{"title": "A Bifurcation Theory Approach to the Programming of Periodic Attractors in Network Models of Olfactory Cortex", "book": "Advances in Neural Information Processing Systems", "page_first": 459, "page_last": 467, "abstract": null, "full_text": "A BIFURCATION THEORY APPROACH TO THE \nPROGRAMMING OF PERIODIC A TTRACTORS IN \nNETWORK MODELS OF OLFACTORY CORTEX \n\n459 \n\nBill  Baird \n\nDepartment  of  Biophysics \n\nU.C.  Berkeley \n\nABSTRACT \n\nA  new  learning  algorithm  for  the  storage  of  static \nand  periodic  attractors  in  biologically  inspired \nrecurrent  analog  neural  networks  is  introduced. \nFor  a  network  of  n  nodes,  n  static  or  n/2  periodic \nattractors  may  be  stored.  The  algorithm  allows \nprogramming  of  the  network  vector  field  indepen(cid:173)\ndent  of  the  patterns  to  be  stored.  Stability  of \npatterns,  basin  geometry,  and  rates  of  convergence \nmay  be  controlled.  For  orthonormal  patterns,  the \nl~grning operation  reduces  to  a  kind  of  periodic \nouter  product  rule  that  allows  local,  additive, \ncommutative,  incremental  learning.  Standing  or \ntraveling  wave  cycles  may  be  stored  to  mimic  the \nkind  of  oscillating  spatial  patterns  that  appear \nin  the  neural  activity  of  the  olfactory  bulb  and \nprepyriform  cortex  during  inspiration  and  suffice, \nin  the  bulb,  to  predict  the  pattern  recognition \nbehavior  of  rabbits  in  classical  conditioning  ex(cid:173)\nperiments.  These  attractors  arise,  during  simulat(cid:173)\ned  inspiration,  through  a  multiple  Hopf  bifurca(cid:173)\ntion,  which  can  act  as  a  critical  \"decision  pOint\" \nfor  their  selection  by  a  very  small  input  pattern. \n\nINTRODUCTION \n\nThis  approach  allows  the  construction  of  biological  models  and \nthe  exploration  of  engineering  or  cognitive  networks  that \nemploy  the  type  of  dynamics  found  in  the  brain.  Patterns  of  40 \nto  80  hz  oscillation  have  been  observed  in  the  large  scale  ac(cid:173)\ntivity  of  the  olfactory  bulb  and  cortex(Freeman  and  Baird  86) \nand  even  visual  neocortex(Freeman  87,Grey  and  Singer  88),  and \nfound  to  predict  the  olfactory  and  visual  pattern  recognition \nresponses  of  a  trained  animal.  Here  we  use  analytic  methods  of \nbifurcation  theory  to  design  algorithms  for  determining  synap(cid:173)\ntic  weights  in  recurrent  network  architectures,  like  those \n\n\f460 \n\nBaird \n\nfound  in  olfactory  cortex,  for  associative  memory  storage  of \nthese  kinds  of  dynamic  patterns. \n\nthe  normal  form. \n\nThe  \"projection  algorithm\"  introduced  here  employs  higher \norder  correlations,  and  is  the  most  analytically  transparent \nof  the  algorithms  to  come  from  the  bifurcation  theory  ap(cid:173)\nproach(Baird  88).  Alternative  numerical  algorithms  employing \nunused  capacity  or  hidden  units  instead  of  higher  order  corr(cid:173)\nelations  are  discussed  in  (Baird  89).  