INFORMATION TO USERS This manuscript~has boen fmm the miao(im master. UMI film the text di* frwn ttio original or ooqy suhitW. Thus, sotne lhe& and dissertation ayWs are in typewbr face, while oîhm rnay be f i any type d cornputer printer. The qurlity d thk nproductiorr k dopandont upm the d th. copy submiüed. Broken or indistinct pria cdored or par quality illustrations and phdographs, print UeWüwwgh, WManclard margins, and imgro~sr alignment can advenely affect aiproduction. In the unlikely event lhat Ihe euthor did mt serd UMI a comptete rnanusaipt and there are misshg pages, these will be noted- Also, if unauthorired copyrÏght material had to be removed, a note will indkate the deletion- Oversize materials (e-g., rnaps, drawings, charts) are reproduced by sectiming the original, begiming at the upper lefthand oomer and conü~ing fmm left to rigM in equal sectioc~s with small wehaps. Photographs induded in the original ma-@ have been reproduced xeqraphically in thb mpy, Higher qwlity 6. x W ûhck and white photographie prints are availabie for any photographs or illustrathmi amring in this copy for an adâiil aiarge. Contad UMI directly to ardef. 8811 & Howell Information and Leaming 300 North Zeeb Rosd, Ann Arbor, MI 481S1346 USA ôûû-521-
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INFORMATION TO USERS
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NOTE TO USERS
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UMI
AN ASYMMETRIC BILATERAL MODEL OF STRETCH REFLEXES
A thesis submitted to the
Faculty of Graduate Studies and Research
in partial fulfillment of the requirernents for the degree of
Master of Engineering
Department of Biomedical Engineering
McGili UniversiSr: Montréal
August, 1997
w a t i a n a Nikitina, 1997
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Abstract
-4 bilateral stretch reflex model was built based on m e t n c agonist - antagonist
reflex connections at the ankle. The overd dynamics were analyzed using Mason's
rule algebra. The model predicts the characteristic agonist and antagonist responses
to stretch. Simulations with pulse displacements and random perturbations are in
qualitative agreement with experimental data and suggest a mechanism whereby the
mean absolute veiocity of the perturbations could depress the stretch reflex at the
spinal level. This implïes that spinal neural circuits can calculate and carry out their
o m integrative h c t i o n s and, in particular, modulate the stretch reflex gain.
Résumé
Cette thèse présente un modèle bilatéral du réflexe d'étirement qui a été construits en
se basant sur les circuits réflexes asymétriques agonistes - antagonistes au niveau de la
cheville. Le modèle a été analysé en utilisant les règles d'algèbre de blason. Ce modèle
peut prédue certains comportements caractéristiques agonistes et antagonistes. Les
résultats des simulations faits avec des impulsions de déplacement et des perturbations
aléatoires sont en accord avec les donne& expérimantales. Il y aurait existence d'un
méchanisme qui régit l'étirement au niveau de la colonne vertébrale en fonction de la
vitesse moyenne absolue des perturbations. Cela implique que les circuits neuronux
au niveau de la colonne ont la capacité de calculer et d'appliquer Ieur propre fonctions
intégratives, en particulier la modulation du réflexe d'étirement-
Acknowledgment s
1 would like to express my sincere appreciation to Dr. R.E. Kearney and Dr. H.L.
Galiana for their invaluable support, guidence and encouragement. 1 am very gratefull
for the opportunity to study under their direction.
1 am indebted to many members of the Biomedical Engineering Department for
t heir stimdating conversation and helpfull suggestions.
1 wodd like to thank Sunil Kukreja for his assistance on countless occasions. His
technicd expertise and perpetual enthusiasm have helped me tremendousZyY
Findy, 1 also Nish to thank my family for their support.
This work was supported by grants fiom the Natural Science and Engineering
Research Council of Canada (NSERC) and Medical Research Council (bLRC), and
Fonds pour la Formation de Chercheurs et l'Aide à la Recherche (FCAR).
Page(s) not included in the original manuscript are unavailable from the author or university. The
manuscript was microfilmed as received.
This reproduction is the best copy available.
UMI
the perturbations is the most important variable in depressing the reflex [53, 131.
Most modeling efforts to date have assumed that the bilateral structure of the
motor control system serves merely t o compensate for the fact that muscle can ody
"pull" and not "push." Therefore, d a t e r a i lumped models have been used such
models do not account for the asymmetry or the modulation of the stretch reflex.
Meanwhile, a bilateral model for another sensory motor system - vestibdo-ocular
reflex- - has desonstrated that interactions between central neural pathways ac-
count for many behavioral aod neurophysiological findings [50]. ilnalytical studies
of vestibule-ocular reflex models show that the key requîrements are reciprocal com-
missural connections and feedback from neural low-pas Slters on each side of the
system. Reciprocal inhibition and Renshaw cells could provide analogous functions
in the neuromuscular system. Therefore, modeling the peripheral neuromuscular sys-
tem as a bilateral asymmetric structure may provide valuable insight into the nature
of motor control. The objective of this work was to develop such a model and to
investigate whether it could account for the properties of the stretch reflex at the
ankle, in particular the modulation of reflex gain with ongoing movement.
1.2 Outline
Chapter 2 presents anatomy and physiology relevant to the neuromuscular control of
the ankle. Properties of skeletal muscle and its proprioceptive receptors are reviewed.
The agonist-antagonist muscle org;tnization of a joint and its spinal connectivity is
presented-
Chapter 3 gives a brief review of the literature concerning the modeling of the
neuromuscular control system including peripheral receptors, muscle, and neurons.
Chapter 4 discusses the structure of the model and the dynamics of each element.
Mason's d e s are used in a theoretical anaiysis of the dynarnics of the overall model.
The parameters of the model are then chosen based on the known physiology and
anatomy of the ankle.
Chapter 5 presents the results of simulation studies with this bilateral asymmetric
model of the stretch reflexes at the d e . Responses to step and pulse displacements
in isolation, and in combination with random inputs are presented and their physio-
logical implications are discussed.
Chapter 6 gives a siimmary of the work, discusses the physiological basis for the
essential elements of the model, and makes suggestions for future work.
Chapter 2
Anat omical and P hysiological
Background
2.1 Introduction
This chapter describes the anatomy and physiology relevant to the peripheral neu-
romuscular control of the ankle. Skeletal muscle and its proprioceptive receptors are
reviewed first . Then the hierarchy of mechanisms controllhg voluntary movements
is discussed. Finally, the agonist-antagonist organization of adde muscles and their
spinal reflex circuits are reviewed.
2.2 Skeletal Muscle
The term muscle describes a number of muscle fibers bound together by connective
tissue. There are three types of muscle in the body: cardiac, smooth and skeletal.
This discussion d be limited to skeletal muscles which produce movement and are
under voluntaxy control.
Skeletal muscles are linked to bones by bundles of collagen fibers, tendons, Iocated
at each end of the muscle (Figure 2.1). Muscle fibers have a characteristic banding,
which results £rom the presence of numerous thick and thin filaments in the cytoplasm
of the myofibrils. One unit of this repeating pattern is known as a sarcomere. The
~trCnH)FUYElin
Figure 2.1: Organization of the muscle. Adapted fiom [55].
thick filaments are composed entirely of the contractile protein rnyosin. The thin
filaments contain the contractile protein actin as weil as other proteins - tropin and
tropomyosin.
The space between adjacent thick and thin filaments is bridged by projections
known as cross bridges, portions of myosin molecules that extend fkom the surface of
the thick filaments tonrard the thin filaments. During muscle contraction the cross
bridges make contact with the thin filaments and exert force on them. Propelled by
the forces on cross bridges the thick and thin filaments move past each other. This
is known as the slidzng-filament mechankm of muscle contraction.
Figure 2.2: Length-tension relation of the muscle. A - Experimental setup: steady state tension measurements are taken at series of hxed muscle lengths- B - The dotted line is due to the passive stiffness of the muscle. The solid line is the length-tension curve for the same muscle as it is stimulated to produce maximal tetanic tension. Total tension is the sum of passive and active tension. Redrawn from (161.
The sliding filament theory explains the force - length relation of a muscle as
shown in Figure 2.2. The active tension developed during contraction depends on the
degree of overlap between thick and thin filaments. The Iength a t which the fiber
develops the greatest tension is termed the optimal length, 1,. When the sarcomere
is stretched to approximately 1.75 l,, the filaments do not overlap, and the number
of possible cross bridge attachment points is very low. Consequently at this point
the force generated is nlmost zero. The tension is also decreased when a sarcomere is
shortened beyond a certain limit. In this case the two sets of thin filaments Tom the
opposite ends of the sarcomere interfere with the cross bridges' abiliq to bind and
esert force. Between 1.75 1, and 1, the tension increases in proportion to the degree
of overlap of thick and thin filaments since the number of potential cross bridge sites
increases.
2.3 Motor Units and Neural Gradation of Muscle
Force
Each muscle fiber receives its innervation £rom the axon of a single motor neuron
located in the spinal cord. A motoneuron, its axon and the muscle fibers it supplies
are called collectively a motor unit, which is the smallest functional unit. There are
two main types of motor units: red and white. Twitch contractions of the red fibers
are slow, relatively small in amplitude, but fatigue resistant- In contrast, white fibers
produce larger tetanic tension, are faster, but are easily fatigued [32].
Changing the number of active muscle units will change the overall muscle force;
this mechanism of neural gradation of force is called recruitment. Motor units are
recruited according to the Henneman "size principle": small units are recruited first
and larger units are recruited progressively as the level of contraction increases [20].
The second mechanism for the neural gradation of force is rate coding. Repetitive
action potentials can produce tetanic tensions three- to five times latger than in a
single twitch. Therefore, varying the hequency of action potentials provides a way
to make three- to fivefold adjustments in the tension of the recruited units. The
relative importance of the two control mechanisms is hard to assess and varies with
the muscle. In general, smaller muscles rely primarily on firing rate coding while
larger muscles rely primarily on recruitment to modulate their force [38].
2.4 Proprioreceptors
The state of muscle is monitored constantly by the CNS, using information from
difFerent types of proprioreceptors embedded within the muscle. These provide infor-
mation about muscle tension, its length and velocity. Two receptors are particdarly
important for motor control: muscle spindles and Golgi tendon organs. As shown
in Figure 2.3 spindes lie in pardel with the muscle fibers and provide the CNS with
information about muscle length and stretching velocity. Golgi tendon organs are
arranged in series with the muscle and provide feedback about the tension developed
in the muscle.
2.4.1 Muscle Spindles
The main components of the muscle spindle are shown in the Figure 2-4. Sensory
endings of afferent nerve fibers are wrapped around modified muscle fibers, called
intrafusal muscle fibers. There are three types of intrafusal muscle fibers: dynamic
nuclear bag, static nuclear bag, and nuclear chah fibers [16]. Several such intrafusal
muscle fibers are enclosed in a connective tissue capsule to form a muscle spindle
having two distinct stretch receptors. There is a single primary ending per spindle
supplied by a large diameter (Ia group) myelinated axon. It innervates all three types
of the intrafusal fibers. There are also multiple secondary (II group) endings that
innervate the static nuclear bag and chah fibers. Distortion of these stretch receptors
generates action potentials in the derent fibers.
Intrafusal fibers also receive efferent innervation kom gamma y, or fusimotor
fibers. The contraction of the intrafusd muscle fibers produces little tension, but
does modulate the response of the spindle. For example, in a shortened muscle the
stretch receptors would normdy become sIack and inactive, but gamma drive to the
intrafusal fibers can maintain the tension in the central portion of the spindle to keep
the stretch receptors active. There are two types of fusimotor fibers. Static fusimotor
fibers have a low resting rate and are tonicdy active during on-going movements,
while dynamic fusimotor fibers have a relatively high resting rate and show strong
Figure 2.3: Anatomical view of the spindle and Golgi tendon organ. Redrawn from
Figure 2.4: Main components of the muscle spindle. For explanation see text. Re- d raw from [16].
phasic modulation. S tatic gamma-neurons enhance the steady state discharge from
the primary spindles, while dynamic gamma-neurons markedy enhance the high-
fkequency burst of spindles during the dqnamic phase of stretch 1161.
2.4.2 Golgi Tendon Organs
Golgi tendon organs, shown in Figure 2.3, are another type of proprïoreceptor located
in the tendons near their junction with the muscle. Each tendon organ is innervated
by a single group Ib axon that looses its myelination after it enters the capsule and
branches into many fine endings. These are mapped axound the filaments of the ten-
don. When the muscle contracts, the tendon filaments straighten, distort the endings
of the Golgi tendon organ, and so generate action potentials. Golgi tendon organ
receptors discharge moderately in response to passive forces or changes in muscle
length, but are very sensitive to active forces developed in response to a-motoneuron
firing [4]. This is due to the fact that each tendon organ is attached to relatively
few muscle fibers, and senses ali forces generated by these fibers. In contrast, only
a fraction of the passive force will be seen by one tendon organ, because the passive
force is divided between aU fibers in the muscle.
