Musculoskeletal geometry accounts for apparent extrinsic representation of paw position in dorsal spinocerebellar tract

A classic experiment concluding that many dorsal spinocerebellar tract neurons encode paw position rather than joint angles has been cited by many studies as evidence for high-level computation occurring within a single synapse of the sensors. However, our study provides evidence that such a computation is not required to explain the results. Using simulation, we replicated many of the original results with purely random connectivity, suggesting that a reinterpretation of the classic experiment is needed.

Abstract

Proprioception, the sense of limb position and motion, arises from individual muscle receptors. An important question is how and where in the neuroaxis our high level “extrinsic” sense of limb movement originates. In the 1990s, a series of papers detailed the properties of neurons in the dorsal spinocerebellar tract (DSCT) of the cat. Despite their direct projections from sensory receptors, it appeared that half of these neurons had consistent, high-level tuning to paw position rather than to joint angles (or muscle lengths). These results suggested that many DSCT neurons compute paw position from lower level sensory information. We examined the contribution of musculoskeletal geometry to this apparent extrinsic representation by simulating a three-joint hindlimb with mono- and biarticular muscles, each providing a muscle spindlelike signal, modulated by the muscle length. We simulated neurons driven by randomly weighted combinations of these signals and moved the paw to different positions under two joint-covariance conditions similar to the original experiments. Our results paralleled those experiments in a number of respects: 1) Many neurons were tuned to paw position relative to the hip under both conditions. 2) The distribution of tuning was strongly bimodal, with most neurons driven by whole-leg flexion or extension. 3) The change in tuning between conditions clustered around zero (median absolute change ~20°). These results indicate that, at least for these constraint conditions, extrinsic-like representation can be achieved simply through musculoskeletal geometry and convergent muscle length inputs. Consequently, they suggest a reinterpretation of the earlier results may be required.

NEW & NOTEWORTHY A classic experiment concluding that many dorsal spinocerebellar tract neurons encode paw position rather than joint angles has been cited by many studies as evidence for high-level computation occurring within a single synapse of the sensors. However, our study provides evidence that such a computation is not required to explain the results. Using simulation, we replicated many of the original results with purely random connectivity, suggesting that a reinterpretation of the classic experiment is needed.

there is an intimate interplay between somatosensation and the control of movement that is evident in the movement deficits that result when somatosensation is lost. Patients make large errors of extent while making reaching movements in different directions (Gordon et al. 1995) and are largely unable to control the dynamics of the limb (Sainburg et al. 1995). Part of the complexity of limb movement control arises from the redundancy that allows us to place our hand or foot in a particular location using an infinite variety of limb configurations. To complete a simple reaching movement, a high-level command, e.g., “reach for that book,” must be transformed into commands for dozens of individual muscles. The cerebellum monitors and processes the movement commands themselves, as well as somatosensory feedback from the limb in a process that makes the movement execution fluid and effortless. An important question in motor neurophysiology has been how these sensorimotor signals are represented by populations of neurons and how these representations are transformed through the central nervous system.

This question has received far more attention in the motor system than it has in the somatosensory system. In a highly cited classic study, Georgopoulos and colleagues (1982) found that neurons in primary motor cortex (M1) appear to represent arm kinematics in an extrinsic, hand-based coordinate frame, where each neuron has broad tuning to a particular direction of hand movement. Such a system would require subsequent transformation of this high level control signal into signals appropriate to activate muscles.

Subsequent experiments examined the question of representation more closely, taking advantage of the limb’s redundancy to alter limb posture and change the relation between hand movement and joint rotation (Caminiti et al. 1990; Kakei et al. 1999; Morrow et al. 2007; Scott and Kalaska 1997). Most of these studies found many M1 neurons with tuning that was more closely related to the “intrinsic” coordinates of joint rotations or muscle lengths. Using similar methods, there is now evidence that much of the transformation from extrinsic to intrinsic coordinates occurs between the premotor and motor cortices (Kakei et al. 2001; Shen and Alexander 1997) and is completed between motor cortex and the spinal cord (Yanai et al. 2008).

