{"title": "Temporal coding in the sub-millisecond range: Model of barn owl auditory pathway", "book": "Advances in Neural Information Processing Systems", "page_first": 124, "page_last": 130, "abstract": null, "full_text": "Temporal coding \n\nin the sub-millisecond range: \n\nModel of barn  owl auditory pathway \n\nRichard Kempter* \n\nWulfram Gerstner \n\nInstitut fur  Theoretische  Physik \n\nPhysik-Department der TU  Munchen \n\nD-85748 Garching bei  Munchen \n\nInstitut fur  Theoretische  Physik \n\nPhysik-Department der TU  Munchen \n\nD-85748  Garching bei Munchen \n\nGermany \n\nGermany \n\nJ.  Leo van  Hemmen \n\nInstitut fur  Theoretische  Physik \n\nPhysik-Department der  TU  Munchen \n\n0-85748 Garching bei  Munchen \n\nGermany \n\nHermann Wagner \nInstitut fur  Zoologie \n\nFakultiit fur  Chemie und  Biologie \nD-85748 Garching bei  Munchen \n\nGermany \n\nAbstract \n\nBinaural  coincidence  detection  is  essential  for  the  localization  of \nexternal  sounds  and  requires  auditory signal  processing  with  high \ntemporal precision.  We present an integrate-and-fire model of spike \nprocessing in the auditory pathway of the barn owl.  It is shown that \na temporal precision in the microsecond range can be achieved with \nneuronal  time constants  which  are  at least  one  magnitude longer. \nAn  important  feature  of our  model  is  an  unsupervised  Hebbian \nlearning rule which leads to a  temporal fine  tuning of the neuronal \nconnections. \n\n\u00b7email:  kempter.wgerst.lvh@physik.tu-muenchen.de \n\n\fTemporal Coding in  the Submillisecond Range:  Model of Bam Owl Auditory Pathway \n\n125 \n\n1 \n\nIntroduction \n\nOwls  are  able  to locate acoustic  signals  based  on  extraction  of interaural  time dif(cid:173)\nference  by  coincidence  detection  [1,  2].  The spatial resolution of sound localization \nfound  in  experiments corresponds  to a  temporal  resolution  of auditory signal  pro(cid:173)\ncessing  well below one millisecond.  It follows that both the firing of spikes and their \ntransmission  along  the  so-called  time pathway of the  auditory system  must occur \nwith  high temporal precision. \n\nEach  neuron  in  the  nucleus  magnocellularis,  the  second  processing  stage  in  the \nascending  auditory pathway,  responds  to signals in  a  narrow  frequency  range.  Its \nspikes  are  phase  locked  to  the  external  signal  (Fig.  1a)  for  frequencies  up  to  8 \nkHz  [3].  Axons  from  the  nucleus  magnocellularis project  to the  nucleus  laminaris \nwhere  signals from  the  right  and  left  ear  converge.  Owls  use  the interaural  phase \ndifference for azimuthal sound localization.  Since barn owls can locate signals with a \nprecision of one degree of azimuthal angle,  the temporal precision of spike encoding \nand  transmission must be at least  in  the range of some 10 J.lS. \n\nThis  poses  at  least  two severe  problems.  First,  the  neural  architecture  has  to  be \nadapted  to operating  with  high  temporal precision.  Considering the fact  that the \ntotal delay from  the ear to the nucleus magnocellularis is approximately 2-3  ms [4], \na  temporal  precision  of some  10  J.lS  requires  some  fine  tuning,  possibly  based  on \nlearning.  Here we suggest that Hebbian learning is an  appropriate mechanism.  Sec(cid:173)\nond, neurons must operate with the necessary  temporal precision.  A firing precision \nof some 10 J.ls  seems truly remarkable considering the fact  that the membrane time \nconstant is probably in the  millisecond range.  