{"title": "Exact differential equation population dynamics for integrate-and-fire neurons", "book": "Advances in Neural Information Processing Systems", "page_first": 205, "page_last": 212, "abstract": null, "full_text": "Exact  differential  equation population \n\ndynamics  for  Integrate-and-Fire neurons \n\nJulian Eggert  * \n\nHONDA  R&D  Europe  (Deutschland)  GmbH \n\nFuture Technology Research \n\nCarl-Legien-StraBe 30 \n\n63073  Offenbach/Main, Germany \n\njulian. eggert@hre-ftr.f.rd.honda.co.jp \n\nBerthold Bauml \n\nInstitut fur  Robotik und Mechatronik \n\nDeutsches  Zentrum fur  Luft  und Raumfahrt  (DLR) \n\no berpfaffenhofen \n\nBerthold.Baeuml@dlr.de \n\nAbstract \n\nIn  our  previous  work,  integral  equation  formulations  for \n\nMesoscopical,  mathematical  descriptions  of  dynamics  of  popula(cid:173)\ntions of spiking neurons  are getting increasingly important for  the \nunderstanding  of large-scale  processes  in  the  brain  using  simula(cid:173)\ntions. \npopulation dynamics  have been derived for  a  special  type of spik(cid:173)\ning neurons.  For  Integrate- and- Fire type neurons, these formula(cid:173)\ntions  were  only  approximately  correct.  Here,  we  derive  a  math(cid:173)\nematically  compact,  exact  population  dynamics  formulation  for \nIntegrate- and- Fire type  neurons.  It can  be  shown  quantitatively \nin  simulations that the numerical  correspondence with  microscop(cid:173)\nically modeled neuronal populations is  excellent. \n\n1 \n\nIntroduction and  motivation \n\nThe goal of the population dynamics approach is to model the time course of the col(cid:173)\nlective activity of entire populations of functionally and dynamically similar neurons \nin a compact way,  using a  higher descriptionallevel than that of single neurons and \nspikes.  The usual observable at the level of neuronal populations is  the population(cid:173)\naveraged instantaneous firing  rate  A(t),  with  A(t)6.t being the number of neurons \nin the population that release a spike in an interval [t, t+6.t).  Population dynamics \nare formulated  in  such  a  way, that they match quantitatively the time  course  of a \ngiven  A(t), either gained experimentally or by microscopical, detailed simulation. \n\nAt  least  three  main  reasons  can  be  formulated  which  underline  the  importance \nof  the  population  dynamics  approach  for  computational  neuroscience.  First,  it \nenables the simulation of extensive networks involving a massive number of neurons \n\n\fand connections, which is  typically the case when dealing with biologically realistic \nfunctional  models  that go  beyond  the single neuron level.  Second,  it increases the \nanalytical understanding of large-scale neuronal dynamics, opening the way towards \nbetter control and predictive capabilities when dealing with large networks.  Third, \nit enables  a  systematic  embedding  of the  numerous  neuronal  models  operating at \ndifferent  descriptional scales into a  generalized theoretic framework, explaining the \nrelationships,  dependencies  and derivations of the respective models. \n\nEarly efforts on population dynamics approaches date back as early as 1972, to the \nwork of Wilson  and  Cowan  [8]  and  Knight  [4],  which  laid  the  basis  for  all  current \npopulation-averaged graded-response  models  (see  e.g.  [6]  for  modeling  work  using \nthese models).  More recently, population-based approaches for spiking neurons were \ndeveloped,  mainly by Gerstner [3, 2]  and Knight  [5].  