Chapter 2. Controlling the statistics of action in obstacle avoidance
2.6 Appendix
Given an n-dimensional variable x, with mean x and covariance P ^ , the
Controlling the statistics of action in obstacle avoidance: Appendix
distribution o f y = / ( x ) . To do this we construct a set o f 2«+l points Jf, each o f which
is assigned a weighting fVi
= X IVq= K j { n + k)
X , = x + ( V ( n + K -)P „ )_ W , = \ l l { n + K )
+
W,^^=\!2(n + K)
where |-^ /(n T r)P ^ j is the i* row of the matrix square root o f {n + K )V ^.
These points are chosen so that they give an unbiased representation o f the mean and
covariance o f the original distribution. Each point is transformed through the non-linear
function/ to give y . = / ( x .) . The new mean ÿ and covariance can be found as:
/=o /=o
The new mean and covariance can be used to define a new set o f points X, and
the process is iterated for every time step.
In the case o f the arm model, the state variable x has 12 dimensions:
* = where q\ is the joint angle, tq\ is the torque
and u\ = motor command at the i^*’ joint. The non-linear function / was the forward
model o f the muscles and dynamics o f the arm. We assume that the distribution o f x ( k )
is Gaussian, and so following Julier and Uhlmann (1995) chose ( « + k ) = 3 . At the start o f
movement the covariance of the state is a matrix o f zeros. Variability in the state arises
because at each time step the standard deviation of the motor commands is set to be
linearly related to the absolute value o f the motor commands, with the constant o f
The sources of signal dependent noise during isometric force production: Abstract
Chapter 3.
The sources of signal dependent noise
during isometric force production
3.1 Abstract
This chapter confirms the presence of SDN during isometric force production and determines how central and peripheral components contribute to this feature of motor control. Peripheral and central components were distinguished experimentally by comparing voluntary contractions to those elicited by electrical stimulation of the extensor pollicis longus muscle. SDN was evident in voluntary isometric contractions as a linear increase in the standard deviation of force with increases in the mean force, showing a constant coefficient of variation. However, during electrically stimulated contractions to the same force levels, there was no relationship between force variability and the mean force produced.
In order to investigate other factors of motor-unit physiology that may contribute to SDN, a muscle model was constructed and its output compared to the empirical data. The modelling results demonstrate that SDN arises due to the basic physiological organization of the motor-unit pool, in particular orderly recruitment of motor neurons over a range of twitch amplitudes and range of recruitment thresholds. The magnitude of SDN variability is correlated to the variability and synchronization of motor neuron spiking. We conclude that the presence of SDN during voluntary isometric contractions is a natural by-product of the organization of the motor-unit pool, and is not dependent on a noisy motor program being relayed to the motor neurons.
The sources of signal dependent noise during isometric force production: Introduction
3.2 Introduction
It has previously been shown (Chapter 2, Harris and Wolpert, 1998) that the
TOPS framework can account for human arm movement performance in a variety of
tasks, including point-to-point arm movements, Fitts law and obstacle avoidance
movements. A fundamental premise o f TOPS is the assumption that the motor
command to the muscles exhibits signal-dependent noise with a constant coefficient of
variation (SDN), There is evidence for such noise in studies of isometric force
production (Schmidt et al., 1979; Galganski et al., 1993; Enoka et al., 1999; Slifkin and
Newell, 1999; Laidlaw et al., 2000). The aim o f this chapter is to understand why this
noise is present in the muscle and associated motor neuron pool.
Several possible sources of variability are present in the muscle and motor
neurons. One hypothesis might be that the contraction of muscle fibres is inherently
variable and introduces noise into muscle force generation. There is also evidence (Del
Castillo & Katz 1954) for stochastic release of neurotransmitter at the neuromuscular
junction, which could contribute to motor noise. Previous studies o f variability in
continuous isometric force production have suggested that the statistical variability and
synchrony in the discharge o f motor neurons supplying the muscle might be responsible
(Semmler and Nordstrom, 1998, Laidlaw et al., 2000, Yao et al., 2000). The orderly
recruitment o f small motor units before larger ones (Hennemann, 1957) might also
influence the type o f noise observed in force production. Finally, central processes such
as variability in the output o f the motor cortex or in movement planning could
contribute to motor noise.
In order to distinguish peripheral (muscle fibre and neuromuscular junction)
noise from neuronal (firing rate and recruitment) noise, we have compared the
The sources of signal dependent noise during isometric force production: Introduction
elicited by neuromuscular electrical stimulation (NMES). NMES is a methodology that
uses a series o f electrical pulses to generate a muscle contraction (Baker et ah, 2000.
The contraction is elicited indirectly through stimulation o f the motor axons with the
electrodes usually located over the muscle motor point. The contractions studied were
extensions o f the distal phalanx o f the thumb, a movement that is produced by the
action o f only one muscle, the extensor pollicis longus (EPL). By using NMES we
could experimentally estimate the noise generated by the peripheral machinery and if
that noise had signal-dependent features.