2.5 Agonist- Antagonist Organizat ion of the Joint
Muscles
-4 contracting muscle c m exert an active force in only one direction, it can "pull",
but not "push". Therefore, active control of a joint requires at Ieast two muscles,
pulling in opposite directions, the so called agonist - antagonist pair. In reality more
than two muscles usudy act around a joint, providing redundant control and often
movement about more than one axis of rotation. For any motion, the muscles can
be divided into two groups: synergists and antagonists, with one muscle in each
group being the dominant.
For example, during plantadexion of the ankle the dominant muscle, that p r e
duces most of the torque, is Triceps Surae, which consists of two parts - Gastroc-
nemius and Soleus as s h o w in Figure 2.6. Synergists, producing much l e s torque
include Plantaris, Tibialis Posterior, Flexor Hallucis Longus and Flc~or Digitorium
Longus. Dorsiflexïon is maialy produced by Tibialis Anterior with the synergistic ac-
tion of Extensor Digitoriwn Longus, Extensor Hallucis Longus and Fibularis Tertius
(Figure 2.5).
2.6 The Motor Control Hierarchy
The neural systems that control body movement are arranged in a hierarchical man-
ner. The highest level determines the general intention of the action. It encompasses
many regions of the brain. The middle level specifies the postures and movements
needed to carry out the intended action. The commands at the middle level are con-
stantly updated depending on feedback information fiom proprioreceptors. Neurons
of the middle level are found in the sensomotor cortex, cerebellwn and brainstem
nuclei. Local centers, in the spinal cord, finaUy determine which a-motoneurons will
Figure 2.5: Muscles of the lower leg: anterior view. Redrawn £rom [6] .
Figure 2.6: Muscles of the lower leg: posterior view. Redrawn from [6].
be activated. Because motor control areas have so many reciprocal connections and
generally are very cornplex, it is very difEicult to assign specific tasks to given centers.
Despite much research effort, knowledge about the relative roles of the difîerent levels
of motor control is very lùnited [9].
2.7 Spinal Circuits
Spinal connections of the motor control system have been and rernain the subject of
extensive research efforts. Figure 2.7 presents the major connections of an agonist-
antagonist muscle pair (see [23] for a complete review). As discussed in Section 2.3
the a-motoneuron axon is the only efferent input to muscle fibers. -4U descending
commands, sensory feedback and various interneuronal connections converge and are
processed a t the level of the a-motoneurons. Because of this, Sherrington named
them the "hd common path" fiom the spinal cord to the muscle.
What are the sjnaptic inputs to the cr-motoneuon ? First, aMns receive monosy-
naptic (direct) excitatory input from the spindle. When a muscle is stretched, the
spindle afFerents discharge (section 2.4); because of the excitatory monosynaptic con-
nection, the firing rate of aMn increases, causing the muscle to contract. This is
the so-cded stretch reflex. The gain of the stretch reflex can be modulated by
many mechanisms. In particular, fushotor control through 7-motoneurons can al-
ter the state of the spindle and modulate the strength of the synaptic input to a
a-motoneuron. Despite its apparent simplicity, the function of the stretch reflex
rernains controversial.
Spindes also make excitatory connections to interneurons (INs) , which in t um
inhibit the a-motoneuron of the antagonist. Increasing the firing of the spindle of
the agonist will thus produce inhibition of the antagonist a-motoneuron, causing the
ant agonist t O relax, and vice-versa. This is the reciprocal inhibition demonstrated
by Sherrington. Reciprocal inhibition is mediated by interneurons, which receive
descending commands, so the CNS can control the balance between agonist-antagonist
muscles. For example, when precise movement is needed, reciproca. inhibition can be
Figure 2.7: Schematic diagram of spinal connections. Peripheral receptors are rep- resented by boxes, inhibitory neurons by shaded circles, excitatory neurons by white circles. ALPHA - a-motoneuron, RC - Renshaw cell, GAMMA - -/-muron, HTA - high threshold afferents.
inhibited, so that agonist and antagonist CO-contract and result in a stiffer joint.
There is another diqmaptic spinal pathway that converges on antagonist muscle
motoneurons during agonist contraction [44]. Interneurons mediating this pathway
are called Ib interneurons. These inhibit homologous a-motoneurons and excite an-
tagonistic a-motoneurons. In addition, fi interneurons inhibit non-antagonistic a-
motoneurons, and other Ib interneurons. The main input to Ib Interneurons is £rom
Golgi tendon organ receptors, dthough 40 % of Ib Interneurons are &O exited by
Ia derents [25]. These interneurons also receive inputs from high threshold &rents
and cutaneous and joint receptors. Therefore, nerve impulses in tendon organs of a
muscle may affect motoneurons of practically any muscle of a limb.
In the same way that Ia interneurons mediate the reflex actions of prima.ry spinde
afferents, there are interneurons in pathways fiom secondâry spindle afferents (II
group) as well. However, unlike Ia interneurons these do not appear to be under
the inhibitory control of Renshaw cells (see below). They are thought to coordinate
activity of a greater variety of muscles than Ia inhibitory interneurons since they send
connections to a-motoneurons of several muscles acting a t a joint [24].
Group II muscle spindles as well as group Ia spindle and group Ib Golgi tendon
organ afferents, âppear to be inhibited presynapticdy by separate populations of
interneurons (ornitted for cla.rity in the Figure 2.7)- These are divided into a separate
category based on functional rather than anatomical critena. They can receive inputs
from all types of proprioreceptors, but their common firnction is to modulate relative
importance of these inputs. For example, presynaptic inhibition can reduce the effects
of input from the prina.ry spindle, and increase the weight of the Ib input. In summary
interneurons mediating presynaptic inhibition can gate the input from sensory fibers
to the a-motoneurons 1231.
Another class of inhibitory neurons are Renshaw cells. Their main excitatory
input is from the a-motoneuron, but they also c m receive inputs from free nerve
endings of groups II, III, N, and from high threshold afferents [23] '. Renshaw cells
'Small-diameter muscle group III, IV afferents and non-spindle group II play an important role during muscle fatigue [57]. In response to metabolites released by fatiguing muscles they act to
retuni inhibitory connections to the same neuron that excites them, Their action is
referred to as recurrent inhibition. Renshaw cells also make inhibitory connections
to ant agonist int erneurons, and so disinhibit antagonist motor neurons. Renshaw cells
also inhibit a-motoneurons innervating synergistic muscles. Therefore, the effect of
recurrent inhibition is distributed to all muscles around the joint.
The connections in Figure 2.7 are for a pair of muscles; there are &O pathways
from the contralateral side of the body. These connections form the basis for the
flexion withdrawal reflex. Flexion withdrawal r o m noxious stimulus is a pro-
tective reflex involving coordinated muscle contractions a t multiple joints through
polysynaptic reflex pathways. It involves reciprocal innervation: flexor muscles of the
stimulated limb are contracted, while the extensor muscles are inhibited. Along with
flexion of the stimulated limb, the reflex produces an opposite effect in the contralat-
eral limb: extensor muscles are excited and f le~or muscles are inhibited. This crossed
extension reflex serves to enhance postural support during withdrawal £rom painful
situation. The connections of ipsilateral FRA (flexion reflex afferents) are presented
in Figure 2.8, while contralateral FRA wiIl produce the opposite pattern (omitted for
clarity) .
The Ia inhibitory interneurons, excited monosynaptica.lly from flexor nerves, re-
ceive stronger net excitation from activity in ipsilateral FRA than do extensor coupled
interneurons. Mthough the FRA interneurons, which excite and inhibit in parallel
corresponding a-motoneurons and Ia iahibitory neurons, are drawn as separate neu-
rom, it is possible that they are actually the same neurons [22].
In conclusion it is important to note that the schematic diagram of Figure 2.7
should be thought of as a lumped representation of spinal circuits. DXerent recep-
tors do not always send connections to all neurons in a pool; sometimes they excite
only a fiaction of it. Moreover, the neurons in a pool may have interconnections be-
tween themselves; for example, self-inhibition has been shown for a-motoneurons [57].
Therefore, the anatomy and physiology of the neuromuscdar control mechanisms of
a joint is very complex and presents a major challenge to a modeler.
in hibi t homonyrnous and synergistic muscles.
Figure 2.8: Ipsilateral FRA reflex connections. Inhibitory neurons are represented as shaded circles, and excitatory as white circles. Group II, group III muscle afferents and afferents from skin and joints are collectively c d e d FRA (flexor reflex afferents).
Chapter 3
Modeling the Neuromuscular
System
3.1 Introduction
The bais of any mode1 is howledge about the system. There are two basic and quite
different ways to obtain such knowledge: one can either study the laws of nature that
govern the system and deduce its behavior, or one can observe its extemal behavior
under different conditions. Accordingly, there are two ways of modeling a system: a
priori, or morphological modeling, and a posteriori or, "black box", modeling. Both
are difficult and challenging if the system in question is biologicd.
Unlike physical laws, biological laws are not well-known or well-defined; and bio-
logical systems, even small ones, are much more complex than human-designed sys-
tems. For example, an information flow diagram for the dynamics GE a single joint,
presented in Figure 3.1, cont ains several subsysterns - muscles, spindles, neurons,
etc. and various nested loops. Moreover, these subsystems interact in nontrivial
ways. Notwithstanding these problems, extensive work has been done in the area of
neuromuscular modeling [l, 2, 31. The advantage of morphological modeling is obvi-
ous: the mode1 reflects the interna1 structure of the real system and, is thus, more
LLtrustworthy", at l e s t to m q physiologists and physicians.
As for "black boxy' modehg, also known as the system identification method,
Figure 3.1: Information flow in the peripheral neuromuscular control system. Re- dranm £rom [29].
biological systems are very di£Ecult to observe, because in general every system is a
subsystem of another biological system. For example to observe the behavior of a
muscle one would have to isolate it from the body, which, generdy speaking, v.dl
change the properties of the muscle so the resulting observations would not be valid
for muscle in its natural state. Still this method is somewhat easier to apply and, if
the goal of the modeling is to acquire quantitative models of physiological function,
it is perfectly suitable.
In modeling a biological system, one should not be limited to any one approach,
but rather try to take advantage of the strengths of both methods. Construction of a
cornplex model based on first principles may be very attractive, but validation of such
a model on all levels poses a difficult problem. Without adequate model validation,
any suggested morphological structure remains speculative. In contrast, "black box"
models, while quantitatively credible, may provide little functional insight. However,
they can be used as a reference against which morphological models can be validated.
Thus, an ideal mode1 would combine the physiological insight of morphological rnodels
with predictive ability of "black box" models.
3.2 Modeling proprioceptive receptors
As discussed in Chapter 2 proprioceptive receptors transduce mechanicd variaoles
into action potentials. Figure 3.2 illustrates three cascade subsystems assumed in
most morphological model of a mechanoreceptor [21].
Figure 3.2: Essential components of mechanoreceptor model. Redrawn fiom 1211.
The first block, c d e d mechanical filtefing, converts a mechanical variable, for ex-
ample muscle length, into eautension of the nerve ending. The second block, transducer
is responsible for generation of action potential and the last, encoder, for impulse
activity. Such morphological models are too complex to include in neuromuscular
control models, therefore lumped models, having a mechanical variabIe as an input
and firing rate as an output, are used more often.
3.2.1 Spindles
Spindles are thought to contribute the most to reflex dynamics and have been the
subject of many modeling efforts. Houk [21] proposed the following transfer function
to describe spindle's mechanical filtering:
where F is the output firing rate of spindle, L is spindle Iength, 77 is an overshoot
factor, Cl is a constant, and T is the time constant of decay of spindle response to a
small step input. Iii this model linear dynamics are followed by a static nonlinearity
(y = x2) and saturation.
Poppele applied systern identification methods (431 to spindles and formulated
modeIs of both primary and secondaxy responses in the absence of fusimotor activa-
tion. He found two static non-linearities: sensitivity of spindle as function of stretch
amplitude as shown in Figure 3.3, and as a function of muscle length as shown in
Figure 3.4.
Figure 3.3: Static nonlinear sensitivity of a spindle as a h c t i o n of stretch amplitude. Reciranm from [43]. Data obtained for sinusoidal stretching of 1Hz. Both primary and secondary spindIes are more sensitive for small amplitudes of stretch.
Figure 3.4: Static nonlineat sensitivity of spindles as a function of muscle length. Redrawn from [43]. Data obtained for sinusoidal stretching of 1Hz. Amplitude of sinusoidal stretching vas five time bigger for secondary spindles. The sensitivity of both changed about five-fold as the muscle was pulled £rom slack to taut.