In our own previous work  [1], \nwe  have  developed  a  theoretical framework  which enables to systematize and sim(cid:173)\nulate  a  wide  range  of models  for  population-based  dynamics.  It  was  shown  that \nthe equations of the framework produce results that agree quantitatively well  with \ndetailed  simulations  using  spiking  neurons,  so  that  they  can  be  used  for  realistic \nsimulations  involving  networks  with  large  numbers  of spiking  neurons.  Neverthe(cid:173)\nless,  for  neuronal populations composed of Integrate-and-Fire (I&F)  neurons, this \nframework was only correct in an approximation.  In this paper, we  derive the exact \npopulation  dynamics  formulation  for  I&F  neurons.  This  is  achieved  by  reducing \nthe  I&F  population  dynamics  to  a  point  process  and  by  taking  advantage  of the \nparticular properties of I&F neurons. \n\n2  Background:  Integrate-and-Fire dynamics \n\n2.1  Differential form \n\nWe  start  with  the  standard Integrate- and- Fire  (I&F)  model  in  form  of the  well(cid:173)\nknown differential equation [7] \n\n(1) \n\nwhich  describes  the  dynamics  of the  membrane  potential  Vi  of a  neuron  i  that  is \nmodeled  as  a  single  compartment  with  RC  circuit  characteristics.  The  membrane \nrelaxation time is in this case T  =  RC with R being the membrane resistance and C \nthe  membrane  capacitance.  The resting  potential  v R est  is  the  stationary potential \nthat is  approached in  the no-input  case.  The input arriving from  other neurons  is \ndescribed in form  of a  current ji. \n\nIn  addition  to  eq.  (1),  which  describes  the  integrate  part  of the  I&F  model,  the \nneuronal  dynamics  are  completed  by  a  nonlinear  step.  Every  time  the  membrane \npotential Vi  reaches a fixed  threshold ()  from  below,  Vi  is  lowered by a fixed  amount \nLl  > 0,  and from the new value of the membrane potential integration according to \neq.  (1)  starts again. \n\nif Vi(t)  =  ()  (from  below)  . \n\n(2) \n\nAt  the  same  time,  it  is  said  that  the  release  of a  spike  occurred  (i.e.,  the  neuron \nfired),  and  the  time  ti  =  t  of this  singular  event  is  stored.  Here  ti  indicates  the \ntime of the  most recent spike.  Storing all the last firing  times, we  gain the sequence \nof spikes  {t{}  (spike ordering index  j, neuronal index i). \n\n\f2.2 \n\nIntegral  form \n\nNow  we  look  at  the  single  neuron  in  a  neuronal  compound.  We  assume  that  the \ninput  current  contribution  ji from  presynaptic  spiking  neurons  can  be  described \nusing the presynaptic spike times tf, a response-function ~ and a connection weight \nW\u00b7  . \n',J \n\nji(t)  =  l: Wi ,j l: ~(t - tf) \n\n(3) \n\nIntegrating the  I&F  equation  (1)  beginning  at the  last  spiking  time  tT,  which  de(cid:173)\ntermines  the  initial  condition  by  Vi(ti)  =  vi(ti  - 0)  - 6.,  where  vi(ti  - 0)  is  the \nmembrane potential just before the neuron spikes,  we  get  1 \n\nj \n\nf \n\nVi(t)  = vRest  + fj(t  - t:) + l: Wi ,j l: a(t - t:; t - tf)  , \n\nj \n\nf \n\nwith the  refractory  function \n\nfj(s)  =  -\n\n(v Rest  - Vi(t:))  e- S / T \n\nand the  alpha-function \n\na(s; s')  =  r  ds\" e-[sf -S\"J/T ~(s\") . \n\nSf \n\nJSI_S \n\n(4) \n\n(5) \n\n(6) \n\nIf we  start the integration at the time ti*  of the spike before the last spike,  then for \nti*  :::;  t < ti the membrane potential is  given by an expression like eq.  (4),  where ti \nis  replaced by t:i* .  