{"title": "Reorganisation of Somatosensory Cortex after Tactile Training", "book": "Advances in Neural Information Processing Systems", "page_first": 82, "page_last": 88, "abstract": "", "full_text": "Reorganisation of Somatosensory Cortex after \n\nTactile Training \n\nRasmus S. Petersen \n\nJohn G. Taylor \n\nCentre for Neural Networks, King's College London \n\nStrand, London WC2R 2LS, UK \n\nAbstract \n\nTopographic maps in  primary areas of mammalian cerebral cortex reor(cid:173)\nganise as  a result of behavioural  training.  The nature  of this  reorgani(cid:173)\nsation  seems  consistent  with  the  behaviour  of competitive  neural  net(cid:173)\nworks,  as  has  been  demonstrated  in  the  past  by  computer  simulation. \nWe model tactile training on the hand representation in primate somato(cid:173)\nsensory  cortex,  using  the  Neural  Field  Theory  of Amari  and  his  col(cid:173)\nleagues.  Expressions for  changes in  both receptive field  size  and  mag(cid:173)\nnification factor are derived,  which  are consistent with  owl  monkey ex(cid:173)\nperiments and make a prediction which goes beyond them. \n\n1.  INTRODUCTION \nThe primary  cortical  areas  of mammals are  now  known  to  be  plastic  throughout  life;  re(cid:173)\nviewed  recently  by  Kaas(1995).  The  problem  of how  and  why  the  underlying  learning \nprocesses  work  is  an  exciting  one,  for  which  neural  network  modelling  appears  well \nsuited.  In  this contribution, we  model  the  long-term effects  of tactile training  (Jenkins et \nai,  1990)  on  the  functional  organisation  of monkey  primary  somatosensory  cortex,  by \nperturbing a topographic net (Takeuchi and  Amari,  1979). \n\n1.1  ADAPTATION IN ADULT SOMATOSENSORY CORTEX \n\nLight  touch  activates  skin  receptors  which  in  primates  are  mapped,  largely  topographi(cid:173)\ncally,  in  area 3b.  In  a series  of papers,  Merzenich  and  colleagues describe  how  area  3b \nbecomes reorganised following peripheral nerve damage (Merzenich et ai,  1983a;  1983b) \nor digit amputation (Merzenich et ai,  1984).  The underlying learning processes may  also \nexplain the phenomenon of phantom limb \"telescoping\" (Haber,  1955).  Recent advances \nin  brain  scanning  are  beginning  to  make  them  observable  even  in  the  human  brain \n(Mogilner et ai,  1993). \n\n1.2  ADAPTATION ASSOCIATED WITH TACTILE TRAINING \n\nJenkins et al  trained owl  monkeys to maintain contact with  a rotating disk.  The apparatus \nwas  arranged  so  that  success  eventually  involved  touching  the  disk  with  only  the  digit \ntips.  Hence  these  regions  received  selective  stimulation.  Some  time  after  training  had \nbeen  completed  electro-physiological  recordings  were  made  from  area  3b.  These  re(cid:173)\nvealed  an  increase in Magnification Factor (MF) for  the stimulated  skin  and  a decrease in \n\n\fReorganization of Somatosensory  Cortex  after Tactile Training \n\n83 \n\nthe size of Receptive Fields (RFs) for that region.  The net territory gained for light touch \nof  the digit tips came  from  area 3a and/or the face  region  of area  3b,  but details of any \nchanges in these representations were not reported. \n\n2.  THEORETICAL FRAMEWORK \n\n2.1  PREVIOUS WORK \n\nTakeuchi  and  Amari(1979),  Ritter  and  Schulten(1986),  Pearson  et  al(1987)  and  Grajski \nand  Merzenich( 1990)  have  all  modelled  amputationldenervation  by computer simulation \nof  competitive  neural  networks  with  various  Hebbian  weight  dynamics.  Grajski  and \nMerzenich(1990)  also  modelled  the  data  of  Jenkins  et  al.  