For s m d amplitude inputs he found the following transfer f'ctions for primarq-
and secondary engings
where input L is muscle length and output F is firing rate of spinclle- Bode plots for
both transfer functions are presented in Figure 3.5. Note that the linear portion of
the response of primary and secondaq fibers are quite similar: both are high - p a s
with cutoff frequency a t 1 Hz. However, primary spindes are much more sensitive to
srnalier changes in the muscle length, and respond to acceleration of the muscle at
kequencies greater than 7 Hz [43].
Hasan [19] constructed a morphological spindle model and tested it against ex-
perimental observations. Be argued that mechanical filtering codd be modeled as a
nonlinear dynamic operation:
where x(t) is muscle length, z(t) is the length of the sensory zone, A and B are
constants that effect the degree of nonlineazïty, and C is the value of the position range
where the muscle spindle ceases to respond to stimuli. The transducer and encoder
blocks were linear subsystems with rate sensitivity. This model was tested mainly
against " r m p and hold" experirnents, giving quantit atively comparable predictions.
The keque~cy response of a hearized Hasan model is presented in Figure 3.5
None of these models included time-varying fusimotor input, which can alter the
overall response as discussed in Chapter 2. Some atternpts were made to model the
fusimotor effect at constant length 121, but as yet there is no multi-input model of
spindle response.
3.2.2 Golgi Tendon Organs
Historically Golgi tendon organ responses have received l e s attention, and al1 models
(reviewed in [2]) are similar. The Anderson model, discwed in [2], serves as an
example:
Figure 3.5: Gain portion of Bode plots for different models of spindle. Note that for high frequencies the dynamics of ail models are similar to a diflerentiator (dotted line) .
This transfer function models changes in the firing rate of receptor with changes in
active tension, developed in the muscle. The kequency response of the Anderson
mode1 is plotted in Figure 3.6.
Recent experimental studies of Golgi tendon organs have shown that in addition
to transducting the mechanical stimulus, dynarnic response of the receptors has a
transient element, related to the rate of rise of the receptor potential. However, this
cornponent of the response appears to affect the frequency of the discharge of only
the first two or three action potentials [12].
Figure 3.6: Gain and phase portions of the fkequency response of the Anderson mode1 of Golgi tendon organs.
3.3 Muscle
Muscles are the actuators of the neuromuscular system, and have two main inputs:
length and neural activation, whose effects need to be modeled. Contractile mechan-
ics detennine the forces evoked in response to length changes with constant neural
activation; activation dynarnics define the changes associated with changes in level
of neural activation. Interactions between these two mechanisrns occur when both
inputs change at the same tirne. A variety of muscle models both "black-box" and
morphological have been developed- The literature dealing with these muscle models
is vast so no attempt at a comprehensive review will be made; only relevant work dl
be cited.
3.3.1 HiIlModel
The classic lumped-parameter Hill model forms the foundation for most "black box",
or macroscopic models of muscle, and is presented in Figure 3.7.
Figure 3.7: Classical structures for the Hill muscle model. Redrawn from [60]. CE - contractile element, SE - series elasticity, and PE - pardel elasticity. For e-xplanation see text.
The main components of this mode1 are a contractile element (CE), a series elastic-
ity (SE) and parallel elasticity (PE). The contractile element representing actomyosin
cross bridges is assumed to generate some force. This force is considered to be mod-
ulated by spring - dashpot systems befure appearing at the tendon as the tensile
force. The series spring models the passive elasticity of the cross-bridge stmcture,
while the pardel elasticity corresponds to that of the connective tissue surrouncling
the sarcornere. Series e las t ic i~ and parde l elasticity can be arranged in the two
forms shown in Figure 3.7. However, if the elements are linearized, the modeLs are
equivdent [39].
The contractile element is characterized by tnro nonlinear relationships: CE force-
velocity and CE tension-length, and is modulated by activation level. The force-
velocity relation reflects the fact that muscles exert lower forces when shortening,
then at constant length. Moreover, the faster they shorten, the less force they produce.
This behavior can be described by a hyperbolic equation introduced by Hi1T:
where F is exerted tension, F,, is isornetric force
of shortening, and a and b are Hill constants. It can
where a l - " is Fmaz
Figure
at optimum length, v is velocity
be rewritten in normalized forrn:
a dimensionless parameter which specifies the
3.8.
r 1
hyperb oLic concavity
Figure 3.8: Effect of parameter al on hyperbolic HiIl equation (shortening to left). Redrawn from [60].
Muscle will lengthen at a constant speed if the applied force is greater than the
isometric force. During muscle lengthening, the force-velocity relationship expressed
by equation 3.6 does not hold. The speed of lengthening under these conditions is
much less than that predicted by HU'S equation. It has been found experimentally
that the hyperbolic concavity parameter, a/, for lengthening is approximately six
times greater than for muscle shortening 1391. Lengthening contractions d l produce
Iarger forces then isometric, and then muscle will veld". The force-velocity curve
also varies with the initial length of the muscle.
The length-tension relation is another characteristic of the CE used to describe
the observation that isometric muscle force reaches a maximum at an intermediate?
so-called "optimum", length, and decreases for shorter or longer lengths. This is
usually a t t ~ b u t e d to the degree of overlap of myosin and actin filaments, as discussed
in the sliding filament theory of muscle contraction in Chapter 2- The experimental
length-tension relation presented in Figure 2.2 has been descnbed by many empirical
fits. For example, Hatze 1611 described the length-tension relationship for a single
sarcomere as
where 0.58 < Xs < 1.8pm is the sarcomere length, and O < c < 1 is the dimensionless
control input.
The two basic non-linearities in the CE scale Mth activation level, so the force
produced by CE, Fce7 is assumed to be a product
where Fm, is the maximal CE force, A is normalized activation, v,, is the rate
of chânge of length, lce is length of muscle, Ffi(vCe) and Fu(l,) are dimensionless
func tions. In t his relation normalized force-velocity and length-tension are assumed
t O be independent static nonlinearities-
The Hill force-velocity relation has been shown to adequately describe nearly all
Figure 3.9: Hatze empuical fit to active length-teasion relation. Compare to experi- mental curve redram in Figure 2.2.
muscles thus far examined, including cardiac and smooth muscle as weU as skeletal
muscle [39]. Nevertheless, there are some major difEculties with his overail model.
First, some simulation studies [46] suggest that muscle does not have a unique in-
stantaneous force-velocity characteristic. Second, the length-tension c w e as shown
in Figure 2.2 is not the stress-stain curve and does not reflect steady-state force in
response to changing of the length of active muscle [29]. Despite these problems, this
traditional model has been the choice for many modeling studies of muscle, and a
number of new Hill-based modeIs have been developed. For e~ample, Morgan et al.
[41] applied the lumped Hill-based model to represent a sarcornere instead of a whole
muscle. These sarcomere models are then connected in series to simulate a fiber. This
model of intersarcomere dynamics can simulate the main feature of tension creep in
£ked-end contractions.
3.3.2 Morphological models
The original biophysical, or rnicroscopic, mode1 of contraction was introduced by
Huxley in 1957 and assumes that a myosin cross bridge can &st only in one of two
distinct biochemical states: attached to actin or detached. In the attached state
it becomes either stretched or compressed, and produces force. The s u n of forces
developed by all cross-bridges is the force of the muscle. It was postulated by Huxley
that there must be two distinct paths between attachment and detachment in order
to produce net work [60]. Let f (x) be the rate constant for attachrnent and g ( x )
for detachment, and assume both are functions of x - the distance from the neutrd
equilibrium state of a cross bridge, Then a partial differential equation can be written:
where n is a probability distribution
cross bridges [59] . Physical variables
this distribution.
function representing the fraction of attached
like muscle force are computed as moments of
Subsequent developments of this microscopie model have allowed for increased
n u b e r s of rate function and activation-deactivath states, multiple actin sites [Il],
and compet ition between myosin heads [62]. The distribution-moment model of Za-
hdak [62] is a two-state model integrated with a mode1 of electrical stimulation and
calcium-activation dynamics; it is thus a compromise between Hill-based and bio-
physical models. Because of a number of assumptions (for example, a t any given time
each cross bridge interacts with only the closest actin site), it remains mathemati-
cally tractable. Nevertheless it offers quite reasonable predictions for most muscles
in studies to date, both for shortening and stretch, especidly the 'MeIding' prop-
erty of muscles. In general, biophysical models remain extremely complex and their
simulation exerts a heavy computational burden.
An interesting concept was introduced recently by Witers [58], foilowing Feldman's
[14] recommendation that muscle and reflex properties should not be considered sep-
arately as has been done traditionaily in most models. An intrafusal muscle (IF)
spindle model was embedded in synthesized muscle-reflex model. The main extra-
fusal muscle model was Hill-based, Le.. has CE, PE and SE elements. The IF' was
assumed to be skeletal muscle tissue lying in pmalle1 with extrafusal muscle (EF),
and thus also has CE, PE and SE elements of its own. The spiode sensor measures
strain across the intrafusal SE and not relative position error. Winters found that this
Hill-based model provided a more appropnate spinde output, especidy for studies
of posture.
Ln conclusion, it is appropriate to cite J.M. Winters and LStark who, in discussing
different muscle models, wrote [59]: "Each of these modeling approaches has its
advantages and disadvantages, and the type of model used will depend strongly on
the goals of study."
3.4 Modeling Neurons and Synaptic Connections
The local control loop of multi-level movement control consists of the spinal a-
motoneuron pool and its efferent fibers, the muscle, the proprioceptive receptors,
and the deren t fibers going back to the same motoneuron pool. The activity of
the a-motoneuon pool is strongly modulated 'Dy interneurons in the spinal cord as
described in section 2.7. Therefore any mode1 of peripheral neuromuscular control
must include models for neurons and their synaptic connections- The amount of
modeling afTorded to the neuromotor circuitry is relatively small. This is mainly
due to the sheer volume of cells involved and the vast number of interconnections,
which makes the identification of components a nontrivial task. Moreover, adaptive
parameter regdation by the CNS makes it difficult to "set" these parameters with
any confidence.
There are two fundamentally different types of neuron models, havïng different
signal representations. Simulation of nonlinear membrane dynamics makes it possible
to simulate discrete action potentials, wbich is useful in models of neural nets. Such
spike generation models are reviewed in [36]. Another type of model is designed to
produce a continuous output proportional to firing rate. This kind of model is better
suited to the lumped representation of a population of neuroes. An example of such
In this model each input has a transmission time delay, e a , and a synaptic gain,
gi- The algebraic sum of all inputs is fed through low-pass dynamics and an optional
nonhear function to produce an instantaneous neuroa firing rate. The most common
nonlinear function is threshold and saturation, but more elaborate nonlinearities have
been used, such as the sigrnoidal function [47]:
where w sets the maximum h e a r dope, P - p is the maximum firing rate, E position
of nidpoint, and Q corresponds to the onset of saturation. va.rying these parame-
ters appropriately implements a d e t y of nonlinear behavior: saturation, magnitude
dependent gain and bilinear slopes. Figure 3.1 1 shows this sigmoid function in its
general form.
Neuron dynamics, associated with the dispersion effects, can be well modeled by
first order transfer function H(s ) = &$$ , nevertheless a second time constant is
sometimes introduced to account for some degree of adaptation in the neuron f'iring
rate [47].
Figure 3.11: Sigmoid function
3.5 System Identification of the Ankle
System identification studies, as mentioned earlier in this Chapter, can provide "black-
box:' models of system behavior to use as a reference against which the morphological
mode1 can be tested. Therefore, the results of system identification of the adde
obtained in our lab are relevant to this work and are described briefly below.
There are two broad classes of system identification methods: parametric and
nonparametric. Nonparametric identiiication describes the syst em by means of nu-
merical descriptions which provide little direct insight into the system order or struc-
ture. The most commonly used h e a r nonparametric models are fiequency response
models and impulse response models. The fkequency response to torque as input is
c d e d the compliance fiequency response and is formulated as follows :
where e ( j w ) is the Fourier transform of joint position, T( jw) is the Fourier transform
of joint torque, and C ( j w ) is the compliance fiequency response function. Frequency
response with position as input, the stiffness frequency response, is the dynamic
inverse of cornpliance and forxnulated as:
T ( j w ) = S ( j w ) B ( j w )
where S( jw) is the stifiess frequency response function.
Stiffness and compliance functions have counterparts in the time domain: so-cded
impulse response functions (IRE'). The compliance IRF, c(r) is defined by :
and is simply the inverse Fourier transform of compliance frequency response function.
-4nalogously, the stifhess IRF s(r)
Since all functions are related, they
is defined by :
(3.15)
do not, in principle, contain any new information
about the system, but due to noise and estimation problems they may display different
behavior [29].
