{"title": "Computational Structure of coordinate transformations: A generalization study", "book": "Advances in Neural Information Processing Systems", "page_first": 1125, "page_last": 1132, "abstract": null, "full_text": "Computational structure of coordinate \ntransformations:  A  generalization study \n\nZoubin Ghahramani \nzoubin@psyche.mit.edu \n\nDaniel M.  Wolpert \nwolpert@psyche.mit.edu \n\nMichael I.  Jordan \njordan@psyche.mit.edu \n\nDepartment of Brain &  Cognitive Sciences \n\nMassachusetts  Institute of Technology \n\nCambridge, MA  02139 \n\nAbstract \n\nOne  of the fundamental properties  that both neural  networks  and \nthe  central  nervous  system share is  the  ability to learn  and gener(cid:173)\nalize  from examples.  While this  property  has  been  studied  exten(cid:173)\nsively  in  the  neural  network  literature  it  has  not  been  thoroughly \nexplored in human perceptual and motor learning.  We have chosen \na  coordinate  transformation  system-the  visuomotor  map  which \ntransforms visual coordinates into motor coordinates-to study the \ngeneralization effects  of learning new  input-output  pairs.  Using  a \nparadigm of computer  controlled  altered  visual  feedback,  we  have \nstudied  the  generalization  of the  visuomotor  map  subsequent  to \nboth local  and context-dependent  remappings.  A  local  remapping \nof one  or  two  input-output  pairs  induced  a  significant  global,  yet \ndecaying,  change  in  the  visuomotor map, suggesting  a  representa(cid:173)\ntion for  the  map composed of units  with large functional  receptive \nfields.  Our study of context-dependent  remappings indicated that \na  single  point  in  visual  space  can  be  mapped to  two different  fin(cid:173)\nger  locations  depending  on  a  context  variable-the starting  point \nof  the  movement.  Furthermore,  as  the  context  is  varied  there  is \na  gradual  shift  between  the  two  remappings,  consistent  with  two \nvisuomotor  modules  being  learned  and  gated  smoothly  with  the \ncontext. \n\n1 \n\nIntroduction \n\nThe  human  central  nervous  system  (CNS)  receives  sensory  inputs  from  a  multi(cid:173)\ntude  of modalities,  each  tuned  to  extract  different  forms  of information from  the \n\n\f1126 \n\nZoubin  Ghahramani,  Daniel M.  Wolpert,  Michael 1.  Jordan \n\nenvironment.  These  sensory  signals  are  initially  represented  in  disparate  coordi(cid:173)\nnate systems, for  example visual information is  represented  retinotopically whereas \nauditory  information is  represented  tonotopically.  The  ability to  transform  infor(cid:173)\nmation  between  coordinate  systems  is  necessary  for  both  perception  and  action. \nWhen  we  reach  to  a  visually  perceived  object  in  space,  for  example,  the  location \nof the  object  in  visual  coordinates  must be  converted  into a  representation  appro(cid:173)\npriate  for  movement,  such  as  the  configuration  of the  arm  required  to  reach  the \nobject.  The  computational structure  of this  coordinate  transformation,  known  as \nthe visuomotor map, is  the focus  of this  paper. \n\nBy  examining the  change  in  visuomotor  coordination  under  prismatically induced \ndisplacement  and  rotation,  Helmholtz  (1867/1925)  and  Stratton  (1897a,1897b)  pi(cid:173)\noneered the systematic study of the representation  and plasticity of the visuomotor \nmap.  Their studies  demonstrate  both the fine  balance between  the  visual  and  mo(cid:173)\ntor  coordinate  systems,  which  is  disrupted  by  such  perturbations,  and  the  ability \nof the visuomotor map to adapt to the displacements induced  by the  prisms.  