INSERM U483, Université Pierre et Marie Curie, 9 quai Saint-Bernard, F-75005 Paris, France, 1 Present address: Institute of Neurology, Queen Square, London, UK
Emmanuel Guigon, INSERM U483, U.P.M.C., Boîte 23, 9, quai Saint-Bernard, F-75005 Paris, France. Email: guigon{at}ccr.jussieu.fr.
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Abstract |
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Introduction |
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Burnod et al. and Bullock et al. (Burnod et al., 1992; Bullock et al., 1993
) have shown that a linear superposition of independent solutions of the vectorial visuomotor mapping obtained at each arm position by Hebbian synaptic changes leads to the correct transformation. However, these models rely on the construction of a code in which neurons are tuned to specific postures of the arm (a labelled-line code), which is necessary for Hebbian learning to work appropriately. There are two main difficulties with such an approach. First, models using tabular representations suffer in general from exponential complexity, i.e. the number of necessary neurons increases exponentially as the required precision or the number of degrees of freedom increases (Atkeson, 1989
; Olson and Hanson, 1990
). Secondly, there is no experimental evidence for a labelled-line representation of arm position. In fact, single unit recordings at different levels of the somatosensory pathway reveal that neuronal discharge is modulated by static limb posture in a monotonic fashion, with saturation at extreme joint angles [reviewed by Helms Tillery et al. (Helms Tillery et al., 1996
)].
An alternative method is to use a function-approximating network, i.e. a neural network model which uses some optimization-based learning algorithm to approximate any inputoutput function. In the present case, basis functions could be constructed as the product of (monotonic) arm position detectors and visual direction detectors (Fig. 1A), and their combinations could be made to converge toward a set of command units through weights that can be adapted by error correction (Pouget and Sejnowski, 1994
). Learning can become unsupervised and local if training examples from the desired mapping are actually samples of the inverse mapping (Kuperstein, 1988
; Burnod et al., 1992; Bullock et al., 1993
; Salinas and Abbott, 1995
). The main criticism against this model concerns the locus of adaptation: learning occurs through a synaptic reorganization at the level of motor commands. In fact, studies involving the learning of new visuomotor transformations emphasize the existence of a proprioceptive component in the adaptation process (Redding, 1978
; Welch, 1986
; Inoue et al., 1997
). For instance, optical rotation during a visually guided pointing task induces adaptation accompanied by activation of the postcentral gyrus (Inoue et al., 1997
).
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Part of this paper appeared as a conference proceeding (Baraduc et al., 1999).
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Materials and Methods |
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Model of the Arm
A planar, two-link right arm, with limited (160°) joint excursion at the shoulder and elbow was used (Fig. 2; in the following figures, unless otherwise specified, the workspace of the arm is set so that full extension of the shoulder and elbow corresponds to the horizontal on the page).
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Arm Position
The form of proprioceptive inputs is crucial to the functioning of the network (Appendix A). These inputs must have two properties. The first is monotonicity, which is dictated by experimental observations (Helms Tillery et al., 1996) (see also Introduction). The second is nonlinearity, which is required by nonlinear variations of the Jacobian matrix of the coordinate transformation with arm position. Nonlinearity may not be found in static muscle spindles since a fusimotor drive can adjust the range of receptor sensitivity and avoid saturation, but it is present in somatic neurons in the form of variable recruitment thresholds and saturations (Tanji, 1975
; Gardner and Costanzo, 1981
) or more complex dependences (Helms Tillery et al., 1996
). We have chosen variable-threshold linear saturating functions as a model of proprioceptive input (see below). Other nonlinear functions would be applicable [e.g. a single (lower or upper) saturation]. In this representation of proprioception, some neurons signal a highly restricted range of posture at extreme joint angles. Simulations show that removing these neurons leads to degraded performances at the border of the workspace. Thus, afferents from articular receptors which actually discharge at extreme angular positions (Clark and Burgess, 1975
) could provide appropriate information for using full ranges of joint angles.
In the model, limb position was represented by the population activity of Np proprioceptive neurons coding for the lengths of agonist or antagonist muscle at the shoulder and elbow (Fig. 2). To avoid unnecessary complexities, the mechanics of the muscle fiber was likened to a rope-and-pulley system.