Especially we  can now express v( ti - 0)  and therefore the initial \ncondition  for  an  integration  starting at  tT  in  terms  of ti*  and  v(ti*  - 0).  In  this \nway,  we  can proceed  repetitively and move  back into the  past.  After  some  simple \nalgebra this results in \n\nVi(t)  =  vRest + l:ry(t-t{)+ l:Wi,j l:a(oo ;t - tf) , \n\n(7) \n\nf \n\nj \n\nf \n\n~ ~-------y~------~ \n\nvfef(t) \n\nv~yn(t) \n\nwith a  refractory function  wich  differs  in  the scale factor from  that in eq.  (5) \n\n(8) \nThe  components  vref(t)  and  v?n(t)  to  the  membrane  potential  indicate  refractory \nand  synaptic  components  to  the  neuron  i,  respectively,  as  normally  used  in  the \nSpike- Response- Model  (SRM)  notation  [2]. \n\nry(s)  =  -6. e- S / T \n\n\u2022 \n\nBoth  equations  (4)  and  (7)  formulate  the  neuronal  dynamics  using  a  refractory \ncomponent,  which  depends  on  the  own  spike  releases  of a  neuron,  and  a  synaptic \ncomponent,  which  comprises  the  integrated  input  contribution  to  the  membrane \npotential by arrival of spikes from other neurons 2.  The synaptic component is based \non the alpha-function  characteristic of isolated arriving spikes,  with an increase of \nthe membrane potential after spike arrival and a  subsequent exponential decrease. \n\n1 Strictly speaking,  the constants vRest,  T, ()  and  ,6,  and the function 1]( s)  may vary for \neach neuron, so that they should be written with a subindex i  [similarly for  n(s; s') , which \nmay vary  for  each  connection  j  -+  i  so  that we  should write  it  with subindices  i, j].  For \nthe sake of clarity, we  omit these indices here. \n\n2S0  the I&F model can be formulated  as  a  special  case of the Spike- Response- Model, \n\nwhich  defines  the neuronal  dynamics in the integral formulation. \n\n\fThe comparison of the equivalent expressions eq.  (4)  and eq.  (7)  reveals an interest(cid:173)\ning property of the I&F model.  They look very similar, but in eq.  (4), the refractory \ncomponent depends  only on the time elapsed since  the last spike  (thus  reflecting a \nrenewal property, sometimes also called a short term memory for  refractory proper(cid:173)\nties),  whereas in eq.  (7),  it depends on a sum of the contributions of all past spikes. \nThe  simpler  form  of the  refractory  contribution  in  eq.  (4)  is  achieved  at  the  cost \nof  an  alpha-function  that  now  depends  on  the  time  t  - ti  elapsed  since  the  last \nown  spike  in  addition to the times t - tf  elapsed since  the release of spikes  at the \npresynaptic neurons j  that provide input to i.  In eq.  (7) , we  have a  more  complex \nrefractory contribution, but an alpha-function that does not depend on the last own \nspike time any more. \n\n2.3  Probabilistic spike  release \n\nProbabilistic  firing  is  introduced  into  the  I&F  model  eq.  (4)  resp.  (7)  by  using \nthreshold noise.  The spike release of each neuron is  controlled by a  hazard function \n>.(v),  so that \n>.(v)dt =  Prob. that a  neuron with membrane potential v spikes in  [t , t + dt) \n\n(9) \nWhen a  neuron spikes,  we  proceed as usual:  The membrane potential is  reset by a \nfixed  amount  6.  and the I&F dynamics  continues. \n\n3  Population dynamics \n\n3.1  Density description \n\nDescriptions  of  neuronal  populations  usually  assume  a  neuronal  density  function \np(t; X)  which depends  on the state variables  X  of the neurons.  