All  of  these  methods \nprovide  solutions  to  the  problem  of  storing  exact  analog  at(cid:173)\ntractors,  static  or  dynamic,  in  recurrent  neural  networks,  and \nallow  programming  of  the  ambient  vector  field  independent  of \nthe  patterns  to  be  stored.  The  stability  of  cycles  or  equi(cid:173)\nlibria,  geometry  of  basins  of  attraction,  rates  of  convergence \nto  attractors,  and  the  location  in  parameter  space  of  primary \nand  secondary  bifurcations  can  be  programmed  in  a  prototype \nvector  field  -\nTo  store  cycles  by  the  projection  algorithm,  we  start  with  the \namplitude  equations  of  a  polar  coordinate  normal  form,  with \ncoupling  coefficients  chosen  to  give  stable  fixed  points  on \nthe  axes,  and  transform  to  Cartesian  coordinates.  The  axes  of \nthis  system  of  nonlinear  ordinary  differential  equations  are \nthen  linearly  transformed  into  desired  spatial  or  spatio-tem(cid:173)\nporal  patterns  by  projecting  the  system  into  network  coordina(cid:173)\ntes  -\nthe  standard  basis  - using  the  desired  vectors  as  colum(cid:173)\nns  of  the  transformation  matrix.  This  method  of  network  syn(cid:173)\nthesis  is  roughly  the  inverse  of  the  usual  procedure  in  bifur(cid:173)\ncation  theory  for  analysis  of  a  given  physical  system. \nProper  choice  of  normal  form  couplings  will  ensure  that  the \naxis  attractors  are  the  only  attractors  in  the  system  -\nare  no  \"spurious  attractors\".  If  symmetric  normal  form  coef(cid:173)\nficients  are  chosen,  then  the  normal  form  becomes  a  gradient \nvector  field.  It  is  exactly  the  gradient  of  an  explicit  poten(cid:173)\ntial  function  which  is  therefore  a  strict  Liapunov  function \nfor  the  system.  Identical  normal  form  coefficients  make  the \nnormal  form  vector  field  equivariant  under  permutation  of  the \naxes,  which  forces  identical  scale  and  rotation  invariant \nbasins  of  attraction  bounded  by  hyperplanes.  Very  complex \nperiodic  a~tractors may  be  established  by  a  kind  of  Fourier \nsynthesis  as  linear  combinations  of  the  simple  cycles  chosen \nfor  a  subset  of  the  axes,  when  those  are  programmed  to  be \nunstable,  and  a  single  \"mixed  mode\"  in  the  interior  of  that \nsubspace  is  made  stable.  Proofs  and  details  on  vectorfield \nprogramming  appear  in  (Baird  89). \nIn  the  general  case,  the  network  resulting  from  the  projection \n\nthere \n\n\fA Bifurcation Theory Approach to Programming \n\n461 \n\nalgorithm  has  fourth  order  correlations,  but  the  use  of  restr(cid:173)\nictions  on  the  detail  of  vector  field  programming  and  the \ntypes  of  patterns  to  be  stored  result  in  network  architectures \nrequiring  only  s~cond order  correlations.  For  biological  mod(cid:173)\neling,  where  possibly  the  patterns  to  be  stored  are  sparse  and \nnearly  orthogonal,  the  learning  rule  for  periodic  patterns \nbecomes  a  \"periodic\"  outer  product  rule  which  is  local,  add(cid:173)\nitive,  commutative,  and  incremental.  