Neuroscience Vol. 49, No. 1, pp. 209-220, 1992

0306-4522/92 $5.00 + 0.00 Pergamon Press Ltd © 1992IBRO

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VIRTUAL TRAJECTORIES, JOINT STIFFNESS, AND CHANGES IN THE LIMB NATURAL FREQUENCY DURING SINGLE-JOINT OSCILLATORY MOVEMENTS M. L. LATASH Department of Physical Medicine and Rehabilitation, Department of Physiology, Rush-Presbyterian St Luke's Medical Center, 1753 W. Congress Parkway, Chicago, IL 60612, U.S.A. Abstract--ln the framework of the equilibrium-point hypothesis, virtual trajectories and patterns of joint stiffness were reconstructed during voluntary single-joint oscillatory movements in the elbow joint at a variety of frequencies and against two inertial loads. At low frequencies, virtual trajectories were in-phase with the actual joint trajectories. Joint stiffness changed at a doubled frequency. An increase in movement frequency and/or inertial load led to an increase in the difference between the peaks of the actual and virtual trajectories and in both peak and averaged values of joint stiffness. At a certain, critical frequency, virtual trajectory was nearly flat. Further increase in movement frequency led to a 180° phase shift between the actual and virtual trajectories. The assessed values of the natural frequency of the system "limb + manipulandum" were close to the critical frequencies for both low and high inertial loads. Peak levels and integrals of the electromyograms of two flexor and two extensor muscles changed monotonically with movement frequency without any special behavior at the critical frequencies. Nearly flat virtual trajectories at the natural frequency make physical sense as hypothetical control signals, unlike the electromyographic recordings, since a system at its natural frequency requires minimal central interference. Modulation of joint stiffness is assumed to be an important adaptive mechanism attenuating difference between the system's natural frequency and desired movement frequency. Virtual trajectory is considered a behavioral observable. Phase transitions between the virtual and actual trajectories are illustrations of behavioral discontinuities introduced by slow changes in a higher level control parameter, movement frequency. Relative phase shift between these two trajectories may be considered an order parameter.

The equilibrium-point hypothesis (EP-hypothesis), in its original ).-version, considers single-joint voluntary movements as consequences of time shifts of hypothetical centrally specified variables r and c related to position and slope of the joint compliant characteristic (JCC). 7-9'11 These two variables are simple functions of the thresholds of the tonic stretch reflex* for the agonist and antagonist muscles, 2as and )`ant'10 For given values of r and c, the joint behaves like a non-linear spring, its equilibrium state being defined by the external load. A centrally induced change in position and/or slope of the JCC leads to a discrepancy between actual joint position and equilibrium position. This discrepancy leads in turn to muscle activation and generation of joint torque directed to move the joint to a new equilibrium state and, subsequently, to a voluntary movement. During

Abbreviations: EMG, electromyogram, electromyographic;

EP, equilibrium point; JCC, joint compliant characteristic. *The term "tonic stretch reflex" has been traditionally used by Feldman for describing the reflex effects of the peripheral receptors during slow changes in muscle length leading to the experimentally observed shape of the JCCs. 2,6'9,11We shall continue to use this term for the combined reflex effects of the peripheral receptors during fast movements, keeping in mind that, in this case, the word "tonic" should not be taken literally,

central shift of the JCC, there is always a lag between equilibrium position and actual instantaneous position in the joint. This lag is due, in particular, to inertial and viscous properties of the peripheral apparatus. If the limb were massless and without damping, the muscles did not have low-pass filtering properties, and there were no changes in the external load, joint trajectory would be exactly the same as the centrally specified equilibrium trajectory, probably shifted along the time axis due to the conduction delays. Let us use the term "virtual trajectory" for a centrally specified e~uilibrium trajectory that would be followed by such an idealized limb. Imagine Huckleberry F i n n with a dead rat on a rope. If Huckleberry wants to move the rat he can do so only by moving his end of the rope. If the rope is substituted with a short metal rod, the rat will closely follow the virtual trajectory specified by M r Finn's hand. If the rope is substituted with a long elastic piece of rubber, the rat's actual trajectory may be very different from the virtual movements of Huckleberry's hand. Magnitude of these deviations will depend, in particular, on mass of the rat, elasticity of the rubber, and desired speed of the movement. Until recently, virtual trajectory remained an abstraction since the EP-hypothesis was based exclusively on experimental reconstruction of JCCs in "static" conditions but not during voluntary

