Simple Neuromuscular Models

Introduction

• Until now, we have studied neurons working together, and (in the previous unit) capable of sensing inputs from their environment; but to complete the connection between the nervous system and behavior, neurons need to cause actions in their environment.

• In general, neurons do this by activating specialized structures that induce movement. Most of the time, the structures that induce movement are made of muscle, the primary motor for movement.

• In this unit, we will introduce you to basic concepts about how muscles work, and then allow you to study a simulation of a muscle. We will then describe different body plans that incorporate muscle, and then allow you to explore a particular example of a muscular structure, a model of a tongue.

Muscle

• The tissue that is primarily responsible for generating force and motion is muscle.

• There are three major categories of muscle: skeletal muscle, the kind of muscle we find in our arms or legs, smooth muscle, which is found in blood vessels and various organs throughout the body, such as in the bladder and the lungs, and cardiac muscle, the muscle of the heart.

• Although there are significant differences in the organization of these three different forms of muscle, all of them share a key mechanism: the movement of filaments composed of actin and myosin relative to one another to generate force and motion.

• The filaments consisting of actin are referred to as the thin filaments, and those consisting of myosin are referred to as the thick filaments.

• Parts of the myosin molecule in the thick filaments protrude outwards, and these small round structures (the myosin head) can bind to the actin molecules in the thin filaments, forming a crossbridge. The binding requires energy (adenosine tri-phosphate, ATP).

• Once bound, the myosin head can rotate, and this rotation moves the thick filament relative to the thin filament. After the movement, the myosin head releases, and then can repeat the movement.

• The relative positions of the thick and thin filaments are critical for the ability of muscle to generate force. If the filaments are pulled too far apart from one another, or overlap too much, the myosin heads cannot form crossbridges. As a consequence, muscle force is dependent on muscle length. If a muscle shortens or lengthens beyond the optimal length for cross bridge formation, the force that it can generate falls. This relationship between force and length is referred to as the length-tension property of muscle. This property is often studied by using a servomotor, a device that (like a voltage clamp, which also works through negative feedback) can exert force to ensure that the length of the muscle remains constant, while measuring the force that was needed to do so. Measurements of muscle force in response to stimulation of the muscle while the muscle is held at a constant length are referred to as isometric (same length) measurements.

• Crossbridge formation also depends on the rate at which the filaments are sliding past one another, and whether the muscle is lengthening (which can allow an attached myosin head to exert more force), or shortening (which will reduce the force that the myosin head can exert). This property, which is referred to as the force-velocity property of muscle, is also studied using a servomotor, and requires that the muscle be lengthened or shortened at a fixed velocity, and that the force be measured as the muscle reaches a particular length (so that the force can be compared to that observed when the muscle is held fixed at the same length).

• Neurons that innervate muscles are motor neurons. The connection between a motor neuron and a muscle is referred to as the neuromuscular junction. In vertebrate skeletal muscle, the transmitter released by motor neurons is acetylcholine.

• Binding of the transmitter to receptors on skeletal muscle induces an action potential in the muscle fibers, and this in turn leads to an enormous increase in calcium ions released into the muscle, in part through calcium stores within the muscle. The increase in calcium levels triggers the entire process of crossbridge formation, and is referred to as excitation-contracton coupling.

• Because calcium levels are very carefully maintained within muscle, a single action potential in a motor neuron does not create maximal calcium levels. As a consequence, the frequency with which the motor neuron fires determines the rate at which force increases, and the maximal force that is reached, a characteristic of muscles that is referred to as the force-frequency property.

• Thus, most models of muscles predict their output force as a function of the force-freqency, length-tension, and force-velocity properties. In some of the simplest models, equations representing each of these properties are simply multiplied by one another. In more realistic models, attempts are made to capture the statistical properties of the crossbridge formation, and then these properties emerge as a consequence of these underlying molecular reactions.

• For understanding mechanical systems, two additional concepts are helpful: stiffness and mechanical advantage:
  • Stiffness is the extent to which a structure resists deformation. Thus, a rod of steel is far stiffer than a piece of rope of the identical dimensions. If two muscles are strongly co-activated around a joint, even though the limb may assume the same position as it would if neither muscle was activated, the joint will be much stiffer.
  • Mechanical advantage is output force divided by input force. If a device has a high mechanical advantage, relatively small input force can generate large output force; if it has low mechanical advantage, the opposite is true. If a muscle has high mechanical advantage, an antagonist muscle may need to be very strongly activated to oppose its actions.