Nevertheless,  it is  shown  below  that \nneuronal spikes  can  be  transmitted with  the required  temporal precision. \n\n2  Neuron  model \n\nWe  concentrate  on  a  single frequency  channel  of the  auditory  pathway  and  model \na  neuron of the nucleus magnocellularis.  Since synapses are directly  located on the \nsoma,  the spatial structure of the neuron  can  be reduced  to a  single compartment. \nIn  order to simplify the dynamics, we  take an integrate-and-fire unit.  Its membrane \npotential changes  according to \n\nu \n\nd \n-u = -- + 1(t) \ndt \n\nTO \n\n(1) \n\nwhere  1(t)  is  some input and  TO  is  the membrane time constant.  The neuron  fires, \nif u(t) crosses  a threshold {)  =  1.  This defines a  firing time to.  After firing u is reset \nto an  initial value uo  =  O.  Since  auditory neurons  are known  to be fast,  we  assume \na  membrane time constant of 2 ms.  Note that this is shorter  than in  other areas of \nthe brain,  but still a  factor of 4 longer  than the  period of a  2  kHz  sound  signal. \n\nThe magnocellular neuron  receives  input from several presynaptic neurons  1 ~ k  ~ \nJ{.  Each input spike at time t{  generates a current pulse which decays exponentially \nwith a fast time constant Tr  =  0.02 ms.  The magnitude of the current pulse depends \non the coupling strength h. The total input is \n\n1(t) = L h: exp( --=-.!. ) O(t  - t{) \n\ntf \n\nt \n\nk,f \n\nTr \n\n(2) \n\nwhere  O(x)  is  the unit step  function  and  the sum runs over all input spikes. \n\n\fR.  KEMPTER, W. GERSTNER, J.  L. VAN HEMMEN, H. WAGNER \n\n/ \\   h \n\nfoE- T~ \nb \n\n-\nI \nI \nI \n<p \n\nI \nI \n\no \n\n/\\ v v t  \nI \nI \nI \n\nI \nI \nI \n\n21t \n\n126 \n\na) \n\nb) \n\nt \n\nt \n\nFig.  1.  Principles of phase  locking and learning.  a) The stimulus consists of a sound \nwave  (top).  Spikes of auditory nerve  fibers  leading to the  nucleus  magnocellularis \nare  phase-locked  to  the periodic  wave,  that  is,  they  occur  at  a  preferred  phase  in \nrelation  to  the  sound,  but  with  some  jitter  0\".  Three  examples  of  phase-locked \nspike  trains  are  indicated.  b)  Before  learning  (left),  many  auditory  input  fibers \nconverge  to  a  neuron  of  the  nucleus  magnocellularis.  Because  of axonal  delays \nwhich  vary  between  different  fibers,  spikes  arrive  incoherently  even  though  they \nare  generated  in  a  phase  locked  fashion.  Due  to averaging over  several  incoherent \ninputs,  the  total  postsynaptic  potential  (bottom  left)  of a  magnocellular  neuron \nfollows  a  rather  smooth  trajectory  with  no  significant  temporal  structure.  After \nlearning (right)  most connections have disappeared  and only a few  strong contacts \nremain.  Input spikes  now arrive coherently and  the postsynaptic potential exhibits \na clear oscillatory structure.  Note  that firing  must occur  during the rising phase of \nthe oscillation.  Thus output spikes  will  be phase locked. \n\n\fTemporal Coding in the  Submillisecond Range:  Model of Bam Owl Auditory Pathway \n\n127 \n\nAll  input signals belong  to the same frequency  channel  with  a  carrier frequency  of \n2 kHz  (period T  = 0.5 ms), but the inputs arise from  different  presynaptic neurons \n(1  ~ k  ~ K).  Their  axons  have  different  diameter and  length  leading  to a  signal \ntransmission delay  ~k which  varies  between  2  and  3 ms  [4].  