{"title": "Resonance in a Stochastic Neuron Model with Delayed Interaction", "book": "Advances in Neural Information Processing Systems", "page_first": 314, "page_last": 320, "abstract": null, "full_text": "Resonance in a  Stochastic Neuron Model \n\nwith Delayed Interaction \n\nToru Ohira* \n\nSony Computer Science Laboratory \n\n3-14-13 Higashi-gotanda \n\nShinagawa,  Tokyo 141,  Japan \n\nohira@csl.sony.co.jp \n\nYuzuru Sato \n\nInstitute of Physics, \n\nGraduate School of Arts  and  Science,  University of Tokyo \n\n3-8-1  Komaba,  Meguro,  Tokyo  153  Japan \n\nysato@sacral.c.u-tokyo.ac.jp \n\nJack D.  Cowan \n\nDepartment of Mathematics \n\nUniversity of Chicago \n\n5734 S.  University,  Chicago,  IL  60637,  U.S.A \n\ncowan@math.uchicago.edu \n\nAbstract \n\nWe study here a simple stochastic single neuron model with delayed \nself-feedback capable of generating spike  trains.  Simulations show \nthat its spike trains exhibit resonant behavior between \"noise\"  and \n\"delay\".  In  order to gain  insight  into this  resonance,  we  simplify \nthe model and study a  stochastic binary element whose  transition \nprobability  depends  on  its  state  at  a  fixed  interval  in  the  past. \nWith this simplified model we  can analytically compute interspike \ninterval histograms, and show how the resonance between noise and \ndelay  arises.  The resonance  is  also  observed  when  such  elements \nare coupled through delayed interaction. \n\n1 \n\nIntrod uction \n\n\"Noise\"  and  \"delay\"  are  two  elements  which  are  associated  with  many  natural \nand  artificial  systems  and  have  been  studied  in  diverse  fields.  Neural  networks \nprovide representative examples  of information  processing systems  with noise  and \ndelay.  Though  much  research  has  gone  into the  investigation of these  two factors \nin the community,  they  have  mostly  been  separately studied  (see  e.g.  [1]).  Neural \n\n* Affiliated  also  with  the Laboratory for  Information Synthesis,  RIKEN  Brain Science \n\nInstitute, Wako,  Saitama, Japan \n\n\fResonance in a Stochastic Neuron Model with Delayed Interaction \n\n315 \n\nmodels incorporating both noise  and delay are more realistic  [2],  but their complex \ncharacteristics have yet to be explored both theoretically and numerically. \nThe  main  theme  of this  paper  is  the  study  of  a  simple  stochastic  neural  model \nwith delayed interaction which can generate spike trains.  The most striking feature \nof this  model  is  that  it  can  show  a  regular  spike  pattern  with  suitably  \"tuned\" \nnoise  and delay [3].  Stochastic resonance in neural information processing has been \ninvestigated  by  others  (see  e.g.  [4]).  This  model,  however,  introduces  a  different \ntype of such resonance, via delay rather than through an external oscillatory signal. \nIt can be  classified  with models  of stochastic resonance  without  an external signal \n[5] . \nThe novelty of this model is the use of delay as the source of its oscillatory dynamics. \nTo  gain  insight  into  the  resonance,  we  simplify  the  model  and  study  a  stochastic \nbinary  element  whose  transition  probability depends  on  its  state  at  a  fixed  inter(cid:173)\nval  in  the  past.  