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1 Neuroscience and Behavior Program, University of Massachusetts, Amherst, Massachusetts 01003; 2 Department of Computer Science, University of Massachusetts, Amherst, Massachusetts 01003
Submitted 8 September 2003; accepted in final form 24 January 2004
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGMENTSREFERENCES |
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; Schwartz et al. 1988). The activity of the MI neurons was described as varying with the cosine between the direction of hand movement in extrinsic space and the neuron's preferred direction (PD), also expressed in extrinsic space. A variety of subsequent studies suggest that MI neural activity may capture other aspects of movement. By varying the origin of hand position in a three-dimensional center-out task, Caminiti et al. (1990, 1991) described MI as encoding hand movement in a polar-coordinate system centered at the shoulder. Kinematic parameters such as velocity, acceleration, target direction, and target position may also be represented by the activity of an MI neuron—in some cases simultaneously (Ashe and Georgopoulos 1994; Moran and Schwartz 1999; Schwartz and Moran 2000) or sequentially (Fu et al. 1995). Isometric studies suggest that MI may also encode aspects of directional force (Georgopoulos et al. 1992; Sergio and Kalaska 1998).
While an extrinsic representation of movement in MI seems to follow from the results of many of the studies outlined in the preceding text, representations of movement in intrinsic space, such as muscle activity or joint angle deviation, cannot be excluded. If the primate varied its arm configuration while it performed the same hand trajectory, corresponding MI activity was described more accurately by models based on joint kinematics or joint torques than models based on hand-movement direction (Scott and Kalaska 1995, 1997). Other models argued that MI activity could be described by equations based on muscle-shortening velocity (Mussa-Ivaldi 1988) or joint angle deviations (Ajemian et al. 2000). Support for a muscle-based representation of movement in MI arose from studies that showed that MI neural activity (and PDs) changed with arm configuration similar to how muscle electromyographic (EMG) activity did in an isometric task (Sergio and Kalaska 1997) and passive arm movement (Scott 1997). Other studies show that the activity of red nucleus neurons, which may encode movement similarly to MI (Houk et al. 1993; Miller and Sinkjaer 1998), correlated better with muscle activity than other variables (Gibson et al. 1985; Houk et al. 1987; Miller and Houk 1995; Miller and Sinkjaer 1998; Miller et al. 1993). Because MI sends projections directly to the spinal cord, among other areas, the idea that MI represents movement in an intrinsic space is an attractive one.
While these studies suggest that MI best represents movement in a particular coordinate frame, many of them indicate that other coordinate frames may be represented as well (though to a lesser degree). The results of more recent studies (Kakei et al. 1999, 2003) make the possible simultaneous representation of multiple-coordinate frames in MI neural activity more apparent. These studies investigated the variation of primate wrist muscle EMG and MI neural activity during a 2 degrees of freedom wrist movement task and provided evidence showing the existence of two subpopulations of MI neurons: neurons with PDs that capture extrinsic properties of movement and neurons with PDs that encode muscle-activation patterns. The existence of both types of neurons may support the idea that a serial processing scheme (see Fig. 1, left) is implemented within MI [although Scott (2003) offers a different interpretation—see DISCUSSION]. Under this model, MI is actively involved in the transformation of an abstract, extrinsic representation of movement into an intrinsic space (Kakei et al. 1999, 2003).
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). Figure 1 illustrates a serial processing scheme (left) versus a series-parallel processing scheme (right, hereafter referred to as parallel) in which multiple populations of MI neurons that are characterized as encoding movement in different spaces can each directly command muscles. Can an extrinsic-like representation of movement participate in the direct activation of muscles? Modeling work by Kakei et al. (2003) and Salinas and Abbott (1995) suggests that this is possible. Both show that a linear transformation exists between an extrinsic representation of movement and an intrinsic one under specific assumptions of how movement-related variables are encoded. However, the intrinsic representations used deviate from how muscles are recruited.
We examine the feasibility of the parallel processing scheme through a neural network model inspired by the work of Kakei et al. (1999). We show that it is computationally possible for a population of neurons with PDs defined in extrinsic space to directly control muscles appropriately. The model produces muscle activation patterns similar to those recorded during the task used in Kakei et al. (1999; Hoffman and Strick 1999). We train the model with a performance-based optimization procedure and do not impose an a priori representation of intrinsic movement such as those used in Kakei et al. (2003) and Salinas and Abbott (1995). We also use the model to examine the use of correlation methods to make inferences of coding schemes and connectivity. Elements of this work have been presented previously in poster form (Shah et al. 2002).
MI and muscle involvement in the production of wrist movements
Hoffman and colleagues (Hoffman and Strick 1999; Kakei et al. 1999, 2003) described a 2 df step-tracking task in which a human or monkey subject moved a manipulandum with its wrist, fixed in a pronated, supinated, or midrange posture (Fig. 2), to move a cursor on a computer screen from a central point to one of several targets falling on a circle around the starting location. The kinematics of movement, EMG activity from several muscles, and single-neuron activity in MI were recorded. Muscle activity as a function of target direction exhibited a "truncated cosine-like" shape—for values of target direction for which a cosine-like function is negative, the truncated cosine is zero. A cosine of the form B cos(
– C) + D was fit to the muscle activation patterns, with a low weight given to values of muscle activity near zero to account for the truncation. The parameter C defined the muscle's PD for that wrist posture.
