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doi: 10.1152/jn.00279.2012109:1182-1201, 2013. First published 31 October 2012;J Neurophysiol

Hirokazu Tanaka and Terrence J. SejnowskiCartesian spatial coordinatesComputing reaching dynamics in motor cortex with

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Computing reaching dynamics in motor cortex with Cartesianspatial coordinates

Hirokazu Tanaka1,2 and Terrence J. Sejnowski1,3

1Howard Hughes Medical Institute, Computational Neurobiology Laboratory, The Salk Institute for Biological Studies, LaJolla, California; 2School of Information Science, Japan Advanced Institute of Science and Technology, Nomi, Ishikawa,Japan; and 3Division of Biological Sciences, University of California at San Diego, La Jolla, California

Submitted 4 April 2012; accepted in final form 26 October 2012

Tanaka H, Sejnowski TJ. Computing reaching dynamics in motorcortex with Cartesian spatial coordinates. J Neurophysiol 109: 1182–1201,2013. First published October 31, 2012; doi:10.1152/jn.00279.2012.—How neurons in the primary motor cortex control arm movements isnot yet understood. Here we show that the equations of motiongoverning reaching simplify when expressed in spatial coordinates. Inthis fixed reference frame, joint torques are the sums of vector crossproducts between the spatial positions of limb segments and theirspatial accelerations and velocities. The consequences that followfrom this model explain many properties of neurons in the motorcortex, including directional broad, cosinelike tuning, nonuniformlydistributed preferred directions dependent on the workspace, and therotation of the population vector during arm movements. Remarkably,the torques can be directly computed as a linearly weighted sum ofresponses from cortical motoneurons, and the muscle tensions can beobtained as rectified linear sums of the joint torques. This allows therequired muscle tensions to be computed rapidly from a trajectory inspace with a feedforward network model.

motor control; computational model; visually guided reaching; cosinetuning; population vector; spatial representation; joint-angle represen-tation; reference frame

THE MOTOR CORTEX is the final cortical pathway to motor circuitsin the spinal cord. The response properties of neurons in themotor cortex have been correlated with a wide variety ofbehavioral and cognitive variables, including extrinsic, spatialvariables of hand movements such as endpoint position (Geor-gopoulos et al. 1984), endpoint velocity (Georgopoulos et al.1982; Moran and Schwartz 1999), endpoint acceleration (Fla-ment and Hore 1988), and endpoint movement fragments(Hatsopoulos et al. 2007), as well as intrinsic variables such asjoint angles (Thach 1978), joint velocities (Reina et al. 2001),endpoint force (Georgopoulos et al. 1992; Taira et al. 1996),and muscle tensions (Evarts 1968; Fetz and Cheney 1980).Several studies have reported neural activities in the motorcortex consistent with an internal model of muscle forces forboth reaching and isometric force tasks (Sergio and Kalaska1998; Sergio et al. 2005). This has led to an impasse indeciding whether the internal representation in the motor cor-tex is based on extrinsic spatial variables or an intrinsicreference frame.

Many neurons in the motor cortex and other motor-relatedareas exhibit broad, cosinelike extrinsic tuning with respect tomovement direction. Hand movement direction can be recon-

structed with a population vector algorithm either from neuronsthat encode extrinsic endpoint velocity (Georgopoulos 1996;Lukashin et al. 1996; Sanger 1996) or from neurons thatencode intrinsic joint angular velocity (Mussa-Ivaldi 1988).The preferred directions (PDs) of motor cortical neurons ex-hibited a shoulder-centered reference frame when a few partsof the workspace were examined (Caminiti et al. 1990). In amore recent study that systematically investigated the entireworkspace, coexisting multiple reference frames were re-ported, including joint-angle and shoulder-based coordinates aswell as extrinsic coordinates (Wu and Hatsopoulos 2006).Other studies have reported that neuronal activities correlatedwith motor-related cognitive variables such as movement prep-aration (Churchland et al. 2006), mental rotation (Georgopou-los et al. 1989; Lurito et al. 1991), or movement sequences(Carpenter et al. 1999). Given the observed complexity andheterogeneity of neuronal responses, Churchland and Shenoy(2007) have suggested that it may be necessary to discard thenotion that the motor cortex represents movement parameters.

The movements of the arm are governed by muscle forces thatexert torques around joints. Models of arm movements typicallyuse joint-angle coordinates, which require the solution ofnonlinear inverse kinematics (the computation of joint angles)and inverse dynamics (the computation of joint torques) inmotor planning and execution (Flash and Sejnowski 2001).This raises many questions: 1) How are trajectories in Carte-sian space converted by neurons in the motor cortex into jointtorques and muscle tensions (Kalaska 2009)? 2) Are theremultiple reference frames in the motor cortex? 3) How are thedesired joint angles computed from the planned arm trajectory?4) How are the desired torques computed from the joint angles?5) What further transformations are needed to convert thedesired joint torques into muscle tensions (Kawato et al. 1988;Todorov and Jordan 2002)?

For single-joint wrist movements starting from distinct pos-tures there are neurons in the motor cortex that encode wristmovement directions in the extrinsic space or muscle activa-tions, but no neurons have been reported that encode intrinsicmovements in terms of joint angles (Kakei et al. 1999). Formultijoint reaching movements, there are reports of an extrin-sic coordinate system in the motor cortex (Georgopoulos et al.1982), an intrinsic joint-based coordinate system (Scott andKalaska 1997), an intrinsic muscle-based coordinate system(Morrow and Miller 2003), a shoulder-centered coordinatesystem (Caminiti et al. 1990), and evidence against a single-coordinate system representation (Wu and Hatsopoulos 2006).From a computational perspective, the explicit use of a joint-

Address for reprint requests and other correspondence: H. Tanaka, HowardHughes Medical Institute, Computational Neurobiology Laboratory, Salk In-stitute for Biological Studies, La Jolla, CA 92037 (e-mail: [email protected]).

J Neurophysiol 109: 1182–1201, 2013.First published October 31, 2012; doi:10.1152/jn.00279.2012.

1182 0022-3077/13 Copyright © 2013 the American Physiological Society www.jn.org



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angle representation requires computation of the joint anglesfrom the geometry of the arm and the joint torques from theequations of motion (EOMs) (Bullock et al. 1993; Soechtingand Flanders 1989). There are obstacles along this route, firstbecause the computation of joint angles is ill-posed and maynot have unique solutions (Atkeson 1989) and second becausefor moving reference frames based on joint angles the EOMshave many terms (Fig. 1A).

An alternative to the dynamics-based approach is the equi-librium-point hypothesis in a fixed spatial reference frame thatexploits viscoelastic properties of the musculoskeletal systemand solves motor control problems without computing the jointangles and joint torques explicitly (Feldman 1966; Feldmanand Levin 1995). This approach has been extended to thevirtual trajectory control, in which temporally moving equilib-rium points guide the endpoint movements (Bizzi et al. 1984;Hogan 1984), and to the passive motion paradigm, in which thegoal of action is expressed by means of an attractive force field(Mohan and Morasso 2011; Morasso et al. 2010; Mussa Ivaldiet al. 1988). The equilibrium-point control and its extensionsemphasize the control of stiffness for the purpose of stabiliza-tion, whereas the dynamics-based approaches emphasize thecomputation of muscle tensions and explicit dynamical equa-tions. There is no reason why both of these approaches couldbe used by the motor system depending on conditions.

This study addresses the representation problem in the motorcortex for visually guided reaching movements. Several invari-ant characteristics of reaching movements are expressed inspatial variables such as straight paths and bell-shaped velocityprofiles (Abend et al. 1982; Morasso 1981), so it is conceivablethat the motor cortex employs spatial trajectory informationdirectly. We show here that the EOMs for reaching based onthe spatial positions of limbs in Cartesian coordinates (Fig. 1B)are considerably more concise than for joint-based referenceframes and have physically intuitive interpretations (Hinton1984). The dynamics of linked rigid bodies using spatialvectors, known as the Newton-Euler method (reviewed inAPPENDIX A), has been well studied, especially in robotics, but ithas not been used for understanding the functions of motorcortex. The problems in computing the joint angles arise in thejoint-angle representation because only the endpoint position isgiven, which our model resolves by introducing redundantCartesian vectors that are closely related to the joint-anglerepresentation. In robotics and computational modeling ofmotor control, the joint-angle representation is popular becauseit provides a concise description for movement without redun-dancy, but the EOMs expressed in joint angles are substantiallymore complex and scale exponentially with the degrees offreedom. Spatial vectors, on the other hand, yield considerably

A B

C

D

Number of links

Num

ber o

f EO

M te

rms

106

105

104

103

102

101

100

2 3 4 5 6 7 8 9 10

Fig. 1. Motor control in joint-angle and spatialcoordinates. A and B: a link model representedin joint-angle space (A) and in Cartesian space(B). For a 2-link model joint angles (�1, �2) or

Cartesian vectors �X�10, X�20, X�21� were used, andfor a 3-link model joint angles (�1, �2, �3) or

Cartesian vectors �X�10, X�20, X�30, X�21, X�31, X�32�were used. C: the equation of motion (EOM)includes translational (proportional to mass)and rotational (proportional to inertia moment)terms. D: number of terms required in EOMs ofn-link system for joint-angle (solid line) andspatial (dashed line) representations.

1183MOTOR CORTEX SPATIAL REPRESENTATION HYPOTHESIS

J Neurophysiol • doi:10.1152/jn.00279.2012 • www.jn.org



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more concise and structured expressions for the EOM, butconsistency among the spatial vector must be maintained.

