{"title": "Spatial Representations in the Parietal Cortex May Use Basis Functions", "book": "Advances in Neural Information Processing Systems", "page_first": 157, "page_last": 164, "abstract": null, "full_text": "Spatial Representations in  the Parietal \n\nCortex May Use Basis Functions \n\nAlexandre Pouget \n\nalex@salk.edu \n\nTerrence J.  Sejnowski \n\nterry@salk.edu \n\nHoward  Hughes  Medical  Institute \n\nDepartment of Biology \n\nUniversity of California, San Diego \n\nThe Salk Institute \nLa Jolla,  CA 92037 \n\nand \n\nAbstract \n\nThe  parietal  cortex  is  thought  to  represent  the  egocentric  posi(cid:173)\ntions  of objects  in  particular  coordinate systems.  We  propose  an \nalternative  approach  to  spatial  perception  of objects  in  the  pari(cid:173)\netal  cortex from  the  perspective  of sensorimotor  transformations. \nThe responses of single parietal neurons can be modeled as  a gaus(cid:173)\nsian  function  of retinal  position  multiplied by  a  sigmoid function \nof eye  position,  which  form  a  set of basis functions.  We  show  here \nhow  these  basis functions  can  be  used  to generate  receptive  fields \nin  either  retinotopic or head-centered  coordinates by simple linear \ntransformations.  This raises the possibility that the parietal cortex \ndoes  not  attempt to compute the positions of objects  in  a  partic(cid:173)\nular  frame  of  reference  but  instead  computes  a  general  purpose \nrepresentation  of the retinal  location and eye  position from  which \nany  transformation can  be  synthesized  by  direct  projection.  This \nrepresentation  predicts that hemineglect,  a  neurological syndrome \nproduced  by  parietal lesions,  should  not  be  confined  to egocentric \ncoordinates,  but should be observed in multiple frames of reference \nin  single patients,  a  prediction supported  by several experiments. \n\n\f158 \n\nAlexandre Pouget,  Terrence J.  Sejnowski \n\n1 \n\nIntroduction \n\nThe temporo-parietal junction in  the human cortex and its equivalent in monkeys, \nthe inferior parietal lobule,  are thought to playa critical role  in spatial perception. \nLesions  in  these  regions  typically result  in  a  neurological syndrome,  called  hemine(cid:173)\nglect,  characterized by a lack of motor exploration toward the hemispace contralat(cid:173)\neral  to  the  site  of the  lesion.  As  demonstrated  by  Zipser  and  Andersen  [11),  the \nresponses  of single cells in the monkey parietal cortex are  also  consistent  with this \npresumed role in spatial perception. \n\nIn the general case,  recovering the egocentric position of an object from its multiple \nsensory  inputs  is  difficult  because  of the  multiple  reference  frames  that  must  be \nintegrated .  In  this  paper,  we  consider  a  simpler  situation  in  which  there  is  only \nvisual input and all  body parts are fixed  but the eyes,  a  condition which  has  been \nextensively  used  for  neurophysiological  studies  in  monkeys.  In  this  situation,  the \nhead-centered  position  of an  object,  X,  can  be  readily  recovered  from  the  retinal \nlocation,R,  and current eye  position, E,  by vector  addition: \n\n(1) \n\nIf the parietal cortex contains a representation of the egocentric position of objects, \nthen  one  would  expect  to find  a  representation  of the  vectors,  X,  associated  with \nthese  objects.  There  is  an  extensive  literature  on  how  to  encode  a  vector  with \na  population  of neurons,  and  we  first  present  two  schemes  that  have  been  or  are \nused  as  working hypothesis  to study  the parietal  cortex.  The first  scheme  involves \nwhat  is  typically  called  a  computational map, whereas  the second  uses  a  vectorial \nrepresentation  [9]. \n\nThis paper  shows  that none  of these  encoding schemes  accurately  accounts  for  all \nthe  response  properties  of single  cells  in  the  parietal  cortex.  