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changes in the hypothetical central commands r and

c'2"sa'6"'2"22'31Asaresult'theEP'hyp°thesiswascriti- r cized as being unable to account for movement dynamics. 3.26 We have recently introduced a method of reconstructing JCCs during discrete single-joint voluntary movements at a variety of speeds. 23,25The method is based on comparing the changes in joint torque and kinematics during relatively slow changes in the external conditions of the movement execution, i.e. loading. It relies upon the ability of the subjects to reproduce the same motor command ("to do the same") when the external conditions change. Time shifts of JCCs enable reconstruction of time changes in joint stiffness and equilibrium position, i.e. virtual trajectory thus making it an observable of human motor behavior. During slow elbow flexion movements, the virtual trajectory was monotonic and slightly ahead of the actual joint trajectory, while joint stiffness did not demonstrate visible changes, During fast single-joint movements, the virtual trajectory was non-monotonic, N-shaped, and joint stiffness had a transient increase in the middle of the movement. The existence of the N-shaped virtual trajectories implies an independent central control upon the process of braking fast movements with a delayed shift of the control variable for the antagonist muscle, 2ant.24 A different control pattern for fast movements has been suggested by Feldman and his colleagues. L~ It does not include any delayed central events and considers braking a consequence of purely feedback reflex action. This basic difference in the two hypothesized control patterns for discrete movements has also been reflected in the control patterns suggested for single-joint oscillatory movements, Feldman7 has suggested two types of control patterns for single-joint oscillatory movements. The first one is used at relatively low frequencies and consists of discrete shifts of 2s of the flexor and extensor muscles leading to a smooth combination of discrete flexion and extension movements. At higher frequencies, stiffness of the joint is being held relatively constant at a certain level dependent upon the desired movement frequency. This is accomplished by controlling the slope of the with the help of the c variable. The r variable is shifting back and forth between certain limits corresponding to the desired amplitude of the oscillatory movement (Fig. 1A). On

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the other hand, if one considers an oscillatory movement as a smooth combination of targeted flexion and extension movements, the experiments with reconstruction of N-shaped virtual trajectories suggest a different control pattern (Fig. 1B). In particular, they imply phasic changes in joint stiffness at a doubled frequency with a peak during each flexion and each extension movement, and the possibility of N-shaped non-monotonic changes in r during both flexions and extensions, There are two aspects that make it important to reconstruct virtual trajectories and joint stiffness

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~ Fig. 1. Two hypothetical control patterns for fast singlejoint oscillatory movements. (A) According to Fetdman,7'8 the coactivation variable, c, which defines the slope of the JCC stays constant, while the other variable, r, which defines position of the JCC oscillates between two extreme values. (B) According to our recent findings, c may be expected to change at a doubled freqtmncy while r may demonstrate N-shaped form during both flexion and extension phases of the oscillatory movement. changes during oscillatory movements. First, it will support one (or neither) of the two types of control patterns hypothesized in the framework of the 2model. Second, it will provide a link between the hypothetical control patterns and physical properties of the system "joint + reflexes". Discrete movements and corresponding control patterns are significantly affected by the transient processes of initiation and termination of the movement, while oscillatory movements may be considered sustained regimes of functioning of the motor system. Changing inertial load is a simple way of studying the effects of changes in the natural frequency upon the control patterns, while changing movement frequency is a way to study the effects of the difference between the actual and natural frequencies upon the control patterns. The following experiments were performed reconstructing JCCs during single-joint elbow movements at different frequencies and against different inertial loads (a preliminary report can be found in Ref. 21). EXPERIMENTALPROCEDURES

Subjects Five neurologically healthy male volunteers, aged 29-51, participated in the study after giving an informed consent according to the procedure approved by the Human Investigation Committee of the Rush-Presbyterian-St Luke's MedicalCenter. All the subjects were right-handed.

Apparatus

The subjects sat comfortably in a chair, adducted the right shoulder 90°, and positioned the right forearm on a horizontal, low friction, light weight manipulandum (moment of inertia is 0.086 Nm s2/rad). The axis of rotation of the lower arm corresponded to the center of the elbow joint. A video monitor, positioned about 1.5 m in front of the subject,continuously displayed position of the limb and two referencepositions (1° wide each). The nominal amplitude of the movements was kept constant in all the series and equal to 36° with a mid-point at 90° of flexion in the elbow joint. A metronome specified the desired frequency of the oscillatory movements. A digital computer controlled the experiment and digitized and recorded joint angle, acceleration, torque, and electromyograms (EMGs) at I000 Hz. Elbowangle was measured by a variable capacitance transducer mounted on the shaft of the manipulandum. Tangential acceleration was measured by an accelerometer mounted on the distal end of the manipulandum. A strain~gauge transducer was mounted on the vertical shaft of the manip-

