------------------------------------------------------------------ Abbreviations: CNS, central nervous system; CV, control variable; EMG, electromyogram; EP, equilibrium point; IC, invariant characteristic; IN, interneuron; MN, motoneuron. ------------------------------------------------------------------
1. Introduction
The traditional view of motor control suggests that the central nervous system (CNS) plans and executes movement in terms of biomechanical variables (movement trajectory, joint angle, final joint position, movement amplitude, velocity, amount of muscle activation or EMG activity, stiffness, force or torque, etc.). The limitations of this approach are illustrated by the following analogy.
Imagine that aliens came to the earth and tried to understand the mechanism of the behavior of, in their view, its most numerous creatures - cars. First, they quantified the interaction of cars with the environment in terms of speed, displacement, forces, power, tire and road quality, and other directly measurable external variables. The observations of cars sliding on icy roads or accidentally colliding convinced the aliens that cars were unable to directly specify their movement trajectory or traction force to negotiate the environment. After years of such empirical studies they concluded that the explanation of the behavior in terms of external variables represented the description of effect without the indication of the cause. Looking inside the car, one alien discovered some levers but it took him some time before he could conclude that they might be the key to the solution of the problem. Finally, he deduced that the active behavior of cars occurred due to the independent specification of the levers' positions by an intrinsic part of the car, the driver. Unlike aliens, we, neuroscientists, are more interested in finding control variables (CVs) underlying the active motor behavior of the driver. For the most part, research has predominantly focused on analyzing active movements in terms of external variables without looking more deeply to reveal the independent "levers" which are used by the CNS to produce these movements. Even though we can directly record neuronal activity in the CNS of vertebrates during different motor tasks, do such studies provide answers to the question regarding the nature of CVs which produce active movement? An intensive discussion of this problem in several target articles has been confounded by a lack of a consistent definition and experimental verification of CVs (Berkinblit et al. 1986; Bizzi et al. 1992; Gottlieb et al. 1989; Stein 1982). A distinctive definition of CVs (also called central commands) has been developed in the framework of the lambda model (Feldman 1966 a; Feldman & Levin 1993; Houk & Rymer 1981; Latash 1993) and will be discussed in detail in this target article. We begin by suggesting that CVs, first of all, characterize the ability of the CNS to produce voluntary actions and are defined as follows:
(1) CVs are specified by the nervous system broadly independently of current external conditions and biomechanical variables. The terms "external conditions" and the "environment" are used synonymously and are associated with external forces (including inertial, reactive and gravitational) acting on the body as well as their time and space characteristics. The term "independent" implies that the CNS may specify a constant value or a particular timing of a CV and maintain it regardless of the movement and of changes in external conditions or/and afferent feedback from muscle proprioceptors. On the other hand, the system has the option whether or not to use this information to change the control pattern. As usual, the system's freedom of choice is limited. In other words, the system may be forced to change CVs under specific external conditions. For example, if somebody hits us on the back, some changes in CVs may be triggered to prevent falling. However, even in such cases, the possibility of varying the triggered control pattern still remains. In the above example, we can take a step forward, turn, fight back or run away. Once the decision to change the control pattern has been made, the new pattern may, again, be performed independently of the current afferent feedback.
(2) CVs may influence biomechanical variables. The latter are non-CVs since they are dependent on external conditions and thus do not meet criterion (1). For example, when we squeeze a cylinder, the length of hand and finger muscles remains the same regardless of our efforts. The dependence of muscle length and other kinematic variables on external conditions categorizes them as non-CVs (see also Section 2).
Note that the notion of independent and dependent variables in mathematics is conditional and as such these variables may usually be rearranged. However, rearranging variables associated with active motor behavior may not be appropriate since this may be equivalent to a rearrangement of cause and effect. Thus, the definition of CVs implies a functional hierarchy in motor regulation. We can conceptualize two functional levels: control and subordinate (previously called executive, see Feldman & Levin 1993). The first level specifies CVs whereas the second continuously regulates biomechanical variables and other non-CVs as a function of CVs, afferent feedback and external forces. This functional hierarchy may not correspond to an anatomical hierarchy in the CNS. In principle, the same CNS structures may contribute to the generation of CVs and non-CVs.
Now we turn our attention to another basic concept of the lambda model. In a recent interview Isaac Stern revealed the secret of his performance saying that a musician "should play music in a frame of reference". This expression may touch upon the strings of motor control specialists who want to know how the CNS generates frames of reference within which to specify coordinates for movements which are appropriately oriented in space. The problem of the frame of reference is closely related to the CV concept (Section 3). It has also attracted much attention in numerous theoretical and experimental studies especially since Bernstein's work (Bernstein 1967; Bloedel 1992; Gielen & van Zuylen 1986; Paillard 1991; Pellionisz & Llin s 1982; Pellionisz 1985; Soechting & Flanders 1992). We will illustrate, in physiological terms, the central idea of the lambda model that the CNS organizes positional frames of reference or systems of coordinates for the motor apparatus and produces active movement by shifting the frames in space (Feldman & Levin 1993). MN threshold properties and proprioceptive feedback may be cardinal components of the mechanism which defines such frames of reference (Section 3). Specifically, these components make MN recruitment dependent on muscle length when the latter exceeds a threshold length, lambda. Parameter lambda is, thus, the point of origin for the positional frame of reference for the generation of active muscle force. By modifying lambda, the CNS specifies a new origin point of the positional frame of reference for force generating and neuronal components of the sensorimotor system. In this way, the system is forced to find a steady state (see Sections 10 and 11) which results in a new equilibrium body configuration in the new frame of reference. The hypothesis that intentional movements are produced by shifting the frame of reference will be extended to multi-muscle and multi-degrees of freedom systems (Sections 7 & 11.3).
The lambda model also implies that, depending on external conditions, the same control process may be associated with different patterns of muscle activation and kinematics. These patterns are thus not programmed but emerge from the dynamic interaction of the system's components and external forces within the designated frame of reference. They may be a manifestation of the system's tendency to reach an equilibrium or steady state. The notion of steady states is the one defined in physics, particularly, the theory of dissipative systems in which irreversible processes play a central role (Andronov et al. 1959; Glansdorff & Prigogine 1971; Kugler & Turvey 1988; Saltzman & Kelso 1987). Like living systems, such systems are open and exchange energy and matter with the outside world to establish a macroscopic internal order. The fact that the system is open means that the environment may have an impact on the steady state of the system that is no less important than that of CVs. The equilibrium point (EP) concept is just one of the descriptors of the steady state emphasizing this constraint. For a single joint, the EP is a combination of the equilibrium joint position and the net muscle torque at this position. If an EP exists, it is defined by time-independent parameters (constants) of the system provided that the current values of CVs are fixed ("frozen"). Since, in the equilibrium state, the net muscle torque equals the external (e.g., gravitational) torque, the EP characterizes the interaction of the joint with the environment and thus emphasizes that they act as equal partners in movement. The initiative to modify this interaction by changing CVs voluntarily rests with the organism. Since the EP depends on external conditions it is not a CV (see also Section 10). However, it is an essential dynamic variable responsible for movement production.
The lambda model, originally published in Russian and later translated into English, has been popularized in the West. The idea of movement production via EP shifts, isolated from the other concepts of the lambda model, was used by Bizzi et al. (1984) in their version of the EP hypothesis. By considering the level of muscle activation as a variable underlying shifts in the system's EP, cause and effect in movement production were rearranged and thus the original formulation of the EP concept was misinterpreted (For further discussion see Feldman 1986, 1992). More important, the principles of sensorimotor integration underlying the lambda model were not comprehended by Bizzi's group (in contrast, see Houk & Rymer 1981). Criticism directed at the EP hypothesis did not, until recently, distinguish between the two versions. The tradition of confusion has been compounded by the recent target article by Bizzi et al. (1992) in which they trivialized the lambda model by making it quite similar and even subordinate to their model (see also McIntyre & Bizzi 1993). We feel that it is time to summarize the lambda model in order to discuss its ideas, possibilities and limitations without the confusions accumulated since its original formulation. It is quite natural in science to look for alternatives to existing views and hypotheses. However, attempts to "disprove" the lambda model without actually understanding it (e.g., Agarwal 1992; Bizzi et al. 1992; Bock & Arnold 1993; Burgess 1992) hardly seems to be a productive approach.
Insert Figure 1 near here
2. Pseudo-control models.
Our approach to CVs will be both theoretical and experimental. We proceed with a discussion of models which try to explain movement production in terms of variables which we consider to be non-CVs. We therefore call these models pseudo- control models. In all examples the system is presumed to be intact.
2.1. Torque, force, stiffness, kinematic and inverse-dynamic models. Suppose we lift a weight from position a to b as shown in Fig. 1 A. At position a, we create a greater muscle torque and begin to lift the weight. When the weight approaches position b, we decrease the muscle torque and decelerate the movement until it stops in the desired position, b. From a physiological point of view, the description of this behavior in terms of classical mechanics disregards a specific problem. Note that the net muscle torque (equalling the gravitational torque of the arm) is the same in the two static positions in Fig. 1 A, i.e. the two states of the system are indistinguishable in terms of muscle torques. However, clearly, the nervous system discriminates between them: after a short external perturbation, say, in position b, the arm returns to the same position but not to position a. What variable(s), then, has predetermined position a in one case and position b in the other? The question cannot be answered in terms of torque control models. Similarly, force control models (e.g., Bock 1990; Bock & Arnold 1993; Meyer et al. 1982; Schmidt et al. 1979) are unable to explain how the CNS chooses between different positions under isotonic conditions. Models treating muscle stiffness as a control variable have similar problems since stiffness is related to muscle force (e.g., Feldman & Orlovsky 1972) and may also be the same in different positions (for a definition of stiffness see Section 4).
Hollerbach & Atkeson (1987) and Soechting & Terzuolo (1986) suggest that the nervous system chooses desired movement kinematics such as effector trajectory, and then computes the muscle torques and muscle activations (inverse dynamics) necessary to implement the program. Supporting this idea is the observation that some kinematic relationships are relatively stable in spite of considerable changes in external conditions, endpoint effectors, and EMG patterns (Bernstein 1967; Fitts 1954; Hogan 1984; Lacquaniti 1989; Soechting & Terzuolo 1986; Viviani & Terzuolo 1982). However, kinematic invariants may be an inherent property of some dynamic systems rather than a programmed property (Bullock & Grossberg 1988; Glansdorff & Prigogine 1971; Kugler & Turvey 1988; Latash 1993; Schner et al. 1990; van Emmerik 1992; see also Section 11). The idea of inverse computations originated from robotics for which it may be practical. It has also successfully been used to improve the description of complex, multi-segmental movements in living systems in terms of mechanics. The idea, however, may be misleading when applied to control processes in such systems (see Hasan 1991 and Loeb 1989). In essence, such models are a variety of torque or force "control" models which, as exemplified in Fig. 1 A, add little to the understanding of how the nervous system chooses between different positions.
2.2. Electromyographic models. The amount of muscle activation or EMG activity as well as variables dependent on it (e.g., muscle force and stiffness) are also not CVs since they are affected by segmental reflexes (Doemeges & Rack 1992; Ghez et al. 1990; Granit 1970; Hoffer & Andreassen 1981; Houk & Rymer 1981) and, consequently, depend on kinematic variables. In addition, if EMG activity were a CV, it could be regulated independently of static muscle force (a non-CV). This assumption obviously conflicts with the well-known monotonic EMG/force relationship described for some muscles (Bilodeau et al. 1990; Jongen et al. 1989; Milner- Brown & Stein 1975). Consequently, depending on the external force, the EMG level in one static position of the joint may be greater than, less than or the same as that in another static position. The case in which the level of tonic EMG activity may be the same in two different positions ("isoelectric" conditions; e.g., for arm positions b and c in Fig. 1 B; see also Feldman 1986) is of specific interest: it shows that the level of muscle activation is not the primary variable which permits the CNS to choose between the two positions.
Let us try to explain such a process in basic neurophysiological terms. Supposing that there is a tonic descending facilitation of ` and/or gamma MNs so that the arm is stabilized at position b with a specific level of muscle activation (Fig. 1 C). To change this position, the CNS presumably specifies a new level of descending facilitation of ` MNs (Fig. 1 D). For simplicity, the descending inflow to ~ MNs and interneurons (INs) remains the same. Is there any physiological variable which may be used as a measure of the facilitation of MNs? At first glance, the resulting increase in MN output or EMG activity can be such a measure. This could satisfy proponents of central programming of EMG patterns. Consider, however, the behavior of the system in more detail. The descending facilitation gives rise to an increase in muscle activation and force resulting in muscle shortening. Because of properties of afferent feedback mediated by muscle afferents and segmental INs, the muscle shortening gives rise to a decrease in muscle activation. Thus, the surplus excitation of ` MNs introduced by the descending facilitation will finally be nullified at a new joint position and the movement will cease. Now compare the initial and the final states of the system (Fig. 1 C and D). MN activity is the same in both positions. The only difference is that the tonic level of descending inputs to the MNs is greater for the final position (c) associated with a shorter muscle length. From this example, it is obvious that the CVs which affect the equilibrium position are defined by the descending control influences to MNs regardless of the level of MN output (the amount of muscle activation).
