Abstract
Many sensorimotor neurons in the CNS encode global parameters of limb movement and posture rather than specific muscle or joint parameters. Our investigations of spinocerebellar activity have demonstrated that these secondorder spinal neurons also may encode proprioceptive information in a limbbased rather than jointbased reference frame. However, our finding that each foot position was determined by a unique combination of joint angles in the passive limb made it difficult to distinguish unequivocally between a limbbased and a jointbased representation. In this study, we decoupled foot position from limb geometry by applying mechanical constraints to individual hindlimb joints in anesthetized cats. We quantified the effect of the joint constraints on limb geometry by analyzing jointangle covariance in the free and constrained conditions. One type of constraint, a rigid constraint of the knee angle, both changed the covariance pattern and significantly reduced the strength of jointangle covariance. The other type, an elastic constraint of the ankle angle, changed only the covariance pattern and not its overall strength. We studied the effect of these constraints on the activity in 70 dorsal spinocerebellar tract (DSCT) neurons using a multivariate regression model, with limb axis length and orientation as predictors of neuronal activity. This model also included an experimental condition indicator variable that allowed significant intercept or slope changes in the relationships between foot position parameters and neuronal activity to be determined across conditions. The result of this analysis was that the spatial tuning of 37/70 neurons (53%) was unaffected by the constraints, suggesting that they were somehow able to signal foot position independently from the specific joint angles. We also investigated the extent to which cell activity represented individual joint angles by means of a regression model based on a linear combination of joint angles. A backward elimination of the insignificant predictors determined the set of independent joint angles that best described the neuronal activity for each experimental condition. Finally, by comparing the results of these two approaches, we could determine whether a DSCT neuron represented foot position, specific joint angles, or none of these variables consistently. We found that 10/70 neurons (14%) represented one or more specific jointangles. The activity of another 27 neurons (39%) was significantly affected by limb geometry changes, but 33 neurons (47%) consistently elaborated a foot position representation in the coordinates of the limb axis.
INTRODUCTION
Since Bernstein's original formulation (Bernstein 1967), principles of simplifying control systems having multiple degrees of freedom, like those for animal limbs, have become central issues in motorcontrol research. Various lines of investigation have suggested that one possible strategy employed by the CNS to avoid controlling the muscles or joint angles individually might be to control endpoint kinematics instead. For example, the neurophysiological finding that the population activity of motor cortical neurons can predict the kinematics of the limb endpoint would be compatible with this view (Schwartz 1994). Similarly, behavioral studies of cat posture have pointed to a specific CNS control of limb endpoint position in the coordinates of limb axis length and orientation as a strategy for maintaining stance in quadrupeds (Lacquaniti et al. 1990). Likewise, human gait analysis has shown that limb axis length and orientation are invariant across gait speeds. This kinematic invariance corresponds to minimization of energy expenditure, suggesting that these variables, which determine limb endpoint during gait, may effectively be controlled by the CNS (Bianchi et al. 1998).
Although some aspects of the control strategies may rely entirely on feedforward mechanisms, it is clear that sensory information plays a role in refining motor strategies to the ongoing demands of a behavioral task. Neurophysiological studies of the primary sensory cortex in behaving monkeys showed that cortical neurons are broadly tuned to the direction of movement and to the position of the hand (Prud'homme and Kalaska 1994). These properties are, in fact, similar to those of neurons located in motor areas of the brain (Schwartz et al. 1988), suggesting that the processing of sensory information can be congruent with the motor strategy. Furthermore similar features also have been found in secondorder sensory neurons projecting to the cerebellum from the spinal cord, i.e., at very early stages of sensorymotor processing (Bosco and Poppele 1993, 1997). The firing rates of the dorsal spinocerebellar tract (DSCT) neurons are broadly tuned for movement direction and foot position in limbcentered coordinates. For example, during passive positioning of the cat hindlimb, the activity of DSCT neurons relates linearly to the limb axis length and orientation, illustrating that loworder sensory neurons may represent the same kinematic variables that are likely to be controlled during stance or gait (Lacquaniti et al. 1990).