There are important differences between the activation o f motor units in the
NMES condition and the voluntary condition. Voluntary activation leads to orderly
recruitment o f motor units (Hennemann, 1957) that fire with variable spike trains
(Calvin and Stevens, 1967; Nordstrom and Miles, 1991). In contrast, NMES recruits
units in a random order, and causes all active motor units to fire synchronously in
response to the stimuli (25 - 30 pulses per second). During NMES, increases in mean
force output are achieved by additional random recruitment alone, i.e. there is no rate
coding.. The synchronous discharge o f the motor units in the NMES condition would
tend to increase the force variability, but at the stimulation rates used, it is likely that
most motor units will be contracting tetanically (Nathan and Tavi, 1990; Thomas et al.,
1991). The random recruitment order arises due to the relationship between
percutaneous stimulation and the range o f alpha motor axon diameters in human nerve.
While the recruitment o f axons to electrical stimulation is biased so that larger axons
will be excited at lower stimulus amplitudes, this is true primarily in the condition
where the nerve is in intimate contact with the electrodes. Percutaneous stimulation, as
in the present study, will recruit the alpha motor axons in a generally random order.
The sources of signal dependent noise during isometric force production: introduction
velocity o f alpha m otor axons, a m easure o f axon diam eter, and tw itch force output in hum ans (B igland-R itchie et ah, 1998).
To sum m arise, our N M ES paradigm is sensitive only to peripheral noise generated during tetanic contractions sum m ed across active m otor units. The voluntary condition will include variability from these sources and also noise due to the influence o f orderly recruitm ent, variability in m otor neuron firing or variability in central m otor processes. Thus, differences in force variability at the sam e m ean force level betw een N M ES and voluntary force generation can reveal the origins o f SDN. Since m any o f the variables o f interest w ere not directly available for testing and m anipulation in hum an experim ents, we also used a m odel o f the m otor-unit pool to deten n in e how factors such as o f the order o f recruitm ent, rate coding and the statistics o f m otor
Voluntary
Muscle fibres
Neuromuscular junction Motor unit recruitment Motor neuron firing Central motor command
NMES
Force Transducer
EPL
F ig u re 3 -1 . P o ssib le so u rc e s o f n oise. A p o o l o f m o to r n e u ro n s in n erv a tin g e x te n s o r p o llic u s lo n g u s is sh o w n , and five p o ss ib le so u rc e s o f no ise are listed. V o lu n tary force g e n era tio n (b lu e a rro w ) w ill in clu d e v a ria b ility from all five so u rc es, b u t N M E S in d u ce d force (g reen a rro w ) w ill in clu d e o n ly v a ria b ility d u e to n o ise in m u sc le fibres a n d th e n e u ro m u s c u la r ju n c tio n . T h e fo rce tra n s d u c e r is sh o w n o v e r the distal jo in t o f th e th u m b , an d E P L is th e o n ly m u sc le actin g to e x te n d th is jo in t and g e n e ra te th e re co rd e d force.
The sources of signal dependent noise during isometric force production: Methods
neuronal firing may contribute to SDN.
3.3 Methods
Experimental method
Five healthy young adults participated in the study (three men and two woman,
age range 26 to 37). An additional two subjects were tested with similar results but as
they were not tested in all conditions, we have excluded them from this report. All
participants indicated that they were right-hand dominant and gave informed consent to
the experimental procedures. The experiments were approved by the Committee for
Ethics in Human Experimentation at University College London.
The participants sat with their right arm resting on a table. The forearm was
positioned midway between pronation and supination and was stabilised with a vacuum
splint. The ulnar surface of the hand rested on the table and the hand was fixed in place
between a specially shaped stationary support against the palmer surface and a foam
covered sliding support applied to the dorsum o f the hand. The thumb rested on the
specially shaped palmer support with the distal phalanx extending beyond the edge o f
the support. The metacarpal and proximal phalanx o f the thumb were secured to a
foam-backed aluminium splint that was individually fit to each subject and extended
onto the forearm along the radius. The immobilisation o f the hand and forearm allowed
isolated movement o f the distal phalanx o f the thumb by the actions o f the flexor and
extensor pollicis longus (FPL, EPL) muscles.
With the hand and forearm secured in position, a force transducer was
positioned to contact the proximal nail bed o f the thumb while the thumb was relaxed.
The force transducer (NanolT, ATI Industries) had a 16-bit resolution over a range of
The sources of signal dependent noise during isometric force production; Methods
extension o f the distal phalanx o f the thumb. The digital output o f the force transducer
was sampled to disk at a rate o f 500 Hz.
An Odstock 4-channel Neuromuscular stimulator (NMES, Dept. Medical
Physics and Biomedical Engineering, Salisbury District Hospital, Salisbury, UK) was
modified for PC control. The modification consisted o f replacing the analogue
potentiometers controlling pulse amplitude with digital ones (Dallas Semiconductor,
D S 1267-50) and interfacing this to the PC via the standard parallel port. The
connections from the computer were optically coupled (Fairchild Semiconductors,
740L6000) to maintain subject isolation. Modulation o f force output with NMES can
be achieved by changes in pulse duration and/or amplitude, which controls recruitment,
and pulse frequency modulation that controls the firing rate o f motor units. We used a
fixed pulse width o f 300 psec, a fixed pulse frequency o f 25 - 30 pps (pulses per
second) and varied the stimulus amplitude to control the strength o f the contraction. To
circumvent the well-known problem o f fatigue during NMES, we used a 1:3 duty cycle,
i.e. 7 sec on followed by 21 sec off.
