Testing hypotheses and the advancement of science: recent attempts to falsify the equilibrium point hypothesis

Exp Brain Res (2005) 161: 91–103 DOI 10.1007/s00221-004-2049-0 RESEARCH ARTICLE Anatol G. Feldman . Mark L. Latash Testing hypotheses and the advan...
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Exp Brain Res (2005) 161: 91–103 DOI 10.1007/s00221-004-2049-0

RESEARCH ARTICLE

Anatol G. Feldman . Mark L. Latash

Testing hypotheses and the advancement of science: recent attempts to falsify the equilibrium point hypothesis

Received: 7 January 2004 / Accepted: 9 June 2004 / Published online: 15 October 2004 # Springer-Verlag 2004

Abstract Criticisms of the equilibrium point (EP) hypothesis have recently appeared that are based on misunderstandings of some of its central notions. Starting from such interpretations of the hypothesis, incorrect predictions are made and tested. When the incorrect predictions prove false, the hypothesis is claimed to be falsified. In particular, the hypothesis has been rejected based on the wrong assumptions that it conflicts with empirically defined joint stiffness values or that it is incompatible with violations of equifinality under certain velocity-dependent perturbations. Typically, such attempts use notions describing the control of movements of artificial systems in place of physiologically relevant ones. While appreciating constructive criticisms of the EP hypothesis, we feel that incorrect interpretations have to be clarified by reiterating what the EP hypothesis does and does not predict. We conclude that the recent claims of falsifying the EP hypothesis and the calls for its replacement by EMG-force control hypothesis are unsubstantiated. The EP hypothesis goes far beyond the EMGforce control view. In particular, the former offers a resolution for the famous posture-movement paradox while the latter fails to resolve it. Keywords Motor control theories . Threshold control . Posture-movement problem . Equifinality . Stiffness

A. G. Feldman (*) Neurological Science Research Center, Department of Physiology, University of Montreal and Rehabilitation Institute of Montreal, 6300 Darlington Avenue, Montreal, Canada e-mail: [email protected] Tel.: +1-514-3402078/2192 Fax: +1-514-3402154 M. L. Latash Department of Kinesiology, The Pennsylvania State University, University Park, PA, USA

Introduction Since the mid-1960s, the equilibrium point (EP) hypothesis, specifically the λ model for motor control, has become a theoretical framework used by a steadily growing number of scientists who analyze the production of movement in humans. Among various successes achieved through the use of the λ model, it has offered a physiologically feasible solution to the classical posturemovement paradox of how a movement can occur without triggering resistance from posture-stabilizing mechanisms (for details see Ostry and Feldman 2003). Sherrington (1910) recognized this problem, and his principle of reciprocal inhibition dealt with the issue of why physiological activation of a muscle group does not trigger stretch reflexes in the antagonist muscles. Von Holst and Mittelstaedt (1950) formulated a reafference principle, which implies that posture-stabilizing mechanisms, including homonymous and heteronymous reflexes of the agonist and antagonist muscles involved in the task, are readdressed to a new posture rather than inhibited when an intentional movement is produced. Specific physiological mechanisms and variables underlying the readdressing were unclear until human studies indicated that central control levels are able to change a component (λ) of the threshold length value at which the activity of muscle is initiated (Fig. 1), and that the readdressing is achieved by shifting the thresholds of appropriate muscles (Asatryan and Feldman 1965; Feldman 1966). By shifting muscle activation thresholds, the system readdresses posturestabilizing mechanisms to a new joint position. The previous position becomes a deviation from the newly specified one, and the same posture-stabilizing mechanisms generate forces that tend to move the joint to the new position. Therefore, the system not only eliminates resistance to movement from the previous posture but takes advantage of the posture-stabilizing mechanisms to move to the new posture. Note that central shifts in the activation thresholds are just ways that the nervous system uses to produce movement (or, if the movement is blocked, isometric