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). MI neurons were labeled extrinsic-like, muscle-like, or joint-like, depending on how their PDs rotated as wrist posture rotated 180°. Because categorization of neurons was based solely on how their PDs rotated, the suffix "-like" was explicitly included. Fifty percent of the MI neurons recorded were labeled extrinsic-like because their PDs changed by only a small amount. The depth of modulation of some neurons in this category varied with wrist posture. Thirty-two percent of the MI neurons were labeled muscle-like because their PDs shifted similarly to muscle PD shifts. None of the neurons were labeled "joint-like," and the rest were not easily classified. The presence of both extrinsic- and muscle-like neurons led Kakei et al. (1999) to suggest that a serial processing scheme was implemented in MI. This implies that only intrinsic neurons (e.g., muscle-like MI neurons) can be pyramidal tract neurons (PTNs). If PTNs consisted of only intrinsic neurons, one might expect a spatial and temporal distinction between the intrinsic and extrinsic neurons. However, Kakei et al. (1999) found no such differences. |
METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGMENTSREFERENCES |
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The results of Kakei et al. (1999) can be interpreted to support the serial processing scheme. However, is it computationally feasible for each population of MI neurons recruited for a task—including extrinsic-like neurons—to directly command muscles? One could approach this question through the implementation of a model in which both intrinsic- and extrinsic-like neurons command muscles. However, in doing so, one would have to make assumptions about the origin of the intrinsic responses. In addition, it would not be clear what the resulting contribution of the extrinsic population would be in the movement generation process. To avoid these difficulties, we instead choose to ask a stronger question: can extrinsic-like neurons alone produce the appropriate muscle responses? To answer this question, we have created a neural network model in which a population of extrinsic-like neurons directly activate muscles. The model is based on the task and experimental findings of Kakei et al. (1999). The model presented in this paper is an extension of the model presented in Fagg et al. (2002), which directly finds muscle activation patterns that satisfy optimization criteria; the resulting muscle activation patterns are similar to those reported in Hoffman and Strick (1999) for the wrist movement task.
Model architecture
The architecture of this network is shown in Fig. 3. In the following paragraphs, vectors and matrices are in bold type while scalars and the elements of the vectors and matrices are in italics. For example, the array of MI neurons is denoted by m, a column vector of which element mi refers to either m neuron i or the activation level of m neuron i. (For simplicity, the term "neuron" will be used to refer to a unit from the m array hereafter.) By design, the extrinsic-like neurons behave like those recorded in Kakei et al. (1999): their PDs are expressed in extrinsic space, but their activation levels are modulated by wrist posture. Some neurons are more active when the wrist is pronated than when it is supinated and vice versa. Each neuron sends projections (K) directly to each of five muscles (a), which correspond to the muscles from which Hoffman and Strick (1999) and Kakei et al. (1999) recorded.
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), each modeled muscle contributes to the endpoint of wrist movement through its activation level and pulling direction, which depends on wrist posture. Hoffman and Strick (1999) determined the pulling direction of a muscle by individually stimulating the muscle and observing immediate wrist movement. Average pulling directions (in extrinsic space) for five muscles and three postures are shown in Fig. 4 (D. S. Hoffman, personal communication). The activation of the muscles determines extrinsic movement of the wrist (x), analogous to cursor movement in the task.
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i N). To create the extrinsic-like behavior found in Kakei et al. (1999), we defined the activity of an m neuron to take on a Gaussian-like shape around its PD and included a wrist posture term (wi) to modulate its depth
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is the target direction, and 
is wrist posture. For i = 1,..., N, wi() = 0, 1/4, or 1/2 for = pronated, midrange, or supinated, respectively, while wi+N () = 1/2, 1/4, or 0. Thus m1 through mN are more active in the pronated wrist posture while mN+1 through m2N are more active in the supinated wrist posture. 
defines the width of the Gaussian; this model uses = 74.5°, which produces a width at half-maximum of 120°. While a narrow tuning function has some advantages in coding one-dimensional features (Amirikian and Georgopoulos 2000; Zhang and Sejnowski 1999), we use a relatively broad tuning function in the interest of generalization. The function [·]+ returns 0 if its argument is <0 —this sets the minimum possible activation level of any mi to 0.
The hard-wired pattern of activity of the neurons behave similarly to the extrinsic-like neurons modulated by wrist posture recorded in MI by Kakei et al. (1999). Figure 5 shows how the activity level of one neuron varies with extrinsic target position. Although the activity of this neuron is modulated by wrist posture, its PD does not change with wrist posture. Figure 6 shows the activity of the array of neurons in m when the target is at = 180°; the activation pattern is different for the three different wrist postures. Both extrinsic target direction and wrist posture are represented in the activation of the neuron array. While the modulation by wrist posture captures some intrinsic information, a muscle coordinate frame is not represented in the activity of the m array.
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). Negative connections can be accounted for with inhibitory interneurons; however, because the abstraction level of this model is high, we choose instead to allow K to contain both positive and negative elements. Muscle activity is a linear function of m
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Each muscle is represented as a unit vector in the direction of its pulling direction, indicated in Fig. 3 as the matrix P, which changes with wrist posture, but is not plastic. We assume that muscles contribute to the endpoint of movement along their vector of action, where the length of the vector is proportional to the muscle's activation level, ai (analogous to the muscle EMG level). The muscles are also assumed to contribute to the movement endpoint independently of one another. Thus the endpoint of movement, x, is determined by the weighted sum of the muscle pulling directions as follows
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i denotes the pulling direction of muscle i when in posture and A is the set of all muscles. Note that only the endpoint of wrist movement, not the path of movement, is simulated.
Selecting the MI 
muscle connections
We wish to show that extrinsic-like neurons can directly generate realistic muscle activation patterns. We do this in the context of the model by showing that a K exists such that for each target position and wrist posture, the wrist reaches the target and muscles only pull (and never push). In Fagg et al. (2002), we discuss the redundancy in muscle activation patterns and show that the introduction of a minimum effort criterion applied to the muscles results in a cosine-like recruitment of muscles similar to that observed experimentally (Hoffman and Strick 1999). Thus we include a third criterion: a minimal degree of effort is used in making the movement.