One issue that has received attention regarding CNS coding of sensorymotor parameters is the extent to which neurons encode global parameters such as limb endpoint or local variables such as joint angles or muscle length or force (e.g., Evarts 1968;Georgopulous et al. 1982; Humphrey 1972; see also review by Donoghue and Sanes 1994). We tentatively resolved this issue for DSCT coding in favor of an endpoint representation (Bosco and Poppele 1997). However, it remains unclear whether this is primarily an explicit representation determined by neural circuitry or an implicit representation determined primarily by limb biomechanics (Bosco et al. 1996). Even though we were able to show that the kinematics of the hindlimb endpoint were consistently better predictors of DSCT activity than were individual joint angles for example, we could not directly rule out biomechanical factors. The reason is that the passive hindlimb is constrained by a biomechanical coupling across joints leading to a covariation of joint angles. Thus instead of the 3 degrees of freedom expected for independent joint motion, the joint interdependence leads to just 2 degrees of freedom. Because the foot position relative to the hip also has 2 degrees of freedom (limb axis length and orientation) and each foot position is determined by a unique combination of joint angles, an unambiguous distinction cannot be made between foot position and jointangle representations. In other words, DSCT activity may simply relate to an individual joint angle or to a particular combination of joint angles and, because of the coupling between joints, appear to encode foot position. Alternatively, the neurons may actually represent foot position by appropriately integrating sensory information from different hindlimb areas.
In the present study we overcame this ambiguity by decoupling hindfoot position from overall limb geometry. We constrained joint motion at the ankle or knee so we could compare DSCT responses to endpoint positions throughout a parasagittal workspace in a constrained and unconstrained limb. We found with this approach that the responses of many DSCT neurons were invariant with foot position, illustrating that at least these cells actually do represent foot position.
METHODS
We report results from experiments on six adult cats anesthetized with barbiturate (Nembutal, Abbott Pharmaceuticals; 35 mg/kg ip supplemented by intravenous administration to maintain a surgical level of anesthesia throughout the experiment). The animals were placed in a stereotaxic apparatus with the hips fixed in position by pins in the iliac crests. Limb kinematics (Fig.1 A) were recorded by means of a video camera (Javelin Model 7242 CCD camera; 60 frames/s) and digitized offline using a motion analysis system (Motion Analysis, Santa Rosa, CA, model VP110; see following text for details). The left hindfoot was attached to a small platform connected to a computercontrolled robot arm (Microbot Alphall+, Questech, Farmington Hills, MI) (see also Bosco et al.1996), that moved the limb passively through a series of 20 foot positions (Fig.1 B, grey circles). Each foot position was held for 6 s (see also Bosco and Poppele 1997; Bosco et al. 1996).
Joint constraints
In order to restrict joint movements without introducing excessive sensory stimuli to the skin or muscles, we applied constraints between surgically implanted bone pins. One pin was placed in the femur ∼5 cm from the femur head and another in the tibia ∼6 cm from its distal end (Fig. 1 B). In a preliminary study reported earlier (Bosco and Poppele 1998), we applied elastic constraints (rubber bands) from the hip to the femur pin (hip constraint), between bone pins (knee constraint), from the tibia pin to the robot platform (ankle constraint), or from the femur pin to the robot (combined knee and ankle constraint). These constraints were generally not very effective in uncoupling joint covariation although they did change the orientation of the covariance plane. Thus in this study we adopted two constraints that were most effective altering joint covariance. One was the elastic ankle constraint illustrated by the wavy line in Fig. 1 B, which affected primarily motion at the ankle joint and we refer to it as “ankle elastic” constraint. The other was a rigid Plexiglas strip fixed between the femoral and tibial pins (straight line in Fig. 1 B). Because this constraint restricted motion at the knee joint almost completely, we will refer to it as the “kneefixed” constraint. We used the ankle elastic constraint exclusively in one experiment and together with the kneefixed constraint in one other experiment. The other four experiments employed only the kneefixed constraint.
Kinematic measurements
Reflective markers were placed on the skin over the hip, knee, ankle, and lateral metatarsalphalangeal joint of the foot. The digitized positions of markers in an image plane approximately parallel to the plane formed by the hip, knee, and ankle markers were corrected for skin slippage at the knee and for outofplane positions using an algorithm that is described in detail in the . Note that this represents a substantial procedural difference in data collection and analysis from that described in our previous papers (Bosco and Poppele 1997; Bosco et al. 1996). Previously the alignment of the image plane was not carefully controlled, and we corrected only the position of the knee marker for skin slippage. We assumed that the image plane projection of markers provided a reasonably close indication of joint positions even though the cat hindlimb does not normally lie in a single plane (which has been the common practice for studies like this; e.g.,Brustein and Rossignol 1998; Goslow et al. 1973; Shen and Poppele 1995).
