{"title": "Phase-coupling in Two-Dimensional Networks of Interacting Oscillators", "book": "Advances in Neural Information Processing Systems", "page_first": 123, "page_last": 129, "abstract": null, "full_text": "Phase-coupling in Two-Dimensional \nNetworks  of Interacting Oscillators \n\nErnst  Niebur,  Daniel M.  Kammen,  Christof Koch, \n\nDaniel Ruderman!  &  Heinz  G.  Schuster2 \n\nComputation and  Neural Systems \n\nCaltech  216-76 \n\nPasadena,  CA  91125 \n\nABSTRACT \n\nCoherent oscillatory activity in large networks of biological or artifi(cid:173)\ncial neural units may be a useful mechanism for coding information \npertaining to a single perceptual object or for  detailing regularities \nwithin  a  data set.  We  consider  the  dynamics  of a  large  array  of \nsimple  coupled  oscillators  under  a  variety  of connection  schemes. \nOf particular  interest  is  the  rapid  and  robust  phase-locking  that \nresults  from  a  \"sparse\"  scheme  where  each  oscillator  is  strongly \ncoupled  to a  tiny,  randomly selected, subset of its neighbors. \n\nINTRODUCTION \n\n1 \nNetworks of interacting oscillators provide an excellent model for numerous physical \nprocesses ranging from the behavior of magnetic materials to models of atmospheric \ndynamics to the activity of populations of neurons in a  variety of cortical locations. \nParticularly prominent in the neurophysiological data are the 40-60 Hz oscillations \nthat have long been  reported in  the rat and rabbit olfactory bulb and cortex on the \nbasis of single-and  multi-unit  recordings  as  well  as EEG  activity  (Freeman,  1978). \nIn  addition,  periodicities  in  eye  movement  reaction  times  (Poppel  and  Logothetis, \n1986),  as well  as oscillations  in  the  auditory evoked  potential in  response  to single \nclick  or  a  series  of  clicks  (Madler  and  Poppel,  1987)  all  support  a  30  - 50  Hz \nframework  for  aspects of cortical  activity.  Two groups  (Eckhorn  et  al.,  1988,  Gray \n\n1 Permanent address:  Department  of Physics,  University  of California, Berkeley,  CA  94720 \n2 Permanent address:  Institut fiir Theoretische Physik, Universitat Kiel,  2300 Kiell, Germany. \n\n123 \n\n\f124 \n\nNiebur, Kammen, Koch, Ruderman, and Schuster \n\nand  Singer,  1989;  Gray  et  al.,  1989)  have  recently  reported  highly  synchronized, \nstimulus specific oscillations in  the 35 - 85  Hz range in  areas  17,  18  and PMLS of \nanesthetized  as  well  as  awake  cats.  Neurons  with similar orientation  tuning up  to \n7  mm  apart  show  phase-locked  oscillations  with  a  phase shift  of less  than  1  msec \nthat  have  been  proposed  to  play  a  role  in  the  coding  of visual  information  (Crick \nand  Koch,  1990,  Niebur  et  al.  1991). \nThe  complexity  of networks  of even  relatively  simple  neuronal  units  - let  alone \n\"real\"  cortical  cells  - warrants  a  systematic  investigation  of the  behavior  of two \ndimensional systems.  To address  this  question  we  begin  with  a  network  of mathe(cid:173)\nmatically simple limit-cycle oscillators.  While the dynamics of pairs of oscillators are \nwell  understood (Sakaguchi,  et  al.  1988, Schuster and Wagner,  1990a,b), this is  not \nthe case for  large networks  with nontrivial  connection schemes.  Of general interest \nis  the phase-coupling that results in  networks of oscillators  with  different  coupling \nschemes.  We  will  summarize some generic features  of simple nearest-neighbor  cou(cid:173)\npled models, models where each oscillator receives input from a large neighborhood, \nand  of \"sparse\"  connection  geometries  where  each  cell  is  connected  to only  a  tiny \nfraction  of the  units  in  its  neighborhood,  but  with  large  coupling  strength.  