The terms in the EOMs in a fixed spatial reference frame arevector cross products between spatial positions and velocitiesor accelerations of the limbs and have many of the propertiesof neurons in the motor cortex. The geometric nature of vectorcross-product terms could explain why broad, cosinelike tun-ing curves are so ubiquitous in motor-related areas (Georgo-poulos et al. 1982; Johnson et al. 1999; Kalaska et al. 1983),although a recent study reported directional tuning consider-ably narrower than cosine tuning and even bimodal directionaltuning (Amirikian and Georgopoulos 2000). Moreover, thevisual system provides input to the motor system in spatialcoordinates, and vector cross products can be computed fromthe spatial representation in the parietal cortex in a feedforwardneural network with a single layer of connections (Pouget andSejnowski 1997).

A model for smooth reaching motions of a three-joint arm ina two-dimensional plane was simulated with a Cartesian spatialrepresentation, and the terms in the model were compared withrecordings from neurons in the motor cortex during similarmovements. There was a close correspondence in the distribu-tion of PDs, dependence on the workspace, and the rotation ofthe population vector. Finally, we show how vector crossproducts allow the muscle tensions for agonist and antagonistmuscles in an arm to be computed based on a minimumsquared tension criterion. In this scheme, instead of the explicitcomputation of joint angles, we introduce a set of redundantspatial vectors for the computation of reaching dynamics.Although this study focused mainly on point-to-point reachingmovements in the horizontal plane, the computational schemealso applies to general movements in three dimensions.

MATERIALS AND METHODS

Equations of motion for a two-link system. The EOMs can bederived in different coordinate systems. We first derive the equationsin the traditional way, using joint angles as the coordinate system, andthen compare the complexity of the equations with those derived withfixed spatial coordinates. To simplify the derivation, we considerdegrees of freedom only in the horizontal plane without gravity. Thederivation here for a two-link system (Fig. 1, A and B) can bestraightforwardly extended to general n-link systems. Another deri-vation based on the Newton-Euler method, which is directly related tospatial coordinates without using joint angles, is given in APPENDIX A.

With the Euler-Lagrange method, the Lagrangian for a two-linksystem is given by

L��i�, ��i�� 1

2�i�1

2

mi�Xi2 � Yi

2� �1

2�i�1

2

Ii��j�1

i

� j2

(1)

where mi and Ii are the ith segment’s mass and inertial moment aroundthe z-axis, respectively. The first and second terms in the Lagrangian are,respectively, the translational and rotational kinetic energies of the sys-tem. The Cartesian coordinates of center-of-mass position of the ithsegment, Xi and Yi, are explicitly determined by the joint angles as

X1 � r1 cos �1

Y1 � r1 sin �1(2)

X2 � l1 cos �1 � r2 cos ��1 � �2�Y2 � l1 sin �1 � r2 sin ��1 � �2�

(3)

where li is the ith segment’s total length and ri is the distance from the(i � 1)th segment endpoint to the ith segment center of mass (Fig. 1A).

The EOMs can be derived with the Euler-Lagrange equation:

�i �d

dt� �L

��i �

�L

��i�i � 1, 2� . (4)

where �i (i � 1,2) are joint torques on the shoulder and elbow,respectively.

With the Lagrangian in Eq. 1, the EOMs for a two-link model in thehorizontal plane in the joint-angle representation are

�1 � �I1 � I2 � m1r12 � m2r2

2 � m2l12 � 2m2l1r2 cos �2��1

� �I2 � m2r22 � m2l1r2 cos �2��2 � m2l2r2�2�1 � �2��2 sin � ,

(5)

�2 � �I2 � m2r22 � m2l1r2 cos �2��1 � �I2 � m2r2

2��2

� m2l1r2�12 sin �2

(6)

These EOMs, which are the joint-angle representation, contain Co-riolis and centrifugal terms because the reference frame defined withrespect to the links changes as the arm moves. The intrinsic joint-angle representation in Eqs. 5 and 6 provides no physical intuitionabout how limb segments move in the extrinsic coordinates. Further-more, physical parameters such as mass (m) and moment of inertia (I)appear in a wide range of combinations. A more physically intuitiverepresentation can be obtained by deriving the EOMs in spatialcoordinates where the physical parameters are more transparent.

If the terms are reorganized with respect to masses and moments ofinertia, we obtain

�1 � m1r12�1 � I1�1 � m2��l1

2 � r12 � 2l1r2 cos �2��1

� �r22 � 2l1r2 cos �2��2 � l1r2�2�1 � �2��2 sin �2�

� I2��1 � �2�(7)

�2 � I2��1 � �2� � m2��r22 � l1r2 cos �2��1 � r2

2�2 � l1r2�12 sin �2� .

(8)Each of these terms can be rewritten as mass or inertia moment timesa vector cross product:

�X�10 � A�10�Z� r1

2�1

�X�21 � A�21�Z� r2

2��1 � �2��X�21 � A�20�Z

� �r22 � l1r2 cos �2��1 � r2

2�2 � l1r2�12 sin �2

�X�20 � A�20�Z� �l1

2 � r22 � 2l1r2 cos �2��1 � �r2

2 � 2l1r2 cos �2��2

�l1r2�2�1 � �2��2 sin �2

(9)

where X�ji�j � i; j � 1, 2; i � 1� and X�j0�j � 1, 2� are the center-of-mass positions of the jth link measured from the ith link endpoint and

from the origin (defined at the shoulder position, 0), respectively (see

Fig. 1B). The velocities (V�ji, V�j0) and accelerations (A�ji, A�j0) are the

1184 MOTOR CORTEX SPATIAL REPRESENTATION HYPOTHESIS




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first and second temporal derivatives of the positions (X�ji, X�j0), respec-

tively. The operator �X��Z extracts a z-component from vector X� andrestricts motion to the X–Y plane. These spatial vectors specify thecenter-of-mass positions and can be computed by appropriately rescalingspatial vectors connecting the joints (shoulder, elbow, or wrist) and theendpoint. Substituting Eq. 9 into Eqs. 7 and 8 yields much simpler andmore structured EOMs expressed in an extrinsic spatial representation:

�1 � �m2X�20 � A�20 �I2

r22 X�21 � A�21 � m1 X�10 � A�10

�I1

r12 X�10 � A�10�

Z

.(10)

�2 � �m2X�21 � A�20 �I2

r22 X�21 � A�21�

Z

. (11)

Note that all the terms in Eqs. 10 and 11 consist of vector crossproducts of position vectors and their derivatives. The physics of thetwo-link model can be read from the equations: Each term propor-tional to a mass represents the translational movement of the center ofmass of a limb segment, and each term proportional to a moment ofinertia represents the rotational movement around the center of massof a limb segment. For example, see Fig. 1C for a graphical expla-nation of the action of �2 (Eq. 11).

Equations of motion for a general n-link system. The EOMs in aCartesian spatial representation for a general n-link system are

�i � �ji

n �mjX� j,i�1 � A� j,0 �Ij

rj2 X� j,j�1 � A� j,j�1�

Z

�i � 1, . . . , n�(12)

and are derived with the Newton-Euler method in APPENDIX A. We listbelow some of their properties in comparison with the joint-anglerepresentation.

1) The EOMs in a spatial representation are shorter and moreconcise than in an intrinsic representation: As n increases the numberof terms required in the joint-angle representation grows exponen-tially, whereas the number of equations for the spatial representationonly increases quadratically (Fig. 1D).

2) Torques that operate on a joint influence the translational androtational movements of segments that are more distal. For example,the equation containing �2 has two terms representing the changes tothe translational and rotational motions of segment 2 (Fig. 1C), andthis generalizes for �1. This intuitive physical interpretation is appar-ent in the spatial representation.

3) When some dynamical inertial parameters such as mass orinertia are modified, the coefficients in multiple terms in the jointrepresentation must be fine-tuned simultaneously. If, for example, themass of the second link m2 is modified, 11 terms in Eqs. 5 and 6 mustaccordingly be adjusted. In contrast, EOMs in the spatial representa-tion (Eqs. 10 and 11) contain only two terms that need adjustment.

4) In a spatial representation the EOMs can be easily generalized tothree dimensions (APPENDIX A):

��i � �j�i�

n �mjX� j,i�1 � A� j0 � I j

X� j,j�1 � A� j,j�1

rj2

� �X� j,j�1 � V� j,j�1

rj2 � �I j

X� j,j�1 � V� j,j�1

rj2 � (13)

where Ij is a 3 � 3 inertia matrix of the jth segment. The terms

quadratic in V� take nonzero values when the angular velocity and theangular momentum are not parallel to each other, and disappear whenthe movement is restricted to the horizontal plane, reducing to Eq. 12.

5) One disadvantage in using a spatial representation is that itexplicitly requires the n(n � 1)/2 center-of-mass positions measuredfrom the origin (shoulder) or from the edge of more proximal

segments. In contrast, the joint-angle representation requires only nindependent angular variables and all values of the joint angles withinthe limits set by mechanical constraints yield a valid hand position.The more concise description of state in the joint-angle representationmakes it attractive for robotics and models of motor control.

6) The redundant set of spatial vectors in our model must beconstrained to provide a valid configuration of limb. In contrast, in thejoint representation, joint angles can provide a concise description oflimb without any constraints.

In summary, although mathematically these two representations areequivalent, they differ in what aspects of computation are emphasized.The joint-angle representation gives a concise, nonredundant basis fordescribing body movements, but the EOMs become complicated andunstructured. In contrast, the spatial representation has a much shorterand structured description for EOMs, with the computational price ofmaintaining consistency between all the spatial vectors.

Contribution of muscle viscosity to the EOMs. The activities ofmotoneurons mainly reflect temporal changes of muscle lengths,which can be approximated by joint-angle velocities, �i. Therefore, inorder for motor cortex to control the arm motion, viscosity terms(Bi�i) must be included on the right-hand sides of the EOMs. Note thatthese viscous terms do not represent a mechanical property of limbsbut rather the change of muscle lengths.