We employ a gradient-descent method to identify an appropriate K matrix. This is accomplished as follows: the initial elements of K are randomly chosen from a uniform distribution between –1/2 and 1/2. For a given target and wrist posture, a random K yields a random muscle activation pattern produced by m and hence a random (and inaccurate) endpoint of wrist movement. As in Fagg et al. (2002), we define the error as follows
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is a regularization parameter set to 0.02. represents a trade-off between target error and muscle activation. Because all movements are of unit magnitude, and x is a linear summation of muscle activity, ||a||2 is on the order of 1 for most movements. Thus = 0.02 represents allowable errors on the order of 2% of movement magnitude. The first term in E(xtarg,a,) represents movement accuracy (1st criterion); the second term represents total muscle activation (effort, 3rd criterion). The error gradient with respect to aj is
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To constrain muscle activation to nonnegative levels (2nd criterion), ej is set to minimize E(xtarg,a,) only if aj 
0; otherwise, ej is set to bring aj toward 0.
The connection matrix K is modified as follows
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is a learning rate set to 0.02 in this model and e is a column vector representing the error terms for all five muscles. Note that this is not intended as a biologically based learning approach. A K is chosen such that all ajs are nonnegative (2nd criterion); within this constraint, a K is selected such that it minimizes E(xtarg,a,), which includes the first and third criteria. The gradient descent method was applied for each of the three wrist postures and 12 targets equally spaced along a unit circle surrounding the central starting position, yielding a total of 36 distinct tasks; the 36 tasks constitute one training epoch. Note that each task, by construction, is represented by a unique m activation pattern (Fig. 6). K was updated iteratively on each of the 36 tasks until the mean target error over all 36 tasks was <0.05.
Thirty independent training runs were executed, each with a unique initial K. Over the 30 runs, the iterative procedure described above converged to a solution within an average of 210,150 ± 7,137 (SD) training epochs. For each run, the SD in target error (over the 36 tasks) was computed. The mean SD in target error over the 30 runs was 0.041 ± 4.13 x 10–4. The mean maximum change in any single muscle activation on the last epoch over all 36 tasks was 0.028 ± 3.19 x 10–4, while the mean maximum change in any single element of K was 0.0014 ± 1.95 x 10–5. Thus termination of a run was accompanied by convergence of the K matrix. For an individual task, the muscle activation pattern (a, the 5-element vector describing the activation levels of the 5 muscles) at convergence deviated from the average over all 30 runs by 0.011 ± 7.4 x 10–3, regardless of initial conditions (the initial K matrix). The K matrix describing the weights of the five projections from each m neuron, on the other hand, deviated from the average over all 30 runs by 0.52 ± 0.15. However, there was a general pattern across the 30 runs (see Fig. 10, B and C). Although there is no unique solution for K, the different Ks do yield the same muscle activation patterns for each of the 36 tasks. The existence of unique muscle activation patterns for each of the 36 distinct tasks is discussed in detail in Fagg et al. (2002). The length of the muscle activation vector, averaged over all 36 tasks, was ||a|| = 1.20 ± 0.38.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGMENTSREFERENCES |
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The following analysis uses the results from 1 of the 30 runs. All other runs have very similar results. Figure 7 shows normalized model muscle activation versus target direction as polar plots for muscles ECRL, ECRB, FCR, and ECU in the midrange wrist posture. Hoffman and Strick (1999) analyzed these four muscles from a monkey subject; their data (Hoffman and Strick 1999; D. S. Hoffman, personal communication) are included in Fig. 7 as dashed lines for comparison. The function B cos( – C) + D, where the PD of a muscle is defined to be the parameter C, was fitted to each muscle activation pattern. Because muscle activation patterns followed a "truncated cosine" (due to the constraint that all ai 0), low muscle-activation levels (<0.05 in our analyses) were given a zero weight in the fitting process. The muscle-activation patterns as produced by this model are similar to those reported in Fagg et al. (2002) that used a gradient descent method to directly find muscle activation patterns that satisfy the same three criteria as this model. The model presented in this paper, in contrast, finds a connection matrix K to transform the activity of the m array into the appropriate muscle activation pattern.
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). This difference is the result of the uneven distribution of pulling directions of the muscles (Fagg et al. 2002). If there is a large gap between the pulling directions of two muscles, then the muscles have to devote additional effort to pulling against each other to reach a target located within the gap. Thus the PD of a muscle will tend toward this gap. The similarity between the muscle activation patterns as produced by this model and those recorded experimentally demonstrate that it is possible for an extrinsic-like representation to be translated (via a linear transformation) into an intrinsic representation of movement (muscle activation patterns). Correlation analysis
Inspired by studies that used a correlation analysis to suggest that neurons in the red nucleus and MI directly encode muscle activity (Gibson et al. 1985; Houk et al. 1987, 1993; Miller and Houk 1995; Miller and Sinkjaer 1998; Miller et al. 1993), we have calculated the correlation coefficient for each neuron-muscle pair over all 36 tasks in our model. The formula used to calculate the correlation coefficient (corrij) between m neuron i (mi) and muscle j (aj) is
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is the index of task in 
(the set of 36 tasks), mi() is the activation of mi for task , aj() is the activation of aj for task , µmi is the average activation of mi over all 36 tasks, and µaj is the average activation of aj. The correlations between each neuron and each muscle ranged between –0.68 and 0.88. While a high positive correlation might be interpreted as the neuron coding in a muscle coordinate frame, in this model it merely indicates that the neuron and the muscle happen to have similar PDs—they are highly active over the same range of tasks. Figure 8 plots, as a function of target direction and for each of the three wrist postures, the activities of the neuron and muscle that have the highest corrij over all neuron-muscle pairs and the activities of the neuron and muscle that have the lowest corrij. The highest correlation (corrij = 0.88) was found between neuron number 49 (of 2N = 96) and muscle ECRL. m49 hasaPDof7.5°, while ECRL has a PD of 342.7° in the pronated wrist posture, 2.2° in the midrange posture, and 17.6° in the supinated posture. Because the neurons and muscles are active over a wide range of target directions, the PDs of m49 and ECRL are considered to be similar (Fig. 8A). The lowest correlation (corrij =–0.68) was found between m39 and muscle ECU. Neuron m39 hasaPD of 292.5°, whereas ECU has a PD of 89.3° in the pronated wrist posture, 119.7° in the midrange posture, and 155.4° in the supinated posture. m39 and ECU were active over opposite target directions (Fig. 8B).
