Biological Cybernetics

Biol. Cybern. 55, 403420 (1987)

9 Springer-Verlag 1987

Muscle Models: What Is Gained and What Is Lost by Varying Model Complexity J. M. Winters and L. Stark Group in Bioengineering, University of California, Berkeley, California, USA Chemical, Biomedical, and Materials Engineering, Arizona State University, Tempe, AZ 85287, USA

Abstract. Three structurally different types of models have evolved over the years to describe muscle-joint systems. The first, based on an input-output analysis of a given task, results in a simple second-order differential equation description that is adequate over a certain movement operating range. The second, based on the classic structural model of Hill (1938), results in a higher-order nonlinear model described by ordinary differential equations. The third, based on an analysis of the biophysical contractile mechanism, results in a complex partial differential equation description. The advantages and disadvantages of each type of model are considered, based on the criteria of identifying the simplest model that can adequately simulate any fundamental type of human movement without modifying model parameters for different tasks. It is shown that an eighth-order Hill-based antagonistic muscle-joint model is able to satisfy these criteria for a given joint if each of the four basic mechanicallysignificant non-linearities of the system are included in the model. This same model structure has been used successfully for eight different muscle-joint systems, ranging in size from knee flexion-extension to eye rotation - the only difference between the models is in the parameter values. Second-order models are shown to be task-specific special cases of the input-output behavior of the eighth-order model, while the more complex biophysical models are hypothesized to have insignificant advantages and many disadvantages over the Hill-based model during normal human movement.

what is known about the system under study. Of all of the "design" decisions required for creating a given model, the most important is the determination of the model s t r u c t u r e . This choice will literally define the limits of the potential vision of the modeler into the system under study. Too simple a model can result in a lack of depth of understanding of important physiological system properties, and often, unfortunately, to inaccurate visions of system behavior. Conversely, too complex a model may lead to an inability of the modeler to gain sufficient insight into system behavior due to the tendency to get lost in model details such as parameter identification or data overload. Biological and physiological systems are in fact complex, being inherently higher-order and nonlinear. The traditional way of handling such a system is to start with a simple model and then gradually evolve to a more complex model as increased understanding is obtained and new questions are raised. Unfortunately, in biological/physiological research, modeling is often a side adventure to experimental work, and often an early model ends up being the only model due to the cost involved in model development. A perfect example of modeling evolution is that of muscle and muscle-joint modeling. Muscle models have developed from three somewhat separate foundations. Each of these modeling approaches has its advantages and disadvantages, and the type of model used will depend strongly on the goals of the study. The fundamental goal here is to discern what type of model is best for simulating a wide variety of human movements, given our current ability in modeling techniques and our present state of physiological knowledge.

1 Introduction

One of the best ways of understanding a given physiological system is to model it. Modeling a given system not only provides insight into internal system behavior but also forces the modeler to state explicitly

2 Muscle Modeling Evolution: Three Directions

Since the classic work of Hill in the late thirties (e.g. Hill 1938), the range of models that have been put forward for simulating human movement has been

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Fig. la--e. Schematics of the basic structural foundations for the three general types of muscle-joint models, a Second-order "systems" model, b Hill series structure for a muscle, connected in parallel to a joint with inertial and parallel elastic properties, c Huxley-based sliding filament mechanism, showing the cross-bridge concept. This element would be attached to another in series, and then finally to a

tendon with series elastic properties incredibly broad. They range from the muscle-joint system being considered as a simple torque generator with negligible muscle dynamics to being a complex, very high order nonlinear system with up to 50 parameters required to describe the motion of a single joint (Hatze 1978). From a purely structural viewpoint, the types of models fall into three broad categories, as defined in Fig. 1.

2.1 Simple Second-Order "Systems" Models The first type are those that do not include the series elastic element introduced by Hill (1938 ). This assumption considerably reduces, at least structurally, the complexity of the model (Fig. la). The muscle-joint structure can then be treated as a second- or even firstorder system with model parameters that vary as a function of the task and the movement operating range. The basic equation of motion for a second-order model is:

