{"title": "Collective Oscillations in the Visual Cortex", "book": "Advances in Neural Information Processing Systems", "page_first": 76, "page_last": 83, "abstract": null, "full_text": "76 \n\nKammen, Koch and Holmes \n\nCollective Oscillations in the \n\nVisual  Cortex \n\nDaniel Kammen &  Christof Koch \n\nPhilip J.  H oImes \n\nComputation and  Neural Systems \n\nDept.  of Theor.  &  Applied Mechanics \n\nCaltech 216-76 \n\nPasadena, CA 91125 \n\nCornell  University \nIthaca,  NY 14853 \n\nABSTRACT \n\nThe  firing  patterns  of populations  of cells  in  the  cat  visual  cor(cid:173)\ntex can  exhibit  oscillatory  responses  in  the  range  of 35  - 85  Hz. \nFurthermore,  groups  of neurons  many  mm's  apart  can  be  highly \nsynchronized  as  long  as  the  cells  have  similar  orientation  tuning. \nWe investigate two basic network architectures that incorporate ei(cid:173)\nther nearest-neighbor or global feedback interactions and conclude \nthat non-local feedback plays a  fundamental role in the initial syn(cid:173)\nchronization and dynamic stability of the oscillations. \n\nINTRODUCTION \n\n1 \n40  - 60  Hz  oscillations  have  long  been  reported  in  the  rat  and  rabbit  olfactory \nbulb  and  cortex  on  the  basis  of single- and  multi-unit  recordings  as  well  as  EEG \nactivity (Freeman,  1972;  Wilson &  Bower 1990).  Recently,  two groups (Eckhorn  et \nai.,  1988 and Gray et ai.,  1989)  have reported highly synchronized, stimulus specific \noscillations  in  the 35  - 85  Hz  range in  areas  17,  18  and  PMLS  of anesthetized  as \nwell as awake cats.  Neurons with similar orientation tuning up to 7 mm apart show \nphase-locked  oscillations,  with a  phase shift  of less  than 3  msec.  We  address  here \nthe  computational architecture necessary  to  subserve this  process by investigating \nto what extent two neuronal architectures,  nearest-neighbor coupling and feedback \nfrom a  central  \"comparator\", can synchronize neuronal oscillations in a  robust and \nrapid  manner. \n\n\fCollective Oscillations in the Visual Cortex \n\n77 \n\nIt was argued in earlier work on central pattern generators (Cohen  et al., 1982), that \nin studying coupling effects among large populations of oscillating neurons,  one can \nignore the details of individual oscillators and represent each one by a single periodic \nvariable:  its  phase.  Our  approach  assumes  a  population  of neuronal  oscillators, \nfiring  repetitively  in  response  to  synaptic  input.  Each  cell  (or  group  of tightly \nelectrically  coupled  cells)  has  an  associated  variable  representing  the  membrane \npotential.  In  particular,  when  (Ji  = 7r,  an  action  potential  is  generated  and  the \nphase is  reset  to its initial value (in our case  to -7r).  The number of times per unit \ntime (Ji  passes through 7r,  i.e.  d(Ji/dt,  is  then proportional to the firing  frequency  of \nthe neuron.  For a  network of n + 1 such oscillators,  our basic model is \n\n(1) \n\nwhere Wi  represents the synaptic input to neuron i  and I, a  function of the phases, \nrepresents the coupling within the  network.  Each oscillator i  in  isolation (i.e.  with \nIi  =  0),  exhibits  asymptotically  stable  periodic  oscillations;  that  is,  if the  input \nis  changed  the  oscillator  will  rapidly  adjust  to a  new  firing  rate.  