The explicit use of the joint-angle velocities might seem toimply the indispensability of an explicit representation of jointangles given that they depend on the inverse Jacobian matrix J ofthe coordinate transformation:

��

� J�1��, X��V� (14)

Fortunately, the joint-angle velocity has a much simpler spatial

representation composed of vector cross products of position (X�) and

velocity (V�):

�i � �X�i,i�1 � V�i,i�1

ri2 �

X�i�1,i�2 � V�i�1,i�2

ri�12 �

Z

�i 2� (15)

and

�1 � �X�10 � V�10

r12 �

Z

. (16)

The resulting viscous terms

Bi�X�i,i�1 � V�i,i�1

ri2 �

X�i�1,i�2 � V�i�1,i�2

ri�12 �

Z

�i 2� (17)

and

B1�X�10 � V�10

r12 �

Z

(18)

are included in the EOMs. These cross-product representations are anefficient method for computing the joint angular velocities bypassingthe Jacobian matrix and its inverse.

For an n-link system, the ith torque equation in a fixed spatialreference frame is

�1 � �j�1

n �mjX� j,0 � A� j,0 �Ij

rj2X� j,j�1 � A� j,j�1�

Z

� B1�X�10 � V�10

r12 �

Z

(19)

�1 � �j1

n �mjX� j,i�1 � A� j,0 �Ij

rj2X� j,j�1 � A� j,j�1�

Z

� Bi�X�i,i�1 � V�i,i�1

ri2 �

X�i�1,i�2 � V�i�1,i�2

ri�12 �

Z

�i 2�(20)





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where X�j,i and X�j,0 are location vectors in the Cartesian coordinates ofthe jth segment measured with respect to the ith segment endpoint and

to the shoulder and V� and A� are their velocity and acceleration,respectively. Equations 19 and 20 directly relate the kinematic vari-ables (spatial positions and derivatives on the right-hand side) todynamic variables (joint torques on the left-hand side), thereby di-rectly transforming the visual trajectories of limb segments into thejoint torques without computing joint angles. The first acceleration-dependent term on the right-hand side represents the mechanicalproperties of the limb, and the second velocity-dependent term rep-resents the viscous properties of the muscle.

Vector cross products as neuronal firing rates. All of the terms inthese EOMs are vector cross products in a fixed spatial referenceframe, which greatly simplifies the computation of joint torques. Thissuggests that, instead of explicitly computing the joint angles, the jointtorques can be computed directly from vector cross products. We thuspostulate that the terms in the vector cross products would be identi-fied with the firing rates of neurons in motor cortex:

RA �X� � A��Z or RV �X� � V��Z (21)

Thus each term is the multiplicative response of the spatial position ofa limb and a velocity or acceleration, variables that are represented byneurons in motor cortex that have been characterized as either “ac-celeration cells” (RA) or “velocity cells” (RV). Because firing rates ofcortical neurons are constrained to a narrow range, the terms in Eq. 21should not be identified with single neurons but rather with neuralpopulations having similar preferences. The magnitudes of the crossproducts could be implemented similarly as the number of motorcortex neurons encoding the same cross product. The vector crossproducts yield cosine tuning curves (Fig. 2A) and explain a work-space-dependent change of PD of a cell (Fig. 2B). The geometry oflimb configurations determines the distribution of PDs of modelneurons. The assumption that each cross-product term corresponds toactivity in a single neuronal population is consistent with a recent

study showing that single movement parameters dominated individualneuronal activities (Stark et al. 2007).

Network computation of vector cross products. We show here thata simple neural network model can compute vector inner and crossproducts. Let a group of neurons have activities that are multiplica-

tively modulated by a limb position vector, X�, and a limb velocity

vector, V�. Without loss of generality, assume that these vectors lie inthe horizontal plane. Then the firing rate should be given by

Rij�X�, V�� fX��, �i� fV��, � j� (22)

where � and � are hand position and velocity direction, respectively,and �i and �j are the preferred hand position direction and handvelocity direction of this neuron. A single-layer neural network witha sum of multiplicatively formed basis functions of the input variablesallows general, nonlinear sensorimotor transformations to be com-puted (Pouget and Sejnowski 1997). Such multiplicative responsescan be realized in a recurrently connected network (Salinas andAbbott 1996).

Assume cosine tuning for hand position and velocity direction anda linear dependence on vector magnitude:

fX��, �i� � �X�� cos �� i�fV��, � j� � �V�� cos �� j�

(23)

where ��X��,�� and ��V��,�� are polar coordinate representations of thehand position and velocity vectors. Sinusoidal activity modulation forstatic limb positions has been reported in the motor cortex and area 5of the parietal cortex (Georgopoulos et al. 1984; Wang et al. 2007), aswell as for velocity in the motor cortex (Moran and Schwartz 1999;Wang et al. 2007).

A weighted sum over these multiplicative response activities isgiven by

Movement direction

A

B

Fig. 2. Cosine tuning with respect to move-ment direction and its workspace depen-dence. A: cosine tuning due to the geometricproperty of vector cross product (left) and theresulting tuning curve (right). B: rotation ofpreferred direction as the hand moves fromthe left to the right workspace.





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R�X�, V�� i,j

wijRij�X�, V�� d� ' d� ' w�� ' , � '� fX��, � '� fV��, � '�

(24)

where the sum is approximated by an integral over preferred handdirections and velocity directions, �i and �j are replaced with contin-uous variables �= and �=, and the weight coefficients wij become acontinuous weight function w(�=, �=). If the weight function is acosine function, w(�=, �=) � cos(�= � �=), or a delta function forvelocity, w(�=, �=) � �(�= � �=), then Eq. 24 becomes

R�X�, V�� X��V�� cos �� X� · V� (25)

which is the inner product between X� and V�. If instead the weightfunction is a sine function, w(�=,�=) � sin(�= � �=), then

R�X�, V�� X��V�� sin �� X� � V��Z (26)

is the z component of the cross product. Other components of the crossproduct can be computed similarly. Therefore, with the known prop-erties of cortical motoneurons, a neural network can perform thevector inner and cross products required for motor control.

Thus the computation of torques is reduced to a weighed sum ofmultiplicative responses. Furthermore, the linear coefficients can beeasily learned with Widrow-Hoff regression (Schwartz et al. 2006).

Modeling neural activity for reaching movements with a three-linkmodel. Reaching movements will be simulated for a three-link modelcorresponding to the shoulder, elbow, and wrist joints (Fig. 1B). Thecorresponding EOMs for the torques are

�1 � �m3X�30 � A�30 �I3

r32 X�32 � A�32 � m2X�20 � A�20 �

I2

r22 X�21 � A�21

� m1X�10 � A�10 �I1

r12 X�10 � A�10 � B1

X�10 � V�10

r12 �

Z

(27)

�2 � �m3X�31 � A�30 �I3

r32 X�32 � A�32 � m2X�21 � A�20 �

I2

r22 X�21 � A�21

� B2�X�21 � V�21

r22 �

X�10 � V�10

r12 �

Z

(28)

�3 � �m3X�32 � A�30�I3

r32 X�32 � A�32 � B3�X�32 � V�32

r32 �

X�21 � V�21

r22 �

Z

(29)For comparison of complexity, an explicit expression of the sameEOMs Eqs. 27, 28, and 29 expressed in the joint-angle representationis provided in APPENDIX B. In these equations, there are 12 independentvector cross-product terms corresponding to a model with 12 popu-lations of neurons in the motor cortex:

R1A � �m1X�10 � A�10�Z, R2

A � �m2X�20 � A�20�Z, R3A � �m3X�30 � A�30�Z,

R4A � �m2X�21 � A�20�Z, R5

A � �m3X�31 � A�30�Z, R6A � �m3X�32 � A�30�Z,

R7A � � I1

r12 X�10 � A�10�

Z

, R8A � � I2

r22 X�21 � A�21�

Z

, R9A � � I3

r32 X�32 � A�32�

Z

,

R10V � �B1

r12 X�10 � V�10�

Z

, R11V � �B2

r22 X�21 � V�21�

Z

, R12V � �B3

r32 X�32 � V�32�

Z

.

(30)

To model reaching, we first computed a point-to-point trajectory ofthe center of mass of the third segment according to the minimum jerkcriterion (Flash and Hogan 1985):

X�30�t� � X�initial � �X�final � X�initial��6� t

tf5

� 15� t

tf4

� 10� t

tf3�(31)

where an initial position X�initial and a final position X�final were assumedto lie in the horizontal plane. Although we used minimum jerk togenerate arm trajectories, the following results are valid even whenthe trajectory is computed from sensory feedback signals. For thesimulations of directional tuning, PDs, and muscle tensions (Figs. 3,4, 7 and 8), the movement duration tf was fixed at 200 ms and theamplitude was 3 cm. For the simulations of population vectors (Figs.5 and 6), the movement duration was fixed at 500 ms and theamplitude was 5 cm. For a cloverleaf drawing simulation (Fig. 8E),the movement duration was 1,000 ms, an initial and final position wasfixed at (x, y)�(0 cm, 15 cm), and four via-points (and velocity there)were taken for four leaves, respectively: (10 cm, 25 cm) and (�40cm/s, �40 cm/s) for top right, (�10 cm, 25 cm) and (�40 cm/s, �40cm/s) for top left, (�10 cm, 5 cm) and (�40 cm/s, �40 cm/s) forbottom left, and (10 cm, 5 cm) and (�40 cm/s, �40 cm/s) for bottomright. The wrist angle (�3) was assumed to be fixed to zero, as in mostexperiments, so that the shoulder and elbow angles (�1 and �2) were

uniquely determined. The segment-position vectors X�10, X�20, X�21, X�31,

and X�32 were computed from X�30. Therefore, the six vectors alwaysgive a valid posture. We assumed that the consistency between the

spatial vectors is maintained and concentrated on the problem offinding the joint torques or muscle tensions needed to obtain a desiredendpoint trajectory, which is a difficult inverse dynamics problem.