It  reduces  to  the  usual \nHebb-like  rule  for  static  attractors. \n\nt  . \n\n11 \n\nS \n\nh \n\n1 (ej  +  wt) \n\n\u2022  1  2 \n,  J- , \n\n\"1  \" \ncyc  e  ,  r  Xj  e \n\nCYCLES \nThe  observed  physiological  activity  may  be  idealized  mathe-\n, ... ,n.  uc  a \nma  1ca  y  as  a \ncycle  is  ~ \"periodic  attractor\"  if  it  is  stable.  The  global \namplitude  r  is  just  a  scaling  factor  for  the  pattern  ~ ,  and \nthe  global  phase  w  in  e 1wt  is  a  periodic  scaling  that  scales  x \nby  a  factor  between  \u00b1  1  at  frequency  w  as  t  varies. \nThe  same  vector  XS  or  \"pattern\"  of  relative  amplitudes  can \nappear  in  space  as  a  standing  wave,  like  that  seen  in  the \nbulb,  if  the  relative  phase  eS1  of  each  compartment  (component) \nis  the  same,  eS1+,  - eS1 ,  or  as  a  traveling  wave,  like  that  seen \nin  the  ~repyriform cortex.  if  the  relative  phase  components  of \n~s  form  a  gradient  in  space,  eS 1+1  - 1/a  e\\.  The  traveling  wave \nwill  \"sweep  out\"  the  amplitude  pattern  XS \nin  time,  but  the \nroot-mean-square  amplitude  measured  in  an  experiment  will  be \nthe  same  ~s,  regardless  of  the  phase  pattern.  For  an  arbitrary \nphase  vector,  t~~se  \"simple\"  single  frequency  cycles  can  make \nvery  complicated  looking  spatio-temporal  patterns.  From  the \nmathematical  point  of  view,  the  relative  phase  pattern  ~  is  a \ndegree  of  freedom  in  the  kind  patterns  that  can  be  stored. \nPatterns  of  uniform  amplitude  ~ which  differed  only  in  the \nphase  locking  pattern  ~  could  be  stored  as  well. \nTo  store  the  kind  of  patterns  seen  in  bulb,  the  amplitude \nvector  ~ is  assumed  to  be  parsed  into  equal  numbers  of  excita(cid:173)\ntory  and  inhibitory  components,  where  each  class  of  component \nhas  identical  phase.  but  there  is  a  phase  difference  of  60  -\n90  degrees  between  the  classes.  The  traveling  wave  in  the \nprepyriform  cortex  is  modeled  by  introducing  an  additional \nphase  g~adient into  both  excitatory  and  inhibitory  classes. \n\nPROJECTION  ALGORITHM \n\nThe  central  result  of  this  paper  is  most  compactly  stated  as \nthe  following: \n\n\f462 \n\nBaird \n\nTHEOREM \nr S  x.s  e1(9js  +  wst)  of \nAny  set  S,  s  - 1,2,  ... , n/2  ,  of  cycles \nlinearly  independent  vectors  of  relative  comJonent  amplitudes \nxS  E  Rn  and  phases  ~s  E  Sn,  with  frequencies  wS  E  R  and  global \namplitudes  r S  E  R,  may  be  established  in  the  vector  field  of \nthe  analog  fourth  order  network: \n\nby  some  variant  of  the  projection  operation  : \n\nTij  ...  Emn  Pim  J mn  P  nj  ,  T \n\n-1 \n\nijk1\u00b7 \n\nEPA   p-1.  p-1 \nnk \n\nmn \n\nim \n\nmn \n\nmJ \n\np-1 \n\nn1' \n\nwhere  the  n  x  n  matrix  P  contains  the  real  and  imaginary  com(cid:173)\nponents  [~S  cos  ~s  ,  ~s  sin  ~S]  of  the  complex  eigenvectors \nxS e 19s  as  columns,  J  is  an  n  x  n  matrix  of  complex  conjugate \neigenvalues  in  diagonal  blocks,  Amn  is  an  n  x  n  matrix  of  2x2 \nblocks  of  repeated  coefficients  of  the  normal  form  equations, \nand  the  input  bi  &(t)  is  a  delta  function  in  time  that  establ(cid:173)\nishes  an  initial  condition.  