Control of oscillatory movements ulandum and measured rotational torque around its axis. Its output was low-pass filtered at 25 Hz. Inertial torque (Ti~) was calculated by multiplying the acceleration by an estimated value for the moment of inertia of the manipulandum and the limb (cf. Ref. 27). The estimated moment of inertia for different subjects varied by less than 10%. Therefore, in the experiments with low inertial load, a total moment of 0.185 Nm s2/rad was used for all the subjects. Three subjects took part in the experiments with a higher inertial load where the nominal total moment of inertia was estimated as 0.57 Nm s2/rad, In three subjects, EMGs of biceps, brachioradialis, and lateral and long heads of triceps were recorded. Pediatric electrocardiographic self-adhesive electrodes were taped over the muscles and used for a bipolar EMG recording, The EMGs were amplified (1600x), band-pass filtered (60-500 Hz), and digitized with 12 bit resolution at 1000 Hz. A torque motor provided the constant extending bias torque (4.5 Nm in all the experiments). A non-zero extending bias torque was used in the studies of discrete movements to assure reproducible initial values of the hypothetical control variables. 23,25A similar bias was used in this study in order to be able to compare the findings during oscillatory and discrete movements. The same motor provided time-varying external torque. Each series consisted of 2.5 rain of oscillatory movement at a metronome-specified frequency. During this time, twenty 2-s episodes were recorded, during 10 of which a ramp change in the torque over 800 ms took place. The other 10 episodes were unperturbed. The perturbations were delivered by the computer pseudo-randomly, i.e. unpredictably but with a balanced number of different final torque values. The new level of torque was maintained for 1.1 s and returned to the bias value. The final values of torque in perturbed episodes were approximately 0, 1.5, 3, 6, and 7.5 Nm. Each value was repeated twice during one series.

211

were always performed first, and the experiments with the high inertial load were performed at a later day. All individual 2-s episodes from a series were reafigned in time to an origin corresponding to the point of maximal flexion (in most of the series) or to the point of maximal extension. The method of alignment depended upon the phase of the oscillatory movement when most of the perturbations started and was kept the same for all the episodes within one series. The point of alignment (to)could be easily detected by visual inspection of individual episodes on the computer monitor. We assume that at t o the values of r and c were the same for each episode of a series (r o and co). Three aligned episodes, loaded, unloaded, and unperturbed, are shown schematically in Fig. 2. Because of different changes in the external torque (Tc~),joint trajectories (~)are different although the central patterns (r and c) are presumably the same. Arbitrary functions are used to illustrate the r(t) and c(t) functions in Fig. 2. The method of analysis is based on a presumption that at any time t~ > to, there exists a linear JCC, characterized by two parameters related to its position and slope, r i and ci (Fig. 2). External torque, angle, and acceleration were measured each 50 ms after to until the time to + 1500 ms (each 25 ms for the movements at 2.1 and 2.5 Hz until the time to + 750 ms). Measurements at each t~ corresponded to the same pair (ra, q) or to the same position and slope of the JCC. If the subject was successfully reproducing the same motor command in successive episodes, plotting the values of total torque (calculated as a sum of external and inertial components, T = Tc~ + T~) and angle measured at the same t~ in different episodes of one series would give a set of points pertaining to the same JCC but measured at different levels of external torque.

Instruction and training The subjects were instructed to perform smooth oscillatory movements in the elbow joint between two reference positions shown at the monitor screen at a frequency specified by the metronome. They were told not to pay attention to accuracy, and to consider the reference positions as points. The subjects were instructed "to practice a movement until it becomes automatic" and given about 3 min for the practice before the first series. In the experiments with the low inertial load, the movement frequencies were 0.5, 1, 1.25, 1.67, and 2.1 Hz. In the experiments with the high inertial load, the frequencies were 0.5, 0.75, 1, and 1.25 Hz. In one subject, movements at 2.5 Hz against the low load and at 1.67 Hz against the high load were also recorded and analysed. At higher frequencies, the subjects reported being tired towards the end of the 2.5-min

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The general instruction was "not to intervene voluntarily" or "do the same, no matter what the motor does". The subjects were given additional explanation not to correct their movements even if they start missing the reference positions because of a change in the external torque and had one practice series of movements at 0.5 Hz which was not analysed. All our subjects had previous experience of participation in the experiments requiring "not to intervene voluntarily". They all agreed that "doing the same" during oscillatory movements was subjectively easier than during discrete single-joint movements,

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to ti Fig. 2. The recorded episodes were aligned according to a certain phase of the oscillatory movement. The time of alignment, t o presumably corresponds to the same values of r and c (to and co) in all the episodes. After alignment, the hypothetical control functions r(t) and c(t) overlap, while joint trajectories (a) may be different due to different external torques (T,~). Three episodes are illustrated, unperturbed, loaded, and unloaded corresponding to different joint trajectories but presumably the same central patterns of r and c. Measurements of joint angle and calculations of torque at a time t~ after to in different episodes will give values of joint angle in different loading conditions but corresponding to the same r a and Ca, i.e. to the same JCC.

