Abstract
Dorsal spinocerebellar tract (DSCT) neurons transmit sensory signals to the cerebellum that encode global hindlimb parameters, such as the hindlimb endpoint position and its direction of movement. Here we use a population analysis approach to examine further the characteristics of DSCT neuronal responses during continuous movements of the hind foot. We used a robot to move the hind paw of anesthetized cats through the trajectories of a step or a figure8 footpath in a parasagittal plane. Extracellular recordings from 82 cells converted to cycle histograms provided the basis for a principalcomponent analysis to determine the common features of the DSCT movement responses. Five principal components (PCs) accounted for about 80% of the total variance in the waveforms across units. The first two PCs accounted for about 60% of the variance and they were highly robust across samples. We examined the relationship between the responses and limb kinematic parameters by correlating the PC waveforms with waveforms of the joint angle and limb axis trajectories using multivariate linear regression models. Each PC waveform could be at least partly explained by a linear relationship to jointangle trajectories, but except for the first PC, they required multiple angles. However, the limb axis parameters more closely related to both the first and second PC waveforms. In fact, linear regression models with limb axis length and orientation trajectories as predictors explained 94% of the variance in both PCs, and each was related to a particular linear combination of position and velocity. The first PC correlated with the limb axis orientation and orientation velocity trajectories, whereas second PC with the length and length velocity trajectories. These combinations were found to correspond to the dynamics of muscle spindle responses. The first two PCs were also most representative of the data set since about half the DSCT responses could be at least 85% accounted for by weighted linear combinations of these two PCs. Higherorder PCs were unrelated to limb axis trajectories and accounted instead for different dynamic components of the responses. The findings imply that an explicit and independent representation of the limb axis length and orientation may be present at the lowest levels of sensory processing in the spinal cord.
INTRODUCTION
Sensorimotor interactions within the vertebrate spinal cord may be partly organized for some level of kinematic control of the limbs. In lower vertebrates, for example, isolated spinal circuits are capable of controlling precise placements of the limb endpoint in response to cutaneous stimuli (Giszter et al. 1989; Robertson et al. 1985). The resulting motor activity has been characterized by modules of “motor primitives” that appear to have a kinematically based organization (Kargo and Giszter 2000a,b; Tresch et al. 1999). Recent evidence from spinally transected cats also demonstrates that postlesion training can recover their ability to walk on a treadmill (Belanger et al. 1996; Brustein and Rossignol 1998) or stand unsupported on a platform (Fung and Macpherson 1999) using kinematic strategies that are almost indistinguishable from those of normal animals. Such results raise questions about the underlying organization of spinal sensory circuits and how they might interact with motor components.
Dorsal spinocerebellar tract (DSCT) neurons have provided a model system for studying the processing of proprioceptive information in the spinal cord. These cells transmit to the spinocerebellum signals that represent at least one result of sensory processing by spinal cord circuitry. Our studies to date have focused primarily on the postural responses of DSCT neurons and they showed that their activity encodes global hindlimb kinematics (Bosco and Poppele 1993,1997; Bosco et al. 1996).
The kinematic parameters that account for most of the response activity are the position of the hindlimb endpoint and its direction of movement (Bosco and Poppele 1997; Bosco et al. 1996). DSCT activity is broadly tuned with respect to foot position leading to a linear gradient of activity levels over the parasagittal workspace of the hind foot. The activity is also broadly tuned with respect to the directions of foot movements in the workspace. Generally, however, these cells exhibit a low sensitivity to movement speed as evidenced by their similar responses to limb movements along a given foot path at speeds that differ by as much as fourfold (Bosco and Poppele 1999). In fact we found that movement speed accounted for only about 15% of the response variance across cells, while movement direction accounted for more than 40%. This result led us to suggest that the speed and directional components of the movement velocity vector may be processed separately in the DSCT circuitry. In part, this behavior may also reflect the behavior of muscle spindles, which are very sensitive to even small changes in muscle length (i.e., sensitive to movement) but have a relatively low sensitivity to the stretch velocity (Houk et al. 1981). However, there is also the possibility that the behavior might result from neural processing in the spinal cord.
Here we examine this issue in more detail by applying a population analysis to the behavior of DSCT neurons during continuous movements of the hind foot. This analysis allows us to summarize the diversity of response behaviors among neurons and thereby provide a better overall view of DSCT behavior. The previous work focused on the response behavior of individual cells and aimed to identify commonalities across cells. However, we found that specific response properties tend to be distributed in the population, suggesting therefore a highly distributed organization. The population analysis we applied in this study allowed us to examine the characteristics of this distributed organization.