Compared with the motor system, the representation of proprioception, the sense of body position, movement, and related forces has received far less experimental attention. Proprioception arises from muscle spindles, situated parallel to muscle fibers and sensitive to muscle length and length change, and from Golgi tendon organs, lying in series with muscles and sensitive to active muscle force, and even cutaneous receptors that respond to skin deformation and stretch (Kandel et al. 2012). However, psychophysical experiments in humans (and common experience) suggest that perception of the arm is focused on hand position or overall limb orientation rather than individual joint angles or muscles lengths (Fuentes and Bastian 2010), thus matching vision and the other exteroceptive senses. Consistent with these studies, proprioceptive neurons in primary somatosensory cortex (S1) have been thought to exhibit extrinsic tuning (Prud’homme and Kalaska 1994) quite similar to that suggested for M1 neurons by the experiments of Georgopoulos and colleagues (Georgopoulos et al. 1982). Thus, the system mediating proprioception has the opposite requirement of the motor system: low level sensors in muscles must ultimately generate high level perception in extrinsic coordinates.

Remarkably, experiments involving recordings from neurons in the dorsal spinocerebellar tract (DSCT) of the cat, a pathway carrying proprioceptive information from the cat’s hindlimb to the cerebellum, suggested that such a transformation had already occurred at that level (Bosco and Poppele 1997; Bosco et al. 1996, 2000). As the hind paw of an anesthetized cat was moved throughout a range of positions in the sagittal plane, DSCT neurons tended to represent the limb position not in terms of individual joint angles or muscle lengths, but rather as a vector drawn from the hip to the paw. For approximately half of DSCT neurons, this extrinsic representation persisted even in the presence of a constraint that fixed the knee angle and changed how the joints covaried during hindlimb movement.

This result was surprising for a number of reasons. For one, the significant amount of computation implied by this transformation must occur very early in sensory systems, at DSCT neurons, which receive inputs directly from sensory afferents. Furthermore, the posited cerebellar role in the coordination of intersegmental dynamics during movement (Cooper et al. 2000; Sainburg et al. 1995) might well profit from muscle-related feedback, rather than a simple representation of paw position. Finally, arguments for the need for this transformation based on characteristics of our conscious perception are much less convincing in the context of cerebellar processing than they are for signals in the cerebral cortex.

The idea that neurons in the DSCT encode the hindlimb state in terms of the paw position has been well accepted into proprioceptive literature (Daley and Biewener 2006; Fuentes and Bastian 2010; Kim et al. 2010; Morton and Bastian 2006; Ting and Macpherson 2005; Weber et al. 2007). However, the original experiment did not explore the contribution of musculoskeletal geometry to the tuning of neural activity. While the knee-fixed constraint changed the hindlimb joint covariance, hindlimb muscle length changes do not have a simple linear relationship with joint rotation. As such, the effect of this constraint on convergent muscle length inputs remains unclear. Consequently, we replicated the experiment in simulation, with neurons that drew their activity directly from randomly weighted combinations of spindlelike muscle length inputs. Surprisingly, the results of this simple neural model closely paralleled the earlier experimental results. This outcome suggests a very different interpretation of the representation of limb state by DSCT neurons. These results suggest that there is little evidence that the DSCT neural signals have been transformed to represent limb state in anything other than simple muscle coordinates. This outcome also highlights the importance of considering the properties of the musculoskeletal system when interpreting signals recorded in the central nervous system.

METHODS

Musculoskeletal model.

To simulate the cat hindlimb, we used a four-segment, three-joint musculoskeletal model, with segment lengths and muscle insertion points adapted from the musculoskeletal model used by Bunderson et al. (2010), based on an anatomical study performed by Burkholder and Nichols (2004). We used eight muscles (five monoarticular and three biarticular) and constrained the limb to the sagittal plane. Figure 1 shows a schematic representation of the model.

Fig. 1.

Schematic of simulated leg, showing segments (black), joints (blue), and muscles (red). BFP, biceps femoris posterior; BFA, biceps femoris anterior; IP, iliopsoas; RF, rectus femoris; VL, vastus lateralis; MG, medial gastrocnemius; Sol, soleus; TA, tibialis anterior. We conducted the simulation by moving the paw to different positions in the sagittal plane in two conditions: elastic joint and knee-fixed.

Our simulation experiment included two constraint conditions meant to simulate those of the original experiment: an elastic constraint that approximated normal limb mechanics and a knee-fixed constraint. We implemented the elastic constraint by placing identical springs at each joint to mimic the passive elastic effects of muscles and fascia, thus defining a minimum energy state for any given paw position. This constraint fully determined the hindlimb posture for any given paw position. As in the baseline condition with the anesthetized cat (Bosco and Poppele 1997; Bosco et al. 2000), the joints covaried essentially linearly, with a plane accounting for 94% of the joint covariance. In our second constraint condition we fixed the knee angle, forcing a linear joint covariance different from that of the elastic condition. The consistency of neural tuning to paw position between these two conditions can be considered a measure of how nearly a given neuron is tuned to extrinsic coordinates.

