LETTER
Communicated by Emmanuel Todorov
Modeling Reaching Impairment After Stroke Using a Population Vector Model of Movement Control That Incorporates Neural Firing-Rate Variability David J. Reinkensmeyer
[email protected] Department of Mechanical and Aerospace Engineering and Center for Biomedical Engineering, University of California at Irvine, Irvine, CA 92697, U.S.A.
Mario G. Iobbi mario
[email protected] Department of Physics and Center for Biomedical Engineering, University of California at Irvine, Irvine, CA 92697, U.S.A.
Leonard E. Kahn
[email protected] Sensory Motor Performance Program, Rehabilitation Institute of Chicago, Chicago, IL 60611, U.S.A., and Department of Biomedical Engineering, Northwestern University, Evanston, IL 60208, U.S.A.
Derek G. Kamper
[email protected] Sensory Motor Performance Program, Rehabilitation Institute of Chicago, Chicago, IL 60611, U.S.A., and Department of Physical Medicine and Rehabilitation,Northwestern University Medical School, Evanston, IL 60208, U.S.A.
Craig D. Takahashi
[email protected] Department of Mechanical and Aerospace Engineering, University of California at Irvine, Irvine, CA 92697, U.S.A.
The directional control of reaching after stroke was simulated by including cell death and ring-rate noise in a population vector model of movement control. In this model, cortical activity was assumed to cause the hand to move in the direction of a population vector, dened by a summation of responses from neurons with cosine directional tuning. Two types of directional error were analyzed: the between-target variability, dened as the standard deviation of the directional error across a wide range of target directions, and the within-target variability, dened as the standard deviation of the directional error for many reaches to a single target. c 2003 Massachusetts Institute of Technology Neural Computation 15, 2619–2642 (2003) °
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Both between- and within-target variability increased with increasing cell death. The increase in between-target variability arose because cell death caused a nonuniform distribution of preferred directions. The increase in within-target variability arose because the magnitude of the population vector decreased more quickly than its standard deviation for increasing cell death, provided appropriate levels of ring-rate noise were present. Comparisons to reaching data from 29 stroke subjects revealed similar increases in between- and within-target variability as clinical impairment severity increased. Relationships between simulated cell death and impairment severity were derived using the between- and within-target variability results. For both relationships, impairment severity increased similarly with decreasing percentage of surviving cells, consistent with results from previous imaging studies. These results demonstrate that a population vector model of movement control that incorporates cosine tuning, linear summation of unitary responses, ring-rate noise, and random cell death can account for some features of impaired arm movement after stroke. 1 Introduction Approximately 50% of stroke survivors have chronic arm movement impairment (Nakayama, Jorgensen, Raaschou, & Olsen, 1994). Although increasingly more detailed descriptions of the kinematic and dynamic features of arm movement after stroke are becoming available (Beer, Dewald, & Rymer, 2000; Kamper, McKenna, Kahn, & Reinkensmeyer, 2002; Levin, 1996; Reinkensmeyer, McKenna, Cole, Kahn, & Kamper, 2002), little is known about the neuronal mechanisms by which these features arise, and few neural models have been proposed to explain these features (Goodall, Reggia, Chen, Ruppin, & Whitney, 1997). The purpose of this study was to test whether a population vector model of the control of hand movement, damaged to simulate the consequences of stroke, could account for directional reaching errors commonly observed after stroke. Population vector models of movement control are based on the observation that the ring rates of many motor cortical cells vary as a function of the hand’s movement direction (Georgopoulos, Kalaska, Caminiti, & Massey, 1982). These direction-sensitive cells have a peak mean ring rate corresponding to one preferred movement direction but are broadly tuned across a range of directions. To form a population vector for a given movement, the vector representing each cell’s preferred direction is scaled by the cell’s mean ring rate, and a vector sum is computed across all scaled vectors. As was shown rst for constrained planar reaches and later for free reaching in 3D space, the population vector typically points in the same direction as the hand’s movement direction (Georgopoulos et al., 1982; Georgopoulos, Kettner, & Schwartz, 1988; Moran & Schwartz, 1999). Neural activity consistent with population vector coding has been identied for the primary
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motor cortex (Georgopoulos et al., 1982, 1988), premotor cortex (Caminiti et al., 1991), area V of the parietal cortex (Kalaska, Caminiti, & Georgopoulos, 1983), and the cerebellum (Fortier, Kalaska, & Smith, 1989). One interpretation of these observations is that neurons in these areas are functionally connected to muscles in such a way as to move the hand in the direction of the population vector (Georgopoulos, 1996). For example, if it is simply assumed that each pyramidal tract neuron contributes additively to activation of a muscle group, then the population vector corresponds to hand movement direction, due in part to the musculoskeletal mechanics (Todorov, 2000). In this article, we assume that there exists some such control scheme, which translates cortical activity into hand movement in the direction of the population vector. Stroke often causes neuronal damage in motor cortical areas, as well as in white matter projections from these areas to the spinal cord. Perhaps the most visible result of this destruction is weakness in the contralesional limbs (Gandevia, 1993). However, detailed kinematic analysis has identied several other