We represented limb kinematics in the coordinates of the limb axis and the joint angles. The limb axis is the segment joining the hip and foot positions, and it defines the foot position in polar coordinates by its orientation (O), the angle measured clockwise from the horizontal to the axis, and length (L) in cm (Fig.1 A). The joint angles also are defined in Fig. 1 Aas the angles measured clockwise between two limb segments. Although the revised kinematic analysis described in the may generate some discrepancies between the jointangle data reported previously (Bosco and Poppele 1997) and those presented here, it is not expected to effect the limb axis parameters because they depend on a single marker (metatarsalphalangeal joint of the foot) given that the hip marker undergoes little or no movement.
Neuronal activity
We recorded unit activity from 72 DSCT axons in the dorsolateral funiculus at the T_{10}–T_{11}level of the spinal cord using insulated tungsten electrodes (5 MΩ, FHC, Brunswick, ME). Units were identified as spinocerebellar by antidromic activation from the white matter of the cerebellum or from the restiform body. Neuronal activity was recorded continuously during series of passive limb movements through the 20 positions of the limb's workspace. We used two to four different movement patterns, each designed to stop at all 20 positions but approaching the positions from different directions. Thus a data set could contain 40–80 trials, with two to four repetitions for each position. Except for the edge positions, each trial was in a different movement direction (seeBosco and Poppele 1997). We aligned the neuronal activity to the onset of limb movement and used only the activity recorded between 4 and 5 s after movement onset. The rationale for using this interval came from the previous finding (Bosco and Poppele 1997) that most of the variance in neuronal activity in the fifth second after movement onset could be accounted for by foot position, even though significant relationships to movement direction still could be observed.
Data analysis
KINEMATIC DATA.
We represented the limb geometry for each foot position by the joint angles in a threedimensional joint space where each joint angle for a given position is plotted as the difference from the mean angle across the 20 positions. The data points in this representation fall within a plane that explains a large fraction of variance in the data set. The joint covariance illustrated by this result implies that jointangle motion is strongly coupled by biomechanical constraints that reduce the limb degrees of freedom (Bosco et al. 1996). Therefore we quantified limb kinematics in the constrained and unconstrained conditions by fitting leastsquare planes separately to each data set and comparing plane orientations, defined by the direction cosines of the vector normal to the plane, and the fraction of the variance explained by each plane.
NEURONAL DATA.
We determined whether a neuron's activity was significantly modulated by foot position by regressing the firing rate (F) recorded for each trial and averaged over the interval between 4 and 5 s after movement onset against foot position expressed in the polar coordinates of the limb axis length (L) and orientation (O)
We also addressed the question of whether there was an invariant relationship between firing rate and a given set of joint angles. For this purpose, we applied the following regression model for each experimental condition
For each cell we also determined the preferred direction corresponding to the direction of the maximal gradient of neural activity in the space defined by limb length and orientation. For this purpose, it was necessary to transform unit activity into a footcentered coordinate system to determine a gradient that did not depend on the units of measure for length and orientation. We did this by using the following transformation (Bosco and Poppele 1997; Kettner et al. 1988)
We used this gradient function separately for each experimental condition and compared the preferred direction angles, G, by computing the cosine of their difference between the constrained and unconstrained conditions.
RESULTS
Kinematic data
One interesting feature of the cat hindlimb is that movements throughout the workspace are accompanied by a linear covariation among the joint angles that effectively reduces the degrees of freedom of the limb. Although the basis for this covariance pattern is likely to reside in the biomechanics of the limb (Bosco et al. 1996), neural strategies are also likely to actively modify and even increase the joint coupling in behaving animals (Lacquaniti and Maioli 1994a,b). In this study, we altered the biomechanical coupling by applying external constraints to the ankle and knee joints. The effect of the constraints is to alter the relationship among joint angles for a given position of the limb endpoint. This is illustrated in Figs. 2,3 and summarized in Table1.