The results presented here are primarily based on a principal component analysis, which identifies factors that are robust response components across cells and across animals. The analysis provides additional support for the hypothesis that DSCT activity encodes a representation of limb position and movement. It also implies that there is an explicit representation of the kinematic velocity profile of the movement but that it is separated into components that correspond to the length and orientation coordinates of the hindlimb axis. Some of the results reported here have also been presented elsewhere in abstracts (Bosco et al. 2000b; Poppele and Bosco 2000) or short reports (Valle et al. 2000).
METHODS
Results are reported from experiments using 12 adult cats anesthetized with barbiturate [pentobarbital sodium (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 frame with the hips fixed in position and the left hind foot attached to a small platform connected to a computer controlled robot arm (Microbot AlphaII+, Questech, Farmington Hills, MI) (see also Bosco et al. 1996). The robot arm moved the limb passively through footpaths modeled either on the locomotion step cycle (see Fig. 1 B) or a figure 8 in a parasagittal plane (Fig. 1 C) (see alsoBosco and Poppele 1999). We used a video camera (Javelin Model 7242 CCD camera; 60 frames/s) to record limb movements, and we digitized limb marker positions offline using a motion analysis system (Motion Analysis, Santa Rosa, CA, model VP110; see following text).
Neural data recording
We recorded unit activity from a total of 163 DSCT axons in the dorsolateral funiculus at the T_{10}–T_{12} 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 and/or the inferior cerebellar peduncle. To generate cycle histograms, we aligned the neural activity with respect to an arbitrary point in the footpath (the beginning of the straight segment directed downward and forward for the figure 8, and the beginning of the forward swing for the step cycle, see Fig. 1, blueyellow border). The histograms, typically based on 10–20 stimulus cycles, provided estimates of spike density resolved to 33.3ms binwidths over a movement cycle. We used the WaldWolfowitz runs test (P < 0.05) to assess whether the activity of each neuron was significantly modulated about the mean level of activity over a hindlimb movement cycle. We recorded the activity from 82 units in eight cats that were modulated by both step and figure8 cycles and from 81 units in another four cats that were modulated by step cycles.
Kinematic measurements
Reflective markers were placed on the skin over the hip, knee, ankle, and lateral metatarsalphalangeal joint of the foot. The positions of the markers in an image plane approximately parallel to the plane of the limb were recorded at 60 frames/s. Digitized marker positions were corrected for skin slippage at the knee and for outofplane rotations of the limb segments using measurements of thigh, shank and foot lengths (Bosco et al. 2000a). As indicators of the hindlimb kinematics, we determined the trajectories of the anatomical joint angles and of the limb axis, which is the segment joining the hip position with the foot position (Fig.1 A, dashed line). The joint angles are defined for the ankle and the knee as the outside angles between the foot and the shank for the ankle and between the shank and the thigh for the knee. The hip angle is measured between the thigh and the horizontal. The limb axis is represented in polar coordinates by its orientation (O) or the angle measured clockwise from the horizontal to the axis, and its length (L) in cm. We estimated the velocities by differentiating the position estimates. The position and velocity data were resampled at 30/s to correspond to the 33.3ms binwidth of the neural data.
Because the hip position was fixed and the foot was attached to the robot platform, the limb axis trajectory was determined entirely by the robot. The joint angles, however, were free to assume whatever trajectory was dictated by the limb biomechanics. Therefore to determine the consistency of trajectories across animals we expressed each angle trajectory as a deviation from the mean angle. The average jointangle trajectories across animals and the velocities determined from the averages were used for the analysis.
Data analysis
PRINCIPALCOMPONENT ANALYSIS.
We analyzed the responses of each sample cell population through the use of principalcomponent analysis (PCA). PCA provides a concise representation of a large data set of response waveforms by finding principal components (PCs) that are common factors across waveforms. The PCs are an ordered set of mutually independent (orthogonal) factors that summarize the total waveform variance across the data set. Any waveform in the original data set can be wholly reconstructed from an appropriately weighted sum of PCs. The set of weighting coefficients used to reconstruct the original response waveform is obtained by computing the Pearson product moment correlation between each response and the PC waveforms.
This property of the PCA also allowed us to represent the response waveforms in a PC vector space in which the PCs formed a set of orthogonal axes. In this representation, each response histogram was depicted by a single point determined from its coefficients of correlation with respect to each PC waveform. Neighboring points in these plots represent similar response waveforms (Osborn and Poppele 1992).