The \nnumerical work was performed on a CM-2 Connection Machine and involved  16,384 \noscillators in  a  128  by  128 square grid. \n\n2  The Model \nThe  basic  unit  in  our  networks  is  an  oscillator  whose  phase  (Jij  is  21r  periodic  and \nwhich  has  the  intrinsic  frequency  Wij.  The  dynamics  of an  isolated  oscillator  are \ndescribed  by: \n\nd(J\u00b7 . d;)  = Wij. \n\n(1) \n\nThe influence of the network  can  be expressed  as  an  additional  interaction term, \n\n(2) \n\nThe  coupling  function,  lij  we  used  is  expressed  as  the  sum  of  terms,  each  one \nconsisting of the product of a  coupling strength  and  the sine of a  phase  difference \n(see  below,  eq.  3).  The  sinusoidal  form  of the  interaction  is,  of course,  linear  for \nsmall differences. \n\nThis system, and numerous variants, has received a considerable amount of attention \nfrom solid state physicists (see, e.g.  Kosterlitz and Thouless 1973, and Sakaguchi et \nal.  1988), although primarily in the limit of t  -\n00.  With an interest in the possible \nrole  of  networks  of oscillators  in  the  parsing  or  segregating  of incident  signals  in \nnervous  systems, we  will  concentrate on short  time, non-equilibrium, properties. \n\nWe  shall  confine ourselves to two generic network  configurations described  by \n\nd(Jij  L '  \n- = w .. + a \nJ .. 1:lszn((J\u00b7\u00b7  -\ndt \n&) \n&), \n\n&) \n\n(J1:I) \n\n, \n\n1:1 \n\n(3) \n\n\fPhase-coupling in Two-Dimensional Networks of Interacting Oscillators \n\n125 \n\nwhere  0'  designates  the global  strength  of the interaction,  and  the geometry of the \ninteractions is  incorporated in  Jij ,kl' \n\nThe networks are all defined on a square grid  and they are  characterized as follows: \n\n1:  Gaussian  Connections.  The  cells  are  connected  to  every  oscillator  within  a \nspecified neighborhood  with  Gaussian  weighted  connections.  Hence, \n\nJ \nij ,kl = 271'0' exp \n\n1 \n\n( (i - k) 2 + (j - I) 2 ) \n\n20'2 \n\n. \n\n(4) \n\nWe  truncate this function  at  20',  i.e.  Jij,kl = 0 if (i - k)2 + (j _1)2 > (20')2.  While \nthe  connectivity  in  the  nearest  neighbor  case  is  4,  the  connectivity  is  significantly \nhigher  for  the  Gaussian  connection  schemes:  Already  0'  = 2  yields  28  connections \nper cell,  and the largest network  we  studied, with 0' = 6,  results in  372  connections \nper cell. \n\n2:  Sparse  Gaussian  Connections.  In  this  scheme  we  no  longer  require  sym(cid:173)\nmetric connections, or that the connection pattern is  identical from unit to unit.  A \ngiven cell is connected to a fixed  number, n, of neighboring cells, with the probability \nof a  given  connection determined  by \n\n'\" \nrij,kl =  271'0' exp \n\n1 \n\n(  (i - k? + (j - I? ) \n\n20'2 \n\n. \n\n(5) \n\nJij,kl  is  unity  with  probability  Pij,kl  and  zero  otherwise.  This  connection  scheme \nis  constructed  by  drawing for  each  lattice site  n  coordinate  pairs from  a  Gaussian \ndistribution,  and  use  these  as  the  indices  of the  cells  that  are  connected  with  the \noscillator  at  location  (i,j).  Therefore,  the probability of making  a  connection  de(cid:173)\ncreases  with  distance.  If a  connection is  made,  however,  the weight  is  the same as \nfor  all  other connections.  