The 15 biomechanical parameters in the EOMs (mi, Ii, li, and ri)were based on data from monkeys (Macaca mulatta) (Cheng andScott 2000), summarized in Table 1. Viscous coefficients (Bi) wereincluded not because of mechanical properties of the limbs butbecause muscle shortening dominates activities of the motoneu-rons, and their values were adjusted so that the activity levels ofthe velocity cells were about the same as those of the accelerationcells. The choice for the values of viscous coefficients was criticalin simulations reconstructing temporal changes of population vec-tors, which implies that the effects of muscle shortening in mo-toneuron activities dominate the inertial properties of limbs. Wehave also simulated the two-link model (APPENDIX C) and confirmedthat essentially the same results were obtained.

Table 1. Model parameters for three-link model

Mass(mi), kg

InertialMoment

(Ii), kg �m2

ViscousCoefficient(Bi), kg � s

TotalLength(li), m

COMLength(ri), m

Segment 1 0.25 4 � 10�4 0.080 0.15 0.075Segment 2 0.18 4 � 10�4 0.080 0.15 0.060Segment 3 0.030 4 � 10�5 0.050 0.05 0.020

COM, center of mass.





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Computing muscle tensions directly from vector cross-productterms. Limb movements are produced by activating muscles, which canpull but not push. Therefore, the motor cortex needs eventually tocompute not joint torques but muscle tensions. We next asked whetherthe computation of muscle tensions simplifies in a spatial representation.To model muscle tensions, six monoarticular (model muscles 1, 2, 3, 4, 7,and 8) and four biarticular (model muscles 5, 6, 9, and 10) muscles wereintroduced (see Fig. 7A). The joint torques are composed of inertial terms(proportional to the mass and the inertial momentum) and the viscousterms (proportional to the viscous coefficients):

�i � �iinertial � �i

viscous �i � 1, 2, 3� (32)

The computation of muscle tensions may, accordingly, be decom-posed into inertial and viscous terms:

f i � f iinertial � f i

viscous �i � 1, . . . , 10� (33)

Viscous terms of muscle tensions are rectified temporal changes ofmuscle lengths (Winter 2009; Zajac 1989), i.e., muscles are resistantto lengthening their lengths but not to shortening,

f 1viscous <��1= , f 2

viscous <��1= , f 3viscous <��2= , f 4

viscous <��2= ,

f 5viscous <��1 � �2= , f 6

viscous <��1 � �2= , f 7viscous <��3= , f 8

viscous <��3= ,

f 9viscous <��2 � �3= , f 10

viscous <��2 � �3= ,

(34)

where <x= denotes rectification, in which negative values are set to zero. Themuscle tensions have length-dependent, elastic terms, but for simplicity wedo not include them. These viscous terms can be readily expressed by vectorcross products between limb position and velocity (or velocity cells), namely,

�1 � �X�10 � V�10

r12 �

Z

R10V , �1 � �2 � �X�21 � V�21

r22 �

Z

R11V ,

�1 � �2 � �3 � �X�32 � V�32

r32 �

Z

R12V .

(35)

However, the computation of inertial terms is nontrivial. Joint torquescan be computed as weighted sums of muscle tensions of monoar-ticular and biarticular muscles (fi):

�� 1

inertial

� 2inertial

� 3inertial� �

�a1 �a1 0 0 a3 �a3 0 0 0 0

0 0 a2 �a2 a4 �a4 0 0 a6 �a6

0 0 0 0 0 0 a5 �a5 �a7 a7��

f 1inertial

É

f 10inertial�

(36)

where the ais are coefficients derived from the moment arms ofcorresponding muscle insertions and the tensions fis are nonnegative(fi 0) because the muscles can pull but not push. For the followingsimulations of muscle tensions (see Figs. 7 and 8), the values of themoment arms were assumed to be constant, as summarized in Table 2.Computing the muscle tensions required to produce given jointtorques is an ill-posed problem because the muscle tensions havemore degrees of freedom than the joint torques, and a solution isusually found by imposing an optimization criterion such as minimi-zation of a squared sum of muscle tensions.

Here, an alternative, computationally efficient method for obtainingmuscle tensions with vector cross products is proposed. We assumethat muscle tensions can be represented by rectified sums of vectorcross products:

f iinertial f i

inertial � �j�1

8

cijRjA (37)

where RAs are acceleration neurons

R1A � �X�10 � A�10�Z, R2

A � �X�20 � A�20�Z, R3A � �X�30 � A�30�Z, R4

A��X�21 � A�20�Z,

R5A��X�31 � A�30�Z, R6

A��X�32 � A�30�Z, R7A � �X�21 � A�21�Z, R8

A � �X�32 � A�32�Z.(38)

and cijs are coefficients to be adjusted shortly. Eight accelerationneurons instead of nine acceleration neurons in Eq. 30 were usedbecause the coefficients multiplying the cross products were irrelevant

and thus �m1X�10 � A�10�Z and �I1

r12X�10 � A�10�

Z

were equivalent.

Equation 37 can be considered a feedforward neural network with onelayer of weights. A similar neural network that transforms directionallytuned neurons into endpoint force was proposed previously (Georgopou-los 1996). We further assumed that agonist and antagonist musclesreceive reciprocal innervations from the motor cortex:

� f i, agonistinertial � �

j�1

8

cijRjA

f i,antagonistinertial � ��

j�1

8

cijRjA

(39)

There were 5 pairs of agonist and antagonist model muscles, andaccordingly 40 coefficients to be optimized. Temporal profiles of joint

torques were computed with Eqs. 27–29 for a given point-to-pointtrajectory based on minimum jerk (Eq. 31). Temporal profiles of muscletensions (fi) were obtained for 16 movement directions starting from 18starting postures by optimizing a quadratic cost function ��i�1

10 f i2�

subject to the nonnegative constraint (fi 0) and Eq. 36. The choice ofthe minimum squared muscle tension is a computational convenience,and many other optimization criteria produce qualitatively similar resultsfor approximating the dynamics of the lower limb (Prilutsky and Zatsi-orsky 2002). The 40 coefficients were then chosen to minimize the costfunction, defined as the temporal integral of the squared differencebetween muscle tensions (fi) and corresponding approximations (fˆi)averaged over all muscles, initial postures, and movements:

Table 2. Moment arms

a1, m a2, m a3, m a4, m a5, m a6, m a7, m

0.02 0.0125 0.014 0.0175 0.01 0.012 0.012





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�i�muscles

�initial

postures

�movementdirections

�time

points

� f i � f i�2(40)

The goal is to find the coefficients (cij) in fis to approximate the fis.The initial hand positions in the polar coordinates were 14, 20, or 26cm for the distance between hand and shoulder and 0°, 30°, 60°, 90°,120°, or 150° for the angle of hand position relative to the horizontalaxis, giving 18 distinct starting postures. The approximate muscletensions were then substituted into the EOMs to see how well thislinear-sum approximation worked. Also, using the same optimizedvalues of coefficients, we tested a cloverleaf trajectory with a largeramplitude and high curvature (Viviani and Flash 1996).

Modeling neural activity for reaching movements in extrinsic andintrinsic coordinates. For a comparison with conventional views ofthe motor cortex, we considered kinematic and dynamic neuralrepresentations. According to the kinematics viewpoint, the activ-ities in motor cortex M1 reflect the endpoint kinematics in theexternal coordinates. These velocity- and acceleration-tuned cellsare modeled as

Rv��v, � jv� � �V�� cos ��v � � j

v�Ra��a, � j

a� � �A�� cos ��a � � ja�

(41)

where ��V��,�v� and ��A��,�a� are polar coordinate representations of thehand velocity and acceleration vectors. We used 10 acceleration- and10 velocity-tuned model neurons whose PDs were uniformly distrib-uted, ranging from 0° to 360°, thereby a total of 20 model neurons.

Alternatively, in an intrinsic, joint-angle reference frame the motorcortical activity may represent movements and is modeled as angularvelocity and angular acceleration tuning as

�Rv��v, � jv� � � �

�� cos ��v � � j

v�

Ra��a, � ja� � � �

�� cos ��a � � j

a�(42)

where ��

� ��1 �2�T and ��

� ��1 �2�T and ��,�v� and ��,�a� are polarcoordinate representations of the angular velocity and angular accel-eration vectors.

Georgopoulos (1996) generated endpoint forces by using a linearsum of these model neurons. Similarly, using either the extrinsic orthe intrinsic reference frame, we approximated muscle tensions asrectified sums of motor cortical activities as

� f i.agonistinertial � �

j�1

10

cija Rj

a � �j�1

10

cijv Rj

v

f i.antagonistinertial � ��

j�1

10

cija Rj

a � �j�1

10

cijv Rj

v

(43)

There were 20 coefficients for an agonist-antagonist muscle pair,yielding 100 coefficients for the 5 muscle pairs. These coefficientswere optimized globally by minimizing the cost function in Eq. 40.

RESULTS

Broad, cosinelike tuning and preferred directions of modelneurons. Using the trajectory in Eq. 31, we derived the timecourses of neuronal activities for straight trajectories in 16uniformly distributed directions (d�k, k � 1, . . . ,16) and thenaveraged the activity of the i-th neuron (R

�ik) in Eq. 30 over the

initial 100 ms for each trajectory. The tuning curve (Fig. 3A)was obtained by plotting R

�ik as a function of movement direc-

tions. The firing rates of the model neurons as a function ofmovement direction were sinusoidally modulated with respect

to each limb’s center-of-mass position due to vector crossproduct (Fig. 2A), consistent with recordings from neurons inmotor cortex (Georgopoulos et al. 1982).