Nonlinear nonparametric methods for systern identification include quasi-lùiear
models which yield time vairying compliance or stiffness functions. Wiener or Volterra
functional expansions can also be used to describe non-linear dynamic systems (see
[56] for review of nonlinear single- and multipleinput systems identification ).
In cont rast to nonparametric methods, parametric models describe systems as a
set of analytical expressions having a finite, usudy small number of parameters. The
key to success of such a model is having a correct set of a priori assumptions about
system structure. Such assumptions can be derived on the basis of nonparametric
descriptions.
For example, the nonparametric stifhess frequency response function, plot ted
in Figure 3.12, displays the characteristic features of a second-order system: gain
is constant at low frequency, has a resonant valley at intermediate kequencies, and
increases at 40 dB/decade at higher frequencies. The phase shift starts at -180 degree
for low frequency? decreases to -90 degree at the resonant frequency, and approaches
O at high frequencies. Consequently, joint dynamics are modeled by:
Figure 3.12: Stifbess kequency response of ankle. Redrawn fiom [29].
where T ( t ) is torque, 0(t) is angular position, I is the inertial parameter, B is the
viscous parameter, and K is the elastic parameter [29]. It can also be rewritten in a
different form, in the frequency domain:
where G is the static gain, w, is the natural frequency, and is the damping param-
eter.
Generally, this parametric mode1 provides a very good description of experimen-
ta1 data for a particular operating conditions [29]. Note that before proceeding to
parametric identification, an assumption about the order of the system was made on
the basis of the results of nonparametric identification.
The parameter values of 1, B, and K in equation 3-16 change dramaticdy with
the operating point, defmed by mean torque, mean position, perturbation amplitude,
etc. To reflect these findings equation 3.16 can be rewritten in quasi-linear form:
where X defines the operating state of the system. Changes in the joint dynamics
with the operating point found in our lab can be briefly summarized as foilows [29]:
Mean torque. The inertiai parameter I remains constant, while the elastic, K,
and viscous, B , parameters increase progressively with the level of contraction.
Mean position. K rernains constant over the mid-range of position, but increases
dramatically toward the extremes of the range of motion. B changes too, in
such a way as to keep damping parameter in equation 3.17 constant over the
range of motion.
Perturbation arnplztude. Stf iess is larger for small disptacement amplitudes
than for large.
a Co-contraction will distort the dependence of the joint dynamics on torque,
because the forces developed by agonist and antagonist will oppose each other
while their contribution to the stiffness will add.
Another hding, relevant to bilaterai modeling, concerns the asymmetric orga-
nization of flexors and extensors. System identification methods applied to human
triceps surae suggested that stretch reflexes in the extensor can be modeled as a
uni-directional rate-sensitive nonlinearity [27]. On the other hand, analogous meth-
ods applied to tibialis anterior [28] led to the conclusion that a direction-dependent
nonlinearity was not an important feature of the flexor stretch reflex.
Finally, the properties of stretch reflex are known to change with ongoing move-
ment. Extensive research has been donc to investigate the modulation of stretch
reflexes under passive and active conditions for both normal and injured subjects
[U, 31, 13, 341.
One approach is to assess the effect of superimposing random perturbations onto
pulse inputs (similar to tendon jerks) [53]. In these recent experiments, pulse inputs
were chosen as inputs because they d o w temporal separation of intrinsic and reflex
components of the mechanical response. Since d natural movements involve
ongoing perturbations of some kind, pulses in some experiments were superimposecl
on random perturbations of Merent amplitude and fiequency content. The magni-
tude of the response was iound to depend on the properties of the pulses, such as
amplitude, direction, and duration of the pulse in highly nonlinear manner. Thus,
both dorsiflexing and plantadexing pulses produced refiex in the same direction. An-
kle position and tonic activiw of the muscle also idiuenced the reflex responses. A
sigmificant time dependence of the dynamics of the ankle was found to be present
for around 1 sec. For example, an extension of the ankle dec ted the response to a
subsequent flexion for up to 1 s [53].
In the combination input experiment, it was found that while intrinsic torque was
not si,lgnificantly dected by random perturbations, even s m d random perturbation
greatly reduced reflex EMG and reflex torque. ïncreasing either the amplitude or
baadwidt h of perturbation depressed the reflex gain. Moreover, when reflex torque
and EMG were plotted as functions of velocity, the two effects behaved similarly,
suggesting that the velocity of a perturbation is the main factor in depressing reflex.
A second set of e-qeriments recorded torques and EMG during passive walking.
The characteristical pattern of walking was recorded and then applied to the passive
ankle. -4t ten points during the walking cycle pulses were applied to the ankle, and
elicited torques and EMG signals were recorded [34]. The modulation of the reflex
torque during this passive walking movement was quantitatively sirnilar to that re-
ported for active wdking. The amplitude of the reflex EMG increased progressively
from heel strike throughout stance, reaching a maximum near full dorsifiexion, there-
after decreasing rapidly and remairiing low till the next w a h g cicle. Recent studies
suggest that the reflex modulation during such passive movements are most likely
due to spinal mechanisms [7].
Similar to [53], time dependence of the stretch responses was observed by Gollhofer
and Rapp in their double stimulus experiments [181. Two identical dorsifiexing dis-
placements were applied to seated and standing subjects. The duration between
the two pulses varied between 100 and 400 ms. Stretch responses were related to
the stretch velocity but not amplitude. Moreover, following the second stimulus the
stretch responses of GS'were very s m d for a 100 ms interstimulus delay, medium
for 200 ms, and recovered after 400 ms. The recovery curve was similar for both
standing and sitting positions, however TA responses of seated subjects either t o t d y
disappeared or were very smaU.
In general three broad causes of reflex modulation have been proposed: fusimotor,
synaptic, and mechanical [51]. hisimotor modulation involves static and dynamic
fusimotor neurons modulating the sensitivity of primaxy and secondary spindles, as
discussed in section 2 -4. Synap tic modulation takes place when a-motoneurons receive
converging inputs from many interneurons and afEerents. Mechanical factors arise
when the same displacements are applied a t different mechanical states of the muscle,
For example, a background force will increase the intrinsic stifkess of the muscle due
to contraction, consequently the length change wiil be smaller and the immediate
force w2.l be larger compared to the same displacement of the relaxed muscle.
-4lI three mechanisms may in principle modulate the gain of the transmission of
the stretch impulse to the muscle. In other words, stretch reflex response to the same
input could be very different depending on the fusimotor, synaptic and mechanical
states of the agonist - antagonist pair. Synaptic mechanism of the modulation appears
to be the most cornmon, and includes such factors as covariation of the reflexes with
the net level of activity in the neural pool and presynaptic inhibition [51].
In summary, system identification methods provide a wealth of concise descriptions
against which a morphological mode1 can be tested.
3.6 Bilateral Effects
Most modeling efforts to date have assumed that the bilateral structure of the motor
system serves merely to compensate for the fact that muscle c m "pull" but not
"push." Therefore, unilateral lumped models have been used. However, given the
asymmetry in stretch reflexes in flexors and extensors and in the reflex connections
with agonists and antagonists, modeling the neuromuscular control system as an
asymmetric bilateral structure may provide additional insight.
Some efforts in this direction have been already made. For example, McRuer's
[40] neuromuscular actuation system model involved an agonist - antagonist muscle
pair, thus providing some bilateral structure. In particular, the overd force-velocity
relationship resulted from interactions between force - velocity relationships of agonist
and antagonist muscles. However, the rest of the model remained lumped, so that
bilat eral interactions at the spinal level were not considered.
Another group of investigators - De Luca and Mambrito [35]- measured the
myoelectric activiw of several motor units of an agonist - antagonist muscle pair
acting on a interphalangeal joint : the human flexor poilïcis longus and e-xtensor
pollicis longus muscles. Myoelectric activity of several motor units was collected wMe
subjects produced isometric forces to track various trajectorïes requiring both flexion
and extension contractions, i-e. d a g CO-contraction. Cross-correlation techniques
showed a "cornmon drive" - a high correlation between firing rates of motor units of
agonist and antagonist, with essentially no time shift. To e-uplain this, De Luca et al.
suggested a control scheme wïth three cornmand channels: fiex, e-utend and coactivate
as shown in Figure 3.13 . According to these authors, reciprocal inhibition between
agonist and antagonist muscles, together with these three central commands, could
provide control for force reversal and switching between proportional and reciprocal
coactivation of the muscles. Although stretch reflexes and Renshaw cell feedback
were presented in this diagram and it was mentioned that they had "the potential of
biasing the net excitatory or inhibitory drive to the motoneuron pool", De Luca did
not investigate possible effects of spinal segmental control in any detail.
Meanwhile, a bilateral model for another sensory motor system - the vestibulo-
ocular reflex (V0R)- has shown that interactions between central neural pathways
can account for a number of behavioral and neurophysiological findings. In particular
it was shown that modulation of commissural gains can cause a switch between a
compensatory position tracking mode and velocity tracking mode [15], causing fast
Figure 3.13: A suggested mode1 for agonist - antagonist control of the thumb inter- phalangeal joint. Redrawn from [35].
or slow movements of the eye. Moreover, the bilateral structure of the VOR could
account for the observation that the linear range of the system exceeded that of its
components [SOI. Many arrangements are possible but analytical studies have shonm
that the key requirements for the model, to predict observed behavior, are reciprocal
commissural connections and feedback £iom neural low-pass Uters on each side of the
system. These are remarkably analogous to the neuromuscular control system.
Figure 3-14 compares the neural organization of the \-OR reflex to the spinal
circuitry of peripheral motor control. Prepositus hypogrossi cells are analogous to
Renshaw cells in providing negative feed-back on each side of the system. Intercon-
nections between two sides across the midline at the level of vestibular nuclei are
similar to reciprocal inhibition of the agonist - antagonist pair in the spinal cord.
However, the stretch control system is more complicated than the VOR, since the
sensory inputs (spindles) are part of the feedback loop and there is another level to
the structure (Ia interneurons). Nevertheless, the similarity between the two systems
suggests that interactions between agonist and antagonist muscles of a joint at the
spinal level merit a detailed study.
Figure 3.14: A cornparison between neural organization of VOR and spinal circuits of motor control system. Only selected neurons are shown. White circles represent e x i t atory neurons, while shaded circles represent inhibitory neurons. Redratvn from (491. Interconnections across midline in VOR system are similar to interconnections between agonist and antagonist interneurons.
Chapter 4
Structure of the model
4.1 Input and Output of the Mode1
The first question to be considered is what should be the main input to the model?
Tn experiments assessing joint dynamics three approaches are possible:
No explicit input - torque and position are measured during normal motor
activity.
Torque is the input and position is the output-
Position is the input and torque is the output.
The relative merits of these approaches were discussed in [29], where it Rias shown
that in the ideal noise-free situation it should make no difference which experimental
paradigrn is used. However: with noise present estimates wïll depend on what is
considered to be the main input to the system. For example, estimates obtained with
position inputs wiU be better at high frequencies, while torque inputs dl produce
better estimates at low freqencies. Based on this analysis [29] the ankle actuator
system was developed [30] to apply displacements to the joint at fiequencies up to
100 Hz. Given the extensive body of research with position as the input, ankle
position was chosen to be the main input to model, so the results of simulation can
be compared directly to experimental data.
A simplified block diagram of the model is presented in Figure 4.1 and its com-
ponents are described below.
4.2 Muscle Spindles
Figure 3-5 shows that at high frequencies, models of spindles behave as dïfferentiators.
Because the simulations were done for ramp and pulse displacements, only high stretch
velocities were of interest, so spindle dynamics were modeled as differentiators. This
assumption was used successfdly in simulatiou studies of peripheral loop stabiüty [?].
T-4 and GS spindles are modeled as a Wiener series : linear dynamics followed by a
s tatic nonlinearity-
Spindles have different sensitivity to positive (dorsifkxion) and negative (plan-
tadesion) stretches [?]. This was approxbated hear ly by setting the sensitivity to
positive stretch k p = 3, to be three times larger then that to shortening (negative
stretch) kn, as shown in Figure 4.2.
Thus, the transfer functions, between angular position of the joint and the outputs
of the agonist and antagonist primary spindles were as follows:
Table 4.1: Spindle response with respect to the position.
Contributions from other peripheral receptors were not included in the model.
Force feedback gain is known to be small in the stretch reflexes [?], therefore Golgi
tendon organ receptors were not considered. The effects of the other sensory propri-
oreceptors such as joint and cutaneous receptors are not documented weIl enough to
be modeled s u c c e s s ~ y . Given the dominance of spindle activity over other receptors,
it seemed reasonable as a first step to investigate whether spindle feedback alone is
sufficient to explain the EMG and force responses to high velocities inputs.