We  typically used  n  = 5,  and in  all  cases  2 ~ n  < 10. \nFor all networks, the sum of the weights of all connections with a given oscillator i, j \nwas conserved and chosen as  0' 2::kl Jij,kl  = 10 * w,  where w is  the average frequency \nof all  N  oscillators  in  the  system,  w =  k 2::ij Wij.  By  this  procedure,  the  total \nimpact of the interaction term is  identical in  all  cases. \n\n3  RESULTS \n\nPerhaps  the  most  basic,  and  most  revealing,  comparison  of  the  behavior  of the \nmodels  introduced  above  is  the  two-point  correlation  function  of phase-coupling, \nwhich  is  defined  as \n\n(6) \n\nwhere  R  is  defined  as  the  separation  between  a  pair  of cells,  R  =  hj - nIl.  We \ncompute and then average C(R, t)  over  10,000 pairs of oscillators separated by R in \nthe array.  In all cases, the frequencies Wij  are chosen randomly, with a Gaussian dis(cid:173)\ntribution with mean 0.5 and variance  1.  In Figure 1 we  plot C(R, t) for separations \n\n\f126 \n\nNiebur, Kammen, Koch, Ruderman, and Schuster \n\nof R =  20,  30,  40,  50,  6,  and  70  oscillators.  Time is  measured  in  oscillation  peri(cid:173)\nods of the mean oscillator frequency, w.  At t = 0,  phases are  distributed randomly \nbetween  0 and  27r  with  a  uniform distribution.  The case  of Gaussian  connectivity \nwith u = 6 and hence 372 connection  per cell is  seen  in  Figure  l(a),  and the sparse \nconnectivity  scheme  with  u  = 6  and  n  = 5  is  presented in  Figure  l(b).  The  most \nstriking  difference  is  that  correlation  levels  of over  0.9  are  rapidly  achieved  in  the \nsparse scheme for  all  cases,  even  for  separations of 70  oscillators  (plotted as  aster(cid:173)\nisks,  *),  while  there  are  clear separation-dependent differences in  the phase-locking \nbehavior  of the  Gaussian  model.  In  fact,  even  after  t  = 10  there  is  no  significant \nlocking over  the  longer  distances of R  =  50,60, or  70  units.  For  local  connectivity \nschemes,  like  Gaussian  connectivity  with  u  = 2 or  nearest  neighbors  connections, \nno  long-range order evolves  even  at  larger times (data not shown). \nData in  Fig.  1 were  computed with  a  uniform  phase  distribution  for  t  = o.  An  in(cid:173)\nteresting and robust feature of the dynamics emerges when the influence of different \ntypes of initial phase distributions are examined.  In  Figure 2 we  plot the probabil(cid:173)\nity  distribution  of phases  at  different  early  times.  In  Figure  2( a)  the  distribution \nof phases  is  plotted  at  t  = 0  (diamonds),  t  = 0.2  (\"plus  signs,  +)  and  at  t  = 0.4 \n(squares) for  the sparse scheme with a  uniform initial distribution.  In  Figure  2(b), \nthe  evolution  of a  Gaussian  initial  distribution  centered  at  ()  =  7r  of the  phases \nis  plotted.  Note  the  slight  curve  in  the  distribution  at  t  =  0,  indicating  that  the \nGaussian  initial seeding  is  rather  slight  (variance  u  =  27r) .  Remarkably,  however, \nthis  has  a  dramatic  impact  on  the  phase-locking  as  after  two-tenth  of an  average \ncycle time (\" plus\" signs) there is  already a pronounced peak in  the distribution .  At \nt = 0.4  (squares)  the  system that started  with  the  uniform  distribution  begins  to \nonly  exhibit  a  slight  increase  in  the phase-correlation  while  the system with Gaus(cid:173)\nsian  distributed  initial  phases  is  strongly  peaked  with  virtually  no  probability  of \nencountering phase values  that differ significantly from the  mean. \n\n4  DISCUSSION \nThe power of the sparse connection scheme to rapidly generate phase-locking through(cid:173)\nout  the  network  that  is  equivalent,  or superior,  to  that  of the massively  intercon(cid:173)\nnected  Gaussian  scheme  highlights  a  trade-off in  network  dynamics:  massive  av(cid:173)\neraging  versus  strong,  long-range,  connections.  