{"title": "Neural Implementation of Bayesian Inference in Population Codes", "book": "Advances in Neural Information Processing Systems", "page_first": 317, "page_last": 323, "abstract": null, "full_text": "Neural  Implementation of Bayesian \n\nInference in  Population  Codes \n\nSi  Wu \n\nComputer Science Department \n\nSheffield  University,  UK \n\nShun-ichi  Amari \n\nLab.  for  Mathematic Neuroscience, \n\nRIKEN Brain Science Institute,  JAPAN \n\nAbstract \n\nThis study investigates a  population decoding paradigm, in  which \nthe  estimation  of  stimulus  in  the  previous  step  is  used  as  prior \nknowledge for consecutive decoding.  We analyze the decoding accu(cid:173)\nracy of such a  Bayesian decoder (Maximum a Posteriori Estimate), \nand  show  that  it  can  be  implemented  by  a  biologically  plausible \nrecurrent  network,  where  the  prior  knowledge  of stimulus  is  con(cid:173)\nveyed by the change in recurrent interactions as a result of Hebbian \nlearning. \n\n1 \n\nIntroduction \n\nInformation in the brain is  not processed by a single neuron, but rather by a  popu(cid:173)\nlation of them.  Such a coding strategy is  called population coding.  It is  conceivable \nthat population coding has advantage of being robust to the fluctuation  in a  single \nneuron's  activity.  However,  people  argue  that  population  coding  may  have  other \ncomputationally desirable properties.  One such property is  to provide a framework \nfor  encoding  complex  objects by  using  basis  functions  [1].  This  is  inspired  by  the \nrecent progresses in nonlinear function approximation, such as, sparse coding, over(cid:173)\ncomplete representation and kernel regression.  These methods are efficient and show \nsome  interesting  neuron-like  behaviors  [2,3].  It is  reasonable  to think that  similar \nstrategies  are used  in  the  brain under  the support  of population  codes.  However, \nto  confirm this  idea,  a  general suspicion has to be clarified:  can the brain perform \nsuch  complex  statistic  inference?  An  important  work  towards  the  answer  of  this \nquestion was done by Pouget and co-authors [4,5].  They show that Maximum Like(cid:173)\nlihood (ML)  Inference, which is  usually thought to be complex, can be implemented \nby a  biologically plausible recurrent network using the idea of line attractor. \n\nML  is  a  special  case of Bayesian inference when the stimulus is  (or assumed to be) \nuniformly distributed.  In case there is prior knowledge on the stimulus distribution, \nMaximum a  Posteriori  (MAP)  Estimate has better performance.  Zhang et al.  has \nsuccessfully  applied  MAP  for  reconstructing  the  rat  position  in  a  maze  from  the \nactivity  of  hippocampal  place  cells  [6]. \nIn  their  method,  the  prior  knowledge  is \nthe  rat's position  in  the  previous  time  step,  which  restricts the  variability  of rat's \nposition in the current step under the continuity constraint.  It turns out that MAP \nhas  a  much  better  performance than  other  decoding  methods,  and  overcomes  the \ninefficiency of ML  when information is  not sufficient  (when the rat stops running). \n\n\fThis  result  implies  that  MAP  may  be  used  by  the  nervous  system.  So  far,  in  the \nliterature  MAP  has  been  mainly  studied  as  a  mathematic  tool  for  reconstructing \ndata, though its potential neural implementation was  pointed out by  [1 ,6]. \n\nIn the present study, we  will firmly  show how to implement MAP in a biologic way. \nThe same kind  of recurrent  network for  achieving  ML  is  used  [4,5].  The decoding \nprocess  consists of two  steps.  In the first  step when there is  no  prior knowledge of \nthe  stimulus,  the  network  implements  ML.  Its  estimation  is  subsequently  used  to \nform  the prior  distribution of stimulus for  consecutive  decoding, which  we  assume \nis  a  Gaussian  function  with  the  mean  value  being  the  estimation.  It  turns  out \nthat this prior knowledge  can be naturally conveyed by the change in the recurrent \ninteractions  according to  the Hebbian learning rule.  