Generation of neural activity.

In our simulation experiments we moved the paw in the sagittal plane through 100 equally spaced locations in a 10×10 polar grid centered on the hip and spanning angles from 245 to 300° with radii from 18 to 22 cm. This matched the grid pattern used in the original experiments (Bosco and Poppele 1997; Bosco et al. 1996, 2000), and resulted in joint angle excursions of 40, 49, and 50° for hip, knee, and ankle, respectively. These excursions are similar to those reported in the original experiments (25–50, 45–60, 60–80° excursions for hip, knee, and ankle joints across several experiments). Varying the stiffness of each joint in the model (±25%) altered the simulated joint excursions but did not affect the overall conclusions reported here (see results).

For each paw position, we found the corresponding hindlimb configuration and calculated the length of each muscle from origin to insertion from the musculoskeletal model. The maximum change in muscle length for a given paw displacement corresponds to the muscle’s pulling direction. Muscle lengths were normalized to lie between 0 and 1, corresponding to the minimum and maximum lengths achieved throughout the full workspace. We transformed the resultant muscle lengths into simulated neural discharge by taking a weighted sum of the normalized lengths to simulate the convergence of length-sensitive afferents onto a given DSCT neuron (Fig. 2). Each neuron was thus characterized fully by its eight muscle weights, which we drew randomly from a standard normal distribution (µ = 0, σ = 1). The zero mean allowed negative (inhibitory) weights. This raw neural activation was passed through a sigmoidal function to produce an average firing rate for each neuron that ranged between 0 and 60 spikes/s. Lastly, we used this average firing rate as the intensity parameter of a Poisson process to simulate spike trains for 10,000 neurons as the paw was held for 2 s in each of the 100 positions of the 10×10 workspace.

Fig. 2.

Block diagram for generating neural activity. Muscle lengths were first normalized to values between 0 and 1, then linearly combined with random weights drawn from a standard normal distribution. Output was then passed through a sigmoid function and a Poisson process to simulate neural activity.

Given the muscle model used to generate the activity of these neurons, the comparison across conditions reveals how consistent paw position tuning might be if DSCT neural responses were driven directly by muscle sensor outputs, where the only neural computation is the summation of randomly scaled inputs. While it is unlikely that real neurons in the DSCT have fully random connectivity from muscle lengths, the use of random weights allowed us to probe the limits of the potential role of musculoskeletal geometry, as opposed to more complex learned weights, on the neural representation of limb state in DSCT.

Analysis of neural representation.

Unless noted otherwise, we analyzed the simulated firing rates using the methods from the original experimental paper (Bosco et al. 2000). To find tuning in Cartesian coordinates, we fit the simulated firing rates to the x and y coordinates of paw position. These planar fits yielded a gradient direction (θ_G), which denotes the direction of maximal change in firing rate, and a gradient magnitude (ρ_G), which denotes the overall sensitivity of the firing rate to displacement from the center, shown in Eq. 1:

$$f={\alpha }_0+{\rho }_{\text{G}}\cdot d\cdot \text{cos}\left(\theta -{\theta }_{\text{G}}\right)$$ (1)

where

$$d=\sqrt{x^2+y^2},\hspace{0.17em}\hspace{0.17em}\theta ={\tan }^{-1}\frac{y}{x}$$ (2)

In Eq. 1, d represents the Euclidean distance from the center of the grid, while θ represents the direction of displacement from the center, defined in Eq. 2. If a neuron changes its Cartesian representation between experimental conditions, one or both of these parameters would change in some way. A change in θ_G might be considered more interesting than a change in ρ_G, as altered sensitivity could be caused simply by global effects on the neurons in the DSCT that result in an overall increase or decrease in activity. On the other hand, a change in θ_G represents a differential change in sensitivity to the two cardinal axes of movement. Such a change resulting from different biomechanical conditions (as in the two conditions in this study) would contradict the hypothesis that the neuron encodes paw position. For this reason, our analysis focuses on changes in θ_G between the two experimental conditions.

We only considered neurons that were well tuned in both the elastic and knee-fixed conditions, as the θ_G is otherwise undefined. As in the original experiment, we considered a neuron to be tuned if it satisfied two conditions: first, using an F-test, the fit of Eq. 1 needed to be significant (P < 0.05) compared with a constant firing rate model; second, the R² of the model fit by Eq. 1 needed to be greater than 0.4 (Bosco et al. 2000).

RESULTS

Tuning of intrinsic neurons to paw position.