We examined the consistency of DSCT response components by deriving the PC waveforms from different data sets. For each of the 82 cells in the first sample group, we concatenated the figure8 and step response histograms to form a single histogram waveform as the input to the PCA. This group was also divided into two groups of 41 cells each by random selection, and each group was analyzed separately. Additional PCAs were performed for the total sample of 82 cells using histogram waveforms representing either the figure 8 or step responses separately.
Finally, a separate group of 81 cells having only the step cycle responses was used to control the consistency of response behavior across animals. We performed a separate PCA on this set of responses for comparison with the step response PCA described in the preceding paragraph.
REGRESSION ANALYSIS.
The PCs or factors derived from the PCA were tested for a relationship between their waveforms and the waveforms of various kinematic trajectories of the hindlimb. For this purpose, the joint angle and limb axis trajectories formed independent variable sets in a multivariate linear regression analysis. Each PC waveform was used as a dependent variable for this analysis. Two separate regression models were tested (see also Bosco et al. 1996)
We also analyzed the effect of limb kinematics on PC weighting with another regression model to determine the combinations of PCs that best represented the limb position or velocity waveforms
We determined both the coefficient of determination (R
^{2}), or fraction of the total variance explained by each regression model (Wilkinson 1990) and the partial coefficients of determination (r
^{2}) or fraction of the explained variance that was not accounted for by other predictors. Ther
^{2} is given by
DYNAMICS ANALYSIS.
The analyses described in the preceding text treated the data as a set of parameter values defined for each point in the movement trajectory. In addition, we also examined the timedependent behavior of the responses to test the hypothesis that the muscle receptors contribute to response dynamics. Muscle receptors act as dynamic filters on a muscle stretch signal. A simple linear model of the muscle spindle filter is described by a transfer function that predicts the time course of the spindle response to an arbitrary stretch. We determined the impulse response for the spindle transfer function given byChen and Poppele (1978) and convolved it with the waveforms of the limb axis trajectories to estimate the responses of muscle spindles that may have been stretched with those waveforms. We used a 200ms integration interval that could resolve time dependent behavior with time constants longer than about 400 ms.
RESULTS
We recorded from 82 single axons in the DSCT while a robot moved the hind foot along foot paths patterned after a step cycle and a figure 8 in a parasagittal plane (Fig. 1, B andC). The two movement paths tested the behavior of the cells for different excursions along the orientation and length dimensions of the limb axis as well as for oppositely directed foot movements in overlapping parts of the workspace. The speed profile of the foot along the step path was similar to that observed during a slow walk (Bosco and Poppele 1999), with the foot speed being highest at beginning of the forward swing and lowest during the back swing. The foot speed over the figure8 path was nearly constant, but the rates of change along the axes of length and orientation varied periodically for both paths (see also Bosco and Poppele 1999) (Fig. 2).
The joint angle trajectories were very similar in each animal as illustrated by the SD across animals illustrated in Fig. 2. The joint angle excursions at the knee were similar for the two footpaths, and they were somewhat smaller for the hip and larger for the ankle during the figure 8 than they were during the step (Fig. 2). Each of the joint angles went through a single cycle of flexion or extension during the step, while the knee and ankle went through two cycles during the figure 8. The trajectories of the limb axis showed two types of cyclic behavior. Limb axis orientation underwent a single cycle over each path while the limb axis length underwent two cycles in each (Fig. 2).
Response waveforms
Cycle histograms based on as few as four or five trajectory cycles showed that DSCT responses were highly reproducible across trials (Fig.3). The cells were generally activated during either the forward or backward swing of the step cycle (yellowred or greenblue segments in Fig. 3) and mostly during the downward or upward stroke of the figure 8 (yellow and green or red and blue segments). Moreover, there did not seem to be a simple relationship between the step and figure8 responses because units with similar step responses could have quite different figure8 responses (e.g., Fig. 3, A and B).
To summarize the variety of response waveforms produced by the cells, we used a PCA to find the PC factors that were common in the population of response waveforms. The PCA identifies an ordered set of mutually independent PCs that summarize the total waveform variance across the data set. We used the composite waveform representing the concatenated figure8 and step responses for each cell in the analysis to represent the cell's total response behavior. The first five PCs, accounting for about 80% of the total variance in the waveforms across units, are illustrated in Fig. 4.
The first two PCs accounted for about 60% of the total variance, and they were quite robust across samples. The higher order PCs, each accounting for <8% of the variance tended to depend on the subset of cells included. This is illustrated in Fig. 4 by a comparison between the PCs determined for the total 82 cells (black traces) and those determined for two subsets (gray traces) obtained by randomly dividing the total into two sets of 41 cells each. Note that the third through fifth PCs had somewhat different waveforms, even though the first two PCs were basically the same for the two subsets.