We  build  on  this  research \nwithin  the  Neural  Field  Theory  framework  (Amari,  1977;  Takeuchi  and  Amari,  1979; \nAmari,  1980) of the Neural Activity Model of Willshaw and von der Malsburg(1976). \n\n2.2  NEURAL ACTIVITY MODEL \n\nConsider a  \"cortical\"  network  of simple,  laterally  connected  neurons.  Neurons  sum  in(cid:173)\nputs  linearly  and  output a  sigmoidal  function  of this  sum.  The  lateral  connections  are \nexcitatory  at  short  distances  and  inhibitory  at  longer ones.  Such  a  network  is  competi(cid:173)\ntive:  the steady state consists of blobs of activity centred around those  neurons locally re(cid:173)\nceiving the greatest afferent input (Amari,  1977).  The range of the competition is limited \nby the range of the lateral inhibition. \n\nSuppose now that the afferent synapses adapt in  a Hebbian manner to  stimuli that are lo(cid:173)\ncalised  in  the  sensory  array;  the  lateral  ones  are  fixed.  Willshaw  and  von  der  Mals(cid:173)\nburg(1976)  showed  by  computer  simulation  that  this  network  is  able  to  form  a  topo(cid:173)\ngraphic  map  of the  sensory  array.  Takeuchi  and  Amari( 1979)  amended  the  Willshaw(cid:173)\nMalsburg model slightly: neurons possess an adaptive firing  threshold in order to  prevent \nsynaptic  weight explosion,  rather than  the  more  usual  mechanism  of weight  normalisa(cid:173)\ntion.  They proved that a topographic mapping is stable under certain conditions. \n\n2.3  TAKEUCHI-AMARI THEORY \nConsider a one-dimensional model.  The membrane dynamics are: \n\nau(~y,t) = -u(x,y,t)+ f s(x,y' ,t)a(y- y')dy'(cid:173)\n\nso(x,t)ao + f w(x-x')f[u(x' ,y,t)]dx'-h \n\n(1) \n\nHere u(x,y,t) is the membrane potential at time I  for point x when a stimulus centred at y is \nbeing presented;  h  is a positive resting potential; w(z) is  the lateral  inhibitory  weight be(cid:173)\ntween  two  points  in  the neural  field  separated  by  a distance z - positive  for  small  Izl  and \nnegative  for  larger  Izl;  s(x,y,t)  is  the  excitatory synaptic  weight  from  y  to  x  at time  I  and \nsr/X,I) is an inhibitory weight from a tonically active inhibitory input aD  to x at time t - it is \nthe adaptive firing  threshold .  f[u]  is  a binary threshold  function  that maps positive mem(cid:173)\nbrane potentials to  1 and non-positive ones to O. \n\nIdealised,  point-like  stimuli  are  assumed,  which  \"spread  out\"  somewhat  on  the  sensory \nsurface or subcortically.  The spreading process is assumed  to be independent of y  and  is \ndescribed  in  the  same  coordinates.  It  is  represented  by  the  function  a(y-y'),  which  de(cid:173)\nscribes the effect of a point input at y  spreading to the point y'. This is a decreasing, posi(cid:173)\ntive,  symmetric function  of Iy-y'l.  With  this  type  of input,  the  steady-state activity  of the \nnetwork is a single blob,  localised around the neuron with maximum afferent input. \n\n\f84 \n\nR. S. PETERSEN, J.  O. TAYLOR \n\nThe afferent synaptic  weights  adapt in  a  leaky  Hebbian  manner  but  with  a  time constant \nmuch larger than that of the membrane dynamics (1).  Effectively this means that learning \noccurs  on  the  steady  state  of the  membrane  dynamics.  The  following  averaged  weight \ndynamics can be justified (Takeuchi and Amari,  1979; Geman 1979): \n\nas( x, y, t) \n\n( \n\nat  =-s x,y,t)+b  p(y'  a Y-Y')f u(x,y'  dy' \n\nJ) ( \n\n[A \n\n)] \n\naso(~y,t) =-so(x,y,t)+b' aoJ p(y')f[u(x,y')]dy' \n\n(2) \n\nwhere r1(x,y')  is the steady-state of the membrane dynamics at x given a stimulus at y' and \np(y') is the probability of a stimulus at y '; b, b' are constants. \n\nEmpirically,  the  \"classical\"  Receptive Field  (RF) of a  neuron  is  defined  as  the region  of \nthe input field  within  which  localised  stimulation causes change in  its  activity.  