Antagonist Spindle - k,s -kps
1 -4gonist Spindle Flex (positive position) ~ P S Ext end (negative position) 1 ks
GS EMC
-i1 torquc F-
m spindle spindle I T I
position
Figure 4.1: Asymmetric model: each neural comection on both sides has a different n-eight associated with it. Sf and Se are the flexor (TA) and extensor (GS) spindles firing rates in spikes/sec respectively. Al and A. are the T.4 and GS motoneuron pre-threshold firing rates in spikes/sec respectively. Spindks are assumed to be three times more sensitive to lengthening than to shortening. By convention dorsiflexion is considered to have positive velocity, therefore there is a negative sign in the transfer function for the T-4 spindles. H(s) is defined by the equation 4.7.
Figure 4.2: Sensitivity of TA spindle (dotted line) and GS spindle (solid h e ) nrith respect to ankle velocity.
Muscle
..At the ankle joint, force was subdivided into three components: a reflex component,
a "non reflex" muscle component, and a component that required to move the
mass of the foot.
The component required to move the m a s of the foot can be found through
Newton's second Law. It was assumed that the ngid limb rotates about a single Lxed
aisis, therefore, the limb dynamics are given by
where T(t] is the torque, required to move the foot through the trajectory 8( t ) , and
1 is the moment of inertia of the joint.
The contractile mechanics of the muscle were modeled as a Hill structure, Le.
a dash-pot in series with a spring. If the dash-pot constant is Khshpot and spring
constant is KSvin,, then the torque of each muscle is given by
Thus the overd "non-reflex" transfer function is a second order system:
Equating 4.3 with the intrinsic torque model obtained through system identifka-
t ion techniques [29], results in foIlowing values
Activation dynamics, Le. reflex torque, were approximated by a linear, second-
order, low-pas transfer function :
The parameters Km = .3719sp/mV/sec, 2, = 0.7 and Wm = 22 were estimated by
fitting a second order impulse response function to experimental impulse response
functions, obt ained by nonpararnetrïc system identification [3lj.
4.4 Spinal Neurons
The interneuronal connections presented in Figure 2.7 are too complex to be included
in a simple model, and so some simpiifications had to be made. Unfortunately. there
is little information about which elements within the central nervous system are
most responsible for controlling the gain of the stretch reflex. The rationale for the
connectivity assumed for the model is described below.
In section 4.2 primary spindle afferents were chosen as the dominant input a t
the spinal level. Therefore, the direct excitatory connections of spindle afferent to
a-motoneurons were included in the model. It is generally acknowledged that t h b
monosynaptic component is active in al1 kinds of movements. The synaptic weights
of the TA and GS spindle connections to their corresponding a-motoneurons are pf
and p, respectively (see Fiagure 4.1). As discussed in section 2.7, spindle feedback is
mediated polyspaptically by several populations of interneurons. The most impor-
tant is the Ia interneuronal pool, the main convergence neurons for primary spindles.
Therefore they were also included in the model with synaptic weights Zj for TA spin-
dle afferents and 1, for GS spindle afferents. Primaq spindle afferents also make
connections with Ib Intemeurons; these were not included in the model, since the
main input to Ib interneurons are Golgi tendon organs.
Ia interneurons are known to be under strong inhibitory control from Renshaw cells
under normal conditions ['XI. Moreover, Renshaw ce& also provide low-pass feedback
for agonist and antagonist reflex pathways [10], a feature shown to be essential for
gain regulation in vestibule-ocdar control. Therefore Renshaw cells were included in
the model. The TA a-motoneurons excite the T-4 Renshaw cell with the synaptic
weight a f , Renshaw cells inhibit the TA a-motoneuron with the synaptic gain kf,
and homologous Ia interneurons with synaptic gain rf-
Finally, Ia iatemeuron of the flexor was assumed to inhibit GS Ia interneuron and
GS a-motoneuron with the synaptic gain of gf and if respectively. Analogously, Ia
interneuron of the extensor inhibit TA Ia interneuron and TA a-motoneuron with the
spap t ic gain of g, and i, respectit-ely. Gains gf, g,, i,, and if represent reciprocal
inhibition between agonist and antagonist muscles, and as discussed in Section ??
may be important for peripheral control.
As for dynamic component of neurons and synapses, the tirne for an impulse to
synapse on the next neuron is of the order of 1 ms. Synaptic delays and neuron
dynamics therefore have negligible delay postural control, and were ignored in the
model. In contrast conduction delays associated with propagation through the spindle
afferents dl depend on the length and conduction velocity of the fibers, and c m be
rather large for muscle of the lower extremity It was assumed in this model that the
delay of reflex torque with respect to intrinsic torque is mainly due to the
conduction delay of primary alferents. Therefore, the delay was set to 0.05 sec to
correspond to experimentai data of [53].
In srimmary, the spinal level of the model structure was limited to Ia interneu-
rons and Renshaw cell pools, and their connections. The Ia internewons and a-
motoneurons were modeled as weighted summers of their synaptic inputs. The Ren-
shaw cells of agonist and antagonist muscles were modeled as low-pas iilters of fbst
order with a time constant around 0.01 sec, correspondhg to the 10 Hz cut off fre-
quency found in the literature [IO]. Dynamics of Ia intemeuons and motoneurons
mere assumed to be negligible-
4.5 Analysis of the Overall Mode1
Signal-flow graphs and Mason's nile were used to derive the overall model dynamics
as described in Appendix A. Because of their sirnilariw to neural whîng, signal-flow
graphs provide a clear diagram of neural control in a complex physiological system.
They have been successfully used in the analysis of the vestibulo-oculomotor system
- A flow diagram of the skeleto-muscular system in presented in Figure 4.3. halysis
was done using Laplace transforrns whîch wÏll appear in bold letters to distinguish
them from time functions.
Figure 4.3: A signal-flow graph of the system. Junction points represent the variables, branches represent gains and transfer functions. Signal can transmit only in the direction described by the m o w of branch.
4.5.1 D y n d c s of the Mode1
Mason's d e yields transfer functions-with two poles and two zeros for each side:
where
Af = firing rate of TA a-motoneuron (sp/sec),
A, = firhg rate of GS a-motonewon (sp/sec),
Sf = firing rate of TA primary spindle (sp/sec),
Se = firing rate of GS primary spindle (sp/sec), and
( a$ peaere) z 3 = - -+--- Tete
Since the Inverse Laplace transform of
equations 4.8 and 4.9 can be nrritten in the time domain as ' :
'Note t bat * signifies convolution
Introducing new notations for gains2:
*In these notations fast and slow mode gains are denoted as G and H respectively. For euample, G"/tands for fast mode gain between extensor spindle and flexor a-motoneuron
the equations 4-51 and 4.11 can be rewritten as follows 3:
The TA spindle output, SI, and GS s p i d e output firing rate, S., in response to a
step displacement wiU be impulse like signals. Therefore, the time domain response
of the TA cr-motoneuron, AI, to a position step will be the sum of two exponentials
- one fast, e ( p - q l t , and one slow, e@+qIt. The gain of the slow exponential will be a
sum of two independent gains H l and H{ , and the gain of fast exponential be
alço a çum of two independent gains Gf and G:. Similady, the time domain response
of the GS a-motoneuron, A,, to a position step will be the sum of two exponentials,
reflecting two modes, which can have very dXerent dynamics, depending on loop
gains. Asymmetric loop gains wiil allow the time constants of the two modes and the
gains of TA and GS responses to be set to any value. Thus, a ~ ~ e t r i c loop gains
will result in different responses for agonists and antagonists.
3Note that * signifies convolution and multiplication
As c m be inferred fkom equations 4.20 and 4.21, the responses of TA and GS would
not eauhibit directional sensitivity, if the spindle responses were Linear. However,
a simple nonlinearity at the spindle Ievel can account for aspnmetric directional
sensitivity of the two modes, and consequently for the fact that both positive and
negative displacements produce reflex torque in the same directions.
For exampie, for a positive (dorsifiexîng) step displacement (kom Table 4.1) Sf = d P - knx and Se = k p s , where P is position of the ankle. Therefore the gain of the
slow mode for At is HI - kp - HF - kn. On other hand, for a negative (plantarfleuing)
step displacement Sf = - k p s and S. = k n z , therefore the gain of the slow mode
for .4f is H/ - kn - HF - kp. The same applies to the fast mode of Al and both modes
of A,. Obviously, selecting different values for kn and kp will d o w the gains for
dorsiflexing and plantar£iexing displacements to be set independently, and so provide
for directional sensitivity.
4.5.2 Resting Rates of the Neurons
All neurons in the model were modeled as being modulated about some resting rate-
Thus, there are eight resting rates in the model:
RAf - the resting rate of the TA a-neuron,
RA, - the resting rate of the GS a-neuron,
Rs, - the resting rate of the T.4 spindle,
Rs, - the resting rate of the GS spindle,
RRf - the resting rate of the TA Renshaw cell,
RR, - the resting rate of the GS Renshaw cell,
RI, - the resting rate of the TA interneuron,
RI, - the resting rate of the GS interneuron.
Because the actuaI resting rates of the neurons di depend on the parameters
of the model, we have to introduce a background input to a.llow the resting rates to
be set properly. These background resting rates correspond p hysiologically to tonic
descending input from the higher centers of the CNS.
Let R;, , E;, , RA,, and RRf denote constant background resting inputs to spin-
de, Intenieuon, a-Motoneuron and Renshaw cell of TA respectively, and Rke, Rie,
RAe, and RJ& denote constant background resting inputs to spindle, interneuron,
a-rnotoneuron and Renshaw cell of GS respectively. The actual resting rates will
equal:
RL,
Rk
RI,
RI.
Rh f
Rke
RA.
The formulas in these transposed columns were found using Mason's rule algebra
as follows. If zero position input is applied (i-e. the ankle is at rest) the actual
resting rate of, for example, extensor a-motoneuron, A,, will reach its steady state
in a certain time depending on the systern poles. This steady state can be found
by calculating the transfer function between background resting input to the flexor
spindle Rk, and extensor a-motonewon output A. using Mason d e . Taking the limit
of this transfer function at s -, oo wiU yield the scale factor between flexor spindle
Rkf and extensor a-motoneuron output RA. i-e. the k t member in equation 4.29.
The other scde factors were obtained analogously. To set actual resting rates to
desired values pnor to simulations, equations 4.22-4.29 were solved simultaneously
for each set of parameters using the Matlab pseudoinverse function.
4.6 Choice of Parameters
Figure 4.4 shows the detailed Simulink mode1 used in the simulations. Parameter
values used in the simulations are summarized in Table 4.2 and are discussed below.
Figure 4.4: Simulink implementation of the model. Shaded boxes represent back- ground resting rates.
Parameter
Rlr
-- -
1 0.7 1 damping parameter 1
Value 3 1 40 sp/sec 40 sp/sec
1 - 7 -
1 22 ] natural frequency 1
P hysiological meaning sensitivity of spinde to positive stretch sensitivity of spinde to negative stretch resting rate of TA spindle ( R b in 4.22 - 4.29) resting rate of GS spinde (RkC in 4.22 - 4.29)
-3'719 sp/mV/sec 1 static gain
- 1 , - I Tp I 0.01 sec
L -
- 1 TF 1
1 0.008 sec
~nertial~arameter gain between aMn firing rate and EMG time constant of GS RC transfer function t h e constant of T-4 RC transfer function gain of &Mn projection to RC for TA
Viscous paramet er Elast ic paramet er
Kdashpot Kspring
-
1 0.005 1 gain of @Mn projection to RC for GS
.I Nrn/rad/s 1 Nm/rad
- - .
1 0.0013 1 gain of RC projection to aMn for T-4 1 - - -
1 10 1 gain of RC projection to aMn for GS I p gf Se
, if ie
rf Te
t hreshold R21 R3r R31 R3r Ra1 R5r
1 0.0259 gain of spindle projection to aMn for TA 1 1.1579e4 gain of spindle projection to aMn for GS 0.5678 gain of spindle projection to LW for TA -12-5433 gain of spindle projection to n\J for GS 0.007 gain of TA IN projection to GS IN 55.9862 gain of GS IN projection to TA IN 10 gain of TA IN projection to GS aMn -0.0013 gain of GS IN projection to TA ctMn le3 gain of TA RC projection to TA IN -0.2 gain of GS RC projection to GS N 50 sp/sec the threshold value for TA and GS aMns 1.0389e5 sp/sec background firing rate of IN TA (Ri, in 4.22-4.29) 530.7145 sp/sec background firing rate of IN GS (Rip in 4.22-4.29) 100.5126 sp/sec background £king rate of TA RC (Rkr in 4.22-4.29) 100.7449 sp/sec background E n g rate of GS RC (Rh in 4.22-4.29) 59.6269 sp/sec background firing rate of TA crMn (Rk, in 4.22-4.29) -4.6160e05 sp/sec background firing rate of GS orMn (Rac in 4.22-1.29)
Table 4.2: Values of parameters used in simulation of biiaterai asymmetrical model for all inputs. Note that analogous parameters for flexor (TA) and extensor (GS) have different values. As discussed in the section 4.5.2 these background firing rates d o w actual resting rates in the model to be set to 100 sp/sec for Renshaw ce&, 40 sp/sec for spindies, 50 and 60 sp/sec for GS and TA a-motoneurom respectively.
