{"title": "Where Does the Population Vector of Motor Cortical Cells Point during Reaching Movements?", "book": "Advances in Neural Information Processing Systems", "page_first": 83, "page_last": 89, "abstract": null, "full_text": "Where does the population vector of motor \n\ncortical cells point during reaching movements? \n\nPierre Baraduc* \n\npbaraduc@snv.jussieu.fr \n\nEmmanuel Guigon \n\nguigon@ccr.jussieu.fr \n\nYves Burnod \n\nybteam@ccr.jussieu.fr \n\nINSERM U483, Universite Pierre et Marie Curie \n9 quai St Bernard, 75252 Paris cedex 05, France \n\nAbstract \n\nVisually-guided  arm  reaching  movements  are  produced  by  distributed \nneural networks within parietal and frontal regions of the cerebral cortex. \nExperimental  data  indicate  that (I) single  neurons  in  these  regions  are \nbroadly tuned to parameters of movement; (2) appropriate commands are \nelaborated by  populations of neurons; (3) the coordinated action of neu(cid:173)\nrons can be  visualized  using a neuronal population vector (NPV).  How(cid:173)\never,  the  NPV  provides only  a rough  estimate  of movement parameters \n(direction, velocity) and may even fail  to reflect the parameters of move(cid:173)\nment when arm posture is changed.  We designed a model of the cortical \nmotor command to  investigate the relation between the desired direction \nof the  movement, the  actual direction of movement and  the direction of \nthe NPV in  motor cortex. The model is a two-layer self-organizing neural \nnetwork  which  combines  broadly-tuned  (muscular)  proprioceptive  and \n(cartesian) visual information to calculate (angular) motor commands for \nthe  initial  part  of the  movement of a  two-link  arm.  The  network  was \ntrained  by  motor babbling  in  5  positions.  Simulations  showed  that  (1) \nthe  network produced appropriate movement direction over a  large part \nof the  workspace;  (2) small  deviations of the  actual  trajectory  from  the \ndesired  trajectory existed  at the extremities of the  workspace;  (3)  these \ndeviations were accompanied by  large deviations of the NPV from both \ntrajectories. These results suggest the NPV does not give a faithful image \nof cortical processing during arm reaching movements. \n\n\u2022 to  whom correspondence should be addressed \n\n\f84 \n\nP.  Baraduc,  E. Guigon and Y.  Burnod \n\n1 \n\nINTRODUCTION \n\nWhen  reaching  to  an  object,  our brain  transforms  a  visual  stimulus  on  the  retina  into  a \nfinely  coordinated  motor  act.  This  complex  process  is  subserved  in  part  by  distributed \nneuronal  populations  within  parietal  and  frontal  regions  of the  cerebral  cortex  (Kalaska \nand  Crammond  1992).  Neurons  in  these  areas  contribute  to  coordinate  transformations \nby  encoding  target  position  and  kinematic  parameters  of reaching  movements  in  multi(cid:173)\nple frames of reference and  to  the elaboration of motor commands by  sending directional \nand  positional  signals  to  the  spinal  cord  (Georgopoulos  1996).  An  ubiquitous feature  of \ncortical  populations is  that  most  neurons  are  broadly  tuned  to  a  preferred  attribute  (e.g. \ndirection)  and  that  tuning  curves  are  uniformly  (or  regularly) distributed  in  the  attribute \nspace (Georgopoulos  1996).  Accordingly, a  powerful tool  to  analyse cortical  populations \nis  the NPV  which describes the behavior of a  whole  population by  a single vector (Geor(cid:173)\ngopoulos  1996).  Georgopoulos et al.  (1986) have shown that the NPV calculated on a set \nof directionally tuned  neurons in  motor cortex points approximately (error  f\"o.J  15\u00b0) in  the \ndirection  of movement.  However,  the  NPV  may  fail  to  indicate  the  correct direction  of \nmovement when  the  arm  is  in  a  particular posture  (Scott and  Kalaska  1995).  These data \nraise two important questions: (1) how populations of broadly tuned neurons learn to com(cid:173)\npute a correct sensorimotor transformation? Previous models (Burnod et al.  1992; Bullock \net al.  1993; Salinas and Abbott 1995) provided partial solutions to this problem but we still \nlack  a  model  which  closely  matches  physiological  and  psychophysical  data  on  reaching \nmovements; (2) Are cortical processes involved in  the  visual guidance of arm movements \nreadable  with  the  NPV  tool?  