Within either joint covariance condition, the relationship between the intrinsic and extrinsic coordinate systems is locally linear (Bosco et al. 1996; Mussa-Ivaldi 1988). In motor systems, it has been common to impose a postural perturbation as a means of disambiguating these two possible representations (Caminiti et al. 1990; Morrow et al. 2007; Oby et al. 2013; Scott and Kalaska 1997). While less common in the somatosensory system, this approach has been used to suggest that at just one synapse removed from the afferent receptors at the level of the dorsal spinocerebellar tract, neurons encode limb state in extrinsic coordinates (Bosco and Poppele 1996; Bosco et al. 2000). We sought to better understand the origin of this remarkable result. To this end, we conducted a simulation to determine the conditions under which muscle length-sensitive neurons might exhibit apparent tuning to paw position in extrinsic coordinates.

Figure 3A shows the activity of a well-tuned neuron as a function of paw position in the elastic condition. When fit with Eq. 1, R² was 0.64 (P < 0.001). θ_G for this neuron (indicated by the arrow) was 80°. Sixty-one percent of all simulated neurons were well tuned to paw position (R² > 0.4, P < 0.001) in the elastic condition.

Fig. 3.

Heat map diagram showing activity of an example simulated neuron at different foot positions during elastic joint (A) and knee-fixed (B) conditions. Red indicates high activity, while blue indicates low activity. Red arrows show the “gradient direction” (θ_G) of this neuron in both conditions. For this neuron, θ_G changed by 15°.

Among this group of well-tuned neurons, the distribution of θ_G was strongly bimodal. Figure 4A shows this distribution in the elastic constraint condition, with a significant mean axis running from 82 to 262° (circular mean test P < 0.0001), which approximately matched the axis running between the hip and paw (which we call the “limb axis”), indicating that neurons were much more likely to be tuned to whole leg extension and withdrawal, than the orthogonal direction of movement (the “orientation axis”). This distribution was remarkably similar to that found in the studies of Bosco and colleagues (Fig. 4B). Because the tuning of our simulated neurons resulted directly from that of the muscles, this nonuniformity in the distribution of θ_G suggests that the hindlimb muscles must also have their greatest length change for paw movement along this axis. Intuitively this is quite reasonable; while the workspace spans a greater range along the orientation axis, only the hip muscles are strongly affected by orientation changes. In contrast, paw movement along the limb axis affects all of the muscles, leading to greater overall muscle length change and a greater sensitivity of the neurons to these movements.

Fig. 4.

Distribution of gradient directions for tuned neurons in simulation (A) and in empirical recordings of previous studies (B). In both cases, the distribution was mostly bimodal along the axis of the leg (the “limb axis”). B has been redrawn from Bosco and Poppele (2001). Dashed lines represent the primary axes of the distribution’s lobes.

Tuned and untuned neurons received essentially the same magnitude weight of inputs from all muscles (medians 2.75 and 2.65, respectively). While we did not find any clear biases in tuned neurons for higher weights on certain muscles than others, we do note that some combinations of inputs (for example, equal inputs from antagonistic muscles with opposing pulling directions) contribute no net neural modulation, while others are much more effective. These combinations of inputs were quite different for tuned and untuned neurons. We computed a net input vector for each neuron, calculated as the vector sum of all muscle pulling directions weighted by each muscle’s input strength for that neuron. The length of this vector, normalized by the scalar sum of the input weights, provides a measure of the “effectiveness” of the inputs. The input to well-tuned neurons was much more effective than that for untuned neurons (a 36% difference for the elastic condition and 48% for knee-fixed).

Stability of cross-condition gradient directions.

Among our simulated neurons, 61% had statistically significant tuning to paw position in the elastic condition, and 69% were tuned in the knee-fixed condition. Fifty-two percent were tuned in both conditions. For these neurons to be considered extrinsically tuned, θ_G must be invariant across these two conditions, for which the mappings between paw position and joint angles differ. This is indeed what Bosco and colleagues found: θ_G for many DSCT neurons remained largely unchanged across the elastic and knee-fixed conditions (Bosco et al. 2000). We made a similar comparison here, examining θ_G among the 52% of simulated neurons that were well tuned to paw position under both conditions.

The θ_G of the example neuron of Fig. 3 rotated from 80° in the elastic condition to 95° in the knee-fixed condition. (Fig. 3B). Unexpectedly, among neurons tuned to both conditions, the majority had similarly small changes in θ_G between conditions. Figure 5A shows a histogram of the cosines of these Δθ_G, a typical measure of the alignment of two vectors. Values close to 1 represent neurons that had very small absolute Δθ_G, while numbers close to 0 represent neurons with nearly orthogonal θ_G. In our data, the median of the cosine of Δθ_G was 0.94, corresponding to |Δθ_G|  = 20°. For comparison, Fig. 5B shows the corresponding original experimental result: a median change in the cosine of Δθ_G of 0.91 (23°) for all tuned neurons (Bosco et al. 2000).