Instead,  we  propose \nan alternative hypothesis which  does  not aim at representing X per se;  instead,  the \ninputs  Rand E are  represented  in  a  particular basis  function  representation.  We \nshow  that  this scheme  is  consistent  with  the  way  parietal neurons  respond  to  the \nretinal position of objects  and eye  position,  and we  give  computational arguments \nfor  why this might be an efficient  strategy for  the cortex. \n\n2  Maps and  Vectorial Representations \n\nOne way to encode a two-dimensional vector is  to use a lookup table for  this vector \nwhich,  in  the  case  of  a  two-dimensional  vector,  would  take  the  form  of  a  two(cid:173)\ndimensional neuronal map.  The parietal cortex may represent  the egocentric  loca(cid:173)\ntion of object,  X,  in a  similar fashion.  This predicts that the visual receptive  field \nof parietal neurons have a  fixed  position with respect  to the head  (figure  IB).  The \nwork  of Andersen  et  al.  (1985)  have  clearly  shown  that  this  is  not  the  case.  As \nillustrated in figure  2A,  parietal neurons have retinotopic receptive fields . \n\nIn  a  vectorial representation,  a  vector  is encoded  by  N units,  each  of them coding \nfor  the projection  of the  vector  along its preferred  direction.  This entails that the \nactivity,  h,  of a  neuron is  given  by: \n\n\fSpatial Representations in  the  Parietal Cortex May  Use  Basis  Functions \n\n159 \n\nA \n\nB  Map Representation \n\n~ V \n\nv x \n\nC  Vectorial Representation \n\n0 \n\no \no \n\n\u00b0io \n0 j[ \n\n\u00b71110 \n\n0 \n\n-'JO \n90 \na (I)'gr\") \n\n180 \n\nFigure  1:  Two neural  representations  of a  vector.  A)  A  vector  if in  cartesian  and \npolar coordinates.  B)  In  a map representation, units have a narrow gaussian tuning \nto the horizontal and vertical components of if . Moreover,  the position of the peak \nresponse is directly related to the position of the units on the map.  C) In  a vectorial \nrepresentation ,  each  unit encodes  the  projection of if  along  its  preferred  direction \n( central  arrows) .  This results in  a  cosine  tuning to the  vector  angle,  ()  . \n\n(2) \n\nWa  is  usually called  the preferred  direction of the cells  because  the activity is  max(cid:173)\nimum whenever  ()  =  0;  that  is,  when  A points  in  the  same  direction  as  Wa.  Such \nneurons  have  a  cosine  tuning to the  direction  of the egocentric  location of objects, \nas shown  also in figure  lC. \n\nCosine  tuning  curves  have  been  reported  in  the  motor  cortex  by  Georgopoulos  et \nal.  (1982) , suggesting that the motor cortex  uses  a  vectorial  code for  the  direction \nof hand movement in extrapersonal space.  The same scheme has  been  also used  by \nGoodman and Andersen (1990),  and Touretzski  et  al.  (1993) to model the encoding \nof egocentric position of objects in the parietal cortex.  Touretzski et al.  (1993) called \ntheir representation  a  sinusoidal array instead of a  vectorial representation. \n\nUsing  Eq.  1,  we  can  rewrite Eq.  2: \n\n(3) \n\nThis second  equation is  linear  in  Ii  and if and uses  the same vectors , Wa ,  in  both \ndot  products.  This leads to three  important predictions: \n1)  The visual receptive  fields  of parietal neurons should  be planar. \n\n2)  The eye  position receptive fields  of parietal neurons should  also  be  planar;  that \nis,  for  a  given  retinal  positions,  the  response  of parietal neuron  should  be  a  linear \nfunction  of eye  position. \n\n\f160 \n\nAlexandre Pouget,  Terrence J.  Sejnowski \n\nA \n\nB \n\n@  \u2022 \n\no \n\n-101'--'---'--~-'--'--'---'---' \n40 \n\n-40 \n\n20 \n(Deg) \n\n-20 \nRetinal Position \n\n0 \n\nFigure 2: Typical response of a neuron in the parietal cortex of a monkey.  A)  Visual \nreceptive  field  has  a  fixed  position  on  the  retina,  but  the  gain  of the  response  is \nmodulated by eye  position (ex).  (Adpated from Andersen  et  al.,  1985) B)  Example \nof an eye  position receptive field,  also called gain field,  for  a  parietal cell.  The nine \ncircles indicate the amplitude of the response to an identical retinal stimulation for \nnine  different  eye  positions.  Outer  circles  show  the  total  activity,  whereas  black \ncircles  correspond  to  the  total response  minus spontaneous  activity  prior to visual \nstimulation.  (Adpated from  Zipser  et  al.,  1988) \n\n3)  The preferred  direction for  retinal location and eye  position should  be  identical. \nFor  example,  if the  receptive  field  is  on  the  right  side  of the  visual  field ,  the  gain \nfield  should also increase  with eye  positon to the right side. \n\nThe visual receptive fields  and the eye  position gain fields  of single parietal neurons \n[2].  In  most  cases,  the  visual \nhave  been  extensively  studied  by  Andersen  et  al. \nreceptive  fields  were  bell-shaped  with  one  or  several  peaks  and  an  average  radius \nof  22  degrees  of visual  angle  [1],  a  result  that  is  clearly  not  consistent  with  the \nfirst  prediction  above.  We  show  in  figure  2A  an  idealized  visual  receptive  field  of \na  parietal  neuron.  The  effect  of eye  position  on  the  visual  receptive  field  is  also \nillustrated.  The eye  position clearly  modulates the gain of the visual  response. \n\nThe prediction  regarding  the receptive field  for  eye  position has been  borne out by \nstatistical analysis.  The gain  fields  of 80% of the  cells  had a  planar component  [1 , \n11] .  One  such  gain field  is  shown in figure  2B. \n\nThere is  not enough data available to determine whether or not the third prediction \nis valid.  However, indirect evidence suggests that if such a correlation exists between \npreferred direction for retinal location and for eye position, it is probably not strong. \nCells  with  opposite  preferred  directions  [2,  3]  have  been  observed.  Furthermore, \nalthough each  hemisphere  represents  all  possible  preferred  eye  position  directions, \nthere is  a  clear  tendency  to overrepresent  the contralateral retinal hemifield  [1]. \n\nIn conclusion, the experimental data are not fully consistent with the predictions of \nthe vectorial code.  The visual receptive fields,  in particular, are strongly nonlinear. \nIf these nonlinearities are computationally neutral,  that is,  they are averaged out in \nsubsequent stages of processing in the cortex,  then the vectorial code  could capture \n\n\fSpatial Representations in  the  Parietal Cortex May  Use  Basis  Functions \n\n161 \n\nthe essence of what the parietal cortex computes and, as such,  would provide a valid \napproximation of the  neurophysiological  data.  We  argue  in  the  next  section  that \nthe nonlinearities  cannot be disregarded  and we  present  a  representational scheme \nin  which  they have a  central computational function. \n\n3  Basis Function  Representation \n\n3.1  Sensorimotor Coordination and Nonlinear Function \n\nApproximation \n\nThe  function  which  specified  the  pattern  of muscle  activities  required  to  move  a \nlimb, or the body, to a specific  spatial location is  a highly nonlinear function of the \nsensory  inputs.  The cortex is not believed to specify  patterns of muscle activation, \nbut  the  intermediate  transformations  which  are  handled  by  the  cortex  are  often \nthemselves nonlinear.  Even  if the transformations are actually linear,  the nature of \ncortical representations often makes the problem a nonlinear mapping.  For example, \nthere exists in the putamen and premotor cortex  cells  with gaussian head-centered \nvisual  receptive  fields  [7J  which  means  that  these  cells  compute  gaussians  of A \nor,  equivalently,  gaussians  of R + E,  which  is  nonlinear  in  Rand  E.  There  are \nmany other examples of sensory  remappings involving similar computations.  