Here and in the following, the activity of a model neuron is equated with its mean firing rate. To each of the four muscles corresponded the activity of the same number of units (Np/4). The firing of a proprioceptive neuron k (noted pk) was defined by a piecewise linear sigmoid of muscle length [for details, see Baraduc et al. (Baraduc et al., 1999)]. Recruitment thresholds were set so that any muscle stretch is signaled by at least one (moderately) strong activity and so that no neuron (except those recruited at extreme muscle lengths) fires for a narrow band of length.
Desired Direction of Movement
Psychophysical and electrophysiological studies suggest that arm movement trajectories are initially specified by the direction and amplitude of the hand-to-target vector (Gordon et al., 1994; Georgopoulos, 1995
; Vindras and Viviani, 1998
). The frame of reference and coordinate system in which the vector is represented are still debated. In the model, movement direction was described in Cartesian coordinates by a unit visual vector V parallel to the hand-to-target vector. The term visual employed here is conventional and gives no information on the origin of the directional signal. The vector V entered the network as a distributed neuronal representation over a set of Nv unit vectors, Vj, uniformly distributed in Cartesian space
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Representation of desired movement direction as the handtarget vector in a body-centered Cartesian reference frame was motivated by its simplicity but is not essential. In fact, the network implements a generic vectorial coordinate transformation scheme, and the coordinate system of the visual input can be changed (e.g. to an oculocentric code) as long as this input remains vectorial.
Motor Command
Reaching movements are produced by complex coordinated patterns of muscular activity. Descending commands that drive the arm toward a target are elaborated in part by motor cortical circuits (Hoffman and Strick, 1995) and, based on anatomical and physiological arguments, their initial effect can be described as a weighted combination of muscle activations (Schwartz et al., 1988
). These activations result in angular displacements, which can be taken as the initial contribution of a command. This functional representation of the motor command relies on the hypothesis that the command system is linear, that is, the resulting effects of commands combine vectorially. We make the assumption that sources of nonlinearity that exist between the command level and arm displacement can be suppressed by dedicated mechanisms (Bullock and Grossberg, 1991
).
In the model, commands were emitted by a layer containing Nc neurons, which contributed to the initial direction of movement by a displacement along a direction in joint space. The individual influence of a command neuron is proportional to its discharge level. The collective effect of the layer is
![]() | ((1)) |
The model authorizes arbitrary variations in command directions with posture, as long as the Jacobian matrix of the transformation (see Appendix A) can be correctly approximated from proprioceptive inputs. The choice of the CDs influences the distribution of directional properties of command neurons (preferred directions; PDs) and the way PDs change with arm position (equation A4). It is thus constrained by experimental observations. First, PDs are uniformly distributed in a central part of the workspace (Caminiti et al., 1990
, 1991
). Secondly, PDs shift in an orderly fashion with the upper arm (Caminiti et al., 1990
, 1991
; Sergio and Kalaska, 1997
). Thirdly, population vectors calculated at a remote posture deviate from movement direction (Scott and Kalaska, 1995
). The use of CDs which are invariant in angular space is appropriate to meet these requirements (see Results).
In the model, the CDs were independent of posture and selected such that the distribution of PDs is uniform for a central position of the arm (Pref, Fig. 2). The method is explained in Appendix B.
Functioning of the Network
The theoretical principle of the model was implemented in the neural network shown in Figure 3. The structure and functioning of the network are described and then compared to the theory.
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![]() | ((2)) |
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In a second step, the activity in the somatic layer was combined with the visual directional information in an Nc x Nv multimodal layer (Fig. 3). The multimodal layer realized a recurrent thresholded additive somatovisual combination, which approximates a multiplication (Salinas and Abbott, 1996
)
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Third, command neurons summed rows of the multimodal layer
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The network implementation differs from the theory by one aspect. Exact multiplication between somatic and visual information was replaced by a pseudo-multiplication. There are two reasons for this: (i) there is no need to invoke cellular or subcellular mechanisms for neuronal multiplication; and (ii) the multimodal layer has interesting and realistic physiological properties which remain hidden at a dendritic level if sigma-pi neurons (neurons that compute the sum of the products sijvj) are assumed to combine visualand proprioceptive inputs.