cardinal features of stroke-affected arm movement, including impaired directional control of reaching (Beer et al., 2000; Kamper et al., 2002; Levin, 1996; Reinkensmeyer et al., 2002). The ability to steer the arm in the desired movement direction is impaired, and the directional variability associated with multiple reaches is increased. The ring-rate variability of individual cortical neurons was recently found to be a power function of their mean ring rate for multiple cortical areas (Lee, Port, Kruse, & Georgopoulos, 1998). We hypothesized that such neural noise, incorporated into a population vector model of movement control, which was modied to reect widespread neural destruction, might account for the increased directional errors in reaching following stroke. Previous reports have demonstrated a decrease in population vector coding accuracy with decreased population size when randomness is included in cell tuning curves or preferred direction distributions (Salinas & Abbot, 1994; Tanaka, 1994). This study extends these previous studies by incorporating physiologically identied levels of ring-rate variability, simulating uniformly random cell destruction to mimic the putative consequences of broad neural damage due to stroke, analyzing two distinct types of directional error in simulated reaching (between- and within-target variability), and comparing the predicted levels of error to experimental data from stroke subjects. 2 Methods 2.1 Population Vector Model of Movement Control. The directional tuning of individual neurons was modeled using the functional form suggested for the motor cortex of rhesus monkeys (Georgopoulos et al., 1982): R D b C k cos µp
(2.1)
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where R represents the neuron’s mean ring rate in a movement direction and µp is the angle between the vector describing the hand’s movement direction and the vector pointing in the preferred direction, and b and k are tting coefcients chosen to be 15 spikes per second and 6.3 spikes per second, respectively, based on the mean values found originally for monkeys (Georgopoulos et al., 1982; see Figure 1A). Each cell’s preferred direction
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Figure 1: Population vector model of stroke-impaired reaching. (A) Directional tuning curve of simulated cell. Error bars show standard deviation of ring rate due to inherent ring-rate variability. (B) Each cell’s preferred direction was modeled as a unit vector in three-dimensional space. The preferred directions for the neuronal population were uniformly distributed across the space. A stroke was simulated by randomly eliminating a given percentage of the cells. (C) The population vector (dashed line) was formed by taking the sum of the preferred vectors weighted by their ring rates and projecting the resultant vector onto the horizontal plane of movement. A cortical movement control scheme was assumed to exist, which translates cortical activity into hand movement in the direction of the population vector. Note that the population vector pointed approximately in the desired movement direction (straight ahead for the shown example) for an undamaged neuronal population, but deviated from the desired direction with simulated cell destruction.
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was modeled as a unit vector in three-space. A three-dimensional model was used in order to allow comparison to three-dimensional reaching data obtained from stroke subjects (see below). The neuronal population was initially chosen arbitrarily to consist of 328 of these unit vectors uniformly distributed across the surface of a unit sphere with a spacing of about 0.2 rad between tips (see Figure 1B). The uniform distribution was created by dividing a sphere into N latitudinal circles based on a chosen spacing for the vectors tips (e.g., d D 0:2 rad), then dividing the circumference of each latitudinal circle into M longitudinal divisions, where M was chosen to yield the chosen tip spacing .M D 2¼ sin.µ /=d, where µ D 0 was the “north pole” of the sphere. The tip spacing, and thus number of unit vectors, was varied to determine the effects of population size on directional errors. Strokes were simulated by randomly eliminating a given percentage of the unit vectors (see Figure 1B). The rationale for randomly eliminating preferred directions was that damage from stroke is rarely precisely localized. In addition, preferred directions do not appear to be topographically organized in any straightforward, modular manner (Ben-Shaul et al., 2003). Thus, stroke likely eliminates a broad range of preferred directions. Inherent ring-rate variability for each neuron was modeled using the relationship identied in rhesus monkeys for the primary motor cortex and parietal areas 2=5 (Lee et al., 1998): ¾ D cRd ;
(2.2)
where ¾ is the standard deviation in the mean ring rate, R is the mean ring rate, c D 1:2, and d D :59. Specically, the actual mean ring rate Ra of a cell was modied to include a noise component RN that was selected at random from a normal distribution with zero mean and standard deviation ¾ (see Figure 1A), Ra D R C RN :
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Note that equations 2.2 and 2.3 correspond to the variability of one cell’s discharge rate in a single direction. To simulate a reach for a selected healthy or damaged “virtual cortex,” the E for the reach and the preferred inner product of the movement direction D vector PE for each cell in the virtual cortex was computed: E ² P: E cos µp D D
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Equations 2.1 through 2.3 were then used to calculate the mean ring rate (Ra) of each neuron for the desired reach direction. The resulting scaled vectors were summed together to form a population vector that was assumed to point in the initial direction of movement. Initial movement direction