The density  quan(cid:173)\ntifies the likelihood that a  neuron picked out of the population will  be found  in the \nvicinity of the point X  in  state space, \n\np(t; X) dX = Portion of neurons  at time t  with state in  [X, X + dX) \n\n(10) \n\nIf we  know p(t; X) , the population activity A(t)  can be easily calculated.  Using the \nhazard function >'(t; X), the instantaneous population activity (spikes per time)  can \nbe calculated by computing the spike release averaged over the population, \n\nA(t)  = J dX >.(t; X) p(t; X) \n\n(11) \n\nThe  population  dynamics  is  then  given  by  the  time  course  of the  neuronal  den(cid:173)\nsity  function  p(t; X),  which  changes  because  each  neuron  evolves  according  to  its \nown  internal  dynamics,  e.g.  after  a  spike  release  and  the  subsequent  reset  of the \nmembrane potential. \n\nThe main challenge for the formulation of a population dynamics resides in selecting \na  low-dimensional state space  [for  an easy calculation of A(t)]  and a  suitable form \nfor  gtp(t; X). \nAs  an example, for  the population dynamics for  I&F neurons it would  be straight(cid:173)\nforward  to use  the membrane potential v  from  eq.  (1)  as  the state variable X.  But \nthis leads to a complicated density dynamics, because the dynamics for  v(t)  consist \nof  a  continuous  (differential  equation  (1))  and  a  discrete  part  (spike  generation). \nTherefore, here we  concentrate on an alternative description that allows a  compact \nformulation  of the desired I&F density dynamics. \n\n\f3.2  Exact  population dynamics for  I&F  neurons \n\nWhich  is  the best state space for  a  population  dynamics  of I&F neurons?  For the \nformulation of a  population dynamics,  it is  usually assumed that the synaptic con(cid:173)\ntributions to the membrane potential are identical for  all neurons.  This is  the case \nif we  group all neurons of the same dynamical type and with identical connectivity \npatterns into one  population.  That is,  we  say  that neurons  i  and i'  belong  to  the \nsame  population  if  Wi,j  =  Wi',j  for  all  j  (for  simulations  of  realistic  networks  of \nspiking  neurons,  this  will  of course  never  be  exactly the  case,  but it  is  reasonable \nto  assume  that  a  grouping of neurons  into  populations  can  be  achieved  to  a  good \napproximation) . \nIn our case, looking at eq.  (4),  we  see that, since o:(s, s')  depends on s =  t - ti and \ntherefore  on  the  own  last  spike  time,  the  synaptic  contribution  to  the  membrane \npotential differs according to the state of the neuron.  Thus we  regard eq.  (7).  Here, \nwe  see  that for  identical  connectivity  patterns Wi,j,  the synaptic  contributions  are \nthe same for  all  neurons,  because 0:(00, s')  does  not depend on the own  spike time \nany more.  Which are then the state variables of eq.  (1)  for  the density description? \nWe  see  that,  for  a  fixed  synaptic  contribution,  the  membrane  potential Vi  is  fully \ndetermined  by  the  set  of the  own  past  spiking  times  {tf}.  But  this  would  mean \nan  infinite-dimensional  density  for  the  state  description  of a  population,  and,  ac(cid:173)\ncordingly, a computationally overly expensive calculation of the population activity \nA(t) according to eq.  (11). \n\nTo avoid this we take advantage of a particular property of the I&F model.  Accord(cid:173)\ning to eq.  (8),  the  single  spike refractory contributions 'TJ(s)  are exponential.  