{"title": "Multi-Electrode Spike Sorting by Clustering Transfer Functions", "book": "Advances in Neural Information Processing Systems", "page_first": 146, "page_last": 152, "abstract": null, "full_text": "Multi-electrode spike sorting \nby clustering transfer functions \n\nDmitry Rinberg \n\nHanan Davidowitz \n\nN aftali Tishby* \n\nNEe Research Institute \n\n4 Independence Way \nPrinceton, N J  08540 \n\nE-mail:  {dima,hanan, tishby }<Dreseareh. nj . nee . com \n\nCategories:  spike sorting,  population coding,  signal  processing. \n\nAbstract \n\nA  new  paradigm  is  proposed  for  sorting  spikes  in  multi-electrode \ndata using ratios of transfer functions  between cells and electrodes. \nIt  is  assumed  that  for  every  cell  and  electrode  there  is  a  stable \nlinear  relation.  These are  dictated  by  the properties of the tissue, \nthe electrodes  and  their relative  geometries.  The main  advantage \nof the method is that it is  insensitive to variations in the shape and \namplitude of a  spike.  Spike  sorting  is  carried  out  in  two  separate \nsteps.  First,  templates  describing  the statistics of each spike type \nare generated by clustering transfer function  ratios then spikes are \ndetected  in  the  data  using  the  spike  statistics.  These  techniques \nwere  applied  to  data generated  in  the  escape  response  system  of \nthe cockroach. \n\n1 \n\nIntroduction \n\nSimultaneous  recording of activity from  many  neurons can greatly expand our un(cid:173)\nderstanding of how  information  is  coded  in  neural  systems[l].  11ultiple electrodes \nare often used to measure the activity in neural tissue and have become a  standard \ntool in neurophysiology [2, 3,4].  Since every electrode is in a different position it will \nmeasure a  different contribution from  each of the different  neurons.  Simply stated, \nthe problem is this:  how can these complex signals be untangled to determine when \neach  individual  cell  fired?  This  problem  is  difficult  because,  a)  the  objects  being \nclassified  are very similar and often noisy,  b)  spikes  coming  from  the same cell  can \n\n\u00b7Permanent  address:  Institute of Computer Science  and  Center  for  Neural  Computa(cid:173)\n\ntion,  The Hebrew University,  Jerusalem,  Israel.  Email:  tishby<Des.huji.ae.il \n\n\fTransfer Function Spike Sorting \n\n147 \n\nvary  in  both  shape  and  amplitude,  depending on  the  previous  activity  of the cell \nand c) spikes can overlap in time, resulting in even more complex temporal patterns. \n\nCurrent  approaches  to  spike  sorting  are  based  primarily  on  the  presumed  consis(cid:173)\ntency  of the spike  shape and  amplitude  for  a  given  cell  [5,  6].  This  is  clearly  the \nonly possible basis for sorting using a single electrode.  Multiple electrodes, however, \nprovide  additional  independent information through the  differences  in  the way  the \nsame  neuron  is  detected  by  the  different  electrodes.  The same  spike  measured  on \ndifferent  electrodes  can  differ  in  amplitude,  shape  and  its  relative  timing.  These \ndifferences  can  depend  on  the  specific  cell,  the  electrode  and  the  media  between \nthem.  They  can  be  characterized  by linear  transfer  functions  that are invariant to \nchanges  in  the  overall  spike  waveform.  In  this  paper  the  importance of this  infor(cid:173)\nmation  is  highlighted  by  using  only the differences  in  how  signals  are  measured on \ndifferent electrodes.  It is  then shown that clusters of similar  differences  correspond \nto the same neuron.  