If the \nparietal cortex is to have a role in these remappings, the cells should respond to the \nsensory  inputs  in  a  way  that can  be  used  to  approximate the  nonlinear  responses \nobserved elsewhere. \n\nOne possibility would be for  parietal neurons to represent  input signals such  as  eye \nposition and retinal location with basis functions.  A basis function  decomposition is \na  well-known  method for  approximating nonlinear functions  which  is,  in  addition, \nbiologically  plausible  [8J. \nIn  such  a  representation,  neurons  do  not  encode  the \nhead-centered  locations of objects,  A;  instead,  they compute functions of the input \nvariables,  such  as  Rand E,  which  can  be  used  subsequently  to  approximate any \nfunctions  of these  variables. \n\n3.2  Predictions of the Basis  Function Representation \n\nNot all functions are basis functions.  Linear functions do not qualify, nor do sums of \nfunctions  which,  individually, would  be  basis functions,  such  as  gaussian functions \nof retinal location plus  a sigmoidal functions  of eye  position.  If the parietal cortex \nuses  a  basis function  representation,  two  conditions have  to be met: \n\n1)  The  visual  and  the  eye  position  receptive  fields  should  be  smooth  nonlinear \nfunction  of Rand E. \n2)  The selectivities  to Rand E should interact  nonlinearly \nThe  visual receptive fields  of parietal neurons  are typically smooth and  nonlinear. \nGaussian  or  sum  of gaussians  appear  to  provide  good  models  of their  response \nprofiles  [2].  The eye position receptive field  on the other hand,  which is represented \nby the gain field,  appears to be approximately linear.  We believe,  however,  that the \npublished data only demonstrate that the eye  position receptive field  is  monotonic, \n\n\f162 \n\nAlexandre Pouget,  Terrence J.  Sejnowski \n\nHead-Centered \n\nRetinotopic \n\no \n\no \n\nFigure  3:  Approximation of a  gaussian  head-centered  (top-left)  and  a  retinotopic \n(top-right)  receptive  field,  by  a  linear combination of basis function  neurons.  The \nbottom 3-D  plots show  the  response  to all possible  horizontal  retinal  position,  r x , \nand  horizontal  eye  positions,  ex,  of four  typical  basis  function  units.  These  units \nare  meant to model actual  parietal neurons \n\nbut  not  necessarily  linear.  In  published  experiments,  eye  position  receptive  fields \n(gain fields)  were sampled at only nine points, which makes it difficult to distinguish \nbetween  a  plane  and  other  functions  such  as  a  sigmoidal function  or  a  piecewise \nlinear  function.  The  hallmark of a  nonlinearity  would  be evidence  for  saturation \nof activity within working range of eye  position.  Several published gain fields  show \nsuch  saturations [3,  11],  but a  rigorous statistical analysis would be desirable. \n\nAndersen  et  al.  (1985) have  have shown that the responses  of parietal neurons  are \nbest  modeled by  a  multiplication between  the retinal  and eye  position selectivities \nwhich  is consistent  with the requirements for  basis functions. \n\nTherefore, the experimental data are consistent with our hypothesis that the parietal \ncortex  uses  a  basis function  representation.  The  response  of most gain-modulated \nneurons  in  the  parietal  cortex  could  be  modeled  by  multiplying a  gaussian  tuning \nto retinal position by  a sigmoid of eye  position, a function which qualifies as a  basis \nfunction. \n\n3.3  Simulations \n\nWe  simulated  the  response  of  121  parietal  gain-modulated  neurons  modeled  by \nmultiplying a  gaussian of retinal position,  rx , with  a sigmoid of eye position,  ex : \n\n\fSpatial Representations in  the  Parietal Cortex May  Use  Basis  Functions \n\nh,o=-----\n\ne,,;-e,l'j \n\ne-\n1 +e-\n\n(rz-rra):il \n\n~ .. ~ \n\nt \n\n163 \n\n(4) \n\nwhere the centers of the gaussians for retinalloction rxi  and the positions of the sig(cid:173)\nmoids for  eye postions  exi  were  uniformly distributredo  The widths of the gaussian \n(T  and the sigmoid t  were  fixed.  