J2(t) + B2(t) + Kx(t) = Gu(t),

(1)

where the parameters J, B, and K are typically, but not always, mechanically analogous to system inertia, viscosity, and elasticity, respectively, and G is a gain. This equation can be re-arranged as:

x(t) + 2~x(t) + oJZx(t) = (G/J)u(t) ,

(2)

where ~ is the damping ratio, and co, is the natural frequency. The output, x, may represent either joint position or torque, depending on the particular application. The input, u, may represent either neural input or external loading, also depending on the application. This second-order approximation was made by many of the early "systems" analysts of human movement, including one of the authors (Stark 1968; Agarwal et al. 1970). Its use is motivated by the fact that extending the "Hill model" to the total musclejoint system immediately creates fairly high-order

models, which might be considered as undesirable when compared to the theoretical beauty and simplicity of the second-order model. From a practical viewpoint, this approach treats the muscle-joint system as a "black box" in which the contents of the box can be approximated by a second-order linear system over a given operating range. This classic "systems" model is also commonly used for non-biological actuator-filtering systems such as in robotics work (discussed in Winters 1986). If the goal is to unravel the mysteries of human movement, then these types of models have some severe limitations: i)the model inherently has only one independent structural location for an input (Fig. la); ii) secondary input signals (e.g. neural antagonistic co-activation during an external impulse) must be represented indirectly via manipulation of model parameter values rather than as explicit inputs; iii) the basic state (x) and input (u) variables may both change as a function of the type of task (e.g. compare isometric versus external loading tasks); and iv) the phenomenologically-obtained model parameter values change as a function of task or even for different operating ranges of the same task.

2.2 The Hill-Based Lumped-Parameter Model The second fundamental type of muscle-joint model is based on the structural model of A.V.Hill (1938), deduced from his well-controlled phenomenological studies of isolated muscle tissue. This classical lumpedparameter model consists of a "contractile element" (CE) in series with what is essentially a series viscoelastic element (SE) (Fig. lb). This basic model structure has been supported by countless groups subsequently, both for isolated muscle [-reviewed in Close (1972) and in Winters (1985)] and for muscle-joint systems (reviewed in Winters 1985). Ordinary differential equations are required to solve this type of model and all of its offsprings (e.g. Cook and Stark 1968). A consistent

405 finding is that each of the two basic muscle dements located in series is inherently nonlinear. The mathematical descriptions of these nonlinear elements vary somewhat from group to group. The two most popular forms follow. The CE force-velocity relation for shortening muscle during maximal contraction (Hill 1938), in a convenient form for parameter description, is: (F + A*Fo) (v + A* Vm) = AFo Vm(1+ A),

(3)

where F is muscle force, v is contractile element velocity, and A, V,,, and Fo are constants defining the dimensionless hyperbolic shape factor, the maximum unloaded CE velocity (x-axis intercept) and the maximum isometric force (y-axis intercept), respectively. With re-arrangement, this equation can be reformulated into a form that shows explicitly the "passed" force versus the force "lost" to heat via a parallel viscosity (Cook and Stark 1967): Fm= Fh -- Fb = Fh -- bhVh, F~(1 + A) , ; (vh + A V,,) Fh = (activation level) *Fo. bh--

(4) (5)

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where F and x are the series element force and extension, respectively, and the constants S, Fo, and Xo define the dimensionless shape parameter and the peak force and extension parameters, respectively. 2.3

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The Huxley-Based Distributed-Parameter Models

The final general type of model was initiated by the classic work of Huxley and colleagues (e.g. Huxley 1957). In this approach, instead of considering muscle as a "black box" system with phenomenological properties, an attempt is made to identify and then model the actual contractile mechanism (e.g.T.L. Hill 1975; Wood and Mann 1981). This arena of muscle modeling stresses the "sliding filament" concept for muscle contraction and models the contractile process as a distributed parameter system requiring partial dif-

(8)

where the variable "n" is a probability distribution function representing the fraction of attached actin/ myosin cross-bridges, and the independent variables are time and length. The functions f ( x ) and g(x) represent, respectively, the forward (bonding) and backward (unbonding) rate parameters, and v(t) represents the speed of shortening of a half sarcomere. The values of the rate parameters are typically piecewise linear functions of x, which is the distance from the equilibrium position of the cross-bridge. It is the builtin length-dependency that distinguishes this type of model. Once n(x, t) has been determined, more physically useful macroscopic variables can be computed as moments of this distribution. For example, assuming a linear force-displacement relation for cross-bridges, the muscle force is (Zahalak 1981): ~Ao:msk~ ~ x n ( x , t ) d x

where K1 and K2 are constants. This can be integrated, and then, using appropriate boundary conditions and reformulation of the describing parameters, we have (Winters and Stark 1985a): F = Fo/(e x~ (e(s/x~ -- 1),