The PCs also did not depend on the movement considered, whether it was the combined step and figure8 paths or either path alone. Although the PCs were basically the same in those cases, their relative weightings were not, as we will examine further in a later section. The differences that did occur between these sets of PCs were once again more evident in the third through fifth PCs (Fig.5, black and gray traces). In fact, some of the difference may simply reflect a different ordering of PCs [e.g., figure8 PC5 (black) becomes PC3 (gray) for figure 8 alone]. The PC waveforms were also quite consistent across animals. We compared the step response PCs with those determined from the responses of 81 different cells from four animals that were not included in the rest of this study. The waveforms of the first five PCs determined for this sample population (Fig. 5, thin black traces) were very similar to those determined for the 82 cells in this study (Fig. 5, gray traces), indicating that these factors are robust contributors to DSCT activity in general.
Relationship to limb kinematics
With this summary of the DSCT behavior in the form of PCs, we could then address the question of how the population behavior relates to limb kinematic parameters. For this purpose, we used the same analytical approach we adopted previously to show the relationship between singleunit activity and the static limb kinematic parameters (Bosco et al. 1996). This time we correlated the combinedresponse PC waveforms with waveforms of the joint angle and limb axis trajectories using the multivariate linear regression models given in methods (Eqs. 1 and 2 ). In each case, we used both the position variables and their rates of change as the independent variables. The latter were included because our previous studies showed limb movement to be a robust contributor to DSCT responses.
The independent variables chosen for these analyses may not have been entirely independent though because of the biomechanical coupling across limb segments (Bosco et al. 1996;Lacquaniti and Maioli 1994a). However when the step and figure8 paths were considered together, joint angle correlations were not as strong as those observed with limb postures. In particular, the knee and hip angle trajectories were weakly correlated (r = −0.24,R ^{2} = 0.22, P < 0.0001), but the linear correlation did not account for much of the variance because of a systematic deviation from linearity. The hip and ankle angle trajectories were basically uncorrelated (r= 0.004, R ^{2} = 0.003, P< 0.9) as were the knee and ankle angle trajectories (r = 0.07, R ^{2} = 0.003,P < 0.2), and limb length and orientation trajectories (r = 0.03, R ^{2} = 0.002,P < 0.7). The results of the multivariate analyses are presented in Tables 1 and2, which show the regression coefficients of each model predictor indicating also their statistical significance levels.
Each PC waveform could be at least partly explained by a linear relationship to all three jointangle trajectories (Table 1). For the first PC, the most significant predictors were the hip angle and the hip and knee angle velocity trajectories. To determine the relative importance of the various independent variables in predicting the PC waveforms, we also computed the partial coefficients of determination (r ^{2}s). These coefficients indicate the fraction of the variance explained by any given predictor (i.e., any individual joint angle and its rate of change) that is not accounted for by the remaining predictors (see methods). According to this analysis, the hip and hip angle velocity alone accounted for 93% of the explained variance (Table 1, bottom), whereas other independent variables accounted for a much smaller fraction of the variance (about 20%). This is illustrated graphically in Fig.6 A, where the PC waveforms predicted by all the joint angle trajectories (gray traces) are compared with the waveforms predicted by the hip angle trajectories alone (thin traces).
The second PC was not quite as well explained as the first PC by joint angle kinematics, particularly for the step trajectory (Fig.6 B, gray trace). All three joint trajectories were highly significant predictors and they each accounted independently for between 40 and 60% of the explained variance (Table 1,bottom). In this case, the ankle angle trajectory accounted for the most of the explained variance (Fig. 6 B, thin traces), although the knee angle trajectory also accounted for about half of the explained variance.
The knee and ankle joint angle predictors were significant in explaining both the third and fourth PC waveforms. The ankle joint trajectory accounted for most of the explained variance for the third PC (35%), while the knee angle trajectory accounted for most of the explained variance for the fourth (63%). Thus with the possible exception the first PC, the joint angle trajectories were generally not very good predictors of the PC waveforms and more than one joint angle was required to adequately account for them.
The waveforms of both the first and second PCs were related more closely to the waveforms of the limb axis parameters, however. A linear regression model with limb axis length and orientation trajectories as predictors (Eq. 2 ) accounted for 94% of the waveform variance for both of these PCs (see Table 2).