This  is  an interesting finding \nand suggests a  new role of Hebbian learning.  In the second step, with the changed \ninteractions,  the  network  implements  MAP.  The  decoding  accuracy  of MAP  and \nthe optimal form  of Gaussian prior are also  analyzed in this  paper. \n\n2  MAP  in  Population Codes \n\nLet  us  consider  a  standard  population  coding  paradigm.  There  are  N  neurons \ncoding for  a  stimulus x.  The population activity is  denoted by r  = {rd.  Here ri  is \nthe response of the ith neuron,  which is  given by \n\n(1) \n\nwhere  fi(X)  is  the tuning function  and fi  is  a  random noise. \n\nThe encoding process of a population code is specified by the conditional probability \nQ(rlx)  (i.e.,  the  noise  model).  The  decoding  is  to  infer  the  value  of  x  from  the \nobserved r. \n\nWe consider a general Bayesian inference in a population code, which estimates the \nstimulus by maximizing a  log posterior distribution, In P(xlr) , i.e. , \n\nargmaxx \nargmaxx \n\nIn P(xlr) , \nInP(rlx) + InP(x), \n\n(2) \n\nwhere P(rlx) is  the likelihood function.  It can be equal to or different from the real \nencoding  model  Q(rlx) ,  depending  on  the  available  information  of  the  encoding \nprocess  [7].  P(x)  is  the  distribution  of x, representing  the  prior  knowledge.  This \nmethod is  also called Maximum a  Posteriori (MAP).  When the distribution of x  is, \nor is  assumed to be (when there is  no  prior knowledge)  uniform, MAP is equivalent \nto ML. \n\nMAP could be used in the information processing of the brain in several occasions. \nLet us consider the following scenario:  a stimulus is  decoded in multiple steps.  This \nhappens  when  the  same  stimulus  is  presented  through  multiple  steps,  or  during \na  single  presentation,  neural  signals  are  sampled  many  times.  In  both  cases,  the \nbrain successively  gains  a  rough  estimation of the  stimulus  in  each  step  decoding, \nwhich  can serve to be the prior knowledge  when  further  decoding is  concerned.  It \nis  therefore  natural  to  use  MAP  in  this  situation.  Experiencing  slightly  different \nstimuli  in  consecutive  steps  as  studied  in  [6],  or  more  generally,  stimulus  slowly \nchanges with time (multiple-step diagram is a discreted approximation), is  a similar \nscenario.  For simplicity, we  only consider that stimulus is  unchanged in the present \nstudy. \n\n\f2.1  The  Performance of MAP \n\nLet us  analyze the performance of MAP. Some notations are introduced first.  De(cid:173)\nnote  Xt  a  particular estimation  of the stimulus  in  the  tth  step,  and  0;  the  corre(cid:173)\nsponding variance.  The prior distribution of x  in the t +  lth step is  assumed to be \na  Gaussian with the mean value  X\"~ i.e., \n\nP(xIXt)  =  _1_ exp-CX-Xt)2 /2r;, \n\n(3) \nwhere  the  parameter  Tt  reflects  the  estimator's  confidence  on  xt,  whose  optimal \nvalue  will  be calculated later. \nThe posterior distribution of x  in  the t +  lth step is  given by \n\n.,J2irTt \n\nP(  I ) =  P(rlx)P(xlxt) \n, \n\nP(r) \n\nxr \n\nand the solution of MAP  is  obtained by solving \n\n\\7 In P(Xt+1 Ir) \n\n\\7lnP(rlxt+l) - (Xt+l-Xt)/T;, \nO. \n\n(4) \n\n(5) \n\nWe  calculate  the  decoding  accuracies  iteratively.  In  the  first  step  decoding,  since \nthere is  no prior knowledge on x, ML is used,  whose decoding accuracy is  known to \nbe  [7] \n\n02- \u00ab\\7lnP(rlx))2> \n\n(6) \n\n1 - < -\\7\\7lnP(rlx) >2' \nwhere the bracket < . >  denotes averaging over Q(rlx). \nNote that, to get the above result, we  have considered that ML is  asymptotically or \nquasi-asymptotically (when an unfaithful model is  used)  efficient  [7].  This includes \nthe cases when neural responses are independent, weakly correlated, uniformly cor(cid:173)\nrelated, correlated with strength proportional to firing  rate  (multiplicative correla(cid:173)\ntion),  or  the fluctuation  in  neural  responses  are  sufficiently  small.  