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The usual expression for the SE force-extension relation is based on the following observation (Glantz 1974): Stiffness = K I F + K 2 = dF/dx,

ferential equations for solution (Fig. lc) that are of the form:

(9)

where A is the cross-sectional area, a is the normalized level of activation, and the other four parameters depend on the microstructural characteristics of the contractile tissue. The main evolutionary trend over the past dozen years with this type of model has been toward even more complex models involving increasing numbers of rate functions and activation-deactivation states (e.g., T.L. Hill 1975; Wood and Mann 1981). The mathematical complexity of this approach is great, and consequently it is very difficult to simulate anything but simple, stereotyped types of movement such as isovelocity stretches. Furthermore, there is little data that can be used to determine the parameters required for these types of models for different human movement systems- especially the rate functions. In general, the direction of both past and present research in this area has been toward continued improvement in understanding the contractile mechanism, regardless of whether there is a mechanical significance to all of the parts of the process. The possible advantage of using this type of model versus the classic Hill-type model is that the former explains the full range of lengthening muscle phenomena while the latter, even with length-dependency built in artificially through nonlinear elements, cannot describe all aspects of the transient lengthening response seen in isolated muscle studies. However, at present, because of their mathematical complexity Huxley-based mechanism models have yet to be used for any practical problem involving human movement.

406

An important compromise between the Huxleybased mechanistic models and the Hill-based phenomenological models, introduced by Zahalak (1981), deserves separate treatment. The rationale behind this approach is that the mechanistic model requires parameters and mathematical structure that are not needed to describe macroscopic muscle mechanical behaviors such as force or stiffness. The resulting "distribution moment" (DM) model retains the fundamental features of the Huxley-based model, but, by assuming that the distribution function n(x, t) is a normally distributed function of length, is able to simulate the general transient lengthening (eccentric) behavior fairly well with a set of three ordinary differential equations replacing the original partial differential equation, each with the form:

seen in actual human movements involving a single joint, subject to the constraint of establishing invariant parameter values. A Hill-based model will be used for reasons that will become evident later. The goal of this section is to describe this model, presenting a more brief form than that in Winters and Stark (1985a). Results with this model will then be compared to those of the other two fundamental types of models, as well as to other lumped-parameter models, in later sections.

3.1 Model Structure The basic model structure, described in detail elsewhere (Winters and Stark 1985a), is essentially an extension of the sixth-order eye movement model of Cook and Stark (1968). The model consists of"equivalent" muscles that insert on either side of the so-called hinge "joint", as can be seen from the pendulum analogy of Fig. 2a. The range of the neural inputs is on the interval (0, 1), with "1" representing maximal excitation. The passive plant, which represents all passive (non-contractile) inertial and viscoelastic muscle-joint system influences, also connects at this node. An input-output view of the model is presented in Fig. 2b. Notice that there are three inputs (two neural, one from the external environment) occurring at structurally different locations. The output for the case in which external torque is specified is joint velocity

dMi -- b i - Fi(Mo, Ma, M 2 ) - iv(t)Mi, (10) dt where i= 1, 2, 3 and the functions F~ depend on the assumed forms of the rate functions. These first three moments represent the muscle force, stiffness, and energy - all useful quantities. 3 The Lumped-Parameter Nonlinear Antagonistic Muscle Model Our goal is to develop the simplest model that can adequately represent the whole range of phenomena ////l

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Fig. 2a--e. Schematic of the model used here. a Pendulum analogy of the model, showing how contraction of antagonistic muscles can cause pendulum rotation, b Input-output block diagram of the model, showing the different structural locations of the inputs. c Structural view of the muscle model itself, showing the five fundamental elements for each muscle (the intertial and parallel elastic properties of muscle are lumped with the passive plant). The delayed sensory feedback pathways are also not shown