Fig. 5.

Histogram of cosine of change in θ_G between experimental conditions in simulation (A) and in previous empirical recordings (B). Both distributions were heavily biased toward 1, indicating that most neurons did not change the gradient direction substantially. For simulated neurons (A), the median value was 0.94, corresponding to a change of ±20°, and for recorded neurons (B), the median value was 0.91, corresponding to a change of ±23°. Additional second x-axis label shows the corresponding absolute change in θ_G in degrees. B has been redrawn from Bosco and Poppele (2000).

Combined with the observation of a bimodal distribution of θ_G along the limb axis, the stability of θ_G between experimental conditions constitutes a surprisingly close match between actual and simulated DSCT neurons. These results beg the question of what mechanism generates the consistent endpoint tuning across the joint coupling conditions. Because the neural activity in our simulation was generated from randomly weighted combinations of muscle lengths, it weakens the argument that the apparent endpoint tuning arises from a specific weighting of inputs.

One potentially simple explanation for this stability of extrinsic tuning is the effect of multiarticular muscles. Because they span multiple joints, their length changes might be more directly related to paw position than individual joint angles. However, when we removed the biarticular muscles from our musculoskeletal model, the stability of endpoint representation largely persisted: 46% of neurons were well tuned to paw position in both constraint conditions, and the median |Δθ_G| was 15°. Thus, while there were fewer well-tuned neurons without the biarticular muscles, the tuning direction change was actually smaller. Evidently, there is a different explanation for the stable extrinsic tuning that is related to how muscle lengths are affected by paw position.

Effect of muscle tuning across constraint conditions.

To investigate the role of musculoskeletal geometry in the bimodal tuning distribution and stability of θ_G across constraint conditions, we examined muscle length tuning in both conditions. Figure 6A shows the tuning to paw position of all muscles in the elastic condition. The length of each vector denotes the sensitivity (ρ_G) of the muscle length to paw movement, and its direction denotes the θ_G. As shown graphically by the ellipse, the principal axis of these vectors, calculated using PCA, was between 92 and 272°, only 10° away from the axis of Fig. 4A. This principal axis, accounting for 80% of the total variance of these tuning vectors, forms the major axis of the ellipse. Likewise, the minor axis corresponds to the magnitude of the second principal component. Thus, in the elastic condition, the hindlimb muscles changed length mostly for paw movements along the limb axis.

Fig. 6.

Muscle tuning vectors in elastic joint (A) and knee-fixed (B) conditions. Ellipses in both panels illustrate the variance of the muscle tuning vectors along the principal axes. Tuning vectors were strongly bimodal along the limb axis, in essentially the manner of the neurons. In the knee-fixed condition, VL, a monoarticular knee muscle, became unmodulated, while the ankle muscles (TA, Sol, MG) changed modulation and hip muscles (IP, BFA) changed tuning direction to compensate. Despite this, the overall distribution of muscle tuning did not change, leading to only small changes in neural tuning direction.

Figure 6B shows the corresponding muscle tuning for the knee-fixed condition, which altered the tuning vectors of individual muscles. Vastus lateralis, a monoarticular knee extensor, could not change length and thus lost all relation to paw movement. In compensation for the lack of knee movement, the ankle needed to move through a much greater range in order for the paw to reach all the positions in the workspace. Consequently, the tuning vectors of soleus (Sol) and tibialis anterior (TA), monoarticular ankle extensors and flexors, respectively, increased considerably in length, as did that of the medial gastrocnemius (MG), a knee-ankle biarticular muscle. Additionally, the monoarticular hip muscles, iliopsoas and biceps femoris anterior changed in tuning direction to reflect the change in hip movement under the knee-fixed constraint. All three ankle-related muscles (MG, TA, Sol) took on essentially identical tuning direction (disregarding sign). Likewise, all hip-related muscles, including the hip-knee biarticular muscle rectus femoris, had nearly the same tuning directions. This grouping of muscle tuning directions in the knee-fixed condition reflects the fact that all muscles were essentially reduced to monoarticular flexors or extensors; the two groups of tuning vectors represent the influence of the hip and the ankle rotation on paw position.