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Fig. 1 Threshold control of muscle activation in the λ model. Note that the activation threshold (λ*) has a component (λ) that is determined by central control influences that are independent of proprioceptive feedback as well as components that are dependent on it. Intentional actions result from changes in the first component. Mechanical perturbations can elicit both short-latency (reflex-like) and/or relatively long-latency, intentional changes in the threshold. These aspects of the model are essential to explanations of equifinality and its violation in some cases of transient perturbation of motion

forces). As such, the threshold values or their central components should not be considered to be an internal representation of a goal (target) of motor actions: the goal is identified by the nervous system in the environment using adequate physical variables and does not need to be recomputed into internal variables. Similarly, the difference between the actual and the threshold muscle lengths does not represent a movement error. The error is identified by comparing the actual position of the effector to its target position in the external space. The controller may change λs until the error is nullified, if this is the goal of the action. This style of control is reminiscent of that used when driving a car: the steering wheel, gas and break pedals are tools (analogous to λs) that the driver uses to reach a desired location in the environment. These tools can be used to achieve various destinations without needing to pre-compute all of the actions with the car controls—which is impossible to do anyway, mainly because the driver shares the road with other cars. As in this example, and unlike the EMG-force control formulations, the nervous system does not need internal models or computations to achieve motor goals. By offering a solution to the posture-movement paradox, the λ model remains unique since other models of motor control have been unable to solve this paradox (Ostry and Feldman 2003). The λ model offers a solution to another problem that has not been addressed in a comprehensive way in the physiological literature—the problem of co-activation of opposing muscle groups. Consider, for example, a single joint (for multiple joints see St-Onge and Feldman 2004) in the absence of a net external torque. Control levels may specify a common

threshold angle (r) for all of the muscles spanning the joint. At this position, the muscles will be inactive. If a joint is moved passively from position r, muscles stretched by the motion will be activated whereas the opposing (antagonist) muscles will be activated when the joint is moved passively in the opposite direction. By changing threshold values for the two muscle groups in opposite directions, control levels may surround position r with a zone in which all muscles may be co-active. The absolute changes in the thresholds for these groups need not be identical as long as they do not influence the net (zero) torque at position r (see Feldman 1993). Therefore, in the λ model, the co-activation (c) command is defined in terms of muscle activation thresholds of opposing muscle groups, not in terms of EMG activity levels as typically assumed in electrophysiological studies. Movement to a new position is produced by unidirectional (in the joint space) shifts in the thresholds of the opposing muscle groups (by a change in the r position), so that the coactivation zone is automatically shifted with it, eliminating any resistance to the deviation from the initial position (Levin and Dimov 1997). Re-addressed to a new arm position, muscle co-activation contributes to the speed of transition to this position while increasing the damping of the system and so suppressing terminal oscillations (Feldman and Levin 1995). Specifically, the c command facilitates both agonist and antagonist motoneurons. However, during a fast joint movement, this command initially results only in an increased agonist EMG burst, whereas for antagonist motoneurons, the efficacy of this facilitation is reduced by an inhibitory influence from the simultaneous change in the r command. Only later, when the lengthening antagonist muscles are activated by the stretch reflex action, does the c command act to enhance the antagonist EMG burst. As a result, the c command contributes to both movement acceleration and deceleration, allowing the system to dissipate kinetic energy and reach the final position with minimal terminal oscillations. The role of the c command in increasing the speed of the movement is not unique: the rate of change in the r command also contributes to the movement speed (StOnge et al. 1997). In some subjects with hemiparesis, the spatial-temporal organization of c commands is deficient, complicating movement production in these subjects (Levin and Dimov 1997). A common misconception about the EP hypothesis is that the r command to a joint specifies its equilibrium position while the c command specifies its stiffness. It is more appropriate to say that, because of its threshold nature, command r influences the location of a spatial zone in which muscles can be activated. Non-central components of activation thresholds (ρ and f(t) in Fig. 1) also influence the location of this zone during the movement. There is natural variability in the steady-state properties of these components representing intermuscular interactions and history-dependent properties of neuromuscular elements, respectively. This variability may lead, in particular, to disparities between the final threshold values and r, and these differences may vary over movement