407 (or position). Alternatively, if the velocity is constrained (e.g., zero, constant, sinusoidal), then the joint torque becomes the output. This input-output behavior is analogous to the current-voltage variables in electrical systems. As seen in Fig. 2c, there are six basic structural elements in the model, representing the fundamental processes of the muscle-joint system: 1. The neural excitation~de-excitation (NEX) process, which represents neural dynamics and converts an idealized neural input into an output signal that resembles the electromyogram (EMG). 2. The CE neuro-muscular activation-deactivation transduction (ACT) process, representing temporal dynamics between the E M G and the contractile process, which is rate-limited by calcium activationdeactivation. 3. The CE active muscle torque-angle (ATA) process, which represents the maximal contraction as a function of angle - this value is multiplied by the output of the activation process. 4. The CE torque-velocity (CETV) relation, which represents the well-known observation that the torque "passed" by the muscle is a function of the velocity (Hill 1938). The force across the dashpot represents energy lost to heat during contraction. 5. The series element (SE), which represents the "stiffness" of muscle tissue in series with the contractile element as a function of force. This element is significant when the stiffness is low (compliance high). 6. The second-order passive plant (Fig. 2a), which consists of a parallel arrangement of linear inertial and viscous elements and a non-linear elastic element that becomes stiffer as movement operating range endpoints are reached. This same model structure, with very different parameter values, has been used to simulate movements of seven different muscle-joint systems, ranging in size from eye rotation to knee flexion-extension (Winters and Stark 1985a). A final area requiring modeling, neural feedback, is riddled with controversy, in no small part due to the fact that it is difficult to isolate neural components without violating the integrity of the system. Many models exist [-see Agarwal and Gottlieb (1985) for a review], and the best approach at present allows virtually all of the postulated possibilities to be available via gain modification, resulting in the following general structure: N I = {K 10 + K2v + Ksa + K40(v ~ + Ks(F.,)} x {G~ U ( t - tl) + G2 U ( t - t2)},

(11)

where 0, v, a are position, velocity and acceleration, Fm is muscle force, ta and t2 are the short- and long-

loop time delays, and Ki and Gi are gains. There is also a small (5 ms) lag filter time constant (not shown). For the present analysis, all feedback gains are either zero or set conservatively, depending on the task - pure mechanical behavior is of primary interest here. 3.2 Model Parameter Identification The key to a good model is to be able to identify model parameters - parameter identification is typically the chief limitation encountered in modeling. For physiological systems, the ideal situation is to develop a set of parameters that never need to be varied, regardless of task. After all, actual human movements use the same muscles for tasks! This criteria for success requires not only a sufficient model structure but also a straightforward parameter identification process, based either on direct experimental measurement or on an analytical process employing anatomical data. In the present work, two of the six basic elements (NEX and ACT) were assumed to be linear first-order processes that could be adequately described by a single time constant. The magnitude of these time constants for the various "equivalent" muscles is assumed to be a function of fiber composition and the size of the synergistic muscles, with larger muscles weighed more heavily (Winters and Stark 1985a). This resulted in the following ranges: 20 ms < Tnox< 50 ms; 20 m s < Tdact< 60 ms.

5 ms < T,r t < 15 ms ; (12)

Two other relations (ATA and PP) could be obtained directly from experimental muscle-joint data. The former is determined simply by asking a subject to contract maximally at a number of different angles - a wealth of data is available for this type of task for many different joints in the body, and approximations of these experimental data, extracted to "small-tonormal" adult males (height 170 cm, weight 70 kg), were fit by a skewed gaussian distribution for each joint in Winters and Stark (1985a). The latter relation involves three sub-elements. The first of these, the system inertia, which can be well approximated here as constant, can be estimated from available literature for any part of the body (see Winters 1985). The second, the passive viscous element, is assumed to be constant an adequate approximation since sensitivity analysis work suggests that this element is of little significance for dynamic human movements. The final subcomponent, the elastic element (PE) can be well represented by a summation of a linear term and an exponential term, with the linear term of greater weight during the long, very compliant toe region and the nonlinear exponential term indirectly defining the joint operating range (Winters and Stark 1985a).

408 The last two elements, namely the CE torquevelocity and the SE element, use the basic relationships described in Sect. 2.2. The individual relations were determined by algorithms that combined knowledge of muscle as a material (e. g. peak unloaded CE velocity or peak SE extension as functions of fiber composition) with geometrical knowledge such as muscle origininsertion, pennation angle, moment arm, etc. (e.g. Winters and Stark 1985a). The final "equivalent" relation used was a weighed function of the individual contributions of each muscle (e. g. Fig. 3, Winters and Stark 1985a). Of special importance is the part of the CE torquevelocity curve for lengthening muscle, which the Hilltype lumped parameter model cannot model as accurately as the Huxley-type cross-bridge-based distributed model. The current fit is basically just an

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4.1.1 Isometric "Impulse" Response. In isometric contraction, the primary input is the neural signal, while the secondary input, joint velocity, is set to zero. Given this velocity condition, the output torque will be an explicit function of the neural input. It turns out that the neuromuscular "action potential", which has a pulse width of 1-2 ms, can be well approximated as an M