The highest firing rate of a neuron occurs in its preferreddirection (PD) (arrows in Fig. 3A), obtained by taking anormalized weighted sum,

PD� i ��k�1

16

Rikd�k

�k�1

16

Rikd�k

(44)

The PDs in the model were nonuniformly distributed (Fig. 3B),in agreement with the experimental distribution in motor cortex(Fig. 3C), which was concentrated in the second and fourthquadrants (Naselaris et al. 2006; Scott et al. 2001; Scott 2005).The skewness of the distribution is a consequence of con-straints on possible geometric configurations of the limb.

The model also predicted that the PDs should rotate whenthe workspace is displaced because the cross products dependon the starting location of the hand (Fig. 2B). In experimentalrecordings, the PDs of most motor cortex neurons consistentlyrotated clockwise along the vertical axis when the workspaceshifted from the left side, to the middle, and to the right side ofthe body (Caminiti et al. 1990). The model neurons exhibitedPDs that rotated similarly (Fig. 3D). We further predicted thatwhen the workspace is vertically displaced the PDs shouldrotate along the horizontal axis. These posture-dependentchanges of the PDs cannot be explained by the extrinsic codinghypothesis, which posits that the cells represent endpointvelocity, acceleration, or the sum of velocity and acceleration.

The dependence of the PD of a neuron on position in theworkspace reveals its reference frame (Ajemian et al. 2000,2001; Wu and Hatsopoulos 2006). Figure 4 depicts the PDs oftwo representative neurons at various locations in the work-

space. A neuron whose activity varies with �m3X�30 � A�30�Z, for

example, contains activity modulated by hand position (X�30)

and acceleration (A�30) relative to the shoulder. The PD of thisneuron becomes rotated by the amount of shoulder rotation(Fig. 4A) and remains invariant if the elbow angle is alteredwith the shoulder-to-hand direction kept constant (Fig. 4C),thereby exhibiting a shoulder-centered preference. PDs in theshoulder-centered frame stay invariant over the workspaceposition, whereas PDs in the joint-angle or Cartesian framesvary considerably (Fig. 4E).

Another model neuron representing �m3X�31 � A�30�Z has hand

position dependence relative to the elbow (X�31) and hand accel-

eration dependence relative to the shoulder (A�30). The PD of thisneuron gets rotated by the amount of both shoulder rotation (Fig.4B) and elbow rotation (Fig. 4D), thereby exhibiting a joint-basedreference frame. PDs in the joint-angle frame change over theworkspace less than those in other frames (Fig. 4F). Other neu-rons have intermediate reference frames between the shoulder-centered and joint-based frames. The neurons in the model exhib-ited the same range of preferred reference frames as those reportedexperimentally in motor cortex (Kakei et al. 1999; Wu andHatsopoulos 2006).

Spatiotemporal properties of population vectors. The popu-lation vector of movement direction has been used to estimate





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the direction of hand motion from a weighted average of PDsfrom cortical neurons (Georgopoulos et al. 1988):

PV��t� � �i�1

12

Ri�t�PD� i (45)

The population vector for the model neurons had systematicdeviations, following an oblique ellipse (Fig. 5A) that closelymatched experimental results for the same workspace position(Fig. 5B) (Scott et al. 2001). The directional errors predicted bythe model were almost the same with those reported in theexperiment (Fig. 5C). These deviations stem from the nonuniformdistribution of PDs that depends on posture (Fig. 3, B and C),consistent with a previous analysis (Sanger 1996). The modelpredicts a workspace dependence of the deviations with much lessskew in the right workspace (Fig. 5D) and skewed toward the

opposite direction in the left workspace (Fig. 5E). Differences inthe bias of population vectors (Georgopoulos 2002; Scott 2005)may therefore arise from postural differences.

The directions of population vectors change during reaching(Fig. 6A) (Georgopoulos 1988), and qualitatively similar shiftshave been observed in the model (Fig. 6B). For the populationvectors to track the limb velocity, the values of viscous coeffi-cients in the model needed to be chosen so that the activity levelsof acceleration cells and velocity cells were comparable, consis-tent with a dominant contribution of muscle length change tomotor cortex activity. There was uncertainty in the choice ofviscous coefficients (B1, B2, and B3 in Table 1) used for thesimulations. To evaluate how sensitive our results were, werepeated the same simulations with the values that were halved ordoubled from the standard values. Most results were not highly

A

B Model ExperimentC

D Left Center Right

Direction of Movement (deg)U

nit A

ctiv

ity (N

· m

)

0.1

0

-0.10 45 90 135 180 225 270 315 360

Fig. 3. Cosine tuning and distribution ofpreferred directions. A: tuning curves for 12model neurons and preferred directions(PDs) indicated by arrows for 6 of them.Inset shows the starting posture and 16movement directions. B: distribution of PDspredicted by the model in a polar plot withradius scaled by activity amplitude (max ac-tivity � min activity) with separate vectorsfor the positive and negative peaks. Musclescan pull but cannot push, so separate neuronswhose PDs were oppositely directed to PDsof model neurons in Eq. 21 were also in-cluded. C: distribution of PDs reported formotor cortex neurons (from Naselaris et al.2006; Scott 2005; Scott et al. 2001). D: di-rections of the 6 PDs shown in A rotatedclockwise over the 3 corresponding work-spaces in C.





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sensitive to the parameter values, except for the temporal profilesof population vectors. When the value of the viscosity was doubled,the population vectors more accurately reflected the endpoint velocity(Fig. 6C). When the viscosity was halved, reversals in the directionsof population vectors reversed at the beginning or end of movement(Fig. 6D). These simulations suggest that, for the population vectoralgorithm to be able to accurately reconstruct a movement trajectory,motor cortex needs to have neurons whose firing rates reflect thetemporal changes in muscle lengths.

Computing muscle tensions from spatial representations.The motoneurons in the final common pathway of the motorsystem create the muscle tensions that together produce thejoint torques needed for reaching. The vector cross productsrepresented in the motor cortex should therefore form a basisfor computing muscle tensions. We explored this transforma-tion in a three-link model with 10 muscles (shoulder flexor andextensor, elbow flexor and extensor, wrist flexor and extensor,

and biarticular flexors and extensors) (Fig. 7A). Because thenumber of muscles exceeds the number of joint torques,inverting Eq. 36 is underdetermined and an additional con-straint must be imposed such as energy minimization (Fagget al. 2002; Prilutsky and Zatsiorsky 2002) or reciprocalactivation of agonist and antagonist muscles (Thoroughmanand Shadmehr 1999). None of these constraints is biologicallyplausible because they require iteration between the estimatesof the torques and muscle tensions to ensure they are satisfied.We show here that the vector cross products form a computa-tionally efficient basis for approximating muscle tensions.

The resulting approximation in Eq. 39 was excellent for all10 muscles, not only for the monoarticular muscles (Fig. 7, B,C, and E) but also for the biarticular muscles (Fig. 7, D and F).The resulting movement trajectories computed from theseapproximate muscle tensions almost perfectly matched thecorresponding minimum-jerk trajectories over the workspace

Cartesian

Shoulder

Joint

C D

A B

E F

x (cm) x (cm)

y (c

m)

y (c

m)

-20 0 20 -20 0 20

20 20

Fig. 4. Multiple coordinate systems in model neurons. The PD of model neurons �X�30 � A�30�Z (A) and �X�31 � A�30�Z (B) changes when the shoulder is rotated with

a fixed elbow. The PD of model neurons �X�30 � A�30�Z (C) and �X�31 � A�30�Z (D) changes when the elbow angle is rotated with a fixed shoulder-to-hand direction.

The PD changes over the workspace locations according to shoulder-centered (�X�30�A�30�Z) (E) and joint-based (�X�31 � A�30�Z) (F) reference frames. Arrowsindicate PDs of the neurons at various locations in the workspace. Insets show PDs represented in the Cartesian (red), joint-based (blue), and shoulder-based(green) frames.





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[exemplar trajectories for 8 directions starting from (0 cm, 26cm) are shown in Fig. 8A]; the direction and the amplitudeerrors measured at the endpoint averaged over the entireworkspace were �0.8 � 10�5° (SD 0.053°) and 8.8 � 10�4

cm (SD 1.6 � 10�3 cm), as shown in Fig. 8D.In contrast, if extrinsic (Eq. 41) or intrinsic (Eq. 42) move-

ment neurons were used instead of the cross products, theoptimized linear approximation was a poor approximation: thedirection and the amplitude errors were �2.87° (SD 51.9°) and0.72 cm (SD 0.17 cm) for extrinsic movement neurons (exem-plar trajectories in Fig. 8B) and 0.42° (SD 15.6°) and 0.12 cm(SD 0.66 cm) for intrinsic movement neurons (exemplar tra-jectories in Fig. 8C). Therefore, a simple feedforward networkthat transforms directional motor commands to muscle activa-tion (Georgopoulos 1996) does not work if the coefficients areoptimized globally.

Using the same globally optimized values of the coefficients,the linear approximation of muscle tensions using vector crossproducts smoothly traced a cloverleaf pattern that involved move-

ment with large amplitudes and high curvatures (Viviani andFlash 1995) (Fig. 8E). Note that the coefficients in Eq. 39 wereoptimized globally over the workspace but nonetheless producedgood local approximations. Thus, with our simplified musclemodel and the optimality criterion of minimizing squared muscletensions, muscle tensions can be computed rapidly and withoutiteration by a rectified linear sum of neural activities from motorcortex neurons that represent vector cross products.