With  this  model,  we  can analytically  compute interspike  interval \nhistograms,  and show  how  the  resonance  between noise  and  delay  arises.  We  fur(cid:173)\nther show that the resonance also  occurs when such stochastic binary elements are \ncoupled  through delayed interaction. \n\n2  Single Delayed-feedback Stochastic  Neuron Model \n\nOur model is  described by  the following  equations: \n\nd \n\nJl dt Vet) \n\n\u00a2(V( t)) \n\n-Vet) + W\u00a2(V(t - r)) + eL(t) \n\n2 \n\n1 + e-1)(V(t)-9) \n\n-1 \n\n(1) \n\nwhere 11  and ()  are constants,  and V  is  the membrane potential of the neuron.  The \nnoise  term eL  has the following  probability distribution. \n\npee =  u) \n\n1 \n(-L 5:  u 5:  L) \n2L \no  (u<-L,u>L) , \n\n(2) \ni.e.,  eL  is  a  time  uncorrelated  uniformly  distributed noise in the range  (-L, L).  It \ncan be interpreted as a fluctuation that is much faster than the membrane relaxation \ntime Jl.  The model  can be  interpreted  as  a  stochastic  neuron  model  with delayed \nself-feedback of weight W, which is an extension of a model with no delay previously \nstudied using the Fokker- Planck equation  [6]. \nWe  numerically study the following  discretized  version: \n\nVet + 1) = 1 + e-1)(V(t-T )- 9)  - 1 + eL \n\n2 \n\n(3) \n\nWe  fix  11  and ()  so  that  this  map has  two  basins  of attractors of differing  size  with \nno  delay,  as  shown  in  Figure  l(A) .  We  have  simulated  the  map  (3)  with  various \nnoise widths and delays and find  regular spiking behavior as  shown in Fig l(C) for \ntuned noise width and delay.  In case the noise width is  too large or too small given \nself-feedback delay,  this  rhythmic  behavior does  not  emerge,  as  shown in  Fig1(B) \nand  (D). \n\nWe  argue that  the delay changes the effective shape of the basin of attractors into \nan  oscillatory  one,  just  like  that  due  to  an  external  oscillating force  which,  as  is \nwell-known, leads to stochastic resonance with a tuned noise width.  The analysis of \nthe  dynamics  given  by  (1)  or  (3),  however,  is  a  non- trivial  task,  particularly  with \n\n\f316 \n\nT.  Ohira,  Y.  Sato and J.  D.  Cowan \n\nrespect  to the spike  trains.  A previous analysis  using  the Fokker-Planck equation \ncannot capture this emergence of regular spiking behavior.  This difficulty motivates \nus to further simplify our model,  as  described  in  the next section. \n\n(A) \n\ncp \n, \n\n(E) \n\n(0) \n\n(b) \n\n... \n\n200 \n\nm \n\n.oo \n\n1000 t \n\n(8) \n\nV(t) \n\n(e) \n\nV(t) \n\n(F) \n\nX(t) \n\n.. , \n\n(G) \n\nX(t) , \n, \n\nI ... \n\n.~. ., \n\n,.p \n\n100 \n\nt \n\nL - -\n\n, \n\n.. \n\n.. \n\n, \n\n, \n,>--- I - I - L.. ~ I - L.. ' - -\n\n\" \n\n(D) \n\nV(t) \n\n(8) \n\nX(t) \n\nFigure 1:  (A)  The shape of the sigmoid  function  4>  (b)  for\", =  4  and  0 =  0.1.  The \nstraight line  (a)  is  4>  =  V  and the crossings of the two lines indicate the stationary \npoint of the dynamics.  