It should be emphasized  that in  a  full  treatment this transfer \nfunction  information will  be combined with other cues  to sort spikes. \n\n2  Spikes,  spectra and  noise \n\nThe basic assumption  behind the spike sorting approach described  here is  that the \nmedium between each neuron-electrode pair can be characterized by a linear system \nthat remains fixed  during the course of an experiment.  This assumption is  justified \nby the approximately linear dielectric properties of the electrode and its surrounding \nnerve tissues. \n\nLinear  systems  are  described  by  their  phase  and  amplitude  response  to  pure  fre(cid:173)\nquencies ,  namely,  by  their  complex  transfer  function  H(w)  =  O(w)j I(w),  where \nI(w)  and  O(w)  are  the  complex  spectra  (Le.  Fourier  transform,  henceforth  called \nspectrum)  of the input and output of the system,  respectively.  In the experiments \ndescribed  here  the  input  signal  is  the  spectrum of the  action  potential  generated \nby  cell  j,  denoted  by  Sj(w)  and  the output  signal  is  the  spectrum of the voltage \nmeasured  at  electrode  Il,  denoted  by  VJ.L(w).  The  transfer  function  of the system \nthat links  Sj(w)  and VJ.L(w)  is  then defined  as  Hf(w)  =  VJ.L(w)jSj(w). \n\nIf the transfer functions are fixed  in time,  the ratio between the complex spectrum \nof any  spike  from  cell  j  as  detected  by  electrodes  Il  and  v ,  VJ.L (w)  and  V II (w ),  is \ngiven  by, \n\nHf(w) \nHj(w)  , \n\n(1) \n\nwhich is  independent of the cell  action potential spectrum Sj(w),  provided that the \nspike was  detected  by  both electrodes. \nThus,  even  if  a  spike  varies  in  shape  and  amplitude,  Tjll (w)  will  remain  a  fixed \ncomplex  function  of  frequency.  This  ratio  is  also  invariant  with  respect  to  time \ntranslations  of the  spikes.  In  addition,  the  frequency  components  are  asymptot(cid:173)\nically  un correlated  for  stationary  processes,  which justifies  treating  the  frequency \ncomponents as statistically independent[7] .  The idea behind the approach described \nhere is  shown  in  Figure 1. \nIn  real  experiments,  however,  noise  can  corrupt  the  invariance  of Tjll.  There  are \nseveral  possible  sources of noise  in  experiments of this  kind:  a)  fluctuations  in  the \ntransfer  function,  b)  changes  in  the spike  shape,  <;j  and  c)  electrical  and  electro(cid:173)\nchemical  noise,  nJ.L. \n\n\f148 \n\nD.  Rinberg, H.  Davidowitz and N  Tishby \n\ncell-1 \n\ncell-1 \n\ncell-2 \n\nQ) \nu \n\n::l -a. \n\nE \nctl \n\ntime \n\ntime \n\ntime \n\nQ) \nu \n::l ::: \na. \nE \nctl \n\ntime \n\ntime \n\ntime \n\nI~ ~ \n\nfrequency \n\nfrequency \n\nfrequency \n\nc:: \n\nc:: \n0 \n\n-u \n::::I  0 --~  ctl \nQ)  ~ -(/) \n~ -\n\nc:: \nctl \n\nFigure  1:  The  idea  behind  spike  sorting  by  clustering  of transfer  function  ratios. \nTwo spikes from  the same cell  (cell-I) may vary in shape/ amplitude during bursting \nactivity,  for  example.  Although the spike shapes  may differ,  the transfer  functions \nrelating  them  to  the  electrodes  do  not  change  so  the  transfer  function  ratios  are \nsimilar  (two  left  columns).  A  different  cell  (cell-2)  has a  different  transfer function \nratio even  though the spikes shapes themselves  are similar to those of cell-l  (right \ncolumn). \n\nIf Hf  varies slowly with time,  the transfer function  noise  is  small relative to <;j,  n Y \nand n Y \u2022  Try  can then be expanded to first  order  in  <;j,  nJ.l.  and n Y  as \n\n(2) \n\nwhich is independent of <;j.  Since the noise,  nJ.l.,  is  un correlated with the spike signal, \n5 j ,  the  variance  at  each  frequency  component  can  be  considered  to  be  Gaussian \n\nwith equal variances on the real and imaginary axes.  