For the first PC, the coefficients for limb axis orientation and orientation velocity were highly significant and the coefficient for length was also significant at the P < 0.005 level (Table 2, top). However the orientation predictors alone accounted for 94% of the variance explained by this model (Table 2,bottom), and even though the contribution of length was significant in the model, it had a negligible influence on the model predictions. This is illustrated in Fig. 6 C, which compares the model prediction with length and orientation predictors (gray traces) and with just the orientation predictors (thin black traces). Note that the orientation predictors alone were more successful in modeling the figure 8 and step PCs than were the hip angle predictors (compare Fig. 6, A and C).
The second PC was highly correlated with limb axis length and length velocity and also with orientation. However, for this PC, the length predictors accounted for nearly all the explained variance. The orientation predictors accounted for only 23% of the variance not explained by the length predictors. Although the regression coefficient for orientation was highly significant it was also very small. Consequently its effect on the prediction of the second PC waveform was minimal and mostly limited to the part of the PC waveform corresponding to the step trajectory (Fig. 6 D, thin black traces).
The higherorder PCs were mostly uncorrelated with the limb axis trajectories. Unlike the joint angle predictors, the limb axis length and orientation trajectories accounted for less than 20% of the PC waveform variance, and with one exception they were insignificant predictors.
The general success of these regression models suggests a tight coupling between the PCs representing the responses to steps and figure8 movements and the kinematics of the movements. The first two PCs, which account for more than 60% of the total variance in the combined step and figure8 response waveforms, are linearly related to the limb axis orientation and length trajectories, respectively. Given that the PCs are mutually independent, these results further suggest that the waveforms of limb axis length and orientation may be independently represented in the responses of the DSCT neurons. The finding that the first PC may also be well predicted by the hip angle trajectory may reflect a consistent correlation between hip angle and limb axis orientation (r = −0.36,R ^{2} = 0.90, P < 0.0001).
Distribution of unit responses in the population
The PCA also allows us to readily compare responses and to determine their distribution in the population (Osborn and Poppele 1992). Because each unit response waveform may be reconstructed from a weighted sum of the PCs, a response waveform may be represented as a vector in a PC vector space where the PCs define the coordinates. Each vector is defined by the set of correlation coefficients for a response waveform and the PCs.
We constructed a representation of this distribution as a scatterplot of response vectors defined by the first two PCs (Fig.7 A). The response of each cell is represented in this plot by a single point, and neighboring points represent responses having similar waveforms. The dominant response patterns of these cells fell roughly into four groups corresponding to a pairing of the positive or negative second PCs each with the positive and negative first PCs. Corresponding response waveforms representative of each group are plotted in 7, B–E. The pattern of a more or less even division between positive and negative pairings of the first and second PCs is also evident in Fig.8 B.
It is also noteworthy that about half of the combined response waveforms (39/82 or 48%) were at least 85% accounted for by just the first two PCs. This fraction was even larger for the responses to either the step or figure 8 alone (Figs. 8 A, open and gray filled circles, and 8B). However, about half of the cells (38/82, 46%) fit this pattern for both their combined and separate responses (Fig. 8 A, black circles). Altogether only about a third of the responses to either step or figure 8 (59/164, 36%) were more than 15% accounted for by the higherorder PCs. These responses are represented by points within the inner rings in Fig. 8 B.
Although the waveforms of the first and second PCs did not depend on whether they were determined separately for the figure8 or step responses or from the combined responses, they did account for different percentages of the total variance in each case (Fig. 5). For the step responses, the first PC accounted for 58% of the variance, while it accounted for only 37% for the figure8 responses. Conversely, the second PC accounted for 29% of the variance for the figure8 responses and only 16% for the steps. This shift in the weighting of these components may also be seen in the distributions plotted for figure8 and step responses in Fig. 8 B. The greater weighting of the first PC in the step responses can be seen as a clustering of response waveforms near the first PC axis. For the figure8 responses, the responses are distributed more evenly between the first and second PCs.
These differences in weightings appear to reflect differences in the excursions of limb axis length and orientation for the two kinds of footpath. The total excursion in orientation for the step was 1.40 times that of the figure 8 (42 vs. 30°, see Fig. 1). The excursion in length for the step, however, was 0.49 times that for the figure 8 (2.9 vs. 5.9 cm). Interestingly, the ratios of the PC weightings for the step and figure8 PCs were quite similar, 1.57 for the first PC and 0.55 for the second (see Fig. 8 C).
