Springs and Muscles, Part 6

[From Bruce Abbott (2015.02.18.1950 EST)]

Part 5 of this series gave a brief overview of the “computational” approach to motor control, which envisions a number of cognitive processes at work in the brain that contribute to motor control, culminating in the execution of a motor “plan” using both feedforward and feedback controllers. The processes envisioned were developed initially for application in robotics, where digital processors would carry out the complex computations needed to determine such things as optimization, state estimation, and inverse kinematics and dynamics. Critics of the application of this approach to biological systems doubted that such computations could be carried out with sufficient speed and precision by the nervous system, especially given the rather imprecise and variable output produced by biological sensors and effectors (muscles), as compared to their robotic counterparts.

In this section I review a strong alternative to the computational approach that began its development with the work of Russian physiologist Anatol Feldman and his colleagues in the early 1960s (Asatryan & Feldman, 1965) and which came to be known as the equilibrium point or EP hypothesis.

The Equilibrium Point Hypothesis of Motor Control

It is possible to develop a model of motor control that treats each component as a “black box” – an input-output function only loosely tied (if at all) to anatomy and physiology. For example, the model may include a function that relates the angle of the elbow joint to a number representing the average rate of firing in the afferent neurons, whose activity is presumed to reflect the angle of the joint. The function relating joint angle to neural activity level in the model may have been chosen by the modeler on the basis of simplicity or convenience if there are no physiological data to guide the choice. In addition, presumed circuitry envisioned in the model may lack a firm anatomical basis. For example, in the computational approach, where in the brain or spinal cord the motor system actually carries out the computation of inverse dynamics, how this is done, involving what structures, may be left to the imagination, or at least to future research. Such models are validated by comparing simulation results to the behavior of the modeled system. If the model performs well, then this is taken as evidence that something approximating the model system exists in the biological system being modeled.

This is definitely not the approach to modeling taken by Feldman and colleagues in developing the equilibrium point hypothesis and its associated models of motor control. Instead, these researchers have invested considerable time and effort to identify the anatomical components involved in a given example of motor control (such as control of arm position), the physiological mechanisms underlying their operation, and the connections among those components.

Origin of the Equilibrium Point (EP) Hypothesis

image00177.jpgThe equilibrium point hypothesis was developed initially to account for the results of simple “partial unloading” experiments (e.g., Asatryan & Feldman, 1965; Matthews, 1959). Asatryan and Feldman (1965) tested human subjects using the apparatus shown in the figure at right (their Figure 1). The forearm was strapped to a narrow platform hinged at the position of the elbow, which allowed the forearm to swing freely about elbow in the horizontal plane. (This arrangement eliminates the influence of gravity on elbow angle.) Two weights on strings were attached to the platform so as to exert a force on the forearm. Various combinations of weights were tested, with the sum of the two weights always equaling 3 Kg.

The subject moved the forearm against the applied force to a specified elbow-joint angle and closed his or her eyes. One of the weights was then released by switching off the power to an electromagnet. The person had been instructed not to voluntarily resist any changes in forearm position that might follow the dropping of the weight. What happened was that the arm moved to a new, stable position.

The experiment tested three initial elbow angles (30, 120, and 150 degrees) and dropped weights ranging from 0.5 kg to 5 kg. The data that emerged were highly regular and consistent with what might be expected of a spring having a given resting length and stiffness. For example, imagine that the pivot of the platform were replaced by a spiral torsion spring, with no arm strapped to the platform. With the two weights pulling from the right (as in panel “a”), the platform would be deflected clockwise until the counterforce that developed in the torsion spring would be in equilibrium with the force developed by pull of gravity on the two weights. If one weight were now dropped, the torsion spring would now rotate the platform to a new equilibrium position at which the reduced force pulling the arm clockwise was again balanced by the lessened counterforce of the spring. The subject’s arm behaved in just this way.

image0058.jpgFigure 2 from Asatryan and Feldman (1965) is reproduced at left. The table’s three main columns give the initial joint-angles and the columns, values of “j” indexing the change in weight following the drop. (Negative numbers are for remaining weights suspended on the left, which tended to pull the forearm counterclockwise.) The numbers within the table give the average change in joint angle. The graphs below the table show the dynamics of these changes in arm position following release of the weight. Clearly visible is some spring-like oscillation of the forearm, especially at the higher changes in applied force.