The  vector  field  of  the  dynamics \nof  the  global  amplitudes  rs  and  phases  -s  is  then  given  exactly \nby  the  normal  form  equations  : \n\nr s  ==  Us  r s \n\n\u2022 \n\n\"T \n\nIn  particular,  for  ask  >  0  ,  and  ass/akS  <  1  ,  for  all  sand  k, \nthe  cycles \ns  - 1,2, ... ,n/2  are  stable,  and  have  amplitudes \nrs  ;;  (us/ass )1I2,  where  us\u00b7  1  -\nNote  that  there  is  a  multiple  Hopf  bifurcation  of  codimension \nn/2  at  \"T  = 1.  Since  there  are  no  approximations  here,  however, \nthe  theorem  is  not  restricted  to  the  neighborhood  of  this \nbifurcation,  and  can  be  discussed  without  further  reference  to \nbifurcation  theory.  The  normal  form  equations  for  drs/dt  and \nd_s/dt  determine  how  r S  and  _s  for  pattern  s  evolve  in  time  in \ninteraction  with  all  the  other  patterns  of  the  set  S.  This \ncould  be  thought  of  as  the  process  of  phase  locking  of  the \npattern  that  finally  emerges.  The  unusual  power  of  this  al(cid:173)\ngorithm  lies  in  the  ability  to  precisely  specify  these  ~ \nlinear  interactions.  In  general,  determination  of  the  modes  of \nthe  linearized  system  alone  (li  and  Hopfield  89)  is  insuf(cid:173)\nficient  to  say  what  the  attractors  of  the  nonlinear  system \nwill  be. \n\n\fA Bifurcation Theory Approach to Programming \n\n463 \n\nPROOF \nThe  proof  of  the  theorem  is  instructive  since  it  is  a  constru(cid:173)\nctive  proof,  and  we  can  use  it  to  explain  the  learning  algori(cid:173)\nthm.  We  proceed  by  showing  first  that  there  are  always  fixed \npoints  on  the  axes  of  these  amplitude  equations,  whose  stabil(cid:173)\nity  is  given  by  the  coefficients  of  the  nonlinear  terms.  Then \nthe  network  above  is  constructed  from  these  equations  by  two \ncoordinate  transformations.  The  first  is  from  polar  to  Car(cid:173)\ntesian  coordinates,  and  the  second  is  a  linear  transformation \nfrom  these  canonical  \"mode\"  coordinates  into  the  standard \nbasis  e1,  e2,  ... ,  eN'  or  \"network  coordinates\".  This  second \ntransformation  constitutes  the  \"learning  algorithm\",  because \nit  tra\"nSfrirms  the  simple  fixed  points  of  the  amplitude  equa(cid:173)\ntions  into  the  specific  spatio-temporal  memory  patterns  desi(cid:173)\nred  for  the  network. \n\nAmplitude  Fixed  Points \nBecause  the  amplitude  equations  are  independent  of  the  rota(cid:173)\ntion  _,  the  fixed  points  of  the  amplitude  equations  charact(cid:173)\nerize  the  asymptotic  states  of  the  underlying  oscillatory \nmodes.  The  stability  of  these  cycles  is  therefore  given  by  the \nstability  of  the  fixed  points  of  the  amplitude  equations.  