Note that Sherrington's metaphor that the MN is the "final common path" (Sherrington 1906/1947) is usually interpreted to mean that all neuronal influences on MNs can be measured in terms of their activation. The above discussion rather favors the Bernsteinian view that descending control influences to MNs cannot be measured in terms of muscle activation (Bernstein 1935).
Another class of models is based on experiments in deafferented and curarized preparations in which actual or fictive movements including locomotion have been observed. This has led to the concept of the central pattern generator (Brown 1914; Grillner 1975; Orlovsky & Shik 1976; Pearson 1985; Rossignol et al. 1988; Stein 1984; Taub et al. 1975). Taking into account the substantial motor deficits in deafferented subjects (Blouin et al. 1993; Ghez et al. 1990), the concept can hardly be interpreted in the sense that movements basically arise from central pattern generators with afferent feedback playing only a modulatory role. However, some models are intrinsically based on such an interpretation. In these models, MNs are considered linear summators such that muscle activation (`) would consist of two additive components: a reflex component (`r) which is dependent on proprioceptive feedback and a central one (`c) which is independent of it:
` = `r + `c (1)
Eq. (1) shows that MNs can be activated in the absence of reflexes as observed in deafferentation experiments. The results of such experiments can hardly be generalized to the intact system if one takes into account the immediate and long-term consequences of deafferentation such as sprouting and synaptic plasticity (Goldberger & Murray 1974; Hellgren & Kellerth 1989; Kaas 1991). Eq. (1) is obviously inconsistent with some properties of the intact system. It is known that sudden unloading produces a silent period in the tonic MN firing of a normally innervated muscle (e.g., Angel 1973; Feldman 1986; Forget & Lamarre 1987; Gerilovsky et al. 1990; Hugon et al. 1982). According to Eq. (1), unloading should only lead to the disappearance of the reflex component `r but experiments show that all activity (`) is suppressed. The explanation may be simple: since MNs are threshold elements, one input may become subthreshold for producing MN firing if the other input is withdrawn. Thus, although the idea of decomposing the output activity of MNs into reflex and central components underlies several motor control models (e.g., Bizzi et al. 1992; Bullock & Grossberg 1988; Gottlieb et al. 1989; McIntyre & Bizzi 1993; Loeb & Levine 1990; Zajac & Gordon 1989), it may misrepresent the process of sensorimotor integration at the level of MNs.
A plausible description of higher brain functions and motor production may be obtained in neuronal network models (Alexander et al. 1992; Bullock & Grossberg 1988; Douglas & Martin 1991; Fetz 1992; Grossberg & Kuperstein 1989). The problem of CVs has still to be resolved in the framework of such models. Those treating motor control in terms of the specification of kinematic variables or EMG activity seem less promising.
3. Control variables and positional frames of reference for
motoneurons
We can interpret the CV concept physiologically by considering the effects of a muscle stretch or a voluntary movement on the MN membrane potential. Some changes may be produced monosynaptically by muscle afferents or polysynaptically by spinal and supraspinal neurones participating in reflex loops. Whatever the specific details for different muscles, we assume that there is an increase in MN depolarization during passive quasi-static muscle lengthening. We also assume that there is a component of change in the membrane potential, kVc, produced by descending systems which may be independent of muscle afferent feedback. This component is thus considered as a CV (see footnote 1).
Insert Figure 2 near here
Interestingly, this electrical measure may be represented in terms of a variable having positional dimensions. In Fig. 2 (left panel), V is the initial membrane potential of the MN (vertical axis) at the initial muscle length, x, when descending control signals are fixed. Because of proprioceptive feedback, the state of the MN depends on muscle length (horizontal axis). A quasi- static stretch of the muscle results in an increasing depolarization of the MN as a function of x (diagonal line) because of predominant facilitation from length sensitive afferents (i.e. muscle spindle afferents in most muscles). The slope of this line represents the positional sensitivity of the stretch reflex at the subthreshold level. The threshold membrane potential, V+, and consequently, the recruitment of the MN, will be reached at a muscle length lambda. The effect of a change in the tonic control signal (Fig. 2, right panel) can be measured by a decrement (kVc) in the membrane potential at the initial muscle length. Now, muscle stretch will result in MN recruitment at a shorter muscle length. Thus the same control signal is expressed as a decrement (klambda) of the threshold muscle length at which the MN is recruited. This decrement is independent of the actual muscle length, x and thus is a CV.
The above considerations illustrate that MN threshold properties and proprioceptive feedback may be essential components of the mechanism which relates basic electrical parameters of the MN to space variables so that the current and threshold membrane potentials are associated with the current and threshold muscle lengths, respectively. As a result, MN functioning becomes associated with external space. Not the actual muscle length but the difference between the actual and threshold length is essential for MN recruitment. By modifying lambda, the control level "tells" the MN in which part of the physiological range of muscle length to work in order to compensate the load.
These theoretical results may be understood in terms of physics. Movement of a body is defined as a change in its position with respect to another object, frame of reference or system of coordinates. Inherent to the concept of the frame of reference is Galileo's principle of relativity of motion: Movement can be produced by shifting the frame of reference. The threshold lambda may be considered as the origin point of the frame of reference for positional recruitment of MNs. By modifying lambdas, the control level specifies a new reference point for positional recruitment of MNs (Fig. 2, right panel) and thus produces movement (Sections 10, 11). Thus shifts in the positional frame of reference may underlie movement control whereas MN activation and force production may be a consequence of this process. This notion is fundamental in the lambda model and may be applied to multi-muscle and multi-degrees-of-freedom control (see Sections 7, 11.3).
We continue to discuss some possible physiological mechanisms of lambda regulation. In the model, any type of length sensitive afferent feedback is sufficient to provide a basis for the regulation of lambda via independent inputs to ` motoneurons. However, gamma innervation of muscle spindles may be an important additional source of such regulation. It is known that for skeletal muscles there are three types of MNs: `, gamma and beta MNs. The first and the third innervate extra- and intra-fusal muscle fibres, respectively, whereas the second supplies both types of fibres (Boyd et al. 1977; Matthews 1981). MNs innervating intrafusal fibres are subdivided into static and dynamic MNs affecting position and velocity gains of muscle spindle afferents, respectively. Muscle afferents may provide feedback not only to ` MNs but also to static and dynamic gamma MNs (Appelberg et al. 1986; Grillner 1969; Wadell et al. 1991). Control inputs to ` and gamma MNs can be coordinated in different ways depending on the motor task (Prochaska et al. 1985). In the lambda model, the decrement klambda results from both direct influences of the control signals on the ` MN and indirect influences mediated by active static a and gamma MNs, muscle spindle afferents, and INs. The direct and indirect components of the CV are denoted by k lambda(d) and k lambda(i). We assume that they are additive in terms of changes in lambda (Fig. 2, right panel):
k lambda = k lambda(d) + k lambda(i) (2)
Both the inputs produce a broad range of shifts in the positional frame of reference for MN recruitment. As a consequence, the range of regulation of actual muscle length and force is also maximized (Feldman 1986).
It is presumed that the lambda is affected by a change in the level of presynaptic inhibition of ` MNs by descending systems combined with opposite changes in the level of postsynaptic facilitation. It has been shown experimentally that the reflex threshold may be diminished by stretching the antagonist muscle or stimulating its nerve (Feldman 1979; Feldman & Orlovsky 1972; Matthews 1959; Nichols 1989; see Fig. 5 C). Thus some changes in lambda may result from reflex intermuscular interactions (Feldman 1986; Feldman et al. 1990) mediated, in particular, by Ia reciprocal inhibitory interneurons (McCrea 1992). As a consequence, only the independent change in lambda (klambda) but not lambda itself, may be considered as a CV. For a more detailed description of how reflex intermuscular interaction is represented in the lambda model see Feldman 1992 and Section 8.
Insert Figure 3 near here
The properties of the lambda model were addressed in early experiments on humans (Asatryan & Feldman 1965; Feldman 1966 a,b; 1979). Since CVs are independent variables, they can be held constant during postural control tasks in spite of perturbations in the external load which result in changes in joint position, force and EMG signals. In these experiments, subjects specified the same initial elbow joint angle against a load opposing flexion created by a weight attached to a horizontal forearm manipulandum (Fig. 3, filled dots). Thus, they specified the same combination of initial torque and position (EP) in each trial. With vision blocked, part of the load was suddenly removed by an electromagnetic device. The change in the load was varied from trial to trial. The subject were instructed "not to intervene" or "not to correct the arm deflections". Position, torques and EMG activity of elbow flexor and extensor muscles were recorded. Results of such experiments revealed a monotonic relationship between the final (static) values of muscle torque and joint angle (termed the torque/angle characteristic). The tonic level of EMG signals was not constant but was related to torque (Fig. 3). This implied that the spring-like behavior of the joint was mediated not by active muscles per se (as is the case in other models, e.g., Hogan 1984) but by the whole system, possibly including supraspinal structures (see Adamovich 1992 for further discussion). Similar behavior has also been observed in more recent work (Davis & Kelso 1982; Gielen et al. 1984; Gottlieb & Agarwal 1988; Latash 1993; Levin et al. 1992; Vincken et al. 1983).
Fig. 3 shows that when the subject changed the initial combination of torque and position, unloading revealed a different torque/angle characteristic. Although the characteristics were similar in shape, they diverged from different points of the passive torque/angle characteristic of the joint (Fig. 3, dashed line). The divergence point of the active characteristic corresponded to the threshold length, lambda, for MN recruitment (Feldman 1986; Feldman & Levin 1993). Note that when the subject changed the initial combination of the torque and angle, he made an active movement. In other words, he produced an action which could only have been made by a change in CVs. The fact that the lambdas were different for different ICs indicates that klambda is a CV.
Insert Figure 4 near here
Three additional experiments have been used to test the hypothesis that this behavior is associated with invariant values of CVs. Two such tests were concerned with the magnitude of the load and the timing of loading. In such experiments, the magnitude of the load depended on the arm position. The coefficient of this dependency is called the positional load gradient. It was shown that the muscle torque/angle characteristic was insensitive to changes of the gradient of the load (from positive to negative values) and to the timing of unloading (single or double steps; Fig. 4 A-D; Feldman 1979). The reversibility of these effects was tested by re-introducing the load. If the reloading was gradual, the arm returned to the initial position (equifinality shown by position traces in Fig. 4 E, F). These three properties suggested that the subject did not change CVs during the experiments and therefore the torque/angle characteristic was called the invariant characteristic (IC).
The unloading-reloading test has also been used to demonstrate that specific load perturbations may break equifinality regardless of instruction (Feldman 1979). When, in such a test, reloading was abrupt, the arm undershot the initial position (Fig. 4 G). Similar findings of the absence of equifinality in perturbation experiments have led some authors to conclude that the IC concept was not valid (e.g., Gottlieb & Agarwal 1988). However, rather than negating the EP hypothesis, these findings may merely reveal that there are limits to the subject's ability to hold CVs constant despite the instruction not to intervene. A similar idea has been suggested by Kay et al. (1991) and van Emmerik (1990) who reported that external perturbations during rhythmical arm movement and handwriting, respectively, resulted in phase resetting. Interestingly, Lackner and DiZio (1993) observed reaching errors (inequifinality) when subjects produced arm movements in a rotating room. They associated these errors with the action of Coriolis force. This force is zero once an arm movement ends and, according to the EP hypothesis, should not affect the end arm position. The errors, however, may be associated with a change in perception of the target position in the rotating room rather than with Coriolis or centrifugal forces. For example, during rotation of the body, the head tends to passively rotate in the opposite direction leading to a change in its position relative to the body. This may result in a gaze shift and, as a result, in the incorrect specification of CVs.
The absence of equifinality after deafferentation (Day & Marsden 1982) may be predicted by the lambda model which assigns a fundamental role to proprioception in the normal functioning of the motor system. In particular, deafferented unlike normally innervated muscles display a significant hysteresis (Nichols & Houk 1976) which may lead to non-equifinality of movement regardless of a specific control strategy.
Insert Figure 5 near here
Additional support for the lambda model comes from experiments in decerebrated cats. It was shown that lambda can be modified by a change in the activity of ~ MNs by selective anaesthesia of their axons (Feldman 1979; Feldman & Orlovsky 1972; Matthews 1959). Feldman and Orlovsky (1972) tonically stimulated different descending systems at the level of the brainstem (Fig. 5), imitating independent descending influences on ` and ~ MNs. Under these background conditions, ankle extensors were then stretched at a slow velocity and EMG signals, muscle force and length were recorded. At a given level of tonic descending activation, muscle EMG activity arose at a specific threshold length lambda and increased with muscle stretch. Increasing tonic stimulation of the Deiter's nucleus (Fig. 5 A) gave rise to a decrease in the lambda resulting in a shift of the static force/length characteristic to the left. On the other hand, stimulation of the pyramidal tract (Fig. 5 B) or medial reticular formation (not shown) had the opposite effect. Nichols & Steeves (1986) obtained similar results observing, in addition, an independent change in the slope of force/length characteristics during stimulation of the red nucleus.