The data reveal that the muscles affecting the angle of the elbow joint (both flexor and extensor) show spring-like behavior, but there is something more interesting going on here. For a given initial load and joint-angle, partial unloading results in joint-angle changes that might be expected of a spring with a given resting length and stiffness. Change the initial force that the subject must exert against the pull of the weights to maintain a given initial joint angle, and you get results consistent with a spring with a different resting length. It appears that the subject accommodated to different initial forces by voluntarily changing the effective resting length of the muscles – the length at which there would be no neural impulses being sent to them via the alpha motor neurons.

Muscle contraction is determined by the level of activity of the pool of alpha motor neurons connected to the muscle at its end plates. An alpha motor neuron will begin sending impulses to the muscle only when its inputs bring its resting potential to a threshold value. According to the EP hypothesis, motor commands arriving at the alpha motor neurons, together with feedback from sensory receptors such as the muscle spindles and Golgi tendon organs, set the threshold for activation of the motor neuron and thus set the length of stretch at which the muscle will begin to resist being stretched further – effectively the zero-point of the spring.

image0069.jpgThe figure at right, from Latash (2008), Figure 1, illustrates the basic ideas behind the EP hypothesis. Each graph shows the force being exerted by the muscle on the vertical axis and the length of the muscle on the horizontal axis. At any point along this curve, the forces tending to stretch the muscle are in equilibrium with the counterforce produced by muscle contraction. The location of this curve is determined by the motor neuron activation threshold, lambda, which defines a muscle force-length characteristic curve. In the figure, lambda is represented not as a neural membrane threshold but as the length of the muscle at which the neurons begin to generate impulses.

In panel A, this threshold length is λ1. At this length the muscle is generating no contractile force. If we impose a steady load L on the muscle, the muscle will be stretched until feedback to the alpha motor neuron produces sufficient activation to generate an equal counterforce. This is the equilibrium point, EP1, and the length at which the equilibrium point is reached is l1. Any points lying off the characteristic curve, such as one located at length l2 and L, will result in changes that converge on this equilibrium point, as illustrated by the filled circles.

Panel B shows how active movement is generated by changing the activation threshold from length λ1 to λ2. This changes the position of the muscle characteristic curve, thereby changing the equilibrium point for this load from EP1 to EP2 and the muscle will shorten a new equilibrium point at length l2, altering joint-angle and thus changing the position of the arm without a change in load.

Panel C illustrates what is expected to happen if the force changes without any voluntary intent to move the limb. Here the tonic stretch reflex comes into play. With the load reduced to L2, the muscle shortens to length l2 and joint-angle changes involuntarily.

The EP model, as developed to simulate control of joint angle at the elbow, includes a representation of both the flexor (biceps) and extensor (triceps) muscles and their inhibitory cross connections: inputs that contract one muscle also exert an inhibitory influence on the opposing muscle, which reduces interference by allowing one muscle to contract while its opponent relaxes. But this effect can be overridden in the model, just as it is in the real system, as when you perform isometric exercises. The model assumes the existence of two kinds of motor command, the “R” command, which contracts one muscle while tending to inhibit contraction of its opponent, and the “C” command, which produces co-contraction and also sets the level of muscle “tone.” The R command produces rotation of the joint, whereas the C command determines the level of co-contraction while still allowing the R command to produce joint rotation. Increasing the level of co-contraction “stiffens” the muscles (raises the apparent spring constant), which reduces the effect of external forces on joint angle, so that the actual angle more closely matches the commanded angle. Both commands work by setting the lambda values of each muscle.