Despite these changes, the overall biased distribution of muscle tuning vectors remained largely unchanged across constraint conditions. The orientation of the covariance ellipse with the knee constraint (Fig. 6B) was very similar to the orientation under the elastic condition (Fig. 6A); the two principal axes differed by only 5°. The similar distribution of preferred neural tuning vectors in the two constraint conditions (Fig. 4) and the small change in tuning directions for individual neurons (Fig. 5) are both likely a direct consequence of the preserved biased distribution of muscle tuning vectors. Neurons with significant tuning in both conditions were very likely to have had a tuning vector along the principal axes shown in Fig. 6; because the direction of this principal axis didn’t change substantially across the two conditions, the preferred directions of most neurons didn’t change either.

Sensitivity to simulation parameters.

Muscle tuning vectors reflect the changes in muscle length resulting from displacements of the paw. The relationship between paw displacement and joint rotation is given by the limb Jacobian: dx = Jdq. Given a set of joint angle changes we can find the resulting paw displacement. However, because the limb is redundant, the opposite relationship is ill posed and only becomes well defined by the addition of constraints, such as the joint stiffnesses included in the elastic condition, reflecting contributions from passive musculoskeletal properties. The further effect of the knee constraint therefore depended on the particular joint stiffnesses used in our simulation. As a trivial example, if the knee joint stiffness were much higher than the stiffness of the other joints, then fixing the knee would have minimal effect on neural tuning vectors. We examined the extent to which our results depended on the particular values of stiffness by repeating the analyses after varying the stiffness of each joint by ± 25%. The number of neurons tuned to both conditions varied only slightly, between 52 and 56%, and median |Δθ_G| for the tuned neurons varied only between 20 and 25°.

For a neuron to change its tuning across constraint conditions, it had to have statistically significant modulation of its tuning across paw positions. These statistical tests are dependent on the noise inherent in the neural discharge and the confidence with which we could measure tuning. We therefore examined whether the main results depended on the amount of uncertainty in neural tuning. To change the measurement uncertainty, we varied the measurement time from 2 s, the length of time used to compute firing rates in most of the original analyses, to a range between 0.1 and 5 s. Longer measurement times meant less noise. As we increased the measurement time, the number of tuned neurons increased tremendously from 7.7% at 0.1 s to 69% at 5 s. However, despite this wide range in number of tuned neurons, the median |Δθ_G| remained largely unchanged, ranging from 16 to 21°. For simulations in which more than 20% of neurons were tuned (measurement times >0.5 s), the median |Δθ_G| ranged between 19 and 21°. This indicates that Poisson noise, apart from determining how many neurons were well tuned to paw position, did not change the stability of their tuning or the major conclusions of the study.

DISCUSSION

Summary.

Because of the nearly linear relation between joint angles and paw position in any restricted workspace, determining the coordinates in which proprioceptive neurons are tuned requires an experiment that alters this relation. Bosco and colleagues (Bosco et al. 2000) altered this relation by fixing the knee and thereby altering the joint covariance. They found relatively little change in neural tuning with respect to paw position across the naturally constrained limb and the knee-fixed limb and reasoned that neurons with this property should be considered extrinsically tuned.

Unexpectedly, we found that their analyses applied to our simulated neurons, which received only random combinations of muscle length inputs, resulted in tuning characteristics that closely matched those of the actual DSCT neurons. We focused on two specific analyses used in the original experiments. The first determined the distribution of neural tuning directions, which was heavily biased to be along the limb axis direction. The second determined the overall change in this tuning direction between two different joint covariance conditions, which was generally small for most neurons. The results of both analyses closely matched the earlier empirical results.

In addition to these analyses, Bosco and colleagues (Bosco et al. 2000) performed a set of regression analyses, which they used to classify neurons into groups that responded consistently to paw positions, joint angles, or neither. We chose not to replicate these analyses, which would have been sensitive to a number of parameters (e.g., measurement errors using skin markers, noise characteristics of DSCT firing rates, etc.) that we had no means to estimate accurately. Since we did not want simply to tweak the simulation parameters to achieve a match with their results, we sought to assess the overall tuning characteristics of neurons without actually classifying them. As a result, while we could not compare the actual classifications of neurons, we were able to show that the overall tuning patterns in the original study were consistent with DSCT neural activity being driven by muscle length.

Our results serve a dual purpose. First, they provide necessary context for the previous experiments in the cat DSCT. The substantial signal transformation implied by those results was quite surprising, given the close proximity of DSCT to the periphery. Apparently, despite its seeming severity, the knee-fixed constraint did not cause a sufficient change in musculoskeletal mechanics to determine whether DSCT neurons were extrinsically tuned. Consequently, a simple linear convergence of muscle spindlelike inputs, given the musculoskeletal geometry, provides an adequate explanation for the apparent tuning to paw position.