This  article  provides answers  to  these  questions  through  a \nphysiologically inspired model of sensorimotor transformations. \n\n2  MODEL OF THE VISUAL-TO-MOTOR TRANSFORMATION \n\n2.1  ARM GEOMETRY \n\nThe  arm  model  has  voluntarily  been  chosen  simple.  It  is  a  planar,  two-link  arm,  with \nlimited (160 degrees) joint excursion at shoulder and elbow.  An agonist/antagonist pair is \nattached at each joint. \n\n2.2 \n\nINPUT AND OUTPUT eODINGS \n\nNo cell  is  finely  tuned  to  a  specific  input or output  value  to  mimic  the  broad  tunings  or \nmonotonic firing characteristics found in cortical visuomotor areas. \n\n2.2.1  Arm position \n\nBy  analogy  with the  role of muscle spindles,  proprioceptive sensors are  assumed to code \nmuscle  length.  Arm  position  is  thus  represented  by  the  population  activity  of NT  =  20 \nneurons coding for the length of each agonist or antagonist. The activity of a sensor neuron \nk  is defined by: \n\nTk  =  adLn(k)) \n\nwhere LIl (k)  is  the length of muscle number n(k) , and ak  a piecewise linear sigmoid: \n\nSensibility thresholds Ak  are uniformly distributed in [Lmin , L max],  and the dynamic range \nis  Ak  - Ak  is taken constant, equal to  Lmax - L min . \n\nL  ~ Ak \nAk  < L  < Ak \nL? Ak \n\n\fPopulation Coding of Reaching Movements \n\n85 \n\n2.2.2  Desired direction \n\nThe  direction  V  of  the  desired  movement  in  visual  space  is  coded  by  a  population  of \nN x  =  50 neurons with cosine tuning in  cartesian space.  Each visual neuron j  thus fires  as: \n\nVj  being  the  preferred  direction  of the  cell.  These  50  preferred  directions  are  chosen \nuniformly distributed in  2-D space. \n\nXj  = V\u00b7 Vj \n\n2.2.3  Motor Command \n\nIn attempt to model the existence of muscular synergies (Lemon  1988), we identified mo(cid:173)\ntor  command with joint movement rather than  with  muscle contraction .  A  motor neuron \ni  among  Nt  =  50  contributes  to  the  effective  movement M  by  its  action  on  a  synergy \n(direction in joint space) Mi. This collective effect is  formally expressed by: \n\nM .=  LtiMi \n\nwhere  ti  is  the  activity  of motor neuron i.  The 50 directions  of action  Mi  are  supposed \nuniformly distributed in joint space. \n\n3  NETWORK STRUCTURE AND LEARNING \n\n3.1  STRUCTURE OF THE NETWORK \n\nInformation concerning the position of the arm and the desired direction in  cartesian space \n\ndesired \n= ....... ~(visual) \ndirection \n\n0-\n\n~, cY \n\nCf \n\nt; \n\n~~~~----~ \n\n. . . . . \n~~~~t\u00ae\u00ae\u00ae \n\nmotor synergy \n\nFigure  1:  Network Architecture \n\nis  combined  asymmetrically  (Fig.  I).  First,  an  intermediate  (somatic)  layer  of neurons \n\n\f86 \n\nP.  Baraduc.  E.  Guigon and Y.  Bumod \n\nforms an internal representation of the arm position by a combination of the input from the \nNT  muscle sensors and the lateral interactions inside the  population.  Activity in  this layer \nis expressed by: \n\nSij  = L Wijk Tk  + L ljp Sip \n\n(1) \n\nwhere the lateral connections are: \n\nljp = cos (27r(j - p)/NT \n\n) \n\nk \n\np \n\nEquation  1 is  self-referent;  so  calculation  is  done  in  two  steps.  The  feed-forward  input \nfirst  arrives at time zero  when there is  no activity in  the layer; iterated action of the lateral \nconnections comes into play when this feed-forward input vanishes. \n\nThe activity  in  the  somatic layer is then  combined with  the  visual  directional information \nby the output sigma-pi neurons as follows: \n\nti  = L Xj  Sij \n\n3.2  WEIGHTS AND LEARNING \n\nj \n\nThe  only  adjustable  weights  are  the  Wijk  linking  the  proprioceptive layer  to  the  somatic \nlayer.  Connectivity is  random and not complete:  only  15% of the  somatic neurons receive \ninformation on arm position.  The visuomotor mapping is learnt by  modifying the internal \nrepresentation of the arm. \n\nMotor commands issued by  the  network are correlated with the visual effect of the move(cid:173)\nment  (\"motor  babbling\").  