It is possible to set a command to a joint angle beyond the physical limits of the joint. Doing this under dynamic conditions will increase the strength of muscle contraction by increasing the difference between the current muscle length and the (impossible to reach) equilibrium point and thereby increase the speed of the motion. The same technique is used to produce muscle forces in excess of those obtainable through the setting of “reachable” equilibrium points, obviating the need for a separate force-control system.

image0089.jpgThe implementation of an EP-based model involves a number of terms that take account such things as cross-inhibitory connections between opponent muscles, the strongly nonlinear relation between motor neuron activation level and muscle contraction, resolution of the R and C commands into appropriate lambda values for each muscle, computation of the effective lambda value, given feedback from the muscle spindles and other receptors, and the proper representation of the physical system being moved by the muscles, including such things as centers of mass and rotation, changes in the moment of force with joint angle, and many others. The illustration at left is one version designed to model the system operating a single joint (Pilon & Feldman, 2006, Figure 6).

To test the properties of this model, I created a simulation of control of the elbow joint that allows you to set the values of the R and C commands by adjusting slider controls. You can watch a stick-figure of the arm move in real time under the influence of these commands, beginning with the elbow in the fully flexed position, and see the dynamics being played out on a graph showing joint angle as a function of time. The screen also displays the values the variables involved in computing joint angles during the simulation run. You can download a copy of the simulation at https://sites.google.com/site/perceptualcontroldemos/home/other-demos – it’s called “EPModel.zip.” As usual, it runs on Windows or on Macs that can simulate the Windows operating system.

image0102.jpg

image0131.jpg

Above are two screenshots taken during model runs. The one at left shows what happens when R is set to a value that holds the elbow angle at about 90 degrees and C is set to zero (no co-contraction). Compare this graph against the Asatryan and Feldman (1965) plot shown earlier. Note the oscillation around the final angle that eventually dies out. The screenshot at right shows the results when using the same value of R but with C set to a moderate value. Increasing the C-value has the effect of contracting the opposing muscles, which increases the effective spring constant and damping as the two muscles pull against each other. As the graph shows, the arm comes to the same joint angle as before but does so with little overshoot and no apparent oscillation.

Lambda versus Alpha Versions of the EP Hypothesis

An alternative version of the EP hypothesis was developed by Emilio Bizzi and colleagues based on deafferentation studies on monkeys, which demonstrated that the monkeys could regain control over their arms after losing sensory feedback from sensory receptors in the muscles, tendons, skin, etc. It has been called the “alpha” model to distinguish it from Feldman’s “lambda” model and to emphasize its reliance on direct activation of the muscle by alpha motor neurons. Although the ability to control limb position and dynamics is compromised after loss of feedback, the monkeys do learn to substitute visual cues to guide reaching. However, the loss of the tonic stretch reflex eliminates the automatic, reflexive adjustments to changes in load. The muscles do retain some springiness due to the passive elasticity of the muscle and tendon tissue, leading to a muscle characteristic curve that is similar in some respects to that seen in the intact system, but changes in input to the alpha motor neurons produces a shift in the slope of the function rather than its position along the length axis. Mark Latash (2010) describes the alpha model as a special case of the lambda model that applies when proprioceptive feedback has been eliminated.

The lambda version of the EP hypothesis has been elaborated to take into account movements involving multiple joints, the dynamics observed when subjects are asked to move a hand to a given position as rapidly as possible (including acceleration and deceleration phases), the development of so-called “motor synergies,” and many other movement-related phenomena. (An excellent tutorial has been provided by Mark Latash, 2010 in his book, Fundamentals of Motor Control.) In 2009 Feldman asserted that the basic EP hypothesis had been established as fact, and suggested that, with its newer developments, it deserved to be called by new name and christened it Threshold Control Theory, or TCP. Thus far, however, motor control researchers by and large have continued to call it the EP hypothesis.

As an approach to motor control, the EP hypothesis shares much in common with the Powers model based on perceptual control theory. Yet in a series of emails exchanged in 2012 between proponents of the EP hypothesis (Anatol Feldman and Mark Latash) and Bill Powers, Feldman and Latash reacted strongly against Powers’ assertion that the two models were practically identical at the muscle-control level. Up next: Just how similar are these two views of motor control?