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repetitions. Therefore, even in the absence of external forces, the final equilibrium position may be somewhat different from r and it is definitely different when these forces are present. The c command defines effectiveness of the external load in moving the joint away from position r. As such, the equilibrium position of a joint acting against a non-zero load is influenced by both r and c. The λ model has also offered a potential solution to the problem of motor redundancy (Lestienne et al. 2000; Balasubramaniam and Feldman 2003; St-Onge and Feldman 2004). The EP hypothesis has been invoked in analysis of the stability of the vertical posture (Zatsiorski and Duarte 1999, 2000; Micheau et al. 2003), hand trajectories (Won and Hogan 1995) and in simulation of trajectories, velocity profiles, stiffness, damping, and EMG patterns of arm point-to-point movements (Feldman et al. 1990; Flanagan et al. 1993; Latash 1993, 1994; StOnge et al. 1997; Gribble et al. 1998). Recently, Günther and Ruder (2003) simulated human walking based on the control principles of the λ model, a pioneer contribution that advances the understanding and modeling of human locomotion beyond the traditional, biomechanical framework. Another recent study has demonstrated that the EP control makes internal inverse and forward models unnecessary in the explanation of learning and adaptation of arm movements to different force fields (Gribble and Ostry 1999; 2000). The EP hypothesis has proved to be helpful when explaining sense of effort (Feldman and Latash 1982; Toffin et al. 2003). Muscle and cutaneous afferent signals per se carry ambiguous information about the positions of body segments. For example, during isometric force production, the activity of muscle spindle afferents increases but the relevant body segments are perceived as motionless (Vallbo 1974). According to the λ model, a correct position is identified based on the measurement of afferent signals relative to their referent values related to the control signals (muscle activation thresholds) underlying the specification of posture. Different kinesthetic illusions elicited by vibration of muscles or tendons are explained by an interference of the vibration-induced afferent inflow with the central or/and afferent components of positional sense (Feldman and Latash 1982). The λ model suggests that central resetting of muscle activation thresholds underlies active muscle force, movement and posture regulation (reviewed in Feldman and Levin 1995). It has been hypothesized that a reduction in the range of threshold regulation might be a primary cause of weakness, spasticity, and deficits in inter-joint coordination in some patients with neurological movement disorders. This prediction has been confirmed for subjects with hemiparesis and cerebral palsy (Jobin and Levin 2000; Levin et al. 2000). Those scientists who actively use the EP hypothesis in their research are indeed well aware of repeated claims of its rejection. They are also aware of the existence of an alternative, EMG-force control model, according to which control levels are directly involved in the specification of EMG activity and muscle torques obtained from transfor-

mations of the desired kinematics via hypothetical inverse and forward models of the neuromuscular system interacting with the environment (Hollerbach 1982; Wolpert et al. 1998; for a recent critical review see Ostry and Feldman 2003). Advocates of the EP hypothesis continue their activity despite this awareness. Why? They do so because they are also aware that the claims of rejection of the EP hypothesis are unconvincing (Feldman et al. 1998) and that the alternative, EMG-force control theory is not physiologically feasible since, in particular, it fails to resolve the most basic, posture-movement paradox in motor control, the problem of co-activation, and even conflicts with the empirical relationship between EMG activity and force (Ostry and Feldman 2003). Some claims of rejection of the EP hypothesis have been addressed in the past (Feldman et al. 1998). Recently, however, new claims have emerged, particularly in several publications from the groups of Rymer and Milner (Popescu et al. 2003; Popescu and Rymer 2000; 2003; Hinder and Milner 2003) who state explicitly that the EP hypothesis has been refuted and that the motor control community has no choice but to accept the idea of direct involvement of neural control levels in EMG-force programming and specification. We appreciate criticisms of the EP hypothesis in general and particularly those coming from groups of these prominent scholars. We feel, however, that it is necessary to address these criticisms in depth and to reassess their validity. This process has led us, unfortunately, to the unpleasant situation of pointing at essential problems inherent to these criticisms and conclusions. In particular, it has been emphasized that linear springs are inadequate models of muscles within the EP hypothesis (Feldman 1986; Feldman and Levin 1995). This point has been disregarded by opponents of the EP hypothesis, leading to major confusion over what this hypothesis does and does not predict, and as a consequence to unsubstantiated rejections of the hypothesis. We need once again to clarify the relationship between the EP hypothesis and mass-spring models.