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The JCCs were obtained by linear regression analysis of the datapoints from one series corresponding to a value of ti. Linear regression analysis was used for simplicity and also because it yielded very high coefficients of correlation (see the Results). Each regression line was characterized by two parameters: slope and intercept: k~ I

records. The averaged EMGs were integrated over one cycle and then normalized with respect to the cycle duration (multiplied by a ratio of 1000 ms to the cycle duration for each movement frequency). The EMG measures were also normalized, for each of the subjects, with respect to the highest value which was always observed during the movernents at the highest frequency (2.1 Hz).

where T is total torque, ~t is joint angle, and k~ and k 2 are constants for a given t,. If the limb can be described during voluntary movement as a controllable linear spring, then there should be a high correlation coefficient for the regression between T and ~t. If there is a reproducible modulation of those spring properties during voluntary movements, then k~ and k 2 can also be expected to consistently vary with t,. JCC can be characterized by an equation:

RESULTS

ct = r + f ( c ) T , (2) wherefis a monotonically decreasing function. Equation (2) is identical in form to equation (I). Therefore, we can equate the coefficients in equations (I) and (2): k~ 1 r = andf(c)=-~ . (3) -

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The values of ct and T for each t, for the episodes of one series were plotted on separate graphs (total torque vs angle). Linear regression analyses were performed and the values of slope and intercept were calculated and later plotted versus t~. Changes of the intercept in angle units (equivalent to the virtual trajectory, see Introduction) were compared to the average trajectory of the aligned unperturbed episodes (actual trajectory) for each experimental condition. The EMGs of the aligned unperturbed episodes were rectified, filtered, and averaged with a 25 ms moving window. Peak EMG values were measured in the averaged

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The values of j o i n t angle a n d torque (calculated as a sum of the external a n d inertial c o m p o n e n t s ) for each t~ in each set o f the analysed episodes of oscillatory movements demonstrated high coefficients of linear correlation for all the subjects, in all the experimental conditions. Examples of datasets a n d regression lines for the times 250, 500, 750, a n d 1000 ms after to for one of the subjects are shown in Fig. 3. The total n u m b e r of datasets analysed was 1170. In approximately 80% of these, the correlation coefficients were over 0.9. W h e n the coefficient of linear correlation was u n d e r 0.7, the dataset was discarded a n d not incorporated into the future analysis. This h a p p e n e d in less t h a n 5 % of all the datasets. It must be r e m e m b e r e d that the regression equations have been assumed to represent instantaneous positions of the J C C at times t,. Their coefficients were used for calculating slope a n d intercept of the J C C [see e q u a t i o n (3)]. The slope of the J C C has been considered to represent j o i n t stiffness, while the intercept in angle units has been equated with virtual trajectory.

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Control of oscillatory movements Averaged unperturbed (actual) and virtual trajectories for one of the subjects for oscillatory movements at three frequencies against the low inertial load are illustrated in Fig. 4A. At the lower frequency, the virtual trajectory was nearly in-phase with the actual trajectory (the left graph in Fig. 4A). An increase in the movement frequency led to an increase in the difference between the peaks of the two trajectories without a change in the phase relation. At a certain, critical frequency, the virtual trajectory was nearly flat (the middle graph in Fig. 4A). Further increase in movement frequency led to an abrupt phase shift of 180 ° between the actual and virtual trajectories (the right graph in Fig. 4A). This phase shift occurred in four subjects between 1.25 and 1.67 Hz, and in the fifth, "non-typical" subject, between 1 and 1.25 Hz. An increase in movement frequency could lead to a visible "notch" at times of maximal joint velocity (the fight graph in Fig. 4A) resembling the N-shaped virtual trajectory (cf. Fig. 1B). The difference between the peak values of actual and virtual trajectories (A - V) measured at the same times increased monotonically with movement frequency (Fig. 5A). The critical frequency can be assessed from this graph assuming that it occurred at a hypothetical movement frequency when the virtual trajectory was absolutely fiat. Since nominal movement amplitude was always 36 °, it theoretically corresponds to a frequency when (A - V) = 18°. For four of the subjects (black dots, thin lines), the critical frequencies were in a relatively narrow range 1.35-1.45 Hz (shown by a bar on the abscissa in Fig. 5A), while for the fifth subject (open circles, bold line)

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Joint stiffness changes The regression coefficient k2 related to joint stiffness has demonstrated a pronounced modulation within a period of the oscillatory movements (Fig. 4B). It reached peaks at approximately times of maximal velocity during both flexion and extension phases. Both peak and average values of kz increased with movement frequency (Fig. 5B). For quantitative assessments of changes in joint stiffness and virtual trajectory, we have used the data from the first cycles in the analysed 2-s episodes because all our subjects demonstrated some degree of yielding towards the end of the analysis epoch. This yielding was obvious in the decreasing phasic changes in k2 and was more pronounced at the lower frequencies (cf. the left and the fight graphs in Fig. 4B).