Indeed, these stimulation experiments are artificial. It is not obvious that in the intact nervous system, descending signals are independent of peripheral ones. Supraspinal structures including the motor cortex and cerebellum may participate, possibly via transcortical loops (e.g., Goodin et al. 1990), in positional recruitment of MNs and therefore in the generation of force/length characteristics. Nevertheless, it is reasonable to assume that at least some component of these descending signals may be independent in intact subjects and may influence the threshold lambda. The results of these experiments are consistent with this hypothesis.
4. Positional reference frame for a single motoneuronal pool. Static activation area
How are the positional frames of reference of different MNs related? Since MNs are recruited according to the size-principle (Henneman 1981), there should be a strong correlation between the two geometrical factors (the size of a MN and its threshold length) with the natural conclusion that the smallest-sized MNs should be recruited at the lowest threshold lengths. Under static conditions, MNs are active in the muscle activation area defined by the inequality
x - lambda r 0 (3)
where x is the actual muscle length and lambda is the threshold muscle length (Fig. 2 A). In the suprathreshold area, muscle activation (`), i.e., some measure of the number of active MNs and their firing rates, is an increasing function of x - lambda. Each MN has its own threshold, lambda(i), for recruitment: lambda(1), lambda(2),..., lambda(n) where lambda(1) = lambda is the threshold of the first recruited MN, superscripts denote the rank of MNs, and n is the total number of MNs in the pool. We assume that CVs affect thresholds of all MNs in a single MN pool simultaneously. In the simplest case this regulation may be monoparametric such that changes in all thresholds are a single- valued function of changes in lambda. It is assumed that, in the suprathreshold area, MN firing increases as a function of the difference between the actual and threshold muscle lengths. The input facilitation may bring the MN to a state of stable depolarization (plateau potential; see Gutman 1991; Hounsgaard et al. 1988) which may be associated with a secondary range of MN firing (Granit et al. 1966).
In Feldman et al. (1990), it was argued that the monoparametric hypothesis is consistent with the reversal of the recruitment order of MNs (e.g., Smith et al. 1980). However, other data show that a single muscle may have compartments subserved by independent MN pools (Denier van der Gon et al. 1991; Loeb 1985; Tax et al. 1989; Tax & Gielen 1993). The principle of rank-ordered recruitment of MNs is presumably valid for each compartment. These compartments may behave as different muscles which are recruited in an order specific to the motor task.
Two gradients or gains of MN recruitment and firing can be distinguished - subordinate gain (partial derivative of ` with respect to x) when movement velocity is zero and CVs are constant and control gain (- partial derivative of ` with respect to lambda) when x is constant. These gains characterize the system's sensitivity to external perturbations and control signals, respectively. Replacing ` with force or torque in the expression for the subordinate gain we receive the definition of muscle stiffness. For the measurement of stiffness, it is essential that the change in force over the change in position is measured under static conditions and CVs are held constant. Hitherto, these principles have not always been adhered to leading to confusing and contradictory data on stiffness regulation. Stiffness can also be defined as the slope of the IC at the point specified by the current values of x and muscle force. As a result, stiffness can be regulated as a function of lambda and these biomechanical variables. It is thus not a CV although numerous studies have identified it as one (for recent discussions on stiffness or gain control, see Bennett et al. 1992; Capaday & Stein 1986; Feldman 1992; Hasan 1992; Hore et al. 1990; Kernell & Hultborn 1990; Latash 1993; Neilson & Neilson 1992; Nichols 1992; Prochazka 1989; Smeets 1992; Stein 1992). The possibility exists that the CNS may provide different values of stiffness at a given operating point defined by the muscle force and length. This would mean that as yet undefined CVs, independent of lambda, may also influence muscle stiffness. We leave this possibility open for discussion.
Attempts to dismiss the lambda model are sometimes based on the belief that the gain of the stretch reflex is too low to provide postural stability and control (Bizzi et al. 1992; Burgess 1992). This view may have developed from Merton's hypothesis (1953) which postulated that the positional gain was high in order to minimize the influence of the load on position. In contrast, regardless of its gain, the stretch reflex in the lambda model forms part of the subordinate level which brings the system to a new EP no matter how much it is deflected from the initial one (cf. Houk & Rymer 1981). For this task, a moderate positional gain may be sufficient (Feldman 1986). A servo-control expert may say that this mechanism is imperfect because it functions with a "positional error". However, this "imperfection" may be valuable when considered in a broader biological context. When we stumble over an obstacle, we may not fall if the system finds a new equilibrium configuration of the body. No doubt, trauma may result if the system's efforts were directed to restoring the initial body configuration. The control level may initiate a transition to the initial body configuration when the emergency is over. The idea that, in the lambda model, movement results from a shift in the positional frame of reference does not permit it to be reduced to a servo-control model (for review of servo-control models see Houk & Rymer 1981; Matthews 1972).
Consider some testable predictions of the model. Inequality (3) indicates that proprioceptive feedback to MNs may, in intact systems, be a necessary condition for maintained (tonic) discharges of MNs associated with a steady posture. Thus, the model predicts that even at maximal background levels of muscle contraction, presumably mediated by strong descending inputs to MNs, tonic MN activity can be silenced briefly by unloading since unloading temporarily interrupts this feedback. This result would be an additional argument against the idea that EMG is programmed by the CNS.
Although the lambda model is based on experimental data from limb muscles, the same hypothetical principles of sensorimotor integration may be applied to other skeletal muscles such as those of the neck and jaw. In other words, we can hypothesize that using the same "do not intervene" paradigm, tonic and phasic stretch reflexes, as well as the presence and the control of the threshold lambda can be demonstrated in these muscles. It has been suggested that, at least in the cat, proprioceptive feedback to MNs of neck muscles is weak or even absent (Richmond & Loeb 1992). However, this hypothesis is based on the effects of electrical stimulation of muscle nerves which may be an inadequate method for the investigation of stretch reflexes in an intact system.
The possibility also exists that the lambda model can be extended to extraocular muscles in spite of the absence of muscle spindles in some species (e.g. cats and dogs; spindles are present in the extraocular muscles of the pig, goat, giraffe, white-tailed gnu, chimpanzee and man; Matthews 1972). Although the firing frequency of MNs of extraocular muscles is (e.g. in the cat) insensitive to muscle length variations (Robinson 1989), a positional threshold mechanism for MN recruitment based on proprioceptive feedback with or without muscle spindles, may exist in this system. The possible application of the lambda model to the oculomotor system is worth investigating.
5. Control variables for muscles of a single joint (R and C
commands).
The reciprocal (R) and the coactivation (C) commands define two types of shifts in the positional frames of reference for agonist and antagonist muscles by changes in lambdas. These definitions offer a more specific representation of R and C commands than previously (Feldman 1980 a,b). Before considering how these commands may act in multi-joint systems, we will review how they may behave at a single joint.
Insert Figure 6 near here
R and C commands have two functionally different effects in terms of shifts in ICs illustrated in Fig. 6. In the absence of the C command, the torque/angle ICs of agonist and antagonist muscle groups are presumed to be joined at their threshold points (i.e., they have a common threshold angle). In the rest of this paper, the symbol R will be used not only as an abbreviation for the reciprocal command, but also to denote the common threshold angle for all muscles of the joint. The correspondence between threshold angles remains during any change in the R command (Fig. 6 A). Neurophysiologically, this change would be associated with reciprocal control inputs to agonist and antagonist MNs (hence the term "reciprocal command"). A change in the R command gives rise to a shift in the EP of the joint. The EP is a combination of the load-dependent net joint torque and position at equilibrium. Therefore, the R command is not simply a positional control variable. For example, the same change in the R command will produce a new joint angle in isotonic conditions (Fig. 6 A) but a new joint torque in isometric conditions.
Unlike the R command, at least in statics, the C command can be characterized by the width of the angular range in which agonist and antagonist muscles are simultaneously active, i.e., in this range, individual muscle torques and the net joint stiffness increase with the strength of the C command (Fig. 6 B). In dynamics, stiffness also depends on the C command. However, this relationship is not single-valued since stiffness like muscle activation also depends on the R command and changing proprioceptive feedback from reciprocal muscle groups. By increasing muscle activity, the C command may also affect the damping of the system (DeSerres & Milner 1991; Feldman 1979; Hogan 1984; Latash 1993). The C command may be negative and provide a zone of inactivity (relaxation) of the same muscles. The C command does not affect the EP of the joint (Fig. 6 B) and consequently cannot be used in isolation to produce movement. However, this command may effectively contribute to the net joint torque when used in combination with the R command (Fig. 6 C). Thus, one may assume that a given R command may be combined with increasingly larger C commands to produce faster movement. In principle, these commands may proceed in parallel, except that the R command furnishes the final equilibrium position which must be maintained whereas the C command is necessary only during the movement, after which it may be gradually reduced (see Section 11.1). It has also been assumed that any movement is produced by a weighted combination of R and C commands (Feldman 1980 a,b). Experimental data supporting the existence of separate neuronal systems for the reciprocal activation and coactivation of antagonist muscles have been demonstrated from cortical recordings in monkeys (Humphrey & Reed 1983) and from studies in man and animals (DeLuca & Mambrito 1987; Drew & Rossignol 1987; Feldman 1980 a,b; Lacquaniti 1992; Lacquaniti et al. 1991; Levin et al. 1992).
6. Kinesthesia: sensorimotor integration
A true integrated model of motor control should also address the aspect of kinesthesis. The lambda model permits us to make an initial step in this direction. Perception of position is likely based on signals from muscle, joint and skin afferents (Gandevia & Burke 1992). Integration of afferent signals has been described at the level of neurons of ascending tracts (e.g., ventral spinocerebellar tract, Arshavsky et al. 1985; Lundberg 1975; McCrea 1992) which may convey positional information to supraspinal levels. However, this information may not be unambiguously represented in afferent signals. For example, during isotonic muscle contractions, spindle afferents may fire at the same frequency regardless of position (Hulliger et al. 1982). It is possible that the decrease in the spindle afferent firing rate normally resulting from muscle shortening is compensated for by an increase in the activity of ~ efferents. Although the activity of proprioceptive afferents remains unchanged, the subject may perceive an alteration in position. On the other hand, during isometric contractions, the activity of muscle spindle afferents increases with increasing torque under the influence of ~ efferents (Vallbo 1974) although the arm is correctly perceived as being motionless. These and other examples have led to the hypothesis that central commands in some way participate in the perception of position (Feldman 1992; Feldman & Latash 1982; Gandevia & Burke 1992; Holst & Mittelstaedt 1950; McCloskey 1978).
In the lambda model, since CVs have positional dimensions, they may indeed be involved in position sense. In this way, the control level becomes an active participant of the sensory process. Two different schemas can be suggested. In the direct perception schema, adequate position sense results from an integration of control and proprioceptive signals (cf. Gandevia & Burke 1992; McCloskey 1978). In the inverted schema, the brain is the primary generator of position sense (Melzack 1989) and it uses control and proprioceptive signals to correlate perception with reality. Both hypotheses are based on the properties of the lambda model illustrated for a single joint in Fig. 6 (lower part). The figure shows that equilibrium position i may be represented by two additive components: central (to position R) and afferent (the distance between R and i). This distance is a function of the net torque at equilibrium and we denote it by psi(T). Thus,
i = R + psi(T) (4)
For given R and C commands, there is a single-valued relationship (IC) between position i and muscle forces or torques. Consequently, both position and force sensitive afferents can provide information about the relative distance, psi(T). From this point of view, it is not surprising that afferent signals from receptors of different modalities may converge on the same spinal INs projecting to MNs (Jankowska 1992). Central reafference from the R command and peripheral afferent signals may converge at the kinesthetic level. We assume that slow isotonic flexor movement is produced by a slow decrease in parameter R whereas afferent activity will be approximately the same (since T is about the same in isotonic contractions). According to Eq. (4), flexion of the joint will be associated with a decrease in the R command and will be perceived despite a constant level of afferent signals. For comparison, when movement is prevented, isometric flexor torque exertions may also be produced by a decrease in parameter R resulting in a concomitant increase in afferent signals (due to the increase in torque). As a result, the two components of position sense will be balanced and the joint will be perceived as motionless.
The hypothesis that CVs are a component of position sense has been used to explain kinaesthetic illusions arising during muscle vibration (Feldman & Latash 1982). Vibration increases the activity of muscle spindle afferents which may result in the sensation of muscle stretch. Concurrently, via their central actions, the increase in afferent signals may provoke a change in the central component of position sense which can illicit the opposite illusion, one of muscle shortening (Lackner & Dizio 1992). A change in the central component of position sense may underlie the classic child's illusion of the arm rising "ghostlike" after pressing it tonically (shoulder abduction) against the wall (Loeb, personal communication).
Note that representation of position sense as the sum of reafferent and proprioceptive signals is also consistent with the second, inverted scheme of position perception. This may explain the phantom limb phenomenon in which a limb continues to be perceived even after it has been amputated (Melzack 1989).