This con(cid:173)\ncept can be modelled in neural field theory as:  the RF of a neuron at x is  the portion of the \ninput field  within  which a  stimulus evokes a positive membrane potential  (inhibitory RFs \nare not considered).  If the neural field  is a continuous map of the sensory surface then the \nRF of a neuron is fully  described by its two borders rdx), rix), defined formally: \n\ni = 1,2 \n\n(3) \n\nwhich are illustrated in  figure  1. \n\nLet RF size and RF position be denoted respectively by the functions rex)  and m(x),  which \nrepresent experimentally measurable quantities.  In terms of the border functions they can \nbe expressed: \n\nr(x) = r2 (x) - r1 (x) \nm(x) = -} (rl {x} + r2 (x)) \n\ny \n\n~--------------------------- x \n\n(4) \n\nas \n\nRF \nFigure \n1: \nboundaries \na \nfunction  of position \nin  the  neural  field, \nfor  a \ntopographi(cid:173)\ncally  ordered  net(cid:173)\nwork.  Only  the  re(cid:173)\ngion \nin-between \nrdx)  and  rix)  has \nsteady-\npositive \nstate \nmembrane \nr1(x,y). \npotential \nrdx)  and  rix)  are \ndefined \nthe \ncondition \nr1(x,r;(x))=O \ni=J,2. \n\nfor \n\nby \n\nUsing (1), (2) and  the definition (3), Takeuchi and  Amari(1979) derived dynamical equa(cid:173)\ntions for  the change in  RF borders due  to  learning.  In  the case of uniform stimulus prob(cid:173)\nability,  they  found  solutions  for  the  steady-state  RF  border  functions.  With  periodic \nboundary conditions, the basic solution is a linear map with constant RF size: \n\n\fReorganization  of Somatosensory  Cortex  after Tactile Training \n\n85 \n\nr(x) =  ro  = const \nm(x) = px ++ro \n\n) \n\nx  = px \n\nuni  ( \nrl \nr~tni (x) = px+ ro \n\n(5) \n\nThis  means  that  both  RF size  and  activity  blob size  are  uniform  across  the  network  and \nthat  RF position m(x)  is  a  linear function  of network  location.  (The  value of p  is  deter(cid:173)\nmined  by  boundary  conditions;  ro  is  then  determined  from  the joint equilibrium  of (I), \n(2\u00bb.  The inverse of the RF position function, denoted by  m-l(y), is the centre of the corti(cid:173)\ncal  active region caused by a stimulus centred at y.  The change in m-l(y) over a unit inter(cid:173)\nval  in  the  input  field  is,  by  empirical  definition,  the  cortical  magnification  factor  (MF). \nHere we model MF as  the rate of change of m-l(y).  The MF for  the  system described  by \n(5) is: \n\n_I ( )  \n\nd \n-I \n-m  y  =p \ndy \n\n(6) \n\n3.  ANALYSIS OF TACTILE TRAINING \n\n3.1  TRAINING MODEL AND ASSUMPTIONS \n\nJenkins et aI's training sessions caused an increase in the relative frequency  of stimulation \nto  the  finger  tips,  and  hence  a  decrease  in  relative  frequency  of stimulation  elsewhere. \nOver a  long  time,  we  can express  this  fact  as  a  localised  change  in  stimulus  probability \n(figure 2).  (This is not sufficient to cause cortical reorganisation - Recanzone et al( 1992) \nshowed that attention to  the stimulation is  vital.  We consider only attended stimulation in \nthis  model).  To  account  for  such  data  it  is  clearly  necessary  to  analyse  non-uniform \nstimulus probabilities, which demands extending the results of Takeuchi  and  Amari.  Un(cid:173)\nfortunately,  it  seems to  be hard  to  obtain general  results.  