Bruce

image00239.jpg

image00331.jpg

[From Bruce Abbott (2015.02.18.1950 EST)]

Part 5 of this series gave a brief overview of the “computational� approach to motor control, which envisions a number of cognitive processes at work in the brain that contribute to motor control, culminating in the execution of a motor “plan� using both feedforward and feedback controllers. The processes envisioned were developed initially for application in robotics, where digital processors would carry out the complex computations needed to determine such things as optimization, state estimation, and inverse kinematics and dynamics. Critics of the application of this approach to biological systems doubted that such computations could be carried out with sufficient speed and precision by the nervous system, especially given the rather imprecise and variable output produced by biological sensors and effectors (muscles), as compared to their robotic counterparts.

In this section I review a strong alternative to the computational approach that began its development with the work of Russian physiologist Anatol Feldman and his colleagues in the early 1960s (Asatryan & Feldman, 1965) and which came to be known as the equilibrium point or EP hypothesis.

The Equilibrium Point Hypothesis of Motor Control

It is possible to develop a model of motor control that treats each component as a “black boxâ€? – an input-output function only loossely tied (if at all) to anatomy and physiology. For example, the model may include a function that relates the angle of the elbow joint to a number representing the average rate of firing in the afferent neurons, whose activity is presumed to reflect the angle of the joint. The function relating joint angle to neural activity level in the model may have been chosen by the modeler on the basis of simplicity or convenience if there are no physiological data to guide the choice. In addition, presumed circuitry envisioned in the model may lack a firm anatomical basis. For example, in the computational approach, where in the brain or spinal cord the motor system actually carries out the computation of inverse dynamics, how this is done, involving what structures, may be left to the imagination, or at least to future research. Such models are validated by comparing simulation results to the behavior of the modeled system. If the model performs well, then this is taken as evidence that something approximating the model system exists in the biological system being modeled.

This is definitely not the approach to modeling taken by Feldman and colleagues in developing the equilibrium point hypothesis and its associated models of motor control. Instead, these researchers have invested considerable time and effort to identify the anatomical components involved in a given example of motor control (such as control of arm position), the physiological mechanisms underlying their operation, and the connections among those components.

Origin of the Equilibrium Point (EP) Hypothesis

Asatryan & Feldman (1965) Figure 1.jpgThe equilibrium point hypothesis was developed initially to account for the results of simple “partial unloading� experiments (e.g., Asatryan & Feldman, 1965; Matthews, 1959). Asatryan and Feldman (1965) tested human subjects using the apparatus shown in the figure at right (their Figure 1). The forearm was strapped to a narrow platform hinged at the position of the elbow, which allowed the forearm to swing freely about elbow in the horizontal plane. (This arrangement eliminates the influence of gravity on elbow angle.) Two weights on strings were attached to the platform so as to exert a force on the forearm. Various combinations of weights were tested, with the sum of the two weights always equaling 3 Kg.

The subject moved the forearm against the applied force to a specified elbow-joint angle and closed his or her eyes. One of the weights was then released by switching off the power to an electromagnet. The person had been instructed not to voluntarily resist any changes in forearm position that might follow the dropping of the weight. What happened was that the arm moved to a new, stable position.

The experiment tested three initial elbow angles (30, 120, and 150 degrees) and dropped weights ranging from 0.5 kg to 5 kg. The data that emerged were highly regular and consistent with what might be expected of a spring having a given resting length and stiffness. For example, imagine that the pivot of the platform were replaced by a spiral torsion spring, with no arm strapped to the platform. With the two weights pulling from the right (as in panel “a�), the platform would be deflected clockwise until the counterforce that developed in the torsion spring would be in equilibrium with the force developed by pull of gravity on the two weights. If one weight were now dropped, the torsion spring would now rotate the platform to a new equilibrium position at which the reduced force pulling the arm clockwise was again balanced by the lessened counterforce of the spring. The subject’s arm behaved in just this way.

Asatryan & Feldman (1965) Figure 2.jpgFigure 2 from Asatryan and Feldman (1965) is reproduced at left. The table’s three main columns give the initial joint-angles and the columns, values of “j� indexing the change in weight following the drop. (Negative numbers are for remaining weights suspended on the left, which tended to pull the forearm counterclockwise.) The numbers within the table give the average change in joint angle. The graphs below the table show the dynamics of these changes in arm position following release of the weight. Clearly visible is some spring-like oscillation of the forearm, especially at the higher changes in applied force.