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was chosen for analysis because this parameter reects descending, feedforward control of voluntary movement, and is unaffected by later-acting, segmental reexes that may be rendered abnormal by stroke (Fellows, Kaus, & Thilmann, 1994). Randomized, virtual subjects were generated with varying percentage destruction of the simulated neuron population. The percentages of destruction simulated were 99, 95, 90, 80, 70, 60, 50, 40, 30, 20, 10, and 0. The destroyed cells were chosen at random from a uniform distribution for each virtual subject. Simulations were performed with these virtual subjects that reproduced the multiple target and single target experiments described in the following section, but with a higher number of subjects (50) at each destruction level to estimate the statistical variables of interest more accurately. 2.2 Experimental Data. To evaluate the population vector model of movement control, its predictions of initial direction error were compared with data from 29 stroke subjects obtained in two experiments. In the multiple target experiment, 16 hemiparetic stroke subjects reached one time toward each of 75 targets (Kamper et al., 2002; Reinkensmeyer et al., 2002). The targets were arranged in 5 latitudinal rows spanning 48 degrees and 15 longitudinal columns spanning 168 degrees on a vertical screen arranged in a semicircle around the seated subject. Subjects reached at a comfortable pace from their lap to a point as close as possible to the target without displacing their trunk. An electromagnetic sensor measured hand position at a sampling rate of 100 Hz. In the single-target experiment, 13 different subjects reached 20 times each to a single target (Takahashi & Reinkensmeyer, 2003). The target used in the single-target experiment was approximately the center target used in the multitarget experiment. The subjects’ hands were attached to a lightweight robotic arm (PHANToM 3.0, SensAble Technologies, Inc.) through a customized orthopedic splint. Each subject raised his or her hand to a start target positioned several inches above the lap, then reached at a comfortable pace to a nish target positioned directly in front, just inside the outer edge of his or her reaching workspace. The robot measured hand position at a sample rate of 200 Hz. For both experiments, an experienced evaluator assessed the motor impairment of the subjects with the Chedoke-McMaster Stoke Arm Assessment score. This assessment is a seven-point scale that has high inter- and intra-rater repeatability, as well as strong correlation with the Fugl-Meyer score (Gowland et al., 1993) and a variety of altered reaching parameters following stroke (Kamper et al., 2002). A Chedoke score of 1 represents complete paralysis, and a score of 2 indicates a trace level of elbow or shoulder movement. Scores 3 to 6 mark progressively improved range, coordination, and speed of movement, with a score of 7 indicating an unimpaired arm. For the multiple-target experiment, the distribution of subjects was [(Chedoke score, number of subjects): (2, 5); (3, 4); (4, 4); (5, 2); (6, 1); (7,4)]. For the
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single-target experiment, the distribution of subjects was [(2, 1); (3, 4); (4, 0); (5, 5); (6, 2); (7,1)]. In selecting subjects, no exclusionary criteria were placed on the type (ischemic or hemmorhagic) or location of stroke, and thus the subjects had a diverse lesion prole. 2.3 Data Analysis. For simplicity, the analysis in this article is focused on movement direction projected onto the horizontal plane (the “yaw” Euler angle). However, the simulated pitch error data (calculated by projecting onto a vertical plane) followed similar trends because the model is rotationally symmetric (i.e., the denition of “horizontal” and “vertical” is arbitrary in model coordinates—there is no way to distinguish “up” and “down” in Figure 1B from the plot of the vectors themselves). In addition, similar trends of increasing pitch error movement variability with increasing impairment severity were observed for the experimental data projected onto the sagittal plane. The directional error for the population vector model was dened as the difference between the desired movement direction and the population vector direction, both projected onto the horizontal plane. For the experimental data, the initial movement direction was dened as the direction of the vector between the start position and a point along the hand trajectory when the hand had moved 5 cm, projected onto the horizontal plane. The distance-based cut-off was chosen instead of a temporal cut-off because subjects moved at different velocities. The directional variability for a slow movement would have been greater than that for a fast movement if a temporal cut-off had been used, even if the movements had the same inherent directional variability. To account for possible systematic variation in initial movement direction during normal movement, the directional error was dened as the difference between the initial movement direction of the contralesional and ipsilesional arms for each subject. The ipsilesional arms were ostensibly normal and used regularly by the subjects for activities of daily living. Data from left arms were transformed in a mirror symmetric fashion. An estimate of the between-target variability in directional error was obtained from the multiple-target experiment, since this experiment measured reaches to many targets. Between-target variability was calculated by taking the standard deviation of the directional error across the target directions. An estimate of the within-target deviation of directional error was obtained from the single-target experiment, since this experiment measured multiple reaches to one target. Within-target variability was calculated by taking the standard deviation of the directional error across the multiple reaches to the one target. A curve-tting procedure was used to model the dependence of betweenand within-target variability on cell death for the simulations and on the clinical severity of stroke for the experimental data. A function consisting of the sum of two exponential functions t the data well, using the GaussNewton method (see Figures 2A, 2B, 2D, 2E).