DISCUSSION

The disparate properties of neural activities in motor cortexcan be explained parsimoniously by the spatial representationhypothesis. Joint torques can be computed without explicitlycomputing the joint angles, and muscle tensions simplifycompared with traditional approaches that require iterativeoptimization. Because the muscle tensions required to make areaching movement can be accurately approximated by alinearly weighted sum of motoneuron activities, the muscletensions could be computed directly from the cross products

C

A B

D E

Err

or (d

eg)

Direction of Movement (deg)

0°337.5°315°

45°

247.5°225°

180°135°

90°67.5°

270°

Fig. 5. Spatial properties of population vec-tors constructed from the model neurons.Deviations are shown between the handmovement directions (blue lines) and popu-lation vectors (red arrows) predicted by themodel (A) and found in motor cortex neu-rons (B) (Scott et al. 2001). In A, the shoul-der and elbow angles at the starting postureare 70° and 50°, respectively. C: differencebetween the direction of population vectorand the direction of hand movement pre-dicted by the model (black circles) and re-ported in the experiment (red triangles). Dand E: the model further predicts that thedeviations are posture dependent in differentworkspaces (D: shoulder 0°, elbow 90°;E: shoulder 90°, elbow 90°). Insets representthe postures on initiating movements usedfor the simulations and the experiment. Inthese simulations population vectors over500 ms (time step 5 ms) were calculated forminimum-jerk trajectories (average popula-tion vectors in the initial 50 ms are shown).





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represented by neurons in the motor cortex with our simplifiedmuscle model. This could explain how the neurons in primatemotor cortex that project directly to lower motoneurons in thespinal cord can effectively control muscle tension activities.Most electrophysiological studies have correlated the firingrates in the motor cortex with endpoint kinematics. However,in the proposed framework, the spatial vectors of all limbsegments are required for the computation of cross products.Therefore, to test our model, a simultaneous recording oftrajectories of all limb segments and neuronal activities isnecessary.

The essential feature of the spatial coordinate frame is that itis fixed with respect to the world to avoid the centrifugal andCoriolis terms that proliferate in a moving coordinate system,

such as joint-based coordinates. Motor structures based onspatial coordinates evolved early, such as the optic tectum, asensorimotor structure in lower vertebrates used for orientingthe body in space. Since sensory information is represented inspatial coordinates, and the body is oriented in space, interact-ing with the world simplifies in a common spatial referenceframe. In mammals the superior colliculus controls eye move-ments in a spatial reference frame, and the motor cortex mayhave evolved to elaborate on this strategy.

Related models. A previously proposed optimal controlapproach to motor control using a biomechanical skeletomus-cular model that minimized the sum of squared control signalsfor a reaching task and an isometric task reproduced experi-mental findings such as broad directional tuning, nonuniform

A

C

D

B

Pop

ulat

ion

vect

orP

opul

atio

n ve

ctor

Pop

ulat

ion

vect

or

Pop

ulat

ion

vect

orP

opul

atio

n ve

ctor

Pop

ulat

ion

vect

or

Time (ms) Time (ms)

90°

0°

45°

0 500 0 500

M M

Fig. 6. Temporal properties of population vectors constructed from the model neurons. A and B: temporal changes of population vectors computed from motorcortex neuron activities (A) (Georgopoulos 1988) and the model’s population vectors for 90° and 45° directions (B). The horizontal axes are aligned to movementonset (indicated by “M” in A). Minimum-jerk trajectories were used to compute the model neural activities, and corresponding population vectors were computedevery 5 ms over 500 ms. C and D: the model’s population vectors when the viscous coefficients were doubled (C) or halved (D).





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distribution of PDs, skewed movement directions in populationvectors, and some temporal changes of firing rates (Guigonet al. 2007; Trainin et al. 2007). These biomechanically de-tailed models, however, did not consider how spatial informa-tion in Cartesian coordinates could be transformed into thejoint-angle space. For an isometric force task, they attributedshifts of PDs and changes of neuronal gains for endpoint forceto the Jacobian transformation from the force to the joint-torque space (Ajemian et al. 2008). The crucial feature of thesemodels is the geometric configuration of segmented limb,which determines the stiffness ellipse of the limb and theJacobian matrix. Although the equations and interpretations ofthe results are quite different, these models share with ourmodel the importance of the biomechanical properties of theupper limb in understanding the neuronal activities in themotor cortex.

Equally important for the motor cortex is how dynamicsis stabilized by controlling the impedance of the limbs,which has been emphasized in equilibrium-point controlmodels and their extensions (Feldman 1966; Feldman andLevin 1995). The motor system could have two separate yetinteracting mechanisms for computing dynamics and stiff-ness, respectively. Indeed, recent studies of human neuro-imaging suggest that reciprocal activation (net joint torque)

and coactivation (stiffness) of wrist muscles can be recon-structed from blood oxygen level-dependent signals (Ga-nesh et al. 2008) and that the activity in the dorsal premotorcortex was correlated with reciprocal activation of muscleswhereas the activity in the ventral premotor cortex wascorrelated with the level of cocontraction (Haruno et al.2012). Our proposed model of dynamics and the equilibri-um-point approaches are therefore not mutually exclusivebut rather are complementary.

Reconciling kinematic and dynamic perspectives. The spa-tial representation model analyzed here reconciles an apparentdisagreement about the contribution of neurons in the motorcortex to arm movements. From a dynamics perspective, neu-rons have been found in motor cortex that compute jointtorques and muscle tensions (Evarts 1968; Fetz and Cheney1980), but other neurons in the motor cortex carry kinematicspatial information that encodes hand movement direction andvelocity (Georgopoulos et al. 1982, 1986). Even in the kine-matic perspective, whether the motor cortex represents anintrinsic joint-angle coordinate or an extrinsic Cartesiancoordinate is an open issue (Kalaska 2009). We have shownin a spatial representation that movement trajectories andmuscle tensions can be reconstructed at the same time frommodel neuronal activities identified with vector cross prod-

Time (ms) Time (ms)

Time (ms)Time (ms) Time (ms)

Mus

cle

Tens

ion

(N)

Mus

cle

Tens

ion

(N)

Mus

cle

Tens

ion

(N)

Mus

cle

Tens

ion

(N)

Mus

cle

Tens

ion

(N)

A B C

D E F

Fig. 7. Muscle tensions derived from spatial representation. A: a 3-link model with 10 muscles [shoulder extensor (f1) and flexor (f2), elbow flexor (f3) and extensor(f4), biarticular flexor (f5) and extensor (f6) for shoulder and elbow, wrist flexor (f7) and extensor (f8), biarticular flexor (f9) and extensor (f10) for elbow and wrist].Sixteen movement directions of the hand are color coded in inset. B–F: muscle tensions obtained with a quadratic cost function [fi(i � 1, . . . , 10), solid colorlines] and corresponding approximations [fi(i � 1, . . . , 10), black dashed lines] for shoulder flexor (f1, B), elbow flexor (f3, C), shoulder-elbow biarticular flexor(f5, D), wrist flexor (f7, E), and elbow-wrist biarticular flexor (f9, F). Best fits for flexor muscles are shown here; similarly good fits were obtained for extensormuscles. The starting posture is shown in A, and the same colors for movement directions are used in B–F.





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ucts (Fig. 9A). This is consistent with evidence for bothkinematic and dynamic signals in the motor cortex.

We have shown that muscle tensions can be reconstructedfrom a rectified sum of weighted cross-product responses forour simplified muscle model, but this is not possible frommodel neurons tuned either to extrinsic (endpoint movement)or intrinsic (joint angles) movements with constant coefficientsover the global workspace. In particular, a linearly weightedsum of model neuron activities directionally tuned to endpointmovements cannot generate the muscle tensions needed toperform accurate reaching, as previously proposed (Georgo-poulos 1996). This does not rule out the use of extrinsic orintrinsic neurons for computing muscle tensions, since thespinal cord could well perform a more complex transformationof these activities. However, we can conclude that vector crossproducts could be a more computationally efficient basis thaneither extrinsically or intrinsically tuned neurons.

A previous model reproduced some properties of motorcortex activities by assuming that motor cortex encodes the

dynamics of linear movement and suggested that directiontuning in the motor cortex might be an epiphenomenon (Todo-rov 2000). However, the linear dynamics approximation usedin that study was limited to a small portion of the workspaceand did not represent the actual structure of a multijointedlimb. Directional tuning with respect to movement directionappeared in the model simply because the model assumeddirectional tuning with respect to endpoint force in the lineardynamics. The spatial hypothesis provides a more generalframework in which the representation of spatial movements isan indispensable intermediate step in computing the torquesfrom dynamical equations and therefore integrates both thedynamical and spatial views.

Whether activities of the motor cortical neurons representsingle movement variables or weighted sums of movementvariables is an open question. One possibility is that singleneurons compute a linear sum of these cross-product terms,consistent with a previous study reporting multiparameterresponses in individual neuronal activities (Ashe and Georgo-

x (cm)

y (c

m)

y (c

m)

y (c

m)

y (c

m)

x (cm) x (cm) x (cm)

A B C

D E

x (cm)y

(cm

)

Fig. 8. Hand trajectories computed from approximate muscle tensions. A–C: computed trajectories using the rectified sum approximation from vector crossproducts (A), extrinsic cells (B), and intrinsic cells (C). Solid color lines indicate desired movement paths (based on the minimum-jerk criterion) of 8 directionsstarting from (x, y) � (0 cm, 26 cm) in the workspace compared with dashed color lines for the trajectories computed from approximate muscle tensions (fi).In A, the solid and dashed lines overlap almost completely. D: desired and approximated trajectories using the cross-product approximation. E: cloverleafdrawing. Gray solid lines and black dashed lines are desired minimum-jerk trajectories and approximate trajectories computed from approximate muscle tensions,respectively.





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poulos 1994). However, it is difficult to dissociate multiplemovement parameters in a center-out reaching task as used inthis and other studies because the movement parameters areoften temporally correlated. A recent study resolved this issueby designing continuous movement tasks and consideringtemporal correlations among kinematic parameters and re-ported that most neurons correlated with single dominantparameters (Stark et al. 2007). Among the kinematic parame-ters, velocity was the most dominant (80%) compared withposition (9%) or acceleration (11%).