7. The united frame of reference for multi-muscle systems and
the hypothesis of anatomical correspondence.
Multi-muscle control has been analyzed in terms of muscle forces, kinematics, EMG signals and single motor unit activity, as well as matrix and tensor transformations of these variables and associated frames of references or systems of coordinates (Flanders et al. 1992; Gielen & van Zuylen 1986; Hasan & Karst 1989; Lacquaniti 1992; Loeb & Levine 1990; Mussa-Ivaldi et al. 1988; Nichols 1989; Pellionisz & Llin s 1982). However, the concept of CVs has not been used in these studies leaving the question of how the CNS produces the observed or postulated coordinations (synergies) and coordinate transformations unans- wered. A key to the solution of the problem may lie in the organization of positional frames of reference with regard to multi-muscle systems (Feldman & Levin 1993).
We have seen that there may be a united positional frame of reference for recruitment in a single MN pool (Section 4). Now we will try to formulate a testable hypothesis on the existence of a united frame of reference in which recruitment of the body's skeletal muscles takes place. This may seem ambitious given the increasing evidence that muscles may be recruited in different combinations and time sequences depending on the motor task (e.g., Loeb 1985; Tax et al. 1990; Tax & Gielen 1993). Using the analogy of the behaviour of gas molecules in a hermetically sealed vessel, we can imagine that each molecule moves relatively independently of the others but that the overall movement is constrained by the vessel itself. The vessel can be viewed as a natural frame of reference for the gas molecules. This vessel or frame of reference can be moved and consequently, all the molecules will follow the shift while remaining relatively independent. Similarly, a united frame of reference governing the recruitment of seemingly independent muscles may exist.
Insert Figure 7 near here
We define the united frame of reference in the following way. Let us suppose that, in the absence of the C command, threshold angles, R1, R2,.., Rn, are specified for all degrees of freedom of the body (n is the number of degrees of freedom). This set of threshold angles is associated with a particular body configuration which may be considered as a referent body schema. We will show that this body schema also represents a frame of reference for recruitment of muscles. Let us consider how the threshold lengths, lambdas, of different muscles are controlled within the referent body schema. The number of muscles is obviously greater than the number of degrees of freedom. A solution to this redundancy problem rests in the definition of the R command as the threshold angle common for all muscles of a joint (Feldman et al. 1990; Smeets 1991; Section 5). Imagine a situation where actual and referent body configurations coincide (i.e., actual joint angles are equal to their corresponding threshold angles). At these threshold angles, all muscles are at their threshold lengths. In other words, at this configuration, lambda(i) = x(i) for any muscle, i, where x(i) is the actual muscle length. Thus, the centrally organized relationship between threshold muscle lengths mirrors the anatomical relationship between the actual lengths of muscles spanning the joints. In this way the redundancy problem is solved. We call this solution the hypothesis of anatomical correspondence.
Let us apply this hypothesis to the set of mono- and bi- articular muscles acting on a limb. The actual muscle length of a bi-articular muscle is a function of two joint angles, i1 and i2:
xb = fb(i1, i2) (5)
The lower boundary condition for activation of a bi-articular muscle in statics is similar to that of a single-joint muscle: the actual length (xb) of the bi-articular muscle should coincide with its threshold length (lambda(b)). This condition is met at the referent limb configuration when i1 = R1 and i2 = R2. As a result, the threshold length lambdab is specified unambiguously by the R commands:
lambda(b) = fb (R1, R2) (6)
For other limb configurations the threshold length may no longer coincide with that defined by Eq. (6): if the limb is deflected from the referent configuration, lambdab may be modified due to a change in the intermuscular interaction mediated by spinal INs and proprioceptive afferents (Section 3). We, however, temporarilly disregard these effects. For a given lambdab, according to Eq. (5), the threshold angle at which the boundary condition of muscle activation is reached will depend on the value of the other joint angle. This dependency is exemplified by a diagonal line denoted lambdab on the angle-angle plane in Fig. 7A. The area to the left of the line represents the static activation area for this muscle. In contrast, the length of a monoarticular muscle subserving, say, joint 1, is a single-valued function of the joint angle and thus the boundary condition for muscle activation (x(1) = lambda(1)) occurs at a specific joint angle R1 irrespective of the configuration of adjacent joints. Therefore the activation areas of mono-articular muscles (hatched areas) have vertical or horizontal borderlines specified by angular thresholds (R1 and R2 in Fig. 7; the angles correspond to threshold lengths lambda(1) and lambda(2) respectively). The diagram emphasizes the differences in muscle function not in force but in spatial terms i.e., their activation in different parts of the joint workspace.
Changes in lambdas by CVs are visualized by shifts of the activation borders for each muscle (Fig. 7 B-D). The set of threshold angles (R1, R2,..) specifies a single point on the angle-angle plane at which the borderlines of all activation areas of mono- and bi-articular muscles intersect (Fig. 7 A, open circle). This point may be considered as the origin of the reference frame for recruitment of MN pools in the joint space. It is assumed that, using different sets of R commands, the nervous system may shift the origin point in any direction in the joint space. Thereby, the frames of reference for all muscles of the limb preserve a common origin point whether a single- or multi-joint movement is made (Fig. 7 B-D). This property is preserved desite reflex intrmuscular interactions (Section 8).
The concept of muscle activation area may be used to extend the C command to multi-muscle systems. Such a command may be represented by an increase in the area of joint space in which antagonist muscles are simultaneously active under static conditions (Fig. 7 E). All borderlines are shifted in E compared to D indicating that the area of coactivation is increased (curvilinear triangle). A change in the C command does not influence the equilibrium configuration of the limb or of the body and thus it is functionally independent of the R command. The two commands can be superimposed.
Insert Fig. 8 near here
Some consequences of the hypotheses formulated in this Section can be experimentally tested. Using anatomical data for selected joints and muscles, one can plot, for specific values of lambdas, borderlines for each muscle as we have done in Fig. 7. Take, for example, the single-joint soleus (ankle extensor) and the double-joint lateral gastrocnemius (ankle extensor and knee flexor). For soleus, the borderline of the activation area is a vertical line on the ankle versus knee angle diagram (Fig. 8). For the other muscle, the borderline ascends diagonally to the right. For both muscles their activation occur in the areas situated to the left of their borders. It can be seen that in zones S + G of the diagram both muscles are simultaneously tonically active. In zone S only soleus and in zone G only gastrocnemius is active. Let the initial limb configuration coincide with the intersection point of the two borderlines. An active change in the limb configuration visualized by a shift in this point or a passive change in the joint angles may reveal that the muscles may work either as synergists or antagonists depending on the direction of movement in joint space. Similar direction-dependent variations in EMG patterns may be predicted for a pair of biarticular muscles although anatomically they may be antagonists for one of the two joints. For fast movements, the dynamic expression for muscle activation area (Section 10) should be taken into account during testing.
Consistent with our hypothetical principle of muscle recruitment are experimental data showing that muscle activation patterns may depend on where a movement is performed in the workspace (Hasan & Karst 1989). The hypothesis of anatomical correspondence has been used for the simulation of intermuscular coordination, electromyographic and kinematic patterns during fast pointing movements in the double-joint arm controlled by 6 muscles, 2 of which were bi-articular (Feldman et al. 1990; Flanagan et al. 1993; see also Section 11.2). Additional experimental studies of EMG patterns may be undertaken with the purpose of verifying the hypothesis.
8. An hypothetical role of reflex intermuscular interaction.
It was assumed (see Section 7) that the control level specifies a set of R commands which represents a single reference point for recruitment of muscles in multi-joint space. As a consequence, appropriate threshold lengths (lambdas) for different muscles are set according to the anatomical relationship between actual muscle lengths. By what physiological mechanism may the anatomical correspondence of lambdas be accomplished by the nervous system? One may suggest that the control level uses a geometrical model of the limb or of the whole body to compute the set of lambdas corresponding to a desired reference point (R1, R2,..) as defined by Eq. (6). However, our opinion diverges from this view. It is likely that the anatomical correspondence is dynamically supported by INs of segmental reflex loops. Related hypotheses that segmental reflex connections may mirror anatomical organization of muscles have been proposed in a number of studies (Loeb & Levine 1990; Miller et al. 1992; Nichols 1989; Pratt et al. 1991).
It can be seen from Fig. 7 A-D that the reference point is the only point associated with the limb configuration at which the active forces are zero and mechanical muscle interaction is minimal. A passive deflection from the referent point is associated with lengthening of some muscles above their activation thresholds. This process may lead to an alteration of the reflex intermuscular interaction measurable in terms of changes in the threshold lengths of antagonists and synergists. Experimental findings showing that intermuscular reflexes affect stretch reflex thresholds are consistent with this assumption (Feldman & Orlovsky 1972; Matthews 1959; Nichols 1989; see also Fig. 5 C). Indeed, inhibitory and facilitatory interaction may be provided by Renshaw cells, Ia inhibitory INs and other segmental INs (Jankowska 1992). Due to reflex modification of threshold lengths, the referent body configuration will no be longer associated with the point at which active forces are zero. We, however, assume that the zero point does not disappear but just shifts from its original position, in terms of joint coordinates, in the direction which is opposite to the direction of the passive displacement. In such a way, reflex intermuscular interaction obviously amplifies the active forces counteracting deflections from the referent body configuration. Mathematically, let the passive deflection be described by vector [rho1, rho2,..., rhon] in joint coordinates (rhoi is its projection on ii). Assuming that the shift in the zero point is proportional and opposite to the displacement vector, the coordinates (ii0, i = 1,..., n) of the new zero point are defined as:
ii0 = Ri - a rhoi (7)
where a is a positive constant. If we substitute these coordinates for R1 and R2 in Eq. (6) we get the value of the threshold muscle length which accounts for the reflex interaction.
The system of reflex interaction in the model resists any deflection from the referent body configuration at which muscle forces are zero and mechanical interaction between muscles is minimal. It also provides a single origin point for recruitment of muscles in joint space even if the deflection is significant. In such a way, an anatomically appropriate relationship between lambdas remains during any change in the limb configuration. In terms of effects on MNs, the reflex interaction gives rise to reciprocal changes in activity of antagonist MNs or simultaneous facilitation of synergist MNs. The two reflex patterns can be switched when the functional role of muscles varies. These effects may be predicted by analysing the shifts in the borderlines (for example, using Fig. 8) resulting from changes in the limb configuration when the limb is moved in different directions away from the initial, referent point.
INs of segmental reflex loops may also play a role in shaping C commands. However, these commands would be mediated by other INs which, in contrast to those mediating the R command, tend to maximize mechanical interaction between muscles (Feldman 1993).
The hypothesis that the principles of minimal or maximal interaction may play a key role in motor control was formulated by Gelfand and Tsetlin (1971). The same hypothetical principles may also play a role during development of neuronal connections. Let us suppose that the R and C commands mature simultaneously. At early stages of motor development, EMG activity produced by spontaneous R commands will likely be minimized by reflex intermuscular interactions. As a result, coactivation EMG patterns produced by the C command may be dominant (see Westerga & Gramsbergen 1993). Indeed, the prediction that common dynamic principles underlie both neuronal development and motor control is worth testing.
Insert Figure 9 near here
A neuronal network model which produces agonist-antagonist coactivation without affecting the joint EP has recently been proposed for a single joint (Fig. 9; Feldman 1993). The model predicts specific connections of segmental INs predominantly mediating the effects of Ib tendon organ afferents to MNs and may be tested electrophysiologically. In the model, descending systems deal with antagonist muscles as if they were symmetrical structures. Systems producing C commands may thus initiate identical shifts in agonist and antagonist lambdas. In the hypothetical neuronal schema (Fig. 9), the C command is mediated by C INs having excitatory inputs from descending systems which in turn facilitate flexor and extensor MNs. In the case when only agonist muscles are initially active (e.g., holding a load), a symmetrical C command may increase their activity whereas antagonist MNs may still be in a subthreshold state resulting in a shift in the EP. Therefore, the symmetrical commands may be transformed into asymmetrical, reflex-dependent signals by INs in spinal proprioceptive pathways before reaching their target MNs. This is a function of inhibitory A INs in the model. When antagonist Ib afferents are silent, the inputs from C INs and agonist Ib afferents activate A INs. Thus, agonist MNs receive not only facilitatory synaptic inputs from C INs but also inhibitory ones from A INs. These inputs cancel each other and thus the C command does not affect the level of agonist activity unless antagonist MNs are simultaneously active. When Ib afferents from antagonists signal that these muscles are active, they presynaptically terminate the activity of agonist A INs. This allows B INs to presynaptically adjust the strength of the inputs from C INs to simultaneously active agonist and antagonist MNs. As has been shown (Feldman 1993), this attenuation should be proportional to the level of active muscle torque.
The schema in Fig. 9 also takes into account that muscle stretch may sometimes evoke reflex coactivation of antagonist muscles (Lacquaniti et al. 1991). To explain this, we assume that C INs may also be triggered by strong activation of different afferents including muscle and skin afferents.