As seen from equations 4.8 and 4.9 twelve parameters completely descnbe the
linear behavior of the system. These are four gains:
six zeros:
and two poles:
P l = P + q
P 2 = P - Q
where
These twelve parameters are nonlinear functions of 14 synaptic gains and two time
constants Te and Tf . In order to set these parameters to physiologicdy appropriate
values (see below), equations 4.30 - 4.41 must be satisfied simultaneously. However,
equations 4.34 and 4.37 were excluded from the set because they depend only on time
constants Tf and Te- Moreover, algebraic analysis showed that only 8 equations are
independent. These are equations 4.30, 4.31, 4.32, 4.33, 4.35, 4.36, 4.37, and 4.38.
SolWig this set of equations for p ,poZ ,Ze,gf ,geYT yre yields:
Note that the parameters a and k for both TA and GS occur in these results
as products only (afkf and aeke), which implies that the dynamics of the system
depend on the recurrent inhibition loop gains in general, but not on particular values
for a and k. For clarity, recurrent loop gains for TA and GS were denoted F and E
respectively.
Next, the gains and zeros were expressed as functions of H;, H I , HL, HE, G{ , G j,
GL, and GE by solving the set of equations 4.12-4.19:
The values for H;, ~ f f , H i , Hz, G{, G;, G!, and G: were chosen on the basis of
the observed behavior of the ankle. Using equations 4.20 and 4-21, gains were set so
that the TA response was dominatecl by the fast mode, while GS responded ni th both
modes. The rationale for this is that the "silent" penod (a brief penod of decreased
eucitability) observed in GS responses, is absent £rom TA responses [Bi. Moreover,
the gain of the fast mode for GS was made 10 times that of the TA fast mode. The
rationale for this assumption comes fiom the experimentally observed ciifference in
the magnitude of the reflex responses of the TA and GS [BI. TA impulse response
functions were much s m d e r than those of TS when expressed as a level of modulation
activity.
To reproduce obsemed behavior, the weights of neural connections were selected
to produce widely separated time constants in the firing rate of &Mn. The faster
cornponent is the phasic response of the ctMn pool to stretch and affects the dynamics
of the reflex torque- It was set to -2 sec to account for the e.uperimenta1 observation
that the spectrum of the transfer function between velocity and EMG is flat up to 50
Hz (pole pl=-5.05 rad/s).
On other hand, the slower component in GS activity was assumed to correspond
to the tonic change in the balckground finng rate of the &Mn pool. Because it is set
to be always below the threshold, it does not affect the dynamics of its EMG, but can
modulate the gain of the stretch reflex. While the time constant for the fast mode
can be chosen directly from experimental data, the time constant for the slow mode
was deduced fiom the experimentally observed time dependence of the reflex gain-
Because modulation of the reflex gain lasts for approximately 1 sec [53, 181, the time
constant of the slow mode was set to 8 ms (pole p2=-114 rad/s).
Substituting the values for p l , ~ 2 , H j , f?!? HI, Hz, G;? Gj, G:, and G: into the
set of equations 4.49-4.56 yielded values for gains and zeros, which in their turn were
substituted into equations 4.42 - 4.48.
Finally t h e constants for Renshaw c e k were estimated koom [IO], and were al-
lowed to m e r sfightly for TA and GS muscles:
Tf = Te + 6, where 1 6 1< 0.002
The final set of equations is as follow:
There are some additional conditions that had to be satkfied. First, the reciprocal
inhibition gains g, and gr m u t be positive. The recurrent loops gains e and f also
must be positive, as well as gains G1 and G3. From equation 4.65, it is clear that g,
can be positive only if il and i, are of opposite çign. Since = 0.3633$, either lf
or 1, &O has to be negative. The set of pammeters presented in Table 4.2 satisfies
a.ll these conditions and equations 4.58 - 4-65.
Positive signs for excitatory and negative signs for inhibitory connections were
accounted for in the signal-flow graph and analysis. However, as discussed above,
solving the equation fields negative values for i,, Te and le, which in physiological
terms wodd translate into an inhibitory projection fiom GS spindle to GS IN, an
excitatory projection fiom GS RC to GS LN, and an excitatory projection from GS
IN to TA aMn. Section 6.6 will discuss the physiological basis of these assumptions.
As discussed in section 3.5 system identification methods suggested that the
stretch reflex in the extensor could be modeled as a uni-directional, rate-sensitive
nonlinearity. It riras assumed that decreased tonic drive of the higher centers of the
CNS to extensor a-motoneurons causes GS EMG response to shortening rapidly sat-
urate. Therefore, to mode1 the uni-directional sensitivity of GS to positive stretch,
steady state firing rate of the GS aimotoneuronal pool was set to 50 sp/sec which is
lower than that of TA (60 sp/sec). Because steady state for TA was higher than that
of GS, both shortening and lenthening of the TA caused visible (not saturated) mod-
ulation of the T-4 a-motoneuronal pool rate. In contrast, because GS a-motoneurons
were modulated about a lower steady state firing rate, their negative modulations
rapidly saturated, and the EMG response of the GS was absent during shortening.
For simplicity, the other rates were kept symmetric for both agonist and antagonist:
namely 40 spisec for spindles [48], 100 sp/sec for Renshaw cells [47], and 50 spisec
for Interneurons.
Chapter 5
Result s
Simulations were performed on a DEC AXP workstationl using the OSF/1 operating
system. S I M U L I N K ~ ~ - an extension to MATLABTM, was used for simulating dy-
namic system performance. The Runge-Kutta integration method was used because
it provides the best performance for highly nonlinear systems. The smallest time con-
stant in the rnodel mis 0.008 sec; so the step time was set to 0.002 sec- Therefore: the
step time was one quarter of the srnalest time constant, to ensure that all transients
mere tracked effectively.
5.1 Step Displacements
Responses to step displacements were simulated h t . The output hring rates of TA
and GS spindles were simply scaled derivatives of a step as shonm in Figures 5.1
and 5.2 for dorsifiexing and plantarfiexîng displacements respectively. Dorsiflexing
the ankIe stretches GS and shortens TA muscle. Note the directional asymmetry
in spindle responses. As shom in Figure 5.1B, the GS spindle response (solid line)
to stretching was three timeç iarger in amplitude than the response of TA spindle
(dash-dot line) to shortening of the TA muscle.
Since the sphdle outputs are impulse-like signals, the time domain equations de-
scribed in section 4.5 should hold for the pre-threshold f i g rates of a-motoneurom - .
' Digital Equipment Corporation, Maynard, MA
of GS and TA. The tenn pre-threshold firing rate refers to the sum of the presynap . - .
tic inputs from aU neurons converging onto a-motoneurons. When the sum of the
presynaptic inputs is high enough for the membrane potential to reach threshold,
the act ual spike is generated. Therefore, post-threshold f i g rate corresponds t O
the actud spike rate of the a-motoneurons. Indeed, as expected from the dynamic
analysis, pre-threshold Fing rate of GS a-motoneurons consists of two exponentials:
fast positive and slow negative. The slow negative mode is below threshold, so the
GS a-motoneuron post-threshold firing rate, shown as a solid line in Figure 5-1C: is
a pulse-like signal-
GS EMG is proportional to the post-threshold GS a-motoneuron firing rate so the
slow mode has no effect on it either. The TA cr-motoneuron firing rate does not have a
slow component, and therefore both the pre- and post-threshold cr-motoiieuron firing
rates are the same as can be seen from Figure 5.1D. The simulated EMG response of
the GS muscle is in good agreement with experimental observations [26].
time (sec) time (sec) time (sec)
Figure 5.1: Simulated response of the model to a step flexion of the ankle. A: positiont B: GS (solid line) and TA (dash-dot line) spindles firing rates, C: pre- (dash-dot line) and post-threshold (solid lïne) GS a-Mns firing rates, D: TA a-Mns firing rates, E: GS EMG, and F: TA EMG.
Figure 5.2 shows the simdated response to a plaatadexing step which shortens
GS and stretches TA. The spindle outputs are again impulse-like signals, but Nith
the reverse polariw: the response of the TA spindle to stretching of TA muscle is
three times larger than the response of GS spindle to shortening (Figure 5.2B). The
response to plantadexhg step displacement also reflets the directional asymmetry.
In response to a change in the direction of the displacement the fast component of
the pre-threshold GS a-motoneuron firing rate changes its sign, but the slow com-
ponent r e m a b negative. This is due to the nonlinearity of the spindes as shown
analyticdy in section 4.5. The slow cornponent is always negative, independent of
the displacement's direction. It WU be shown later in this Chapter that this slow
tonic component can change reflex gain, after even brief movements of the ankle, by
changing the operating point. Both fast and slow components are below threshold
level for the negative ramp, so there is no GS EMG, which is in full agreement Mth
experimental hdings [8].
While GS responses correspond to that observed experimentally, there is a dis-
crepancy between the simulated and observed experimentdy TA EMG [26], which
frequently consists of two distinct bursts of activity- The fist is thought to be a
monosynaptic stretch reflex (MSR) component, while the second burst starts at about
75 ms and lasts for approximately 40 ms. The present mode1 predicts oniy a single
burst of activity corresponding to the MSR.
Figure 5.3 shows the resdts of applying the same stretch as in Figure 5.1 during
tonic activity of the GS muscle. Tonic activity was modeled as constant input to the
a-motoneuronal pool. As can be seen from the right panel, GS EMG drops below
baseline for a short period of time after the MRS burst. This corresponds to the
"silent" period, documented in 1331.
time (sec) n'me (sec) time (sec)
Figure 5.2: Simulated response of the mode1 to a step extension displacement to the ânkle. A: position, B: GS (solid h e ) and TA (dash-dot line) spindes firing rates, C: pre- (dash-dot fine) and post-threshold (solid line) GS a-Mns firing rates, D: TA a-Mns firing rates, E: GS EMG, and F: TA EMG.
time (sec) time (sec)
Figure 5.3: Sirnulated response to a step flexion of the d e during tonic contraction of the GS: pre- (dash-dot line) and post- (solid line) threshold (lep) GS a-Mns firing rates , and GS EMG (right). Note bnef (80 ms) period of decreased activity.
5.2 Flexion and Extension Pulses
Figure 5.4 shows the results of applying a brief pulse displacement to the model. The
Figure 5.4: Simulated response of the model to a bnef flexion (solid line) and extension ( dash-dot line) to the anlde. A: position, B: gastrocnemius-soleus electromyogram (GS EMG), C: reflex torque (with dc level removed), D: tibialis anterior electromyo- gram (TA EMG).
eariy component, occurring before the reflex delay of 50 ms is symmetrical for the
dorsiflexing and plantarfiexbg pulses, and is due to intrinsic mechanisms. The later
components reflect torques arising £rom the stretch reflex.
Bot h dorsifiexing and plant arfkxing displacements produced reflex torques t hat
plantarfiex the ankle, although the amplitude of the response to a plantadexhg pulse
is srnaller and later than that in response to a flexing pulse. This is in agreement
Nith the experimental data [53] reproduced in Figure 5.5. Note that in the model
simulations, all variables retuni to t,heir desired resting rates as chosen in section 4.6.
Figure 5.5: Ensemble averages of position (top), torque (middle), and gastrocnemius- soleus electromyogram (GS EMG; bottom) in response to bnef flexions (solid line) and extensions (dotted line) of the ankle joint. Records were zeroed to the mean values before the stimulus. Reprinted from [53].
Figure 5.6 show pre- and post-threshold fixing rates of GS a-Mn for the same
simulation. The GS a-& response to 'the first negative-going edge of the pulse is
below the threshold, and, so, does not evoke a reflex response. However, it does move
the operating point of the system further from threshold, and depresses the response
to the positive-going edge of the dorsiflexing displacement.
Figure 5.6: Simulated response to plantadexhg pulse displacements: pre- (dash-dot line) and post- (solid line) threshold GS a-Mns £king rate.
5.3 Pulses of DWerent Widths
Responses to pulses of different widths were simulated to detennine if the dynamics of
the model's two modes could also explain the dependence of reflex gain on the width
of pulses as described in [53]. The slow component of GS activity, corresponding
to a tonic change in background firing rate of the a-Mn pool, is negative for both
dorsiflexhg and plantadexhg movements. Therefore, the negative-going edge of
the plantadexing pulse decreases the tonic mode and, so decreases amplitude of the
response to a positive displacement. The Iaïger the amplitude of the slow mode, the
more the reflex response to the positive-going edge is attenuated. Figure 5.7 shows
the results of this simulation. The reflex and EMG responses to a longer pulse is
Zarger than that of the shorter puise.