Training
The network was trained by correlating its motor commands with the visual effect of the movement (motor babbling) (Kuperstein, 1988; Bullock et al., 1993
). For a given starting position of the arm, movements were made in random directions. The directions in joint space (as given by an efferent copy of the command) were then associated with directions in visual space (visual feedback) and the current arm position. The requirement of uniformly distributed training examples in visual space (equation A6
) is problematic since these examples are not chosen freely but result from random commands in angular space. This constraint is relieved by replacing equation (A10)
by a different learning rule (Baraduc and Guigon, 2001). The learning scheme translates into the algorithm, which is a repetition of the following cycle:
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![]() | ((5)) |
Training was stopped once the mean absolute error measured at the five positions and for 16 uniformly distributed movement directions stabilized. It took ~20,000 iterations.
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Results |
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We first tested if the network solves the coordinate transformation with reasonable accuracy. We calculated the direction of the movements produced by the network for each of 16 uniformly distributed directions in Cartesian space. This was done for 21 starting positions of the arm (recall that one position in space corresponds to only one position of the arm). The results are displayed in Figure 4. The bold arrow shows the initial direction of the hand in response to the desired direction 0°. Accurate learning should result in an isotropic distribution of the arrows (as are the desired directions in Cartesian space); their deviation from the desired direction can be grossly appreciated by judging the horizontality of the reference arrow (bold).
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Global performance depends on the size of the learning set. Learning in only one position was clearly not sufficient to obtain a correct behavior on the whole workspace. However, learning in the 21 test positions did not give the best results. Trying to reduce the errors on an extreme position (such as when the arm is nearly fully extended) deteriorated the performance at the center of the workspace. As movements are generally executed in front of the body, the learning positions were placed in the central zone. Using more than five positions in this zone did not lead to a significant improvement, so this number was retained for all simulations.
The distributed coding of the information in the network was expected to provide robustness to lesion and noise. This was confirmed by random suppression of units or addition of noise to the inner layers. This robustness indicates that performance is not due to the selectivities of critical neurons whose death would be fatal to the operation of the network.
Single Neuron Discharge
We will now focus on the firing properties of isolated model neurons. Discharge of the somatic layer units over the work-space was analyzed to understand the nature of the somatic recoding. For multimodal layer units, directional tuning revealed no surprise as they derived directly from the properties of the visual layer neurons. We thus only delineated their quasi-multiplicative behavior. In the command layer of the network, directional selectivities were studied for each unit. As in numerous experimental studies, they were computed in Cartesian space through a multilinear regression.
Three vectors can be associated with single command neurons. The first is the preferred direction, defined by the direction of the hand for which the neuron fires maximally. The second vector is the command direction (CD, introduced in Materials and Methods): it is defined as the direction in joint space in which the cell drives the arm. For example, a command neuron equally facilitating the shoulder and elbow flexors will have a CD oriented at 45° in {shoulder, elbow} joint space. Lastly, the same command cell has a direction of action (DA). This DA is the direction in extrinsic Cartesian space in which the cell displaces the hand. This direction obviously depends both on the CD and on arm posture: the effect of an elbow flexion changes with the orientation of the upper arm. It is generally believed that the PD of a neuron is the direction in which this cell alone would produce the hand movement (the DA); however, this need not be true.
Proprioceptive Neurons
The properties of proprioceptive neurons are close to those of a fraction of somatic neurons (see below). They will not be described further.
Somatic Neurons
The activity of somatic layer neurons changes in a monotonic fashion with hand position over the whole workspace. It should be emphasized that the model equations do not force the somatic unit to fire monotonically. Indeed, the weights Wijk are not constrained to be positive, and a broad selectivity for a given arm position could potentially emerge. This was, however, never the case.
Six broad classes of neurons can be delineated. Determination was done de visu and is only an attempt to structure a continuum of response properties, as in physiological studies. In the Off-center radial (Fig. 5A) as in the Radial (Fig. 5B
) type, the discharge increased or decreased in concentric rings, roughly centered at the shoulder, or at the position of the elbow when in full extension. The activity was roughly a weighted combination of shoulder and elbow angles. Other units behaved the same way, but were silent in the central part of the workspace and fired only for large flexion (Left, Fig. 5C
) or extension (Right, Fig. 5D
) of the shoulder. These features reflect the properties of proprioceptive inputs (Table 1
).