Since \nany sum of exponential functions  with common relaxation constant T  can be again \nexpressed  as  as  an  exponential  function  with  the same  T ,  we  can  write  instead  of \nvrf(t) from  eq.  (7) \n\n(12) \nNow  the membrane potential Vi(t)  only  depends  on the time of the last own  spike \nti  and the refractory contribution amplitude modulation factor at the last spike 'TJi . \nThat is,  we  have transferred the effect  of all  spikes previous to the last one into 'TJi. \nIn addition,  we  have  to care about  updating of ti and 'TJi  when  a  neuron spikes  so \nthat we  get  3 \n\n(13) \n\n'TJi \nti \n\n--+ \n--+ \n\n'TJi  =  1 + 'TJie-(t-tn!T  , \nti =  t  . \n\nThe effect of taking into account more than the most recent spike ti in the refractory \ncomponent  vief(t)  leads  to  a  modulation  factor  'TJi  greater than  1,  in  particular  if \nspikes  come in a  rapid succession so  that refractory contributions can accumulate. \nInstead  of  using  a  modulation  factor  'TJi  the  effect  of previous  spikes  can  also  be \ntaken into account by introducing an  effective last spiking time ii. \n\nvi\"f(t)  =  'TJ(t  - in =  'TJi'TJ(t  - tn  , \n\n(14) \n\nwhere ii and 'TJi  are connected by \n\ni; =  t; + TIn'TJi \nThe  effect  of i*  is  sort  of funny.  Because  of 'TJi \n:::::  1  it  holds  for  the  effective  last \nspiking time ii :::::  ti.  This means,  that, while  at a  given time t  it is  allways ti :::;  t, \nit happens that ii :::::  t,  meaning the neurons  behave  as  if they would  spike  in  the \nfuture. \n\n(15) \n\n3Here, the order of reemplacement matters; first  we  have to reemplace 1]:, then ti. \n\n\fFor the membrane potential we  get now  instead of eq.  (7) \n\nVi(t)  =  vRest  + ry(t  - tn + 2..: Wi ,j 2..: 0:(00; t - t;) \nand for  the update rule for  the effective last spiking time t;  follows \n\nf \n\nj \n\ntA* \ni -+ i=  \n\ntA* \n\nf (t  tA*) \n\n' i '  \n\nwith \n\n(16) \n\n(17) \n\n(18) \n\nTherefore we  can regard the dimensionality of the state space of the I&F dynamics \nas  1-dimensional  in  the  description  of eq.  (16).  The  dynamics  of the  single  I&F \nneurons  now  turns  out  to  be  very  simple:  Calculate the  membrane  potential Vi(t) \nusing  eq.  (16)  together  with  the  state  variable  t;,  and  check  if Vi(t)  exceeds  the \nthreshold.  If not,  move  forward  in  time  and  calculate  again.  If the  membrane \npotential exceeds threshold, update t;  according to eq.  (17)  and then proceed with \nthe calculation of Vi(t)  as  normal. \nUsing  this  single  neuron dynamics ,  we  can  now  proceed  to gain  a  population dy(cid:173)\nnamics  using  a  density  p(t; t*).  The  time  t  is  here  the  explicit  time  dependence, \nwhereas t*  denote the state variable of the population.  By fixing  t*  and the synap(cid:173)\ntic  contribution  vsyn(t)  to  the  membrane  potential,  the  state  of a  neuron  is  fully \ndetermined and the hazard function  can be written as  ,X[vsyn(t); t*]. \n\nThe dynamics of the density p(t; t*) is  then calculated as follows.  Changes of p(t; t*) \noccur  when  neurons  spike  and  t*  is  updated  according  to  eq.  (17).  The  hazard \nfunction controls the spike release, and, therefore, the change of p(t; t*).  For chosen \nstate  variables,  p(t; t*)  decreases  due  to  spiking  of the  neurons  with  the  fixed  t*, \nand  increases  because  neurons  with  other  t'*  spike  and  get  updated  in  just  that \nway that after updating their state variable falls  around t*.  This occurs  according \nto the reemplacement rule eq.  (17)  when \n\nf(t, t'*)  =  t*  . \n\nTaking all together the dynamics of the density p(t; t*)  is  given by \ndecrease due to same state t*  spiking \n\nA \n\n'-,X[vsyn(t); t*]p(t; t*)' \n\n-ftp(t;t*) = \n\n1+ 00 \n\n+  -00  dt'*  8[J(t, t'*) - t*] ,X[vsyn(t); t'*] p(t; t'*) \n\nincrease due to spiking of neurons with other states t'* \n\nThe  population  activity  can  then  be  calculated  using  the  density  according  to \neq.  (11)  as  follows \n\n1+00 \n\nA(t) = \n\n- 00  dt* ,X[vsyn(t); t*] p(t; t*) \n\n(19) \n\n(20) \n\n(21) \n\n(22) \n\nRemark  that  the  expression  for  the  density  dynamics  (eq.  20)  automatically  con(cid:173)\n\nserves the norm of the density,  so  that 1+00 \n\n- 00  dt* p( t ; t*)  =  const  , \n\nwhich  is  a  necessary  condition because the number of neurons  participating in the \ndynamics must remain constant. \n\n\f4  Simulations \n\nThe  dynamics  of  a  population  of  I&F  neurons ,  represented  by  the  time  course \nof  their  joint  activity,  can  now  be  easily  calculated  in  terms  of  the  differential \nequation  (20) , if the neuronal state density  of the neuronal population p(t; i*)  and \nthe synaptic  input  vsyn(t)  are  known.  This  means  that  all  we  have to store is  the \ndensity  p(t; i*)  for  past  and  future  effective  last  spiking  times  i*  4 .  Favorably for \nnumerical  simulations,  only  a  limited  time  window  of i*  around the  actual time  t \nis  needed for  the dynamics.  The activity A(t) only appears as  an auxiliary variable \nthat is  calculated with the help of the neuronal density. \n\nIn figure  1 the simulation results  for  populations  of of spiking  neurons  are  shown. \nThe neurons are uncoupled and a  hazard function \n\nA(V)  =  ~ e2,B(v-e)  , \n\nTO \n\n(23) \n\nwith spike rate at threshold  liTO  = 1.0ms- 1 ,  a  kind of inverse temperature (3  = 2.0, \nwhich controls the noise  level,  and the threshold e =  1.0.  The other parameters of \nthe model  in  eq.  (1)  are:  resting potential vRest  =  0,  jump in  membrane potential \nafter spike release  ~ =  1 and time constant T  =  20ms.  This parameters are chosen \nto be biologicaly plausible. \n\nA (spikes/ms) \n\n0.14 \n\n0.12 \n\n0.1 \n\n0.08 \n\n0.06\n\n0.04 \n\n0.02 \n\nr-------\n\nvsyr'i-' _ _  1_00 __  15_0 __  20_0  _ _  2_50 __  30_0 ----,1 (ms) \n\no~ I c) \n\n100 \n\n150 \n\n200 \n\n250 \n\n300 \n\nI (ms) \n\n: \n= \n\nb) \n\nII \" \" \nn \n\" \" \n:: :: \n!l \n: ~ \n! \\_, .. ----: \n: \n_______ J \n\nI \n1\\ \n\n1l \n* : \\  \n! \\_ .. ----2  '-1 \n! ....... j  ! \n~ .. ' \n\n~,' \n\nI:, \n\nr-------\n\nFigure  1:  Activity  A(t)  of  simulated  populations  of  neurons.  The  neurons  are \nuncoupled  and to each neuron the same synaptic field  vsyn(t),  ploted in c)  and d), \nis  applied.  a)  shows  the  activity  A(t)  for  a  population of I&F  neurons  simulated \non the one hand as  N  =  10000 single neurons  (solid  line)  using eq.  (7)  and on the \nother  hand  using  the  density  dynamics  eq.  (20)  (dashed  line).  In  b)  the  activity \nA(t) of a population ofI&F neurons (dashed line)  and a population of SRM neurons \nwith renewal  (solid line)  are compared.  For all simulations the same parameters as \nspecified in the text were used. \n\nThe  simulations  show  that  the  density  dynamics  eq.  (20)  reproduces  the  activity \nA( t)  of a population of single I&F neurons almost perfect, with the exception of the \nnoise in the single neuron simulations due to the finite  size  effects.  