{"title": "Spiral Waves in Integrate-and-Fire Neural Networks", "book": "Advances in Neural Information Processing Systems", "page_first": 1001, "page_last": 1006, "abstract": null, "full_text": "Spiral Waves  in Integrate-and-Fire \n\nNeural Networks \n\nJohn G.  Milton \n\nDepartment of Neurology \nThe University of Chicago \n\nChicago, IL 60637 \n\nPo Hsiang Chu \n\nDepartment of Computer Science \n\nDePaul University \nChicago, IL 60614 \n\nJack D.  Cowan \n\nDepartment of Mathematics \nThe University of Chicago \n\nChicago, IL 60637 \n\nAbstract \n\nThe formation of propagating spiral waves is studied in a  randomly \nconnected neural network composed of integrate-and-fire neurons \nwith  recovery  period and  excitatory  connections  using  computer \nsimulations.  Network  activity is  initiated by  periodic stimulation \nat a single point.  The results suggest that spiral waves can arise in \nsuch a  network via a  sub-critical Hopf bifurcation. \n\n1 \n\nIntroduction \n\nIn neural networks activity  propagates through populations, or layers,  of neurons. \nThis propagation can be monitored as  an evolution of spatial patterns of activity. \nThirty years ago, computer simulations on the spread of activity through 2-D ran(cid:173)\ndomly connected networks demonstrated that a variety of complex spatio-temporal \npatterns can be generated including target waves  and spirals  (Beurle,  1956,  1962; \nFarley and Clark,  1961; Farley,  1965).  The networks studied by these investigators \ncorrespond to inhomogeneous excitable media in which the probability of interneu(cid:173)\nronal connectivity decreases exponentially with distance.  Although travelling spiral \nwaves can readily be formed in excitable media by the introduction of non-uniform \n\n1001 \n\n\f1002 \n\nMilton, Chu, and Cowan \n\ninitial  conditions  (e.g.  Winfree,  1987),  this  approach is not suitable for  the study \nand classification of the dynamics associated with the onset of spiral  wave  forma(cid:173)\ntion.  Here we  show that spiral waves  can \"spontaneously\"  arise from target waves \nin a  neural network in which  activity is  initiated by  periodic stimulation at a  sin(cid:173)\ngle  point.  In particular, the onset of spiral wave  formation appears to occur via a \nsub-critical Hopf bifurcation. \n\n2  Methods \n\nComputer  simulations  were  used  to  simulate  the  propagation of activity  from  a \ncentrally placed source in a  neural network containing 100 x  100 neurons arranged \non a  square lattice with  excitatory interactions.  At t = 0  all  neurons were  at rest \nexcept  the source.  There were free  boundary conditions and all  simulations were \nperformed on a  SUN SPARC 1+ computer. \n\nThe network was constructed by assuming that the probability, A,  of interneuronal \nconnectivity was an exponential decreasing function of distance, i.e. \n\nA = (3 exp( -air!) \n\nwhere a  = 0.6, {3  = 1.5 are constants and Ir I is the euclidean interneuronal distance \n(on  average each neuron makes 24 connections and '\" 1.3 connections per neuron, \ni.e.  multiple connections occur).  Once the connectivity was determined it remained \nfixed  throughout the simulation. \n\nThe dynamics of each neuron were represented by an integrate-and-fire model pos(cid:173)\nsessing  a  \"leaky\"  membrane potential and an  absolute  (1  time step)  and  relative \nrefractory or  recovery  period as  described  previously  (Beurle,  1962;  Farley,  1965; \nFarley and Clark, 1961):  the membrane and threshold decay constants were, respec(cid:173)\ntively,  k m  = 0.3 msec- 1 ,  ko  = 0.03 msec- 1\u2022  The time step of the network was taken \nas  1 msec and it was  assumed that during this time a  neuron transmits excitation \nto all other neurons connected to it. \n\n3  Results \n\nWe illustrate the dynamics of a  particular network as a  function of the magnitude \nof the excitatory interneuronal excitation, E,  when all  other parameters are fixed. \nWhen E  < 0.2 no activity propagates from the central source.  For 0.2 < E  < 0.58 \ntarget waves  regularly emanate from  the centrally  placed source  (Figure  1a).  For \nE  ~ 0.58  the  activity  patterns, once established,  persisted even  when  the source \nwas  turned off.  