Also,  the typical dynamics of V (t) from  the map model are \nshown as  we  change noise  width  L.  The values  of L  are (B)  L = 0.2,  (C)  L = 0.4, \n(D)  L  =  0.7.  The  data  is  taken  with  T  =  20,  '\"  =  4.0,  0  =  0.1  and  the  initial \ncondition  V(t)  = 0.0  for  t  E  [-r,O].  The  plots  are shown  between t  = a to  1000. \n(E)  Schematic  view  of the single  binary  model.  Some  typical  dynamics  from  the \nbinary  model  are  also  shown.  The values of parameters  are r  =  10,  q  =  0.5,  and \n(F) p = 0.005,  (G) p  =  0.05,  and (H)  p = 0.2. \n\n3  Delayed Stochastic Binary N enron Model \n\nThe model  we  now  discuss  is  an  approximation  of the  dynamics  that  retains  the \nasymmetric stochastic transition  and  delay.  The state X(t)  of the system  at time \nstep t  is  either -lor 1.  With the same noise eL,  the model is  described as follows: \n\nX(t + 1) \n\nO[f(X(t - T\u00bb + eLl, \n1 \nf(n)  =  2\u00aba + b) + n(a - b\u00bb, \nO[n] \n\n(0  ~ n), \n\n(4) \nwhere  a and b are  parameters such  that  lal  ~ L  and  Ibl  ~ L,  and r is  the  delay. \nThis model is  an approximate discretization of the space of map (3) into two states \n\n-1  (0) n), \n\n1 \n\n\fResonance in  a Stochastic Neuron Model with Delayed Interaction \n\n317 \n\nwith a and b controlling the bias of transition depending on the state of X  r  steps \nearlier.  When a  i- b,  the transition between the two states is  asymmetric,  reflecting \nthe two differing sized  basins of attractors. \n\nWe can describe this  model more concisely in probability space (Figure  I(E)).  The \nformal  definition is  given as follows: \n\nP(I, t + 1) \n\nP(-I,t+l) \n\np \n\nq \n\nX(t - r) = -1, \np, \n1- q,  X(t - r) = 1, \nX(t - r) = 1, \nq, \n1- p,  X(t - r) = -1, \n1 \n2(1 + L)' \n1 \n2(1 - L)' \n\nb \n\na \n\n(5) \n\nwhere P(s, t) is  the probability that  X(t)  =  s.  Hence,  the transition  probability of \nthe model depends on its state r  steps in the past, and is a  special case of a  delayed \nrandom walk  [7]. \nWe randomly generate X(t) for  the interval t  =  (-r, 0).  Simulations are performed \nin  which  parameters  are  varied  and  X(t)  is  recorded  for  up  to  106  steps.  They \nappear to be qualitatively similar to those generated by the map dynamics (Figure \nI(F),(G),(H)). ;,From the trajectory X(t),  we construct a  residence time histogram \nh( u)  for  the system to be in  the state  -1 for  u  consecutive  steps.  Some  examples \nof the histograms  are  shown in Figure  2  (q  =  1 - q  = 0.5,  r  = 10).  We  note  that \nwith  p  \u00ab  0.5,  as  in Figure  2(A),  the model  has a  tendency  to switch or  spike  to \nthe X  =  1 state after  the  time step interval of r.  But the spike  trains  do not  last \nlong and result  in a  small peak in the histogram.  For the case of Figure 2(C) where \np  is  closer  to  0.5,  we  observe  less  regular  transitions  and  the  peak height  is  again \nsmall.  With appropriate p  as in Figure 2(B), spikes tend to appear at the interval T \nmore frequently,  resulting in higher peaks in  the histogram.  This is  what  we  mean \nby  stochastic resonance  (Figure  2(D)).  