Four of these functions are shown at the bottom of \nfigure  3. \n\nWe  used  these  basis functions  as  a  hidden  layer to approximate two  kinds of out(cid:173)\nput functions:  a  gaussian  head-centered  receptive  field  and  a  gaussian  retinotopic \nreceptive  field .  Neurons  with  these  response  properties  are  found  downstream  of \nthe parietal cortex in the premotor cortex [7]  and superior colliculus, two structures \nbelieved  to be involved in the control of,  respectively,  arm and eye  movements. \n\nThe weights for  a  particular output were obtained by using the delta rule.  Weights \nwere  adjusted  until the  mean  error  was  below  5%  of the  maximum output  value. \nFigure 3 shows our best approximations for both the head-centered  and retinotopic \nreceptive  fields.  This demonstrates  that  the  same pool of neurons  can  be used  to \napproximate several  diffferent  nonlinear functions. \n\n4  Discussion \n\nNeurophysiological data support our hypothesis  that the parietal cortex represents \nits  inputs,  such  as  the  retinal  location  of objects  and  eye  position,  in  a  format \nsuitable to non-linear function  approximation, an operation central to sensorimotor \ncoordination.  Neurons have gaussian visual receptive fields modulated by monotonic \nfunction of eye position leading to response function that can be modeled by product \nof gaussian and  sigmoids.  Since  the product of gaussian  and sigmoids forms  basis \nfunctions,  this representation  is  good for  approximating nonlinear functions of the \ninput  variables. \n\nPrevious  attempts  to  characterize  spatial  representations  have  emphasized  linear \nencoding  schemes  in  which  the  location of objects  is  represented  in  egocentric  co(cid:173)\nordinates.  These  codes  cannot  be  used  for  nonlinear function  approximation  and, \nas such,  may not  be  adequate for  sensorimotor coordination  [6,  10].  On  the other \nhand,  such  representations  are  computationally interesting  for  certain  operations, \nlike addition or rotation.  Some part of the brain more specialized in navigation like \nthe hippocampus might be using such  a scheme  [10]. \n\nIn  figure  3,  a  head-centered  or a  retinotopic  receptive  field  can  be  computed from \nthe same pool of neurons.  It would  be  arbitrary to say  that these  neurons encode \nthe  positions  of objects  in  egocentric  coordinates.  Instead,  these  units  encode  a \nposition  in  several  frames  of reference  simultaneously.  If the  parietal  cortex  uses \nthis  basis  function  representation,  we  predict  that  hemineglect,  the  neurological \nsyndrome which results from lesions in the parietal cortex, should not be confined to \nany particular frame of reference.  This is precisely the conclusion that has emerged \nfrom recent studies of parietal patients [4].  Whether the behavior of parietal patients \ncan  be fully  explained  by  lesions of a  basis function  representation  remains  to  be \ninvestigated. \n\n\f164 \n\nAlexandre Pouget,  Terrence J.  Sejnowski \n\nAcknowledgments \n\nWe  thank  Richard  Andersen  for  helpful  conversations  and  with  access  to  unpub(cid:173)\nlished  data. \n\nReferences \n\n[1]  R.A.  Andersen,  C.  Asanuma, G.  Essick,  and  R.M.  Siegel.  Corticocortical  con(cid:173)\nnections of anatomically and physiologically defined subdivisions within the in(cid:173)\nferior  parietal lobule .  Journal  of Comparative  Neurology,  296(1):65-113, 1990. \n[2]  R.A. Andersen,  G.K.  Essick,  and  R.M.  Siegel.  Encoding of spatial location by \n\nposterior  parietal neurons.  Science,  230:456-458 , 1985. \n\n[3]  R.A.  Andersen  and  D.  Zipser.  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