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Changes in discharge frequency with arm position (positional selectivity) are generally quantified by a preferred axis, i.e. the direction of displacement that leads to a posture at which the discharge frequency is maximal (Kalaska et al., 1983; Kettner et al., 1988
). We calculated distributions of positional selectivities for the somatic units for comparison with observed distributions in sensorimotor regions. Multiple regression analysis was used to calculate best fitting planes over the workspace (Kalaska et al., 1983
). Joint angles were restricted to 20140° to avoid nonlinear effects at extreme positions and allow comparisons with experimental data which generally concern limited parts of the workspace. Mean R2 over the population was 0.49 (n = 2500). The distribution of positional selectivity, calculated over neurons for which R2 > 0.7 (n = 904, 36%), was bimodal, with a preferred axis along 40220°.
Multimodal Neurons
Multimodal neurons combine in a pseudo-multiplicative way the activities of somatic and visual neurons. As the somatic units exhibit monotonic firing profiles, the global effect on multimodal neurons is roughly a gain field. Figure 6 illustrates more precisely this pseudo-multiplicative property. The multimodal neuron has the same preferred direction as its visual input neuron, and its peak activity scales linearly with the somatic input, as for an exact multiplication. However, this multimodal interaction does not reduce to a gain effect as an increase in the somatic input leads to a decrease in the multimodal neuron discharge for the non-preferred direction. In other words, the tuning width of such a neuron is not fixed, but depends on arm posture. The visualproprioceptive interaction in such a multimodal neuron can thus be described as an arm position-dependent modulation of visual selectivity.
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As expected from the theory (equation A3), command neurons were broadly tuned to movement direction in Cartesian space. This is shown for one neuron in Figure 7A
. At each of the 21 tested positions, 95% of the neurons were directionally tuned (linear regression; mean R2 = 0.93). Preferred directions rotated with the upper arm, as shown for the same neuron in Figure 7
. In this case, the PD shifted clockwise as the arm extended.
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The same results were true for the population. For a half-extended elbow, the PDs closely followed the rotation of the shoulder (Table 2). We tested the model on the paradigm used by Sergio and Kalaska (Sergio and Kalaska, 1997
). Shift of PDs between a central position (Pcen) and eight peripheral hand locations uniformly distributed over a circle of 8 cm radius was calculated. Shifting was clockwise (mean 9°) for rightward targets and counterclockwise (mean 6°) for leftward targets. These results are qualitatively consistent with those of Sergio and Kalaska (Sergio and Kalaska, 1997
).
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Results obtained in the command layer at the single cell level were collected to obtain a view of the population activity.
Neural Population Vector
A population vector, defined as
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Movement Direction
Actual movement direction is in general close to the desired direction, while the NPV errs substantially (Fig. 9). The behavior of the NPV is dictated by the distribution of PDs. Errors in NPV are found whenever the distribution is not uniform (Figs 8A and 9A
). These errors do not preclude a correct calculation of movement direction. This is explained for a theoretical case in Figure 10
. We generated theoretical distributions of PDs (equation A4
) and DAs (equation A5
) for a 20-neuron output layer (to obtain a more legible plot). Arm position was chosen to obtain a clear-cut effect. The distributions are shown in Figure 10A,B
. Reconstruction of the NPV for a given desired direction fails since individual contributions of neurons along their PDs tend to cluster (Fig. 10C
). Contributions along the DAs are also not in the desired direction of movement, but they are organized so as to nullify their effect on an axis orthogonal to this direction (Fig. 10D
). In fact, command neurons must fire more to produce the movement in directions where theDA vectors are sparse and short. This links PDs and DAs in such a way that the preferred directions cluster along an axis for which DAs are sparse and thus the PD of a cell is nearly orthogonal to its motor effect (equation B1
).
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Discussion |
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We also need to explain why the observed properties are truly emergent characteristics of the model and not simple consequences of the specific architecture of the network. Discharge properties of somatic units are a by-product of the acquisition of the vectorial visuomotor transformation. Multimodal units are constrained to perform a multiplication, but their modulation by arm position derives from the somatic activities. Last, the discharge behavior of command units is dictated by the non-trivial relationship between the PDs and CDs.