In  our  model Wi \nis  assumed  to derive from  neurons  in  the  lateral geniculate  nucleus  (LG N)  and  is \npurely excitatory. \n\n2  FREQUENCY AND PHASE LOCKING \nAny realistic model of the observed, highly  synchronized, oscillations must account \nfor  the fact  that the individual neurons oscillate at different frequencies in isolation. \nThis is  due  to variations in the synaptic  input, Wi,  as  well as  in  the intrinsic  prop(cid:173)\nerties  of the cells.  We  will  contrast the abilities  of two markedly different  network \narchitectures to synchronize these oscillations.  The \"chain\" model (Fig.  1 top) con(cid:173)\nsists  of a  one-dimensional array of oscillators  connected  to their nearest neighbors, \nwhile  in  the alternative  \"comparator\"  model  (Fig.  1  middle),  an array of neurons \nproject to a single unit, where the phases are averaged (i.e.  (lin) L~=o Oi(t)).  This \naverage is  then feed  back to every neuron  in  the  network.  In  the  continuum limit \n(on  the unit interval)  with all Ii = I  being identical,  the two models  are \n\n80(x, t) \n\n8(J(x, t) \n\n8t \n\n8t \n\n(2) \n\nn  x \n\n(Chain  Model) \n\n(Comparator  Model) \n\nW(x) + .!.. 88f (4)) \nw(x) + 1((J(x, t) -10 1 (J(s, t)ds),  (3) \nwhere  0  < x  < 1  and  4>  is  the  phase  gradient,  4>  = ~M. In  the  chain  model,  we \nrequire that I  be an odd function (for simplicity of analysis only)  while our analysis \nof the comparator model holds for  any continuous function I.  We use two spatially \nseparated  \"spots\"  of width 6 and amplitude Q'  as visual input (Fig.  1 bottom).  This \npattern was chosen as a  simple version of the double-bar stimulus that (Gray  et  al. \n1989)  found  to evoke coherent oscillatory  activity in  widely  separated  populations \nof visual cortical cells. \n\n\f78 \n\nKammen, Koch and Holmes \n\n-\u2022 \n-~ \u2022 \n\n\u2022 \n\n\u2022\u2022\u2022 \n\n\u2022 \n\n\u2022 \n\n00(0) \n\n8i=n(t) \n\n+ \nm(n) \n\nmen) \n\n00(0) \nm(x) \nI \n\nI ta \n\nIta \n\nx \nFigure  1:  The  linear  chain  (top)  and  comparator  (middle)  architectures.  The \nspatial pattern of inputs is  indicated  by Wj(x).  See  equs.  2 &  3 for  a  mathematical \ndescription of the models.  The  \"two spot\"  input is  shown at bottom and represents \ntwo parts of a  perceptually extended  figure. \n\nWe determine  under  what  circumstances  the  chain  model  will  develop  frequency(cid:173)\nnecessarily at the same  time),  i.e.  8 2 (} /8x8t = O.  We prove (Kammen,  et  al.  1990) \nlocked  solutions,  such  that  every  oscillator  fires  at  the  same  frequency  (but  not \nthat frequency-locked solutions exist as long as  In(wx- fo:17  w(s)ds)1  does not exceed \nthe  maximal  value  of  I,  Imax  (with  w = f; w(s)ds  the  mean  excitation  level). \nThus,  if the  excitation  is  too  irregular  or  the  chain  too  long (n  \u00bb  1),  we  will  not \nfind  frequency-locked  solutions.  Phase coherence between the excited regions is  not \ngenerally  maintained  and  is,  in  fact,  strongly  a  function  of the  initial  conditions. \nAnother  feature  of the  chain model  is  that  the  onset  of frequency  locking  is  slow \nand  takes time of order Vii. \nThe location of the stimulus has no effect on phase relationships in the comparator \nmodel  due  to  the  global  nature  of the  feedback.  The  comparator  model  exhibits \ntwo distinct regimes of behavior depending on the amplitude of the input, a.  In the \ncase of the two spot input (Fig. 1 bottom ), if a  is  small, all neurons will frequency(cid:173)\nlock  regardless  of location,  that  is  units  responding  to  both  the  \"figure\"  and  the \nbackground  (\"ground\")  will  oscillate  at  the  same  frequency.  