The data reveal that the muscles affecting the angle of the elbow joint (both flexor and extensor) show spring-like behavior, but there is something more interesting going on here. For a given initial load and joint-angle, partial unloading results in joint-angle changes that might be expected of a spring with a given resting length and stiffness. Change the initial force that the subject must exert against the pull of the weights to maintain a given initial joint angle, and you get results consistent with a spring with a different resting length. It appears that the subject accommodated to different initial forces by voluntarily changing the effective resting length of the muscles – the length at which theree would be no neural impulses being sent to them via the alpha motor neurons.

Muscle contraction is determined by the level of activity of the pool of alpha motor neurons connected to the muscle at its end plates. An alpha motor neuron will begin sending impulses to the muscle only when its inputs bring its resting potential to a threshold value. According to the EP hypothesis, motor commands arriving at the alpha motor neurons, together with feedback from sensory receptors such as the muscle spindles and Golgi tendon organs, set the threshold for activation of the motor neuron and thus set the length of stretch at which the muscle will begin to resist being stretched further – effectively the zero-point of thee spring.

<image005.jpg>The figure at right, from Latash (2008), Figure 1, illustrates the basic ideas behind the EP hypothesis. Each graph shows the force being exerted by the muscle on the vertical axis and the length of the muscle on the horizontal axis. At any point along this curve, the forces tending to stretch the muscle are in equilibrium with the counterforce produced by muscle contraction. The location of this curve is determined by the motor neuron activation threshold, lambda, which defines a muscle force-length characteristic curve. In the figure, lambda is represented not as a neural membrane threshold but as the length of the muscle at which the neurons begin to generate impulses.

In panel A, this threshold length is λ1. At this length the muscle is generating no contractile force. If we impose a steady load L on the muscle, the muscle will be stretched until feedback to the alpha motor neuron produces sufficient activation to generate an equal counterforce. This is the equilibrium point, EP1, and the length at which the equilibrium point is reached is l1. Any points lying off the characteristic curve, such as one located at length l2 and L, will result in changes that converge on this equilibrium point, as illustrated by the filled circles.

Panel B shows how active movement is generated by changing the activation threshold from length λ1 to λ2. This changes the position of the muscle characteristic curve, thereby changing the equilibrium point for this load from EP1 to EP2 and the muscle will shorten a new equilibrium point at length l2, altering joint-angle and thus changing the position of the arm without a change in load.

Panel C illustrates what is expected to happen if the force changes without any voluntary intent to move the limb. Here the tonic stretch reflex comes into play. With the load reduced to L2, the muscle shortens to length l2 and joint-angle changes involuntarily.

The EP model, as developed to simulate control of joint angle at the elbow, includes a representation of both the flexor (biceps) and extensor (triceps) muscles and their inhibitory cross connections: inputs that contract one muscle also exert an inhibitory influence on the opposing muscle, which reduces interference by allowing one muscle to contract while its opponent relaxes. But this effect can be overridden in the model, just as it is in the real system, as when you perform isometric exercises. The model assumes the existence of two kinds of motor command, the “R� command, which contracts one muscle while tending to inhibit contraction of its opponent, and the “C� command, which produces co-contraction and also sets the level of muscle “tone.� The R command produces rotation of the joint, whereas the C command determines the level of co-contraction while still allowing the R command to produce joint rotation. Increasing the level of co-contraction “stiffens� the muscles (raises the apparent spring constant), which reduces the effect of external forces on joint angle, so that the actual angle more closely matches the commanded angle. Both commands work by setting the lambda values of each muscle.

It is possible to set a command to a joint angle beyond the physical limits of the joint. Doing this under dynamic conditions will increase the strength of muscle contraction by increasing the difference between the current muscle length and the (impossible to reach) equilibrium point and thereby increase the speed of the motion. The same technique is used to produce muscle forces in excess of those obtainable through the setting of “reachable� equilibrium points, obviating the need for a separate force-control system.