Response dynamics
The analysis presented here is basically a static analysis that considers unit activity at any moment in time to be determined solely by the limb position and its instantaneous velocity at the same moment. The analysis does not explicitly account for any time dependence. Thus although it is clear from the regression analysis that the first two PCs relate significantly to both the positional and velocity components of the limb axis trajectory (Table 2), it is not clear how this behavior relates to the timedependent behavior of muscle receptors, for example. Receptor activity is best described by rate constants that predict a characteristic adaptive transition between movement and positionrelated activity rather than the instantaneous transition predicted by the regression model. The simplest model that adequately describes the behavior of muscle spindles is a linear filter that acts on the timedependent stretch applied to the spindle to produce a timedependent modulation of its firing rate (e.g., Chen and Poppele 1978). If spindle dynamic behavior was responsible for the movement sensitivity of the DSCT neurons, then the spindle filter should also predict the waveform of the first two PCs that relate explicitly to the limb axis trajectory.
We tested this by applying a simple linear model for the muscle spindle (Chen and Poppele 1978) to the waveform of the limb axis trajectories to predict the response of a receptor stretched with this same waveform. If a muscle like a hip flexor, for example, were to be stretched with the waveform of the limb axis orientation trajectory, the spindles in that muscle would undergo a periodic change in length having the waveform similar to that of the limb axis orientation change over a movement cycle. The same considerations would apply to the muscle receptors in the muscles that are periodically stretched by changes in limb axis length.
The model we tested predicts the actual time course of the response adaptation to a 1mm step change in muscle spindle length (Fig.9 A) (Chen and Poppele 1978). The response waveforms predicted by the model with the limb axis orientation waveform applied as the input signal agree reasonably well with the waveforms of the first PC for the step and figure8 responses. Figure 9 B shows the relationships among the input waveform (thin black traces), the waveform predicted by the spindle model (gray traces), and the PC waveforms (thick black traces). The waveforms of the second PC are also reasonably predicted by applying the same linear muscle spindle model to the waveforms of the limb axis length trajectories (Fig. 9 C).
These results suggest that DSCT response components may be accounted for simply by the waveforms of the limb axis length and orientation trajectories that have been filtered by the dynamical properties of the muscle receptors. Thus it is not necessary to postulate that the spinal circuitry generates any additional dynamical components.
The results might also be taken to suggest that the DSCT responses are driven primarily by sensory receptors having dynamical properties like those of the muscle spindles. However, other receptors are also likely to contribute to the DSCT activity, and we showed previously that about a third of the DSCT cells that are modulated by limb position have a very low sensitivity to limb movement (Bosco and Poppele 1999). A question raised by the PCA then is whether it can also account for the diversity of dynamical behavior that is evident in the individual responses. For example, the responses illustrated in Fig. 3,C and D, have waveforms that are comparable to the negative limb axis orientation and negative length velocity trajectories respectively, yet none of the PCs could be predicted by the position or velocity trajectories alone.
We explored this by mapping the position and velocity waveforms in PC space, that is, by determining the coefficients for combinations of PCs that best represent the limb position or velocity trajectory waveforms. For this purpose, we used the regression model described by Eq.3 (methods). The representation of limb axis length maps out a locus of points in a fourdimensional PC space as the length trajectory is varied from a purely positional representation through various position and velocity combinations to a purely velocity representation. Each combination required nearly the same coefficients for the first, second, and fourth PCs, and only the coefficient of the third PC varied as different combinations of position and velocity were modeled. This is illustrated in Fig.10, which shows the locus of points projected onto the plane of the third and fourth PCs (□). This result implies that the dynamic mix of position and velocity components of a response to limb length changes is represented by different weightings of the third PC. A similar result was found for the relationship between the fourth PC and the limb orientation dynamics. This is illustrated also in Fig. 10 as a locus of points forming a line nearly parallel to the fourth PC axis (●). In this case, there was also a systematic change in the coordinates of the fifth PC (not illustrated), implying that different weightings of the fourth and fifth PCs may account for the varied dynamics of orientation responses.
DISCUSSION
The basic result reported here is that the population responses of DSCT neurons to hindlimb movement can be well accounted for by two principal components. These are independent response components that correlate, respectively, with the limb axis length and orientation trajectories, suggesting that these global parameters may be somehow parsed from the sensory input by the spinal circuitry.
PCA
Although we found a variety of responses to the step and figure8 movements from the neurons we studied, a PCA revealed that their waveforms could be accounted for by a summation of only a few component waveforms. In fact, two PCs accounted for at least 85% of the waveform variance for about half of the cells sampled. We tested the hypothesis that the PCs relate to limb kinematics and found that the limb axis orientation and length trajectories explained well over 90% of the PC waveform variance in these two components. While the orientation and length trajectories were both highly significant predictors of the first two PC waveforms, it is clear in Fig. 6 that the orientation trajectory alone can account for the first PC and that length alone can account for the second PC. Moreover, no single joint angle trajectory was as successful in predicting the PC waveforms. It is not unreasonable, therefore to conclude that the first two PCs, which are by definition separate and independent components of the DSCT responses are indistinguishable from neural representations of the limb axis orientation and length respectively. These first two PCs are highly robust across cell samples.