Another model (Bullock & Grossberg 1988) has also incorporated the idea of symmetrical facilitation of agonist and antagonist MNs originally proposed in the lambda model (Feldman 1966; 1980 a,b). In contrast to the lambda model, it suggests that recurrent and Ia reciprocal inhibition play a major role in adjusting these signals according to the biomechanical asymmetries of muscle action. However, the model is incomplete since only the case when flexor and extensor muscles are simultaneously active has been considered and the threshold properties of MNs are ignored.
Our model is consistent with the recent finding that a large number of Ib INs also receive inputs from Ia muscle spindle afferents (Jankowska 1992; McCrea 1992). In the model, different afferent inputs may converge on the same INs. For example, for a given lambda there is a single-valued relationship (IC) between torque (or force) and angle (muscle length). Consequently, force and position signals are, with some reservations, equivalent and may be conveyed to the same INs. In the model, it is essential that selection of alternative pathways to MNs depends on whether the muscles generate an active force (torque) or not. Ib afferents are specifically sensitive to the level of active force (Jami 1992) whereas muscle spindle afferents may have background activity even if the muscle is not active. Thus, in the present model, Ib inputs to such INs are dominant.
9. Intra- and extrapersonal frames of reference for motor
control and locomotion.
In Sections 7 and 8 we suggested that the control level may specify recruitment thresholds (R1, R2, ...) in joint angle space (to be more specific, in the space of all degrees of freedom). This set of threshold angles may provide a frame of reference for different body configurations (the referent body scheme). This also implies that changes in the referent scheme may result in movement. How these processes may be organized at a neuronal level is considered.
Insert Figure 10 near here
Note that while the configuration of the body can vary widely, some of its parameters (e.g., segment lengths) remain invariant. This geometric invariance may underlie the neuronal body scheme. Neuronal structures may be characterized not only by their temporal activity but also the spatial distribution (topology) of this activity (e.g., Munoz et al. 1991; Smolianinov 1980; Stein 1992). Fig. 10 shows an example of an artificial one- dimensional neuronal network which dynamically maintains the distance between two centres of activity: if the position of one centre is shifted, the second centre begins to move in order to restore the initial distance between them. This tendency to maintain the inter-centre distance dynamically may be a characteristic of the neuronal body scheme. Based on this reasoning, we assume that every steady state of the neuronal body scheme may be geometrically similar ("topologically isomorphic") to a particular body configuration and that the control level may change the scheme by relocating the neuronal activity associated, say, with the end point of the arm. This forces the neuronal structure to find a new steady-state distribution of activity which restores the correct geometric relationships associated with the new body configuration. Related changes in neuronal activity may also produce individual R commands which alter the actual body configuration. Indeed, these ideas are consistent with findings of multiple and plastic projections to and from the cortex (e.g., Merzenich et al. 1988).
Note that the referent body scheme is an intrapersonal frame of reference. It defines a referent body configuration in terms of threshold angles without specifying the body's orientation in extrapersonal space. For example, astronauts can easily vary their body configuration regardless of its orientation in space. However, on earth, our actions usually require a specific orientation of the body in extrapersonal space. Shaking our friend's hand implies a vertical body position and a face-to-face orientation at a specific distance. Indeed, many other motor tasks may be primarily controlled in terms of extrapersonal rather than internal, joint space (Flanagan et al. 1993; Georgopoulos et al. 1986; Hogan & Flash 1987; Kalaska & Crammond 1992; Lacquaniti 1992; Stein 1992). For such control to take place, the referent body scheme should be combined with one associated with external space. A solution proposed by the lambda model, is that the set of threshold angles is combined with independent variables representing the localization and orientation of the body in space.
The lambda model indicates how, in principle, a frame of reference can be organized for any sensorimotor system (e.g., the vestibular one). Such an organization is based on the threshold properties of MNs or neurons projecting to them, afferent feedback and inputs to these cells from independent, control structures. In particular, the extrapersonal frame may be organized with the use of afferent feedback from the vestibular or/and visual system. Thus, taking a step may result from translation of the frame of reference associated with external space leading to the establishment of the initial (or, in case of perturbation, a new) body configuration in another part of space. This simple principle may underlie locomotion although associated mechanisms of its realization may be complex.
The ideas outlined in this Section are, indeed, preliminary and speculative but they show that the hypothesis on movement production via shifts in the frames of reference may be generalized to control levels associated with other afferent systems. These ideas are elaborated here in order to provoke further discussion and to outline future directions for the development of the lambda model.
10. Control variables and the system's steady states; movement
production
In the lambda model, the notion that CVs produce movement by shifting the positional frames of reference is related to the idea that CVs create steady (stationary) states or EPs of the system, modify them and affect their stability. This notion will be used in the present Section to discuss dynamic events leading to movement.
In the model, when CVs specify a new steady state, the initial state of the system may be considered as a deflection from the projected one. If the system is stable, the deflection will elicit changes in neuronal activity which will accelerate and then decelerate the transition of the system to the new steady state. Depending on the damping of the system at the neuronal and biomechanical levels, the new steady state will be reached after these two phases or after additional decaying terminal oscillations. In intact systems, afferent feedback to MNs and muscle properties may play a dominant role in this process. In the case of muscle deafferentation, the remaining central and peripheral mechanisms may provide sufficient damping to meet the stability requirement of the steady state. EMG and kinematic patterns of fast movement such as the tri-burst pattern and bell-shaped velocity profiles (e.g., Forget & Lamarre 1987; Gottlieb et al. 1989; Wallace 1981) are thus natural consequences of shifts in the steady state at neuronal levels not only in intact but also in deafferented subjects. Moreover, muscle activation patterns are a function of EP shifts but not vice versa.
Insert Figure 11 near here Fig. 11 illustrates this notion for an intact muscle interacting with a constant load. MN recruitment and the development of active muscle force (solid line) begins when the muscle length exceeds the threshold length, lambda. The initial EP (filled circle) is the point at which the solid line (i.e. initial IC) and the load characteristic (dashed horizontal line) intersect. To balance the load at the equilibrium position (x) some tonic level of MN activity (`1) is necessary. Let us suppose that the system is initially at position x. To produce a movement to another position, the nervous system can shift the threshold by klambda moving the IC to the left (dotted line). The old EP is replaced by a new one (open circle). The initial position may be considered as a deflection from the new equilibrium position. Subsequent events are consequences of this deflection. Before movement begins, according to the new IC, muscle activation will start to change from `1 to a greater level, `3, resulting in muscle shortening. and the eventual establishment of a new tonic level, `2, associated with the new EP. Note that the change in lambda by the control level results in a new EP before any changes in muscle activation occur. After the change in lambda, the new EP remains the same irrespective of muscle activation (`1
`3 `2) in contrast to the suggestion of some other versions of the EP hypothesis (e. g., Bizzi et al. 1992). The system may even oscillate about the final position and produce rhythmical bursts of EMG while the new EP is stable.
Fig. 11 allows us to illustate the notion that neither the EP nor the EMG signals are directly specified by CVs. Imagine that the movement was arrested at the initial position, x, by a mechanical device. In other words, the isotonic load was replaced with an isometric one which can be represented as a vertical line (a load with an infinite positional gradient, see Section 3). Now the new EP will be the point at which the final IC intersects the new load characteristic. This EP is associated with level `3 of muscle activation. Thus although the control pattern klambda may be the same, a high level of the final tonic EMG activity will be generated when an isotonic load is unexpectedly replaced with isometric one.
Insert Figure 12 near here
To describe the origin of EMG patterns in more detail, the concept of the dynamic muscle activation area has been used (Feldman 1986; Feldman et al. 1990). This concept takes into account the dynamic sensitivity of afferent feedback to MNs. Figure 12 A (right panel) shows that the threshold muscle length under dynamic conditions (lambda*) is a decreasing function of velocity dx/dt (velocity is positive if muscle length increases). In a linear approximation,
lambda* = lambda - f dx/dt (8)
where f is a coefficient. The condition of muscle activation, Eq. (3), can be generalized to dynamics in the form:
x - lambda* r 0 (9)
The boundary condition for MN activation, x - lambda* = 0, represents a line in muscle length/velocity coordinates, i.e., on a phase plane (Fig. 12 B). The position of the boundary line is specified by parameter lambda. The slope is associated with the coefficient f of velocity and may reflect the dynamic sensitivity of muscle spindle afferents (Feldman 1986; Feldman et al. 1990). In the model, this coefficient influences the system's stability and may be modified independently by the CNS in which case it becomes a new CV having the dimension of time.
The level of muscle activation is an increasing function of the value x -lambda* which is equal to the horizontal distance (s) between the threshold line and the point shown by the filled circle in Fig. 12 B. This point represents the combination of kinematic variables at time t - t(prime) where t(prime) is a reflex delay. This describes the dynamic activation area, the area in muscle length and velocity coordinates, in which the muscle is active. Altering CVs causes a change in lambdas depicted by a shift in the borders of the activation area, leading to a change in s, i.e. EMG modifications and movement. These properties of the activation area have been used for the qualitative explanation of EMG patterns in single- and double-joint movements (Feldman et al. 1990; Flanagan et al. 1993). Examples of numerical simulations of kinematic and EMG patterns during fast single-joint movements are shown in Figs. 13 and 14.
Insert Figures 13 & 14 near here
In spite of its simplicity, the concept of activation area integrates, in a compact form, several physiological mechanisms: the non-linear, threshold properties of ` MNs; their rank-ordered recruitment; the effects of central and afferent influences to `, ~ and ~ MNs; position and speed sensitivity of muscle spindles and their regulation by ~ MNs; the basic properties of the tonic stretch reflex, the phasic stretch reflex and the unloading reflex (Feldman 1986; Feldman et al. 1990; Latash 1993). Decay properties of spindle afferents (Windhorst et al. 1976) and properties of reciprocal inhibition (Jankowska 1992) have also been integrated in this concept (Feldman et al. 1990; Feldman 1992). However, in its present form, the concept is still incomplete. For example, history-dependent behavior of MNs associated with refractory periods, afterhyperpolarization, etc., are not accounted for by the model. Indeed, these additional dynamic properties of MNs and associated systems have yet to be integrated in the concept.
11. Invariants specifying motor behaviors or the conservative
control strategy.
It has been argued that the changes in elbow joint position and in EMG activity in response to unloading ("do not correct paradigm", see Section 3) result from reactions of the auxiliary level to changes in the load when CVs are held constant. This is an example of the conservative control strategy. When the subject is given the opposite instruction, to maintain arm position for example, he has to modify CVs. In this and similar paradigms, the control level presumably minimizes the number of modifiable CVs and maintains the parameters of these changes (e.g., the rate of change) invariant. This is a more general definition of the conservative control strategy. One advantage of this strategy is that by keeping, for example, the rate of change in the R command constant, the nervous system can initiate movement without pre- determining its distance: the system can arrest the shift earlier or prolong it if the target position changes using temporal coding for movement distance. The other advantage of this strategy is that, by specifying invariant CVs or invariant rates of change of CVs, the participation of the control level in activities at the auxiliary level is minimized. Interestingly, one consequence of the conservative strategy may be the minimization of the number of degrees of freedom used in a motor task - the idea underlying the solution of the redundancy problem (see below).
The conservative strategy will be discussed for discrete single-joint movements, planar double- and triple-joint arm pointing movements.
11.1. Changes in position at a single joint. According to the conservative strategy, a self-paced discrete movement may be achieved by a constant rate of shift in the R command during the time necessary for it to specify the desired movement distance (Fig. 13). For another distance, the rate may be the same but the duration of the R command may change. This may also be the mechanism of movement production during the fastest movements for which the rate of the R command is likely already maximal being constrained by the velocity of associated changes in neuronal activity. Thus, the only way to regulate the movement distance is by changing the duration of the command. The kinematic consequences of such a conservative control strategy are obvious: If two (e. g., fast) movements of different amplitudes are aligned, their kinematics will initially coincide and then diverge as has been observed systematically in experimental studies (Adamovich et al. 1984; Gottlieb et al. 1989; Levin et al. 1992; Wadman et al. 1979). This divergence point corresponds to that of the ramp R commands. Thus, if A is the amplitude of the smaller movement and g is the duration of the coincident movement paths, then the rate (nu) of the R command is given by nu = A/g (Fig. 15 A). The value of ~ for fast elbow movements has been found to be in the range of 500 - 700 x/s (Fig. 15 B top). Consequently, for a movement of 60x, it takes about 100 ms to shift the R command to its final level. This means that the change in the R command ends near the movement's peak velocity, although EMG bursts and the movement itself will continue for at least 150 ms (Fig. 15 B). This stresses, once again, the proposal of the model, that EMG patterns are not programmed. The process of termination of the movement may also be completed by the auxiliary level under the minimal supervision of the control level (constant R command).
Insert Figure 15 near here
From this hypothesis it follows, that terminal oscillations usually observed during fast movements are not represented by CVs. They simply reflect the underdamped dynamic reaction of the auxiliary level which is unable to rapidly dissipate the energy acquired from the shift in the EP. This is consistent with the fact that these oscillations disappear if movement is opposed by an acceleration-dependent load which absorbs some of the energy (Fig. 16, Abdusamatov et al. 1988).