Note  that  a  delay  as \nsmall as  0.25 ms shifts the signal by  half a  period. \n\nEach  input  signal  consists  of a  periodic  spike  train  subject  to two  types  of noise. \nFirst,  a  presynaptic  neuron  may  not  fire  regularly  every  period  but,  on  average, \nevery nth  period only where  n  ~ 1/(vT) and v is the mean firing rate of the neuron. \nFor the sake of simplicity, we  set  n  =  1.  Second,  the spikes  may occur slightly too \nearly or too late compared to the mean delay~. Based on experimental results,  we \nassume a  typical shift (1  = \u00b10.05 ms [3].  Specifically  we  assume in  our model  that \ninputs from a  presynaptic  neuron  k arrive  with the probability density \n\nP(  J)  __  1_  ~  [-(t{  -nT- ~k)2l \n\ntk  -\n\n. m= \nv2~(1 \n\nL...t  exp \n\nn=-OO \n\n2 \n2(1 \n\n(3) \n\nwhere  ~k is the axonal transmission delay of input k (Fig.  1). \n\n3  Temporal tuning through  learning \n\nWe  assume  a  developmental  period  of unsupervised  learning  during  which  a  fine \ntuning of the temporal characteristics of signal transmission  takes place  (Fig.  Ib) . \nBefore learning the magnocellular neuron receives  many inputs (K = 50) with weak \ncoupling (Jk  = 1).  Due to the broad distribution of delays the tptal input  (2)  has, \napart from  fluctuations,  no  temporal structure.  After  learning,  the  magnocellular \nneuron  receives  input from two or three presynaptic neurons only.  The connections \nto those neurons  have  become very  effective;  cf.  Fig.  2. \n\na) \n\n30 \n\n20 \n\n10 \n\n<f ,-\n\n0 \n2.0 \n\nc) \n\n30 \n\n20 \n\n10 \n\n<f ,-\n\n2.5 \n\n~[ms) \n\nb) \n\n30 \n\n20 \n\n10 \n\n<f ,-\n\n3.0 \n\n0 \n2.0 \n\nd) \n\n30 \n\n20 \n\n10 \n\n<f ,-\n\n2.5 \n\n~[ms] \n\n3.0 \n\n0 \n2.0 \n\n2.5 \n\n~[ms) \n\n3.0 \n\n0 \n2.0 \n\n2.5 \n\n~[ms) \n\n3.0 \n\nFig.  2.  Learning.  We  plot  the  number of synaptic  contacts  (y-axis)  for  each  delay \n~ (x-axis).  (a)  At the  beginning,  the  neuron  has  contacts  to  50 presynaptic neurons \nwith delays 2ms ~ ~ ~ 3ms.  (b)  and (c)  During learning,  some presynaptic neurons \nincrease  their  number  of  contacts,  other  contacts  disappear. \n(d)  After  learning, \ncontacts  to  three  presynaptic  neurons  with  delays  2.25,  2.28,  and  2.8  ms  remain. \nThe  remaining  contacts  are  very strong. \n\n\f128 \n\nR.  KEMPfER, W. GERSTNER, J. L. VAN HEMMEN, H.  WAGNER \n\nThe constant h:  measures the  total coupling strength between  a presynaptic neuron \nk and  the postsynaptic  neuron.  Values of h:  larger  than one  indicate that several \nsynapses  have  been  formed.  It has  been  estimated  from  anatomical  data  that  a \nfully developed  magnocellular neuron receives inputs from as few  as  1-4 presynaptic \nneurons,  but each presynaptic axon shows multiple branching near the postsynaptic \nsoma and makes up to one hundred synaptic contacts on the soma of the magnocel(cid:173)\nlular neuron[5].  The result of our simulation study is consistent with this finding.  In \nour model, learning leads to a final state with a few  but highly effective inputs.  The \nremaining inputs all have the same time delay modulo the period T  of the stimulus. \nThus,  learning leads  to reduction  of the number of input  neurons  contacts  with  a \nnucleus magnocellularis neuron.  This is the fine  tuning of the neuronal connections \nnecessary  for  precise  temporal coding  (see  below, section  4). \n\nt:: j \n\n0.0 \n\n0.5 \n\no \n\n5 \nX [ms] \n\n10 \n\na) \n\nb) \n\n-X -3: \n\n0.2 \n\n0.0 \n\n1.0 \n\n-X -w  0.5 \n\n0.0 \n\no \n\n5 \nX [ms] \n\n10 \n\nFig.  3.  (a)  Time  window  of learning  W(x).  