Complex spiral  waves  occurred when  0.58  <  E  <  0.63  (Figures \n1b-ld).  In these cases spiral meandering, spiral tip break-up and the formation of \nnew  spirals  (some with  multiple  arms)  occur continuously.  Eventually  the spirals \ntend to migrate out of the network.  For E  ~ 0.63 only disorganized spatial patterns \noccurred without clearly distinguishable wave fronts, except initially (Figures ie-f). \n\n\fSpiral Waves  in  Integrate-and-Fire  Neural Networks \n\n1003 \n\nFigure  1:  Representative  examples  of  the  spatial  pattern of neural  activity  as  a \nfunction of E:(a)  E  = 0.45,  (b - e)  E  = 0.58 and  (f)  E  = 0.72.  Color code:  gray = \nquiescent,  white =  activated, black  =  relatively refractory.  See text for  details. \n\n,~~~~~~~WL \n\nb \n\nc \n\n.1 \n\nD5 \n\n.07 \n\nc.:I \n0 \nZ \n~  .14 \ni:L \nCIl z \n0 \n~ :::> \nU.I z \nz \n0 \n~ \nu \n< \n~ u.. \n\n0 \n\n.2 \n\no \n\no \n\n5 \n\n10 \n\nITERATION  x 102 \n\nFigure 2:  Plot of the fraction of neurons firing  per unit  time for  different  values  of \nE:  (a) 0.45,  (b) 0.58,  and (c) 0.72.  At t  =  0 all neurons except the central source are \nquiescent.  At  t  = 500  (indicated by .J-)  the source is shut off.  The region indicated \nby  (*)  corresponds to an epoch in which spiral tip breakup occurs. \n\n\f1004 \n\nMilton,  Chu, and Cowan \n\nThe temporal dynamics of the network can be examined by plotting the fraction F \nof neurons that fire  as  a  function of time.  As  E  is increased through target waves \n(Figure  2a)  to  spiral  waves  (Figure  2b)  to  disorganized  patterns  (Figure  2c),  the \nfluctuations in F  become less  regular,  the mean value increases and the amplitude \ndecreases.  On closer inspection it can be seen that during spiral wave  propagation \n(Figure  2b)  the  time  series  for  F  undergoes  amplitude  modulation  as  reported \npreviously (Farley,  1965).  The interval of low amplitUde, very irregular fluctuations \nin F  (*  in Figure 2b)  corresponds to a  period of spiral tip breakup (Figure lc). \n\nThe appearance of spiral waves  is  typically  preceded by  20-30 target waves.  The \nformation of a  spiral wave  appears to occur in two steps.  First there is an increase \nin  the minimum  value of F  which  begins at  t  '\" 420  and more abruptly occurs at \nt  '\" 460  (Figure 2b).  The target waves  first  become asymmetric and then activity \npropagates from the source region without the more centrally located neurons first \nentering the quisecent state (Figure  3c).  At this  time the spatially  coherent wave \nfront of the target waves becomes replaced by a disordered noncoherent distribution \nof active  and  refractory neurons.  Secondly,  the dispersed network  activity  begins \nto coalesce  (Figures 3c  and 3d)  until  at  t  '\" 536  the first  identifiable spiral occurs \n(Figure 3e). \n\nFigure 3:  The fraction of neurons firing per unit time, for  differing values of gener(cid:173)\nation  time t:  (a)  175,  (b)  345,  (c)  465,  (d)  503  (e)  536,  and  (f)  749.  At  t  =  0  all \nneurons except the central source are quiescent. \n\nIt was found that only 4 out of 20 networks constructed with the same 0, j3  produced \nspiral  waves  for  E  = 0.58  with periodic central point stimulation  (simulations,  in \nsome  cases,  ran  up  to  50,000  generations).  However,  for  all  20  networks,  spiral \nwaves  could  be obtained by  the use  of non-uniform initial  conditions.  Moreover, \nfor  those  networks  in  which  spiral  waves  occurred,  the  generation  at  which  they \nformed differed.  These observations emphasize that small fluctuations in the local \nconnectivity of neurons likely  play  a  major role in governing  the  dynamics of the \nnetwork. \n\n\fSpiral Waves  in  Integrate-and-Fire Neural  Networks \n\n1005 \n\n4  Discussion \n\nSelf-maintaining spiral waves  can a..rise  in  an  inhomogeneous neural network  with \nuniform initial  conditions.  Initially well-formed  target waves  emanate periodically \nfrom the centrally placed source.  