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Figure 2: Comparison of simulated and experimental directional reaching errors. (A) Simulation data: the between-target variability increased with decreasing percentage of cells surviving. A double exponential curve (thick line) t the simulated directional error well (thin line, barely visible). (B) Experimental data: the between-target variability increased with increasing clinical severity of impairment (a score of 2 denotes severe impairment and 7 no impairment). A double exponential curve (thick line) was used to t the experimental data (thin line, mean ( standard error). (C) Relationship between surviving percentage of cells and impairment level, derived from data in A and B. (D–F) Same as A–C except the performance measure was the within-target variability, that is, the standard deviation of the directional error for multiple reaches to the same target.
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By using the double exponential functions, it was then possible to derive a relationship between the percentage of cells destroyed and the clinical severity of impairment. This relationship was derived for both the between-target variability, obtained from the multiple-target experiment, and the withintarget variability, obtained from the single-target experiment. Specically, for a given clinical impairment level, the corresponding variability measure was calculated using the function t to the experimental data. The percentage of cells destroyed that most closely matched this error was found by inverting the function relating variability to percentage of cells destroyed using a look-up table. Before inversion, the simulation function was modied by adding a constant offset to it. This offset caused the variability corresponding to no cell destruction to equal that for no clinical impairment (Chedoke score = 7). The rationale for adding this constant was that the reaching data from the unimpaired arms displayed greater variability than predicted by the population vector model at 0% cell destruction. This additional noise might be expected in vivo because of “downstream” noise sources in, for example, motor neurons, neuromuscular junctions, and muscle itself (Jones, Hamilton, & Wolpert, 2002). This noise was modeled as an additive constant offset for all percentages of cell death. A sensitivity analysis was performed to determine how the simulation results depended on the population size, the neural tuning parameters (mean ring rate b and amplitude of cosine tuning curve k in equation 2.1), and the neural noise parameters (noise amplitude c and noise exponent d in equation 2.2). The population size was varied across the set f328; 1302; 5218; 20,860; 83,442g. For analysis of the tuning parameters, the index of directional modulation ID was dened as the ratio of the ring-rate range to the mean ring rate itself (ID D 2¤ k=b). This index has been used previously to describe tuning properties of cortical cells and indicates the ring rate modulation about a cell’s mean ring rate (Georgopoulos et al., 1982). If ID D 0, the cell is not directionally sensitive (its ring rate does not change with direction). If ID D 1, then the cell varies its ring rate by an amount equal to its mean ring rate across all directions. ID was made to vary away from its experimentally identied default value of 0.8 through the range 0.1 to 8.0 by varying b between 10 and 50 spikes per second, and k between 5 and 40 spikes per second, both in 5 spikes per second intervals. The parameters c and d were kept constant at their default values as b and k were varied. For analysis of the noise parameters, the noise index IN was dened as the ratio of the noise level corresponding to the mean ring rate and the mean ring rate (IN D cbd =b). IN was made to vary away from its experimentally identied default value of 0.4 through the range 0.0 to 4.1 by varying c between 0 and 2.4, and d between 0 and 1.8, both in 0.3 intervals. The parameters b and k were kept constant at their default values as c and d were varied.
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3 Results As expected, the simulated population vector pointed in approximately the same direction as the desired direction vector for an undamaged neuronal population (see Figure 1C). Small deviations were present due primarily to the inherent ring-rate variability included in the model. The population vector deviated increasingly away from the desired movement direction when cells in the neuronal population were randomly destroyed (see Figure 1C). The between-target variability (the standard deviation of the directional error across multiple targets) increased as the percentage of surviving cells decreased (see Figure 2A). The experimental data showed a similar trend as a function of clinical severity of arm impairment: the variability increased as the impairment severity increased (see Figure 2B). The mean directional error was approximately zero across damage levels, a prediction approximated by previous experimental data (Beer et al., 2000; Reinkensmeyer et al., 2002). The within-target variability (the standard deviation of the directional error for multiple reaches to the same target) also increased with decreasing percentage of surviving cells (see Figure 2D). The actual data from the single target experiment showed a similar trend as a function of clinical severity of arm impairment (see Figure 2E). By using the procedure described in section 3, curves were t to the model and experimental data and then used to derive a relationship between the percentage of surviving cells and the clinical severity of impairment. The predicted clinical impairment severity increased monotonically with decreasing percentage of surviving cells for both the multiple- (see Figure 2C) and single-target experiments (see Figure 2F). The curves were similar despite the distinctly different measures of variability and different stroke subjects used to generate them. A sensitivity analysis was performed to determine how the directional variability and damage-impairment relationships depended on the model parameters. Larger cell populations generated smaller between- and withintarget variability (see Figures 3A and 3C), causing the damage-impairment relationships to shift and become more nonlinear (see Figures 3B and 3D). The variability measures were also plotted against the absolute number of surviving cells rather than the percentage of surviving cells (see Figure 4). Both between- and within-target variability decreased with number of surviving cells, but there was a difference in the behavior of the between- and within-target curves with different initial population sizes. Specically, note that the right-most terminal point of each overlaid curve in Figures 4A and 4B represents a simulation result for an undamaged and therefore uniform population, while all other points of the curve are results for damaged and therefore nonuniform populations. Thus, if the right-most terminal point lies on top of all overlaid curves, then uniform and nonuniform popula-
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Figure 3: Sensitivity of simulation results to population size. Thick dashed line: N D 328, the default value used for the simulations. The arrows indicate N increasing through the set f328; 1302; 5218; 20860; 83442g. (A) Sensitivity of between-target variability. (B) Sensitivity of damage-impairment relationship derived from between-target variability. (C) Sensitivity of within-target variability. (D) Sensitivity of damage-impairment relationship derived from withintarget variability.