More  precisely,  the  learning  algorithm  is  a  repetition  of the \nfollowing cycle: \n\n1.  choice of an  arm position among 5 positions (stars on Fig.  2) \n2.  random emission of a motor command (ti) \n3.  corresponding visual reafference (Xj) \n4.  weight modification according to  a variant of the delta rule: \n\nc:'Wijk  oc  (tiXj  - Sij) Tk \n\nThe  random  commands are  gaussian  distributions of activity  over the  output layer.  5000 \nlearning  epochs  are  sufficient  to  obtain  a  stabilized  performance.  It  must  be  noted  that \nthe  error between the  ideal  response of the  network and the actual  performance never de(cid:173)\ncreases completely to  zero,  as  the constraints of the  visuomotor transformation  vary  over \nthe workspace. \n\n4  RESULTS \n\n4.1  NETWORK PERFORMANCE \n\nCorrect learning of the  mapping was  tested  in  21  positions in  the  workspace in  a pointing \ntask  toward  16  uniformly distributed directions  in  cartesian  space.  Movement directions \ngenerated by  the  network are shown  in  Fig.  2 (desired direction 0 degree is  shown  bold). \nNorm of movement vectors depends on the global activity in the network which varies with \narm position and movement direction. \n\nPerformance of the  network is  maximal near the learning positions.  However, a good gen(cid:173)\neralization is obtained (directional error 0.3\u00b0, SD 12.1\u00b0); a bias toward the shoulder can be \nobserved in extreme right or left positions.  A similar effect was observed in  psychophysical \nexperiments (Ghilardi et a1.  1995). \n\n\fPopulation  Coding of Reaching Movements \n\n87 \n\n90 \n\n180 .0  \n\n270 \n\nFigure 2:  Performance in a pointing task \n\n4.2  PREFERRED DIRECTIONS AND POPULATION VECTOR \n\n4.2.1  Behavior of the population vector \n\nPreferred  directions  (PO)  of output  units  were  computed  using  a  multilinear regression; \na  perfect cosine  tuning  was  found,  which  is  a  consequence of the  exact multiplication  in \nsigma-pi  neurons.  Then,  the  population  vector,  the  effective  movement  vector,  and  the \ndesired  movement were compared (Fig.  3)  for two  different arm configurations A  and  B \nmarked on Fig.  2.  The movement generated by  the  network (dashed arrow) is  close to the \n\n~, \n\n~1' ,  , \n\ndeSired direction \ncontnbution of one neuron \n\npopulation vector ~ \nactual movement ......... ;:., \n\nFigure 3:  Actual movement and population vector in  two arm positions \n\ndesired one (dotted rays) for both arm configurations. However, the population vector (solid \narrow) is  not always aligned with the movement. The discrepancy between movement and \npopulation vector depends both on the direction and  the  position of the arm:  it is  maximal \n\n\f88 \n\nP  Baraduc,  E.  Guigon and Y  Burnod \n\nfor positions near the borders of the workspace as position B.  Fig.  3 (position B) shows that \nthe deviations of the population vector are due to the anisotropic distribution of the PDs in \ncartesian space for given positions. \n\n4.2.2  Difference between direction of action and preferred direction \n\nMarked anisotropy in  the distribution of PDs is compatible with accurate performance.  To \nsee  why,  let  us  call  \"direction of action\"  (DA)  the  motor cell's contribution to  the  move(cid:173)\nment. The distribution of DAs presents an  anisotropy due to the geometry of the arm.  This \nanisotropy is canceled by the distribution of PDs.  Mathematically, if U  is a N  x 2 matrix of \nuniformly distributed 2D vectors, the PD matrix is  UJ-1  whereas the DA matrix is UJT , \nJ  being the jacobian of the angular-to-cartesian mapping.  Difference between DA and PD \nhas  been  plotted  with  concentric arcs  for  four  representative  neurons at  21  arm  positions \nin  Fig.  4.  Sign and magnitude of the difference vary continuously over the  workspace and \n\nneuron number  4 \n\n/ Vi: \n\nDA, \n\n_ \n\nclockwise \n\n= \n\ncounterclockwise \n\n\"  .. \n\nFigure 4:  Difference between direction of action and preferred direction for four units. \n\noften exceed 45  degrees.  It can also be noted that preferred directions rotate  with  the arm \nas was experimentally noted by (Caminiti et a1.  