On \neach  axis  r s'  the  other  components  rj  are  zero,  by  definition, \n\nrj  = rj  (  uj  - Ek  ajk  r k2  )  \u2022  0,  for  rj  \u2022  0,  which  leaves \n\nr s  -\n\nrs  (  Us  - ass  r s 2  ),  and \n\nr s  - 0 \n\nlJ \n\n1 \n\n11 \n\n1 \n\n, \n\n- 2  a ..  r~.  r..... \nJ \n\nlJ \n\n1 \n\nThere  is  an  equilibrium  on  each  axis  s,  at  r s.(us/ass )1I2,  as \nclaimed.  Now  the  Jacobian  of  the  amplitude  equations  at  some \nfixed  point  r~  has  elements \n\nJ 11  =  u.  -\n\n:5  a ..  r~.2  - ~  a ..  r~.2  . \n\nJ . .  -\nFor  a  fixed  point  r~s  on  axis  s,  J ij  \u2022  0  ,  since  r~i  or  r~j  \u2022  0, \nmaking  J  a  diagonal  matrix  whose  entries  are  therefore  its \neigenvalues.  Now  J l1  \u2022  u1  - a is  r~ s 2,  for  i  /.  s,  and  J ss  \u2022  Us  -\n:5  ass  r~/.  Since  r~/ \u2022  us/ass'  J ss  \u2022  - 2  us'  and  J ii  \u2022  ui  - a is \n(us/ass).  This  gives  aisfass  >  u1/us  as  the  condition  for  nega(cid:173)\ntive  eigenvalues  that  assures  the  stability  of  r .... s .  Choice  of \naji/a ii  )  uj/u i ,  for  all  i, j \nall  axis  fixed  points. \n\n,  therefore  guarantees  stability  of \n\n]7-i \n\nlJ \n\nJ \n\nCoordinate  Transformations \nWe  now  construct  the  neural  network  from  these  well  behaved \nequations  by  the  following  transformations, \nFirst;  polar  to  Cartesian,  (rs'-s) \nv2s  = r s  sin  -s \nV 2s-1 \n\nto \n,and  differentiating  these \n\n(v2s-1.v2s)  :  Using \n\n'\"  r s  cos  -s \n\n\f464 \n\nBaird \n\ngives: \n\nV2s-1  \u2022  r s  cos  \"s \n\nby  the  chain  rule.  Now  substituting  cos  tis  \u2022  v2s-1/r s  ' \nand \n\nr s  sin  \"s  \u2022  v2s, \n\ngives: \n\nv2s \n\n- v2s  rs  + \n\n(v2  l/ r  )  .. \ns \n\ns-\n\ns \n\nEntering  the  expressions  of  the  normal  form  for  rs  and  tis' \ngives: \n\nand  since  222  \n\nrs  = v2s-1  +  v2s \n\nv2s-1  - Us  v2s-1  - Ws  v2s  +  E  j  [v2s-1  a sj  - v2s  bsj ]  (v2j-/ +  v2/) \n\nn/2 \n\nSimilarly, \n\nn/2 \n\nv2s  - Us  v2s  +  Ws  v2s-'  +  E  j  [v2s  asj  +  v2s-1  bSj ]  (v2j _/  +  v2/)\u00b7 \n\nSetting  the  bsj  - 0 \nto  get  a  standard  network  form,  and  reindexing  i,j-l,2, ... ,n  , \nwe  get  the  Cartesian  equivalent  of  the  polar  normal  form  equa(cid:173)\ntions. \n\nfor  simplicity,  choosing  Us  -\n\n- T  +  1 \n\nn \n\nn \n\nHere  J  is  a  matrix  containing  2x2  blocks  along  the  diagonal  of \nthe  local  couplings  of  the  linear  terms  of  each  pair  of  the \nprevious  equations  v2s-1  '  v2s  \u2022  with \nseparated  out  of  the \ndiagonal  terms.  The  matrix  A  has  2x2  blocks  of  identical  coef(cid:173)\nficients  a sj  of  the  nonlinear  terms  from  each  pair. \n\n-\n\nT \n\n1 \nw, \n\n- w, \n1 \n\nJ  = \n\n1 \nw2 \n\n- w2 \n\n1 \n\n\" \n\n~ \n\n\" -\n\na'l  a\" \na\"  a1, \n\na 12  a'2 \na'2  a'2 \n\na 21  a 21 \na 21  a 21 \n\na 22  a22 \na 22  a22 \n\n., \n\n\fA Bifurcation Theory Approach to Programming \n\n465 \n\nLearning  Transformation  - Linear  Term \nSecond;  J  is  the  canonical  form  of  a  real  matrix  with  complex \nconjugate  eigenvalues,  where  the  conjugate  pairs  appear  in \nblocks  along  the  diagonal  as  shown.  