These figures show each set of joint angles for 20 foot positions and the corresponding leastsquare covariance planes. Each data point plots the three jointangle values for a given foot position. Note that the orientations of the covariance planes are similar across experiments in the unconstrained condition (Figs.2 A and 3 A). This is documented in Table 1 by the similar direction cosines for each covariance plane and high percentage of variance they explained. The average covariance plane orientation is somewhat different from that reported earlier, however (Bosco et al. 1996). We attribute this discrepancy to our previous failure to account for the nonplanar configuration of the cat hindlimb and the associated errors in determining joint angles (seemethods and ).1
Overall, the effect of the ankle elastic constraint on passive limb biomechanics was a slight rotation of the jointangle covariance plane with respect to the control. In fact, as suggested by the high percentage of variance explained by these planes (90.6 and 93.2%), the ankle constraint did not significantly reduce the strength of the biomechanical coupling, but it did change the orientation of the covariance plane as indicated by the direction cosines in Table 1.
In contrast, the rigid knee constraint fixed between the femural and tibial pins dramatically altered the biomechanical coupling among joint angles (Figs. 2 C and 3 B). This constraint was variably successful in actually fixing the knee joint angle, being most successful in cat 4 (Fig. 3 B). Typically there was 5–10° of rotation at the knee. In all cases, the joint covariance plane rotated significantly with respect to the control condition, and the strength of the biomechanical coupling also was disrupted. The average fraction of variance explained by the leastsquare planes was 47.9%, compared with a mean 85.2% in the control condition.
Our major interest here is to distinguish between a neural representation of limb geometry based on jointangle coordinates and a representation based on the foot position alone. Because the kneefixed constraint was more successful in decoupling the relationship between endpoint and joint angles, those results will be emphasized.
Neuronal data
The activity of a substantial fraction of the DSCT neurons recorded in the lower thoracic spinal cord is broadly tuned with respect to foot position (Bosco and Poppele 1993). In particular, there is a linear relationship between neuronal firing rates and the length and the orientation of the limb axis (which define foot position in polar coordinates). This result provided the rationale for quantifying DSCT positional modulation with a linear model based on limb axis length and orientation as predictors of firing activity (methods, Eq. 1 ). In this series of experiments, the model explained a significant fraction of variance (R ^{2} > 0.4, P < 0.0001) in the firing rate of 70 of 74 DSCT neurons examined (95%).
FOOTPOSITION REPRESENTATION.
To determine whether the footposition representation changed as a result of the applied constraints, we used a regression model in which we introduced binary variables associated with the constrained conditions as firing rate predictors along with the length and the orientation of the limb axis (methods, Eq. 2 ). Coefficients associated with these binary variables measured the effects of a constraint on the intercept and slope of the relationships between firing rates and foot position.
The results of this analysis indicate that the firing rates of some neurons were not affected by the constraints, whereas others were (Figs. 4 and5). Neurons 2576 and2710 are examples that were not significantly affected (Fig. 4, A–F). Regression planes describing their relationships between firing rate and foot position were, in fact, identical for the unconstrained and constrained conditions (compare 4, A and D, with Band E, respectively). In agreement with this qualitative judgement, the regression analysis (Eq. 2 ) showed that the coefficients associated with the binary variables A orK were not significant. The firing rate values predicted by this model are plotted against the firing rate values actually recorded for cells 2576 and 2710 (Fig. 4, C andF) during the unconstrained (open triangles) and constrained (filled circles) conditions overlapped extensively and the corresponding regression lines coincided. Overall, 28 of the 70 modulated neurons (40%) failed to show significant changes in their modulation with foot position after the application of joint constraints (P < 0.01).
The firing rates of another nine neurons (13%) increased or decreased uniformly over the workspace as a result of the limb geometry changes caused by the joint constraints. However, the slope of the relation between firing rate and foot position for these neurons remained unchanged. One example of this group of neurons is represented bycell 2671 (Fig. 4, G–I).
Thirtythree neurons (37%) showed significant changes in their footpositionrelated activity after the application of joint constraints. Two of these (cells 2567 and 2652) are illustrated in Fig. 5. Note that the regression planes for the control (Fig. 5, A and D) and constrained conditions (B and E) are different in both intercept and slope, suggesting that the spatial tuning of the neurons depended on the overall geometry of the limb rather than simply on the position of the endpoint.
The effect of a joint constraint on the activity pattern of a cell was not simply allornone, although we did assess the effect using a very conservative cutoff level (P < 0.01). In some cases the constraint effects were clear, whereas in others they were very small but appeared consistent. For example, cell 2710 shows a small but consistent change in the overall firing level that is evident in Fig. 4 E. One way to examine this issue across the DSCT population is to examine the distribution the tstatistic of the regression across cells. Figure6 A shows a broad distribution of significance values with a median value of 2.57, corresponding to aP value of 0.013. We would estimate from this analysis that about half of the neurons were affected to some extent by the constraint and the other half were more likely unaffected.
