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Limitations of Somatosensory Feedback in Control of Posture and Movement

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Abstract

The sections in this article are:

1 Some Properties of Servomechanisms
1.1 Loop Gain
1.2 Frequency Transfer Function
1.3 Properties of Different Loads
1.4 Vector Representation of Frequency Transfer Function
1.5 Vector Representation of a Delay
1.6 Combination of Different Vectors
1.7 A Velocity‐Sensitive Transducer
1.8 A Low‐Pass Filter
1.9 Stability
1.10 Nonlinear Systems
2 Response of Limbs to Sinusoidal Movements
2.1 Effects of Limb Mass
2.2 Resistance of Muscles to Movement
2.3 The Two Components of Muscle Force
2.4 Muscles as Low‐Pass Filters
2.5 Timing of Reflex Force
2.6 Timing of Muscle Spindle Afferent Activity
2.7 Reflex Delay
2.8 Summary
3 Negative Stiffness and Spontaneous Oscillations
3.1 Factors Affecting Stability of Stretch Reflexes
3.2 Nature of the External Load
3.3 Level of Muscle Activation
3.4 Resistance to Movements of Different Amplitudes
3.5 Comparing the Stretch Reflex With a Man‐Made Control System
3.6 Stability of a Man‐Made System
3.7 Stability and Gain in the Stretch Reflex
3.8 Summary
4 Reflex Responses or Precomputed Activity?
4.1 Response to a Sudden Disturbance
4.2 Tuning of the Neuromuscular System
4.3 Triggered Responses to a Disturbance
4.4 Summary
5 Servo Assistance of Voluntary Movements
6 Precomputed Movements
6.1 Summary
7 Feedback and Learning
8 Forward‐Looking Control Systems
9 Conclusions
Figure 1. Figure 1.

A: schematic diagram of position‐controlling servomechanism. Motor is assumed to generate a force proportional to its input, irrespective of movement. A positive force (tension) causes a negative length change (shortening). B: when same system is moved by forces applied to the load, the actual force required is the sum of forces required to move the load and forces necessary to overcome resistance generated in feedback pathway.

Figure 2. Figure 2.

Frequency transfer function of load that combines elastic resistance k, viscosity η, and mass m. Vector N follows path indicated as frequency of movement increases. Distance of N from origin gives its modulus, and the angle ϕ indicates phase lead of force on position. Points marked along path indicate equal increments in frequency; ω, angular velocity.

Figure 3. Figure 3.

Vector representations of numerous frequency transfer functions. A: mechanical load of elastic stiffness k and mass m, the frictional resistance of which does not change with frequency. When m is zero, the vector has the value y at all frequencies. B: 2 examples of feedback systems in which a force proportional to length occurs at some fixed interval after that length. For smaller circle, interval was 50 ms; for larger, 25 ms. Larger circle indicates greater force per unit displacement. C: combination of reflex stiffness and load stiffness obtained by geometrical addition of vector path in A to smaller circle in B. Solid line shows stiffness of system with its mass; broken line shows it without the mass. D: similar to C, but larger circle of B has been added to A. E: effect of velocity‐sensitive transducer. Delay and load are same as in D (without mass), but reflex activity is assumed to be proportional to velocity of movement. F: again load and delay are same as in D, but reflex activity is assumed to be initiated by muscle spindles with transfer functions Ks(s + 0.44)(s + 11.3)(s + 44.0)/(s + 0.04)(s + 0.816). See reference 69. s, Laplace operator. G, H, and I: diagrams of how vector paths in D, E, and F are modified by effects of low‐pass filter

critically damped second‐order filter with corner frequency 6 Hz 54
Figure 4. Figure 4.

Method of driving human elbow joint through sinusoidal flexion‐extension movements. Forearm is fixed in splints and constrained to move about axis of joint. A crank couples the wrist to an eccentric pin on a rotating wheel. Inset record, frequency was 15 Hz, and tension in crank was highest in flexion.

From Rack 71
Figure 5. Figure 5.

Stiffness of interphalangeal joint of thumb. Joint was driven through sinusoidal movements of ± 1.3° while subject maintained a mean flexing torque of 0.75 N·m. Resistance to movements at 2–15 Hz is represented here as a series of vectors. Some frequencies are marked on figure.

From Brown, Rack, and Ross 10
Figure 6. Figure 6.

Resistance to sinusoidal movement of human elbow. Joint was driven through movements of ± 0.24° (see Fig. 4) while subject maintained a mean flexing torque of 10 N·m. A: limb‐stiffness vectors. B: component of stiffness remaining after subtraction of force required to move mass of forearm and hand. C: stiffness of limb when encased in splints that were equivalent to a mass of 600 g held in the hand. C is original experimental result from which A and B were obtained.

Replotted from Joyce, Rack, and Ross 47
Figure 7. Figure 7.

Response of tetanized muscle to sinusoidal stretching. A: cat soleus muscle stretched through ± 0.8 mm at about 5 Hz. Tetanic stimulation (50 impulses/s) began during record. B: stiffness vectors plotted at 3 different amplitudes: × = ± 1.9 mm; ▪ = ± 0.8 mm; • = ± 0.35 mm. Some frequencies are marked on figure.