The effects of inertial loading Movements against an increased inertial load have shown qualitatively similar patterns of changes in the difference between the peaks of actual and virtual trajectories (A - V) and joint stiffness (k2) with movement frequency (bold lines in Fig. 6). Note, however, that both (A - V) and k2 increased more steeply with movement frequency than for the movements against the low inertial load. A phase shift between the actual and virtual trajectories occurred at lower frequencies (about 0.75-1 Hz for all three subjects, cf. two bars on the abscissa of Fig. 6A) as assessed by equating (A - V) to 18 ° (see above). This

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FREQUENCY Fig. 5. (A) Changes in the difference between the peaks of actual and virtual trajectories (A - V in degrees) and (B) changes in the peak and average values of the coefficient k2 (in Nm/deg) with movement frequency (in Hz) for all five subjects. Linear regression equations and coefficients of linear correlation for the pooled data are shown. The critical frequency was assessed by equating the difference (A - V) to 18°. It is shown by a black bar in A below the abscissa axis. The non-typical subject (open circles, bold lines), had higher values of (A - V), a correspondingly lower value of the critical frequency (black dot under the abscissa axis in A), and lower values of k v

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FREQUENCY Fig. 6. (A) Changes in the difference between the peaks of actual and virtual trajectories (A - V in degrees) and (B) changes in the peak and average values of the coefficient k2 (in Nm/deg) with movement frequency (in Hz) for three subjects moving against the low (thin lines) and high (bold lines) inertial load. The data for the non-typical subject are shown by open circles (cf. Fig. 5). Note the higher values of both (A - V) and k2 for the movements against the high inertial load. The range of critical frequencies is lower for the movements against the high load (a bar above the abscissa axis in A, Fh) than for the movements against the low load (a bar below the abscissa axis in A, Fj),

Control of oscillatory movements difference was statistically significant (P < 0.001 according to the two-tailed paired Student's t-test).

Critical and natural frequencies Phase shift between the actual and virtual trajectories is plotted as a function of movement frequency in Fig. 7. At a certain, critical frequency, the phase shift jumps from in-phase values of about 0 to out-of-phase values of about 3.14 (in radians). Note that the subject who demonstrated a phase shift earlier (open circles in Fig. 7A) also had a correspondingly lower assessment of the critical frequency based on equating the difference (A - V) to 18° as compared with the other four subjects (cf. the black dot and the black bar on the abscissa axis redrawn from Fig. 5A). The ranges of the critical frequency for the movements against the low (black dots, Fig. 7B) and high (open circles, Fig. 7B) inertia also correspond to their previous assessments (cf. the bars on the abscissa axis redrawn from Fig. 6A). The natural frequency of the limb was assessed using values of joint stiffness averaged across the first half-cycle and the assessed values of the moment of inertia. Changes in the natural frequency with movement frequency for the oscillatory movements against the low inertial load for all five subjects are shown in A

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FREQUENCY Fig. 7. Phase shift (in radians) between the actual and virtual trajectories as a function of movement frequency (in Hz). (A) The data for all five subjects moving against the low inertial load. The data for the "non-typical" subject are shown by open circles. (B) The data for three subjects moving against the low (block dots) and high (open circles) inertial load. Critical frequencies, as assessed by equating (A - V) to 18° are shown at the abscissa axes (cf. Figs 5 and 6).

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FREQUENCY Fig. 8. Natural frequency (in Hz) of the system "limb + manipulandum"was assessed using averages across the first half-cycle values of joint stiffness (kz). The upper graph shows the results for all five subjects moving against the low inertial load. The black bar (black dot for the "non-typical" subject) shows the range of movement frequencies when they are equal to the natural frequency of the system. The lower graph shows the data for three subjects moving against the low (open circles) and high (closed circles) inertial load. the arrows show approximately the ranges of movement frequencies when they are equal to the natural frequency of the system. Compare with the assessments of critical frequencies in Figs 5-7. Fig. 8A. Data for the three subjects who took part in the experiments with the low and high inertial loads are presented in Fig. 8B. Let us emphasize the following general features of the behavior of the natural frequency. First, the natural frequency of the limb increases with movement frequency due to an increase in the average value of joint stiffness (cf. Figs 5B and 6B). Second, the natural frequency is equal to movement frequency (points of intersection with the 45 ° solid line in Fig. 8) close to the critical frequencies when the phase transition between the actual and virtual trajectories occurred (cf. Fig. 7). Third, the subject who has demonstrated an earlier phase transition (Figs 5A, 7A) corresponding to a lower critical frequency also had lower values of joint stiffness (Fig. 5B) and correspondingly lower natural frequency of the limb. His data are emphasized in all the figures.