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The purpose of this section is to document the ability of the present Hill-based model to simulate a wide variety of very different types of human movement, with emphasis on "systems-type" tests. Classic "systems" input signals, such as impulses, steps, ramps, or sinusoids, will be given as the primary input, which may be at either a neural or external input location. The neural or external signal that is not the primary input will be classified here as the secondary input. This secondary input usually stays at a constant value throughout the test. Finally, the remaining external node must be the output (Fig. 4). It is interesting that the classic tests in muscle physiology, which go by names such as "isometric", "isotonic", "isokinetic", "forced-oscillation", "fast voluntary", etc., are all in fact "systems" tests. To recognize what explicit and implicit assumptions were made in past work, these classic tests will be cataloged by their inputs and output.

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inverted Hill's equation, going up to a steady-state peak eccentric contraction that is 30 % higher that the peak isometric contraction (Fig. 3a), which gives a curve similar in shape and magnitude to that used by others who have had to confront this lengtheningmuscle issue with a lumped-parameter model (e.g. Hatze 1978).

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Fig. 3a, b. Example of the contractile elementtorque-velocity(a) and series element (b) torque-extension relations for individual muscles and how these are combined to obtain "equivalent" muscle properties, shown here for elbow extension. The final relation for the torque-velocitycase is obtained by summing the individual torques for each velocity (not shown). The final series element relation is obtained by using a weighed average, with stronger muscles weighted more heavily (Winters and Stark 1985a)

Fig. 4. Input-output possibilities for the three fundamental signal ports (the two antagonistic neural signals are assumed to be coupled). In each case, the "primary" input enters the model from the left, the "secondary" input enters from above, and the output leaves from the right

409 Table !. Time constants describing the muscle torque output trajectories during maximal contraction under the isometric condition

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a b Fig. 5a, b. Simulation of isometric (i.e. external velocity equals zero) contractions, a "Twitch response" for the ankle dorsiplantar flexors,using an "action potential" neural impulse input of 2 ms duration, b "Tetanus" response for four flexion-extension systems, with flexion torque trajectories plotted upward and extension trajectories downward. Torques are normalized relative to the maximum contraction torque of all possible joint positions. All contractions shown here occur at the midoperating range for the particular joint, which explainswhy some of the torques reach maximal steady-state values under 1.0

impulse. The physiological literature calls the resulting torque response the "twitch response". An example of a "twitch response" is shown in Fig. 5a for ankle dorsiand plantar flexion. Peak torques and the times of the peaks are almost identical to the experimental work of Buchthal and Schmalbruch (1970).

4.1.2 Isometric "Pulse" Response. An extended set of action potentials at a given average firing rate, for a group of muscle motor units that are not firing in synchrony, defines a neural pulse. The step (pulse of significant width) response for a neural step level of 1.0 (i.e. maximal firing rate) and a zero-velocity secondary input is referred to in the physiological literature as a "maximal isometric contraction" or "tetanus" (e. g. Hill 1938). The resulting torque output transient is a function of the CE, the SE, and the excitationactivation process. A number of experimental studies have obtained the transient dynamics during maximal isometric contraction, both for the case of voluntary effort and the case of electrical stimulation (reviewed in Winters 1985, Chap.IV). In the present model, the latter case means that neural excitation dynamics are bypassed, and hence the model would respond a little faster, just as seen experimentally. As seen in Fig. 5b for simulations of the maximal voluntary effort case, the torque output of the model is similar to the step response of an overdamped second-order model with a neural input and a torque output! This is also seen experimentally. From an input-output perspective, for this task, the eighth-order models would then reduce to a second-order model with the parameter values listed in Table 1. These values agree well with published data

for elbow flexion (Wilkie 1950), knee extension (Hatze 1981), wrist rotation (Houk 1963), and ankle plantar flexion (Gottlieb and Agarwal 1973), with most models slightly "hot" (faster response), undoubtedly due to the neural input of the human not being as perfect as our ideal step input. The only exception is for ankle dorsiflexion (Gottlieb and Agarwal 1973), where the model was 25 % too slow.