Along  the  x-axis  we  plot  the  time \ndifference  between  presynaptic  and  postsynaptic  fiing  x  =  t{  -\ntl:.  The  window \nfunction  W(x)  has  a  positive  and  a  negative  phase.  Learning  is  most  effective,  if \nthe  postsynaptic  spike  is  late  by  0.08  ms  (inset).  (b)  Postsynaptic  potential {(x). \nEach  input  spike  evoked  a  postsynaptic potential which  decays  with  a  time  constant \nof 2  ms.  Since  synapses  are  located  directly  at  the  soma,  the  rise  time  is  very \nfast  (see  inset).  Our learning  scenario  requires  that  the  rise  time  of {(x)  should  be \napproximately  equal to  the  time x  where  W(x)  has  its  maximum. \n\nIn  our  model,  temporal  tuning  is  achieved  by  a  variant  of Hebbian  learning.  In \nstandard  Hebbian  learning,  synaptic  weights  are  changed  if pre- and  postsynaptic \nactivity  occurs  simultaneously.  In  the  context  of temporal  coding  by  spikes,  the \nconcept of (simultaneous activity' has to be refined.  We  assume that a  synapse k  is \n\n\fTemporal Coding in the Submillisecond Range:  Model of Barn Owl Auditory Pathway \n\n129 \n\nchanged,  if a  presynaptic spike t{  and  a  postsynaptic spike to  occur  within  a  time \nwindow W(t{ -to).  More precisely, each pair of presynaptic and postsynaptic spikes \nchanges  a synapse  Jk  by  an  amount \n\n(4) \n\nwith a prefactor ,  =  0.2.  Depending on the sign of W( x),  a contact to a presynaptic \nneuron  is  either  increased  or  decreased.  A  decrease  below  Jk  =  0  is  not  allowed. \nIn  our  model,  we  assume  a  function  W(x)  with  two  phases;  cf.  Fig.  3.  For  x  ~ \n0,  the  function  W(x)  is  positive.  This  leads  to  a  strengthening  (potentiation)  of \nthe  contact  with  a  presynaptic  neuron  k  which  is  active  shortly  before  or  after  a \npostsynaptic  spike.  Synaptic contacts  which  become  active  more  than  3  ms later \nthan the postsynaptic spike are decreased.  Note that the time window spans several \ncycles of length T.  The combination of decrease  and  increase  balances the  average \neffects  of potentiation  and  depression  and  leads  to a  normalization of the  number \nand  weight of synapses.  Learning is stopped  after 50.000 cycles of length T. \n\n4  Temporal coding after learning \n\nAfter  learning  contacts  remain  to  a  small  number of presynaptic  neurons.  Their \naxonal transmission delays coincide or differ by multiples of the period T.  Thus the \nspikes  arriving from  the few  different  presynaptic  neurons  have approximately the \nsame phase and add up to an input signal (2) which retains, apart from fluctuations, \nthe periodicity of the external sound  signal  (Fig.4a). \n\na) \n\nb) \n\n-. \n\n9--+'\" \n\nCJ) \n\n~ \n\no \n\n1t \n\n21t \n\no \n\n1t \n\n21t \n\nFig.  4.  (a)  Distribution  of input  phases  after learning.  The  solid  line  shows  the \nnumber of instances  that  an  input  spike  with phase  <p  has  occured  (arbitrary  units). \nThe  input  consists  of spikes from  the  three  presynaptic neurons  which have  survived \nafter  learning;  cf.  Fig.  1 d.  Due  to  the  different  delays,  the  mean  input  phase \nv(lries  slightly  between  the  three  input  channels.  The  dashed  curves  show  the  phase \ndistribution  of the  individual  channels,  the  solid line  is  the  sum  of the  three  dashed \ncurves.  (b)  Distribution  of output  phases  after learning.  The  histogram  of output \nphases  is  sharply  peaked.  