Is the EP hypothesis a “mass-spring” model? Changes in the arm positions of subjects is graded with the amount of unloading (Asatryan and Feldman 1965; Forget and Lamarre 1987); this is analogous to a physical massspring system. The term “spring-like behavior” refers to this case and implies a similarity in behavior between biological and physical systems that otherwise have nothing in common. In some cases, such an analogy is taken literally, when muscles with or without reflexes are modeled as springs with parameters (most often, stiffness and damping) determined by the level of muscle activation (for a recent example, see Popescu et al. 2003). Spring analogies and models have played both positive and negative roles in motor control research. Starting from Weber (1846), other scientists, including Bernstein (1935) applied the spring analogy to muscles and literally modeled muscles as springs. However, Hill (1938) dem-

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onstrated that the mass-spring model misrepresents basic dynamic properties of muscles, including its energy output. When the EP hypothesis was introduced (Asatryan and Feldman 1965; Feldman 1966), the spring analogy was applied not to isolated muscles but to the intact neuromuscular system. This was based on the identification of a system’s neurophysiological parameter—the threshold muscle length, at which alpha motoneurons begin being recruited due to an integration of proprioceptive and central influences on the membrane of the motoneurons (for details, see Feldman 1986). Without such integration (e.g., in humans or animals with lost proprioceptive sensitivity) this parameter does not exist. Drawing analogies between this parameter and changes in spring properties such as stiffness and zero length was helpful in introducing notions that had been absent in the motor control lexicon—the EP concept, the possibility of spring-like behavior (such as responses of the arm to unloading), and equifinality (see also Sternad and Turvey 1995). However, it was never intended that the analogies should be taken literally as universal, established relations. Despite the recognition that equifinality is not an absolute rule (Feldman 1979), the spring analogy helped to predict that the intact system, including muscles, reflexes and the control levels can display equifinality in a range of transient, smooth perturbations. The empirical confirmations of this prediction (Schmidt and McGown 1980; Jaric et al. 1999; Rothwell et al. 1982) should be considered as supporting evidence for the EP hypothesis and added to the list of its successes. Recent findings of violations of equifinality have narrowed the range of conditions in which equifinality can be observed, as in cases of destabilizing force fields (Lackner and DiZio 1994; DiZio and Lackner 1995; Hinder and Milner 2003), but have not eliminated the phenomenon as such. In recent years, a negative aspect of the mass-spring analogy has become obvious. In simulations of movements, the spring analogy has been taken literally to represent muscle properties (Gomi and Kawato 1996; Schweighofer et al. 1998; Bhushan and Shadmehr 1999; Popescu et al. 2003), although it has been noticed (Feldman et al. 1998) that this leads to misrepresentation of some of the basic properties of the neuromuscular system, resulting in questionable estimations of equilibrium trajectories, for example, in studies by Latash and Gottlieb (1991) and Gomi and Kawato (1996). Our analysis of the model by Popescu et al. (2003) that follows is another illustration of how neuromuscular properties can be misrepresented when the spring analogy is taken literally by using an unrealistic second-order model. A departure point in most arguments against the EP hypothesis is the wrong idea that the hypothesis assumes that the neuromuscular system always behaves like springs, with their characteristic property of equifinality, resulting in a basic misconception that equifinality is a fundamental property of the neuromuscular system in the EP hypothesis (see Lackner and DiZio 1994; Hinder and Milner 2003; Popescu and Rymer 2000). The EP hypothesis was founded on the empirical fact that forces

produced by the neuromuscular system are positiondependent, whereas springs were used as an example of a physical system that has an analogous property. In addition, the spring analogy within the λ model referred to the whole neuromuscular system, not to the muscles per se. Not all systems with position-dependent force generators have the property of equifinality. Even physical springs transiently stretched beyond certain limits can lose this property while retaining position-dependent resistance. This shows that, even if the spring analogy is taken literally, as a model of the system, it would be incorrect to assume that equifinality should be considered to be a fundamental property of motor behavior in the EP hypothesis. Experimental observations of violations of equifinality disqualify this assumption but not the hypothesis. As discussed in earlier publications (Feldman 1986; Latash 1993), muscle force-length properties are highly non-linear. Under certain perturbations, non-linearities (in particular, hysteresis) may be revealed in the properties of sarcomers, muscle fibers, tendons, connective tissues, whole muscle structure, motoneurons, and sensory feedback regulating activity of motoneurons. In addition, these properties change with shifts in the threshold of muscle activation; such shifts can result from descending signals as well as from interneurons that are responsible for intermuscle reflex interactions (Fig. 1). It is a gross simplification to refer to this constellation of factors as defining “stiffness” and “viscosity” in spring models of the muscle-reflex system. The concept of threshold control is especially illustrative of why spring models of the musclereflex system have little validity. Theoretically, threshold systems are non-linear and cannot be considered linear even locally, for small changes in variables. Since activation thresholds shift during voluntary movements, the joint ranges at which local linearity is broken may appear anywhere in the biomechanical range, making the neuromuscular system fundamentally non-linear. Therefore, neither physiologically, nor mathematically, even for small changes in variables, can one justify the standard decomposition of muscle forces into two additive positionand velocity-dependent components, as accepted in all mass-spring models. In summary, the spring analogy has been the cause of much confusion, especially when taken literally, resulting in false rejections of the EP hypothesis. Although the term “spring-like behavior” seems adequate in reference to some motor effects (like changes in the arm posture in response to an unloading), the spring model is highly inadequate in reference to many other aspects of motor behavior, especially those involving a fundamentally nonlinear process—threshold control.