From Rack 70
Figure 8. Figure 8.

Resistance to imposed sinusoidal opening‐closing movement of a monkey mandible. Movement was ± 0.5 mm at incisors. Monkey maintained a mean biting force of 8 N. A: stiffness vectors in intact animal. B: stiffness vectors after interruption of afferents from jaw‐closing muscles. C: vectors in B have been subtracted from those in A to demonstrate amount of stiffness that could be attributed to action of stretch reflex.

Replotted from Goodwin et al. 28
Figure 9. Figure 9.

Resistance to sinusoidal movement of human elbow. Same subject and amplitude of movement as Figure 6. Mean flexing torque was 32 N·m. A: resistance of limb to movements. B: component of stiffness remaining after subtracting component of force due to mass. C: stiffness that was actually measured when limb was encased in splints.

Replotted from Joyce, Rack, and Ross 47
Figure 10. Figure 10.

Spontaneous oscillation at normal elbow joint (same subject as Figs. 6 and 9). Subject flexed with force of 32 N·m against a spring. Spring had a stiffness equivalent to 106 N·m and mass of forearm together with its enclosing splints was 0.149 kg·m2.

Figure 11. Figure 11.

A: resistance to sinusoidal movement at interphalangeal joint of thumb. Movement was through ± 1.3° while subject exerted a mean flexing torque of 0.5 N·m. Solid line joins vectors that denote stiffness of thumb alone. Broken line shows how these vectors are displaced by addition of a mass with moment of inertia 1.3 g·m2. B: spontaneous tremor of same joint; thumb was loaded with a mass of inertia 1.3 g·m2, and subject flexed with a force of 0.5 N·m against a compliant spring (spring stiffness equivalent to 0.08 N·m/rad).

Figure 12. Figure 12.

Effect of mean flexing force on stiffness at thumb interphalangeal joint. The 0.75‐N·m record is from Figure 5; with other mean flexing forces (marked on figure), stiffness vectors follow different paths.

From Brown, Rack, and Ross 10
Figure 13. Figure 13.

Resistance of human elbow joint to different amplitudes of sinusoidal movement. Resistance includes forces required to overcome mass of forearm. Mean flexing torque, 26 N·m. Amplitude of movement for A, ±0.12°; B, ±0.25°; and C, ±1.2°.

Replotted from Joyce, Rack, and Ross 47
Figure 14. Figure 14.

Effect of movement on continuously stimulated muscle. Cat soleus muscle was stimulated by trains of pulses that were distributed sequentially among different groups of motor units so that while each muscle fiber was stimulated at only 3, 5, 7, or 15 pulses/s (pps), action potentials reached the muscle at 5 × that rate. Dotted lines indicate rapid changes in tension that accompanied first part of shortening or lengthening movement.

Figure 15. Figure 15.

Diagram showing effects of cocontraction of antagonist muscles. Forces developed by flexor and extensor muscles (similar to the 15 pps record of Fig. 14) act in opposite directions and so long as there is no movement of the joint, there is no external force. When, however, the joint is subjected to an extending movement, the force in the flexor muscle rises and the force in the extensor falls; the joint resists the displacement with a force that depends on the summed stiffnesses of the 2 muscles.



Figure 1.

A: schematic diagram of position‐controlling servomechanism. Motor is assumed to generate a force proportional to its input, irrespective of movement. A positive force (tension) causes a negative length change (shortening). B: when same system is moved by forces applied to the load, the actual force required is the sum of forces required to move the load and forces necessary to overcome resistance generated in feedback pathway.



Figure 2.

Frequency transfer function of load that combines elastic resistance k, viscosity η, and mass m. Vector N follows path indicated as frequency of movement increases. Distance of N from origin gives its modulus, and the angle ϕ indicates phase lead of force on position. Points marked along path indicate equal increments in frequency; ω, angular velocity.



Figure 3.

Vector representations of numerous frequency transfer functions. A: mechanical load of elastic stiffness k and mass m, the frictional resistance of which does not change with frequency. When m is zero, the vector has the value y at all frequencies. B: 2 examples of feedback systems in which a force proportional to length occurs at some fixed interval after that length. For smaller circle, interval was 50 ms; for larger, 25 ms. Larger circle indicates greater force per unit displacement. C: combination of reflex stiffness and load stiffness obtained by geometrical addition of vector path in A to smaller circle in B. Solid line shows stiffness of system with its mass; broken line shows it without the mass. D: similar to C, but larger circle of B has been added to A. E: effect of velocity‐sensitive transducer. Delay and load are same as in D (without mass), but reflex activity is assumed to be proportional to velocity of movement. F: again load and delay are same as in D, but reflex activity is assumed to be initiated by muscle spindles with transfer functions Ks(s + 0.44)(s + 11.3)(s + 44.0)/(s + 0.04)(s + 0.816). See reference 69. s, Laplace operator. G, H, and I: diagrams of how vector paths in D, E, and F are modified by effects of low‐pass filter

critically damped second‐order filter with corner frequency 6 Hz 54


Figure 4.