4.1.3 Isovelocity "Pulse" Response. In this case the external velocity is constrained to a certain value, rather than to zero as in the isometric case, and again a maximal neural signal is applied. This type of test has been used extensively in recent years due to the wide availability of machines such as the CYBEX, which approximate the isokinetic condition for the limited case of concentric (shortening) contractions after an initial transient acceleration artifact has subsided (Sapega et al. 1982). The common protocol is to try to obtain the concentric torque-velocity relation by repeating the test at various velocities and then remeasuring torque each time. Each test thus gives one point on such a curve. Measurement of both peak torque and the torque at various specified angles has been used to obtain the curves. A controversial aspect has been that some groups find non Hill-shape torquevelocity curves, with peak torque occurring at a lowvelocity concentric loadings such as 30 or 60 deg/s (e. g. Perrine and Edgerton 1978), while others produce curves similar in shape to Hill's equation (e.g. Thorstensson et al. 1976). This paradoxical situation that has not been resolved even by careful experimentation (Yates and Kamon 1983), and is confounded by what is simply a measurement problem, as demonstrated in Fig. 6, where it is shown that torque-velocity curves obtained from measurements of simulation output of the maximal isokinetic elbow contraction task depend on, in addition to the actual system: i) the "velocity clamping" effectiveness of the machine (not shown); ii) the initial conditions for the contraction (Fig. 6b); and iii) the location in the torque trajectory the torque measurement is made (Fig. 6c). In all of these cases, the actual CE torque-velocity relation within the model is, of course, the Hill equation. Notice that the

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measured torque-velocity relations differ from the actual internal relation, providing curves similar to the two experimental extremes mentioned earlier - an indication that using the isokinetic approach, at least as defined by the CYBEX, may not be an appropriate method for obtaining torque-velocity relations for the contractile element! It is also simple to show that it is the series element that is primarily responsible for this inability to determine the CE torque-velocity relation from externally-measured torque-velocity behavior (Winters 1985).

4.2 Primary Input: Neural; Output: Velocity The second basic category are those cases of neurallyinitiated movement in which the external load is a secondary input. It may be zero ("free movement") or constant (isotonic movement).

4.2.1 Anisometric (Free) Pulse Response. There is a good deal of data available for fast eye, head and upper limb movements with no external loads, all for the case of maximal voluntary effort. In all of these cases, the neural signal can be well approximated as a pulse approaching maximal for a time period ranging from 30 ms to 200 ms, depending on the particular system and the size of the desired movement [e. g. Clark and Stark (1974) and Winters et al. (1984) for eye movements, Zangemeister et al. (1981) and Hannaford et al. (1984) for head movements, and Wadman et al. (1980) for arm movements]. This agonist burst is of course

followed by subsequent "clamping" bursts in both muscles. When the experimental (EMG) and simulation pulse input signals are matched (e. g. Fig. 7a), the resulting output trajectories (joint angle, velocity) can be also shown to be similar, both in magnitude and timing. With simulation work, one has more control of the height of the neural input pulse than in human testing, where sub-maximal voluntary effort is hard to quantify. An example of the kinematic output response to neural pulses of varying strength is displayed in Fig. 7b. Notice that superposition does not hold - doubling the neural energy does not double the peak velocity nor the position attained, nor halve the time to these peaks. This is an explicit example of nonlinear behavior. Sensitivity analysis work (Winters 1985) shows quite clearly that the primary source of this input-output nonlinearity early in the movement is the torquevelocity relation (notice the relatively low level of muscle torque during the high-velocity stage) and the primary source toward steady-state conditions are the parallel elastic and active torque-angle relations particularly the former. For most muscle joint systems, the output position trajectories can be fairly well approximated by a second-order fit to a step input. Approximate parameters describing second-order fits of the data for neural input step levels of 0.1 and 0.5 are listed in Table2. Notice that as the input level increases, the "time constants" decrease, i.e. the response is faster. However, due primarily to the parallel

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Fig. 7a-b. Examples of fast movements with maximal agonist neural activity and no external torque loading, a A typical "bangbang" fast head movement. The neural pulse widths and heights are approximated from EMG's seen in Hannaford et al. (1984) and Zangemeister et al. (1981). The resulting position and velocity trajectories match well with the resulting experimental trajectories, both in magnitude and timing, b An example of the effectof varying neuropulse input heights on output trajectories, shown here for head rotation. Notice that superposition does not hold