Comparison with Experimental Data
Performance
The network calculates correct commands over most of the workspace. Poorer performances are found in regions where the transformation is strongly position-dependent. Furthermore, there is a trend for clockwise errors in the right part of the workspace and counterclockwise errors in the left part. This pattern of errors is not a characteristic of the model, but only of the simulations (the theory predicts perfect performances). The errors are due to imperfect approximation of nonlinearly varying coefficients of the Jacobian matrix from quasi-linear proprioceptive inputs, and could be reduced using a more complex proprioceptive code, e.g. multiarticular inputs. It is interesting that misrepresentation of arm position has been invoked to explain similar biases during reaching movements (Ghilardi et al., 1995).
A property of the model is its capacity to learn a global transformation from a restricted training set and thus to generalize appropriately to untrained positions. Where does this property come from? To simplify, suppose that the coefficients of the Jacobian matrix are linear functions of muscle lengths, and all response functions in the network are linear so that reconstruction of a coefficient of the Jacobian matrix from positional inputs following training (equation A8) is linear, whatever the training positions. In fact, knowledge of the Jacobian matrix at a restricted number of positions (i.e. enough to identify the coefficients of the linear relationship) is sufficient to determine how the coefficient of the Jacobian matrix varies with the muscle lengths. Thus the model can learn the appropriate matrix with only a few training positions and exhibits strong generalization capacities. A similar phenomenon occurs in the network, limited by the nonlinearities present at various stages. Thus the generalization comes from the structure of the somatic layer. Its lateral connections that contribute to the accuracy of the population computation play only a minor role in the generalization process per se.
Trying to learn the mapping for each arm posture leads to a decrease in the overall performance. This reveals a destructive interference when learning at the extremes of the arm range. Interestingly, the performance of human subjects follows a similar pattern: training on the right side of the workspace to improve locally the visuomotor mapping leads to a mean increase of the angular error over the workspace (Ghilardi et al., 1995).
Somatic Neurons
The somatic layer builds a new representation of arm configuration from proprioceptive inputs. However, the shape of somatic receptive field (sRFs) is defined by constraints of the visuomotor transformation, but not by the proprioceptive information (see equation A2). As the coefficients of the Jacobian matrix, the theoretical sRFs vary in a monotonic way over the workspace: no broad selectivity for a given arm position indeed emerges in the somatic layer.
Possible cortical regions containing neurons similar to somatic units are primary somatosensory cortex (Gardner and Costanzo, 1981; Cohen et al., 1994
; Prud'homme and Kalaska, 1994
; Helms Tillery et al., 1996
), anterior parietal cortex (Kalaska et al., 1983
; Lacquaniti et al., 1995
), motor cortex (Kettner et al., 1988
; Caminiti et al., 1990
) and premotor cortex (Caminiti et al., 1991
), but also earlier stages in the somato-sensory pathway (Bosco et al., 1996
). An observed difference between the cortical areas is related to their sensitivity to changes in arm position. Distributions of positional selectivities are uniform anterior to the central sulcus (Kettner et al., 1988
; Scott and Kalaska, 1997
), but are biased along an anterior posterior axis in the somatosensory and parietal regions (Kalaska et al., 1983
; Cohen et al., 1994
; Prud'homme and Kalaska, 1994
; Helms Tillery et al., 1996
). Our results showing that the distribution of selectivities in the somatic layer is actually biased along this axis suggest that the somatic layer may be located in anterior regions of the parietal cortex.
A critical test of the model would involve showing that, in monkeys, single neurons modulated by static arm position but unmodulated by visual directional information change their discharge behavior following rotation of the optical display (optical tilt). This is a difficult, but feasible experiment (Wise et al., 1998).
Multimodal Neurons
The multimodal layer contains neurons which are broadly tuned to movement direction and modulated by arm position. The modulation was characterized by an absence of shift in preferred direction and a monotonic effect (gain field) on discharge. This fixed PD is an immediate consequence of multiplying a cosine tuning function (visual) with a monotonic one (somatic) and thus is clearly due to both the preset properties of the model and the emergent features of the somatic layer. Similar neurons have been reported recently in a study of motor cortical cells during wrist movements (Kakei et al., 1999). Our model provides insights into the origin of this discharge behavior and its role in visuomotor processing.