In  other strong \ncorrelation cases,  ML  is  proved to be  non-Fisherian,  i.e,  its decoding error satisfies \na  Cauchy type of distribution with  variance  diverging.  Decoding  accuracy can no \nlonger be quantified by variance in such situations  (for  details, please refer to  [8]) . \nNow  come  to  calculate the  decoding  error in  the  second  step.  Suppose  X2  is  close \nenough to x.  By expanding \\7lnP(rlx2)  at x  in eq.(5),  we  obtain \n\n\\7lnP(rlx) +  \\7\\7lnP(rlx)(x2 - x)  - (X2  - X1)/T;  =  O. \n\n(7) \n\nThe random variable  Xl  can be decomposed  as  Xl  =  x  +  f1,  where  f1  is  a  random \nnumber satisfying Gaussian distribution of zero  mean and variance Oi. \nBy using  the notation of f1,  we  have \n\nA \n\n\\7lnP(rlx)+fdTf \nX2  -x =  l/T; - \\7\\7lnP(rlx)' \n\n(8) \n\nFor  the  correlation  cases  considered  in  the  present  study  (i.e,  those  ensure  ML \nasymptotically  or  quasi-asymptotically  efficient),  - \\7\\7 In P(rlx)  can  be  approxi(cid:173)\nmated as  a  (positive)  constant  according to the law  of large numbers  [7,8].  There(cid:173)\nfore,  we  can define  a  constant variable \n\na  =  T;(-\\7\\7lnP(rlx)), \n\n(9) \n\n\fand a  random variable \n\nR  = \n\n\\71nP(rlx) \n\n- \\7\\71n P(rlx) . \n\nObviously R  satisfies the Gaussian distribution of zero  mean and variance 0I. \nBy using  the notations 0:  and R,  we  get \n\nwhose variance is  calculated to be \n\no:R+fl \nX2- X = - - -\n1+0: \n\nSince  (1  + 0:2)/(1 +  0:)2  :::::  1 holds for  any positive 0:,  the decoding accuracy in  the \nsecond step is  always improved.  It is  not difficult  to check that its minimum  value \nis \n\n(10) \n\n(11) \n\n(12) \n\n(13) \n\n(14) \n\n2 \nwhen  0:  =  1,  or,  the optimal value  of Tl  is \n\n2  -\n\n0 2 - !02 \n1> \n\nTl=----,.......,.....,.. \n- \\7\\71n P(rlx) \n\n1 \n\nWhen a  faithful  model  is  used,  -\\7\\71nQ(rlx)  is  the Fisher information.  Tl  hence \nequals to the variance of decoding error.  This is  understandable. \n\nFollowing the same procedure, it can be proved that the optimal decoding accuracy \nin  the tth step is  0; = tOI  when the width of Gaussian prior being Tl  = tTl. \nIt is interesting to see that the above multiple decoding procedure, when the optimal \nvalues  of Tt  are  used,  achieves  the  same  decoding  accuracy  as  a  one-step  ML  by \nusing  all  N  x  t  signals.  This  is  the  best  for  any  estimator  to  achieve.  However, \nthe  multiple  decoding  is  not  a  trivial  replacement  of one-step  ML,  and  has  many \nadvantages.  One  of them  is  to  save  memory,  considering that  only  N  signals  and \nthe  value  of  previous  estimation  are  stored  in  each  step.  Moreover,  if  a  slowly \nchanging stimulus is concerned, the multiple decoding outperforms one-step ML for \nthe balance between adaptation and memory.  These  properties  are  valuable when \ninformation is  processed  in  the brain. \n\n3  Network Implementation of MAP \n\nIn  this  section,  we  investigate how  to implement  MAP  by  a  recurrent  network.  A \ntwo-step decoding is  studied.  Without loss of generality, we  consider N  ---+  00  and \ndo calculation in the continuous limit. \n\nThe network we  consider is  a fully  connected one-dimensional homogeneous neural \nfield,  in which c denotes the position coordinate, i.e., the neurons' preferred stimuli. \nThe tuning function  of the neuron with preferred stimulus c is \n\nf c(x)  =  _1_ exp-( c- x)2/2a2  . \n\n\"fiifa \n\n(15) \n\nFor simplicity, we consider an encoding process in which the fluctuations in neurons' \nresponses  are  independent  Gaussian  noises  (more  general  correlated  cases  can  be \nhandled similarly), that is, \n\nQ(rlx)  =  ~ exp- ~ j(T c - f c (x))2 dC, \n\n(16) \n\n\fwhere  p is  the  neuron density  and  Z  is  the normalization factor.  A faithful  model \nis  used in both steps decoding,  i.e.,  P(rlx)  =  Q(rlx)  (again, generalization to more \ngeneral cases of P(rlx)  -::/:- Q(rlx)  is  straightforward.). \nFor the above model setting, the solution of ML  in the first  step is  calculated to be \n\nXl  =  argmaxx  J rc!e(x)de, \nwhere the condition J J;(x)de =  const has been used. \nThe solution of MAP  in the second step is \n\nX2  = argmaxx  J rc!e(x)de - (x - xd2/ 2Tf. \n\nCompared with eq.