DIRECTION OF MAXIMAL ACTIVITY GRADIENT.
The slope changes in the relationships between neural activity and foot position indicate changes in sensitivity but not necessarily in the direction of the gradient of positional activity. Consider for example, a case for which the slopes for both limb axis parameters (Land O) change proportionally. This would indicate an overall change in sensitivity to position, but the direction of the gradient of neuronal activity, i.e., the cell's preferred direction, would be unchanged. Instead differential sensitivity changes along the length and orientation dimensions would indicate changes in the preferred direction and possibly sensitivity changes also.
The analysis of significance we presented in the preceding text does not distinguish between these two possibilities. Furthermore because the limb parameters have different units (length and angle), it is difficult to compare changes in their regression slopes directly. Therefore we employed a coordinate transformation that allowed us to determine each neuron's preferred tuning direction with respect to foot position (see methods, Eq. 4 ). In Fig.6 B, we plot the distribution of the cosine of the differences between the preferred directions in the constrained and unconstrained condition for each cell. The result of this analysis shows that the preferred directions did not change significantly for about half of the cells (cosine values near 1.0). The median value was 0.919, corresponding to a difference in preferred directions of 23°. This would correspond to no change if the error in determining the preferred direction by this method were of the order of ±12°.2
JOINTANGLE REPRESENTATIONS.
It appears from the preceding analysis that some fraction, perhaps as many as half of the DSCT neurons do in some way encode the limb endpoint position explicitly. To gain some insight about extent of sensory processing contributing to this behavior, we also examined each cell's behavior with respect to joint angle changes.
For this purpose, we used a regression model based on jointangle coordinates (Eq. 3 ) to determine the set of joint angles that related to the neuronal activity for a given experimental condition. Furthermore by eliminating the insignificant relationships or relationships with joint angles that covaried with other joint angles that were more strongly correlated with cell activity, we could estimate which joint angles most strongly predicted a cell's firing rate. In the following section, we consider again the four specific examples presented in Figs. 4 and 5 to illustrate various strengths of correlation with joint angles and foot positions.
First consider the simple scenario of a cell that responds stereotypically to a given joint angle or a particular combination of joint angles. One example is the behavior of cell 2567,which responded differently in the constrained and unconstrained conditions (Fig. 7; same cell as Fig. 5,A and B). Scatterplots of firing rates against hip angle in Fig. 7, B and C show that this neuron related the same way to hip angle in both conditions. Moreover, plots of both firing activity (A) and hip angle values (D) in the two conditions showed similar systematic deviations about the identity line. We would interpret this result as indicating that this neuron's activity may be tracking the hip angle.
Another cell the activity of which was related primarily to the hip angle in both constrained and unconstrained conditions wascell 2710 (Fig. 8; same cell as Fig. 4, D and E). However, in this case the relationship between firing rate and hip angle was different in the control condition, when it was best approximated by a linear function (B), and in the constrained condition, when it was best fit was a quadratic function (C). In contrast, this neuron showed essentially invariant representations of the foot position (limb axis length and orientation) across conditions as shown by the scatterplot of the firing rates in the control versus constrained conditions (A). All data points are distributed about evenly above and below a line that is parallel to the identity line, whereas the same plot for the hip angle values (D) shows a systematic trend in the data relative to the identity line. This result would argue for a neuronal representation of foot position rather than hip angle.
A more complex possibility is that the activity of DSCT neurons may relate to different sets of joint angles in the passive and constrained conditions. For example, the activity of cell 2652 (Fig.9; same cell as in Fig. 5, Dand E) related most clearly to the hip angle in the unconstrained condition (B) yet the ankle angle became the strongest firing rate predictor in the constrained condition (I). Actually, the regression analysis showed that all three joint angles were significantly related to the firing activity in the control condition, whereas only the hip and ankle angles were the significant predictors in the constrained condition. This neuron, however, did not show an invariant relationship to foot position (A) and therefore did not relate consistently to any of the kinematic variables we measured, although it was clearly modulated by passive limb positioning across the workspace.
