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The Journal of Neurophysiology Vol. 83 No. 5 May 2000, pp. 2931-2945
Copyright ©2000 by the American Physiological Society
1Department of Neuroscience and 2Graduate Program in Biomedical Engineering, University of Minnesota, Minneapolis, Minnesota 55455
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Bosco, G., R. E. Poppele, and J. Eian. Reference Frames for Spinal Proprioception: Limb Endpoint Based or Joint-Level Based?. J. Neurophysiol. 83: 2931-2945, 2000. 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 second-order spinal neurons also may encode proprioceptive information in a limb-based rather than joint-based 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 limb-based and a joint-based 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 joint-angle 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 joint-angle 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 joint-angles. 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.
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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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 motor-control 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
feed-forward 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 second-order sensory neurons projecting to the cerebellum from the spinal cord, i.e., at very early stages of sensory-motor 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 limb-centered 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 low-order 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
sensory-motor 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 joint-angle 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.
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METHODS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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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.
1A) were recorded by means of
a video camera (Javelin Model 7242 CCD camera; 60 frames/s) and
digitized off-line 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
computer-controlled 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.
1B, grey circles). Each foot position was held for 6 s
(see also Bosco and Poppele 1997; Bosco et al.
1996).
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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. 1B). 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 co-variation 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. 1B, 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. 1B). Because this
constraint restricted motion at the knee joint almost completely, we
will refer to it as the "knee-fixed" constraint. We used the ankle
elastic constraint exclusively in one experiment and together with the
knee-fixed constraint in one other experiment. The other four
experiments employed only the knee-fixed constraint.
Kinematic measurements
Reflective markers were placed on the skin over the hip, knee,
ankle, and lateral metatarsal-phalangeal 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 out-of-plane positions using an
algorithm that is described in detail in the APPENDIX. 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.
1A). The joint angles also are defined in Fig. 1A
as the angles measured clockwise between two limb segments. Although
the revised kinematic analysis described in the APPENDIX
may generate some discrepancies between the joint-angle 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 (metatarsal-phalangeal 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 T10-T11
level of the spinal cord using insulated tungsten electrodes (5 M
,
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 (see
Bosco 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 three-dimensional 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 joint-angle
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 least-square 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)
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(1) |
0-2 are
the coefficients and 
is the residual error. The rationale for this
approach came from our earlier observation that most DSCT neurons
modulated by foot position showed linear relationships with the length
and the orientation of the limb axis (Bosco et al.
1996). For the subset of neurons that were significantly
modulated according to Eq. 1 (P < 0.001),
we determined further whether the foot position representation was
invariant across experimental conditions. For this purpose, we modified the model in Eq. 1 to test the hypothesis that plane
parameters describing the relationships between firing rate and foot
position are different in the constrained and unconstrained conditions
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(2) |
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). Thus the coefficients 3 and
6 indicated intercept offsets between control
and constrained conditions. Similarly, interactions between the foot
position terms (L and O) and each of the binary
variables, produced coefficients (4,
5, 7, and
8), indicating changes in the slope of the
plane relative to the L or O axis. Finally, we
used t statistics on the coefficients associated with
A, K, and their interactions with foot-position terms
(L and O) setting a P < 0.01 as
cutoff to indicate significant differences across experimental
conditions. Note that, when only one constrained condition was used
only one binary variable (A or K) was added to
the model.
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
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(3) |
value (<0.01) for predictors to enter the model, we also were able to eliminate predictors that did not
contribute independently because of their relationship with more
significant predictors (Bosco et al. 1996; also
Wilkinson 1990).
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 foot-centered 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)
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(4) |
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RESULTS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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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 Table
1.
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These figures show each set of joint angles for 20 foot
positions and the corresponding least-square covariance planes. Each data point plots the three joint-angle values for a given foot position. Note that the orientations of the covariance planes are
similar across experiments in the unconstrained condition (Figs.
2A and 3A). 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 (see
METHODS and
APPENDIX).1
Overall, the effect of the ankle elastic constraint on passive limb biomechanics was a slight rotation of the joint-angle co-variance 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. 2C and 3B). This constraint was variably successful in actually fixing the knee joint angle, being most successful in cat 4 (Fig. 3B). 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 least-square 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 joint-angle coordinates and a representation based on the foot position alone. Because the knee-fixed 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
(R2 > 0.4, P < 0.0001) in the firing rate of 70 of 74 DSCT neurons examined (95%).
FOOT-POSITION REPRESENTATION. To determine whether the foot-position 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 and 5). Neurons 2576 and 2710 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 B and E, respectively). In agreement with this qualitative judgement, the regression analysis (Eq. 2) showed that the coefficients associated with the binary variables A or K 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 and F) 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).
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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 (L and 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. 6B, 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°.2JOINT-ANGLE 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 joint-angle 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). Scatter-plots 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.
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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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 joint-angle 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
joint-angle 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 joint-angle 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 joint-angle 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 joint-angle 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 joint-based 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 sensory-motor 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
joint-based 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 joint-related 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
joint-angle 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
joint-angle covariance. Any perturbation in this system could be
compensated by sensing a mismatch between an endpoint-related signal
and a joint-angle-related signal, which then could be applied as an
error signal to reestablish the appropriate joint-angle covariance.