Pilon & Feldman (2006) Figure 2.jpgThe implementation of an EP-based model involves a number of terms that take account such things as cross-inhibitory connections between opponent muscles, the strongly nonlinear relation between motor neuron activation level and muscle contraction, resolution of the R and C commands into appropriate lambda values for each muscle, computation of the effective lambda value, given feedback from the muscle spindles and other receptors, and the proper representation of the physical system being moved by the muscles, including such things as centers of mass and rotation, changes in the moment of force with joint angle, and many others. The illustration at left is one version designed to model the system operating a single joint (Pilon & Feldman, 2006, Figure 6).

To test the properties of this model, I created a simulation of control of the elbow joint that allows you to set the values of the R and C commands by adjusting slider controls. You can watch a stick-figure of the arm move in real time under the influence of these commands, beginning with the elbow in the fully flexed position, and see the dynamics being played out on a graph showing joint angle as a function of time. The screen also displays the values the variables involved in computing joint angles during the simulation run. You can download a copy of the simulation at https://sites.google.com/site/perceptualcontroldemos/home/other-demos – it’s called “EPModel.zip.â€? As usual, it runs on Windows or on Macs that can simulate the Windows operating system.

···

<image013.jpg>

Above are two screenshots taken during model runs. The one at left shows what happens when R is set to a value that holds the elbow angle at about 90 degrees and C is set to zero (no co-contraction). Compare this graph against the Asatryan and Feldman (1965) plot shown earlier. Note the oscillation around the final angle that eventually dies out. The screenshot at right shows the results when using the same value of R but with C set to a moderate value. Increasing the C-value has the effect of contracting the opposing muscles, which increases the effective spring constant and damping as the two muscles pull against each other. As the graph shows, the arm comes to the same joint angle as before but does so with little overshoot and no apparent oscillation.

Lambda versus Alpha Versions of the EP Hypothesis

An alternative version of the EP hypothesis was developed by Emilio Bizzi and colleagues based on deafferentation studies on monkeys, which demonstrated that the monkeys could regain control over their arms after losing sensory feedback from sensory receptors in the muscles, tendons, skin, etc. It has been called the “alpha� model to distinguish it from Feldman’s “lambda� model and to emphasize its reliance on direct activation of the muscle by alpha motor neurons. Although the ability to control limb position and dynamics is compromised after loss of feedback, the monkeys do learn to substitute visual cues to guide reaching. However, the loss of the tonic stretch reflex eliminates the automatic, reflexive adjustments to changes in load. The muscles do retain some springiness due to the passive elasticity of the muscle and tendon tissue, leading to a muscle characteristic curve that is similar in some respects to that seen in the intact system, but changes in input to the alpha motor neurons produces a shift in the slope of the function rather than its position along the length axis. Mark Latash (2010) describes the alpha model as a special case of the lambda model that applies when proprioceptive feedback has been eliminated.

The lambda version of the EP hypothesis has been elaborated to take into account movements involving multiple joints, the dynamics observed when subjects are asked to move a hand to a given position as rapidly as possible (including acceleration and deceleration phases), the development of so-called “motor synergies,� and many other movement-related phenomena. (An excellent tutorial has been provided by Mark Latash, 2010 in his book, Fundamentals of Motor Control.) In 2009 Feldman asserted that the basic EP hypothesis had been established as fact, and suggested that, with its newer developments, it deserved to be called by new name and christened it Threshold Control Theory, or TCP. Thus far, however, motor control researchers by and large have continued to call it the EP hypothesis.

As an approach to motor control, the EP hypothesis shares much in common with the Powers model based on perceptual control theory. Yet in a series of emails exchanged in 2012 between proponents of the EP hypothesis (Anatol Feldman and Mark Latash) and Bill Powers, Feldman and Latash reacted strongly against Powers’ assertion that the two models were practically identical at the muscle-control level. Up next: Just how similar are these two views of motor control?