Eventually,  provided that E  is in a  critical range \n(Figures  1  & 3),  the target waves  may break up  and  be replaced  by  spiral waves. \nThe  necessary  conditions  for  spiral  wave  formation  are  that:  1)  the  network  be \nsufficiently  tightly  connected  (Farley,  1965;  Farley  and  Clark,  1961)  and  2)  the \nprobability of interneuronal connectivity should decrease with distance (unpublished \nobservations).  As the network is made more tightly connected the probability that \nself-maintained activity arises increases provided that E  is in the appropriate range \n(unpublished  observations).  These  criteria  are  not  sufficient  to  ensure  that self(cid:173)\nmaintained activity,  including spiral waves,  will  form  in a  given  realization of the \nneural network.  It has previously been shown that partially formed spiral-like waves \ncan arise from  periodic point stimulation in  a  model excitable media in which  the \ninhomogeneity  arises  from  a  dispersion  of  refractory  times,  k;l  (Kaplan,  et  al, \n1988). \n\nIntegrate-and-fire  neural  networks  have  two  stable  states:  a  state in  which  all \nneurons are at rest,  another associated with spiral waves.  Target waves  represent \na  transient  response  to  perturbations away  from  the stable  rest  state.  Since  the \nneurons  have  memory  (i.e. \nk m ), \nthe mean threshold and membrane potential of the network  evolve  with  time.  As \na  consequence  the  mean  fraction  of  firing  neurons  slowly  increases  (Figure  2b). \nOur simulations suggest that at some point,  provided that the connectivity of the \nnetwork  is  suitable,  the rest  state suddenly  becomes unstable  and  is  replaced  by \na  stable  spiral  wave.  This  exchange  of stability  is  typical  of a  sub-critical  Hopf \nbifurcation. \n\nthere is  a  relative  refractory  state with  ko  \u00ab \n\nAlthough  complex,  but organized,  spatio-temporal patterns of spreading activity \ncan readily be generated by a  randomly connected neural network, the significance \nof these phenomena, if any, is not presently clear.  On the one hand it is not difficult \nto  imagine  that  these  spatio-temporal dynamics  could  be related  to  phenomena \nranging from  the generation of the EEG,  to  the spread of epileptic  and migraine \nrelated activity and the transmission of visual images in the cortex to the formation \nof patterns and learning by artificial neural networks.  On the other hand,  the oc(cid:173)\ncurence of such phenomena in artificial neural nets could conceivably hinder efficient \nlearning, for  example,  by  slowing  convergence.  Continued study of the properties \nof these networks will  clearly be necessary before these issues can be resolved. \n\nAcknowledgements \n\nThe authors acknowledge useful discussions with Drs.  G.  B.  Ermentrout, L.  Glass \nand D.  Kaplan and financial support from  the National Institutes of Health (JM), \nthe Brain Research Foundation (JDC, JM), and the Office of Naval Research (JDC). \n\nReferences \n\nR.  L.  Beurle.  (1956)  Properties of a  mass of cells  capable of regenerating pulses. \nPhil.  Trans.  Roy.  Soc.  Lond.  240 B, 55-94. \n\n\f1006 \n\nMilton, Chu,  and Cowan \n\nR.  L.  BeUl-Ie.  (1962)  FUnctional organization in random networks.  In Principles  of \nSelf-Organization,  H.  v.  Foerster and  G.  W.  Zopf,  eds.,  pp  291-314.  New  York, \nPergamon Press. \nB.  G.  Farley_  (1965)  A  neuronal network model and the \"slow potentials\"  of elec(cid:173)\ntrophysiology.  Compo  in Biomed_  Res.  2,  265-294. \nB.  G.  Farley &  W.  A.  Clark.  (1961)  Activity in networks of neuron-like elements. \nIn Information  Theory,  C.  Cherry, ed., pp 242-251.  Washington, Butterworths. \nD.  T.  Kaplan,  J.  M.Smith,B.  E.  H.  Saxberg  &  R.  J.  Cohen. \ndynamics in cardiac conduction.  Math.  Biosci.  90,  19-48. \n\n(1988)  Nonlinear \n\nA.  T.  Winfree. \nPrinceton, N.J. \n\n(1987)  When  Time  Breaks  Down,  Princeton  University  Press, \n\n\f", "award": [], "sourceid": 689, "authors": [{"given_name": "John", "family_name": "Milton", "institution": null}, {"given_name": "Po", "family_name": "Chu", "institution": null}, {"given_name": "Jack", "family_name": "Cowan", "institution": null}]}