(17),  eq.(18)  has one  more term corresponding to the contribu(cid:173)\ntion of prior distribution. \n\nNow  come  to  the  study of using  a  recurrent  network  to  realize  eqs.(17)  and  (18). \nFollowing  the  idea  of  Pouget  et  al.  [4,5],  the  following  network  dynamics  is  con(cid:173)\nstructed.  Let  Ue  denote  the  (average)  internal state of neuron at e,  and We,e'  the \nrecurrent  connection  weights  from  neurons  at  e  to  those  at  e'.  The  dynamics  of \nneural excitation is  governed by \n\n(17) \n\n(18) \n\n(19) \n\n(20) \n\nwhere \n\nJ  0 \ndt =  -Ue +  We ,e' \ndUe \nU; \n\no e  =  ----;;-=--=-\n1 + f..LJU;de \n\ne, de  + Ie, \n\n' \n\nis  the activity of neurons  at e and Ie is  the external input arriving at e. \n\nThe recurrent interactions are chosen to be \n\n(21) \nwhich ensures that when there is no external input (Ie =  0), the network is neutrally \nstable on line attractor, \n\nc,c'  -\n\n- exp-(e-e')2/2a2 \n, \n\nW \n\n(22) \nwhere  the  parameter  D  is  constant  and  can  be  determined  easily.  Note  that  the \nline  attractor  has  the  same  shape  as  the  tuning  function.  This  is  crucial,  which \nallows the network perform template-matching by using the tuning function, being \nas  same as  ML and MAP. \n\n'r:/z, \n\nWhen  a  sufficiently  small  input  Ie  is  added,  the  network  is  no  longer  neutrally \nstable on the line  attractor.  It can be proved that the steady state of the network \nhas  approximately the same shape as  eq.(22)  (the  deviation  is  of the  2nd  order of \nthe  magnitude  of Ie.),  whereas,  its  steady  position  on  the  line  attractor  (i.e.,  the \nnetwork estimation) is  determined by maximizing the overlap between Ie and Oe(Z) \n[4,9]. \nThus,  if  Ie  =  ere  in  the  first  step1,  where  e  is  a  sufficiently  small  number,  the \nnetwork estimation is  given  by \n\n21  =  argmaxz  J reOe(z)de, \n\n(23) \n\n- - - - - - - - - - - - -\n\nused in  [5] , has the same result . \n\nlConsider an instant input, triggering the network to  be initially  at  Oe(t  =  0)  =  r e, as \n\n\fwhich  has  the  same  value  as  the  solution  of  ML  (see  eq.(I7)).  We  say  that  the \nnetwork implements ML. \n\nTo implement MAP in the second step, it is  critical to identify a  neural mechanism \nwhich  can  'transmit'  the  prior  knowledge  obtained  in  the  first  step  to  the  second \none.  We  find  that this is  naturally done  by Hebbian learning. \n\nAfter  the  first  step  decoding,  the  recurrent  interaction  changes  a  small  amount \naccording to the Hebbian rule,  whose  new  value is \n\n(24) \nwhere  TJ  is  a  small  positive  number  representing  the  Hebbian  learning  rate,  and \nOe(,2d  is  the neuron activity in the first  step. \nWith  the  new  recurrent  interactions,  the  net  input from  other neurons  to the  one \nat c is  calculated to be \n\nJ We,el  Oel  dc'  J We,el  Oel  dc'  +TJOe(,2d J Oe/(zd Oe,dc' , \n\n(25) \n\nwhere  1/  is  a small constant.  To get the last approximation, the following facts  have \nbeen  used:  1)  The  initial  state of neuron  in  the  second  step  is  at  Oe(Z1 ),  2)  The \nneuron activity Oe during the second  step is  between  Oe(zd  and Oe(Z2 ),  where Z2 \nis  the position of the steady state; 3)  (Z1  - z2 )2/2a2 \u00ab 1,  considering that neurons \nare widely  tuned as  seen  in  data (a  is  large)  and  consecutive  estimations are close \nenough.  These  factors  ensures  the  approximation,  J Oe/ (zd Oe,dc'  :=;:j  const  to  be \ngood enough. \n\nSubstituting eq.(25)  in  (19),  we  see that the network dynamics in the second  step, \nwhen  compared  with  the  first  one,  is  in  effect  to  modify  the  input  Ie to  be  I~  = \n\u20ac(re  + AOc(zd), where  A  is  a  constant and can be determined easily. \nThus, the network estimation in the second step is  determined  by  maximizing the \noverlap between I~ and Oc(z),  which  gives \n\nZ2  = argmaxz  J rcOc(z)dc + A J Oe(zdO e(z )dc. \n\n(26) \n\nThe first  term in  the right handside is  known to achieve ML.  