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elasticity, the magnitude of the response does not go up proportionally. Consequently, the parameters that mathematically represent stiffness (K) and viscosity (B) increase and the "gain" decreases. Note also that the so-called second-order parameters have no biomechanical meaning, and that they represent nothing more than an exercise in output-trajectory curvefitting. lsotonic Movements. The preceding "free" maximal voluntary movements are in fact just special cases of the isotonic loading situation in which the external torque is zero. This is the classic method that 4.2.2

was used by the early investigators such as Hill (1938) for isolated muscle and Wilkie (1950) for the musclejoint system to determine CE behavior. The basic problem with the method for muscle-joint systems is that the inertia of the system prevents ideal analysis of the CE torque-velocity relation (Wilkie 1950) - in contrast to the isokinetic task in which the SE confounds CE measurement. Simulation results presented elsewhere (e.g. Winters and Stark 1985a) support Wilkie's hypothesis that CE torque-velocity curves can indeed be approximated from externallymeasured torque-velocity data once inertial dynamics effects are subtracted out.

412 Table 2. Second-order step response parameter fits of the joint position output trajectories for "free" movements initiated by neural "steps" of 10 % and 50 % of maximal neural activity Model

PH

T1

T2

(ms) Elbow flexion Elbow extension Wrist flexion Wrist extension Wrist rotation Knee flexion Knee extension Ankle D-flexion Ankle P-flexion Head rotation Eye rotation

0.1 0.5 0.t 0.5 0.1 0.5 0.1 0.5 0.1 0.5 0.1 0.5 0.1 0.5 0.1 0.5 0.1 0.5 0.1 0.5 0.1 0.5

150 100 200 160 180 100 200 100 180 100 250 140 200 140 360 300 180 150 160 120 90 70

150 80 200 160 40 50 40 30 50 40 250 140 60 60 40 40 80 60 160 120 20 20

MAG

J

(deg)

(kg m 2)

92 110 92 120 70 105 70 100 55 100 60 110 38 56 40 60 38 55 52 75 50 75

0.06 0.06 0.06 0.06 0.006 0.006 0.006 0.006 0.003 0.003 0.4 0.4 0.4 0.4 0.02 0.02 0.02 0.02 0.02 0.02 3E-7 3E-7

4.2.3 Neurally-Initiated Sinusoidal Movements. Another fundamental type of movement that can occur is neurally-initiated oscillation, for instance the oscillations seen in the various types of physiological and pathological tremor or during tasks such as brushing one's teeth. The fastest voluntary frequency that a given part of the body can maintain oscillation is called the "freewheeling" frequency (Stark 1968). Simulations, performed by placing neural inputs to the two antagonistic muscles out of phase with each other, result in a signal that, to a first approximation, is a sinusoid in which the gain (compliance) and phase are a function of frequency. Consequently, Bode plots can be obtained (Fig. 8a). Notice that the frequency response behavior would have to be approximated by a higher-order s y s t e m - a second-order approximation could only be used for frequencies under about 2-5 Hz, depending on the system. The Bode plots can be shown to be a function of the amplitude and offset of the neural sinusoids, as would be expected for an inherently nonlinear system. Unfortunately, other than for elbow flexion-extension, where the above data is consistent with Viviani et al. (1976) and Zahalak and H e y m a n (1979), little experimental data exists on voluntary sinusoidal movements. The above data can be indirectly c o m p a r e d to tremor, where the approximate "bandwidths" appear to be consistent with the

B N-m 0.8 1.2 0.6 0.8 0.3 0.4 0.3 0.4 0.08 0.1 3.2 5.7 7.2 9.5 0.6 0.6 0.4 0.5 0.3 0.3 1.7E-5 1.4E-5

K

Gain

N-m 2.7 6.0 1.5 2.7 1.4 2.4 1.2 3.3 0.4 0.8 6.4 20. 28. 48. 1.4 1.7 1.4 2.4 0.8 1.4 1.4E-4 1.2E-4

0.7 0.4 0.5 0.2 1.1 0.8 1.5 1.2 0.4 0.3 0.4 0.5 0.7 0.4 0.1 0.05 0.05 0.03 0.9 0.4 0.12 0.03

ranges of observed tremor frequencies (e.g. Stein and Lee 19 81), suggesting mechanical limitations to tremor frequencies.