With  n  =  5,  the  sparse  scheme \neffectively  \"tiles\"  a  two-dimensional lattice  and tightly phase-locks oscillators even \nat opposite corners of the array.  Similar results are obtained even  with  n  = 2 (data \nnot shown). \n\nIn  many  ways  the  Gaussian  and  sparse  geometries  reperent  opposing  avenues  to \nachieve  global  coherence:  exhaustive  local  coupling  or  distributed,  but  powerful \nlong-range  coupling.  The  amount  of wiring  necessary  to implement  these  schemes \nis,  however,  radically  different. \n\n\fPhase-coupling in Two-Dimensional Networks of Interacting Oscillators \n\n127 \n\nAcknowledgement \n\nEN is supported by the Swiss National Science Foundation through Grant No.  8220-\n25941.  DMK is a recipient of a Weizman Postdoctoral Fellowship.  CK acknowledges \nsupport from the Air Force  Office  of Scientific  Research,  a  NSF  Presidential Young \nInvestigator  Award  and  from  the  James  S.  McDonnell  Foundation.  HGS  is  sup(cid:173)\nported  by  the Volkswagen  Foundation. \n\nReferences \n\nCrick,  F.  and  Koch,  C.  1990.  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A  model  for  neuronal  oscillations  in  the \nvisual cortex 2:  Phase description of feature  dependant synchronization.  Biological \nCybernetics,  64, 83. \n\n\f128 \n\nNiebur, Kammen, Koch, Ruderman, and Schuster \n\n(A) \n\n\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\u2022 \u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\n\u2022\u2022\u2022\u2022 \n\n\u2022 \n\n... 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I \n\nFigure 1:  Tw~point correlation  functions,  C(R, t), for  various  separations,  R,  in \n(a)  the  (T  = 6  Gaussian  scheme  with  372  connections  per  cell  and  (b)  the  spar~e \nconnection  scheme  with  (T  = 6  and  n  = 5  connections  per  cell.  Separations  of \nR = 20  (diamonds),  R = 30  (\"Plus\"  signs,  +), R = 40  (squares),  R = 50  (crosses, \nx), R = 60  (triangles), and R = 70  (asterisks, *)  are shown.  Note the rapid locking \nfor  all  lengths in  the sparse scheme  (b)  while  the Gaussian  scheme (a)  appears far \nmore  \"diffusive,\"  with  progressively  poorer and slower  locking  as  R increases. \n\n\fPhase-coupling in Two-Dimensional Networks of Interacting Oscillators \n\n129 \n\n14~----~----~----~-----r----~--~--~----~----~----~ \n\n(A I \n\n10 \n\nPI' 1 \n\n8 \n\nr. \n\nFigure  2:  Snapshots  of  the  distribution  of  phases  in  the  sparse  scheme \n(n  = 5,0'  =  6)  when  the  system  begins  from  (a)  uniform  and  (b)  a  Gaussian \n\"biased\"  initial distribution.  The figures  show  the probability P(O)  to find  a phase \nbetween  0 and 0 + dO  (bin size  'KIlO).  At t = 0,  the  distribution is fiat  (a) or very \nslightly  curved  (b);  see  text.  The difference  in  the  time  evolution  can  clearly  be \nseen in the state of the system after t = 0.2  (\"plus\"  signs, +) and t = 0.4  (squares). \n\n\f", "award": [], "sourceid": 302, "authors": [{"given_name": "Ernst", "family_name": "Niebur", "institution": null}, {"given_name": "Daniel", "family_name": "Kammen", "institution": null}, {"given_name": "Christof", "family_name": "Koch", "institution": null}, {"given_name": "Daniel", "family_name": "Ruderman", "institution": null}, {"given_name": "Heinz", "family_name": "Schuster", "institution": null}]}