Studying Muscle Properties

You now have enough information to begin to study muscle properties. The link below opens a simple simulation of a linear muscle. In this model, a Merkel cell (mechanoreceptor) synapses directly onto the muscle.

• When the mechanoreceptor is stimulated, it depolarizes and activates the muscle. The Activation time constant is a muscle property that controls how rapidly the muscle responds to depolarizations of the mechanoreceptor.
• When the muscle is activated, it contracts, which both shortens its length and increases the force it applies. Negative force in this simulation corresponds to pulling (shortening).
• One end of the muscle is anchored and cannot move.
• The other end of the muscle is connected to an adjustable spring. The Spring resting position parameter controls the length to which the spring is pulling the muscle. The Spring stiffness parameter controls how strong the spring is.

Question 1: Familiarize yourself with the simulation by completing these exercises.

(a)	Run the simulation with the default parameters. Notice the touch stimulus that is being applied to the mechanoreceptor and how both the mechanoreceptor and the muscle respond. Take pictures. What length does the muscle rest at when it is not activated? What minimum length does it shorten to while it is activated? How does increasing or decreasing the Activation time constant for the muscle affect the muscle contraction?
(b)	Change the Maximum pressure for the first touch stimulus from 8 mN to 0 mN. Take pictures. What length does the muscle rest at? Why does the muscle behave differently compared to part (a)?
(c)	Without resetting the simulation, change the Spring resting position from 6 cm to 4 cm. Take pictures. What is the new resting length for the muscle? Why does changing the Spring resting position affect the resting muscle length?
(d)	Without resetting the simulation, change the Spring stiffness from 0.01 N/cm to 0.001 N/cm. Take pictures. What is the new resting length for the muscle? Why does changing the Spring stiffness affect the resting muscle length?

Question 2: The force a muscle can generate depends on its length at the time of activation. If a muscle is not permitted to shorten or lengthen (e.g., because the muscle is trying to move a heavy load unsuccessfully), then activation of the muscle generates an isometric contraction. The force generated during an isometric contraction depends on the muscle length. A plot of isometric force as a function of muscle length is called a length–tension curve. You will create a length–tension curve for this problem.

(a)	Reset the simulation. To create a length–tension curve, the muscle should be activated with a consistent protocol in every experiment. It will also be easier to take your measurements if you activate the muscle long enough for each contraction to generate its maximum force. To accomplish this, extend the activation time by changing the Stimulus end time for the first touch stimulus from 200 ms to 2000 ms, and run the simulation. Take pictures. What does the Merkel cell (mechanoreceptor) do when the continuous touch stimulus is applied? How would this be different if you were stimulating a Meissner corpuscle instead? (This is review of the previous unit.) What happens to the length of the muscle when it is activated? How much does it change? Can this be classified as an isometric contraction? What is the maximum force the muscle generates under these conditions? (Negative force means the muscle is pulling.)
(b)	You can prevent the muscle from moving by modifying the spring parameters. Without resetting the simulation, increase the Spring stiffness from 0.01 N/cm to 1 N/cm, and run the simulation. Take pictures. What happens to the length of the muscle when it is activated? How much does it change? Can this be classified as an (approximately) isometric contraction? What is the maximum force the muscle generates under these conditions?
(c)	By moving the spring, you can hold the muscle at different lengths. Without resetting the simulation, change the Spring resting position to 10, 9, 8, 7, 6, 5, 4, and 3 cm, and measure the maximum muscle force for each muscle length. Report all your measurements. You do not need to take pictures for this step.
(d)	Plot the data you collected in part (c) with appropriate axes labels and units. How do you interpret your results? At what lengths does the muscle produce the most force? At what lengths does the muscle produce the least force? Why do you think this might be?

Body Plans

• In general, muscles are part of larger structures, forming the body of an animal.

• The three major kinds of body plans that have been described are musculo-skeletal systems, hydrostatic skeletons, and muscular hydrostats.

• We are most familiar with musculo-skeletal systems, since our arms and legs are excellent examples of such structures. These systems consist of muscles attached to hard structures (such as bones) through ligaments or tendons. We are familiar with endoskeletons, that is, the bones are central, and the muscles surround them. Insects also have musculo-skeletal systems, but the muscles are contained within the hard surrounding structure; these are referred to as exoskeletons.