Command Neurons
Command neurons are broadly tuned to movement direction and their PD changes in an orderly fashion with shoulder angle (Caminiti et al., 1990, 1991
; Sergio and Kalaska, 1997
) [for wrist movements see also (Kakei et al., 1999
)]. This is an immediate consequence of the model of command neuron activity. The same principle (combination of nonshifting gain fields and shifting receptive fields to compute coordinate transformations) is found in Salinas and Abbott (Salinas and Abbott, 1995
). In fact, our theory of coordinate transformations (Baraduc and Guigon, 2001) is an extension to the multidimensional case of the one-dimensional case developed by Salinas and Abbott (Salinas and Abbott, 1995
).
Variations in arm posture not only cause rotation of the PDs but more generally modulate their distribution. There is no direct experimental evidence for this. Scott and Kalaska (Scott and Kalaska, 1997) compared distributions of PDs for a natural and an abducted posture. They found that both distributions were nonuniform, although it is generally reported that the former is uniform (Schwartz et al., 1988
; Caminiti et al., 1990a) [but see (Georgopoulos et al., 1982
)]. According to the model, these discrepancies could result from the influence of arm position on PDs.
The model raises a dissociation between the preferred direction of a unit and its direction of action. This is the motor formulation of the distinction between receptive and projective fields in vision (Lehky and Sejnowski, 1988). Neuronal receptive fields are shaped by the spatial distribution of the afferent synaptic weights. In contrast, their projective fields are shaped by the weight distribution of their efferent projections to another layer of retinotopic neurons. Informally, the projective field characterizes the meaning of the cell's discharge for the down-stream layers. Here and similarly, the PD is a property of the receptive field of a command neuron, whereas the DA is the main descriptor of its projective field. The relationship between PDs and DAs is dictated by the choice of the CDs (equations A4 and A5). Accordingly, the two directions need not be the same in general and are not same in the present case. In fact, CDs could be crafted to obtain the same distribution of PDs and DAs. However, this would lead to inappropriate patterns of shift in Pds with arm position. There can be only indirect experimental support to our result since directions of action are not easy to measure in vivo (Lemon, 1988
). One supporting argument is discussed below in relation to the population vector. More generally, the measure of PD distribution could give a hint on the distribution of DAs: it could be checked if the latter is compatible with the known anatomical and physiological properties of the muscles.
Population Vector
Reconstruction of movement direction from neuronal activities relies on the assumption that the neurons contribute to movement along their preferred direction (Georgopoulos et al., 1986), that is, the PDs are the DAs. Since PDs and DAs were not the same in our model, deviations of the NPV from the movement or target direction were observed, particularly at extreme arm postures. We noted previously that an adequate choice of the CDs could equate PDs and DAs and thus remove the errors of the NPV. Using equations (A5) and (A4), we see that the CDs should vary with arm position to satisfy
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Scott and Kalaska (Scott and Kalaska, 1995) reported that the population vector of motor cortical neurons calculated at an unnatural arm posture deviated from the movement direction whereas the same vector calculated at a natural posture was correct. This is an immediate consequence of the nonuniform distribution of PDs (Salinas and Abbott, 1994
). This result could also be interpreted in terms of the dissociation between PDs and DAs suggested by the model. However, alternative interpretations would need to be considered before drawing firm conclusions.
Redundancy
Our learning scheme is a case of direct inverse modeling (Jordan and Rumelhart, 1992), that is, the transformation is learned from samples of its inverse. This technique may be unable to find an appropriate solution for a nonlinear one-to-many mapping since the mean of correct outputs is not necessarily a correct output itself [convexity problem (Jordan and Rumelhart, 1992
)]. There is no such problem here as the mapping is linear. The difficulty of the redundant case is to find a proprioceptive code which efficiently discriminates arm postures. Simulations show that (i) the representation used here is not sufficient; and (ii) non-linear interactions between afferents from different articulations greatly improve the neural coding of posture and allow the visuomotor transformation for a redundant arm to be learned. Details can be found elsewhere (Baraduc, 1999
).
Learning and Locus of Adaptation
The learning rule employed here belongs to the family of error-correction rules. However, it is also an unsupervised rule since there is actually a single source of information for learning [random outputs (Hertz et al., 1991)] and the error term can be computed by the network. The target activity of somatic neurons is the product of visual and command signals, which are available during the training period. A possible biological implementation involves a layer of error units (error layer) that calculate the difference between required and actual somatic activity (Fig. 3
). These neurons would be strongly active during early phases of training and would dictate the postsynaptic activity of somatic neurons. Other implementations could also be considered as there are no experimental data to assess the existence of error units. Note that the visual signal can be either the actual visual effect of the command or a predicted effect provided by an internal (forward) model of the direct mapping (C
V). This mapping is a position-dependent linear transformation which can be learned in the same way as the inverse mapping.