Sub(cid:173)\nsequently,  many studies have further  demonstrated the remarkable plasticity of the \nmap in response  to a  wide variety of alterations in the relationship  between  the  vi(cid:173)\nsual  and motor system (for  reviews  see  Howard,  1982  and Welch,  1986)-the single \nprerequisite for  adaptation seems to be that the remapping be stable (Welch,  1986). \nMuch  less  is  known,  however,  about  the  topological properties of this map. \n\nA coordinate transformation such as the visuomotor map can be regarded as  a func(cid:173)\ntion relating one set  of variables (inputs)  to another  (outputs).  For the  visuomotor \nmap the  inputs  are  visual  coordinates  of a  desired  target  and  the  outputs  are  the \ncorresponding  motor  coordinates  representing  the  arm's  configuration  (e.g.  joint \nangles).  The  problem of learning a  sensorimotor  remapping can  then  be  regarded \nas  a  function  approximation problem.  Using  the  theory  of function  approximation \none  can make explicit the correspondence  between  the representation  used  and the \npatterns  of generalization  that  will  emerge.  Function  approximators  can  predict \npatterns  of generalization  ranging  from  local  (look-up  tables),  through  intermedi(cid:173)\nate  (CMACs,  Albus,  1975;  and  radial basis  functions,  Moody  and  Darken,  1989  ) \nto global (parametric models). \n\nIn  this  paper  we  examine  the  representational  structure  of the  visuomotor  map \nthrough  the  study  of its  spatial  and  contextual  generalization  properties.  In  the \nspatial generalization study  we  address  the  question  of how  pointing changes  over \nthe  reaching  workspace  after  exposure  to  a  highly  localized  remapping.  Previous \nwork on spatial generalization, in a study restricted to one dimension, has led to the \nconclusion  that the  visuomotor map is  constrained  to generalize linearly  (Bedford, \n1989).  We test this conclusion by mapping out the pattern of generalization induced \nby one  and two remapped  points in  two dimensions. \n\nIn the contextual generalization study we  examine the  question of whether  a  single \npoint in visual space can be mapped into two different finger  locations depending on \nthe  context  of a  movement-the start point.  If this  context-dependent  remapping \noccurs,  the  question  arises  as  to how  the mapping will  generalize  as  the  context  is \nvaried.  Studies of contextual remapping have  previously shown  that variables such \nas eye position (Kohler, 1950; Hay and Pick, 1966; Shelhamer et al.,  1991), the feel of \nprisms (Kravitz,  1972; Welch,  1971) or an auditory tone (Kravitz and Yaffe,  1972), \ncan  induce  context-dependent  aftereffects.  The  question  of  how  these  context-\n\n\fComputational Structure of Coordinate  Transformations \n\n1127 \n\ndependent  maps  generalize-which  has  not  been  previously  explored-reflects  on \nthe  possible  representation  of multiple  visuomotor  maps  and  their  mixing  with  a \ncontext  variable. \n\n2  Spatial  Generalization \n\nTo  examine  the  spatial  generalization  of  the  visuomotor  map  we  measured  the \nchange in pointing behavior subsequent to one- and two-point remappings.  In order \nto measure pointing behavior and to confine  subjects  to learn limited input-output \npairs  we  used  a  virtual  visual  feedback  setup  consisting  of  a  digitizing  tablet  to \nrecord  the  finger  position  on-line  and  a  projection/mirror  system  to  generate  a \ncursor  spot  image representing  the finger  position  (Figure  1a).  