The EP hypothesis has no problem with stiffness, but methods of stiffness estimation may have problems In a recent series of papers, the group of Rymer claimed that the EP hypothesis is false, based on their assessments

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of the elbow joint stiffness during voluntary movements. Rymer and his colleagues state that the EP hypothesis requires the joint stiffness provided by the neuromuscular system to be “high” in order to generate the requisite torques based on the difference between the actual and the equilibrium position, and that their finding of “unexpectedly low” elbow stiffness conflicts with this hypothesis (Popescu et al. 2003). High joint stiffness was required in the servo-hypothesis (Merton 1953), which was based on an assumption of a very high gain in the tonic stretch reflex loop, whereas the EP hypothesis, specifically, the λ model, removed this constraint (Feldman 1986). We would like to stress that the EP hypothesis, contrary to the claim by Rymer and his colleagues, has no unique requirements for the values of stiffness and damping. It complies with general rules determining the stability of dynamic systems, to which the neuromuscular system belongs. These rules say, in particular, that the system cannot be stable unless it opposes any deviation from its current posture or movement trajectory by increasing muscle torques depending on deviation and velocity. The coefficients describing this behavior (addressed as stiffness and damping, respectively) should exceed the respective coefficients characterizing the tendency of the external load to move the system away from the posture or trajectory (see Feldman 1986). Indeed, according to these rules, the system cannot be stable if stiffness or/and damping are zero, but in this and other cases instability is usually prevented by an increase in muscle co-contraction, leading to an increase in both stiffness and damping (see Latash 1994; Stokes and Gardner-Morse 2000; Franklin and Milner 2003). Let us emphasize that the criteria of stability described above must be met by any theory of motor control, including the force control model promoted by Rymer’s group. Therefore, contrary to their suggestion, the force control model would not be better off with the low stiffness values these authors reported. The method of stiffness estimation used by Rymer’s group to support the rejection of the EP hypothesis is also questionable. It makes limited sense to assign a complex neuromuscular system a single value of “stiffness” based on its response to a brief perturbation (Latash and Zatsiorsky 1993). Such values, however, can be computed based on a model, but they reflect the functional form of the model, not the actual neuromuscular system. The model used by Popescu et al. (2003) for stiffness assessment is especially illustrative of this point. For example, the joint stiffness in their study remained not only low but practically unchanged from the beginning to the end of the movement, despite the powerful EMG bursts and the substantial growth in the tonic level of coactivation of the agonist and antagonist muscles during the transition of the forearm from the initial to the final position (see Fig. 8 in Popescu et al. 2003). These findings apparently reflect the property of the model but not the real neuromuscular system, for which muscle stiffness generally increases with the level of EMG activity and the net

joint stiffness increases with coactivation of opposing muscle groups. In the model, the torque generated at the elbow joint is a linear combination of position-, velocity-, and acceleration-dependent terms (their Eq. 1), with the net static torque defined in a spring-like manner  ¼ K

(1)

where K is stiffness and Θ is the joint angle (defined as 0° when the forearm is positioned approximately in the middle of the biomechanical range). Negative or positive values of the joint angle represent elbow flexion and extension from this zero angle, respectively. Flexor torque is positive and extensor torque is negative. According to known stability criteria (see Bellman 1960), the stiffness K must be positive (K>0). As a consequence (see Eq. 1), the torque is negative (τ

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