The  Cartesian  normal  form \nequations  describe  the  interaction  of  these  linearly  uncoupled \ncomplex  modes  due  to  the  coupling  of  the  nonlinear  terms.  We \ncan  interpret  the  normal  form  equations  as  network  equations \nin  eigenvector  (or  \"memory\")  coordinates,  given  by  some  diag(cid:173)\nonalizing  transformation  P,  containing  those  eigenvectors  as \nits  columns,  so  that  J \na  p-1  T  P.  Then  it  is  clear  that  T  may \ninstead  be  determined  by  the  reverse  projection  T  _  P  J  p-1 \nback  into  network  coordinates,  if we  start  with  desired  eigen(cid:173)\nvectors  and  eigenvalues.  We  are  free  to  choose  as  columns  in \nP,  the  real  and  imaginary  vectors  [XS  cos  9s  ,  XS  sin  9S]  of  the \ncycles  ~s  e i9s  of  any  linearly  independent- set -S  of  p~tterns \nto  be  learned.  If  we  write  the  matrix  expression  for  the  proj(cid:173)\nection  in  component  form,  we  recover  the  expression  given  in \nthe  theorem  for  Tij , \n\nNonlinear  Term  Projection \nThe  nonlinear  terms  are  transformed  as  well,  but  the  expres(cid:173)\nsion  cannot  be  easily  written  in  matrix  form.  Using  the  com(cid:173)\nponent  form  of  the  transformation, \n\nsubstituting  into  the  Cartesian  normal  form,  gives: \n\nXi  -\n\n(-'T+1)  E j  Pij  (E k  P-1jk  xk)  +  E j  Pij  Ek  J jk  (E I  P-\\I  xl) \n\n+  E j  Pij  (E k  P-1jk  xk)  EI  Ajl  (Em  p-\\m  xm)  (En  p-\\n  xn) \n\nRearranging  the  orders  of  summation  gives, \n\nXi  =  (-'T+1)  Ek  (E j  Pij  P-1jk )  xk  +  EI  (E k  E j  Pij  J jk  P-\\l)  xl \n\n+  En  Em  Ek  EI  E j  Pij  P  jk  AjI  P  1m \n\n( \n\n-1 \n\n-1 \n\np-1 \n\n) \n\nIn  xk  xm  xn \n\nFinally,  performing  the  bracketed  summations  and  relabeling \nindices  gives  us  the  network  of  the  theorem, \n\nxi  = - 'T  xi  +  E j  T1j  Xj  +  Ejkl  Tijkl  Xj  Xk  xl \n\nwith  the  expression  for  the  tensor  of  the  nonlinear  term, \n\n\f466 \n\nBaird \n\nT ijk1  - Emn  Pim  Amn  P  mj  P  nk  P  n1 \n\n-1 \n\n-1 \n\n-1 \n\nQ.E.D. \n\nLEARNING  RULE  EXTENSIONS \nThis  is  the  core  of  the  mathematical  story,  and  it may  be  ex(cid:173)\ntended  in  many  ways.  When  the  columns  of  P  are  orthonormal, \nthen  p-1  \u2022  pT,  and  the  formula  above  for  the  linear  network \ncoupling  becomes  T  = pJpT.  Then,  for  complex  eigenvectors, \n\n- 0  for  a  static \n\nThis  is  now  a  local,  additive,  incremental  learning  rule  for \nsynapse  ij,  and  the  system  can  be  truly  self-organizing  be(cid:173)\ncause  the  net  can  modify  itself  based  on  its  own  activity. \nBetween  units  of  equal  phase,  or  when  9i s  =  9j S \npattern,  this  reduces  to  the  usual  Hebb  rule. \nIn  a  similar  fashion,  the  learning  rule  for  the  higher  order \nnonlinear  terms  becomes  a  multiple  periodic  outer  product  rule \nwhen  the  matrix  A  is  chosen  to  have  a  simple  form.  