Second, these results show the importance of simulation to inform the design of similar experiments, providing the basis for an appropriate null hypothesis. In this case, the gradient direction analysis shows very little change between conditions; thus, in order for DSCT neurons to be considered extrinsic, they must demonstrate even less change than this null hypothesis. An appropriate use of this simulation for future experiments could be to choose a more effective constraint that maximized the change in endpoint tuning given muscle-based neurons.

Contributions of musculoskeletal properties to apparent neural coding.

A key question of our study was to determine why the neural tuning was stable across two seemingly very different joint constraints. We considered whether the stable endpoint tuning of length-sensitive DSCT neurons might have arisen because of the geometry of biarticular muscles, the length of which is not a simple function of individual joint angles. However, when we removed the biarticular muscles from our model, the results remained qualitatively unchanged.

Instead, the invariance appears to be due to the strongly bimodal distribution of the muscle tuning vectors along the limb axis for both constraint conditions (Fig. 6). In the normal, elastic condition, this bimodality is understandable biomechanically, as all three joints rotate during whole-leg flexion and extension, but for anterior-posterior paw movement, muscles other than those at the hip changed very little. This strong bimodality was not substantially affected by the knee constraint. Several muscle tuning vectors changed, as hip and ankle rotation compensated for the lost knee rotation. However, the greatest effect on muscle tuning vectors was on their length, most obviously for the ankle muscles TA, MG, and Sol (gray, pink, and brown vectors, respectively), which all lengthened dramatically. The overall bimodality along the limb axis was preserved.

As context for the empirical results, this simulation makes the discovery of constraint-invariant neurons much less surprising. The musculoskeletal geometry of the hindlimb can explain both the nonuniform distribution of tuning to paw positions and the stability of tuning between the knee-fixed condition and a natural movement condition. As such, the most straightforward explanation for the behavior of DSCT neurons is that it is the result of a convergence of muscle inputs.

It is important to note once again that our analysis, as that of the original study, considered only neurons that were significantly tuned to paw position. Only 61% of the simulated neurons were tuned in the elastic condition, a number that differed greatly from the 95% in the original study. This large difference is at least in part a product of the random connectivity used in our simulations. In the simulation, a neuron could receive equal inputs from antagonistic muscles, essentially nulling the effect of paw movement on the neuron’s firing rate. Indeed, the input to tuned neurons was 35–50% more effective at modulating neural activity as was input to untuned neurons, despite there being no difference in the magnitude of the weights themselves. A significant amount of such weight cancellation is unlikely to occur in the actual DSCT; such ineffective combinations of inputs are likely to be pruned during development, leaving a bias toward tuned neurons.

One caveat to this study, as well as in the original experimental study, is that the relationship between limb configuration and neural activity during normal behaviors can also be altered by fusimotor drive. In the anesthetized conditions of the original experiments, this drive would be significantly reduced (Prochazka et al. 1977), and our simulations matched these experimental conditions by deriving neural activity directly from muscle length. Nonetheless, an important question is how variations in this drive might have changed the apparent representation in DSCT. Normally, both dynamic and static gamma drive appear to be modulated during the step cycle. Although the time course of this modulation remains uncertain, gamma static drive (most relevant for the postures represented in this study) increases abruptly at movement onset, with a sustained, dramatic increase during muscle shortening (Taylor et al. 2000). A further complication is that the particular pattern of gamma activation appears to depend on muscle type, be it flexor, extensor, or biarticular (Loeb 1981). More study is required to understand how this changing fusimotor drive would affect the neural representation in DSCT.

Simulation for experimental design.

In addition to shedding light on the previous empirical results, this simulation can help in designing experiments to test the coordinate frame of DSCT or other neurons. Simulation can be used for this in three separate ways: choosing the best analyses for assessing tuning stability, generating a null hypothesis to test results against, and optimizing the constraint to cause maximum tuning change.

With respect to analytical approaches, the original experiment (and our simulation) examined changes only in the direction, θ_G, of the neural response gradient, not its magnitude, ρ_G. As we suggested above, it is not unreasonable to discount changes in ρ_G as they represent simply an overall change in sensitivity to movement, rather than a differential change in how neurons respond to movement in different directions. However, because the changes imposed by the knee-fixed constraint were focused mostly along the limb axis, neurons with gradient vectors in that direction would be predicted to change more in length than direction. In our simulation, we found a median change in ρ_G of 38% from the elastic to the knee-fixed condition. Hence, for this particular constraint, considering changes in ρ_G as well as θ_G might have clarified the interpretation of whether or not DSCT neurons are extrinsically tuned.