A
Figure 5.7: Simulated response to 25 ms (solid line) and 75 ms (dash-dot line) plan- tarfiexing pulses. A: position, B: gastrocnemius-soleus electromyogram (GS EMG), C: reflex torque, D: tibialis anterior electromyogram (TA EMG) .
The refiex attenuation dmeased as the width of the pulse increased [53]. This
attenuation was predicted qualitatively by the model. Figure 5.8 shows the pre-
threshcld GS &-Mn k g rates for pulse widths of 25 ms and 75 ms. The positive-
Figure 5.8: Sïmulated response to applying plantarfking pulse displacement 25 ms and 75 ms long: GS a-Mn firing rate for shorter pulse (dash-dot h e ) and for longer pulse (dotted line). Solid lines depict pre-threshold firing rates.
going (reflex-producing) edge of the 75 ms pulse starts at an operating point which is
closer to threshold then does that for the 25 ms pulse, and, thus, produces a bigger
reflex
Figure 5.9 sumarizes the results of a series of simulations with pulses of Werent
widths. Torque and GS EMG amplitudes increase monotonically with the pulse width
up to 1 second, but are still less then the corresponding values for positive (flexing)
displacement pulses, which is in full agreement with the experimental curves [53].
O O. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Pulse duration (sec)
Figure 5.9: EEect of the pulse width on the amplitude of the reflex torque and GS EMG. Horizontal lines at the right of the figure correspond to positive pulse displacements.
5.4 Two Pulse Simulations
As discussed in section 3.5, another experimental approach used to investigate reflex
dependence on past events [18] has been to apply two-pulse displacements separated
by a delay. Simulations were perforrned to duplicate this experiment al paradigm.
EMG responses to a series of two-pulse dorsiflexïng displacements with vaxïable delay
are shown in Figure 5.10. As with the simulations of section 5.3, the reflex EMG was
TlME (sec) delay (ms)
Figure 5.10: Simulated EMG response to applyïng two dorsiflexing pulses with the different delays between them.
depressed for approximately one second after the first pulse, with depression of the
reflex decreasing as the time intenal between the pulses increased. In the simulations
the effect is due to the slow tonic mode, as can be seen £rom Figure 5.11.
The tonic mode dies out after 1 second, at this t h e the system has returned to
its "normal'' operating point, and reflexes are fully recovered. In other words if fmo
pulses are applied far enough apart for the tonic mode to have decayed, the gain of
the reflex will be the same. However, if the second stimulus is applied before the
common mode has decayed, the system will be at a dinerent operating point, and the
Figure 5.11: Sixnuiated response of the model to two dorsiflexing pulse displacements of the ankle. The dotted lines depict the t h e course of the input. Note that EMG responses are delayed with respect to the position input due to the conduction delay in spindle aiferents. Both pre- (dash-dot line) and post-threshold (solid h e ) GS a- motoneurons firing rates are plotted. A: delay between pulses is 100 ms. B: delay between pulses is 300 ms. Note that the time scale for A and B are the same for easy comparison.
Figure 5.12: Simulated response of the model to two plantarflexing pulse displace- ments of the ankle. The dotted lines depict the time course of the input. Note that EMG responses are delayed with respect to the position input due to the conduction delay in spindle afEerents. Both pre- (dash-dot line) and post-threshold (solid line) GS a-motoneuron firing rates are plotted. A: delay between pulses is 100 m. B: delay between pulses is 300 ms. Note that the time scale for A and B are the same for easy comparison.
apparent gain of the response to the second pulse will be lower. Theses simulation
results compare well with the results of the two pulse experiments reproduced in
Figure 5.13.
Figure 5.13: The recovery of the second stretch response as a function of the time delay between h t and second dorsifiexion (soleus muscle upper part; gastrocnemius medialis muscle lower part). Reprinted from [Ml.
Figure 5.12 shows similar simulations for negative pulses. Compare Figure 5.12B
and 5.11B. In both simulations the delay between pulses and the amplitude of the
pulses were the same. However, the EMG response to the second plantdexïng pulse
was somewhat smder than the EMG response to the second dorsifking pulse. This -4. .-
is because in plantadexhg simulations,.two effets superimpose to depress the reflex.
First there was a delay dependent depf&ion, analogous to dorsiflexing experiments.
Second as in Section 5.2 the negative-going edge of the first plantarflexïng pulse did
not evoke a reflex, but lowered the operating point-
T h s e two independent factors caused the dope for reflex depression to be Merent
for domifiexhg and plantarflexing pulses. The amplitudes of the reflex torque and
GS EMG as hrnctions of the time interval between pulses are plotted in Figure 5.14.
Reflex torque and EMG decreased progressively as the time i n t e d betnreen the
two pulses decreased. As expected the dopes for the dorsiflexing and plantarfiexing
displacement are different. Once more, the amplitude of the EMG and reflex torque
are almost recovered after 1 second.
Delay between pulses (sec)
Figure 5.14: Effect of the delay between two pulses on the magnitude of the EMG and reflex response to the second pulse. Responses to pulses that flexed (O) and extended (+) are shown. Horizontal lines at the right of the figure correspond to values of the first pulse.
5.5 Effect of Random Perturbations on Pulse Re- ...
sponses.
Simulations were performed to duplicate the experimental paradigm of [53], in which
position pulses were superimposed on random perturbations having a truncated Gaus-
sian amplitude distribution and a flat power spectnim. Because we are dealhg with
a nonlinear system, both the amplitude and fiequency characteristics of the input
can affect the behavior of the system- Therefore, in two different sets of simulations
the amplitude and bandwidth of the perturbation were varied independently. Figure
5.15 shows the amplitude distribution and power spectrums for the two sets.
Perturbations used in the first set of simulations have the same amplitude distri- ' . - -
bution as shown in Figure 5.15A, but Werent fiequency content presented in Figure
5.15C. Signals were designed to have different zero-crossing rates, which reflects the
spectral content of a signal. The second set of signals have the same frequency con-
tent presented Figure 5.15D, but àifferent amplitude distributions as shom in Figure
5.15B. These random perturbations were formed following the experimental design
of [53], and were intended to examine the effects of both amplitude and frequency
content in independent trials.
Figure 5.16 shows the responses to constant pulse displacements with super-
imposed random perturbations of different amplitudes. Random perturbations are
knom to modulate reflex responses, but to have no effect on torques, therefore the
intrinsic torque was subtracted from the total torque, Le./ only reflex torque was ..:..
plotted for combination trial simulations. Both the amplitude of GS EMG and reflex
torque decreased as the amplitude of the perturbations increased.
-5.02 -0.01 O 0.01 0.02 amplitude (rad)
frequency (Hz)
u -- - - -
- - -- - -
-0.01 -0.005 O 0.005 0.01 amplitude (rad)
frequency (Hz)
Figure 5.15: Properties of the superimposed random perturbations. Perturbation used in k t set have the same amplitude distribution (lefi top), but difFerent band- width (left bottom): 3 zero-crossings per second (solid line), 8 zero-crossings per second (dash-dot line), and 11 zero-crossings per second (dotted line). Here zero crossing rate is used as a measure of frequency content. Perturbation used in the second set have different amplitude distribution (right top), but the same frequency content (right bottom). Amplitude of perturbations are: 0.003 rad (solid line), 0.005 (dash-dot line), and 0.006 rad (dotted line).
Figure 5.16: Set 1: reflex torque and GS EMG for position perturbations of different peak-to-peak amplitudes. Values below traces are amplitudes in rad.
Reflex torque &O changes with the bandwidth of the random perturbations as
shown in Figure 5.17. Here. the superimposeci perturbations have the same amplitude
distribution but different zero-crossing rate. Increasing the bandwidth of the pertur-
bations decreases the model's reflex response, as observed experimentally. Findy,
Figure 5.17: Set 2: reflex torque and GS EMG for position perturbations of different bandwidt h. Values below traces are zero-crossing rates.
in Figure 5.18, reflex torque and GS EMG for both sets are replotted as a function
of the mean absolute velocity. The effect of the mean absolute velocity in both sets
nearly superirnposes, which agrees well wit h experiment al results [53] -
Figure 5.18: Similar effect of displacement velocity on reflex torque and GS EMG in two sets-
0.5
g0.4 Y
g 0.3 UJ
g 0.2
, 1 1 1 I 1 I I ï
- + -
- - - - - -
0.1 - -
0 - I 1 1 t I I I 1
0.4 0.6 0.8 1 1.2 1.4 1 -6 1.8 2 MEAN ABSOLUTE VELOCITY(rad/s) x 104
Chapter 6
Discussion and Conclusions
An a s ~ ~ e t r i c bilaterai model for stretch reflexes at the adde joint built from known
physiology reproduces a wide range of observed behavior, in particular the modulation
of the stretch reflex gain with veiocity of passive position perturbations. The effect
relies on asymrnetry in the antagonist-agonist pathways and on nonlinear spindle and
a-motoneuron responses. The functional implications of these essential structural
elements will be summarized in tum.
6.0.1 Summary of Results
1. A bilateral model for stretch reflexes at the adde that incorporates asymmetric
reflex connections and nonlinear spindle responses was developed.
2. Sransfer functions between position and pre-threshold firing rates of the GS and
T-4 a-motoneurons were calculated as functions of the synaptic weights and other
parameters of the model. The system is described by two pales, six zeros and four
gains.
3. This set of nonlinear equations and inequalities was solved andytically7 so the post-
threshold firing rates of the GS and TA a-motomeurons can be set to physiologicdy
valid values. A constant set of the parameters was chosen for a l l simulations.
4. Simulations with pulses exhibit the directional asymmetry of the agonist antagonist
responses. Moreover, the reflex gain was found to depend on the pulse width and
tirnecourse of the input.
5. Simulations with pulse displacements and random perturbations are in qualitative
agreement with experimental data and suggest a mechanism which d o w the mean
absolute velociw of the perturbation to depress the stretch reflex at the spinal level.
The efFect relies on asyrnmetry in the antagonist-agonist pathways and on noniinear
spindle responses. The model suggests that the depression of the stretch reflex a t
higher velocities is caused by the slow mode, which takes longer to corne back to
baseline.
6.1 The Role of Sensory Input
These results suggest that spindle sensory input may plays an important role in co-
ordination and control of movement. Based on the experiments with the decerebrate
cats before and after cutting the dorsal roots, Nichols and Houk suggested that asym-
metry in firing of the primary aEerent of the muscle spindle makes response to stretch
and release more symmetric and consistent [16]. But the present analytical and stim-
ulation st udies suggest t hat the functional significance of spinde activity extends
beyond merely compensating for asymmetry of muscle response. The present model
readily explains the fact that both positive and negative dispIacements produce reflex
torque in the same direction [53]. Such "rectification" of the stretch reflex can result
from spindle nonlinearities. Indeed, it was s h o w analytically and through simula-
tions that such a simple nonlinearity at the spindle level as asymmetric directional
sensitivity can account for 'iectification" of the stretch reflex.
Velocity sensitivity and nonlinearity of spindle responses can dso be a mechanisrn
to adapt the reflex gain during different movement patterns. This is consistent with
recent animal experiments role of reflex in the coordination and control of movement
[42]. These show that afferent inputs are responsible for modification of transmission
in reflex pathway, regdating the timing of locomotor activiw and duration of stance
phase.
6.3 The Role of the Asymmetry in Agonist-Antagonist
Organizat ion
The present model relays on aqmmetric connections between the agonist and an- . -
t agonist . It has been established experimentally that stret ch reflexes are organized
differently in agonist and antagonist. It was suggested that a relatively s m d TA
response is not important functiondy. The GS muscle produces larger torques in
response to perturbation. Physiologicdly this is very usefül because the center of
gravity usually lies forward of the ankle joint, and flexing perturbations are more
likely to cause imbalance. Therefore GS acts against gravity to maintain balance-
Following the suggestion of (501, the present model incorporates the reciprocal in-
hibit ion between agonist and antagonist intemeurons, and pro jections from interneu-
rons to antagonistic a-motoneurons. As expected from a t heoretical analysis [15]
these agonist-antagonist connections significantly affect response dynamics, allow-
ing for two independent poles in the transfer functions between spindle output and
a-motoneuron k g rates. However, a symmetric model of the peripheral neuromus-
cular control could not account for the gros asymmetry of the TA and GS responses
documented experimentally in [28] and [27]. Therefore the loop gains and tirne con-
stants on each side were allowed to vary independently. This asymmetry of the reflex
organization of TA and GS d o m the model to predict difFerent shapes for agonist
and ant agonist responses.
Therefore, in addition to rnaintaining proper posture, the asymmetric organization
may be a mechanism for modulation of the reflex gain. According to Our theoretical
analysis, if the system was symmetric the model could not predict reflex modulation
Nith the velociiy of the perturbation, nor account for asymmetric responses of TA
and GS.