tions produced a similar result. This is true for within-target variability (see Figure 4B) but not between-target variability (see Figure 4A). Plotting the data on log-log axes makes this point more obvious (see Figures 4C and 4D). The lack of dependence of within-target variability on population uniformity is made explicit in Figure 4E, which plots results from simulations using populations of equal initial size with different levels of cell death. Here, the levels were chosen to keep the residual cell population size the same (N D 328). As can be seen, within-target variability did not depend on the starting size of the population, while between-target variability did (see Figure 4E). Put another way, within-target variability did not depend on the percentage of cells surviving, provided the absolute number of cells surviving was equal. Note that the percentage of cells surviving is directly related to the population uniformity because a uniform distribution becomes less uniform as its members are randomly removed from it. For example, Figure 4F shows a standard measure of uniformity for spherical distributions,
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the Rayleigh statistic (Mardia & Jupp, 2000), which is essentially the normalized magnitude of the resultant vector calculated by adding all residual, preferred-direction unit vectors. If these vectors are uniformly distributed, the Rayleigh statistic is equal to zero. If they all point in the same direction, the statistic is equal to one. This uniformity measure varied approximately linearly with cell destruction. Thus, within-target variability did not vary with population uniformity, while between-target variability did. The dependence of both the between- and within-target variability on percentage of surviving cells became more nonlinear for larger indices of directional modulation ID , dened as the ratio of the range of ring rates spanned by the cell to the mean ring rate (see Figures 5A and 5C). As a result, the curvature of the damage/impairment relationship also depended on ID (see Figures 5B and 5D). Thus, the results were dependent on the amount of directional sensitivity incorporated into the cells in the model. Larger noise indices IN created greater errors for all percentages of surviving cells (see Figures 6A and 6C). However, the dependence of withintarget variability on cell death was most pronounced for a moderate range
Figure 4: Facing page. Effect of absolute population size on between- and withintarget variability. In A and B, the different lines correspond to simulations with different population sizes. The starting population size can be determined by the right-side termination of the lines. The lines that do not terminate correspond to population sizes 5218, 20,860, and 83,442; that is, the lines for the larger starting population sizes are truncated, although they follow the same trend. (A) Between-target variability. (B) Within-target variability. (C) Betweentarget variability on log-log scale. (D) Within-target variability on log-log scale. (E) Dependence of between- and within-target variability on cell destruction level, when residual cell population size is equal. The percentage of destruction of the population was varied while keeping its size the same by simulating populations of equal initial size following different levels of cell death. Specically, ve different initial population sizes (N D f328 510 860 1302 83,442g) were simulated, each with a different percentage of surviving cells (% surviving = f100.0, 64.3, 38.1, 25.1, 0.4), where the percentages were chosen such that the number of surviving cells was equal across populations (328). (F) Uniformity is directly related to the fraction of cells surviving. The measure of uniformity N 2 , where n is the number of cells surviving, and R N is is the Rayleigh statistic, nR the mean resultant length calculated by adding all residual, preferred direction unit vectors and dividing by the number of vectors (Mardia & Jupp, 2000). If these vectors are uniformly distributed, the Rayleigh statistic is equal to zero. If they all point in the same direction, the statistic is equal to one. The plot was generated by simulating different levels of cell destruction for a population of initial size 328 and calculating the Rayleigh statistic across 1000 simulations for each level of cell destruction.