However, a perturbation analy(cid:173)\nsis around the uniform probability solution (5) is possible. \n\nTo proceed  in  this way,  we  must be able to  assume  that the change  in  the  stimulus prob(cid:173)\nability density function away from uniformity is small.  This reasoning is expressed by the \nfollowing equation: \n\np(y) = Po  + E p(y) \n\n(7) \n\nwhere pry)  is  the new stimulus probability in  terms  of the  uniform one and  a perturbation \ndue to  training:  E is  a small  constant.  The effect of the perturbation is to ease the weight \ndynamics (2) away from the solution (5) to a new steady-state.  Our goal is  to discover the \neffect of this on the RF border functions,  and hence for RF size and MF. \n\np(y) \n\no \n\nchange \n\nFigure  2:  The  type \nof \nin \nstimulus  probabil(cid:173)\nity  density  that  we \nassume \nto  model \nthe  effects  of  be(cid:173)\nhavioural training. \n\ny \n\n\f86 \n\nR. S. PETERSEN, J. G. TAYLOR \n\n3.2  PERTURBATION ANALYSIS \n\n3.2.1  General Case \n\nFor a small enough perturbation, the effect on the RF borders and  on the activity blob size \nought also  to  be small.  We consider effects to  first  order in  E,  seeking  new solutions  of \nthe form: \n\ni = 1,2 \n\n,{x} = r; {x} - ~ {x} \nm{x} = +(~ (X}+'2 (x}) \n\n(8) \n\nwhere the superscript peT denotes the  new,  perturbed equilibrium and  uni denotes the  un(cid:173)\nperturbed,  uniform probability equilibrium.  Using (1) and  (2)  in  (3) for  the  post-training \nRF borders, expanding to  first  order in  E,  a pair of difference equations  may  be obtained \nfor the changes in RF borders.  It is convenient to define the following terms: \n\nro \n\nAt (x) = J p(y+ px)k(y)dy-b' a~  J p(y)dy \n\nrt '(x) \n\no \no \n\nr,\"no (x) \n\nr;-n' (x ) \n\nA2 {x} =  J p(y + px + TO  )k(y)dy - b'  a~  J p(y)dy \nk(y) = b J a(y - y' )a(y' )dy' \nB = b'  a~p() -k(ro)po > 0 \nC= w(p-tTo)p-t  <0 \n\n(9) \n\nwhere the  signs  of Band  C arise due  to  stability conditions  (Amari,  1977; Takeuchi  and \nAmari,  1979).  In terms of RF size and RF position (4), the general result is: \n\nB~2 ,(X} = ~(~ + I)At (x) - M2 (x) \nBC~2m{X) = (B- C -+ C~)(~+ I}At (x) + (C- B++(C -2B)~)A2 (x) \n\nwhere ~ is the difference operator: \n~ f{ x) = f( x + p - t To) - f( x) \n\n3.2.2  Particular Case \n\n(10) \n\n(11 ) \n\nThe second  order difference equations (l0) are rather opaque.  This  is  partly due to  cou(cid:173)\npling in y caused by the auto-correlation function key):  (10) simplifies considerably if very \nnarrow  stimuli  are  assumed  - a(y)=O(y)  (see  also  Amari,  1980).  For periodic  boundary \nconditions: \n\n(12) \n\nwhere: \n\n\fReorganization  of Somatosensory  Cortex  after Tactile Training \n\n87 \n\nm -I P(W (y) = m -I pre (y) + Em -I (y) \n\n= p-l(y_+ro)+Em-l(y) \n\nand we have used the crude approximation: \n\ndx  m  x  \"\" t;: ~m x - 2\" P  ro \nd  _() \n\n1 \n\n( \n\n1 \n\n_I \n\n) \n\n(13) \n\n(14) \n\nwhich  demands  smoothness  on  the  scale  of  10 .  However,  for  perturbations  like  that \nsketched  in  figure  2,  this  is  sufficient  to  tell  us  about the  constant regions  of MF.  (We \nwould  not expect to  be  able  to  model  the  data in  the  transition  region  in  any  case,  as  its \nform is too dependent upon fine detail  of the model). \n\nOur results  (12)  show  that  the  change  in  RF  size  of a  neuron  is  simply  minus  the  total \nchange in stimulus probability over its RF.  Hence RF size decreases where p(y) increases \nand  vice  versa.  Conversely,  the change in  MF at a  given  stimulus  location is  roughly  the \nlocal average change in  stimulus probability there.  