Second, such a simulation would be useful in generating a null hypothesis based on intrinsic tuning to which empirical results could be compared. For example, in addition to an analysis of Cartesian tuning direction, the original experiments included an analysis in polar coordinates, in which each neuron’s firing rate was fit to the length and orientation of a vector that pointed from the hip to the paw. Polar tuning of approximately half of the DSCT neurons did not change significantly across conditions, which led the authors to conclude that they were extrinsically tuned. However, the t-test used for that analysis should be used only to show that two distributions are significantly different; in this case, that a particular neuron is not extrinsically tuned. The fact that a neuron fails the t-test is not sufficient to conclude that it has the same tuning in both conditions. A more appropriate test would be an equivalence test, which, when passed, shows that two quantities are within some given bound of each other. This bound represents a difference that is scientifically (as opposed to statistically) significant. In the θ_G analysis, the equivalence test bound on the median Δθ_G might be 20°, as that value results from randomly connected DSCT neurons. If an equivalence test shows a median change significantly less than this bound, it provides good evidence that the neurons are extrinsically tuned.

Finally, such a simulation could be used for basic experimental design. Our simulation showed that the knee-fixed constraint did not significantly change how the muscles respond to paw movement compared with the natural elastic constraint condition. Invariance in neural tuning despite a constraint that caused significant changes in muscle tuning along the orientation axis would be much stronger evidence of extrinsic tuning in the DSCT.

Broader implications of these modeling results.

An important question is the extent to which our conclusions apply more broadly than to the particular sagittal plane motions and knee-fixed constraints of this experiment in the somatosensory system. The classic studies of Georgopoulos revealed hand-centered tuning of M1 neurons during reaching (Georgopoulos et al. 1982) that was analogous to the apparent paw-centered tuning of DSCT. However, numerous subsequent studies found clear evidence that the tuning of most M1 neurons is not consistent with a simple endpoint model across different workspaces (Caminiti et al. 1990; Morrow et al. 2007) or limb postures (Cherian et al. 2011; Scott and Kalaska 1997). A recent simulation study demonstrated the importance of a variety of musculoskeletal properties on the distribution of M1 preferred directions (Lillicrap and Scott 2013). Interestingly, they found that limb geometry was more important than the presence or absence of biarticular muscles, an observation that is consistent with ours.

The brain-machine interface (BMI) field currently faces similar difficulties. Remarkably accurate predictions of hand trajectory can be made using simple linear combinations of M1 firing rates (Carmena et al. 2003; Serruya et al. 2002; Taylor et al. 2002; Wessberg et al. 2000). It is likely that BMIs based on such extrinsic coding models will perform poorly when used to predict movements following postural perturbations that dissociate extrinsic from intrinsic variables, as suggested by previous studies (Cherian et al. 2011; Oby et al. 2013). Likewise, recordings from small ensembles of afferent neurons in the dorsal root ganglia in cats have been used to predict the kinematics of the paw in extrinsic coordinates during passive limb displacement under anesthesia (Stein et al. 2004) and walking on a treadmill (Weber et al. 2006), but the success of these methods may be due at least partially to the constrained biomechanics of the tested movements. The results of the present study suggest that properties of the musculoskeletal system being studied should be carefully considered when evaluating the ability of BMI interfaces to predict limb movements.

Conclusion.

We have shown that under the conditions of the original experiments (sagittal plane motion, unchanging gamma drive), limb biomechanics and a simple linear convergence of muscle length inputs can replicate the apparent extrinsic representation of limb state by DSCT neurons. Given that our muscle-based simulation replicated key features of neurons in the DSCT, there is no longer justification to conclude that the signals carried by these neurons have undergone a transformation to produce a high-level representation of limb state.

GRANTS

This research was supported by the National Science Foundation Grant No. DGE-1324585 and National Institute of Neurological Disorders and Stroke Grants NS048845, NS095251, and NS086973.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

R.H.C. performed experiments; R.H.C. analyzed data; R.H.C., M.C.T., and L.E.M. interpreted results of experiments; R.H.C. and L.E.M. prepared figures; R.H.C., M.C.T., and L.E.M. drafted manuscript; R.H.C., M.C.T., and L.E.M. edited and revised manuscript; R.H.C., M.C.T., and L.E.M. approved final version of manuscript; M.C.T. conceived and designed research.