Bruce

[From RIck Marken (2015.02.19.1500)]
 RM: These statements seem to be contradictory. The first implies that the EP hypothesisÂ
was not based on comparing model behavior to the behavior of the actual system (which is

what we do in PCT). Rather, it was based on anatomical (and I presume physiological)Â

considerations. But the second says that the EP model was developed to account for the

results of a behavioral experiment – Â the “partial unloading” experiment. And, indeed, theÂ

description of the test of the EP modes is a comparison of model behavior to the behavior

of the subjects in the “partial unloading” experiment.Â

RM: So it looks to me like the EP model was developed and tested in the same way as the

PCT model: it was developed to account for observed behavior and tested by seeing howÂ

well model behavior matches actual behavior. The only difference I can see is that the PCT

model was developed to account for control behavior while the EP model was developedÂ

to account for behavior where people are specifically asked to try not to control. Â

RM: Am I wrong (as usual;-)?

BestÂ

Rick

···

Bruce Abbott (2015.02.18.1950 EST)–

Â

BA (1): It is possible to develop a model of motor control that treats each component as a “black boxâ€? – an input-output function onnly loosely tied (if at all) to anatomy and physiology…Such models are validated by comparing simulation results to the behavior of the modeled system…This is definitely not the approach to modeling taken by Feldman and colleagues in developing the equilibrium point hypothesis and its associated models of motor control. Instead, these researchers have invested considerable time and effort to identify the anatomical components involved in a given example of motor control (such as control of arm position), the physiological mechanisms underlying their operation, and the connections among those components…

Â

BA (2): The equilibrium point hypothesis was developed initially to account for the results of simple “partial unloadingâ€? experiments (e.g., Asatryan & Feldman, 1965; Matthews, 1959). Â


Richard S. Marken, Ph.D.
Author of  Doing Research on Purpose
Now available from Amazon or Barnes & Noble

[From Bruce Abbott (2015.02.19.1945 EST)]

Rick Marken (2015.02.19.1500) –

Bruce Abbott (2015.02.18.1950 EST)

BA (1): It is possible to develop a model of motor control that treats each component as a “black boxâ€? – an input-output function onlly loosely tied (if at all) to anatomy and physiology…Such models are validated by comparing simulation results to the behavior of the modeled system…This is definitely not the approach to modeling taken by Feldman and colleagues in developing the equilibrium point hypothesis and its associated models of motor control. Instead, these researchers have invested considerable time and effort to identify the anatomical components involved in a given example of motor control (such as control of arm position), the physiological mechanisms underlying their operation, and the connections among those components…

BA (2): The equilibrium point hypothesis was developed initially to account for the results of simple “partial unloading� experiments (e.g., Asatryan & Feldman, 1965; Matthews, 1959).

RM: These statements seem to be contradictory. The first implies that the EP hypothesis

was not based on comparing model behavior to the behavior of the actual system (which is

what we do in PCT). Rather, it was based on anatomical (and I presume physiological)

considerations. But the second says that the EP model was developed to account for the

results of a behavioral experiment – the “partial unloading” experiment. And, indeed, the

description of the test of the EP modes is a comparison of model behavior to the behavior

of the subjects in the “partial unloading” experiment.

RM: So it looks to me like the EP model was developed and tested in the same way as the

PCT model: it was developed to account for observed behavior and tested by seeing how

well model behavior matches actual behavior. The only difference I can see is that the PCT

model was developed to account for control behavior while the EP model was developed

to account for behavior where people are specifically asked to try not to control.

RM: Am I wrong (as usual;-)?

BA: I appreciate your input on this, Rick. Perhaps I was not clear.  The findings of the partial unloading studies lead Feldman to the hypothesis that the angular position of the elbow joint was being determined by setting the threshold for muscle activation (lambda) via the alpha motor neuron pool, which establishes the position of the muscle characteristic curve relating muscle length to force. The “equilibrium point� is then a unique point along this curve: the length at which the counterforce generated by muscle contraction balances the applied force. The EP model was developed based on this hypothesis, using available data on the relevant anatomy and physiology and, where relevant information was lacking, conducting physiological research to fill in the model. This approach contrasts with “computational models� that make assumptions about what sort of processes are being carried out in the nervous system without necessarily tying these processes to any actual anatomy or physiology. Bill’s PCT-based model was also tied to anatomical and physiological knowledge available to Bill in the early 1970s and thus, like the EP model, not in the mold of the “computational� approach.