They  will,  however, \nfire  at different times,  with () Jig 1=  () gnd.  If a  is  above a critical threshold,  the units \nresponding  to  the  \"figure\"  will  decouple  in  frequency  as  well  as  phase  from  the \nbackground while still maintaining internal phase coherency.  Phase gradients never \nexist within  the excited  groups,  no matter what  the input amplitude. \n\n\fCollective Oscillations in the Visual Cortex \n\n79 \n\nWe numerically simulated the chain and comparator models with the two spot input \nfor the coupling function fCf})  =  sin(f}).  Additive Gaussian noise was included in the \ninput, Wi.  Our analytical results were confirmed; frequency and phase gradients were \nalways present in the chain model (Fig.  2A)  even  though the coupling strength was \nten  times  greater  than  that  of the comparator modeL  In  the comparator network \nsmall excitation levels  led  to frequency-locking along the entire array and  to phase(cid:173)\ncoupled activity within the illuminated areas (Fig.  2B), while large excitation levels \nled  to phase and  frequency  decoupling between the  \"figure\"  and the  \"background\" \n(Fig.  2C).  The excited  regions  in the comparator settle  very  rapidly  - within 2  to \n3 cycles - into phase-locked activity with small  phase-delays.  The chain model,  on \nthe other hand, exhibits strong sensitivity to initial conditions as well as a very slow \napproach to coherence that is  still not complete even after 50  cycles  (See  Fig.  2). \n\nA \n\nB \n\nc \n\nFigure 2:  The phase portrait of the chain (A),  weak (B)  and strongly (C)  excited \ncomparator networks  after 50 cycles.  The input,  indicated  by the  horizontal lines, \nis  the  two spot  pattern.  Note  that  the  central,  unstimulated,  region  in  the  chain \nmodel has been  \"dragged along\"  by  the flanking  excited  regions. \n\n3  STABILITY ANALYSIS \nPerhaps the  most  intriguing aspect  of the  oscillations  concerns  the  role  that  they \nmay play in cortical information processing and the labeling of cells responding to a \nsingle perceptual object.  To be useful in object coding,  the oscillations must exhibit \nsome degree  of noise  tolerance both  in the  input signal and  in the  stability of the \npopulation to variation in the firing  times  of individual cells. \n\n.  d b \n\nh \n\n. \n\nII \n\nI \u00b7 \u00b7  d \n\ne popu atlOn  IS  etermme \n\nThe degree to which input noise  to individual neurons  disrupts the synchronization \ny t  e ratio  coupling  strength  =  irT.  or sma  per-\nf th \no \nturbations, wet) = Wo + f(t),  the action of the feedback,  from  the nearest neighbors \nin  the  chain and  from  the  entire  network  in  the  comparator,  will  compensate  for \nthe  noise  and  the neuron will  maintain coherence with  the  excited  population.  As \nf  is  increased first  phase and  then frequency  coherence  will  be lost. \n\ninput  noise  ~ F \n\nIn Fig.  3 we compare the dynamical stability of the chain and comparator models. \nIn  each  case  the  phase,  (J,  of a  unit  receiving  perturbated  input  is  plotted  as  the \ndeviation from the average phase, (Jo,  of all the excited units receiving input WOo  The \nchain in  highly  sensitive to  noise:  even 10%  stochastic  noise  significantly  perturbs \nthe  phase of the neuron.  