Choosing an  appropriate p  is  equivalent  to \n\"tuning\"  the  noise  width  L,  with  other  parameters  appropriately  fixed. \nIn  this \nsense,  our model exhibits stochastic resonance. \n\nThis  model  can  be  treated  analytically.  The  first  observation  to  make  with  the \nmodel is  that given  r,  it  consists  of statistically independent r + 1 Markov chains. \nEach Markov chain has its state appearing at every r+l interval.  With this property \nof the model,  we  label time step  t  by the two integers sand k  as follows \n\n(6) \nLet  P\u00b1(t) ==  P\u00b1(s, k)  be the probability for  the state to be in the \u00b11 state at  time \nt  or (s, k).  Then, it can be shown that \n\n(O::;s,O::;k::;r) \n\nt=s(r+l)+k, \n\na(1 - ,S) + ,s P+(s  =  0, k), \n{3(1- ,S) + ,s P_(s =  0, k), \n\nP+(s, k) \nP_ (s, k) \n\na \n\n{3 \n\n, \n\np \n\nq \n\n, \np+q \n- -, \np+q \n1 - (p + q). \n\n(7) \nIn  the steady  state,  P+(s  --+  oo,k)  ==  P+  =  a  and  P_(s  --+  oo,k)  ==  P_  =  {3.  The \nsteady state residence  time histogram can  be obtained by computing the following \n\n\f318 \n\nT.  Ohira, Y.  Safo and J.  D.  Cowan \n\nquantity,  h(u)  = P(+;-,Uj+),  which  is  the  probability  that  the  system  takes \nconsecutive  -1 states  U  times  between two  + 1 states.  With the  definition  of the \nmodel  and  statistical  independence  between  Markov  chains  in  the  sequence,  the \nfollowing expression  can  be derived: \n\n(1  ~ U  < r) \n\nP(+;-,Uj+) \n\nP+(P_)Up+  =  (,8)U(a)2 \n\n=  p+(p-r(1- q)  = (,8r(a)(1- q) \n=  p+(p-r(q)(1- p)U-T(p) =  (,8)U(p)2 \n\n(8) \n(9) \n(10) \nWith appropriate normalization, this expression reflects the shape of the histogram \nobtained by numerical  simulations,  as  shown in  Figure  2.  Also,  by differentiating \nequation  (9)  with  respect  to  p,  we  derive  the  resonant  condition  for  the  peak  to \nreach maximum height as \n\n(u > r) \n\n(u =  r) \n\nq=pr \n\n(11) \n\nor, equivalently, \n\nL - a = (L + b)r. \n\n(12) \nIn Figure 2(D),  we  see  that  maximum peak amplitude is  reached by choosing  pa(cid:173)\nrameters according to equation (11).  We note that this analysis for the histogram is \nexact in the stationary limit, which makes this model unique among those showing \nstochastic resonance. \n\n(M \"'::L \n\nb(a)  ~  Ol~ \n.\" \n'M. \n\nj  ~ \n\n~ \n\n~  ~ \n\nl~  I ~  ~  I'!. \n\nI ' ! . :O  \n\n\u2022 \n\n(at\n\nhCD) \n\n.\" \n\n\"'::I~ \n....  Ol~ ... . ~. \n\n1  ~ \n\n, \n\n\" \n\n10 \n\n12  ~  I'  17  ~ \n\njO \n\n\u2022 \n\n(et \n\n\"'~~ (I  Ol~ \nb(ot  .. \" ... \n\nI\"  us  UP!>  20 \n\n\u2022 \n\n\", \nh(tt  . \"\" \n\n~ \n\n,', !> \n\n\"\"L \n\n.\" ... . ~. \n\n10 \n\n10 \n\n)0 \n\n\u2022 \n\nto \n\n\"'a .... \" \n\nFigure  2:  Residence  time  histogram  and  dynamics  of X(t)  as  we  change p.  The \nvalues  of p  are  (A)  p  = 0.005,  (B)  p  =  0.05,  (C)  p  = 0.2.  The solid  line  in  the \nhistogram is  from  the analytical expression given in equations  (8-10).  Also,  in (D) \nwe  show  a  plot  of peak  height  by  varying p.  