Insert Figure 16 near here
An alternative hypothesis (Hasan 1986; Hogan 1984; Lan & Crago 1992; Latash 1993; Latash & Gottlieb 1991) suggests that for fast discrete movements, the equilibrium position rapidly shifts towards its future final position but then shifts back and forth, eventually reaching the final equilibrium position (N- shaped pattern). Using such a pattern, these authors have simulated fast single-joint movement kinematics (see, however, St. Onge et al. 1993). The role of N-shaped patterns is unclear since monotonic (ramp) patterns of CVs have already been sufficient to successfully simulate the kinematic as well as the EMG characteristics of fast elbow, mandibular, and double-joint arm pointing movements (Feldman et al. 1990; Flanagan et al. 1992, 1993; St. Onge et al. 1993). Latash & Gottlieb (1991) tried to verify the N-shape pattern experimentally by measuring shifts of torque/angle elbow characteristics during fast movements. However, they did not extrapolate the characteristics to static ones (translations of which could provide more direct data on the pattern of the EP shift) making the conclusion on the N-shape pattern equivocal.
The suggestion of an N-shaped control signal appears to be a compromise between the traditional view that the control level is essentially involved in the programming of EMG patterns and the alternative one that all EMG bursts needed to initiate and perform a movement to a new position may be generated by the auxiliary level in response to a fast monotonic shift in the EP (Figs. 13, 14). Although such a compromise seems, from our point of view, artificial, only empirical studies may allow one to choose between the two hypotheses. These hypotheses make different predictions about the duration of the EP shift for the fastest movements. In the case of the monotonic pattern, the shift is arrested near the peak velocity of the actual movement (see Figs. 13, 14) whereas, in the alternative hypothesis, the shift and the movement end practically simultaneously. It has recently been found that fast 60x elbow flexion movements may be arrested after about 5-10x by a strong opposing load although the shift in the ICs remains the same ("do not correct" paradigm). The movement time is thereby reduced to about 100 ms. This means that the shift of the ICs for this 60x movement does not continue for more than 100 ms, i.e., it is arrested long before the movement offset (Feldman, Adamovich, and Levin; submitted manuscript). The hypothesis of the ramp shape of the R command is associated with several properties of fast single-joint movements which have been verified experimentally. 1) The ramp R command can be represented as the sum of two or more sequentially generated ramp signals of smaller amplitude. If the C command can be decomposed in a similar way or if it is constant, this superposition principle may also be applied to actual movement trajectories (Abdusamatov et al. 1987). In other words, single- joint movements can be represented as the sum of identical movement components initiated sequentially (Fig. 15 B; see also Milner & Ijaz 1990). 2) In response to unexpected shifts of the visual target, the ramp control signal can be stopped earlier or continued further depending on the time of the shift in the target position. Provided that the inter-target time is short, the movement will not differ kinematically from one which was originally planned to the final target, and there will be no inflection point in the movement trajectory. This effect has been demonstrated for both arm and eye movements (Jeannerod 1988; Pllison et al. 1986).
It is known that in fast movement against a viscous load, the antagonist EMG burst can be suppressed (Mustard & Lee 1987; Waters & Strick 1981). This effect is also reproduced in the lambda model (Fig. 13, right panel). The pattern of the R and C commands used to simulate this effect is the same as that eliciting a tri-phasic EMG pattern in the absence of the load.
In the model, a reciprocal pattern of agonist/antagonist EMG signals may change to a coactivation pattern if the C command increases leading to an increase in movement speed. However, an increase in the C command above a specific level may not have any further effects on movement kinematics while it may continue to increase EMG levels (Feldman et al. 1990). Thus, the model reproduces a characteristic feature of human movements which was especially emphasized by Bernstein (1967): movement kinematics may be stable despite variable EMG levels.
Thus, experimental results are consistent with the hypothesis that the conservative strategy is used in the control of single-joint movements.
Insert Fig. 17 near here
11.2. Double-joint pointing movement. Fast pointing movements of the arm to stationary and shifted visual targets have also been simulated. In the model, the double-joint arm is subserved by 6 muscles, 2 of which are bi-articular (Feldman et al. 1990; Flanagan et al. 1993). The model assumes that the control level shifts the positional frame of reference associated with extrapersonal space. This CV may be depicted in this space by the control velocity vector U related to a referent position of the arm endpoint. (Note that vectors are shown in boldface characters whereas their lengths are expressed as scalars: absolute value of vector U = U). When there is no external force, the referent position coincides with the equilibrium position of the end point. In this case, vector U is directed, with possible perceptual errors (cf. de Graaf et al. 1991; Flanders et al. 1992), to the target. The rate, duration and direction (m) of the shift in U may be specified independently. Note that information about the distance to the target is unnecessary for movement initiation. In the case of a change in the target position, vector(U) is rotated accordingly. Because of the vector nature of the command, this action can formally be considered as the superposition of one vector(U) with a second one, directed from the old to the new target (cf. Flash & Henis 1991). We hypothesize that this vector may correspond to the population vector characterizing the activity of motor cortical neurons (Caminiti et al. 1990; Georgopoulos et al. 1983; Kalaska 1991). Vector(U) is then transformed to individual control velocities dR/dt for each joint (Fig. 16; see also Feldman et al. 1990; Flanagan et al. 1993). The decomposition of the control velocity vector into individual joint commands may result from the dynamic neuronal process associated with the functioning of the referent body scheme (see Section 9). This aspect of the model has not yet been elaborated. Instead, we used arm geometry for this decomposition in the following way (Fig. 17 A). For simplicity, external loads were zero and the actual joint angles i1 and i2 initially coincided with the corresponding threshold angles (R1 and R2). Note that rotation in a single joint in isolation shifts the endpoint in the direction perpendicular to the appropriate radius-vector (r1 or r2). For example, for the first joint the rate of this displacement equals the vector product of vector(r1) and d(vector R1)/dt (the latter is the vector directed along the rotation axis of the joint). Shifts of the endpoint in the direction vector(U) are produced by simultaneous changes in the threshold angles according to the parallelogram rule or the vector equation:
vector(r1) X d(vectorR1)/dt + vector(r2) X d(vectorR2)/dt
= vector U (10)
The equation can be solved for R1 or R2 (see Appendix). For example (in scalar form):
dR1/dt = a1 U cos e2 (11)
where a1 equals 1/r1 sin (e2 - e1). Angles e1 and e2 are shown in Fig. 17 A. Note that they are functions of the direction (m) and threshold angles R1 and R2 (see Appendix). Thus the transformation is described by the dynamic equations in which the direction and magnitude of vector(U) are invariant. The central commands are generated regardless of the current values of actual joint angles and their derivatives. Individual lambdas for each muscle are specified based on threshold angles according to the hypothesis of anatomical correspondence (Section 7).
The model also includes commands with more generalized functions such as the C command and the damping f command (see Section 10) which affect dynamic and kinematic characteristics of the movement but not the shifts in the equilibrium position of the endpoint specified by commands. In particular, these commands are essential in diminishing the influence of reactive forces (including centrifugal and Coriolis) on arm endpoint kinematics. The presence of bi-articular muscles in the model facilitates the control of reactive forces. As in the model of single-joint movements, muscle torques are functions of CVs and kinematic variables (see the legend to Fig. 13). These equations, in combination with the Lagrangian equations for the double-joint arm, are numerically integrated (direct dynamics). The model allows one to simulate both the kinematic and EMG patterns of pointing movements in different directions. For a constant control velocity vector, we receive slightly curved actual endpoint trajectories having bell-shaped velocity profiles (Fig. 17 B; Feldman et al. 1990; Flanagan et al. 1993). In the simulation, the movement amplitude, bell-shaped velocity profiles of individual joints as well as the magnitude and duration of EMG signals are all functions of movement direction (Fig. 17 B). Agonist-antagonist EMG patterns are tri-phasic. Similar to empirical data (Hasan & Karst 1989) for specific movement directions, simulated EMG patterns in double-joint muscles may be atypical. For example, in the model, biceps may be active during extension of the elbow in specific movement directions. The model also simulates movements to displaced targets (Flanagan et al. 1990; 1992; 1993).
Other control strategies can be elaborated by the model which reduce or eliminate the need to consider limb geometry. As follows from Fig. 17 A, the direction of arm endpoint shift will be the same if commands dR1/dt and dR2/dt are multiplied by the same coefficient. The price may be a non-constant velocity shift of the endpoint. In such a way, the denominators in the expressions for ai can be dropped to yield:
dR1/dt = a r2 U cos e2; dR2/dt = - a r1 U cos e1 (12)
where a is constant.
Another possibility is that the system may ignore limb geometry and specify constant rates of R commands to shift the limb endpoint in the desired direction (m). The R commands may be staggered or simultaneous. Their rates may be modified after significant errors are accumulated in the process of movement.
Are there any predictions of the lambda model which can be tested in electrophysiological studies of neuronal activity at different brain levels during pointing tasks? We indicate one possibility associated with the assumption that central commands in pointing movements may be independent of external, biomechanical events. In particular, the control velocity vector specifies a referent position of the arm endpoint whereas its equilibrium position is also defined by external forces. In primates, many studies have demonstrated correlations between the activity of cortical neurons with movement kinematics or forces. According to the CV concept, activity of neurons having control functions will indeed correlate with kinematics (direction, velocity, distance, etc,) or, if movement is prevented, with force (e.g., Georgopoulos et al. 1992; Kalaska & Crammond 1992). These correlations may, however, hide the true control nature of neurons: that their activity, in essence, is independent of these external biomechanical variables. For example, suppose a monkey makes movements against a load to a target and then the load is suddenly removed in some trials. Provided that the monkey previously learned "not to correct the arm deflections", one may predict that the patterns of control neuronal activity will remain the same despite the changes in kinematics (including movement direction) and forces. This "CV-test" may help to classify neurons into control and non-control ones. However, some control neurons may simultaneously be part of reflex loops making the classification difficult.
11.3. Triple-joint pointing movements: A solution of redundancy problem. Nobody complains about having too many degrees of freedom but neuroscientists proclaim that it is a critical problem (Bernstein 1967). How does the nervous system coordinate so many degrees of freedom to produce active movements? Redundancy is associated with the ability to select different interjoint coordinations and movement trajectories to the same target, a phenomenon called motor equivalence (Bernstein 1967).
A solution to the redundancy problem for pointing movements offered here does not decrease the ability of the system to move joints in isolation or in combination and to specify any physiologically achievable limb configuration and interjoint coordination (cf. Reeke et al. 1990). At the same time, it is suggested that each time the control level performs the movement, a conservative strategy is used - the brain likely keeps as many CVs as possible invariant (see Section 11).
Insert Figure 18 near here
Consider three-joint planar pointing movements of the arm to different targets (Fig. 18). As for the double-joint arm, we assume that the brain specifies control velocity vector(U) eliciting shifts of the equilibrium position of the arm endpoint. Notice that a minimum of two joints must be coordinated to provide straight line motion of the endpoint in any direction (Fig. 16 A). Thus each pair is a minimal synergy providing movement of the endpoint to the target. In the three-joint arm, the first joint can be coordinated with the second or third or the second joint can be coordinated with the third to produce shifts of the endpoint in the appropriate direction. The control vector(U) may be decomposed into three collinear vectors with coefficients w12, w13 and w23 representing the weights of each joint pair. It is assumed that these weights can be independently specified by the control level and thus they may be an additional set of CVs. The vector command for each joint pair is then transformed into individual rates of shift dRij/dt in the R command for each joint according to the procedure already described for the double-joint model (Fig. 17 A; Feldman et al. 1990; Flanagan et al. 1993). Since the ith joint can be coupled with any jth joint the net rate of change of the R command, dRi/dt, for the ith joint is the sum of component commands dRij/dt for different js (the principle of superposition):
dRi/dt = d dRij/dt j = 1 to 3, j is not equal to i (13)
where 1 to 3 is the number of joints. Thus, to perform the task the nervous system presumably specifies a control velocity vector(U) and weights, wij, characterizing the inter-joint coordinations. This gives rise to a unique solution of the motor task. As a result, the central commands associated with control velocity vector(U) are determined for each joint unambiguously. These commands give rise to actual motion according to dynamic principles already described for single- and double-joint lambda models.
The nervous system may "play" with the set of minimal synergies by changing the weighting parameters and, as a result, the interjoint coordination while preserving the movement of the endpoint in the same direction. For example, all three joint pairs may contribute to the motion of the endpoint if all weights, wij, are positive. If one weight is zero, the interjoint coordination will be constrained although all joints will participate in the motion of the endpoint. If two weights are nullified, the remaining parameter specifies motion in only two joints to move the endpoint. Within the boundaries of actual constraints, the nervous system prefers to minimize the number of joints participating in the movement: In planar pointing arm movements, the motion takes place primarily at the shoulder and elbow while motion at the wrist is insignificant. When all three parameters have some value but their sum is zero, motion of the endpoint is nullified. As a consequence, the arm will rotate about the fixed endpoint. A single-joint movement can be performed in two ways: by moving the control velocity vector along the corresponding arc, or by breaking the symmetry wij = wji and nullifying one of the coefficients. Thus, the model allows flexible control of interjoint coordination and, independently of that, the endpoint trajectory.