The PCA selects linearly independent factors that are common to a set of waveforms like the response histograms. One way such factors might be identified is if they represent separate components that combine differently in the various DSCT response waveforms. We examined the population distribution of response waveforms (Fig. 5) and found that most responses were largely accounted for by a weighted sum of the first two PCs, and the sums occurred in all possible combinations of same sign and opposite signs. Furthermore the ratio of the PC weightings were proportional to the ratio of excursions in orientation and length in the stimulus. These results imply that activity components related to limb length and orientation are manifest in the circuitry in a manner that allows their combination by linear summation in a variety of ways. It suggests that the limb axis representations may be generated independently to provide separate inputs to the DSCT neurons. One interpretation of these results, therefore is that the first two PCs revealed by the PCA might actually represent the response waveforms of presynaptic elements that synapse on DSCT neurons.
Representation of movement dynamics
This interpretation of PCs may help account for the first two PCs, but it fails to explain the responses that were not accounted for simply by the first two PCs. Although higherorder PCs did not account individually for more than about 8% of the variance in the population of response waveforms, they were nevertheless robust and consistently present in the responses of diverse populations of cells. Therefore they do not represent random or idiosyncratic components. Nevertheless, they were not simply related to any of the limb kinematic variables we examined.
A clue about the role of the higherorder PCs in describing the response waveforms comes from an examination of the relationship between the PCs and the response dynamics. The first two PCs, representing the independent coordinates of length and orientation, also appear to correspond to a particular dynamical representation of the length and orientation trajectories, namely one that reflects the dynamic properties of the muscle spindles. While this may have been a common feature of the responses, it was not the only dynamical representation observed in the DSCT responses. The evidence we presented here suggests the PCA may represent the dynamical diversity in the ensemble by isolating factors that modify the basic dynamics represented by the first and second PCs. Thus the higherorder PCs provide one way of accounting for various weightings of positional and velocity trajectory components. If so, it implies that the first two PCs may not fully represent the presynaptic circuitry, and other elements may exist whose activity is not explicitly represented by PC waveforms.
Circuit considerations
Thus there may be no simple interpretation of the PCA results that relates to elements in the DSCT circuitry. In light of previous studies from this laboratory (Osborn and Poppele 1989, 1993), it seems likely that the circuitry is more networklike, where parameters like limb dynamics are widely distributed among the presynaptic elements. The factors isolated by the PCA in such a system may be thought of as statistical representations of likely activity patterns within the circuitry. In fact, we cannot rule out that the representations of limb axis length and orientation are not also distributed in some way; but a more parsimonious interpretation of the evidence is that these parameters are encoded separately and combined at the level of the DSCT neurons.
The evidence favoring this interpretation comes from two basic observations made in this study. The first is the relationship found between PC waveforms and limb axis orientation and length trajectories. The second observation is the distribution of the waveforms described by the PCs. The distributions illustrated in Figs. 5 and 8 suggest a modular organization based on positive and negative versions of the first two principle components. Given the preceding interpretation of the first two PCs, these distributions are consistent with a presynaptic organization consisting of two basic modules representing the limb axis orientation and length respectively. Outputs from these modules would then be combined according to a distributed set of weighting coefficients allowing many possible combinations to be expressed in the DSCT activity patterns. Furthermore, the data suggest that these hypothetical modules may be modulated separately by stimulus parameters related to either orientation or length. Because the relative weightings for orientation and length were found to be proportional to stimulus intensities, it implies that their central weightings were not modulated under these conditions.
Finally our analysis of the higherorder PCs indicated that they were separately influenced by dynamics associated with changes in orientation and changes in length (Fig. 10). Changes in the weighting of the third PC were associated specifically with changes in length dynamics while the weighting of the fourth and fifth PC were associated with changes in orientation dynamics. Thus it appears that a distributed representation of response dynamics may somehow exist separately for length and orientation.