Functional Representation of Directional Visuomotor Transformations
A neuroimaging study showed that a region of the superior parietal lobule activates during early exposure to optical tilt whereas the postcentral gyrus is active during late exposure (Inoue et al., 1997). Since the acquisition curve of tilt adaptation is a negatively accelerated exponential (Ebenholtz, 1966
), a possible interpretation relates the parietal activity to error reduction (in the error layer) in the rapidly varying phase of the curve and postcentral activation to consolidation of learning. The exact significance of the latter activation is unclear. Inoue et al. (Inoue et al., 1997
) trained their subjects for 12 min, even though asymptotic performance during adaptation to optical tilt is reached after 12 h (Welch, 1986
). Thus the cerebral activation might well reflect an ongoing adaptive process instead of a steady state. This interpretation is consistent with a recoding of postural information in a somatosensory region, as used in the model.
Our model can be compared to an approach relying on basis functions (Poggio, 1990; Pouget and Sejnowski, 1994
, 1997
; Salinas and Abbott, 1995
). A common property is the ability to learn directional visuomotor transformations from a restricted training set. In both cases, the network contains three types of broadly tuned neurons: (i) not modulated by arm position; (ii) modulated by arm position with nonshifting PD; and (iii) modulated by arm position with shifting PD. Dissociation of PDs and DAs is a feature of the two types of models. There is, however, a significant difference between the models. Learning is a reconfiguration of inputs in one case and of outputs in the other. This makes little actual difference as long as a single adaptation is used. Whenever several distortions are applied simultaneously (e.g. optical tilt and prismatic deviation), manipulation of the output layer prevents concomitant adaptation to the perturbations. Alternatively, separate recodings of proprioceptive inputs for directional and positional visual information permit simultaneous compensation for tilt and displacement, as expected from psychophysical studies (Redding, 1975
).
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Conclusion |
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Notes |
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Appendix A: Mathematical Bases of the Model |
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![]() | ((A1)) |
Consider distributed neuronal representations v (Nv-dimensional vector) and c (Nc-dimensional vector) of V and C defined by their coordinates
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For a given arm configuration P, equation (A1) defines a linear mapping which can be represented in a distributed manner by the Nc x Nv matrix
![]() | ((A2)) |
From the point of view of neural networks, the matrix J(P) can be considered as weights between an input layer (Nv neurons) and an output layer (Nc neurons). From equation (A2), the activity of an output neuron i is
![]() | ((A3)) |
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![]() | ((A4)) |
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![]() | ((A5)) |
The distributed representation J(P) can be constructed by Hebbian learning from a set of Nex training pairs of unit vectors {V,C
= J(P)V
) provided that the input examples are uniformly distributed (Baraduc and Guigon, 2001), i.e.
![]() | ((A6)) |
![]() | ((A7)) |
Since c = J(P)v holds at each arm position P, a general solution at all positions is obtained by considering the matrix J(P) as neuronal activities
![]() | ((A8)) |
![]() | ((A9)) |
![]() | ((A10)) |
In the model presented in this paper, the somatic layer computes matrix J through the weights W: after learning, we have sij = Jij. The role of the multimodal layer is to compute an approximate mij of the products Jijvj. A few details have been modified from the theory presented here. Lateral connections have been added to provide resistance to noise and permit a large saving in terms of adjustable weights, and transfer functions g constrain the firing rates to be positive. These changes from the present mathematical framework do not affect the operation of the network noticeably.
These theoretical derivations are limited here to the case of cosine tuning curves and uniform distributions. The case where the CDs are not uniformly distributed are treated in this appendix; other cases are discussed elsewhere (Baraduc and Guigon, 2001). This theory extends to the case of a redundant arm using the MoorePenrose inverse of the Jacobian of the direct kinematics in equation (A1) (Baraduc and Guigon, 2001). Again the difficulty is in the mapping defined by equation (A8).
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Appendix B: Invariant Command Directions in Angular Space |
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![]() | ((B1)) |
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References |
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