The  solid  line  is  from  equation  (9). \nThe parameters are r  = 10,  q = 0.5. \n\n4  Delay Coupled Two Neuron Case \n\nWe  now  consider  a  circuit  comprising two  such stochastic binary neurons  coupled \nwith delayed interaction.  We observe again that resonance between noise and delay \n\n\fResonance in a Stochastic Neuron Model with Delayed Interaction \n\n319 \n\ntakes  place.  The coupled  two neuron  model  is  a  simple  extension  of the  model  in \nthe previous section.  The transition probability of each neuron is  dependent on  the \nother neuron's state at a fixed  interval in the past.  Formally, it can be described in \nprobability space  as  follows. \n\nPl(l, t + 1) \n\nPl(-I,t+l) \n\nP2(1, t + 1) \n\nP2(-I,t+l) \n\nX 2(t - 72)  = -1, \nPI! \n1- q!,  X 2(t - 72) = 1, \nX 2(t - 72)  = 1, \nql, \n1 - PI,  X 2(t - 72) = -1, \nXl(t - 7d = -1, \nP2, \n1- q2,  Xl(t - 7d = 1, \nXl(t - 71) = 1, \nq2, \n1- P2,  Xl(t - 71) = -1 \n\n(13) \nPi(S, t)  is  the  probability  that  the  state  of the  neuron  i  is  Xi(t)  =  s.  We  have \nperformed  simulation  experiments  on  the  model  and  have  again  found  resonance \nbetween noise  and delay.  Though more  intricate than the single  neuron model,  we \ncan perform a  similar theoretical analysis  of the histograms and have obtained ap(cid:173)\nproximate results for some cases.  For example, we obtain the following approximate \nanalytical results for  the peak height of the interspike histogram of Xl  for  the case \n71  =  72  ==  7.  (  The peak occurs  at  71  + 72  + 1.) \nH(Pl' P2, qI, q2)  = \n\n(14) \n\n{J.t3(P!, P2, ql, q2 )ql + J.t4(PI, P2, ql, q2)(1 - pd Y \n{J.tl(P!'P2, ql, q2)(qlq2PI  + ql(l - q2)(1  - ql)) \n+J.t2(Pl,P2,ql,q2)((I- pdq2Pl + (1- Pl)(I- q2)(I- qd)} \nh (PI, P2, ql, q2)!2(PI, P2, ql, q2) \n\nJ.tl (PI, P2, qI, q2) \n\nJ.t2 (PI, P2, ql, q2) \n\nJ.t3 (PI, P2, ql , q2 ) \n\nJ.t4 (PI! P2, qI, q2 ) \n\nII (PbP2, ql, q2) \n\n12(PI,P2, qI, q2) \n\nS(PI,P2, ql, q2) \n\nS(PI , P2 , ql , q2) \n\nII (PI ,P2, qI, q2) \nS(PI,P2, q!, q2) \n!2(PI,P2, ql, q2) \ns (PI! P2, ql, q2) \n\n1 \n\nS(PI,P2, ql, q2) \nPI(I - P2) + P2(1  - ql) \nql (1  - q2) + q2 (1  - qd \nP2  + PI (1  - P2  - q2) \nq2  + ql(l - P2  -q2) \nh (PI, P2, ql, q2)!2(PI ,P2, ql, q2) \n\n+ h(PI,P2, ql, q2) + !2(PI,P2, ql, q2) + 1 \n\n(15) \n\n(16) \n\n(17) \n\n(18) \n\n(19) \n\n(20) \n\n(21) \n\nThese  analytical  results  are  compared  with  the simulation experiments,  examples \nof which  are  shown  in  Figure  3.  A  detailed  analysis,  particularly  for  the  case  of \n71  =I  72,  is  quite intricate and is  left for  the future. \n\n5  Discussion \n\nThere are two points to be noted.  Firstly,  although there are examples which may \nindicate  that  stochastic  resonance  is  utilized  in  biological  information  processing, \nit  is  yet  to be explored  if the resonance  between noise  and  delay  has  some  role  in \n\n\f320 \n\nT.  Ohira, Y.  Sato and J.  D.  Cowan \n\n(A) \n\npi \n\n0 . 01 \n\nh(t)  ..... \n\n0 . 008 \n\n(8) \n\n(C) \n\nh(t) ::::~ \n\n..00<  ~ \n\no.oo:z \n\n0 . 00:3: \n\n0 .2 \n\n0 . 3 \n\n.. \n\n0 . 1 \n\n0 , ' \n\n0 . 1 \n\n.. \n\n0.' \n\n0 . 5 \n\nFigure 3:  A plot of peak height by varying P2.  