A somewhat less obvious example, but one that did relate consistently to the foot position, was cell 2576 (Fig.10; same cell as in Fig. 4,A and B). In this case, the knee angle was a significant predictor in both control and constrained conditions (E and F). In the control condition, the stepwise regression analysis eliminated the hip angle as a predictor despite its significant correlation with firing rate (Fig. 10 B) because of its relatively high correlation with the other two angles (knee,r = 0.48; ankle, r = 0.73). Although there was a weak relationship between firing rate and ankle angle (H), it was, nevertheless, significant and not redundant given the low correlation between ankle and knee angles (r = 0.13). In the constrained condition, the correlation strengths between pairs of joint angles were more evenly distributed (r = ∼0.5 for all jointangle pairs) and the relationship between firing rate and ankle angle became weaker (I), making the hip angle (C) the only other significant predictor. Although this argument against a simple jointangle representation may appear subtle, it is substantiated by the observation that the overall relation between firing rate and foot position was unaffected by the constraint (A) even though the individual joint angles for each limb position deviated systematically in the two conditions (D, G, andJ). Taken together these results suggest that this DSCT neuron was integrating sensory information across joints to elaborate a representation of foot position independently from the actual combination of joint angles.
A summary of all the results from the various types of analysis is presented in Table 2. From this summary we may conclude that only 10/70 cells (14%) responding to foot positioning did so by responding consistently to specific joint angles across conditions. The activity of four of these cells also related consistently to the limb endpoint, whereas the activity of the other six did not. Thus of the 33 neurons (47%) for which the activity did not relate consistently to foot position, 27 (39%) did not relate consistently to any of the kinematic variables we measured. The activity of the other 53% of the responsive cells was essentially invariant with respect to foot position when the joint covariance pattern was altered.
DISCUSSION
The principal finding of this study is that the activity of a significant fraction of the DSCT population represents the hindlimb endpoint position as distinct from the specific limb geometry associated with the endpoint. The implication of this result is that the circuitry of the DSCT must somehow compute endpoint position from the sensory information derived from various joints and muscles. It seemed initially that this computation might be assisted in some way by the biomechanical properties of the passive limb which lead to a covariant relationship among joint angles. However, when this relationship was disrupted by means of jointangle constraints, the endpoint representation persisted in about half of the cells we tested.
We used two types of constraints. One was an elastic constraint across the ankle joint. This constraint had only subtle effects on limb kinematics, including a negligible effect on the strength of jointangle coupling, but it did alter the relative coupling among joint angles and also the response to foot position for some cells (e.g., cell 2567). The other constraint, which held the knee angle nearly constant, significantly disrupted both the limb kinematics and the jointangle coupling and was ultimately the most effective in altering responses to foot position. In a preliminary study reported earlier (Bosco and Poppele 1997), we also applied elastic constraints separately to each of the three joints. Similar to the results of this study, we found that at least 30% of the cells had the same response in each constraint condition. Thus the fraction of cells the activity of which correlates consistently with limb endpoint position does not appear to depend on which joint was constrained or the degree to which joint coupling was disrupted.
There are at least two possible mechanisms that could explain this finding. One possibility is a wide sensory convergence from the hindlimb onto the DSCT neurons. This could allow the DSCT circuitry to compute an estimate of foot position independently from the overall limb geometry by redistributing the relative weights of the afferent input. However, another possibility is that the neuronal firing activity may relate primarily to a single joint angle the motion of which is not much affected by the limb constraint. In this case, the activity would be actually monitoring a joint angle but would appear to be related to the limb axis only because these two kinematic variables covary.
To address this issue, we used a regression model based on the combinations of joint angles that determined foot positions in the constrained and unconstrained conditions. The analysis showed that the activity of 20 of the 37 neurons having an invariant representation of foot position (except for intercept changes) was related to different combinations of joint angles across constraint conditions as expected for a computational mechanism based on sensory integration. The activity of remaining 17 neurons related to the same joint angles in the two conditions and therefore could have been determined primarily by biomechanical constraints rather than by sensory integration. However, we found that the activity of a number of these cells did not faithfully mirror the jointangle differences imposed by the constraints (e.g., cell 2710), implying the presence of sensory integration in some of these cases as well.
The same issue was examined for the 33 neurons that did not show invariant relationships with the foot position. A subgroup of six of these neurons did relate to the same joint angles or combination of joint angles in constrained and unconstrained conditions. However, the activity of the remaining 27 DSCT neurons related to different jointangle combinations in different experimental conditions. This result (illustrated for cell 2652 in Fig. 5,D–F) is more difficult to interpret. It suggests that these neurons may be encoding some aspects of limb geometry, and it certainly speaks in favor of sensory convergence.