Comparison  of the  position  of the  maxima  of the  solid \ncurves  in  (a)  and  (b)  shows  that  the  output  is  phase  locked  to  the  input  with  a \nrelative  delay  fl<p  which  is  related  to  the  rise  time  of the  postsynaptic  potential. \n\n\f130 \n\nR.  KEMPTER, W.  GERSTNER, J. L. VAN HEMMEN, H. WAGNER \n\nOutput spikes  of the magnocellular neuron  are  generated  by  the integrate-and-fire \nprocess  (1).  In  FigAb we  show  a  histogram of the phases of the output spikes.  We \nfind  that  the  phases  have  a  narrow  distribution  around  a  peak  value.  Thus  the \noutput  is  phase locked  to the external  signal.  The  width of the phase distribution \ncorresponds  to  a  precision  of 0.084  phase  cycles  which  equals  42  jlS  for  a  2  kHz \nstimulus.  Note  that  the  temporal  precision  of the output  has  improved  compared \nto the input where we  had three channels with slightly different  mean phases and a \nvariation of (T  =  50jls each.  The increase  in the precision  is  due to the average over \nthree  uncorrelated  input signals. \n\nWe  assume that the same principles are used  during the following stages  along the \nauditory  pathway.  In  the nucleus  laminaris several  hundred  signals  are  combined. \nThis improves the signal-to-noise ratio further and a  temporal precision below 10  jlS \ncould  be achieved. \n\n5  Discussion \n\nWe have demonstrated that precise temporal coding in the microsecond range is pos(cid:173)\nsible despite neuronal time constants in the millisecond range.  Temporal refinement \nhas  been  achieved  through  a  slow  developmental  learning rule.  It is  a  correlation \nbased  rule  with  a  time window  W  which  spans  several  milliseconds.  Nevertheless \nlearning leads to a  fine  tuning of the connections supporting temporal coding with \na resolution of 42  jlS.  The membrane time constant was  set  to 2 ms.  This is  nearly \ntwo orders of magnitudes longer than  the achieved  resolution.  In  our model,  there \nis  only one fast  time constant  which  describes  the  typical duration  of a  input cur(cid:173)\nrent  pulse  evoked  by  a  presynaptic spike.  Our  value  of Tr  =  20  jlS  corresponds  to \na  rise  time of the  postsynaptic  potential  of 100  jls.  This seems  to be  realistic  for \nauditory  neurons  since  synaptic  contacts  are  located  directly  on  the  soma  of the \npostsynaptic  neuron.  The  basic  results  of our  model  can  also  be  applied  to other \nareas of the brain and can  shed  new light on some aspects  of temporal coding with \nslow  neurons. \n\nAcknowledgments:  R.K.  holds  scholarship  of  the  state  of  Bavaria.  W.G.  has  been \nsupported by the Deutsche Forschungsgemeinschaft  (DFG) under grant number He 1729/2-\n2.  H.W. is  a  Heisenberg fellow  of the  DFG. \n\nReferences \n\n[1]  L.  A.  Jeffress,  J.  Compo  Physiol.  Psychol.  41, 35  (1948). \n[2]  M.  Konishi,  Trends  Neurosci. 9,  163  (1986). \n[3]  C.  E.  Carr and M.  Konishi, J.  Neurosci.  10,3227 (1990). \n[4]  W.  E. Sullivan and  M.  Konishi, J.  Neurosci.  4,1787 (1984). \n[5]  C.  E.  Carr and R.  E.  Boudreau, J.  Compo  Neurol.  314, 306  (1991). \n\n\f", "award": [], "sourceid": 1157, "authors": [{"given_name": "Richard", "family_name": "Kempter", "institution": null}, {"given_name": "Wulfram", "family_name": "Gerstner", "institution": null}, {"given_name": "J.", "family_name": "van Hemmen", "institution": null}, {"given_name": "Hermann", "family_name": "Wagner", "institution": null}]}