Method of driving human elbow joint through sinusoidal flexion‐extension movements. Forearm is fixed in splints and constrained to move about axis of joint. A crank couples the wrist to an eccentric pin on a rotating wheel. Inset record, frequency was 15 Hz, and tension in crank was highest in flexion.

From Rack 71


Figure 5.

Stiffness of interphalangeal joint of thumb. Joint was driven through sinusoidal movements of ± 1.3° while subject maintained a mean flexing torque of 0.75 N·m. Resistance to movements at 2–15 Hz is represented here as a series of vectors. Some frequencies are marked on figure.

From Brown, Rack, and Ross 10


Figure 6.

Resistance to sinusoidal movement of human elbow. Joint was driven through movements of ± 0.24° (see Fig. 4) while subject maintained a mean flexing torque of 10 N·m. A: limb‐stiffness vectors. B: component of stiffness remaining after subtraction of force required to move mass of forearm and hand. C: stiffness of limb when encased in splints that were equivalent to a mass of 600 g held in the hand. C is original experimental result from which A and B were obtained.

Replotted from Joyce, Rack, and Ross 47


Figure 7.

Response of tetanized muscle to sinusoidal stretching. A: cat soleus muscle stretched through ± 0.8 mm at about 5 Hz. Tetanic stimulation (50 impulses/s) began during record. B: stiffness vectors plotted at 3 different amplitudes: × = ± 1.9 mm; ▪ = ± 0.8 mm; • = ± 0.35 mm. Some frequencies are marked on figure.

From Rack 70


Figure 8.

Resistance to imposed sinusoidal opening‐closing movement of a monkey mandible. Movement was ± 0.5 mm at incisors. Monkey maintained a mean biting force of 8 N. A: stiffness vectors in intact animal. B: stiffness vectors after interruption of afferents from jaw‐closing muscles. C: vectors in B have been subtracted from those in A to demonstrate amount of stiffness that could be attributed to action of stretch reflex.

Replotted from Goodwin et al. 28


Figure 9.

Resistance to sinusoidal movement of human elbow. Same subject and amplitude of movement as Figure 6. Mean flexing torque was 32 N·m. A: resistance of limb to movements. B: component of stiffness remaining after subtracting component of force due to mass. C: stiffness that was actually measured when limb was encased in splints.

Replotted from Joyce, Rack, and Ross 47


Figure 10.

Spontaneous oscillation at normal elbow joint (same subject as Figs. 6 and 9). Subject flexed with force of 32 N·m against a spring. Spring had a stiffness equivalent to 106 N·m and mass of forearm together with its enclosing splints was 0.149 kg·m2.



Figure 11.

A: resistance to sinusoidal movement at interphalangeal joint of thumb. Movement was through ± 1.3° while subject exerted a mean flexing torque of 0.5 N·m. Solid line joins vectors that denote stiffness of thumb alone. Broken line shows how these vectors are displaced by addition of a mass with moment of inertia 1.3 g·m2. B: spontaneous tremor of same joint; thumb was loaded with a mass of inertia 1.3 g·m2, and subject flexed with a force of 0.5 N·m against a compliant spring (spring stiffness equivalent to 0.08 N·m/rad).



Figure 12.

Effect of mean flexing force on stiffness at thumb interphalangeal joint. The 0.75‐N·m record is from Figure 5; with other mean flexing forces (marked on figure), stiffness vectors follow different paths.

From Brown, Rack, and Ross 10


Figure 13.

Resistance of human elbow joint to different amplitudes of sinusoidal movement. Resistance includes forces required to overcome mass of forearm. Mean flexing torque, 26 N·m. Amplitude of movement for A, ±0.12°; B, ±0.25°; and C, ±1.2°.

Replotted from Joyce, Rack, and Ross 47


Figure 14.

Effect of movement on continuously stimulated muscle. Cat soleus muscle was stimulated by trains of pulses that were distributed sequentially among different groups of motor units so that while each muscle fiber was stimulated at only 3, 5, 7, or 15 pulses/s (pps), action potentials reached the muscle at 5 × that rate. Dotted lines indicate rapid changes in tension that accompanied first part of shortening or lengthening movement.



Figure 15.

Diagram showing effects of cocontraction of antagonist muscles. Forces developed by flexor and extensor muscles (similar to the 15 pps record of Fig. 14) act in opposite directions and so long as there is no movement of the joint, there is no external force. When, however, the joint is subjected to an extending movement, the force in the flexor muscle rises and the force in the extensor falls; the joint resists the displacement with a force that depends on the summed stiffnesses of the 2 muscles.

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Peter M. H. Rack. Limitations of Somatosensory Feedback in Control of Posture and Movement. Compr Physiol 2011, Supplement 2: Handbook of Physiology, The Nervous System, Motor Control: 229-256. First published in print 1981. doi: 10.1002/cphy.cp010207