The data presented here suggest that about half of the DSCT neurons we sampled encode information in the reference frame of the limb endpoint kinematics. That is, their spatial tuning direction is basically unaltered by joint constraint. The other half appear to encode information in a reference frame that may be more closely tied to the actual joint angles. It is not likely that this type of dichotomy between a jointbased and limb endpoint representation is unique to the spinocerebellar system, however, because it also has been described in other areas of the CNS involved in sensorymotor integration.
In motor cortex for example, Scott and Kalaska (1997)investigated neuronal activity in a visually guided reaching task performed with two different arm postures. They reported significant changes in the firing properties of most (70%) motor cortical neurons related to overall arm posture. However, many of the significant changes in cell behavior were restricted to overall firing levels or modulation amplitudes so that significant changes in preferred directions occurred for only ∼50% of the neurons. A similar fraction of motor cortical cells showed invariant preferred tuning directions in a recent study that dissociated muscle actions from the directions of wrist movements (Kakei et al. 1999). These results were interpreted to suggest that the ensemble of cells in primary motor cortex having the two types of representation might represent a transformation between muscle or jointbased and endpoint representations.
The presence of the separate representations of joints and endpoint at the early stages of sensory processing in the spinal cord also may suggest other functional roles. For example, it may provide specific sensory information about limb mechanics for a given endpoint position through a comparison of endpoint and jointrelated information. Consider for example the case of maintaining stance, a behavior that does involve the spinocerebellum. Experimental evidence suggests that quadrupeds maintain stance by controlling the length and the orientation of the limb axis, that is, the limb endpoint (Lacquaniti et al. 1990). A possible strategy for accomplishing this was suggested by the finding that the limb axis geometry maintained in stance is associated with a particular jointangle covariance pattern. The advantages of this strategy in simplifying the control of a multidegree of freedom limb were discussed by Lacquaniti and Maioli (1994b). Such a strategy might be implemented by producing muscle activity patterns that appropriately modify the biomechanical coupling among joints. If so, the control task for stance would be to establish a covariant relation among joint angles such that each desired endpoint, in this case, the ones associated with a nearly vertical limb axis, maps onto a set of joint angles in the covariance plane. Maintaining the desired set of endpoints then would be accomplished by maintaining the established jointangle covariance. Any perturbation in this system could be compensated by sensing a mismatch between an endpointrelated signal and a jointanglerelated signal, which then could be applied as an error signal to reestablish the appropriate jointangle covariance.
We might speculate that the spinocerebellar system could play a role in this proposed compensation. If each endpointrelated DSCT cell were matched somehow with a corresponding jointrelated cell having a congruent endpointrelated activity for a given behavioral state (such as quiet stance), then as long as their activities remained congruent, it would indicate a consistent relationship between endpoint and jointangle covariance. Any mismatch in their respective signals would indicate a deviation from the desired jointangle covariance state.3 If the cerebellum were able to detect mismatches in the signals of such paired spinocerebellar neurons, it might initiate some control signal to reestablish the desired jointangle covariance, for example, by modulating limb reflexes.
We should consider, however, that DSCT cells also may encode other than kinematic parameters. In fact, the joint constraints we used imposed external forces on the hindlimb that may strongly influence muscle and skin receptors that project to the DSCT. The implication is that models of neuronal activity such as ours that are based exclusively on limb kinematics may not capture the features of neuronal activity that are primarily sensitive to the distribution of forces across the limb. In the companion paper, we start addressing this issue by studying the effect of activating selective muscle groups on the DSCT representation of limb position (Bosco and Poppele 2000).
Acknowledgments
The authors thank A. Rankin for help and assistance on this project and Drs. M. Flanders and J. Soechting for critical and helpful comments on the manuscript. J. Eian wrote the .
This research was supported by National Institute of Neurological Disorders and Stroke Grant NS21143.
Footnotes

Address for reprint requests: R. E. Poppele, 6145 Jackson Hall, University of Minnesota, 321 Church St. SE, Minneapolis, MN 55455.