Electromyographic patterns EMG recordings and analysis were performed in three subjects during movements against the low inertial load. Both flexor (biceps and brachioradialis)

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triceps) demonstrated clear E M G modulation at the movement frequency (Fig. 9). At the lower frequencies, the extensors seem to increase their activity simultaneously with the flexors (1 and 1.67 Hz in Fig. 9) although at I Hz, the modulation of the E M G s within the cycle was barely visible. An increase in the movement frequency led to the emergence of two distinct bursts in the extensor muscles, in-phase and out-of-phase with the flexor bursts (2.1 Hz, dotted E M G traces in Fig. 9). The lack of the out-of-phase extensor component at the low frequencies was probably due to the fact that the external extending torque was enough to move the limb during the extension phases of the oscillations, so that the subjects just needed to relax the flexors. The changes in the peak and integrated E M G s of the biceps and lateral head of triceps with movement frequency are shown in Fig. 10. The E M G s were integrated in the averaged records over one movement cycle and then normalized with respect to the duration. Since the data for each of the subjects demonstrated similar behavior, the regression lines and coefficients of linear correlation for the pooled data are presented. Both peak and integrated values of the E M G s for all the muscles have demonstrated

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1 2 FREQUENCY Fig. 10. Peak EMG values for the biceps (flexor, open symbols) and lateral head of triceps (extensor, closed symbols) were measured in the averaged unperturbed records (the upper graph). The EMGs were also integrated over one movement cycle and normalized with respect to the cycle duration (the lower graph). All the EMG values were normalized, for each subject, with respect to the highest value which was invariably observed during the 2.1 Hz movements. The data for each of the subjects demonstrated similar behavior, so the regression lines and coefficients of linear correlation are shown for the pooled data~ EMGs of the other flexor (brachioradialis) and the other extensor (long head of triceps) behaved similarly and, therefore, are not illustrated. The vertical lines show the range of the critical frequencies (F~) when the phase transition between the virtual and actual trajectories took place. The EMG scales are in arbitrary units; the frequency scale is in Hz.

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a monotonic increase with movement frequency without any special behavior at the critical frequencies whose range is shown by vertical lines in Fig. 10. Note that all the subjects had maximal values of all the E M G measures during movements at the highest frequency (2.1 Hz). Therefore, the points corresponding to 2.1 Hz in Fig. 10 are superpositions of three closed and three open symbols.

DISCUSSION

1500

TIME Fig. 9. Above: averaged EMGs of the biceps and lateral head of triceps during oscillatory movements at 1, 1.67, and 2.1 Hz. Below: averaged unperturbed joint trajectories. The point of alignment (maximal extension) corresponds to 200 ms, Note the emergence of two distinct triceps bursts at the highest frequency. EMG scales are in #V; angle scale is in degrees; time scale is in ms.

The reconstructed patterns of virtual trajectories and joint stiffness changes during single-joint oscillatory movements are similar to those illustrated in Fig. 1B. The similarities include considerable phasic changes in joint stiffness at a doubled frequency and similar to N-shape virtual trajectories observed at the higher movement frequencies. We have not observed

Control of oscillatory movements reversals of the virtual trajectories that would resemble those reported in an earlier study of very fast discrete elbow flexion movements. 23,25 This is probably due to the difference between joint velocities in the two studies, e.g. peak velocities in the earlier study reached about 500°/s while in the present study, they were always under 300°/s. The general similarity of the predicted and reconstructed patterns supports both our hypothesized control pattern for such movements and the method used for the reconstruction of the virtual trajectories, Mechanisms o f control o f movement frequency

There seem to be two mechanisms associated with an increase in movement frequency. First, there is an increase in the difference between the peak values of the actual and virtual trajectories (A - V). This difference is assumed to be a measure of generated joint torque, and its increase is a reflection of a simple physical notion that in order to increase movement frequency one needs to use higher joint torques, Second, there is an increase in joint stiffness, both average and peak, leading to a corresponding increase in natural frequency of the limb. This is already a more subtle mechanism. If a spring-like system oscillates at its natural frequency, it does not require application of high external forces (only to compensate for the energy dissipation). The further the actual frequency is from the natural frequency, the higher the required external forces to sustain the oscillations. So, an increase in the natural frequency with movement frequency is a mechanism attenuating the difference between the actual and natural frequencies, thus decreasing the torque requirements. From this view, an increase in joint stiffness looks like a very important adaptive mechanism which decreases the necessary difference (A - V) for a given movement frequency. One of our subjects was different from the others in his strategy. He did not increase joint stiffness as much as the other subjects but rather used considerably higher values of (A - V) (his data are stressed in all the figures), Moreover, an increase in joint stiffness counteracts changes in the natural frequency that would otherwise be introduced by an increase in the system's inertia. In the experiments, an increase in the inertial load led to a considerable increase in joint stiffness, This observation is also consistent with the idea that the purpose of the system is to keep the natural frequency as close as possible to actual movement frequency by the accessible means, i.e. by changing joint stiffness. A similar conclusion has recently been drawn by Bingham et al. 5 who studied simultaneous rhythmic movements of two limbs against different inertial loads and observed corresponding differences in the stiffness values for the left and right joints, Note also that the importance of natural frequency for patterns of different types of voluntary movemerits has been stressed by Turvey et al. 29