In our model, the cross-product terms were computed fromkinematic variables with a feedforward neural network andmuscle tensions were computed as rectified linear sums ofcross-product terms. This suggests a hierarchical organization.The primary motor cortex of primates is divided into rostraland caudal subareas defined by cytoarchitectural zones (Geyeret al. 1996), descending pathways (Rathelot and Strick 2009),and functional representations (Sergio et al. 2005). Of partic-ular relevance to our study is the proposal based on data fromvarious recording studies that neuronal activity in the rostralM1 correlates with overall directions and kinematics of end-point motion, whereas activity in the caudal M1 correlates withthe temporal pattern of force production and motor output(Sergio et al. 2005). From the perspective of our model, it is

tempting to identify the rostral M1 with the cross-productrepresentation (Eq. 21) and the caudal M1 with the linear-sumrepresentation of muscle tensions (Eq. 37). This leads to theprediction that there should be functional connections from thekinematic representation in the rostral part to the force repre-sentation in the caudal part.

Motor cortex neuronal activities explained by geometricproperties of cross products. The ubiquity of broad, cosineliketuning curves in motor cortex and the position dependence ofthe PDs follow naturally from vector cross products and thegeometry of space. How motor cortex neuron activities vary indifferent workspaces is not completely understood, but similarresponse modulation by eye or head positions has been de-scribed in parietal visual neurons by the so-called gain fields(Andersen et al. 1985). We have proposed a feedforward neuralnetwork that computes vector cross products from limb posi-tion and velocity or acceleration, similar to a previous modelfor wrist movements with distinct postures in which multipli-cative responses between posture and extrinsic movementreproduced the response of musclelike neurons in the motorcortex (Kakei et al. 2003). Together with our simulation resultsthat linear summation of purely movement-related model neu-rons failed to approximate muscle tensions, our model predictsthat posture and movement variables for multijointed move-ments in the motor cortex should be represented multiplica-tively, in the same way that eye position is represented as again field for visual receptive fields in the parietal cortex(Andersen et al. 1985). The workspace dependence may reflecta general solution for population coding of spatial transforma-tions involving large body movements.

In endpoint force generation, broad, cosinelike tuning couldalso arise as a result of minimizing the detrimental effects ofsignal-dependent multiplicative noise (Todorov 2002). How-ever, in that study the broad tuning curves were derived withrespect to the force direction and not to movement direction.For a linear model that approximates full nonlinear multiseg-mented limb dynamics in a small part of the workspace, a forcedirection may approximate the movement direction, but, ingeneral, a force direction would differ from a movementdirection with full multisegmented limb dynamics. Anothercomputational study reproduced the posture dependence ofPDs at the single-neuron level in an isometric force task bypostulating that the motor cortical neurons are tuned to pre-ferred torque directions, but cosine tuning to torque directionswas assumed rather than derived from a computational princi-ple (Ajemian et al. 2008). Neurons have been reported in themotor cortex for which tuning to isometric force generationand tuning to reaching movement direction are not the same, asexpected from the arguments made here (Sergio et al. 2005).

The model makes the strong prediction that neural activities inmotor cortex should reflect not only hand endpoint movement inthe workspace coordinates but also the center of mass movementsof other limb segments. Moreover, we predict that some neuronsshould have not only shoulder- but also elbow- or wrist-basedreference frames. Some neurons in the dorsal premotor cortexhave visual receptive fields anchored to the hand irrespective ofgaze direction (Graziano et al. 2000), consistent with this predic-tion. It should be possible to reanalyze existing experimental datato confirm these model predictions.

Cross products as motor primitives in motor adaptation. Thethree-link model can be expanded to include more joints and

A

B

Fig. 9. Proposed representation and computation in motor cortex. A: achematicof how both kinematic (hand trajectory) and dynamic (muscle tension) vari-ables can be reconstructed from the cross-product representation. B: steps inplanning and executing body movements. The computation of joint angles isrequired to compute the control signals in the conventional scheme but isunnecessary in the new scheme because the control signals can be computeddirectly from the spatial representation of the trajectory.





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muscles, which requires more cosine-tuned neurons. Also, as thebody grows, and as the masses of the limbs change over time, thedynamics must be continually recalibrated; this can be accom-plished by reweighting the terms in Eq. 37. The constraintsbetween the various terms in this equation could be similarlylearned through experience, since not all combinations of jointangles and limb positions are feasible. An initially uniform dis-tribution of PDs could be pruned by learning to form the nonuni-form distribution in Fig. 3B, reflecting the geometric constraints.Experiments could be designed to test this hypothesis by artifi-cially lengthening limbs and seeing how the tuning curves ofneurons in motor cortex adapt to the changes.

Studies of how motor adaptation at one workspace or direc-tion generalizes to other untrained workspaces or directionsprovide a window as to what basis functions of movements, ormotor primitives, are adopted in the motor cortex. Humanpsychophysical experiments reported that generalization ofadaptation to viscous force field at one workspace occurs in theshoulder-based reference frame (Shadmehr and Mussa-Ivaldi1994). This pattern of generalization can be naturally explainedby the position dependence of cross products: Model neuralactivities identified with cross products remain invariant ifmovement direction is rotated by the same angle at which theshoulder is rotated (Fig. 4). We modeled the force-field adap-tation experiment by adjusting the coefficients of velocity-dependent cross products and reproduced the patterns of gen-eralization in a shoulder-based reference frame.

In contrast, generalization of visuomotor adaptation betweenspatially displaced workspaces occurs in an extrinsic referenceframe (Krakauer et al. 2000; Wang and Sainburg 2005). Thesepsychophysical studies could also be reproduced with the modelunder the assumption that the coefficients of inertia-related cross-product terms underwent adaptive changes under these condi-tions. These adaptation results provide strong support that not onlyreaching dynamics but also motor adaptation uses vector crossproducts as computational basis functions. Motor adaptation in aspatial representation is the focus of a forthcoming paper (Tanakaand Sejnowski, manuscript in preparation).

Computing joint torques and muscle tensions without anexplicit joint-angle representation. It is interesting to note thatKakei et al. (1999) reported neurons encoding wrist movementdirections in the extrinsic space regardless of posture (“extrin-sic-like” cells) and neurons encoding posture-dependent mus-cle activities (“musclelike” cells) but no neurons that exhibiteda joint-angle reference frame (“jointlike” cells). This impliesthat the motor cortex represents both spatial movements andmuscle activities but not joint angles for wrist movements,although some studies have reported neural activity in intrinsicjoint-angle reference frame for arm reaching movements(Reina et al. 2001; Scott and Kalaska 1997; Thach 1978). In aspatial representation the computation of joint torques andmuscle tensions simplifies if movements are expressed withcross products of spatial vectors but not with joint angle, so themotor cortex may have exploited the computational efficiencyof computing limb dynamics with spatial vectors. Also, theabsence of an explicit joint-angle representation (Kakei et al.1999) is inconsistent with a conventional serial scheme ofmovement planning and execution that requires an explicitcomputation of joint angles (Fig. 9B) (Buneo and Andersen2006; Flash and Sejnowski 2001; Hollerbach 1982; Kalaskaand Crammond 1992; Kawato et al. 1987).

Our model postulates a computational scheme that is analternative to schemes with an explicit joint-angle representa-tion. The direct transformation from limb trajectory to muscletensions is consistent with vectorial movement planning (Gor-don et al. 1994). Although the final computation of muscletensions in Eq. 37 as a linear sum of neural activities impliesthat an explicit representation of joint torques may be unnec-essary, neural activities will still be correlated with them.Despite the diverse properties of neurons that have been foundin M1, there may be a simple geometric principle underlyingthe complex properties of neurons in motor cortex based onvector cross products of postural and kinematic variables. Theidentification of vector cross-product terms in the torque equa-tions with single neurons is a minimal representation, which inthe motor cortex could be expanded to provide more diversityand redundancy in computing the final muscle tensions.

Sensory feedback. One limitation of the present model is thatthere is no direct force control in Eq. 21, which would precludecontrol of stiffness when interacting with an unstable environ-ment. Recent human psychophysical studies have shown thatthe control of reaching dynamics involves feedback in thepresence of dynamical and sensory noise (Chen-Harris et al.2008; Liu and Todorov 2007; Nagengast et al. 2009; Todorovand Jordan 2002). In contrast to feedforward control, in whichthe desired trajectory is preplanned before being transformedinto joint torques or muscle tensions, in an optimal feedbackmodel deviations of sensory feedback signals from their esti-mates are used to modify the control signal during the move-ment and thus there is no need for a preplanned trajectory.Although feedforward and feedback control differ in this re-spect, they both need to map sensory signals onto controlsignals. Vector cross products could provide a convenientframework for optimal feedback control since they span acomputationally efficient basis for converting sensory signalsinto control signals. Furthermore, cross products can be com-puted from either visual or proprioceptive signals, so both canbe represented in the same bases without the need for thecomputation of joint angles.

The model avoided the explicit computation of joint anglesat the cost of introducing a redundant representation of spatialvectors. How the brain maintains consistency among all thespatial vectors is a nontrivial problem. Because the proprio-ceptive feedback always provides a valid configuration ofposture, comparing the feedback from proprioceptive inputscan maintain the consistency in the spatial representation. Inthis view, motor planning and sensory feedback are no longerseparable but inherently integrated. The problem of maintain-ing the consistency among spatial vectors is closely related tothe concept of the body schema, a representation of body partsin space used for controlling body movements, which may befound in the parietal cortex (Haggard and Wolpert 2005).Therefore, our model of the motor cortex might be extended tothe parieto-frontal motor areas.

Conclusion. This study focused on arm reaching movementsin the horizontal plane. For three-dimensional movementsthere are additional terms (Eq. 13), but they are all vector crossproducts. Thus similar conclusions will hold for more generalmovements, which can be tested. Knowing how a population ofneurons in the motor cortex represents an action is only a firststep toward understanding how actions are planned and carriedout with sensory feedback. The next step is to generalize the





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model to include both feedback control and proprioceptivefeedback. Work in progress will resolve these limitations.