• Analyzing musculo-skeletal systems requires knowing the distance from the point of attachment of the muscle to the joint (i.e., the moment arm). When the length of the moment arm is multiplied by the force generated by the muscle, one can calculate the torque that a muscle exerts at that joint. In musculo-skeletal systems, muscles exert force by shortening, and so to restore the position of the bones around the joint, muscles are often arranged in antagonistic pairs. The overall torque exerted at the joint is then the sum of the torques exerted by the antagonists around the joint. Much of the mathematics that is used for analyzing jointed robots can be used for analyzing the movements of musculo-skeletal systems.

• A major issue for understanding the control of musculo-skeletal systems is determining how to set the angles and torques at the different joints to reach a particular point in space. To make this concrete, hold your right arm straight out. By moving it using only your shoulder joint, you can rotate it, move it back and forth, or up and down. These different possible motions at the joint are referred to as degrees of freedom. If you now consider the movements of your elbow, wrist, and fingers, you can readily see that you have many more than the minimum number of degrees of freedom necessary to move to any point in three dimensional space. These extra degrees of freedom confer great flexibility, but also raise challenging questions for control.

• Another major body plan is the hydrostatic skeleton, which is typical of animals such as worms. The body consists of outer layers of muscle that surround a fluid-filled central cavity. This body plan allows for a great deal of flexibility, but it is hard to exert precise forces at specific points, which is much easier to do with a musculo-skeletal system.

• The third major body plans are muscular hydrostats. Muscular hydrostats are soft-tissue structures composed primarily of muscle that animals use to manipulate their environments. Examples include the trunks of elephants, the tentacles of cephalopods, and the tongues of many vertebrates. They are comprised of antagonistic muscle groups that work in opposition to one another to change the shape of the structure. Because muscles are largely composed of water, and because water is nearly incompressible except at very high pressures, muscles too are incompressible. Consequently, although muscles can change shape by contracting, their volume remains approximately constant. Incidentally, this is why your biceps bulge when you flex your arms (but the biceps muscle is not a mucular hydrostat because the bulging is not used to generate lateral force). The change in shape in one muscle can alter the size and shape of another muscle that surrounds it. Because muscular hydrostats are not constrained by joints, they have an even larger number of degrees of freedom than do musculo-skeletal systems.

• To make this concrete, picture a longitudinal muscle surrounded by a circumferential muscle (like a hotdog in a bun). When the longitudinal muscle contracts along its length, it expands in the perpendicular direction, and this can act to stretch the surrounding circumferential muscle. In contrast, when the circumferential muscle contracts, it can squeeze the longitudinal muscle, and cause it to lengthen.

• If fibers are helically wound within the muscle, bending and twisting movements become possible. This is what allows an elephant's trunk or an octopuses' tentacle to curve around objects and manipulate them very dexterously.

Studying a Muscular Hydrostat

You now have enough information to study an example of a muscular hydrostat. The link below will open a simulation of the tongue of a South American lizard, the gold tegu. The lizard uses its tongue to lap up water while drinking. During lapping, the tongue rhythmically elongates and shortens. In this model, two muscles work antagonistically to control muscle length. A longitudinal muscle that runs down the length of the tongue contracts to shorten it; a circumferential muscle wraps around the longitudinal muscle, like a tube, and squeezes the tongue to lengthen it. The simulation is based on a published model, which you can find here (the name of the first author may look familiar to you).

The longitudinal and circumferential lengths of the tongue are plotted by the simulation. To see the movements of the tongue that these changes in length correspond to, click the "Toggle Animation" button.

Question 3: As stated above, the firing frequency of motor neurons innervating a muscle has a large effect on the force generated by the muscle. In this simulation, motor neuronal firing frequency is represented simply by the circumferential and longitudinal "neural input" values, which are unitless and can vary in time. The neural inputs are plotted in the third and fourth graphs, and you can directly manipulate them using the simulation controls. Complete the following exercises to gain a better understanding of these controls and the simulation.