Thus the mean of Try  will  be \n\nindependent of 5 j , <;j  and nJ.l.  while its variance will  be inversely  proportional to 5 j . \n\n3  A  model system:  the  escape  response  of the cockroach \n\nThese  techniques  were  tested  on  a  relatively  simple  neural  system  - the  escape \nresponse system of the American cockroach.  The escape behaviour, which has been \nstudied  extensively  [9,  10,  11],  is  activated  when  the  insect  detects  air  currents \n\n\fTransfer Function Spike Sorting \n\n149 \n\n\" it \n!~ \n\nI \n\n:  10 ms \n\n( f\u00b7--, \"-, , \n\n. t  \\  .. \n- ~ \n1 t  \\' \n\\ \n. l \n\n\\ \n\n10.5mv \n50 ms \n\nFigure  2:  A  schematic  representation  of the  experiment.  Typical  raw  data  mea(cid:173)\nsured  on  two  electrodes is  shown  at  right.  Relative  time  delays  are  evident  in  the \ninset,  but  are  not  a  necessary  condition  for  the sorting techniques  described  here. \nAbbreviations are:  p-puffers,  cg-circal ganglion,  c-cerci. \n\nproduced  by  the  movements  of a  predator.  The  insect  detects  the  approach  of a \npredator, determines the direction of approach and starts running in an appropriate \ndirection.  The cockroach  does  this  by  detecting  the movement  of several  hundred \nfine  hairs  located  on  two  appendages,  called  cerci,  protruding  from  the  posterior \nend  of  the  animal.  Each  of  these  hairs  is  connected  to  a  single  neuron.  Axons \nfrom  these cells converge on a  dense neuropil  called  the cercal ganglion  (cg),  where \ndirectional information is  extracted and  conveyed  to the rest  of the body by  axons \nin the abdominal  nerve.  This is  shown schematically in  Figure 2. \n\nThis  system  proved  to  be  well  suited  as  a  first  test of the sorting  technique.  The \nsystem  is  simple  enough  so  that  it  is  not  overwhelming  (since  only  7  neurons  are \nknown  to contribute to the code)  but complex enough  to really  test the approach. \nIn addition, the nerve cords are linear in geometry, easily accessible and very stable. \n\nMale  cockroaches  (Periplaneta  americana)  were  dissected  from  the  dorsal  side  to \nexpose the nerve cord.  The left and right cords were gently separated and two tung(cid:173)\nsten wire electrodes were  hooked onto the connective about 2 mm apart,  separated \nby  abdominal  ganglia.  The  stimulus  was  presented  by  two  loudspeakers  driving \ntwo  miniature  wind  tunnels  pointed  at  the  cerci,  at  90  degrees  from  one  another \nas  shown  in  Figure  2.  Recordings  typically  lasted  for  several  hours.  Data  were \ncollected  with  a  sampling frequency of 2 . 104  Sis which  was  sufficient  to  preserve \nthe high  frequency  components of the spikes. \n\n\f150 \n\n0.5 \n\nD.  Rinberg, H.  Davidowitz and N.  Tishby \n\n1 \n\n-1.5 \n\n-1 \n\nRe(T) \n\n1 \n\nFigure  3:  Real  and  imaginary  parts  of Tr v  a  single  w.  The  circles  have  centers \n(radii)  equal  to  the  average  (variance)  of Tr v  at  w  =  248.7  rad  S-l.  Note  that \nwhile  some  clusters  seem  to overlap  at this  frequency  they  may  be  well  seperated \nat others.  Cluster-l  is  dispersed throughout the complex plane and its  variance  is \nwell  beyond the range of this plot. \n\n4  Clustering and  the  detection of spikes \n\nThe  spike  sorting  algorithm  described  here  is  done  is  two  separate  stages.  First, \na  statistical  model  of  the  individual  spike  types  is  built  from  \"clean\"  examples \nfound  in  the data.  Only then are occurrences of these spikes  detected in the multi(cid:173)\nelectrode  data.  This  two-step  arrangement  allows  a  great  deal  of  flexibility  by \ndisconnecting  the clustering  from  the  detection.  