APPENDIX A

Derivation of Spatial Representation with the Newton-Euler Method

The physics of a linked rigid body under constraints has been studiedfor centuries in physics and more recently in robotics. Our derivation ofthe EOMs is based on standard methods of rigid body dynamics. Thespatial representation for EOMs in Eq. 12 was derived by explicitly usingjoint angles with the Euler-Lagrange method. There is another, recursivemethod in robotics for deriving EOMs for a segmented system, known asthe Newton-Euler method (Luh et al. 1980). Although the two methods

are mathematically equivalent, the Newton-Euler method is far moreintuitive because it uses spatial vectors rather than joint angles and moresystematic because the computation is recursive backward from the distalsegments to more proximal segments. Moreover, the computation ofthree-dimensional motion with the Newton-Euler method is no morecomplicated than that of two-dimensional motion, whereas in the Euler-Lagrange method a two-dimensional movement computation does notstraightforwardly generalize to three dimensions. Although the spatialrepresentation has been known for a long time in physics and robotics, ithas not been used for the purpose of understanding the functions of motorcortex.

The recursive equations for translational forces and rotationaltorques, which are solved backward from distal to proximal seg-ments, are

F�i � F�i�1 � miA�i,0

��i � ��i�1 � miX�i,i�1 � A�i,0 � X�i,i�1e � F�i�1 � Ii�

�i � ��i � �Ii��i�

�i � 1, · · · , n� (A1)

where X�j,j�1e is a vector connecting the (j � 1)th joint to the jth joint,

Ii is the 3 � 3 inertial matrix, and ��i and ��i are angular velocity and

acceleration, respectively. F�i and ��i are force and torque vectors

exerted on the ith joint, respectively. No external force F�n�1 or torque��n�1 at the endpoint is imposed, for simplicity. To illustrate how toderive EOMs, we first consider a two-link (n � 2) segmental model.First we begin with equations of the second segment as

F�2 � m2A�20

��2 � m2X�21 � A�20 � I2��2 � ��2 � �I2��2�(A2)

where we used F�3 � 0 and ��3 � 0. Recursively, the equations of thefirst segment are

F�1 � F�2 � m1A�10

��1 � ��2 � m1X�10 � A�10 � X�10e � F�2 � I1��1 � ��1 � �I1��1�

(A3)

By noting using F�2 � m2A�20 and X�20 � X�10e � X�21, the torque at the

shoulder joint in Eq. A3 now reads

��1 � m1X�10 � A�10 � m2X�20 � A�20 � I1��1 � ��1 � �I1��1� � I2��2

� ��2 � �I2��2� . (A4)

The angular velocities and angular accelerations have the followingspatial representations:

��i �X�i,i�1 � V�i,i�1

ri2 , ��i �

X�i,i�1 � A�i,i�1

ri2 (A5)

so the EOMs Eqs. A2 and A4 in the spatial representation become

��1 � m1X�10 � A�10�I1

X�10 � A�10

r12 ��X�10 � V�10

r12 ��I1

X�10 � V�10

r12

� m2X�20 � A�20 � I2

X�21 � A�21

r22 � �X�21 � V�21

r22 � �I2

X�21 � V�21

r22

(A6)

��2 � m2X�21 � A�20 � I2

X�21 � A�21

r22 � �X�21 � V�21

r22 � �I2

X�21 � V�21

r22

(A7)

If the movement is restricted in the horizontal plane, the termsquadratic in �� vanish because �� and I�� are parallel to each other, andonly the z-components of joint torques take nonzero values. For thisspecial case,

�1 � �m1X�10 � A�10 � I1

X�10 � A�10

r12 � m2X�20 � A�20 � I2

X�21 � A�21

r22 �

Z

(A8)

�2 � �m2X�21 � A�20 � I2

X�21 � A�21

r22 �

Z

(A9)

Here, the �is are z-components of the torque vector, ��i, and Iis are theinertial momentum of the ith segment around the z-axis. These are theEOMs we derived by using the Euler-Lagrange method in Eqs. 10 and11 of the main text. The recursive nature of the Newton-Euler methodallows us to derive a general formula for an n-link system in threedimensions:

��i � �j�i�

n �mjX� j,i�1 � A� j0 � I j

X� j,j�1 � A� j,j�1

rj2

� �X� j,j�1 � V� j,j�1

rj2 � �I j

X� j,j�1 � V� j,j�1

rj2 � (A10)

This is Eq. 13 in the main text. Again, by restricting the movement inthe horizontal plane, we finally arrive at the special case:

�i � �j�i�

n �mjX� j,i�1 � A� j0 � Ij

X� j,j�1 � A� j,j�1

rj2 �

Z

(A11)

which is Eq. 12 in the main text.As already shown for the two-dimensional movements, the com-

putation of torque for three-dimensional movements greatly simplifiesfor spatial vector cross products compared with the expressions whenjoint angles are used, in both the number and the complexity of theterms. Therefore, our computational framework that postulates the useof spatial representation is not restricted to two-dimensional move-ments in the horizontal plane but generalizes to three-dimensionalmovements.





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APPENDIX B

EOMs in Intrinsic Coordinates for a Three-Link Articulated Model

EOMs for the three-link system in the joint-angle representation areshown for comparison with those in a spatial representation. Theseequations are equivalent to Eqs. 27, 28, and 29 in the main text.

�1 � �I1 � I2 � I3 � m1r12 � m2�l1

2 � r22� � m3�l1

2 � l22 � r3

2��2m2l1r2cos�2 � 2m3l2r3cos�3

�2m3l1l2cos�2 � 2m3l1r3cos��2 � �2��1

� �I2 � I3 � m2r22 � m3�l2

2 � r32��m2l1r2cos�2

�m3l1r3cos��2 � �3� � 2m3l2r3cos�3 � m3l1l3cos�2��2

� �I3 � m3r32 � m3l2r3cos�2 � m3l1r3cos��2 � �3��3

� �m2l1r2sin�2 � m3l1l2sin�2 � m3l1r3sin��2 � �3��22

� �m3l2r3sin�3 � m3l1r3sin��2 � �3��32

� �2m2l1r2sin�2 � 2m3l1l2sin�2 � 2m3l1r3sin��2 � �3��1�2

��2m3l1r3sin��2 � �3� � 2m3l2r3sin�3��3�2

��2m3l1r3sin��2 � �3� � 2m3l2r3sin�3��3�1

� B1�1 (B1)

�2 � �I2 � I3 � m2r22 � m3�l2

2 � r32�

�m2l1r2cos�2 � m3l1l2cos�2

� 2m3l2r3cos�3 � m3l1r3cos��2 � �3��1

� �I2 � I3 � m2r22 � m3�l2

2 � r32� � 2m3l2r3cos�3��2

� �I3 � m3r32 � m3l2r3cos�3��3

� �m3l1l2sin�2 � m2l1r2sin�2 � m3l1r3sin��2 � �3��12

� m3l2r3sin�3�32

� 2m3l2r3sin�3�2�3

� 2m3l2r3sin�3�3�1

� B2�2 (B2)

�3 ��I3 � m3r32 � m3l2r3cos�3 � m3l1r3cos��2 � �3��1

��I3 � m3r32 � m3l2r3cos�3��2

��I3 � m3r32��3

��m3l2r3sin�3 � m3l1r3sin��2 � �3��12

�m3l2r3sin�3�22

�2m3l2r3sin�3�1�2

�B3�3

(B3)

APPENDIX C

Two-Link Articulated Model

Reaching movements and neural activities are modeled with thethree-link model in the main text, and qualitatively similar resultswere obtained when a two-link model was used. For an articulatedarm model with two connected links, the spatial representation of theEOMs including the viscosity terms is

�1 � �m2X�20 � A�20 �I2

r22 X�21 � A�21 � m1X�10 � A�10 �

I1

r12 X�10 � A�10

� B1

X�10 � A�10

r12 �

Z

(C1)

�2 � �m2X�21 � A�20 �I2

r22 X�21 � A�21

� B1�X�21 � V�21

r22 �

X�10 � V�10

r12 �

Z

(C2)

There are five acceleration terms and two velocity terms, so a total ofseven model neurons are introduced.

R1A � �m1X�10 � A�10�Z, R2

A � �m2X�20 � A�20�Z, R3A � �m2X�21 � A�21�Z, R4

A � � I1

r12 X�10 � A�10�

Z

,

R5A � � I2

r22 X�21 � A�21�

Z

, R6V � �B1

r12 X�10 � V�10�

Z

, R7V � �B2

r22 X�21 � V�21�

Z

(C3)

With these expressions and a minimum-jerk trajectory (Eq. 31),temporal profiles of model neurons are computed. Essentially thesame results were obtained for cosine tuning (Fig. 3A), nonuniformdistribution of preferred directions (Fig. 3B), multiple coordinatesystems (Fig. 4, E and F), population vectors (Figs. 5 and 6), andmuscle computation (Figs. 7 and 8).

ACKNOWLEDGMENTS

We thank Dr. Shinichi Furuya for inspiring discussions on initiating thiswork, Dr. Hiroshi Imamizu for suggestions on the cortical joint representa-tions, Dr. Geoffrey Hinton for discussion on learning mechanisms of spatialrepresentations, Dr. Stephen H. Scott for discussion on control of reaching, andthree anonymous reviewers for insightful comments.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by theauthor(s).

AUTHOR CONTRIBUTIONS

Author contributions: H.T. and T.J.S. conception and design of research;H.T. performed experiments; H.T. analyzed data; H.T. and T.J.S. interpretedresults of experiments; H.T. prepared figures; H.T. drafted manuscript; H.T.and T.J.S. edited and revised manuscript; H.T. and T.J.S. approved finalversion of manuscript.

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