6.4 Mechanism for Modulation of the Stretch Re-
flex Gain
Spindle nonlinearities and asymmetric reflex connections cause the system to respond
with two modes having dinerent dynamics. The fast mode is responsible for generating
torque. The slower component corresponds to tonic change in the background firing
rate of the a-rnotoneuron pool. This component can be thought of as the operating
point of the system. Because it is negative for both dorsi£ie,xing and plantdexing
displacements, it is always below threshold and produces no torque. However, it
changes the excitability of the a-motoneuron pool and thus the apparent reflex gain.
For example, in the simulations with: pulse displacements, the response of the
system to a negative pulse was smaller than that to a positive one of the same size.
This is because the h-st negative going edge of the negative pulse produces the slower
component, nrhich depresses the response of the system to the following positive edge
of the pulse. The simulations lend support to the suggestion [53] that the reflex
torque to a negative pulse displacement in experiments is due to positive veiocity of
relengthening, and is depressed by a previous shortening of the muscle. The same
mechanism is responsible for depressing the reflex in two-pulse experiments- Here
the slow mode of the response of the qstem to the first pulse depresses the response
to the second pulse. Findy, simulations with tonic inputs suggest that the 'Went"
period is &O due to reflex dynamics; this would explain why the silent period is
absent following deatferentation [54].
The directional asymrnetry or "rectification" of the slow component of the GS
a-motoneuron together with a nonlinearity at the a-motoneuron level is suggested
to be a mechanisrn of the depression of the stretch reflex with the velocity of the
perturbation. Because the slow component is always negative and takes longer to
corne back to baseline, any m e of ongoing movement wiU lower the reflex gain.
Moreover the reflex depression will be graded with the velocity of the perturbations.
The faster the input to the mode1 changes, the less time transients will have to die out,
and the further below threshold system sits, the lower is reflex gain. This inability
of the slow mode to reach steady states makes the system operating point depend
directly on the velocity of perturbations.
6.4 Advantages of Bilateral versus Unilateral Mode1
Bilateral modeling of the agonist antagonist coupling offers several advantages over
unilateral models. First, asymmetry of adde flexor and extensor muscle responses
can be modeled, and its functional sigrdicance for motor control can be explored.
As discussed in section 3.5, EMG responses of flexor and extensor to stretch are
ver). different. This important experimentd observation cannot be accounted for in
a unilateral model, because ody net torque and reflex EMG are modeled.
Second; bilateral models introduce extra degree of freedorn that can be useful in
explainhg the role of peripheral neuromuscular control. As been shown in 1151, an
important characteristic of a bilateral model with reciprocal commissural coupling is
that the system behavior is described by two poles, i-e. system response is represented
by two exponentials instead of one. Moreover, the system responses are affected by
both agonist and antagonist inputs. These factors together with nonlinearity of the
sensors introduce the extra degree of complexity that d o w the system to better
control its overall dynamic behavior at spinal level.
Finally, elaborate asymmetric reflex connections can be accounted for with a bi-
lateral structure. In unilateral models the spinal level is usually represented as a
single block. Due to this inherent limitation, simulations with d a t e r a l models will
inevitable underestimate the capabilities of the spinal level reflex connections, and
therefore underrate the importance of peripheral control of rnovement. Memhi le ,
various investigations have suggested that many physiological functions are controlled
at this level [42,7]. Therefore, by ailowing us to account for both agonist and antag-
onist reflex connections at the spinal level, bilateral models provide further insight
into the nature of motor control. For example, the present model of the asymrnetric
refles connections a t the ankle suggest a mechanism for the modulation of reflexes
with the velocity of random perturbations.
Possible P hysiological Explanat ion for
ing Factors . . ..
The asymmetric model of stretch reflexes is capable of reproducing a wide range
of experimentally observed behavior with one constant set of parameters- However,
the parameter values required by the model are different fiom what they might be
expected. Thus they suggest that the gain of spindle projection to IN for GS, le, is
-12.5433, gain of GS RC projection to GS IN, re is 0.2, and gain of GS IN projection
to TA a-motoneuron, i,, is 0.0013' These values are opposite in sign of what would be
expected from physiology. To what extent can reversed values for these parameters
be justified on a physiologicd basis ? m d &.-.,
The negative projection of 1, - the projection fiom GS primary spindle to GS
internewon- is readîly explained. Ia Interneurons receive excitatory inputs from pri-
mary spindles in both homonymous and antagonist muscles [SI. However, in the
present model the inputs kom the antagonist primary spindles were not modeled.
Thus, the excitatory effects of GS primary spindles on Ia Interneuron of TA could
appear as a negative effect of T-4 primary spindles on Ia Interneuron, because the
firing rate of TA and GS spindles have opposite modulation directions.
As for a positive projection of i, (gain of GS IN projection to TA a-motoneuron),
mutual inhibition of the Renshaw cells documented in [45], is omitted in the model.
However, its effect could make the gain of projections from GS Ia IN to TA a-
motoneuron appear positive. The projection from GS Ia IN to TA a-Mn via GS
a-rnotoneuron, Gs Renshaw cell, TA Renshaw cell, TA a-motoneuron have positive
signs. This path is drawn in bold in Figure 6.1B. Note that the product of the gains
in this path is positive. So it is reasonable to argue that the effect of this positive . .
pathway can appear as a positive sign for i, in the undermodeied structure-
Analogously, as can be seen from Figure 6.1A the projection from GS RC to GS
'These values are from the Table 4.2. Note. that negative signs for the inhibitory connections were accounted for in the analysis. Therefore negative entries for r, and i, in the Table represent positive values for t hese parameters. . .
IN via GS a-motoneuron, TA a-motoneuron, TA Renshaw c d , and TA IN could also
have a positive sign. Follow the bold lin& in Figure 6.1A to note that the product
of the correspondhg gains is positive. This path involves mutual inhibition of the
agonist and antagonist a-motoneuron pools [23] and an excitatory connection fiom
GS IN to GS a-motoneuron [42]. However, because these were not modeled in the
present structure, theïr compound effect can appear as positive value for T,. To - .. .
account for the proper anatornical signs for Z,, i,, and T, these pathways should be
included in future development of the modd.
These additional pathways may also be essentid to predict the poIysynaptic
stretch reflex response. As shown in the Section 5.2 the simulated EMG response
of the GS muscle is in good agreement with experimental data, but there is a discrep
ancy between TA EMG of the mode1 and real data. In particular the polysynaptic
stretch reflex (PSR) response observed with latency of 75 ms is not predicted by the
present model. There are two contrasting hypotheses as to the origin of the PSR. One
suggestion is that the characteristic pattern of agonist - antagonist EMG responses is
preprogrammed and generated by supraspinal regions. An alternative hypothesis is
that this pattern is the result of spinal reflex actions. The present model predicts one
burst at a Iatency of 50 msec, which depends on the velocity of the stretch, therefore
the model accounts ordy for the MSR component. Nevertheless, it is worth noting
that the results of simulations by no means disprove the second hypothesis. Given
the effect of anesthesia of the hand on the PSR component [37] it is plausible that
other afferents, possibly group II spindle or cutaneous, and their synaptic connections
play a role in generating the PSR. Simulations Nith extended models are needed to
further assess these two hypotheses.
6.7 P hysiological Implications
Notwithstanding the Limitations noted above the model predicts a wide range of
known behavior. Because it is built on the known reflex connection at the ankle,
it can suggest possible peripheral mechanisms for motor control. Such speculative
Figure 6.1: Subsets of neural connections of the agonist - antagonist organization. Connections used in the present structure of the mode1 are depicted as solid lines. Connections which could be responsible for reversed values for some parameters are depicted as dashed lines. A: the subset explaining positive projection from GS Ren- shaw ce& to GS interneurons. 8: the subset explaining positive projection £rom GS interneurons to TA a-rnotoneurons. For explanation see text.
models cannot prove that a particdar mechanisrn is occurring, but they d o w us to
e-xplore t hei. possible si&cance.
Corrections of errors and other adjustments in the functioning of the motor system
must be made as quickly as possible. The s m d delay of the spinal circuits allows
these adjustments to be regulated qkckly and efficiently. It has been shown in this
thesis that sensory inputs and spinal circuits are complex enough to be able to deal
with the task of modulation of the reflex g&.
The results Mth the random perturbations imply that spinal neural circuits can
calculate and carry out their own integrative functions, in particular
the stretch reflex gain.
6.8
1. As a
reflexes,
Recommendations for Future Work
first step in these simulatio~ studies, in the bilateral model
modulation of
of the stretch
only major neural connections of the ankle were included. These simplifica-
tions resulted in reversed values for some of the parameters, as discussed in section
6.6. The mutual inhibition of the Renshaw ce&, a-motoneurons, and spindle con-
nections to antagonist IN should be included in the model. Moreover, accounting for
the dynamics of these neurons and synapses may provide insight into mechanisms of
polysynaptic stretch reflexes.
2. Golgi tendon organ receptors are extremely sensitive to active force, and rnake
excitatory connections to Ib Interneurons. This polysynaptic reflex constitutes force
feedback to the muscle. Moreover itAhas been found that Golgi tendon organs play
important roles in the regulation of locomotion, in particular timing of the locomotion
rhythm [42]. Inclucling these receptors in the model may provide future insight into - .
the relative contribution of the peripheral and central pathways in the control of --.
movements. . .
3. The present model uses only one of the mechanisms of neural gradations of
the muscle force - rate coding. However, 'as discussed in 2.3 recruitment of the mo-
toneurons &O controls muscle force. Since it is likely that both mechânisms are used
together, a more complete mode1 of stretch reflexes should include recruitment as well
as rate coding.
4. Finally, stretch reflexes are &O modulated throughout each cycle of periodic
movements such as
reflex gain increases
It then decreases to
. - cycling, wallang, and running. Thus, d m g walking, stretch
throughout stance and reaches a maximum near full dorsïfiexion. - - .
zero during the transition £rom stance to swing and remains low
during the swing phase. A similar pattern was observed in recent experiments, in . .. .
which the ankie was moved passively through the same trajectory as during wallang
[34], suggesting that spinal mechanisms are responsible for this modulation.
To extend the present mode1 for use with low frequency inputs as passive wa.lking
pattern, the velocity component of primary spindles should be put in paralle1 with a
block representing the spinclle's length-dependence. For a h t approximation simple
gain would suffice, since spindle output is proportional to muscle Iength a t loy fre-
quencies. Given the moddation of the operating point of the system with the velocity
of the movement, it is possible that characteristic movements during the walking cycle
dso cause the aMn background exitability to change, thereby modulating the reflex
gain.
Appendix A
A signal-flow graph may be defined as a graphical means of portraying the input-
output relationships between the variable of a set of linear algebraic equations.
When constructing a signal-flow graph, junction points, or nodes are used to
represent the variables. The nodes are connected together by line segments called
branches, according to the cause-and-effect equations. The branches have associated
branch gains and directions. A signal can transmit through a branch only in the
direction of the arrow.
Before discussing Mason's d e s for analysis of signal-flow graphs, several terms
have to be dehed. An input node is a node that has only outgoing branches. An
output node is a node that has only incoming branches. A path is any collection of
a continuous succession of branches traversed in the same direction. -4 forward path
is a path that starts at an input and ends a t an output node, and along which no
node is traversed more than once. A loop is a path that originates and tenninates on
the same node, and along which no other node is encountered more thaa once. The
product of the branch gains encountered in traversing a path is called a path gain.
The forward-path gain is the gain of a forward path. The loop gain is the path gain
of a loop.
The general gain formula for a transfer function between input and output nodes
where y, = input-node viwiable
y,t = output-node miriable . .
M = gain between yin and y,,
N = total number of fommd paths between y h and ymt
iMk = gain of the kth f o f ~ v s d path betmeen yin and y,t
Pm, = gain product of the mth possible combination of the r nontouching loops
(1 r N ) . (Two parts of a signal-flow graph are nontouching if they do not share
In other words,
A = 1- (sum of the gains of all individual loops) + (sum of products of gains of
all probable combinations of two nontouching loops) - (sum of products of gains of
al1 possible combinations of three nontouching loops)+ ...
4, = the A for that part of the signal-flow graph that is nontouching with the
kth forward path.
Consider the signal flow diagram in Figure A.1.
Figure A.1: Signal flow graph of a feedback control system.
There is only one forward path between R(s) and C(s), and the forwar-path gain
is Ml = G(s). There is only one loop, and the loop gain is Pt, = -G(s)H(s). There
are no nontouching loops since there is only one loop. Furthemore, the forward path
is in touch with the only loop. Thus, Al = 1, and A = 1 - Pli = 1 + G(s)H(s). The G s closed loop tramfer function is = = l + G ~ ~ l & ( s ) .
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