(a)	Open the tongue simulation, and press the No neural input button. This turns off the neural input to the muscles of the tongue. Take pictures. In the absence of neural inputs, what is the resting length of the tongue? In the absence of neural inputs, what is the resting circumference of the tongue?
(b)	Now change the Stimulus baseline for the neural input to the longitudinal muscle to 0.1. This represents constant, low frequency firing in the motor neurons innervating the longitudinal muscle. Notice the effect this has on the Neural Input for Longitudinal Muscle plot. Take pictures. What is the new length of the tongue? Did the tongue increase or decrease in length when the longitudinal muscle was activated? What is the new circumference of the tongue? Did the circumference increase or decrease when the longitudinal muscle was activated?
(c)	Press the No neural input button again. Change the Stimulus baseline for the neural input to the circumferential muscle to 0.1. This represents constant, low frequency firing in the motor neurons innervating the circumferential muscle. Again, notice that this change is reflected in the appropriate neural input plot. Take pictures. What is the new length of the tongue? Did the tongue increase or decrease in length when the circumferential muscle was activated? What is the new circumference of the tongue? Did the circumference increase or decrease when the circumferential muscle was activated?
(d)	Now press Lapping simulation button. Notice now that the neural inputs to the two muscles are changing in time. Take pictures. Describe what is happening in the tongue length and circumference plots. Relate this to the geometry of the tongue and the behavior of the animal. What are the maximum and minimum lengths of the tongue during lapping? Notice that the neural inputs to the longitudinal and circumferential muscles are not turned on at precisely the same time. What do you think would happen if the neural inputs were identically timed? Write down your prediction. Then, change the Stimulus delay for the longitudinal muscle to 0 ms and run the simulation. Do you think this new behavior of the tongue will be effective for lapping?

Question 4: If the lizard wants to extend its tongue to some target length and hold it there, the nervous system must activate the antagonistic circumferential and longitudinal muscles in the right proportions. There are, in fact, many combinations of circumferential and longitudinal activation levels that could result in extending the tongue to a particular length. You will investigate these relationships by creating a plot of the activations of the two muscles needed to hold the tongue at specific lengths. You do not need to take pictures for this question.

(a)	Create a table in your lab notebook for organizing your data. Click this link, and copy the table template into your notebook.
(b)	Open the tongue simulation, and press the No neural input button. Set the Stimulus baseline for the neural input to the longitudinal muscle to 1.0. Now, find a value for the circumferential Stimulus baseline that causes the tongue to have a length of 2 cm. Record this in your table (this will be the entry at the bottom of the "2 cm" column). Find an accurate value with two or three significant digits. Repeat this by changing the longitudinal Stimulus baseline to 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, and 0.2 and finding the value for the circumferential Stimulus baseline that causes the tongue to have a length of 2 cm. Again, use two or three significant digits. You will find that it is not possible to make the tongue have a length of 2 cm if the longitudinal Stimulus baseline is set to 0.1, so you can leave this entry in your table blank. Fill in the remaining columns of your table by repeating the steps above with target lengths of 3 cm, 4 cm, and 5 cm. In some cases, it will not be possible to find appropriate values. These cases are represented by dashes in your table, and you do not need to worry about them.
(c)	Create a plot of the data you collected. Put the longitudinal muscle neural input on the x-axis, and put the circumferential muscle neural input on the y-axis. You should plot four series of points, one for each of the target tongue lengths (2 cm, 3 cm, 4 cm, and 5 cm). All four sets of data should be plotted in the same graph.
(d)	The different geometries of the antagonistic longitudinal and circumferential muscles provide them with different mechanical advantages, depending on the length of the tongue. Compared to the other lines you plotted, is the slope of the line for the 2 cm length steep or shallow? As the longitudinal muscle activation increases, does the circumferential muscle need to be activated a little more or a lot more to compensate? What does this imply about the relative mechanical advantages of the two muscles when the tongue is short (2 cm)? Repeat this analysis for when the tongue is 5 cm long. Compared to the other lines you plotted, is the slope of the line for the 5 cm length steep or shallow? As the longitudinal muscle activation increases, does the circumferential muscle need to be activated a little more or a lot more to compensate? What does this imply about the relative mechanical advantages of the two muscles when the tongue is long (5 cm)? Summarize what you have learned about when the longitudinal muscle has a mechanical advantage over the circumferential muscle, and vice versa. Thinking about the shape of the tongue, can you explain why this is true?
(e)	Please focus on the set of neural inputs that hold the tongue at a length of 3 cm. For each combination of neural inputs for the longitudinal and circumferential muscles, the tongue length is the same. What is the functional difference between having low levels of activation in both muscles simultaneously versus having high levels of activation in both muscles simultaneously? What property of the tongue will be different?