1991). \n\n5  DISCUSSION \n\nWe  first  asked  how  a  network  of broadly  tuned  neurons  could  produce  visually  guided \narm  movements.  The  model  proposed  here  produces  a  correct behavior over  the  entire \nworkspace.  Biases were observed at the extreme right and left which closely resemble ex(cid:173)\nperimental data in  humans (Ghilardi et a1.  1995). Single cells in  the output layer behave as \nmotor cortical cells do and the NPV of these cells correctly indicated the direction of move(cid:173)\nment for  hand  positions in  the central  region of the  workspace (see  Caminiti et al.  1991). \nModels of sensorimotor transformations have already been proposed.  However they either \nconsidered motor synergies in  cartesian coordinates (Burnod et a1.  1992), or used sharply \n\n\fPopulation  Coding of Reaching Movements \n\n89 \n\ntuned  units  (Bullock  et  al.  1993),  or motor effects  independent of arm  position  (Salinas \nand  Abbott  1995).  Next,  the  use of the NPV to  describe cortical activity  was questioned. \nA  fundamental assumption in  the calculation of the NPV is  that the  PD of a  neuron is  the \ndirection  in  which  the arm  would  move if the  neuron  were stimulated.  The model shows \nthat the two directions DA and PD do not necessarily coincide, which is  probably the case \nin  motor  cortex  (Scott  and  Kalaska  1995).  It  follows  that  the  NPV  often  points  neither \nin  the actual movement direction nor in  the desired  movement direction (target direction), \nespecially for unusual arm configurations. A maximum-likelihood estimator does not have \nthese  flaws;  it  would  however accurately  predict the  desired movement out of the  output \nunit  activities,  even  for  a  wrong  actual  movement.  In  conclusion:  (l) the  NPV  does  not \nprovide  a  faithful  image  of cortical  visuomotor  processes;  (2)  a  correct  NPV  should  be \nbased on  the DAs,  which cannot easily be determined experimentally; (3) planning of tra(cid:173)\njectories in  space cannot be realized by the successive recruitment of motor neurons whose \nPDs sequentially describe the movement. \n\nReferences \n\nBullock,  D.,  S.  Grossberg,  and  F.  Guenther  (1993).  A  self-organizing  neural  model  of \nmotor equivalent reaching and tool  use by  a multijoint arm.  J  Cogn Neurosci 5(4), 408-\n435. \nBurnod, Y.,  P.  Grandguillaume, I.  Otto,  S.  Ferraina, P. Johnson, and  R  Caminiti  (1992). \nVisuomotor transformations  underlying arm  movements  toward  visual  targets:  a  neural \nnetwork model of cerebral cortical operations. J Neurosci 12(4),  1435-53. \nCaminiti, R, P.  Johnson, C.  Galli,  S. Ferraina, and Y.  Burnod (1991).  Making arm move(cid:173)\nments within different parts of space:  the premotor and motor cortical representation of a \ncoordinate system for reaching to  visual targets.  J Neurosci 11(5), 1182-97. \nGeorgopoulos, A  (1996).  On  the  translation  of directional  motor cortical  commands to \nactivation of muscles  via spinal interneuronal systems.  Brain Res Cogn  Brain  Res 3(2), \n151-5. \nGeorgopoulos, A, A  Schwartz,  and  R  Kettner (1986).  Neuronal  population coding of \nmovement direction . Science 233(4771), 1416-9. \nGhilardi, M. , J.  Gordon, and C.  Ghez (1995).  Learning a visuomotor transformation in  a \nlocal area of work space pr oduces directional biases in other areas.  J NeurophysioI73(6), \n2535-9. \nKalaska,  J.  and  D.  Crammond (1992).  Cerebral cortical  mechanisms of reaching move(cid:173)\nments.  Science 255(5051),1517-23. \nLemon, R  (1988).  The output map of the primate motor cortex.  Trends Neurosci 11 (II), \n501-6. \nSalinas,  E.  and L.  Abbott (1995).  Transfer of coded information  from  sensory  to  motor \nnetworks.  J Neurosci 15(10),6461-74. \nScott, S.  and J.  Kalaska (1995).  Changes in  motor cortex activity during reaching move(cid:173)\nments with similar hand paths but different arm postures.  J Neurophysioi 73(6), 2563-7. \n\n\f", "award": [], "sourceid": 1494, "authors": [{"given_name": "Pierre", "family_name": "Baraduc", "institution": null}, {"given_name": "Emmanuel", "family_name": "Guigon", "institution": null}, {"given_name": "Yves", "family_name": "Burnod", "institution": null}]}