In  the comparator model  (Fig.  3B)  noise  must reach  the \n\n\f80 \n\nKammen, Koch and Holmes \n\n40% level to have a  similar effect on the phase.  As the noise increases above 0.30wo \neven frequency coherence  is  lost in the chain model (broken error  bars).  Frequency \ncoherence is  maintained in the comparator for  f  = 0.60wo. \n\ne \nA) \n\n+0.10  ~ \n\n0.00 \n-G.OS \n-G.IO  ~ \n\n0.0 \n\n20 \n\nB) \n\n............. ~ .... 1t .... :1....  ..~ \n\n\u2022 \n\nl \n\n\"I \n\n40 \n\n60 \n\n0.0 \n\u00a3 (%  of 000) \n\n20 \n\n40 \n\n60 \n\nFigure  3:  The  result  of a  perturbation  on  the  phase,  0,  for  the  chain  (A)  and \ncomparator  (B)  models.  The terminus of the  error  bars  gives  the  resulting  devia(cid:173)\ntion  from  the  unperturbed  value.  Broken bars  indicate  both  phase  and  frequency \ndecoupling. \n\nThe stability of the solutions of the comparator model to variability in the activity \nof individual neurons can easily  be demonstrated.  For simplicity consider  the case \nof a single input of amplitude WI  superposed on a background of amplitude Woo  The \nsolutions in  each region  are: \n\n(4) \n\ndOo \ndt \ndOl \ndt \n\n2 \n\nI  ( 00  - 01) \n1(01  - 00) \n\nwo+ \n\nWI  + \n\n(5) \nWe define the difference in the solutions to be \u00a2(t) = 01(t)-00(t) and Aw = W1-WO. \nWe then have an equation for  the rate the solutions converge or diverge: \n\n2 \n\n. \n\n<P \n2 \n\nd\u00a2 \n- =  Aw + 1(-) - 1(--)\u00b7 \ndt \n\n(6) \nIf the solutions are stable (of constant velocity) then d01/dt = dOo/dt and 01 =  Oo+c \nwith c a  constant.  We  then have  the stable solution \u00a2* = c  d\u00a2* /dt = Aw + I(~)\u00ad\nI( - ~) = O.  Stability of the solutions can be seen by perturbing 01 to 01 = 00 + c + f \nwith If I < 1.  The perturbed solution, \u00a2 = \u00a2* + f,  has  the derivative d\u00a2/dt = df/dt. \nDeveloping I( \u00a2)  into a  Taylor series  around  \u00a2*  and  neglecting  terms  on the order \nof f2  and higher,  we  arrive at \n\n<P \n2 \n\ndf = :. [1'( c:..) \n\ndt \n\n2 \n\nI'( =:..)] \n\n2 \n\n2  + \n\n. \n\n(7) \n\n\fCollective Oscillations in the Visual Cortex \n\n81 \n\nIf f(\u00a2)  is  odd  then  f'(\u00a2)  is even, and eq.  (7)  reduces  to \n\n(8) \nThus, if f'(c/2) < 0  the perturbations will  decay to zero and  the system will main(cid:173)\ntain phase locking within the excited  regions. \n\nd\u20ac  = \u20acf'(:') \ndt \n2  . \n\n4  THE FREQUENCY MODEL \nThe  model  discussed  so  far  assumes  that  the  feedback  is  only  a  function  of  the \nphases.  In particular,  this implies that the comparator computes the average phase \nacross  the  population.  Consider,  however,  a  model  where  the feedback  is  propor(cid:173)\ntional to the average firing frequency of a group of neurons.  Let us  therefore replace \nphase in  the feedback function  with firing  frequency, \n\naO(x, t)  = w(x) \n\nat \n\n+ \n\nf  (ao(x, t)  _  aO(x, t)) \n\nat \n\nat \n\n(9) \n\nwith \u00a5t  =  It J; O(s, t)ds = \u00a5t.  This is  a  very special differential equation as can be \nseen  by setting v(x,t)  =  aO(x, t)/at.  This  yields  an algebraic  equation for  v  with \nno explicit  time dependency: \n\nv(x) = w(x) + f(v(x) - v(x)) \n\nand,  after an integration, we  have, \n\nO(x, t)  = lot v(x)dt = v(x)t + Oo(x). \n\n(10) \n\n(11) \n\nThus, the  phase relationships depend on the initial conditions, Oo(x),  and  no  phase \nlocking occurs.  