Neuronal network models have provided interesting insights into the solution of the redundancy problem (Kuperstein 1988; Reeke et al. 1990). The model by Reeke et al. (1990) is based on the idea of the selection, by training, of gestures of initially unorganized motions. Kuperstein's model is similar, but the performance is based on the discrepancies between actual postures and those computed from sensory information. Learning changes the synaptic weights in the network in order to minimize these discrepancies. Our model may indicate that the CNS, rather than taking care of movement trajectories or postures, may organize specific positional frames of reference for body segments and produce movement by translating them in space. In addition, the conservative strategy may provide a unique inter-joint coordination for every single movement but reserves the possibility to change it in any desirable way in sequential movements.
12. Conclusions
The hypothetical principles of sensorimotor integration are discussed in the framework of the lambda model partly in relation to the weaknesses of existing models which are based on ideas of programming MN activity, muscle forces, stiffness or movement kinematics. We have focused on central aspects of the lambda model - control variables, the notion that the nervous system organizes positional frames of reference for the motor apparatus and produces active movement by translating them in space, the conservative control strategy in motor behavior, etc. One of the experimentally testable outcomes of the lambda model is the finding of a united positional frame of reference for skeletal muscles of the body. This frame specifies MN recruitment in multi-muscle systems as a function of control and kinematic variables. Simultaneously, it establishes a correspondence between control processes and anatomical characteristics of muscles. We also illustrated that the conservative control strategy may give rise to a solution of the redundancy problem in the number of degrees of freedom of the motor apparatus. The model may provoke experimental studies, algorithmic and neuronal network simulations according to the suggested dynamic principles.
APPENDIX Eq. (10) can be solved for R1 and R2 by taking the scalar product of each side with unit vectors collinear with vector (r2) and vector(r1), respectively. The two terms in the left-hand side of Eq. (10) are vectors perpendicular to vector(r1) or vector(r2). Therefore the scalar product yields only one term different from zero. For example,the scalar product with unit vector collinear with r2 yields
r1 dR1/dt cos [(c/2) - (e2 - e1)] = U cos e2 (14)
Using the trigonometric identity theorem yields cos [(c/2) - (e2 - e1)] = sin (e2 - e1) and it follows that dR1/dt is defined by Eq. (11). Angles e1, e2 and radii r1 and r2 are functions of the direction (m) of vector U and commands R1 and R2. For example, e2 = R1 + R2 - m (for details see Flanagan et al. 1993). ----------------------------------------------------------------
Abdusamatov, R. M., Adamovich, S. V., Berkinblit, M. B., Chernavsky, A. V. & Feldman, A. G. (1988) Rapid one-joint movements: a qualitative model and its experimental verification. In: Stance and motion: Facts and concepts, ed. V. S. Gurfinkel, M. E. Ioffe, J. Massion & J. P. Roll. Plenum.
Abdusamatov, R. M., Adamovich, S. V. & Feldman, A. G. (1987) A model for one-joint motor control in man. In: Motor control, ed. G. N. Gantchev, B. Dimitrov & P. Gatev. Plenum.
Adamovich, S. V. (1992) How does the nervous system control the equilibrium trajectory? Behavioral and Brain Sciences 15:704-5.
Adamovich, S. V., Burlachkova, N. I. & Feldman (1984) Wave nature of the central process of formation of the trajectories of change in the joint angle in man. Biophysics 29:130-34.
Agarwal, G. C. (1992) Movement control hypothesis: A lesson from history. Behavioral and Brain Sciences 15:705-6.
Alexander, G. E., DeLong, M. R. & Crutcher, M. D. (1992) Do cortical and basal ganglionic motor areas use "motor programs" to control movement? Behavioral and Brain Sciences 15:656-65.
Andronov, A. A., Witt, A. A. & Haiken, S. E. (1959) Theory of oscillations, Physmat, Moscow. In Russian.
Angel, R. W. (1973) Spasticity and tremor. Effects on the silent period and the after-volley. In: New developments in electromyography and clinical neurophysiology, Vol. 3, ed. J. E. Desmedt. Karger.
Appelberg, B., Johansson, H. & Sojka, P. (1986) Fusimotor reflexes in triceps surae muscle elicited by stretch of muscles in the contralateral hind limb of the cat. Journal of Physiology (London) 373:419-41.
Arshavsky, Y. I., Gelfand, I. M. & Orlovsky, G. N. (1985) The cerebellum and control of rhythmical movements. In: The motor system in neurobiology, ed. E. V. Evarts, S. P. Wise & D. Bousfield. Elsevier.
Asatryan, D. G. & Feldman, A. G. (1965) Functional tuning of the nervous system with control of movement or maintenance of a steady posture: I. Mechanographic analysis of the work of the limb on execution of a postural task. Biophysics 10:925-35.
Bennett, D. J., Hollerbach, J. M., Xu, Y., & Hunter, I. W. (1992) Time varying stiffness of human elbow joint during cyclic voluntary movement. Experimental Brain Research 88:433-42. Berkinblit, M. B., Feldman, A. G. & Fukson, O. I. (1986) Adaptability of innate motor patterns and motor control mechanisms. Behavioral and Brain Sciences 9:585-638.
Bernstein, N. A. (1935) The problem of interrelation between coordination and localization. Archives of Biological Science 38:1-35. In Russian.
Bernstein, N. A. (1967) The coordination and regulation of movements. Pergamon.
Bilodeau, M., Arsenault, A. B., Gravel, D. & Bourbonnais, D. (1990) The influence of an increase in the level of force on the EMG power spectrum of elbow extensors. European Journal of Applied Physiology 61:461-66.
Bizzi, E., Accornero, N., Chapple, W. & Hogan, N. (1984) Posture control and trajectory formation during arm movements. Journal of Neuroscience 4: 2738-44.
Bizzi, E., Hogan, N., Mussa-Ivaldi, F. A. & Giszter, S. (1992) Does the nervous system use equilibrium-point control to guide single and multiple joint movements? Behavioral and Brain Sciences 15:603-13.
Bloedel, J. R. (1992) Functional heterogeneity with structural homogeneity: How does the cerebellum operate? Behavioral and Brain Sciences 15:666-78.
Blouin, J., Bard, C., Teasdale, N., Paillard, J., Fleury, M., Forget, R. & Lamarre, Y. (1993) Reference systems for coding spatial information in normal subjects and a deafferented patient. Experimental Brain Research 93:324-31.
Bock, O. (1990) Load compensation in human goal-directed arm movements. Behavioral & Brain Research 41:167-77.
Bock, O. & Arnold, K. (1993) Error accumulation and error correction in sequential pointing movements. Experimental Brain Research 95:111-117.
Boyd, I. A., Gladden, M. H., McWilliam, P. N. & Ward, J. (1977) Control of dynamic and static nuclear bag fibres and nuclear chain fibres by gamma- and beta-axons in isolated cat muscle spindles. Journal of Physiology (London) 265:133-62.
Brown, T. G. (1914) On the nature of the fundamental activity of the nervous centres: together with an analysis of the conditioning of rhythmic activity in progression, and a theory of the evolution of function in the nervous system. Journal of Physiology 48:18-46.
Bullock, D. & Grossberg, S. (1988) Neural dynamics of planned arm movements: emergent invariants and speed-accuracy properties during trajectory formation. Psychological Reviews 95:49-90.
Burgess, P. R. (1992) Equilibrium points and sensory templates. Behavioral and Brain Sciences 15:720-22.
Caminiti, R., Johnson, P. B. & Urbano, A. (1990) Making arm movements within different parts of space: Dynamic aspects in the primate motor cortex. Journal of Neuroscience 10:2039-58.
Capaday, C. & Stein, R. B. (1986) Amplitude modulation of the soleus H-reflex in the human during walking and standing. Journal of Neuroscience 6:1308-13.
Davis, W. R. & Kelso, J. A. S. (1982) Analysis of "invariant characteristics" in the motor control of Down's syndrome and normal subjects. Journal of Motor Behavior 14:194-212.
Day, B. L., & Marsden, C. D. (1982) Accurate repositioning of the human thumb against unpredictable dynamic loads is dependent upon peripheral feedback. Journal of Physiology (london) 327:393-407.
de Graaf, J. B., Sittig, A. C. & Denier van der Gon, J. J. (1991) Misdirections in slow goal-directed arm movements and pointer- setting tasks. Experimental Brain Research 84:434-38.
DeLuca, C. J. & Mambrito, B. (1987) Voluntary control of motor units in human antagonist muscles: Coactivation and reciprocal activation. Journal of Neurophysiology 58:525-42.
Denier van der Gon, J., Tax, T., Gielen, S. & Erkelens, C. (1991) Synergism in the control of force and movement in the forearm. Reviews of Physiology, Biochemistry and Pharmacology, Vol. 118, Springer-Verlag. pp. 98-124.
DeSerres, S. J. & Milner, T. E. (1991) Wrist muscle activation patterns and stiffness associated with stable and unstable mechanical loads. Experimental Brain Research 86:451-58.
Doemges, F. & Rack, P. M. H. (1992) Changes in the stretch reflex of the human 1st dorsal interosseous muscle during different tasks. Journal of Physiology (London) 447:563-73.
Douglas, R. J. & Martin, K. A. C. (1991) Opening the grey box. Trends in Neurosciences 14:286-92.
Drew, T. & Rossignol, S. (1987) A kinematic and electromyographic study of cutaneous reflexes evoked from the forelimb of unrestrained walking cats. Journal of Neurophysiology 57:1160-84.
Feldman, A. G. (1966a) Functional tuning of the nervous system with control of movement or maintenance of a steady posture. II. Controllable parameters of the muscles. Biophysics 11:565-78.
Feldman, A. G. (1966b) Functional tuning of the nervous system with control of movement or maintenance of a steady posture. III. Mechanomyographic analysis of execution by man of the simplest motor task. Biophysics 11:667-75.
Feldman, A. G. (1979) Central and reflex mechanisms in the control of movement, Nauka. in Russian
Feldman, A. G. (1980a) Superposition of motor programs. I. Rhythmic forearm movements in man. Neuroscience 5:81-90.
Feldman, A. G. (1980b) Superposition of motor programs. II. Rapid forearm flexion in man. Neuroscience 5:91-5.
Feldman, A. G. (1986) Once more on the equilibrium-point (lambda model) for motor control. Journal of Motor Behavior 18:17-54.
Feldman, A. G. (1992) Fundamentals of motor control, kinesthesia and spinal neurons: in search of a theory. Behavioral and Brain Sciences 15:735-37.
Feldman, A. G. (1993) The coactivation command for antagonist muscles involving Ib interneurons in mammalian motor control systems: an electrophysiologically testable model. Neuroscience Letters 155:167-70.
Feldman, A. G., Adamovich, S. V., Ostry, D. J. & Flanagan, J. R. (1990) The origin of electromyograms - Explanations based on the equilibrium point hypothesis. In: Multiple muscle systems: Biomechanics and movement organization, ed. J. M. Winters & S.L.- Y. Woo. Springer-Verlag.
Feldman, A. G. & Latash, M. L. (1982) Interaction of afferent and efferent signals underlying joint position sense: Empirical and theoretical approaches. Journal of Motor Behavior 14:174-93.
Feldman, A. G. & Levin, M. L. (1993) Control variables and related concepts in motor control. Concepts in Neuroscience 4:25- 51.
Feldman, A. G. & Orlovsky, G. N. (1972) The influence of different descending systems on the tonic stretch reflex in the cat. Experimental Neurology 37:481-94.
Fetz, E. E. (1992) Are movement parameters recognizably coded in the activity of single neurons? Behavioral and Brain Sciences 15:679-90.
Fitts, P. M. (1954) The information capacity of discrete motor responses. Journal of Experimental Psychology 67:381-91.
Flanagan, J. R., Feldman, A. G. & Ostry, D. J. (1992) Equilibrium trajectories underlying rapid target-directed arm movements. In: Tutorials in motor behavior II, eds. G. E. Stelmach & J. Requin. North Holland Press. pp.661-75.
Flanagan, J. R., Ostry, D. J. & Feldman, A. G. (1990) Control of human jaw and multi-joint arm movements. In: Cerebral control of speech and limb movements, ed. G. Hammond. Springer-Verlag.
Flanagan, J. R., Ostry, D. J. & Feldman, A. G. (1993) Control of trajectory modifications in target-directed reaching. Journal of Motor Behavior 25:140-52.
Flanders, M., Tillery, S. I. H. & Soechting, J. F. (1992) Early stages in sensorimotor transformation. Behavioral and Brain Sciences 15:309-20.
Flash, T. & Henis, E. (1991) Arm trajectory modifications during reaching towards visual targets. Journal of Cognitive Neuroscience 3:220-30.
Forget, R. & Lamarre, Y. (1987) Rapid elbow flexion in the absence of proprioceptive and cutaneous feedback. Human Neurobiology 6:27-37.
Gandevia, S. C. & Burke, D. (1992) Does the nervous system depend on kinesthetic information to control natural limb movements? Behavioral and Brain Sciences 15: 614-32.