Separate representations of length and orientation
This interpretation raises the question of how the sensory input might become organized as to produce separate length and orientation representations. It seems likely that the sensory receptors encode muscle stretch, joint rotations and localized skin distortions. But even localized sensory signals may carry information related to wholelimb parameters because of mechanical coupling within the limb (Bosco et al. 1996). Mechanical structures that span more than one joint, such as biarticulate muscles, tend to couple the movements of adjacent limb segments. The coupling leads to a specific correlation between limb axis parameters and the joint angles at the hip, knee, and ankle (Bosco et al. 1996). In fact, under static conditions, different limb axis orientations were found to be correlated primarily with changes in hip (r = 0.721) and knee angle (r = 0.535) and to a lesser extent with ankle angle changes (r = 0.357). In contrast, different limb axis lengths were primarily correlated with changes in knee (r = 0.642) and ankle angle (r = 0.432) and poorly correlated with changes in hip angle (r = 0.041). Thus a weighting of sensory projections that favors combinations of inputs from receptors responding to changes in hip and knee angles would also favor a representation of limb orientation. Likewise a weighting that favors combinations from the knee and ankle would also favor a representation of limb length.
It is interesting to note in this regard that the topographic mapping from hindlimb cutaneous receptors to the lumbar spinal cord may help impose the appropriate weightings. The cutaneous organization provides a kind of rostralcaudal mapping of the hind limb receptors within the spinal cord (e.g., Brown et al. 1998), with the anterior hip area being represented more rostrally and the foot and ankle more caudally. An estimation of the locations of dorsal horn neurons in the spinal cord that receive primary afferent input from cutaneous receptors around the hip, knee and ankle taken from Brown et al. (1998) is plotted in Fig. 11. At the L_{5} spinal level, the proportion of cells receiving input from the hip and knee areas dominates over the proportion receiving ankle or foot input. At the L_{6} level, however the projection favors the knee and ankle inputs. Therefore a population response of interneurons at L_{5} would tend to favor a representation of limb orientation, whereas the population response of L_{6} interneurons a limb length representation.
Thus it might only require that these two populations (L_{5} and L_{6}) somehow converge separately onto distinct groups of interneurons that are presynaptic to the DSCT to provide the separate representations of limb length and orientation implied by the PCA. However, it is unclear exactly how this separation might be achieved or maintained.
One reason for considering this example is to illustrate that the observed global sensory representation might result from any of a number of factors including the biomechanical coupling within the limb and the topography of sensory projections. Furthermore, local information about the hip angle can provide a close approximation to limb axis orientation as evidenced by the finding that both these variables were highly significant predictors of the first PC. However, the DSCT representation does distinguish the two because the orientation trajectory was the better single predictor, particularly in the responses to the figure8 footpath. We also cannot exclude that limb biomechanics may play a decisive role in reducing the variance of the sensory input to a twodimensional representation with little or no contribution from the neural circuitry. However, we did show that some DSCT cells can encode an invariant twodimensional representation of limb posture under static conditions when we altered the relationship between joint angles and foot position with a knee angle constraint (Bosco et al. 2000a).
Significance for sensorimotor integration
The possibility of an explicit neuronal representation of limb length and orientation also has some theoretical interest because it could provide the basis for a mechanism to reduce the necessary degrees of freedom for limb control. In fact, a series of studies on the control of posture in cats found that the animals were able to control limb length and orientation independently (Lacquaniti and Maioli 1994a,b; Maioli and Poppele 1991). This could imply a control organization with separate channels for these limb axis parameters (comparable in some way to distance and direction). The findings of this and related earlier studies (Bosco and Poppele 1997, 2000; Bosco et al. 1996, 2000a) about the relationship of DSCT activity to limb length and orientation provide experimental support for this interpretation. The findings imply further that an explicit, independent representation of length and orientation is also present at the lowest levels of sensory processing in the spinal cord.
Conclusion
The data and analyses presented here suggest a population code for the DSCT in which representations of hindlimb kinematics are distributed across the units of the population. The length and orientation coordinates of the hindlimb axis or limb endpoint are represented independently, and within each representation there is a distributed representation of the kinematic parameters of position and velocity. The PCA suggests that these representations are somehow modularized, allowing various combinations to be expressed in the DSCT activity. The first two PCs identify two independent components that correlate with limb kinematics, while higherorder components appear to summarize the response dynamics. Thus the first five principal components, which are highly robust across samples, summarize a populational representation that accounts for between 80 and 90% of the total waveform variance seen the DSCT responses.
Acknowledgments
The authors thank J. Eian for help and assistance on this project.
This research was supported by National Institute of Neurological Disorders and Stroke Grant NS21143.
Footnotes

Address for reprint requests: R. E. Poppele, Dept. of Neuroscience 6145 JH, 321 Church St. SE, Minneapolis, MN 55455 (Email: dick{at}umn.edu).
 Copyright © 2002 The American Physiological Society