While frequency  locking only occurs for  w(x)  =  0  the feedback  can \nlead  to tight frequency  coupling among the excited  neurons. \n\nReformulating the chain model  in  terms of firing-frequencies,  we  have \n\nao(x, t)  =  !. (aw(x) + !. a2 f  (ao(x, t))) \n\nat \n\nn \n\nax \n\nn ax2 \n\nat \n\n(12) \n\nunder the assumption that  f(-x) = -f(x).  With -y(x,t) = a~~~!t), we  again arrive \nat a  stationary algebraic equation \n\n-y(x) =;;-\n\n1  (aw \n\n) \nax + ;;- ax2fb(x)) \n\n1  a2 \n\n, \n\nand \n\n\u00a2(x, t) = lot -y(x)dt = -y(x)t + \u00a2o(x) \n\n(13) \n\n(14) \n\nIn other words, the system will develop a time-dependent phase gradient.  Frequency \nlocked  solutions  of the sort  \u00a5t = 0  everywhere only occur if w(x) = 0  everywhere. \nThus, the chain architecture leads to very static behavior, with little ability to either \nphase- or  frequency-lock. \n\n\f82 \n\nKammen, Koch and Holmes \n\n5  DISCUSSION \nWe  have investigated  the ability of two networks  of relaxation  oscillators  with  dif(cid:173)\nferent  connectivity  patterns  to  synchronize  their  oscillations.  Our  investigation \nhas  been  prompted  by  recent  experimental  results  pertaining  to  the  existence  of \nfrequency- and  phase-locked  oscillations  in  the  mammalian  visual  cortex  (Gray  et \nal.,  1989;  Eckhorn  et  al.,  1988).  While these 35 - 85  Hz oscillations are induced by \nthe visual stimulus,  usually a flashing or  moving bar,  they are not locked to the fre(cid:173)\nquency of the stimulus.  Most surprising is  the finding  that cells  tuned  to the same \norientation,  but  separated  by  up  to  7  mm,  not  only  exhibit  coherent  oscillatory \nactivity,  but do so  with a  phase-shift of less  than 3  msec (Gray  et  al.,  1989).1 \n\nWe have assumed  the existence of a  population of cortical oscillators,  such as  those \nreported in cortical slice preparations (Llimis,  1988;  Chagnac-Amitai and Connors, \n1989).  The issue  is  then how  such a  population of oscillators  can  rapidly  begin  to \nfire  in near  total synchrony.  Two neuronal architectures suggest  themselves. \nAs a mechanism for establishing coherent oscillatory activity the comparator model \nis far  superior to a  nearest-neighbor model.  The comparator rapidly (within 1 - 3 \ncycles)  achieves  phase coherence,  while  the chain model exhibits a  far  slower onset \nof synchronization and  is  highly  sensitive  to the  initial conditions.  Once  initiated, \nthe  oscillations  in  the  two  models  exhibit  markedly different  stability characteris(cid:173)\ntics.  The diffusive  nature of communication in  the chain results  in  little  ability  to \nregUlate  the  firing  of individual  units  and  consequently  only  highly  homogeneous \ninputs  will  result  in collective  oscillations.  The long-range  connections  present  in \nthe comparator, however, result in stable collective oscillations even in  the presence \nof significant  noise  levels.  Noise  uniformly  distributed  about  the mean firing  level \nwill  have little effect  due to the averaging performed by the comparator unit. \n\nA  more  realistic  model  of  the  interconnection  architecture  of the  cortex  will  cer(cid:173)\ntainly have to take both local as well as global neuronal pathways into account and \nthe  ever-present  delays  in  cellular  and  network  signal  propagation  (Kammen,  et \nal.,  1990).  Long  range  (up  to  6  mm)  lateral  excitatory  connections  have  been \nreported  (Gilbert  and  Wiesel,  1983).  