4.3 Primary Input: Torque; Output: Velocity In this type of test, the neural signal is now the secondary input and the external torque is the primary input. It has been found consistently (e.g. Stark 1968; Agarwal et al. 1970; Berthoz and Metral 1975; Zahalak and H e y m a n 1979; Lacquaniti et al. 1982) that the output response to an external loading is a strong function of the "mind-set" of the individual (e.g. c o m m a n d to subject, ability to anticipate). F o r useful simulation work, this "mindset" must be defined more explicitly. The following assumptions have been found to be valuable: the subjective "resist" command, which is known by E M G data to result in coactivation of antagonists (Lacquaniti et al. 1982), is assumed to be correlated to the ongoing (initial) co-contraction level. The kinematic and torque feedback gains are then assumed to be linear functions of this ongoing cocontraction level (Winters 1985).

4.3.1 External "Impulse" Response. The response of the various muscle-joint systems to impulsive loading depends strongly on the task. Since there is a signifi-

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b under three different ongoing co-contraction levels under added inertia conditions similar to Houk (1963). Notice that the output response is a strong function of the ongoing neurally-induced co-contraction level - a result consistently observed whether delayed sensory feedback is present or not. A larger impulse yields a similar result (Fig. 9b). The relation between feedbackinduced neural input transients and the actual muscle torque (Fig. 9c) during the task of Fig. 9b shows quite clearly that the muscle torques do not follow the neural signal transients! As another example, Fig. i 0a simulates the work of Lacquaniti et al. (J 982), in which the forearm is coupled to an external spring that is suddenly released from an extended position. The output trajectories of a variety of simulation external loading conditions can be wellapproximated as underdamped second-order impulse responses, just as was done experimentally. The resulting input-output stiffness and viscosity coefficients, which are strong functions of the ongoing co-

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contraction level, lie within the experimental ranges defined by the "resist maximally" and "do-not-resist" conditions (Fig. 10b). This data suggests that there is indeed a strong correlation between the ongoing cocontraction level and the command to the subject. Even more importantly, since with simulation all internal state variables may be accessed, it is possible to identify the source(s) of observed behavior. It can be clearly seen that there is a fundamental mechanical source for the co-contraction - output impedance coupling that is based on the structural arrangement of the series element and the contractile element (Winters 1985; Winters and Stark 1985b). Furthermore, this effect depends on the nonlinearities of these elements linear elements could not show this effect since the stiffness would be independent of operating ranges and since superposition holds (Winters and Stark 1985b)! The ongoing muscle-torque levels simply set operating ranges on these non-linear relations. Delayed feedback appears relegated to a supporting role with feedback gains directly tied to this ongoing co-contraction level - varying delayed feedback gains without varying the co-contraction level leads to unusual, nonphysiological results (Winters 1985). These results also appear compatible with the experimental results obtained for tasks with non-impulsive inputs such as those of Crago et al. (1976) in which the ongoing

"mind-state" set the response for a significant period of time. It is important to remember that the process described here is a natural consequence of the model and not due to parameter modification with task, as is required for second-order models.

4.3.2 External "Frequency" Response. Another type of "systems" test that has been performed frequently is a sinusoidal analysis, with either torque or velocity as the input and the other variable as the output. In either case, the output signal is approximately sinusoidal, with the gain (compliance) and phase depending on the input frequency. Notice that the frequency response due to external oscillation is different than that due to a neural oscillation input (compare Fig. 8b with 8a). As with preceeding tasks, the experimental results depend on the ongoing muscle activation level, with the low frequency gain decreasing and the resonance frequency increasing with increased voluntkry resistance or with increasing external bias torques (Agarwal and Gottlieb 1977; Zahalak and Heyman 1979; Berthoz and Metral 1975). The latter "bias torque" method, in which the instructions to the subject are the same but the subject is required to hold against varying ongoing constant bias torques during the smaller amplitude oscillation, is an excellent way to discern how neurally-initiated muscle torque operating ranges influence the system

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Fig. 10a, b. Results from simulations of the work ofLacquaniti et al. (1982)for elbow flexion-extensionresponses while coupled to various external springs, a Example of output position, muscle torque, and neural output trajectories for the response to a sudden release from an external spring position as a function of varying co-contraction levels, b Summary of the resulting second-order human elbow flexion-extensionfits as a function of the ongoing co-contraction level for different external springs. The arrows give the parameter ranges found by Lacquaniti et al. (1982) for their various commands to the subjects

as to represent the added inertial and bias torque conditions of "sitting") effects the model response.

5 The Remaining Problem: Contraction of Lengthening Muscle The primary reason given for using Huxley-based distributed-parameter models is that the Hill-based lUmped-parameter model cannot adequately approximate muscle behavior during transient loading. The

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