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i N) have the same PD. The scale bars in Fig. 9, top left, illustrate the relationship between the PDs of the neurons and the index of the neurons (also see METHODS section for the definition of PDi). Also shown for each muscle are their PDs in each of the three wrist postures. The corrijs between each muscle and the neurons vary sinusoidally with the PDs of the neurons. For each muscle, there is some neuron with which it has a high corrij (ranging from 0.77 to 0.88, depending on the muscle). In each case, the PD of that neuron falls within the range of the PDs of the muscle as the wrist rotates from pronation to supination.
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i N, then that muscle's PD in pronation is closer to PDi than its PD in supination (for example, muscle FCU in Fig. 9). Similarly, if the muscle has the highest corrij with a neuron of index N + 1 i 2N, then that muscle's PD in supination is closer to PDi than its PD in pronation (for example, muscle ECRB in Fig. 9). This is the case because the higher activation level of the neuron in one wrist posture over the others contributes more strongly to the correlation measure. Therefore the corrij between a muscle and a neuron is determined by their PDs and how active they are in each of the three wrist postures. Connection strength and correlation
High correlations between the activity of a neuron and the activity of a muscle may be used as evidence that there is a positive connection between that neuron and muscle. However, this is not the case in this model. Figure 10A plots the weight, Kij, of the connection between mi and aj as a function of corrij. The highest correlation associated with each neuron (across the muscles) is highlighted as a square, whereas the lowest correlation associated with each neuron is highlighted as a circle. The corresponding connection weights for both highest and lowest corrijs are normally distributed: the mean Kij (± SD) for the highest corrijs is 0.096 ± 1.36, whereas the mean Kij for the lowest corrijs is –0.049 ± 1.04. Thus corrij is not related to Kij—for some neurons, the connection weight between it and the muscle with which it has a high corrij is highly negative. The variability in the distribution of lowest corrijs (circles) is less than that of the highest corrijs (squares). This is due to the requirement that the linear transform specified by K must produce nonnegative muscle activation levels.
Figure 10, B and C, shows the weight vector (the weights of the connections emanating from a single neuron to each of the 5 muscles) for neurons 49 and 39, respectively, for all 30 runs (gray dots) and the mean of the 30 runs (black dots). Although there is considerable variability among the 30 runs, the weight vector exhibits a general pattern: although the set of all Kijs ranges from –3.2 to 3.7 for the single run focused on in these results, an individual Kij will only vary on the order of 1 across the 30 runs. Figure 10B shows that the Kij between m49 and ECRL (which have the highest corrij for the single run) is near 0, whereas the Kij between m39 and FCU (which have the lowest corrij) is moderately negative. In addition, the Kij between m49 and FCU is fairly high, but their corrij is –0.44 for the single run. The Kij between m39 and FCR is highly negative, but their corrij is 0.74. Thus even given the pattern across the 30 runs, Kij and corrij may be conflicting.
Figure 10 shows that corrij does not predict the connection strength between mi and aj (cf. Engel et al. 1991; König and Engel 1995; Munk et al. 1995). By construction, one neuron will exhibit correlated activity with other neurons. All that is required to produce the observed neuron-muscle correlation is that the net connection strength to the muscle be positive. This effect is possible because there are many more neurons than muscles or controlled degrees of freedom. Figures 9 and 10 also show that each neuron exhibits the full range of corrijs, from highly positive to highly negative, with some muscle.
Sensitivity to noise
To determine the robustness of the model in the presence of noise, signal-dependent noise (Harris and Wolpert 1998) was added to neuron activity. For each neuron, noise was selected from a normal distribution with a mean of zero and a SD of mi
. Simulations were run 30 times each with = 0.01 (low noise) and = 0.1 (high noise), which represent a range of biologically plausible magnitudes of noise (Todorov 2002; van Beers et al. 2003). For both cases, the model converged to a unique K (in contrast to the highly variable Ks found by the 0-noise model). Like the Ks trained without noise, the elements of the Ks trained with a low noise level had a wide range (–4.4 to 4.7). However, for high noise, the elements of K had a narrow range (–0.19 to 0.34). In addition, for low noise, corrij did not predict Kij, but for high noise, there was a weak predictive relationship. The mean Kij for the highest corrijs was 0.06 ± 0.09, whereas the mean Kij for the lowest corrijs was –0.02 ± 0.04. While this tendency was significant (t-test, P < 0.001), some Kijs between a neuron and muscle with a positive corrij were negative (and vice versa). We examine the implications of these results in DISCUSSION.
In neither case did the model reach the termination condition (average target error <0.05) within 1,000,000 training iterations. However, the resulting K produced an endpoint of movement near the target. Model strategy was tested by removing noise and examining movement endpoint and muscle activation. The average target error (over the 36 tasks) was 0.07 ± 0.06 for low noise and 0.12 ± 0.08 for high noise. For both levels of noise, a unique (across the 30 runs) muscle activation vector was found for each of the 36 tasks. To determine if neural noise resulted in a change in muscle recruitment, we computed ||ao() – a()||, where ao() is the muscle-activation vector for task derived from zero-noise training and a() is derived from noisy training. Averaged over the 36 tasks, the deviation in muscle recruitment was 0.15 ± 0.10 for low noise and 0.37 ± 0.18 for high noise. The deviation in movement endpoint, computed in the same manner, was 0.05 ± 0.03 for low noise and 0.09 ± 0.05 for high noise. Thus even though the introduction of noise may result in different muscle activation patterns and movement endpoints for some tasks, the strategy was similar.