Let us  see  the contri(cid:173)\nbution of the second one,  which can be transformed to \n\nJ Oe(zd Oc(z)dc  =  Bexp-CZI-Z)2/4a2, \n\n(27) \nwhere  B  is  a  constant.  Again,  in  the above calculation,  (Z1 - z)2/4a2 \u00ab 1 is  used \nfor  the same argument discussed above. \n\n-B(z - zd 2 /4a 2  + terms  not on z, \n\n:=;:j \n\nCompare eqs.(I8)  and  (27),  we  see that the second term plays the same role as the \nprior knowledge in MAP. Thus, the network indeed implements MAP.  The value of \nA  (or the Hebbian learning rate)  can be adjusted accordingly to match the optimal \nchoice of Tf . \nThe  above  result  is  confirmed  by  the  simulation experiment  (Table.I) ,  which  was \ndone  with  101  neurons  uniformly  distributed  in  the  region  [-3,3]  and  the  true \nstimulus  being  at O.  It shows  that the estimation of the  network agrees  well  with \nMAP. \n\n\fTable 1:  Comparing the decoding accuracies of the network and MAP with different \nvalues  of a  (the corresponding values  of T[  and  A  are adjusted.).  The parameters \nare a =  1,  f-t  =  0.5  and (J2  =  0.01.  The data is  obtained after 100 trials. \n\n4  Conclusion and  Discussion \n\nIn  summary we  have  investigated how  to  implement  MAP  by  using  a  biologically \nplausible recurrent  network.  A  two-step decoding paradigm is  studied.  In the first \nstep when there is  no prior knowledge, the network implements ML, whose estima(cid:173)\ntion is  subsequently used to form  the prior distribution of stimulus for  consecutive \ndecoding.  In the second step, the network implements MAP. \n\nLine  attractor and Hebbian learning are two  critical elements  to implement  MAP. \nThe former enables the network to do template-matching by using the tuning func(cid:173)\ntion, being as same as ML and MAP. The latter provides a mechanism that conveys \nthe prior knowledge obtained from the first  step to the second one.  Though the re(cid:173)\nsults in this paper may quantitatively depend on the formulation of the models, it is \nreasonable to believe that they are qualitatively true, as both Hebbian learning and \nline  attractor are biologically plausible.  Line  attractor comes from  the translation \ninvariance  of network  interactions,  and  has  been  shown  to  be  involved  in  several \nneural computations [10-12].  We expect that the essential idea of Bayesian inference \nof  utilizing  previous  knowledge  for  successive  decoding  is  used  in  the  information \nprocessing of the brain. \n\nWe  also  analyzed  the  decoding  accuracy  of  MAP  in  a  population  code  and  the \noptimal form  of Gaussian prior.  In  the  present  study,  stimulus is  kept  to be fixed \nduring  consecutive  decodings.  A  generalization  to  the  case  when  stimulus  slowly \nchanges over time is  straightforward. \n\nReferences \n\n[1]  A.  Pouget, P.  Dayan &  R.  Zemel.  Nature  Reviews  Neurosci ence,  1,  125-132,  2000. \n\n[2]  B.  Olshausen &  D.  Field.  Nature,  381,  607-609,  1996. \n\n[3]  T.  Poggio  &  F .  Girosi.  Neural  Computation, 10,  1445-1454,  1998. \n\n[4]  A. Pouget &  K.  Zhang.  NIPS,  9, 1997. \n\n[5]  S.  Deneve,  P.  E.  Latham &  A.  Pouget.  Nature  N euroscience,  2,  740-745,  1999. \n\n[6]  K.  Zhang,  1.  Ginzburg,  B.  McNaughton  &  T.  Sejnowski.  J.  Neurophysiol.,  79,  1017-\n1044,  1998. \n\n[7]  S.  Wu,  H.  Nakahara &  S.  Amari.  Neural  Computation, 13,  775-798,  200l. \n\n[8]  S.  Wu,  S.  Amari &  H.  Nakahara.  CNS*Ol  (to appear). \n\n[9]  S.  Wu,  S.  Amari & H.  Nakahara.  Neural  Computation  (in  press). \n\n[10]  S.  Amari.  Biological  Cybernetics,  27,  77-87,  1977. \n\n[11]  K.  Zhang.  J. Neurosci.,  16,  2112-2126,  1996. \n\n[12]  H . Seung.  Proc.  Natl.  Acad.  Sci.  USA , 93, 13339-13344,  1996. \n\n\f", "award": [], "sourceid": 1988, "authors": [{"given_name": "Si", "family_name": "Wu", "institution": null}, {"given_name": "Shun-ichi", "family_name": "Amari", "institution": null}]}