Gelfand, I. M., & Tsetlin, M. L. (1971) On mathematical modeling of mechanisms of central nervous system. In: Models of the structural-functional organization of certain biological systems, ed. I.M. Gelfand, V.S. Gurfinkel, S.V. Fomin & M.L. Tsetlin. MIT Press.
Georgopoulos, A. P., Ashe, J., Smyrnis, N. & Taira, M. (1992) The motor cortex and the coding of force. Science 233:1416-19.
Georgopoulos, A. P., Kalaska, J. F., Caminiti, R. & Massey, J. T. (1983) Interruption of motor cortical discharge subserving aimed arm movements. Experimental Brain Research 49:327-40.
Georgopoulos, A. P., Schwartz, A. B. & Kettner, R. F. (1986) Neuronal population coding of movement direction. Science 233:1416-19.
Gerilovsky, L., Struppler, A., Velho, F. & Niehage, O. (1990) Discharge pattern of tonically activated motor units during unloading. Electromyography and clinical Neurophysiology 30:459- 67. Ghez, C., Gordon, J., Ghilardi, M. F., Christakos, C. M. & Cooper, S. E. (1990) Roles of proprioceptive input in the programming of arm trajectories. Cold Spring Harbor Symposia in Quantitative Biology 55:837-47.
Gielen, C. C. A. M., van der Heuvel, P. J. M. & Denier van der Gon, J. J. (1984) Modification of muscle activation patterns during fast goal-directed movements. Journal of Motor Behavior 16:2-19.
Gielen, C. C. A. M. & Van Zuylen, E. J. (1986) Coordination of arm muscles during flexion and supination application of the tensor analysis approach. Neuroscience 17:527-39.
Glansdorff, P. & Prigogine, I. (1971) Thermodynamic theory of structure, stability and fluctuations. Wiley.
Goldberger, M. E. & Murray, M. (1974) Restitution of function and collateral sprouting in the cat spinal cord: the deafferented animal. Journal of Comparative Neurology 158:37-54.
Goodin, D. S., Aminoff, M. J., & Shih P.-Y. (1990) Evidence that the long-latency stretch responses of the human wrist extensor muscle involve a transcerebral pathway. Brain 113:1075-91.
Gottlieb, G. L. & Agarwal, G. C. (1988) Compliance of single joints: Elastic and plastic characteristics. Journal of Neurophysiology 59:937-51.
Gottlieb, G. L., Corcos, D. M. & Agarwal, G. C. (1989) Strategies for the control of voluntary movements with one mechanical degree of freedom. Behavioral and Brain Sciences 12:189-250.
Granit, R. (1970) The basis of motor control. Academic Press.
Granit, R., Kernell, D. & Lamarre, Y. (1966) Synaptic stimulation superimposed on motoneurones firing in the secondary range to injected current. Journal of Physiology (London) 187:401-15.
Grillner, S. (1969) Supraspinal and segmental control of static and dynamic gamma-motoneurones in the cat. Acta Physiologica Scandinavica Suppl. 327:1-34.
Grillner, S. (1975) Locomotion in vertebrates: central mechanisms and reflex interactions. Physiological Reviews 55:247-304.
Grossberg S. & Kuperstein, M. (1989) Neural dynamics of adaptive sensory-motor control. Pergamon.
Gutman, A. M. (1991) Bistability of dendrites. International Journal of Neural Systems 1:291-304.
Hasan, Z. (1986) Optimized movement trajectories and joint stiffness in unperturbed, inertially loaded movements. Biological Cybernetics 53:373-82.
Hasan, Z. (1991) Biomechanics and the study of multijoint movements. In: Motor control: Concepts and issues, ed. D. R. Humphrey & H.-J. Freund. Wiley.
Hasan, Z. (1992) Is stiffness the mainspring of posture and movement? Behavioral and Brain Sciences 15:756-8.
Hasan, Z. & Karst, G. M. (1989) Muscle activity for initiation of planar two-joint arm movements in different directions. Experimental Brain Research 76:651-5.
Hellgren, J. & Kellerth, J.-O. (1989) A physiological study of the monosynaptic reflex responses of cat spinal `-motoneurons after partial lumbosacral deafferentation. Brain Research 488:149-62.
Henneman, E. (1981) Recruitment of motoneurons: the size principle. In: Progress in clinical neurophysiology, Vol. 9, Motor unit types, recruitment and plasticity in health and disease, ed. J. E. Desmedt. Karger.
Hoffer, J. A. & Andreassen, S. (1981) Regulation of soleus muscle stiffness and viscosity in premammillary cats: mechanical and reflex components. Journal of Neurophysiology 45:267-85.
Hogan, N. (1984) An organizing principle for a class of voluntary movements. Journal of Neuroscience 4:2745-54.
Hogan, N. & Flash, T. (1987) Moving gracefully: quantitative theories of motor coordination. Trends in Neuroscience 10:170-4
Hollerbach, J. M. & Atkeson, C. G. (1987) Deducing planning variables from experimental arm trajectories: pitfalls and possibilities. Biological Cybernetics 56:279-92.
Holst, E., von, & Mittelstaedt, H. (1950) The reafference principle: Interaction between the central nervous system and the periphery. Naturwissenschaften, 37:464-76.
Hore, J., McCloskey, D.I. & Taylor, J.L. (1990) Task-dependent changes in gain of the reflex response to imperceptible perturbations of joint position in man. Journal of Physiology (London) 429:309-21.
Houk, J. C. & Rymer, W. Z. (1981) Neural control of muscle length and tension. In: Handbook of physiology. Section 1: The nervous system, vol. 2, Motor control, ed. J. M. Brookhart, V. B. Mountcastle, V. B. Brooks & S. R. Geiger. American Physiological Society.
Hounsgaard, J., Hultborn, H., Jespersen, B. & Kiehn, O. (1988) Bistability of `-motoneurones in the decerebrate cat and in the acute spinal cat after intravenous 5-hydroxytryptophan. Journal of Physiology (London) 405:345-67.
Hugon, M., Massion, J. & Wiesendanger, M. (1982) Anticipatory postural changes induced by active unloading and comparison with passive unloading in man. Pflgers Archiv 393:292-6.
Hulliger, M., Nordh, E. & Vallbo, ~. B. (1982) The absence of position response in spindle afferents units from human finger muscles during accurate position holding. Journal of Physiology (London) 322:167-79.
Humphrey, D. R. & Reed, D. J. (1983) Separate cortical systems for control of joint movement and joint stiffness: reciprocal activation and coactivation of antagonist muscles. Advances in Neurology 39:347-72.
Jami, L. (1992) Golgi tendon organs in mammalian skeletal muscle: Functional properties and central actions. Physiological Reviews 72:623-66.
Jankowska, E. (1992) Interneuronal relay in spinal pathways from proprioceptors. Progress in Neurobiology 38:335-78.
Jeannerod, M. (1988) The neuronal and behavioral organization of goal-directed movements. Clarendon Press.
Jongen, H. A. H., Denier van der Gon, J. J. & Gielen, C. C. A. M. (1989) Inhomogeneous activation of motoneuron pools as revealed by co-contraction of antagonistic human arm muscles. Experimental Brain Research 75:555-62.
Kaas, J. (1991) Plasticity of sensory and motor maps in adult mammals. Annual Review of Neuroscience 14:137-67.
Kalaska, J. F. (1991) Reaching movements to visual targets: Neuronal representations of sensori-motor transformations. Seminars in the Neurosciences 3:67-80.
Kalaska, J. F. & Crammond, D. J. (1992) Cerebral cortical mechanisms of reaching movements. Science 255:1517-23.
Kay, B. A., Saltzman, E. L. & Kelso, J. A. S. (1991) Steady-state and perturbed rhythmical movements: A dynamical analysis. Journal of Experimental Psychology: Human Perception and Performance 17:183-97.
Kernell, D. & Hultborn, H. (1990) Synaptic effects on recruitment gain: a mechanism of importance for the input-output relations of motoneurone pools? Brain Research 507:176-9.
Kugler, P. N. & Turvey, M. T. (1988) Self-organization, flow fields, and information. Human Movement Science 7:97-129.
Kuperstein, M. (1988) Neural model of adaptive hand-eye coordination for single postures. Science 239:1308-11.
Lackner, J. P. & DiZio, P. (1993) Factors contributing to initial reaching errors and adaptation to Coriolis force perturbations. Society for Neuroscience Abstracts 19:1595.
Lackner, J. R. & DiZio, P. (1992) Gravitoinertial force level affects the appreciation of limb position during muscle vibration. Brain Research 592:175-80.
Lacquaniti, F. (1989) Central representations of human limb movement as revealed by studies of drawing and handwriting. Trends in Neuroscience 12:287-91.
Lacquaniti, F. (1992) Automatic control of limb movement and posture. Current Opinion in Neurobiology 2:807-14.
Lacquaniti, F., Borghese, N. A., & Carrozzo, M. (1991) Transient reversal of the stretch reflex in human arm muscles. Journal of Neurophysiology 66:939-54.
Lan, N. & Crago, P. E. (1992) Equilibrium-point hypothesis, minimum effort control strategy and the triphasic muscle activation pattern. Behavioral and Brain Sciences 15:769-71.
Latash, M. L. (1992) Are we able to preserve a motor command in the changing environment? Behavioral and Brain Sciences 15:771- 73.
Latash, M. L. (1993) Control of human movement. Human Kinetics Publishers, Champaign.
Latash, M. L. & Gottlieb, G. L. (1991) Reconstruction of shifting elbow joint compliant characteristics during fast and slow movements. Neuroscience 43:697-712.
Levin, M. F., Feldman, A. G., Milner, T. E. & Lamarre, Y. (1992) Reciprocal and coactivation commands for fast wrist movements. Experimental Brain Research 89:669-77.
Loeb, G. E. (1985) Motoneurone task groups: coping with kinematic heterogeneity. Journal of Experimental Biology 115:137-46.
Loeb, G. E. (1989) Strategies for the control of studies of voluntary movements with one mechanical degree of freedom. Behavioral and Brain Sciences 12:227.
Loeb, G. E. (1992) Past the equilibrium-point. Behavioral and Brain Sciences 15:774-75.
Loeb, G. E. & Levine, W. S.(1990) Linking musculoskeletal mechanics to sensorimotor neurophysiology. In: Multiple muscle systems: Biomechanics and movement organization, ed. J. M. Winters & S. L.-Y. Woo. Springer-Verlag.
Lundberg, A. (1975) Control of spinal mechanisms from the brain. In: The nervous system, vol. 2. ed. D. B. Tower, Raven.
Matthews, P. B. C. (1959) The dependence of tension upon extension in the stretch reflex of the soleus of the decerebrate cat. Journal of Physiology (London) 47:521-46.
Matthews, P. B. C. (1972) Mammalian Muscle Receptors and Their Central actions. Edward, Arnold, London
Matthews, P. B. C. (1981) Muscle spindles: Their messages and their fusimotor supply. In: Handbook of physiology, Sect. 1: The nervous system, vol. 2, part 1. American Physiological Society.
McCloskey, D. J. (1978) Kinaesthetic sensibility. Physiological Reviews 58:763-820.
McCrea, D. A. (1992) Can sense be made of spinal interneuron circuits? Behavioral and Brain Sciences 15:633-43.
McIntyre, J. & Bizzi, E. (1992) Servo models for the biological control of movement. Journal of Motor Behavior 25:193-202.
Melzack, R. (1989) Phantom limbs, the self and the brain: The D. O. Hebb Memorial Lecture. Canadian Psychology 30:1-16.
Merton, P.A. (1953) Speculations on the servo control of movement. In: The spinal cord, ed. G. E. W. Wolstenholme. Churchill.
Merzenich, M. M., Recanzone, G., Jenkins, W. M., Allard, T. T. & Nudo, R. J. (1988) Cortical representational plasticity. In: Neurobiology of neocortex, ed. P. Rakic & W. Singer. Wiley.
Meyer, D. E., Smith, J. E. K. & Wright, C. E. (1982) Models for the speed and accuracy of aimed movements. Psychological Reviews 89:449-82.
Miller, L. E., Gielen, C. C. A. M., Theeuwen, M. & Doorenbosch, C. (1992) The activation of mono- and biarticular muscles in multijoint movements. In: Control of firm movement in space; neurophysiological and computational approaches, eds. R. Caminiti, P. B. Johnson & Y. Burnod.
Milner, T. E. & Ijaz, M. (1990) The effect of accuracy constraints on three-dimensional movement kinematics. Neuroscience 35:365-74
Milner-Brown, H. S. & Stein, R. B. (1975) The relation between the surface electromyogram and muscular tension. Journal of Physiology (London) 246:259-62.
Munoz, D. P., Plisson, D. & Guitton, D. (1991) Movement of neural activity on the superior colliculus motor map during gaze shifts. Science 251:1358-60.
Mussa-Ivaldi, F. A., Morasso, P. & Zaccaria, R. (1988) Kinematic networks. A distributed model for representing and regularizing motor redundancy. Biolo