We examined the model's ability to reach targets on which it was not explicitly trained. A model with K trained on the original 12 targets and no noise was presented with 144 random targets (taken from a uniform distribution between 0 and 360°). While movement endpoints were in the vicinity of the targets, accuracy was compromised—mean target error increased to 0.22 ± 0.14 over the 432 tasks (144 targets and 3 wrist postures). If the model was trained with random targets for each training epoch (for 1,000,000 epochs) and thus sampled a greater proportion of possible target directions, the mean error over 144 random targets was 0.11 ± 0.08. We also examined the model's sensitivity to the width of the target encoding. When the width of the m neuron tuning function was increased to = 114.5°, a model trained on the original 12 targets and tested with 144 random targets had a mean target error of 0.14 ± 0.10. Thus when trained over a greater variety of targets or with a wider neuron tuning function, the model's ability to generalize increased. Finally, we tested a model with random PDs for the m neurons (as opposed to evenly distributed PDs) for 30 runs. The only notable difference between the two models was that the model with randomly distributed PDs took an average of 343,330 ± 200,060 iterations to reach the termination condition.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGMENTSREFERENCES |
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; Caminiti et al. 1991; Georgopoulos et al. 1982, 1992; Kakei et al. 1999; Miller and Houk 1995; Mussa-Ivaldi 1988; Scott and Kalaska 1997). These studies rely on evidence of MI cell activity correlating with some aspect (or aspects) of an executed movement. Emerging from some of these studies, and from analogies with the robot control domain, is a serial processing scheme (cf. Dum and Strick 2002; Loeb et al. 1999; Scott 2000) in which an extrinsic representation of movement (such as a visual cue) is transformed through multiple stages and brain regions and culminates within MI, where movement may be encoded as an explicit representation of muscle-recruitment levels. However, the results of Kakei et al. (1999) argue against such a specialized role for MI. In particular, a significant subset of the cells observed in this study exhibited what Kakei et al. (1999) termed an extrinsic-like behavior, in which cell PD did not change substantially despite changes in the muscle recruitment pattern. It should be noted that a large subset of these cells did exhibit a change in the depth of modulation as a function of wrist configuration and hence encoded some intrinsic information. In addition, a subset of cells exhibited muscle-like behavior in which cell PD changed in a fashion similar to muscle PD. The existence of both extrinsic- and intrinsic-like populations of cells within MI led Kakei et al. (1999) to support the serial processing scheme and to suggest that a portion of the transformation might take place within MI itself.
The modeling results presented in this paper suggest that populations of MI neurons that encode movement in different coordinate spaces can each be directly involved in the recruitment of spinal motorneurons (we refer to this approach as a parallel processing scheme; Fig. 1). This perspective challenges the pure serial processing scheme in which intrinsically behaving (specifically, muscle-like) MI cells are the only cortical source of motorneuron input. In addition, experimental evidence suggests that premotor areas, which encode movement in a more abstract coordinate frame and likely take part in an earlier stage of the sensorimotor transformation (Kakei et al. 2001, 2003), have direct connections to spinal motorneurons (Dum and Strick 2002; Luppino et al. 1994). Through the use of a model of the Kakei et al. (1999) task, we ask about the computational feasibility of the parallel processing scheme. Rather than implementing a model in which both intrinsic- and extrinsic-like cells activate muscles, we chose instead to include only the extrinsic-like cells. Such an approach asks a more interesting question than the former (because intrinsic-like cells should be capable of driving muscles without the addition of extrinsic-like cells) and also asks a harder computational question. One benefit of making such a simplification is that it avoids the need for making specific assumptions about the origins of the muscle-like cell responses. Nevertheless, this approach is not meant to argue against the existence or utility of intrinsically behaving cells.
Our model shows that a linear transformation exists from a population of extrinsic-like cells to a muscle activation pattern capable of producing movements to a specified target for the task described in this paper. In addition to generating appropriate movements, the model also produces muscle activation patterns similar to the EMG patterns recorded by Hoffman and Strick (1999). Taken together, these results demonstrate that it is computationally feasible for extrinsic-like MI neurons to directly command muscle activation patterns, which in turn supports the plausibility of the parallel processing scheme.
Interpreting MI neural activity
Why do the results of different experimental studies lead to different interpretations of what is being encoded by MI? Todorov (2000) suggests that because muscle activation patterns (and hence forces) are affected not only by the descending motor command but also by the muscle's length, change in length, and acceleration of length, the form of the experimental question can lead to different interpretations about what is being coded by cells of MI. Thus in some tasks, MI activity appears to reflect extrinsic variables, whereas in other tasks, MI activity appears to reflect intrinsic variables. In his paper, the issue of multiple representations being simultaneously exhibited by MI is not addressed (although the issue is also not excluded). An explanation for simultaneous representations is given by Scott (2003). In his review of the role of MI in goal-directed movements, Scott suggests that the different populations of MI neurons, each seemingly representing different facets of movement, control different elements of the task. For example, to move its arm in a certain way, a primate must also control its posture, different types of spinal neurons, and a multitude of other variables often ignored in data analysis and modeling work. The different coordinate frames simultaneously represented in MI neural activity may be due to its control of the myriad variables necessary to accomplish a task. Although this may be the case, we show that it is computationally possible for units that behave in one coordinate frame to directly control units that behave in a different coordinate frame—neurons do not have to correlate their activity with that of the units they command. Therefore neurons that behave differently (e.g., extrinsic- and muscle-like neurons) can control the same variables (e.g., muscle activation) in parallel. In addition, Loeb et al. (1999) discusses how having different populations of neurons commanding the same variables may increase performance and robustness.