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of IN ; in other words, the curve was atter for small and large values of IN (see Figure 6C). The destruction/impairment relationship derived from the between-target variability was relatively insensitive to the noise index (see Figure 6B), while the one derived from the within-target variability was more sensitive (see Figure 6D). Thus, the within-target variability results were most sensitive to the ring-rate noise levels incorporated into the model. 4 Discussion
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The population vector model of movement control used in this study exhibited increased between- and within-target variability of directional error with increasing cell death. Reaching data from stroke subjects showed comparable trends in these variables as a function of the clinical score of motor impairment. Thus, the model provides a possible mechanistic link between
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Figure 5: Sensitivity of simulation results to index of directional modulation ID . Thick dashed line: ID D 0:8, the default value used for the simulations; light lines: ID < 0:5; medium lines: 0:5 < ID < 1:2 dark lines: ID > 1:2. The arrows indicate increasing ID . (A) Sensitivity of between-target variability. (B) Sensitivity of damage-impairment relationship derived from between-target variability. (C) Sensitivity of within-target variability. (D) Sensitivity of damage-impairment relationship derived from within-target variability.
quantied levels of cell destruction and specic features of impaired arm movement after stroke. 4.1 Mechanisms of Reaching Variability in the Population Vector Framework. The predicted increase in between-target variability can be explained by the gradual destruction of the set of preferred directions. Removing cells at random from a uniform distribution of preferred directions causes the distribution to become nonuniform (i.e., there is only an innitesimal probability that it will remain uniform; see Figure 4F). Thus, each individual instantiation of virtual cortex (each single simulated subject) had a nonzero mean directional bias for each target, since its population vector was formed from a nonuniform set of preferred directions (see Figure 7). However, the average directional bias across many instantiations of virtual cortex (i.e., across many simulated subjects) was zero for each target (see Figure 7). This is because the virtual subjects were created by eliminating
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Figure 6: Sensitivity of simulation results to noise index IN . Thick dashed line: IN D 0:4, the default value used for the simulations; light lines: IN < 0:2; medium lines: 0:2 < IN < 0:6 dark lines: IN > 0:6. The arrows indicate increasing IN . (A) Sensitivity of between-target variability. (B) Sensitivity of damage-impairment relationship derived from between-target variability. The curve attens for small and large values of the noise exponent d. (C) Sensitivity of within-target variability. (D) Sensitivity of damage-impairment relationship derived from withintarget variability.
directions selected from a uniform probability distribution, and thus the virtual subjects with biases to the left cancelled the ones with biases to the right, on average. The standard deviation of the directional bias taken across many targets for a given virtual subject was, however, nonzero. The mean standard deviation across many subjects was thus also nonzero. The numerical simulations indicated that this standard deviation increased as the population became smaller. In other words, the statistical distribution of directional biases broadened with decreasing numbers of cells alive in the population (see Figure 7). This nding is consistent with previous reports that have demonstrated a decrease in population vector coding accuracy with decreased population size when randomness is included in cell tuning curves or preferred-direction distributions (Salinas & Abbot, 1994; Tanaka, 1994). In contrast, the generation of within-target variability does not require a nonuniform distribution. Thus, the level of within-target variability was
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Figure 7: Histogram of directional errors for the simulations of the multitarget experiment. (Left) Individual virtual subjects displayed a nonzero, mean directional error across the 75 targets (data shown are for 50% cells surviving for a population size of 328 neurons). (Middle) However, when the subjects were averaged, there was a zero-mean directional error, since cell destruction was simulated using a uniform distribution. (Right) The standard deviation of the directional bias (i.e., the between-target variability) increased as the population became smaller (thin, medium, thick lines have 10%, 50%, and 100% cells surviving, and ¾ D 56:0, 19.8, and 6.4 degrees, respectively) while the mean remains zero. At 100% surviving cells, the cell population distribution is uniform, but the distribution of directional errors has a nonzero width because of the ring-rate noise. The bin width for all histograms was 5 degrees.
similar for a uniform population of size N or a nonuniform population of size N derived from a larger starting population (see Figure 4E). Firing-rate variability in individual neurons is essential. Without this noise, a given instantiation of virtual cortex is a deterministic system, and the model generates the same error for each reach to the same target. Even when individual ring-rate variability is incorporated, however, it might be expected at rst glance that the directional variability of the population vector would increase with larger cell populations. When ring-rate noise is included, the population vector is the sum of independent random variables. The variance of a sum of independent random variables is the sum of the individual variances and therefore increases with increasing number of addends.