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

↵1 Previously we assumed a negligible inward or outward rotation of limb segments and thus did not correct data for outofplane marker positions. However, the cat does not stand with the hindlimb in a single plane. The plane defined by the foot, ankle, and knee markers tends to be rotated laterally out of a parasagittal plane defined by the hip, knee, and ankle markers. This rotation becomes pronounced at the extremes of the workspace where is it also accompanied by outward rotations of the upper leg, leading to inaccurate estimates of joint angles and their covariance plane. It may be consistent with this view that the jointangle covariance planes reported for the ankle elastic constraint (Fig. 2 B) are similar to those described in earlier experiments without constraints. A possible interpretation of this result is that the ankleconstraint restricted inward/outward rotations of the foot at the ankle joint, creating the experimental analogue of the planar movement assumption made previously.

↵2 This level of uncertainty seems consistent with measurements of preferred direction we reported earlier (Bosco and Poppele 1997). In that case, a cosine model was used to summarize responses to centerout or outcenter movements from a given foot position. Typical variance of 10–20° was found over successive measures (for example, Fig. 3 in Bosco and Poppele 1997), suggesting that intertrial variability and measurement accuracy could account for an uncertainty of the order of ±12°.

↵3 For the sake of clarity, we emphasized that comparisons be made between activities in endpoint and jointrelated cells, as if these were clear subpopulations of cells. In fact, as suggested in Fig.6 A, the properties of joint and endpointrelatedness may be relative and distributed through the DSCT population. However, for this proposed comparison with work, it should only be necessary that cells more related to endpoint be consistently paired with cells more related to joints.
 Copyright © 2000 The American Physiological Society
Appendix
In some circumstances it is not convenient or costeffective to measure threedimensional (3D) limb kinematics directly, so it is desirable to have a twodimensional (2D) measurement technique that can accurately portray 3D limb geometry. Such a method is described in this appendix.
Twodimensional measurements often are made on a projection of joint markers onto an imaging plane. This method is accurate only when all of the joints lie in a single plane that is parallel to the imaging plane. Although these conditions may be approximately met for certain hindlimb configurations in the cat, they do not hold in general. For example, the ankle and knee joints may translate out of a parasagittal leg plane under some conditions (see Fig. FA1). Under such conditions, a given knee angle, K, determined from 2D marker data, will appear to increase (K′) as the plane defined by the hip, knee, and ankle markers rotates to a position no longer parallel to the imaging plane. At the same time, distances between hip and knee and between knee and ankle markers appear to decrease. Under such conditions, joint angles will appear to change as the leg plane rotates leading to systematic errors in jointangle estimates as the limb moves.
The method described in the following text corrects for these outofplane rotations by considering the geometry of the experimental setup. This includes the known distances between joint markers (i.e., the lengths of the limb segments) and the location of at least one marker along a line perpendicular to the imaging plane. In our case, the distance of the toe marker from the midsagittal plane of the cat was constant in all trajectories because the robot moved the toe in a parasagittal plane perpendicular to the imaging plane.
The algorithm explanation requires the definition of two reference points: the spatial invariant point and the image invariant point. Given a set of planes parallel to the 2D imaging plane containing the markers to be imaged, a spatial invariant point is defined as the unique point in any plane the image of which is invariant under translations of the point perpendicular to the plane. The location of the image of a spatial invariant point is the image invariant point (s′ in Fig. FA2). The location of the spatial invariant point in the reference plane (Z = 0) is defined as the origin [S(0,0,0) in Fig. FA2] of the 3D coordinates of spatial points,P(X, Y, Z) having corresponding image points, p′(x, y).
Next we account for the fact that the projection of a point,P, onto the image plane, p′, depends on the distance of P from the camera (the perspective problem). We treat this here in terms of a dilation mapping, which maps thezaxis locations of spatial objects into dilation factors,d_{z}
Let r = X/x′ be the calibration relating a distance on plane Z = 0 to its image distance on the imaging plane, then
The task now is to determine P(X, Y, Z) for a given marker in space. Consider the ankle marker in our case which has the coordinates (Xa, Ya, Za) relative to the spatial invariant point S(0,0,0). Using the dilation equation,Eq. EA1, the ankle position in space can be expressed as
Note that because Eq. EA5 is quadratic, it yields two values of Za. The correct solution in our case was obvious from the hindlimb kinematics. In the case where both solutions have the same sign, one solution is always clearly out of the experimental workspace. When the two solutions have opposite signs, then it may be necessary to know whether the marker is in front of or behind the reference plane.