BA: The EP model has of course been tested against the observed behavior of the real system, just as is done in PCT.

BA: In drawing the conclusion you reach in the last sentence of your second paragraph, it seems to me that you are confusing the objectives of one experiment for the objectives of the EP model. In the experiment, subjects were asked to move their elbow joints to specific starting positions (control), after which they were not to voluntarily interfere with any angular changes resulting from unloading. Here the objective was to determine the muscle characteristic curve (which is a function of feedback from muscle spindles). Data following changes in starting position showed that the characteristic curve moved along the length axis consequent to these voluntary position changes. The EP model was then developed to explain how individuals exert voluntary control over movement (by varying lambda) and produce involuntary adjustments to changes in load – movementss of the equilibrium point along the muscle characteristic curve produced by the feedback mechanism that is responsible for the tonic stretch reflex.

Bruce

[From Rick Marken (2015.02.21.0910)]

BA: As an approach to motor control, the EP hypothesis shares
much in common with the Powers model based on perceptual control theory. Yet in
a series of emails exchanged in 2012 between proponents of the EP hypothesis
(Anatol Feldman and Mark Latash) and Bill Powers, Feldman and Latash reacted
strongly against Powers’ assertion that the two models were practically
identical at the muscle-control level.
Up next: Just how similar are these two views of motor control?

RM: I’m looking forward to seeing what you have to say about this. But in the meantime I need some help understanding the EP model. My first problem is with this figure:

image0089.jpg

RM: I don’t understand what it’s saying. Is the Length variable (x axis) equivalent to the length of a spring and the Load variable (y axis) equivalent to a force applied to the mass on the spring (or to the weight of the mass itself?). If so, then why is the function relating length to load non-linear? Wouldn’t that function just be Hooke’s law – x = 1/k*Fa – , which is linear? Is it the damping that introduces the non-linearity? And the difference between active and passive movement kind of puzzles me. It looks like active movement shifts the force/length curve. I presume this is done by changing the spring constant of the muscle? So a constant load is moved (length is changed) by changing the spring constant of the muscle? is that right? Passive movement seems to result simply from changing the load while the muscle spring constant remains the same. Is that right? So the passive movement is what the are studying in their experiment, right? Where the subjects are instructed not to control the position of the arm when the load changes?

RM: My other problem is with the model itself, as shown in this figure:

image0058.jpg

RM: First, it’s not clear from this diagram where the nervous system ends and the environment begins. It looks like it might be between the “generalized thresholds”, R,C, which I think might be neural, and the muscle thresholds, gamma, which seem to be properties of the muscles. So from gamma on to the left I think we are in the muscle (environment) itself until we get to torque, which I assume is a physical effect of muscle contraction; so starting at M we are in the physical environment. Is that right? If it is, then I am confused by what the proprioceptive feedback link is. I think of proprioception as a perceptual (neural), not a physical, variable. But the diagram shows the proprioception entering the muscle rather than the nervous system. This seems pretty sloppy for people who purport to be building a model that is neurophysiologically accurate. Also, it looks like gamma functions as a reference specification for the proprioceptive feedback. But gamma seems to be a characteristic of muscle fiber while the proprioception seems to be a measure of muscle movement; so these are two different kinds of variables. Very confusing.

RM: Finally, it’s not clear to me why this would be of interest to a control theorist. Perhaps you will explain this in the next installment. But it looks like the EP hypothesis applies to a situation where control is involved as little as possible. The subject’s task was to “not” resist disturbances to the position of the arm that result from changes in the load. To the extent that the subjects could do this (which I presume they could do by lowering the gain of their position control systems,; interesting that they can do this) then the observed behavior should be similar to that of a passive system like a damped mass/spring system. But it seems to me that such a finding could only be of interest to people who are interested in studying the behavior of cadavers rather than that of living control systems. So the EP model would be of interest to people like Galvani, who did study the behavior of dead frogs, but I don’t see what the EP contributes to our understanding of how people control their limbs. But I guess I’ll find out from the next installment of this series.

Best

Rick

···

Richard S. Marken, Ph.D.
Author of Doing Research on Purpose.
Now available from Amazon or Barnes & Noble