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The observed decrease occurs because the magnitude of the population vector grows more rapidly than its standard deviation, for moderate values of the noise exponent d. To see this, consider a planar model with uniformly distributed preferred directions (see Figure 8). The population vector is computed as X sin µp Vx N D D V Ra cos µp Vy
(4.1)
µp
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Ra
Thus, the standard deviation increases with increasing population size. However, the standard deviation grows less quickly than the mean magnitude of the population vector, for moderate values of the noise exponent (i.e., parameter d in equation 2.2). Figures 8A through 8C show the mean magnitude of the population vector and its standard deviation ellipse for different size cell populations when the noise exponent equals the value identied for rhesus monkeys (d D 0:6) (Lee et al., 1998). The within-target variability µWT was calculated as the directional error corresponding to the standard deviation ellipse (see Figure 8A). The within-target variability decreases with increasing cell population size because the population vector magnitude grows more rapidly (see Figure 8D). Numerical simulation results verify this analysis (see Figure 8E). As d increases beyond about 2.0, however, the within-target variability no longer depends on the population size (see Figure 8F), because the variability grows faster than the mean population vector magnitude (see Figure 8D). Thus, appropriate levels of ringrate noise are essential to produce the observed dependence of within-target variability on cell death. 4.2 Limitations of the Model and Future Research. One limitation of the proposed model is its lack of anatomical specicity. In particular, is it reasonable to expect that a model nonspecic to lesion anatomy and lacking cortical organizational features such as somatotopy would predict directional errors following diverse kinds of stroke? One possible answer is that stroke is rarely precisely localized; it typically affects both gray and white matter and does not respect traditional neuroanatomical boundaries. Widespread destruction of neurons in multiple brain areas following stroke may act on average to degrade a distributed population vector code for controlling
D. Reinkensmeyer, M. Iobbi, L. Kahn, D. Kamper, and C. Takahashi
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Figure 8: Planar analysis of within-target variability. (A–C) Population vector (arrow) and its standard deviation ellipse (circle) for cell populations of varying sizes with uniformly distributed perferred directions. For these panels, the noise exponent (d) was 0.6, and the standard deviation ellipse was calculated using equation 4.2. The within-target variability (µWT ) is the directional error that corresponds to the standard deviation ellipse: one-half the angle that subtends the N where jVj N and ¾V are the magnitude circle, calculated as µWT D arcsin.¾V =jVj/, and standard deviation of the population vector, calculated from equations 4.1 and 4.2. Note that the axis limits are expanded for the larger populations (B and C), but the aspect ratio remains the same. Thus, the individual cell vectors are visible in A but become illegible in B and C. In addition, the standard deviation ellipses are larger for the larger populations, although the scale change obscures this growth. The within-target variability decreases with increasing population size because the mean magnitude of the population vector grows more rapidly than the standard deviation of the vector. (D) Both the population vector standard deviation (solid line) and the mean magnitude of the population vector (dashed line) grow with increasing number of cells, but the population vector grows more rapidly for noise exponents less than approximately 2. The population vector standard deviation is shown for d D 0:6; 0:9:1:2; 1:5; 1:8 (solid curves with increasing slopes). (E) The prediction of equation 4.2 compares well with simulation results (dashed line). (F) Variability range is dened as the difference between the maximum and minimum variability for cell populations ranging from 10 to 1000 cells—the vertical span of the curve in E. Within-target variability depends most on the population size for moderate noise exponent values.
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hand movement direction. The distributed nature of motor representations is increasingly recognized (Kalaska & Crammond, 1992). For example, multiple distributed representations of the hand have been demonstrated within primary motor cortex (Strick & Preston, 1982), and areas controlling even individual ngers are distributed throughout the hand region of primary motor cortex (Schieber, 1999). Investigation of bilateral pyramidotomy in monkeys suggested that recovery was better predicted by percentage of the bers damaged than by location of the lesion (Lawrence & Kuypers, 1968). Population vector codes have been found to adequately predict hand movement direction for the primary motor cortex (Georgopoulos et al., 1982, 1988), premotor cortex (Caminiti et al., 1991), area V of the parietal cortex (Kalaska et al., 1983), and the cerebellum (Fortier et al., 1989). If many brain regions contribute in parallel to hand movement control, it is possible that stroke will on average damage directional control of reaching in a manner similar to the model predictions. Furthermore, preferred directions do not appear to be topographically organized in a straightforward, modular manner (Ben-Shaul et al., 2003). Thus, it is unlikely that even focused lesions would produce destruction of a narrow range of preferred directions; rather, stroke likely destroys a broad range of preferred directions, consistent with the approach used in the present model. Another possible limitation of the proposed model is that the variability levels it predicts depend on the population size. The corticospinal tract alone is estimated to contain about 1 million bers (Rothwell, 1987). Using a population size of 1 million cells in the current model would be expected to produce negligible change in variability levels across large percentages of cell death (cf. the simulation with 83,442 initial cells; see Figure 3). However, not all cells in the corticospinal tract control arm movement, and not all arm movement cells are independent. Recordings from multi-electrode arrays indicate correlations in ring patterns between neighboring cells (Lee et al., 1998; Maynard et al., 1999). Training of hand movements in primates has been shown to increase the synchronization of cortical cell ring (Schieber, 2002). Results from the development of direct brain interfaces for movement control suggest that spike patterns from several hundred neurons (Wessberg et al., 2000), or even just tens of neurons (Serruya, Hatsopoulos, Paninski, Fellows, & Donoghue, 2002), are sufcient to specify movement direction with some accuracy. Thus, the cortex may perform the specic task of controlling movement direction or reaching with effectively only hundreds or thousands of independently operating neural groups, each of which may in turn consist of many neurons with correlated preferred directions and ring patterns. A striking result of the model is that the derived relationships between cell death and clinical impairment level in Figures 2C and 2F have similar forms, even though they are derived from completely different types of variability (between-target and within-target variability). Clinical scores of motor impairment have been found previously to depend strongly on