This holds even \nfor the peaks occuring at the steps of the applied synaptic field vsyn (t), although the \ndensity dynamics is entirely based on differential equations and one would therefore \nnot expect such an excellent match for  fast  changes in activity. \n\n4 VSYll (t)  only appears as a scalar in the dynamics, so that no integration over time takes \n\nplace here. \n\n\fThe simulations also show that there can be a big difference between I&F and SRM \nneurons  with  renewal.  Because of the  accumulation of the  refractory effects  of all \nformer  spikes  in  the  case  of I&F  neurons  the  activity  A(t)  is  generaly  lower  than \nfor  the  SRM  neurons  with  renewal  and the  higher  the  absolute  actitvity  level  the \nbigger is  the difference between both. \n\n5  Conclusions \n\nIn this paper we  derived an exact differential equation density dynamics for  a popu(cid:173)\nlation of I&F neurons starting from the microscopical equations for  a single neuron. \nThis density  dynamics  allows  a  compuationaly efficient  simulation of a  whole  pop(cid:173)\nulation of neurons. \n\nIn future  work  we  want  to simulate  a  network of connected neuronal  populations. \nIn  such  a  network  of populations  (indexed  e.g.  by  x) ,  a  self-consistent  system  of \ndifferential  equations  based  on  the  population's  p(x, t; i*)  and  A(x, t)  emerges  if \nwe  constrain ourselves to neuronal populations connected synaptically according to \nthe  constraints  given  by  the  pool  definition  [2].  In  this  case,  two  neurons  i  and j \nbelong to pools x  and y, if Wi,j  =  W(x, y).  This allows us to write for  the synaptic \ncomponent of the membrane potential \n\n(24) \n\nvsyn(x,t) = 2: W (x , y) 100 ds'a(oo;s')A(y,t-s') \n\ny \n\n0 \n\nUsing  the  alpha-function  a(oo ; s')  as  introduced  in  (6),  and  a  \"nice\"  response(cid:173)\nfunction  ~ for  the  input  current  time  course  after  a  spike,  we  can  write  eq.  (24) \nusing differential equations that use  A(y, t)  as  input.  This results in a  system that \nis  based entirely on differential equations and is  very cheap to compute. \n\nReferences \n\n[1]  J.  Eggert  and  J.L.  van  Hemmen.  Modeling  neuronal  assemblies:  Theory  and imple(cid:173)\n\nmentation.  N eural  Computation, 13(9):1923- 1974,  200l. \n\n[2]  W.  Gerstner.  Population  dynamics of spiking neurons:  Fast transients,  asynchronous \n\nstates and locking.  Neural  Computation, 12:43- 89,  2000. \n\n[3]  W .  Gerstner  and  J .  L.  van  Hemmen.  Associative  memory  in  a  network  of 'spiking' \n\nneurons.  Network,  3:139- 164,  1992. \n\n[4]  B.  W.  Knight .  Dynamics of encoding in  a  populations of neurons.  J.  Gen.  Physiology, \n\n59:734- 766,  1972. \n\n[5]  B.  W.  Knight.  Dynamics of Encoding  in  Neuron  Populations:  Some  General  Mathe(cid:173)\n\nmatical Features.  Neural  Comput.,  12:473- 518,  2000. \n\n[6]  Z.  Li.  A  neural  model  of contour  integration  in  the  primary  visual  cortex.  Neural \n\nComput. , 10(4):903- 940,  1998. \n\n[7]  H.  C.  Tuckwell.  Introduction  to  Th eoretical  N eurobiology.  Cambridge University Press, \n\nCambridge, 1988. \n\n[8]  H.  R.  Wilson  and  J.  D.  Cowan.  Excitatory  and  inhibitory  interactions  in  localized \n\npopulations of model neurons.  Biophys.  J.,  12:1- 24,  1972. \n\n\f", "award": [], "sourceid": 2076, "authors": [{"given_name": "Julian", "family_name": "Eggert", "institution": null}, {"given_name": "Berthold", "family_name": "B\u00e4uml", "institution": null}]}