The solid line is from  equation (14-\n20).  The parameters are T1  = T2  =  10,  q1  = q2  = 0.5,  (A)P1  = P2,  (B) P1  = 0.005, \n(C) P1  =  0.025. \n\nneural information  processing.  Secondly,  there  are  many investigations of spiking \nneural  models  and their applications  (see  e.g.,  [8]).  Our model  can  be considered \nas  a  new  mechanism  for  generating  controlled  stochastic  spike  trains.  One  can \npredict  its  application  to  weak  signal  transmission  analogous  to  recent  research \nusing stochastic resonance with a larger number of units in series [9].  Investigations \nof the network model with delayed interactions are currently underway. \n\nReferences \n\n[1)  Hertz, J. A.,  Krogh,  A., &  Palmer, R. G.  (1991).  Introduction to  the  Theory  of Neural \nComputation.  Redwood  City:  Addison-Wesley. \n\n[2)  Foss, J., Longtin, A.,  Mensour, B., &  Milton, J . G.  (1996).  Multistability and Delayed \nRecurrent Loops.  Physical Review Letters,  76,  708-711;  Pham, J., Pakdaman, K., Vibert, \nJ.-F.  (1998).  Noise-induced coherent oscillations in randomly connected neural networks. \nPhysical Review  E,  58,  3610-3622;  Kim,  S.,  Park,  S.  H.,  Pyo,  H.-B.  (1999).  Stochastic \nResonance in Coupled Oscillator Systems with Time Delay.  Physical Review  Letters,  8!, \n1620-1623; Bressloff, P. C.  (1999).  Synaptically Generated Wave Propagation in Excitable \nNeural Media.  Physical Review Letters,  8!,  2979-2982. \n[3)  Ohira, T. &  Sato, Y.  (1999).  Resonance with Noise and Delay.  Physical Review Letters, \n8!, 2811-2815. \n\n[4)  Gammaitoni, L.,  Hii.nggi,  P., Jung, P., &  Marchesoni, F.(1998).  Stochastic Resonance. \nReview of Modem Physics,  70,  223-287. \n\n[5)  Gang, H., Ditzinger, T., Ning,  C.  Z., &  Haken, H.(1993) Stochastic Resonance without \nExternal Periodic Force.  Physical Review Letters,  71,  807-810; Rappel, W-J.  &  Strogatz, \nS. H. (1994).  Stochastic resonance in an autonomous system with a nonuniform limit cycle. \nPhysical Review E,  50,3249-3250; Longtin,  A.  (1997).  Autonomous  stochastic resonance \nin bursting neurons.  Physical Review E,  55,  868-876. \n[6)  Ohira,  T.  &  Cowan J . D.  (1995).  Stochastic Single  Neurons,  Neural  Communication, \n7518-528. \n\n[7)  Ohira,  T.  &  Milton,  J.  G.  (1995).  Delayed Random  Walks.  Physical  Review  E,  5!, \n3277-3280; Ohira, T. (1997).  Oscillatory Correlation of Delayed Random Walks,  Physical \nReview  E,  55,  RI255-1258. \n\n[8)  Maas, W.  (1997).  Fast Sigmoidal Network via Spiking Neurons.  Neural  Computation, \n9(2),  279-304;  Maas, W. (1996).  Lower Bounds for  the Computational Power of Networks \nof Spiking Neurons.  Neural Computation,  8(1),  1-40. \n[9)  Locher, M., Cigna, D., and Hunt, E. R. (1998).  Noise Sustained Propagation of a Signal \nin Coupled Bistable Electric Elements Physical Review Letters,  80,  5212-5215. \n\n\f", "award": [], "sourceid": 1656, "authors": [{"given_name": "Toru", "family_name": "Ohira", "institution": null}, {"given_name": "Yuzuru", "family_name": "Sato", "institution": null}, {"given_name": "Jack", "family_name": "Cowan", "institution": null}]}