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motor cell death. For example, the amount of degeneration of descending bers in the cerebral peduncles was strongly correlated with a three-point score of motor function (Warabi, Inove, Noda, & Murakami, 1990). Axonal injury in the posterior limb of the internal capsule assessed via magnetic resonance spectroscopy of N-acetylaspartate, an index of axonal loss, was also strongly correlated with a clinical score of motor impairment following stroke (Pendlebury, Blamire, Lee, Styles, & Matthews, 1999). The current model predicts that the larger the starting population is, the more damage can occur before impairment becomes evident (cf. Figure 3). A third possible limitation of the proposed model is the validity of the population vector movement control hypothesis itself, which forms the basis for the model. Cortical control of movement is incompletely understood, and several theoretical and experimental studies have challenged the concept that the cortex controls hand movement direction using a population vector approach (e.g., Mussa-Ivaldi, 1988; Sanger, 1996; Scott, Gribble, Graham, & Cabel, 2001; Scott & Kalaska, 1995; Todorov, 2000). For example, the model relies on a uniform distribution of preferred directions for accurate prediction of hand movement direction. Recent studies have found nonuniform distributions, particularly when movements are performed with abducted arm postures, producing small but systematic deviations between the population vector and actual movement direction (Scott et al., 2001; Scott & Kalaska, 1995). However, measurement of a nonuniform population does not eliminate the possibility that the nervous system controls movement by drawing from a uniform subpopulation (Scott et al., 2001), and thus this issue awaits further elaboration. Another challenge to the population vector approach is that cortical cell ring rates vary not only with movement direction and speed but also with applied force levels (Evarts, 1968), suggesting that the cortex does not explicitly control hand velocity. These observations have led to the suggestion that the population vector species muscle activation directly and that muscle and environmental mechanics dictate the observed ring-rate correlations (Todorov, 2000). The model used in this article is amenable to this formulation and would be expected to produce similar results, since the population vector is equivalent within a scale factor to the net muscular force driving the arm within this direct cortical control formulation. Another limitation of the model is that it ignores possible contributions of weakness to impaired directional control of reaching. For example, individuals with stroke typically have difculty moving their arms into workspace areas more distal from their bodies, likely due to reduced shoulder exion and abduction strength (Kamper et al., 2002). The proposed model should be restricted to workspace areas where such gross impairments are not present. An interesting direction for future research is to examine the relationship of residual cell population size to muscle force production in the context of the direct cortical control paradigm (Todorov, 2000). In such a paradigm, smaller residual cell populations, coupled with arm musculoskeletal me-
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chanics, may result in distal workspace decits similar to those seen after stroke, as the reduced population vector magnitude corresponds to reduced net muscle force. Put another way, the current study focused only on directional errors; decreases in population vector magnitude with cell destruction may correspond to strength or velocity limitations. A nal problematic aspect of the model is that it predicts that stroke subjects are intrinsically able to move with zero-mean directional error for any direction, as long as some cells remain that span the set of all movement directions. Specically, assume a stroke subject is asked to move in direction D1 and instead moves in direction D2. The model predicts that selection of another desired direction D3 will cause an (erroneous) movement in direction D2. The question arises as to why individuals with stroke do not simply remap their input directions to achieve the desired output directions. One possible answer is that perhaps some can, but it takes preservation of specic neural systems and intensive rehabilitation therapy. Also, there is some interesting but preliminary data that stroke subjects can be taught to correct for systematic errors in directional control rather quickly using appropriately designed robotic force elds (Patton, Mussa-Ivaldi, & Rymer, 2001; Raasch, Musse-Ivaldi, & Rymer, 1997). Specically, if a force eld is applied that amplies a repeatable directional error (i.e., pushes the hand even farther away from the target), then some subjects appear to be able to adapt to the force eld across tens of movements and return to their previous directional error levels. When the eld is unexpectedly removed, the subject then moves more directly to the target (i.e., a benecial “aftereffect” occurs, which reduces the original directional error). This suggests that the subject was intrinsically capable of generating the muscle activation patterns needed to move directly to the target but perhaps was not volitionally capable. The neural processes associated with implicit motor adaptation seem to establish access to the correct muscle activation patterns, although preliminary indications are that the benecial aftereffect lasts for only tens of trials, not permanently. Thus, it may be that there are fundamental limitations in the ability to volitionally remap sensory motor mappings for directional control following many strokes. The high levels of trial-to-trial variability associated with severe stroke may contribute to these limitations, as they are essentially “noise” on any kinematic error signal driving adaptation.
Acknowledgments This work was supported by Whitaker Foundation Biomedical Engineering Research Grant RG-98-0004, Public Health Service Research Grant M01RR00827 from the National Center for Research Resources, and the UCI Undergraduate Research Opportunities Program.
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