By  controlling the \npresence of the cursor spot and its relationship to the finger  position, we  could both \nrestrict visual feedback  of finger  position to localized regions of space and introduce \nperturbations  of this feedback. \n\na) \n\nVGAScreen \n\nPnljecIor \n\n\\  Angerf_ \n\nIi  \\ \n/, \n\n/0:  \\  image \n\n\\ \n\n/ ,  \n/ ,  \n\n/  ,/ ,l \n\n\\ \n\n\\)  Rear profection acreen \n\n-,'  >'\"'!'f'----Vi-rtual-lma-II\"Seml-sliverad mirTOr \n\nDl!Jtizjng Tablel \n\nActual \n\nFIngerPosI1Ion \n\n'.. ~ \n\n;(.. \"'\" \n.... \n\n.. \n'.. \n-\" \n\no \n\no \n\n.......... \n\na \n\n150m \n\nb) \nP_   0 \nFinger Position \n\n0 \n\n0 \n\nActual \n\nAnger Position \n\n~~ \n\n, \n\nC)[J' , \n..... \n\u00b7  .  . \nd)Q' . \ne)lZj' , \n\"  ,t , \n, \n\n\u2022  1 \n\u2022 \n\u2022  T  \u2022 \n\n, \n\n, \n\nFigure  1.  a)  Experimental  setup.  The  subjects  view  the  reflected  image  of  the \nrear  projection  screen  by  looking  down  at  the  mirror.  By  matching  the  screen(cid:173)\nmirror  distance  to the mirror-tablet  distance  all  projected images  appeared  to be in \nthe  plane  of  the  finger  (when  viewed  in  the  mirror)  independent  of head  position. \nb)  The  position  of  the  grid  of  targets  relative  to  the  subject.  Also  shown,  for  the \nx-shift  condition,  is  the  perceived  and  actual  finger  position  when  pointing  to  the \ncentral training target.  The visually  perceived finger position is indicated  by  a cursor \nspot  which  is  displaced  from  the actual finger  position.  c)  A  schematic  showing  the \nperturbation  for  the  x-shift  group.  To see  the cursor  spot  on  the central  target  the \nsubjects  had  to  place  their  finger  at  the  position  indicated  by  the  tip  of the arrow. \nd)  &  e)  Schematics  similar  to  c)  showing  the  perturbation  for  the  y-shift  and  two \npoint  groups,  respectively. \n\nIn the tradition of adaptation studies  (e.g.  Welch,  1986), each experimental session \nconsisted  of three  phases:  pre-exposure,  exposure,  and  post-exposure.  During the \npre- and post-exposure  phases,  designed  to assess  the visuomotor map, the subject \npointed  repeatedly,  without  visual feedback  of his finger  position,  to  a  grid  of tar(cid:173)\ngets over the workspace.  As  no visual input of finger  location was given, no learning \nof the  visuomotor  map  could  occur .  During  the  exposure  phase  subjects  pointed \nrepeatedly  to one or two  visual  target locations,  at which  we  introduced  a  discrep-\n\n\f1128 \n\nZoubin  Ghahramani,  Daniel M.  Wolpert,  Michael!.  Jordan \n\nancy  between  the  actual  and  visually displayed finger  location.  No  visual feedback \nof finger  position was given except when within 0.5 cm of the target, thereby confin(cid:173)\ning any learning to the  chosen  input-output pairs.  Three local perturbations of the \nvisuomotor  map  were  examined:  a  10  cm  rightward  displacement  (x-shift  group, \nFigure  lc),  10  cm displacement towards the body  (y-shift  group,  Figure  Id),  and a \ndisplacement at two points, one  10  cm away from,  and one  10  cm towards the body \n(two point group , Figure  Ie).  For example, for  the  x-shift displacement the subject \nhad to place his finger  10 cm to the right of the target to visually perceive  his finger \nas  being on target  (Figure  Ib).  Separate control subjects,  in which the relationship \nbetween  the  actual  and  visually  displayed  finger  position  was  left  unaltered,  were \nrun for  both the one- and  two-point  displacements, resulting in  a  total of 5 groups \nwith  8 subjects  each. \n\n50 \n\n-\u20ac) \n\n50  __  .......................... \n\n\" \n\n45 \n\n40 \n\nJS \n\n30 \n\n25 \n\n] \n>-\n\nX  \u00ab(,:m) \n\nCl \n\n45  CJ \n\n41) \n\nt9 \n\n-10 \n\n-5 \n\n10 \n\n15 \n\n20 \n\nX  (I.:m ) \n\n25 \n\n20 \n\n55 \n\n50 \n\n45 \n\nc \n\n40 \n\n'0  e \n0.  ~ 35 \n~  >- 30 \nE-\n\n25 \n\n20 \n\n15 \n\nG \n\n4'  __  .......................... \n\n..... \n\n40  __  ............  _ \n\n.. ..';1.'  __  ............ __  \n,. \n\n........ \n\n_ \n\n30 \n\n........ \n\n--.....  ~~ .............. \n\n20 \n\n45 \n\n40 \n\n\\ \n\n\\ \n\n~ 35 \n\n>- 30 \n\n\\ \n\n10 \n\n1 ~ \n\n20 \n\nX  (Lm) \n\nX (em) \n\nI \n\n25 \n\n20 \n\n55 \n\n5(J \n\n45 \n\n30 \n\n25 \n\n2(J \n\n15 \n\n-10 \n\n\".'i \n\n10 \n\n15 \n\n20 \n\n-10 \n\n() \n\n10 \n\n2() \n\nX  (em) \n\nx  (em) \n\nt \n\n,  , , \n\"  ,  \\ \nI '  t \nt \n~  I \n~  tit \n\nI \n\nt \n\nt t \nJ f \n\nI \n\nI \n\nI \nI \n\nI \n\n-15  \u00b7 10 \n\n-5 \n\n0 \n\n5 \n\n10  15  20  2.' \n\n-15  -10\u00b75 \n\nI) \n\n5 \n\n11)  15 \n\n211  25 \n\n-10 \n\n() \n\n!() \n\n2() \n\nx  (em) \n\nX (em) \n\nX (em) \n\nFigure  2.  Results  of  the  spatial  generalization  study.  The  first  column  shows  the \nmean  change  in  pointing,  along  with  95%  confidence  ellipses,  for  the  x-shift,  y-shift \nand two point groups.  The second column  displays  a vector field  of changes obtained \nfrom  the data by  Gaussian  kernel smoothing.  The third  column  plots the proportion \nadaptation  in  the direction  of the  perturbation.  Note that whereas for  the  x- and  y(cid:173)\nshift  groups  the lighter  shading  corresponds  to greater  adaptation,  for  the two point \ngroup lighter  shades correspond  to  adaptation in  the positive  y  direction  and darker \nshades in  the negative  y  direction. \n\n\fComputational Structure  of Coordinate  Transfonnations \n\n1129 \n\nThe  patterns  of spatial  generalization  subsequent  to  exposure  to  the  three  local \nremappings  are  shown  in  Figure  2.  All  three  perturbation  groups  displayed  both \nsignificant  adaptation at the  trained  points,  and significant,  through decremented, \ngeneralization  of this  learning  to  other  targets.  As  expected,  the  control  groups \n(not  shown)  did  not  show  any  significant  changes.  The extent  of spatial  general(cid:173)\nization,  best  depicted  by  the  shaded  contour  plots in  Figure  2,  shows  a  pattern  of \ngeneralization  that  decreases  with distance  away from  the  trained  points.  Rather \nthan inducing  a  single  global  change  in  the  map,  such  as  a  rotation  or  shear,  the \ntwo point exposure  appears to induce two opposite fields  of decaying generalization \nat the intersection  of which there  is  no change in the  visuomotor map. \n\n3  Contextual Generalization \n\nThe goal of this experiment was first  to explore  the possibility that multiple visuo(cid:173)\nmotor maps, or modules,  could  be learned,  and if so,  to determine how  the overall \nsystem behaves  as  the  context  used  in  training each  module is  varied.  To achieve \nthis  goal,  we  exposed  subjects  to  context-dependent  remappings  in  which  a  sin(cid:173)\ngle  visual  target  location  was  mapped  to  two  different  finger  positions  depending \non  the  start  point  of the  movement.  Pointing  to  the  target  from  seven  different \nstarting points (Figure 3)  was  assessed  before  and after an exposure phase.  During \nthis exposure  phase subjects made repeated  movements to the target from starting \npoints  2  and  6  with  a  different  perturbation  of the  visual  feedback  depending  on \nthe  starting  point .  The  form  of these  context-dependent  remappings  is  shown  in \nFigure  3.  For example, for  the open  x-shift group  (Figure  3c),  the  visual feedback \nof the finger  was  displaced  to the right for  movements from  point 2 and to the left \nfrom point  6.  