Note that changes in RF size correlate \ninversely with changes in MF.  Figure 3 is  a sketch of these results for  the perturbation of \nfigure 2. \n\nMF \n\no \n\nRF \n\no \n\ny \n\n\\ \n\nI \n\nI \n\nL.J \n\nFigure 3:  Results of perturbation analysis for  how behavioural training (figure 2)  changes \nRF size and  MF respectively, in  the case where stimulus width can be neglected.  For MF \n- due to  the approximation (14) - predictions do not apply near the transitions. \n\n4.  DISCUSSION \nEquations  (12)  are  the  results  of our  model  for  RF  size  and  MF after  area  3b  has  fully \nadapted to  the behavioural task,  in the case where stimulus width can  be neglected.  They \nappear to be fully  consistent with  the  data  of Jenkins et al  described  above:  RF  size  de(cid:173)\ncreases  in  the region  of cortex selective for  the stimulated body part and  the  MF for  this \nbody  part  increases.  Our  analysis  also  makes  a  specific  prediction  that  goes  beyond \nJenkins et aI's  data,  directly  due  to  the  inverse  relationship  between changes  in  RF  size \nand  those in  MF.  Within the regions  that surrender territory  to  the entrained  finger  tips \n(sometimes the face region),  for which MF decreases, RF sizes should increase. \n\nSurprisingly  perhaps,  these  changes  in  RF  size  are  not due  to  adaptation  of the  afferent \nweights  s(x,y).  The  changes  are  rather  due  to  the  adaptive  threshold  term  six).  This \npoint will be discussed more fully elsewhere. \n\nA limitation of our analysis is the assumption that the  change in stimulus probability  is  in \nsome sense small.  Such an approximation may  be reasonable for  behavioural  training but \nseems less so as regards important experimental protocols like amputation or denervation. \nEvidently a more general analysis would be highly desirable. \n\n\f88 \n\nR. S.  PETERSEN,J. O. TAYLOR \n\n5.  CONCLUSION \nWe have analysed a system with three  interacting  features:  lateral  inhibitory  interactions; \nHebbian  adaptivity of afferent synapses  and  an  adaptive firing  threshold.  Our results  in(cid:173)\ndicate  that  such  a  system  can  account  for  the  data  of Jenkins  et  aI,  concerning  the  re(cid:173)\nsponse of adult somatosensory cortex to  the changing environmental demands imposed by \ntactile  training.  The analysis  also brings  out a prediction of the  model,  that  may  be test(cid:173)\nable. \n\nAcknowledgements \n\nRSP  is  very  grateful  for  a  travel  stipend  from  the  NIPS  Foundation  and  for  a  Nick \nHughes  bursary  from  the  School  of Physical  Sciences  and  Engineering,  King's College \nLondon, that enabled him to participate in the conference. \n\nReferences \nAmari S. (1977) BioI.  Cybern.  2777-87 \nAmari S. (1980) Bull.  Math.  Biology 42339-364 \nGeman S.  (1979) SIAM 1.  App.  Math.  36 86-105 \nGrajski  K.A.,  Merzenich  M.M.  (1990)  in  Neural  Information  Processing  Systems  2 \nTouretzky D.S. (Ed) 52-59 \nHaberW.B. (1955)1. Psychol.  40115-123 \nJenkins W.M., Merzenich M.M.,  Ochs M.T.,  Allard  T.,  Gufc-Robles E.  (1990) 1. Neuro(cid:173)\nphysiol.  63 82-104 \nKaas J.H.  (1995) in The  Cognitive Neurosciences Gazzaniga M.S. (Ed  ic) 51-71 \nMerzenich M.M., Kaas J.H., Wall J.T.,  Nelson R.J.,  Sur M.,  Felleman DJ. 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B194 203-243 \n\n\f", "award": [], "sourceid": 1053, "authors": [{"given_name": "Rasmus", "family_name": "Petersen", "institution": null}, {"given_name": "John", "family_name": "Taylor", "institution": null}]}