However,  their  low  conduction  velocities \n(~  1  mm/msec)  would  lead  to  significant  phase-shifts  in  contrast  to  the  data. \nWhile  the  cortical  circuitry  contains  both  local  as  well  as  global  connection,  our \nresults  imply  that  a  cortical architecture with  one  or  more  \"comparator\"  neurons \ndriven by the averaged  activity of the hypercolumnar cell  populations is  an attrac(cid:173)\ntive mechanism for  synchronizing the observed oscillations. \n\nWe  have  also  developed  a  model  where  the  firing  frequency,  and  not  the  phase  is \ninvolved  in  the  dynamics.  Coding  based  on  phase  information  requires  that  the \ncells  track  the  time  interval  between  incident  spikes  whereas  the  firing  frequency \nis  available  as  the  raw  spike  rate.  This  computation  can  be  readily  implemented \n\n1 Note that this result is obtained by averaging over many trials.  The phase-shift for individual \ntrial  may  possibly  be  larger,  but  could  be  randomly  distributed  from  trial  to  trial  around  the \norigin. \n\n\fCollective Oscillations in the Visual Cortex \n\n83 \n\nneurobiologically  and  is  entirely  consistent  with  the  known  biophysics  of cortical \ncells. \nVon der Malsburg (1985)  has argued that the temporal synchronization of groups of \nneurons  labels  perceptually  distinct  objects,  subserving figure-ground  segregation. \nBoth  firing  frequency  and  inter-cell  phase  (timing)  relationships  of ensembles  of \nneurons  are  potential channels  to encode  the  signatures  of various  objects  in  the \nvisual field.  Perceptually distinct objects could  be coded by groups of synchronized \nneurons,  all  locked  to  the  same frequency  with  the  groups  only  distinguished  by \ntheir phase relationships.  We do not believe, however,  that phase is a  robust enough \nvariable to code this information across  the cortex, A more robust scheme is  one in \nwhich  groups of synchronized  neurons are  locked at different firing  frequencies. \n\nAcknowledgement \n\nD.K.  is  a  recipient of a  Weizman Postdoctoral Fellowship.  P.H. acknowledges sup(cid:173)\nport  from  the  Sherman  Fairchild  Foundation and  C.K.  from  the  Air  Force  Office \nof Scientific Research, a  NSF  Presidential Young Investigator Award and  from  the \nJames  S.  McDonnell  Foundation.  We  would  like  to thank  Francis  Crick for  useful \ncomments and  discussions. \n\nReferences \n\nChagnac-Amitai, Y.  &  Connors, B.  W.  (1989)  1.  Neurophys.,  62,  1149. \nCohen,  A.  H.,  Holmes, P. J. &  Rand  R.  H.  (1982)  1.  Math.  Bioi.  3,345. \nEckhorn, R.,  Bauer, R.,  Jordan, W., Brosch, M., Kruse, W., Munk, M. &  Reitboeck, \n\nH.  J.  (1988)  Bioi.  Cybern.,  60,  121. \n\nFreeman, W.J. (1972)  1.  Neurophysiol.  35,762. \nGilbert, C.  D.  &  T.N. Wiesel (1983)  1.  Neurosci.  3,  1116. \nGray,  C.  M.,  Konig,  P.,  Engel,  A.  K.  &  Singer,  W.  (1989)  Nature 338, 334. \nKammen,  D.  M.,  Koch,  C.  and Holmes,  P.  J.  (1990)  Proc.  Natl.  Acad.  Sci.  USA, \n\nsubmitted. \n\nKopell  N.  &  Ermentrout, G.  B.  (1986)  Comm.  Pure  Appl.  Math.  39,623. \nLlimis,  R.  R.  (1988)  Science  242,  1654. \n\nvon der Malsburg,  C.  (1985)  Ber.  Bunsenges  Phys.  Chem., 89, 703. \nWilson,  M.  A.  &  Bower, J.  (1990)  1.  Neurophysiol.,  in press. \n\n\f", "award": [], "sourceid": 285, "authors": [{"given_name": "Daniel", "family_name": "Kammen", "institution": null}, {"given_name": "Christof", "family_name": "Koch", "institution": null}, {"given_name": "Philip", "family_name": "Holmes", "institution": null}]}