Therefore  the same visual target  was  mapped to two  different  finger \npositions depending on the context of the movement.  To test learning of the remap(cid:173)\nping and  generalization  to other start points we  examined the  change  in  pointing, \nwithout visual feedback,  to the target from the  7 start points. \n\na) control \n\nb) crossed x-shift \n\n~~' \n-' . \n... \n\n.. ' \n.. -\n\n-' \u2022 \n\ne'-\n\nt, \n\n\u2022 t \n\no \n\ndlYL\\ \n\n:,., \n\n.... \n\n1234567   1234567  \n\nc) open x-shift \n\n.. ' . \" \n........... \n\n~ \n\n. .....\u2022 \n\n...\u2022.\u2022. \n\no \n1234567   1234567  \n\n0 \n\no \n\nFigure  3.  Schematic  of  the exposure  phase  in  the  contextual  generalization  exper(cid:173)\niment.  Shown  are  the  actual  finger  path  (solid  line),  the  visually  displayed  finger \npath  (dotted line),  the seven  start  points  and  the  target  used  in  the  pre- and  post(cid:173)\nexposure  phases.  The perturbation  introduced  depended  on whether  the  movement \nstarted form start point 2 or 6.  Note that for the three perturbation groups,  although \nthe subjects saw a  triangle  being  traced out,  the finger  took a  different  path. \n\n\f1130 \n\nZoubin Ghahramani,  Daniel M.  Wolpert,  Michael I.  Jordan \n\nThe results are shown in Figure 4.  Whereas the controls did not show any significant \npattern of change,  the three  other  groups showed  adaptive, start point dependent, \nchanges  in  the  direction  opposite  to  the  perturbation.  Thus,  for  example,  the  x(cid:173)\nopen group  displayed a  pattern of change in the leftward (negative  x)  direction for \nmovements from the left start points  and rightwards for  movements from the right \nstart  points.  Furthermore,  as  the  start  point  was  varied,  the  change  in  pointing \nvaried gradually. \n\na)  U \n\ncontrol  0--0 \ncroned  x\u00b7,bltt  - -\n\nb)  2.0 \n\n0 \n\na 0.' \n!!. \n\" \n.;J  -0.' \n... \nc. \n.. \n><  -u \n\n.\" \n\na \n!!.  1.0 \n\" \n.S! \n;; \n.. \nc. \n.. \n... \n\n0.0 \n\n.\" \n\n~v~ \n\n-2.' \n\n, \n4 \n\nStart point \n\n-1.0 \n\n2 \n\n, \n4 \n\nStart point \n\nFigure 4.  a)  Adaptation in the  x  direction  plotted  as  a function  of starting point  for \nthe control,  crossed x-shift and open x-shift groups (mean and 1 s.e.).  b) Adaptation \nin  the  y  direction  for  the control and y-shift  groups. \n\n4  Discussion \n\nClearly,  from  the  perspective  of  function  approximation  theory,  the  problem  of \nrelearning the visuomotor mapping from exposure to one or two input-output pairs \nis  ill-posed.  The mapping learned,  as  measured  by the pattern of generalization to \nnovel  inputs,  therefore  reflects  intrinsic  constraints  on  the  internal  representation \nused. \nThe results from the spatial generalization study suggest that the visuomotor coor(cid:173)\ndinate transformation is  internally represented  with  units  with  large  but localized \nreceptive  fields.  For  example,  a  neural  network  model  with  Gaussian  radial  basis \nfunction units  (Moody  and Darken,  1989),  which  can  be  derived by assuming that \nthe internal constraint in the visuomotor system is  a smoothness constraint (Poggio \nand Girosi,  1989), predicts  a  pattern of generalization very  similar the one experi(cid:173)\nmentally observed  (e.g.  see Figure 5 for a simulation of the two point generalization \nexperiment).1  In contrast,  previously proposed models for  the representation of the \nvisuomotor map based  on  global parametric representations  in  terms of felt  direc(cid:173)\ntion  of gaze  and  head  position  (e.g.  