A round anode electrode (5 cm diameter, PALS Plus platinum electrode) was
applied to the dorsal surface o f the forearm just proximal to the wrist joint over the EPL
tendon. The motor point for the EPL was located by searching with a round (2 cm
diameter) saline soaked gauze electrode while palpating the EPL tendon until a site was
found that produced a robust extension o f the thumb. The motor point was generally
located when the search electrode was halfway down the forearm toward the ulnar side.
The site was marked and a carbon rubber electrode (cut to 1 X 2 cm) was fixed on the
skin over the motor point to serve as the cathode.
Prior to the start o f the experiment proper, the subjects were asked to slowly
The sources of signal dependent noise during isometric force production; Methods
and hold this maximum for 3 seconds while receiving verbal encouragement. The
average peak force during the last 2 seconds o f the contraction was calculated over the 3
trials. This value was loosely considered the maximal voluntary contraction (MVC) and
all forces are reported with respect to this value.
Following localisation o f the EPL motor point, the relationship between stimulus
amplitude and force output was tested. The minimal stimulus amplitude was set to
produce approximately 20% MVC and the maximal amplitude to produce roughly 70%
MVC. Six stimulus amplitudes were selected to cover the range between 20% and
70%. As the relationship between stimulus amplitude and force output was non-linear
and varied between subjects, the linear spacing o f stimulus amplitudes did not result in a
linear spacing o f force output. However, this non-linearity was o f little concern to the
present study.
A session consisted o f three types o f isometric contractions: voluntary, NMES
and mixed. In the voluntary condition the subjects extended their thumb against the
force transducer to move a visual cursor into a target window. After 3 seconds, visual
feedback o f the target and cursor were removed and the subjects were asked to maintain
a constant effort for an additional 4 seconds; the last 4 seconds o f the force recording
were used for the subsequent analysis. This was repeated 6 times at each of 6 different
target force levels ranging from approximately 20 - 70% MVC. In the NMES condition
the subjects were asked to close their eyes, relax completely and resist the temptation to
interact with the stimulus evoked contraction. The stimulus amplitude was linearly
ramped to a final amplitude over 2 sec followed by stimulation at a constant amplitude
for an additional 5 seconds. The last 4 seconds o f the force recording were used in the
subsequent analysis. This stimulation protocol was repeated 6 times at each o f the 6
The sources of signal dependent noise during isometric force production; Methods
o f voluntary effort onto a stimulus induced contraction to move the cursor into a target
window fixed at approximately 70% MVC. The same 6 stimulus amplitudes were used
as in the NMES condition. However, in this case the subjects had to attend to the
stimulus and produce the extra effort needed to reach the fixed target. Thus the
voluntary effort needed was inversely proportional to the magnitude o f the stimulus-
induced contraction.
The order o f three conditions was randomly presented, and within each
condition the six amplitudes were randomly presented, while ensuring that two identical
conditions were not repeated back-to-back. The time between trials at one target
amplitude was 21 seconds and there was a 2-minute interval prior to the presentation o f
the next condition.
Model of the motor-unit pool
The modelling component o f this study relied heavily on a model o f the motor-
unit pool by Fuglevand and colleagues that has been described in detail (Fuglevand et
al., 1993, Yao et al., 2000). The previous studies using this model considered the output
of both force and EMG, whereas we have used only those portions o f the model
concerned with force production. Where this implementation o f the model closely
follows the previously published version, we will briefly describe the main assumptions
and parameters used. In those areas where our implementation departs from the original
model o f Fuglevand et al. (1993), a more thorough description is given. The model o f
the motor-unit pool was implemented in the MATLAB environment, and is summarised
schematically in Figure 3-2. The duration o f the simulations was 5 seconds with a time
The sources of signal dependent noise during isometric force production: Methods
The model consisted o f a pool o f 120 motor neurons that was excited by an
excitatory drive (E) distributed uniformly across the motor neuron pool and measured in
arbitrary excitation units (eu). For each simulated trial, a motor neuron was activated if
its recruitment threshold (RTE) was greater than E. The distribution o f recruitment
thresholds across the pool was modelled by an exponential relationship resulting in a
pool with a relatively greater number o f low than high threshold motor neurons. Upon
recruitment, each motor neuron o f the pool fired with a minimum mean rate o f 8 pulses
per second (pps). The increase in mean firing rate above 8 pps for an increase in the
excitatory drive was modelled as a linear function with a gain o f 1.5 pps/eu. The mean
firing rate of each motor neuron increased up to its peak rate, which ranged from 45 pps
for the lowest threshold motor neuron to 35 pps for the highest threshold motor neuron
{Eq. 5 Fuglevand et al., 1993 and left hand plot o f Figure 3-2). The relative
contribution o f recruitment and rate-coding to force production was determined by the