For  example,  here  the  clustering \nwas  done  on transfer  function  ratios  while  the detection was  done on  full  complex \nspectra.  These stages are described  below in  more detail. \n\n4.1  The clustering  phase \n\nFirst, the multi-electrode recording is  chopped into 3 ms long frames  using a sliding \nwindow.  Frames  that  have  either  too  low  total  energy  or  too  high  energy  at  the \nwindow edges  are discarded.  This leaves  frames  that  are energetic  in  their  central \n2 ms and  are assumed to carry one spike.  No  attempt is  made to find  all  spikes  in \nthe data.  Instead,  the idea is  to generate a  set of candidate spike types from  clean \nframes. \n\nOnce  a  large  collection  of candidate spikes  is  found,  TrV(w)  is  calculated for  every \n\n\fTransfer Function Spike Sorting \n\n151 \n\nHook #1 \n\ncluster \n\nyCluster \n\nframes \n\n6 \n\n324 \n\n720 \n\n~ 5 \nrvv \n..- -\n&  V \n\n729 \n\n518 \n\n4 \n\nd \n\n3 \n\ncs \n2 \n\n'-\" \n\n748 \n\nHook #2 \n\n-'lr  7E \nca  &  -'\\I  2 \n\nz \n\nR  & \n\n\u2022 \n\nE2\u00a3@\u00a3C9 \n\n.... --\n\nR  g  4V  500~V I \n\n~.I\",  ru@ \n\n~ ~6 I!!!I! \n\n1 \n\n1 w\u00a7 \n\nFigure  4:  Results  of  clustering  spikes  using  transfer  function  ratios.  Note  that \nalthough cluster-5 and cluster-6 are similarly shaped on hook-l they are time shifted \non  hook-3.  Cluster-l  is  made  up  of overlaps which  are  dealt  with  in  the  detection \nphase. \n\nspike.  These  are  then  grouped  together  into  clusters  containing  similar  Ttl! (w). \nResults of the clustering are shown  in  Figure 3 while the corresponding waveforms \nare  shown  in  Figure  4.  Full  complex  spectra  are  then  used  to  build  a  statisti(cid:173)\ncal  model  of the  different  spike types,  {Vj(w), af(w)},  which  represent  each cell's \naction  potential  as  it appears on each  of the electrodes. \n\n4.2  The detection  phase \n\nOnce  the  cluster  statistics  are  determined ,  an  independent  detection  algorithm  is \nused.  The data is  again  broken  into short frames  but now  the idea is  to find  which \nof the spike types (represented by  the different clusters found  in the previous steps) \nbest represents the data in that frame.  Each frame can contain either noise,  a  spike \nor  an overlap of 2 spikes  (overlaps  of more than  2 spikes  are  not  dealt  with).  This \npart  is  not  done  on  transfer  function  ratios  because  dealing  with overlaps  is  more \ndifficult. \n\n5  Concl usion \n\nA  new  method  of spike  sorting  using  transfer  function  ratios  has  been  presented. \nIn effect  the sorting is done on the properties of the tissue between the neuron and \n\n\f152 \n\nD. Rinberg, H  Davidowitz and N.  Tishby \n\nthe electrode and individual spike shapes become less important.  This method may \nbe useful  when dealing with bursting cells where the transfer function  ratios should \nremain  constant  even  though  the  spike  amplitude  can  change  significantly.  This \ntechnique  may prove to be a  useful  tool for  analysing  multi-electrode data. \n\nAcknowledgments \n\nWe are grateful to Bill Bialek for numerous enlightening discussions and many useful \nsuggestions. \n\nReferences \n\n[1]  M.  Barinaga. Listening in  on the brain.  Science  280, 376-378  (1998). \n[2]  M.  Abeles.  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Journal  of Comparative  Physiology  A  121,307-324  (1977) . \n\n\f", "award": [], "sourceid": 1493, "authors": [{"given_name": "Dmitry", "family_name": "Rinberg", "institution": null}, {"given_name": "Hanan", "family_name": "Davidowitz", "institution": null}, {"given_name": "Naftali", "family_name": "Tishby", "institution": null}]}