We might speculate that the spinocerebellar system could play a role in this proposed compensation. If each endpoint-related DSCT cell were matched somehow with a corresponding joint-related cell having a congruent endpoint-related 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 joint-angle covariance. Any mismatch in their respective signals would indicate a deviation from the desired joint-angle 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 joint-angle 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).
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APPENDIX |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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In some circumstances it is not convenient or cost-effective to measure three-dimensional (3-D) limb kinematics directly, so it is desirable to have a two-dimensional (2-D) measurement technique that can accurately portray 3-D limb geometry. Such a method is described in this appendix.
Two-dimensional 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. A1). Under such conditions, a given knee angle, K, determined from 2-D 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 joint-angle estimates as the limb moves.
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The method described in the following text corrects for these out-of-plane rotations by considering the geometry of the experimental set-up. 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 2-D 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. A2). The location of the spatial invariant point in the reference plane (Z = 0) is defined as the origin [S(0,0,0) in Fig. A2] of the 3-D coordinates of spatial points, P(X, Y, Z) having corresponding image points, p'(x, y).
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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 the
z-axis locations of spatial objects into dilation factors,
dz
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Let r = X/x' be the calibration
relating a distance on plane Z = 0 to its image
distance on the imaging plane, then
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X) and not X alone from the image, we may
use these geometric considerations to determine X as follows
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(A1) |
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. A1, the ankle position in space can be expressed as
![(<IT>Xa</IT><IT>, </IT><IT>Ya</IT><IT>, </IT><IT>Za</IT>)<IT>=</IT>(<IT>xad</IT>(<IT>Za</IT>)<IT>, </IT><IT>yad</IT>(<IT>Za</IT>)<IT>, </IT><IT>Za</IT>)<IT>=</IT>(<IT>xa</IT>[<IT>rm</IT>(<IT>Za</IT>)<IT>−</IT><IT>r</IT>]<IT>, </IT><IT>ya</IT>[<IT>rm</IT>(<IT>Za</IT>)<IT>−</IT><IT>r</IT>]<IT>, </IT><IT>Za</IT>)](/content/vol83/issue5/fulltext/2931/img012.gif)
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(A2) |
![(<IT>Xt</IT><IT>, </IT><IT>Yt</IT><IT>, </IT><IT>Zt</IT>)<IT>=</IT>(<IT>xtd</IT>(<IT>Zt</IT>)<IT>, </IT><IT>ytd</IT>(<IT>Zt</IT>)<IT>, </IT><IT>Zt</IT>)<IT>=</IT>(<IT>xt</IT>[<IT>rm</IT>(<IT>Zt</IT>)<IT>−</IT><IT>r</IT>]<IT>, </IT><IT>yt</IT>[<IT>rm</IT>(<IT>Zt</IT>)<IT>−</IT><IT>r</IT>]<IT>, </IT><IT>Zt</IT>)](/content/vol83/issue5/fulltext/2931/img013.gif)
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(A3) |
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(A4) |
![<IT>Xa</IT><IT>=</IT><IT>xad</IT>(<IT>Za</IT>)<IT>=</IT><IT>xa</IT>[<IT>rm</IT>(<IT>Za</IT>)<IT>−</IT><IT>r</IT>] and](/content/vol83/issue5/fulltext/2931/img015.gif)
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![<IT>Ya</IT><IT>=</IT><IT>yad</IT>(<IT>Za</IT>)<IT>=</IT><IT>ya</IT>[<IT>rm</IT>(<IT>Za</IT>)<IT>−</IT><IT>r</IT>]](/content/vol83/issue5/fulltext/2931/img016.gif)
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![<IT>L</IT><SUP><IT>2</IT></SUP><IT>=</IT>(<IT>xa</IT>[<IT>rm</IT>(<IT>Za</IT>)<IT>−</IT><IT>r</IT>]<IT>−</IT><IT>Xt</IT>)<SUP><IT>2</IT></SUP><IT>+</IT>(<IT>ya</IT>[<IT>rm</IT>(<IT>Za</IT>)<IT>−</IT><IT>r</IT>]<IT>−</IT><IT>Yt</IT>)<SUP><IT>2</IT></SUP><IT>+</IT>(<IT>Za</IT><IT>−</IT><IT>Zt</IT>)<SUP><IT>2</IT></SUP>](/content/vol83/issue5/fulltext/2931/img017.gif)
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(A5) |
Note that because Eq. A5 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.
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ACKNOWLEDGMENTS |
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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 APPENDIX.
This research was supported by National Institute of Neurological Disorders and Stroke Grant NS-21143.
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FOOTNOTES |
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Address for reprint requests: R. E. Poppele, 6-145 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 out-of-plane 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 joint-angle covariance planes reported for the ankle elastic constraint (Fig. 2B) are similar to those described in earlier experiments without constraints. A possible interpretation of this result is that the ankle-constraint 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 center-out or out-center 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 joint-related cells, as if these were clear subpopulations of cells. In fact, as suggested in Fig. 6A, the properties of joint- and endpoint-relatedness 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.
Received 30 August 1999; accepted in final form 7 February 2000.
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REFERENCES |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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4
trials per position) plotted against the length and orientation of the
limb axis for the passive unconstrained limb. The plane is the
least-squares regression fit to the data (Eq. 