Related modeling work
Several other modeling studies used error-driven backpropagation to show that a sensorimotor transformation can be computed by a simple network. Anastasio and Robinson (1989, 1990) developed three-layer neural network models of the vestibulo-oculomotor reflex in which the transformation from the vestibular signals to the appropriate oculomotor responses were performed by a hidden layer. The authors found that for a network with an equal number of units in the input, hidden, and output layers, the hidden layer units behaved as tensor theory would predict: the units were maximally active for a specific direction of eye or head rotation that coincided with the direction of motor response of an output neuron. However, when the number of hidden layer units was greater than the number of output units, the PDs and activation magnitudes of the hidden layer units were distributed and did not coincide with the output units. The latter model produced hidden layer units that behaved much more like the vestibular nucleus neurons of the cat, which are thought to perform the sensorimotor transformation (Anastasio and Robinson 1990).
In another study, Xing and Andersen (2000) developed several three-layer network models of varying complexity; the task for all models was to produce the correct motor output in response to the sensory input. The simplest of these models required no coordinate transformations, whereas the most complicated of these models required both integration of multiple coordinate frames and coordinate transformations. In models where coordinate transformations were required, the coordinate spaces of the hidden layer neurons were distributed and did not correspond to any particular coordinate frame represented in the output array. In addition, the hidden units of the more complicated models activated output neurons of different coordinate frames simultaneously.
With a large number of neurons, neurons do not have to behave in a stringent manner to perform a coordinate transformation. A simple linear mapping may be sufficient for the necessary transformation (Sanger 1994). As shown by these studies (Anastasio and Robinson 1989, 1990; Xing and Andersen 2000) and our model, the characteristics of the activities of neurons (e.g., how activity varies with aspects of movement) does not necessarily lead to a causal relationship with the units the neurons command.
Two other studies use models to show that a linear transformation can accomplish a transformation similar to the one examined in our model. Kakei et al. (2003) present a simple model that shares some fundamental architectural features with our model. The model shows that the weighted sum of two cosines, each of the form B cos( – PD) and with different PDs, produces a third cosine, the PD of which depends on the relative weighting of the first two cosines. The first two cosines represent two neurons that can be described as extrinsic-like modulated by wrist posture, whereas the third cosine represents a neuron that can be described as muscle-like. Kakei et al. (2003) suggest that this linear summation can account for the muscle-like neurons found in MI. In another modeling study, Salinas and Abbott (1995) hard-wired an array of sensory neurons [which behave similar to the extrinsic-like MI neurons in Kakei et al. (1999)] and an array of motor neurons [which behave similar to the muscle-like MI neurons in Kakei et al. (1999)] and used a Hebbian learning rule to modify the connections between the two. After training, the sensory array could directly produce the appropriate activation patterns in the motor array. We view the results of these studies as lending support to the parallel processing scheme argued for in this paper.
Although our model investigates a similar transformation as those presented in Salinas and Abbott (1995) and Kakei et al. (2003), our approach allows for the use of an optimization procedure to select the motor output (muscle activation patterns in our model) as opposed to committing to a predefined representation. Salinas and Abbott (1995) and Kakei et al. (2003) assume that the output array follows a Gaussian or cosine form; our method allows for significant flexibility in selecting an arbitrary output pattern. However, in all cases, the simple, linear transformation is made possible by the sparse representation exhibited by the input and output representations. The sparse representation in our model is the result of the optimization procedure and was not explicitly included. In addition, we show how the muscle activation output of our model leads to endpoint of movement; this creates a plausible motor plant that allows us to use a performance-dependent optimization procedure.
The gradient-descent method used in our model has certain advantages. While Kakei et al. (2003) show how the linear summation of two extrinsic-like neurons can produce an arbitrary muscle-like neuron, our model specifically finds how to weigh the contributions of the extrinsic-like neurons to produce the appropriate muscle activation patterns. Salinas and Abbott (1995) employ a Hebbian-style learning rule that, in the limit, tends toward a linear transformation in which connection strength is predictable from input/output correlation. Unlike the learning rule employed in our model, this rule reduces the number of effective degrees of freedom that are exploitable in selecting a connection matrix. Although their learning rule does represent one that is more biologically plausible (in cortex) than the error-driven rule used in our model, it must presume the existence of an external module that is able to independently drive the sensory inputs and motor outputs correctly.
Effects of noise
The introduction of signal-dependent noise in our model leads to a harder problem and thus a reduction in the degrees of freedom afforded by neural redundancy. Although a transformation exists from the extrinsic-like neurons to muscle activation, there are more restrictions on this transformation. For example, because the total amount of noise to a muscle is minimized when averaged over many neurons, it is advantageous to recruit many rather than fewer neurons to provide the same input (cf. Todorov 2002). Thus in the model with high neural noise, neurons with similar PDs will tend to activate a muscle the same way (correlation and connection strength will be related to some degree). However, in a high noise simulation with more MI neurons (192), and thus greater degrees of freedom, the predictive relationship between corrij and Kij was weaker. In addition, to keep muscle activity from growing unreasonably (because neurons are cooperating more), the elements of K must be close to 0, also a property we observe in the model with high neural noise. Although some of the noisy model behavior can be explained by a reduction in the degrees of freedom, some aspects can also be explained by similarities with stochastic gradient search methods (Hassoun 1995; Hoptroff and Hall 1989), which intentionally introduce noise in gradient descent optimization. One of the advantages of this method is that it aids in finding a global minimum (Schoen 1991), seen in the unique Ks found in both of our models with noisy neurons.
Future work
To focus on the issue of coordinate transformations, we kept the level of abstraction of our model very high. In the model presented in this paper and in Fagg et al. (2002), we make the assumption that all muscles pull with equal strength in their pulling directions. Fagg et al. (2002) discusses how muscle behavior changes when relative pulling strengths are changed. In short, the qualitative nature of muscle recruitment does not change. We also use an abstract representation of muscle, in which we do not include time-related aspects of muscle action, in our models. However, the temporal behavior of muscle recruitment is also important (Hoffman and Strick 1999). One area of future research with this model is to incorporate the temporal aspects of movement generation and the dependence on spinal and muscular dynamics (Houk et al. 2002).