Given  our \npresent  ignorance  of  the  full  biophysics  of  intracellular \nprocessing,  it  is  not  entirely  impossible  that  some  dimension(cid:173)\nality  of  the  higher  order  weights  in  the  mathematical  network \ncoul~  be  implemented  locally  within  the  cells  of  a  biological \nnetwork,  using  the  information  available  on  the  primary  lines \ngiven  by  the  linear  connections  discussed  above.  When  the  A \nmatrix  is  chosen  to  have  uniform  entries  Aij  - c  for  all  its \noff-diagonal  2  x  2  blocks,  and  uniform  entries  Aij  - c  - d \nfor  the  diagonal  blocks,  then, \n\nT ijk1  \u2022 \n\nThis  reduces  to  the  multiple  outer  product \n\nThe  network  architecture  generated  by  this  learning  rule  is \n\nThis  reduces  to  an  architecture  without  higher  order  correla(cid:173)\ntions  in  the  case  that  we  choose  a  completely  uniform  A matrix \n(A1j  - c \n\n,  for  all  i,j).  Then \n\n+ \n\n+ \n\n\fA Bifurcation Theory Approach to Programming \n\n467 \n\nThis  network  has  fixed  points  on  the  axes  of  the  normal  form \nas  always,  but  the  stability  condition  is  not  satisfied  since \nthe  diagonal  normal  form  coefficients  are  equal,  not  less, \nthan  the  remaining  A matrix  entries.  In  (Baird  89)  we  describe \nhow  clamped  input  (inspiration)  can  break  this  symmetry  and \nmake  the  nearest  stored  pattern  be  the  only  attractor. \nAll  of  the  above  results  hold  as  well  for  networks  with  sig(cid:173)\nmoids,  provided  their  coupling  is  such  that  they  have  a  Tayl(cid:173)\nor's  expansion  which  is  equal  to  the  above  networks  up  to \nthird  order.  The  results  then  hold  only  in  the  neighborhood  of \nthe  origin  for  which  the  truncated  expansion  is  accurate.  The \nexpected  performance  of  such  systems  has  been  verified  in \nsimulations. \n\nAcknowledgements \nSupported  by  AFOSR-87-0317.  I  am  very  grateful  for  the  support \nof  Walter  Freeman  and  invaluable  assistance  of  Morris  Hirsch. \n\nReferences \nB.  Baird.  Bifurcation  Theory  Methods  For  Programming  Static \n\nor  Periodic  Attractors  and  Their  Bifurcations  in  Dynamic \nNeural  Networks.  Proc.  IEEE  Int.  Conf.  Neural  Networks, \nSan  Diego,  Ca.,pI-9,  July(1988). \n\nB.  Baird.  Bifurcation  Theory  Approach  to  Vectorfield  Program(cid:173)\n\nming  for  Periodic  Attractors.  Proc.  INNS/IEEE  Int.  Conf. \non  Neural  Networks.  Washington  D.C.,  June(1989). \n\nW.  J.  Freeman  & B.  Baird.  Relation  of  Olfactory  EEG  to  Be(cid:173)\n\nhavior:  Spatial  Analysis.  Behavioral  Neuroscience  (1986). \n\nW.  J.  Freeman  & B.  W.  van  Dijk.  Spatial  Patterns  of  Visual \n\nCortical  EEG  During  Conditioned  Reflex  in  a  Rhesus  Monkey. \nBrain  Research,  422,  p267(1987). \n\nC.  M.  Grey  and  W.  Singer.  Stimulus  Specific  Neuronal \n\nOscillations  in  Orientation  Columns  of  Cat  Visual  Cortex. \nPNAS.  In  Press(1988). \n\nZ.  Li  & J.J.  Hopfield.  Modeling  The  Olfactory  Bulb.  Biologi(cid:173)\n\ncal  Cybernetics.  Submitted(1989}. \n\n\f", "award": [], "sourceid": 145, "authors": [{"given_name": "Bill", "family_name": "Baird", "institution": null}]}