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There seem to be two extremes in the way one can control single-joint oscillatory movements. First, not to modulate joint stiffness but to change the difference (A - V) so that the required frequency is iraposed. Second, to modulate joint stiffness so that the required frequency is close to the natural frequency of the limb thus virtually eliminating the necessity to modulate ( A - V). At the natural frequency, the system is supposed to require minimal central interference, and we have indeed observed minimal changes in the virtual trajectory a t the critical frequencies close to the assessed natural frequency of the system. Apparently, our subjects preferred to use an intermediate strategy, that is to modulate stiffness and (A - V ) simultaneously in response to the changes in the inertial load and/or requirements to movement frequency. It is possible that some kind of optimization takes place penalizing very high coactivation levels (very high joint stiffness) as well as very high amplitude changes in the virtual trajectory. Such an optimization approach has recently been used by Lan and Crago 2° based on an earlier model of Hasan. ~4 This approach has led to predictions of phasic changes in joint stiffness and similar to Nshape virtual trajectories during unidirectional discrete single-joint movements similar to the patterns observed in our earlier experiments. 23'25 Virtual trajectory--an observable It seems appropriate to try to strip the notion of "virtual trajectory" of its mystic nature and bring it into the family of observables during human vohintary movements. Let us recall that this notion has been introduced for single-joint movements as a centrally specified equilibrium trajectory associated with a function r(t). Conduction time delays, and inertial and viscous properties of the system make actual limb trajectory quite different from its central template. However, this does not mean that the template cannot be observed or reconstructed. The method we have used in this and earlier studies (more details can be found in Ref. 23), allows reconstruction of virtual trajectories although with certain assumptions and simplifications. We hope that the simplifications have not made the method misleading, our optimism being based on the high reproducibility of the findings and their "making physical sense". However, even if this method is doomed to be shown in future to be too crude and in a need of refinement, it still demonstrates that the virtual trajectory can be reconstructed and, therefore, is an observable. Accepting virtual trajectory as an observable does not mean that one should start a search for a neuron or population of neurons coding the virtual trajectory. Bernstein in 19354 warned against drawing too close parallels between localization and coordination and searching for anatomical or morphological formations directly coding different aspects of motor behavior. Virtual trajectory seems to us an example of a Bernsteinian behavioral observable that may not

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M.L. LATASH

have a clearly defined neurophysiological substrate. Other, more familiar examples include joint acceleration or movement distance that are certainly observables of motor behavior although a neurophysiologist would have a hard time trying to find a population of neurons coding acceleration or distance. Virtual trajectory is certainly harder to measure than, for example, joint angle, but this is rather due to the lack of appropriate instruments, although we are trying to introduce a crude one.

Phase transitions If one considers virtual trajectory a behavioral observable, phase transitions between it and another behavioral observable, actual movement trajectory, are illustrations of behavioral discontinuities of a mass-spring system introduced by small changes in a higher level control parameter, movement frequency, Relative phase shift between these two trajectories may be considered an order parameter, i.e. a parameter that makes the dimensionality of the control space smaller than the dimensionality of the state space. In this context, the present results are illustrations of relatively rapid changes in the order parameter introduced by slow changes in a control parameter (cf. Ref. 30). Observations of an abrupt phase transition between the virtual and actual trajectories (Fig. 7) are somewhat similar to the results of experiments studying phase relations between two limbs during bimanual tasks, t~lg'2s Only two stable states have been found, in-phase and out-of-phase. A small change in the movement frequency led to an abrupt jump from an out-of-phase mode into an in-phase mode, just as in our experiments a small change in movement frequency led to an abrupt jump from an in-phase mode into an out-of-phase mode. Does this similarity imply basic common features of the control mechanisms for single-joint and multi-limb repetitive movements? At the present stage of our knowledge, we seem unable even to speculate about this issue,

Central control and electromyograms Let us consider the following simple physical systern that can demonstrate spring-like oscillatory behavior (Fig. 11). In the framework of the A-model,9 the "flexor" and "extensor" springs shown in Fig. 11 have a complex structure involving both the muscles and their reflex connections from peripheral receptors (tonic stretch reflex). Most of this structure is not directly observable. Force generators of these springs (muscles, M] and M2) are well localized and much more accessible for direct measurement, e.g. for E M G recordings. Central control can be imposed upon such a system in different ways. For example, properties of the springs can be modulated, According to the A-model, which is the basis of the present study, resting length (zero length) of the springs is the only central variable supplied to the system.