Our model includes a well-defined representation of movement based on neurons recorded in Kakei et al. (1999). Why do we see structure in neural activity at all? In continuing work, we have adapted this model to include MI neurons that are not hard-wired to behave a certain way. We are examining the effects of local interactions (not included in the model presented in this paper) and optimization criteria (such as minimization of metabolic energy, Balasubramanian et al. 2001; Levy and Baxter 1996; Schreiber et al. 2002) in the formation of neural behavior. Preliminary results show that the local interactions and optimization criteria produce both extrinsic- and muscle-like MI neurons that are active simultaneously during the wrist task.
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ACKNOWLEDGMENTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGMENTSREFERENCES |
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GRANTS
Preparation of this report was supported by National Institutes of Mental Health Grant MH-48185-10, National Science Foundation Grant EIA 9703217, and National Aeronautics and Space Administration Grant NAG9–1445.
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FOOTNOTES |
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Address for reprint requests and other correspondence: A. Shah, Dept. of Computer Science, University of Massachusetts, 140 Governor's Dr., Amherst, MA 01003 (E-mail: ash{at}cs.umass.edu).
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|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGMENTSREFERENCES |
|---|
Amirikian B and Georgopoulos AP. Directional tuning profiles of motor cortical cells. Neurosci Res 36: 73–79, 2000.[CrossRef][Web of Science][Medline]
Anastasio TJ and Robinson DA. The distributed representation of vestibulooculomotor signals by brain-stem neurons. Biol Cybern 61: 79–88, 1989.[Web of Science][Medline]
Anastasio TJ and Robinson DA. Distributed parallel processing in the vertical vestibulo-ocular reflex: learning networks compared to tensor theory. Biol Cybern 63: 161–167, 1990.[CrossRef][Web of Science][Medline]
Ashe J and Georgopoulos AP. Movement parameters and neural activity in motor cortex and area 5. Cereb Cortex 6: 590–600, 1994.
Balasubramanian V, Kimber D, and Berry MJI. Metabolically efficient information processing. Neural Comput 13: 799–815, 2001.[CrossRef][Web of Science][Medline]
Caminiti R, Johnson PB, Galli C, Ferraina S, and Burnod Y. Making arm movements within different parts of space: the premotor and motor cortical representations of a coordinate system for reaching to visual targets. J Neurosci 11: 1182–1197, 1991.[Abstract]
Caminiti R, Johnson PB, and Urbano A. Making arm movements within different parts of space: dynamic aspects in the primate motor cortex. J Neurosci 10: 2039–2058, 1990.[Abstract]
Dum RP and Strick PL. Motor areas in the frontal lobe of the primate. Physiol Behav 77: 677–682, 2002.[CrossRef][Medline]
Engel AK, König P, Kreiter AK, and Singer W. Interhemispheric synchronization of oscillatory neuronal responses in cat visual cortex. Science 252: 1177–1179, 1991.
Fagg AH, Shah A, and Barto AG. A computational model of muscle recruitment for wrist movements. J Neurophysiol 88: 3348–3358, 2002.
Fu QG, Flament D, Coltz JD, and Ebner TJ. Temporal encoding of movement kinematics in the discharge of primate primary motor and premotor neurons. J Neurophysiol 73: 836–854, 1995.
Georgopoulos AP, Ashe J, Smyrnis N, and Taira M. The motor cortex and the coding of force. Science 256: 1692–1695, 1992.
Georgopoulos AP, Kalaska JF, Caminiti R, and Massey JT. On the relations between the direction of two-dimensional arm movements and cell discharge in primate motor cortex. J Neurosci 2: 1527–1537, 1982.[Abstract]
Gibson AR, Houk JC, and Kohlerman NJ. Relation between red nucleus discharge and movement parameters in trained macaque monkeys. J Physiol 358: 551–570, 1985.
Harris CM and Wolpert DM. Signal dependent noise determines motor planning. Nature 394: 780–784, 1998.[CrossRef][Medline]
Hassoun MH. Global search methods for neural networks. In: Fundamentals of Artificial Neural Networks. Cambridge, MA: The MIT Press, 1995, chapt. 8, p. 417–468.
Hoffman DS and Strick PL. Step-tracking movements of the wrist. IV. Muscle activity associated with movements in different directions. J Neurophysiol 81: 319–333, 1999.
Hoptroff RG and Hall TJ. Learning by diffusion for multilayer perceptron. Electron Lett 25: 531–533, 1989.[CrossRef]
Houk JC, Dessem DA, Miller LE, and Sybirska EH. Correlation and spectral analysis of relations between single unit discharge and muscle activities. J Neurosci Methods 21: 201–224, 1987.[CrossRef][Web of Science][Medline]
Houk JC, Fagg AH, and Barto AG. Fractional power damping model of joint motion. In: Progress in Motor Control: Structure-Function Relations in Voluntary Movements, edited by Latash M. Champaign, IL: Human Kinetics, 2002, vol. 2, p. 147–178.
Houk JC, Keifer J, and Barto AG. Distributed motor commands in the limb premotor network. Trends Neurosci 16: 27–33, 1993.[CrossRef][Web of Science][Medline]
Kakei S, Hoffman DS, and Strick PL. Muscle and movement representations in the primary motor cortex. Science 285: 2136–2139, 1999.
Kakei S, Hoffman DS, and Strick PL. Direction of action is represented in the ventral premotor area. Nat Neurosci 4: 1020–1025, 2001.[CrossRef][Web of Science][Medline]
Kakei S, Hoffman DS, and Strick PL. Sensorimotor transformations in cortical motor areas. Neurosci Res 46: 1–10, 2003.[CrossRef][Web of Science][Medline]
Kalaska JF and Crammond DJ. Cerebral cortical mechanisms of reaching movements. Science 255: 1517–1523, 1992.