Harris,  1965)  or  linear  constraints  (Bedford, \n1989)  do not predict  the decaying patterns of Cartesian generalization found. \n\n1 See  also  Pouget  &  Sejnowski (this  volume)  who,  based on a related  analysis  of neuro(cid:173)\n\nphysiological  data from  parietal cortex,  suggest  that  a  basis function  representation  may \nbe used  in  this  visuomotor  area. \n\n\fComputational Structure  of Coordinate  Transformations \n\n1131 \n\n50 \n\n, \n\n45 \n\n\\ \n\n\\ \n\n\\ \n\n\\ \n\n>-\n\n40  \\  \\  \\ \n~ 35  \\  ~ ! \n\n30  \\ \n25  ~ \n\n~ \n\n20 \n\n\\ \n\nt \n\nt  } ~ \nt \ne \nI  t  t  >-\n\n~ \n\nI \n\nI \n\nI \n\nI' \n\nI' \n\n\u00b7 10 \n\n\u00b7 5 \n\n0 \n\n10 \n\n15 \n\n20 \n\nX  (em) \n\nX (em) \n\nFigure 5.  Simulation of the two point spatial generalization experiment using  a radial \nbasis function  network with 64  units with 5 cm Gaussian  receptive fields.  The inputs \nto the network were the visual coordinates of the target and the outputs were the joint \nangles  for  a  two-link  planar arm  to  reach  the  target.  The network  was first  trained \nto  point  accurately  to the  targets, and  then,  after exposure  to  the  perturbation,  its \npattern of generalization  was  assessed. \n\nThe  results  from  the  second  study suggest  that multiple visuomotor maps can  be \nlearned  and  modulated by  a  context.  A  suggestive  computational model  for  how \nsuch separate  modules can be learned  and combined is  the mixture-of-experts  neu(cid:173)\nral  network  architecture  (Jacobs  et  al. ,  1991).  Interpreted  in  this  framework,  the \ngradual  effect  of varying  the  context  seen  in  Figure  4  could  reflect  the  output  of \na  gating  network  which  uses  context  to  modulate  between  two  visuomotor  maps. \nHowever,  our  results  do  not  rule  out  models  in  which  a  single  visuomotor map  is \nparametrized by  starting location, such  as  one based on the coding of locations via \nmovement vectors  (Georgopoulos,  1990) . \n\n5  Conclusions \n\nThe goal of these studies has been to infer the internal constraints in the visuomotor \nsystem through the study of its patterns of generalization to local  remappings.  We \nhave found that local perturbations of the visuomotor map produce global changes, \nsuggesting  a  distributed  representation  with  large  receptive  fields.  Furthermore, \ncontext-dependent perturbations induce changes in pointing consistent with a model \nof visuomotor learning in which separate maps are learned and gated by the context. \nThe  approach  taken  in  this  paper  provides  a  strong  link  between  neural  network \ntheory and  the  study of learning in  biological systems. \n\nAcknowledgements \n\nThis project was supported in part by a grant from the McDonnell-Pew Foundation , \nby  a  grant from  ATR Human Information Processing  Research  Laboratories,  by  a \ngrant from Siemens Corporation, and by grant  N00014-94-1-0777 from the Office of \nNaval  Research.  Zoubin  Ghahramani and  Daniel  M.  Wolpert  are  McDonnell-Pew \nFellows  in  Cognitive  Neuroscience.  Michael I.  Jordan  is  a  NSF  Presidential Young \nInvestigator. \n\n\f1132 \n\nZoubin  Ghahramani,  Daniel M.  Wolpert,  Michael I.  Jordan \n\nReferences \n\nAlbus,  J.  (1975) .  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(1986).  Adaptation to space perception.  In Boff,  K., Kaufman,  L. , and Thomas, \nJ .,  editors,  Handbook  of perception  and performance, volume  1,  pages  24- 1-24-45. \nWiley-Interscience,  New  York. \n\n\f", "award": [], "sourceid": 948, "authors": [{"given_name": "Zoubin", "family_name": "Ghahramani", "institution": null}, {"given_name": "Daniel", "family_name": "Wolpert", "institution": null}, {"given_name": "Michael", "family_name": "Jordan", "institution": null}]}