscosity will be small and, accordingly, the parallel elastic element will lengthen almost as quickly as the series elastic element. In contrast, when you stretch the spindle rapidly, for example at a rate of 30 mm/s, the viscosity greatly resists stretch of the parallel elastic element and more of the total stretch in the spindle will be taken up by the series elastic element, at least at first. Therefore, during a rapid stretch, the primary muscle-spindle afferent will fire much more than during a slow stretch. These properties create the appearance of a velocity signal, as illustrated both in the model and in recordings from a cat soleus muscle in Figure 10.
the model in Figure
7 suggests, activity of the g-motor neuron contracts the spindle and
affects the lengths of the series and parallel elastic elements.
This contraction, in turn, affects the discharge of the muscle-spindle
afferents. If your muscle spindles
did not have g-motor neurons, their length changes would
simply reflect the length changes in the extrafusal muscle.
What benefit does the nervous system gain by having a muscle-spindle
system in which length depends not only on the extrafusal muscles—and
therefore the angle of joints—but also on the activity of g-motor
neurons? As you see in this
section, activation of g-motor neurons allows the CNS to bias the sensitivity of the primary
spindle afferents and turns them into a sensor that measures movement errors.
Suppose that you decide to flex your elbow.
You CNS activates a-motor
neurons that act on the biceps, producing force in the extrafusal muscles and
shortening the muscle along trajectory x(t).
In this case, the term trajectory applies to the change in muscle length
as a function of time. Of course, trajectory in this sense has an obvious
relationship with trajectory in the more-usual sense of kinematics: e.g.,
the movement of the hand in space as a function of time. Now suppose that the CNS programs the g-motor
neurons based on its expectation of how the biceps should shorten during
this movement. Later chapters deal
with this notion of expectation in detail, but for the present purpose you can
think of this conceptually as the desired (d) trajectory of muscle length
as a function of time: xd(t).
As you activate your a-motor neurons, you also activate the g-motor
neurons—a coupling termed a–g coactivation—and you do
so such that if the muscle changes length according to your expectation, tension
in the muscle spindle will not change, i.e.,g(t). You can find this level
of intrafusal-fiber force by solving for it in Eq. (3):
The above equation tells you that when your biceps is about to shorten, your CNS must activate the g-motor neurons by the amount g(t) to maintain tension in the spindle. Figure 11 shows an example: As the simulated muscle shortens by 0.5 cm, the desired trajectory programs activity in the g-motor neuron, which produces force g(t) in the muscle spindle. This increased tension precisely cancels the decrease in tension due to shortening of the muscle. Therefore, the series elastic element does not change length and, therefore, the primary muscle-spindle afferent does not discharge. However, because of activating the contractile element in the spindle, the parallel elastic element has shortened, which causes reduced discharge in the secondary muscle-spindle afferents.
Now suppose that as
the biceps muscle shortens, an unexpected perturbation prevents it from flexing
the elbow as much or as fast as desired. The
g-motor neurons have activity that corresponds
to the desired trajectory, but—because of this external force—the actual
trajectory falls short. When you
simulate the dynamics of the system under this condition, you see that the
primary muscle-spindle afferents do not remain silent during the perturbed
movement, but strongly increase their activity, instead (Figure
12). In other words, if,
during contraction, the muscle shortens less than expected, primary
muscle-spindle activity increases despite the fact that the muscle shortens.
By modulating the activity of the g-motor neurons, the CNS has gained a mechanism
by which it can detect error in a movement: more or faster shortening than
expected depresses primary muscle-spindle activity; whereas less or slower
shortening results in the activation of these afferents.
Activity in primary muscle-spindle spindle afferents thus signals less
muscle shortening than expected. The
CNS deals with this information very directly: the afferents monosynaptically
neurons of the same muscle. Therefore,
the error signal acts through the quickest possible CNS route to further
activate the extrafusal muscles, produce more force, and further shorten the
muscle. In order for the primary
spindle afferents to act as an error detector, the simulations suggest that the g-motor
neurons must coactivate with a-motor neurons.
In contrast, activity in the secondary spindle afferents does not reflect
an error signal but instead send to the CNS a fairly faithful measure of actual
muscle length—with brief deviations from a pure length signal due to the
viscous component of the parallel elastic element.
These neurons lack monosynaptic input on the a-motor neurons, but instead act on interneurons in the spinal cord, some
of which receive inputs from the brain.
the section above entitled
of passive properties
, you saw how the passive, spring-like properties of muscles provide a
line of defense against
model developed in this chapter shows how afferent feedback from muscles
provides a second line of defense. Errors
and instability provoked by external perturbations can be meliorated, if not
corrected, by a combination of these active and passive muscle properties:
primary muscle-spindle afferents appear to play the largest role during rapid
movements, with secondary afferents doing so during maintenance of steady
summarizes the functions of the muscle-spindle and golgi-tendon-organ afferents
in terms of a feedback control system. Inputs
neurons cause muscles to produce force that acts on tendons, shortens muscles,
and changes joint angles. Muscle-spindle
afferents sense changes in muscle-length, as biased by activity in the g-motor neurons. The golgi
tendon organs sense force changes, with their signal to inhibitory interneurons
modulated by the descending inputs.
feedback control system rejects perturbations and helps ensure that the limb
follows the desired trajectory. Two
locations in the model adjust outputs: one modulates the level of g-motor-neuron
activation, the other does so for the interneurons that golgi tendon organs act
on. Consider a situation in which the load exceeds expectations.
For example, you see a milk carton you assume to be empty, so you decide
to dispose of it. When you begin to lift the carton after grasping it, its
weight—the load on your muscles—is more than expected.
The prime-mover muscles will shorten, but because of the
increased load, they will not shorten as much as expected for an empty milk
You saw earlier that the g-motor neuron activity is set based on
the expected length change in the muscle. When
the muscle shortens less than expected, afferent signals further activate the
a-motor neuron, producing more force, and countering the effect of the load, at
least to some extent. The force
feedback system, mediated by the golgi tendon organ, works similarly. As depicted in Figure
13, inputs signaling force from the golgi tendon organs inhibit
activity in motor neurons via a spinal interneuron.
This influence would have the effect of a negative feedback loop in which
high levels of force result in decreasing the motor command and therefore
moderating the force. By inhibiting
the interneuron that the golgi tendon organ acts upon (unfilled arrow in Figure
13), descending inputs can modulate how effectively force feedback
inhibits the a-motor neurons. Strong
inhibition from the descending inputs imposes a high threshold for the force
feedback pathway. Only if the force exceeds this threshold will the feedback pathway