König P and Engel AK. Correlated firing in sensory-motor systems. Curr Opin Neurobiol 5: 511–519, 1995.[CrossRef][Web of Science][Medline]
Levy WB and Baxter RA. Energy efficient neural codes. Neural Comput 8: 531–543, 1996.[Web of Science][Medline]
Loeb GE, Brown IE, and Chen EJ. A hierarchical foundation for models of sensorimotor control. Exp Brain Res 126: 1–18, 1999.[CrossRef][Web of Science][Medline]
Luppino G, Matelli M, Camarda R, and Rizzolatti G. Corticospinal projections from mesial fronal and cingulate areas in the monkey. Neuroreport 5: 2545–2548, 1994.[Web of Science][Medline]
McKiernan BJ, Marcario JK, Karrer JH, and Cheney PD. Corticomotoneuronal postspike effects in shoulder, elbow, wrist, digit, and intrinsic hand muscles during a reach and prehension task. J Neurophysiol 80: 1961–1980, 1998.
Miller LE and Houk JC. Motor coordinates in primate red nucleus: preferential relation to muscle activation versus kinematic variables. J Physiol 488: 533–548, 1995.
Miller LE and Sinkjaer T. Primate red nucleus discharge encodes the dynamics of limb muscle activity. J Neurophysiol 80: 59–70, 1998.
Miller LE, van Kan PLE, Sinkjaer T, Andersen T, Harris GD, and Houk JC. Correlation of primate red nucleus discharge with muscle activity during free-form arm movements. J Physiol 469: 213–243, 1993.
Moran DW and Schwartz AB. Motor cortical represenation of speed and direction during reaching. J Neurophysiol 82: 2676–2692, 1999.
Munk MHJ, Nowak LG, Nelson JI, and Bullier J. Structural basis of cortical synchronization. II. Effects of cortical lesions. J Neurophysiol 74: 2401–2414, 1995.
Mussa-Ivaldi FA. Do neurons in the motor cortex encode movement direction? an alternative hypothesis. Neurosci Lett 91: 106–111, 1988.[CrossRef][Web of Science][Medline]
Salinas E and Abbott LF. Transfer of coded information from sensory to motor networks. J Neurosci 15: 6461–6474, 1995.
Sanger TD. Theoretical considerations for the analysis of population coding in motor cortex. Neural Comput 6: 29–37, 1994.
Schoen F. Stochastic techniques for global optimization: a survey of recent advances. J Global Optimizat 1: 207–228, 1991.[CrossRef]
Schreiber S, Machens CK, Herz AVM, and Laughlin SB. Energy-efficient coding with discrete stochastic movements. Neural Comput 14: 1323–1346, 2002.[CrossRef][Web of Science][Medline]
Schwartz AB, Kettner RE, and Georgopoulos AP. Primate motor cortex and free arm movements to visual target in three-dimensional space. I. Relations between single cell discharge and direction of movement. J Neurosci 8: 2913–2927, 1988.[Abstract]
Schwartz AB and Moran DW. Arm trajectory and representation of movement processing in motor cortical activity. Eur J Neurosci 12: 1851–1856, 2000.[CrossRef][Web of Science][Medline]
Scott SH. Comparison of onset time and magnitude of activity for proximal arm muscles and motor cortical cells before reaching movements. J Neurophysiol 77: 1016–1022, 1997.
Scott SH. The role of primary motor cortex in goal-directed movements: insights from neurophysiological studies on non-human primates. Curr Opin Neurobiol 13: 671–677, 2003.[CrossRef][Web of Science][Medline]
Scott SH and Kalaska JF. Changes in motor cortex activity during reaching movements with similar hand paths but different arm postures. J Neurophysiol 73: 2563–2567, 1995.
Scott SH and Kalaska JF. Reaching movements with similar hand paths but different arm orientations. I. Activity of individual cells in motor cortex. J Neurophysiol 77: 826–852, 1997.
Scott SH. Role of motor cortex in coordinating multi-joint movements: is it time for a new paradigm? Can J Physiol Pharmacol 78: 923–933, 2000.[CrossRef][Web of Science][Medline]
Sergio LE and Kalaska JF. Systematic changes in directional tuning of motor cortex cell activity with hand location in the workspace during generation of static isometric forces in constant spatial directions. J Neurophysiol 78: 1170–1174, 1997.
Sergio LE and Kalaska JF. Changes in the temporal pattern of primary motor cortex activity in a directional isometric force versus limb movement task. J Neurophysiol 80: 1577–1583, 1998.
Shah A, Fagg AH, and Barto AG. Cortical involvement in the recruitment of wrist muscles. In: Neural Control of Movement Conference. Naples, FL: Society for the Neural Control of Movement, 2002.
Todorov E. Direct cortical control of muscle activations in voluntary arm movements: a model. Nat Neurosci 3: 391–398, 2000.[CrossRef][Web of Science][Medline]
Todorov E. Cosine tuning minimized motor errors. Neural Comput 14: 1233–1260, 2002.[CrossRef][Web of Science][Medline]
van Beers RJ, Haggard P, and Wolpert DM. The role of execution noise in movement variability. J Neurophysiol October 2003; 10.1152/jn. 00652.2003.
Xing J and Andersen RA. Models of the posterior parietal cortex which perform multimodal integration and represent space in several coordinate frames. J Cognit Neurosci 12: 601–614, 2000.[CrossRef][Web of Science][Medline]
Zhang K and Sejnowski TJ. Neuronal tuning: to sharpen or broaden? Neural Comput 11: 75–84, 1999.[CrossRef][Web of Science][Medline]
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, corrijs between the muscle and m neurons 1 through N; 
, corrijs between the muscle and m neurons N + 1 through 2N. N = 48. Vertical lines, the PDs of each muscle when the wrist is pronated (PRO), midrange (MID), or supinated (SUP) postures. Top left: scale bars illustrating the relationship between m neuron index and m neuron PD. The corrij depends on how similar the PD of the m neuron is to the PDs of the muscle.