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Fig. 11. Two springs ("flexors" and "extensors") act on a peripheral load with the help of force generators (muscles, M~ and Ms). These force generators are accessible for measurement, e.g. for EMG recording. According to the EP-hypothesis, central control can change resting lengths of the springs. If the central control is "frozen", the system will oscillate at its natural frequency and demonstrate modulation of the muscle activity due exclusively to the tonic stretch reflex action. Therefore, muscle activity cannot be considered a reflection of central control processes. If properties of the springs in Fig. 11 are held constant, there are no external influences, and the system is given a chance to oscillate about its equilibrium at its natural frequency, the springs will demonstrate modulation of their forces which can be recorded in the output elements, muscles. Note that we are talking now about a passive system without any central control that behaves according to the general laws of physics. This example illustrates that patterns of muscle activation are consequences of action of the tonic stretch reflex with or without central modulation and, therefore, cannot be considered reflections of central control processes. This statement is an antithesis to a large number of models of control of single-joint movements that are based on direct regulation of hypothetical control signals directly to the ~-motoneuron pools, thus prescribing the patterns of muscle activation and muscle forces (for review see Ref. 13). Alternative models of E M G emergence have recently been suggested based on the A-model. ~'H'24 In our experiments, recordings of surface EMGs of two elbow flexors and two elbow extensors showed a monotonic increase with movement frequency of both peak and integrated EMGs without any special behavior at the critical or natural frequencies. Similar results could be expected if one measures spring forces in the system shown in Fig. 11. These monotonic E M G changes are consistent with the idea of the mixed nature of the E M G and once again illustrates the limited value of the E M G recordings in motor control studies. The recorded burst-like muscle activity at the critical frequencies was most likely due exclusively to the reflex action while the central command (virtual trajectory) was nearly constant (flat).

Control of oscillatory movements CONCLUSIONS

Let us return to Huckleberry Finn and a dead rat on a long elastic piece of rubber (see Introduction). If Huckleberry wants the rat to oscillate at the natural frequency of the system "rat + rubber", he does not need to do much, just to supply an initial push and compensate for energy dissipation (nearly flat virtual trajectory). Note that tension in the rubber will be m o d u l a t e d at the m o v e m e n t frequency, just like muscle activity in o u r experiments. I f the desired frequency is different f r o m the n a t u r a l frequency,

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Huckleberry can grab the rubber at a different place thus changing the natural frequency of the system rat + rubber and/or apply a non-trivial pattern of external force to his end of the rubber. We hope that this not-very-serious example illustrates "physical sense" of our observations which is the major source of optimism for the author.

Acknowledgements--Theauthor would like to thank Dr G. L. Gottlieb and Mr Om Paul for their help, Dr J. J. Nicholas for financial support, and Dr S. R. Gutman for many helpful discussions. The study was partially supported by an NIH grant AR 33189.

REFERENCES

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26. Marsden C. D., Obeso J. A. and Rothwell J. C. (t983) The function of the antagonist muscle during last limb movements in man. J. Physiol. 335, 1 13. 27. Miller D. I. and Nelson R. C. (1976) Biomechanics of Sport. Lea & Febiger, Philadelphia. 28. Schmidt R. C., Carello C. and Turvey M. T. (1990) Phase transitions and critical fluctuations in the visual coordinatio~ of rhythmic movements between people. J. exp. Psychol. hum. Percept. Perf. 16, 227-247. 29. Turvey M. L., Schmidt R. C., Rosenblum L. D. and Kugler P. N. (1988) On the time allometry of coordinated rhythmic movements. J. theor. Biol. 130, 285-325. 30. Turvey M. T. (1990) The challenge of a physical account of action: a personal view. In The Natural-Physical Approach to Movement Control (eds Whiting H. T. A., Meijer O. G. and van Wieringen P. C. W.), pp. 57-92. VU University Press, Amsterdam. 31. Vincken M. H., Gielen C. C. A. M. and Denier van der Gon J. J. (1983) Intrinsic and afferent components in apparent muscle stiffness in man. Neuroscience 9, 529-534.

(Accepted 20 January 1992)

Virtual trajectories, joint stiffness, and changes in the limb natural frequency during single-joint oscillatory movements.

In the framework of the equilibrium-point hypothesis, virtual trajectories and patterns of joint stiffness were reconstructed during voluntary single-...
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