{"title": "Application of Blind Separation of Sources to Optical Recording of Brain Activity", "book": "Advances in Neural Information Processing Systems", "page_first": 949, "page_last": 955, "abstract": null, "full_text": "Application of Blind Separation of Sources to \n\nOptical Recording of Brain Activity \n\nHolger Schoner, Martin Stetter, Ingo Schie61 \n\nDepartment of Computer Science \n\nTechnical  University of Berlin Germany \n\n{hjsch,moatl,ingos}@cs.tu-berlin.de \n\nJohn E. W.  Mayhew \n\nUniversity of Sheffield, UK \nj. e.mayhew@sheffield.ac.uk \n\nJennifer S. Lund, Niall McLoughlin \n\nInstitute of Ophthalmology \n\nUniversity College London, UK \n{j.lund,n.mcloughlin}@ucl.ac.uk \n\nKlaus Obermayer \n\nDepartment of Computer Science, \n\nTechnical  University of Berlin, Germany \n\noby@cs.tu-berlin.de \n\nAbstract \n\nIn the analysis of data recorded by optical imaging from intrinsic signals \n(measurement of changes of light reflectance from cortical tissue) the re(cid:173)\nmoval  of noise and  artifacts  such  as  blood  vessel  patterns  is  a  serious \nproblem. Often bandpass filtering is used, but the underlying assumption \nthat a spatial frequency  exists,  which separates  the mapping component \nfrom  other components  (especially  the  global  signal),  is  questionable. \nHere we propose alternative ways of processing optical imaging data, us(cid:173)\ning blind source separation techniques based on the spatial decorre1ation \nof the data.  We  first  perform  benchmarks  on  artificial  data in  order  to \nselect  the  way  of processing, which is  most robust with respect  to  sen(cid:173)\nsor noise.  We then apply it to recordings of optical imaging experiments \nfrom  macaque primary visual cortex. We show that our BSS technique is \nable  to  extract ocular dominance and orientation preference maps  from \nsingle  condition  stacks,  for  data,  where  standard  post-processing pro(cid:173)\ncedures  fail.  Artifacts,  especially  blood  vessel  patterns,  can  often  be \ncompletely removed  from  the maps.  In  summary,  our method for blind \nsource separation using extended spatial decorrelation is a superior tech(cid:173)\nnique for the analysis of optical recording data. \n\n1 \n\nIntroduction \n\nOne approach in the attempt of comprehending how the human brain works is the analysis \nof neural activation patterns in the brain for different stimuli presented to a sensory system. \nAn  example  is  the  extraction  of ocular dominance  or orientation preference  maps  from \nrecordings  of activity  of neurons  in  the primary  visual  cortex  of mammals.  A  common \ntechnique for extracting such  maps  is optical  imaging (01) of intrinsic signals.  Currently \nthis is the imaging technique with the highest spatial resolution (~ 100 J1m) for mapping of \nthe cortex.  This method is explained e.g. in [1], for similar methods using voltage sensitive \ndyes see [2, 3] .  01 uses changes  in  light reflection to estimate spatial patterns of stimulus \n\n\f950  H.  Sch6ner.  M  Stetter. I.  Schiej3l,  J  E.  Mayhew,  J  Lund, N.  Mcloughlin and K.  Obermayer \n\nanswers.  The overall change recorded  by  a CCD or video camera is  the total  signal.  The \npart of the total signal due to local  neural  activity is called the mapping component and  it \nderives from changes in deoxyhemoglobin absorption and light scattering properties of the \ntissue.  Another component of the total  signal is a \"global\" component, which  is also cor(cid:173)\nrelated with stimulus presentation, but has a much coarser spatial  re~olution .  It derives  its \npart from changes in  the blood volume with the time.  Other components are blood vessel \nartifacts, the vasomotor signal  (slow  oscillations of neural  activity),  and  ongoing activity \n(spontaneous,  stimulus-uncorrelated activity).  Problematic for  the  extraction  of activity \nmaps are especially blood vessel  artifacts and  sensor noise, such as  photon shot noise.  A \nprocedure often used for extracting the activity maps  from  the recordings is  bandpass fil(cid:173)\ntering,  after preprocessing  by  temporal , spatial, and  trial  averaging.  Lowpass  filtering  is \nunproblematic,  as  the spatial resolution of the mapping  signal  is limited by  the scattering \nproperties of the brain tissue,  hence everything above a limiting frequency  must be  noise. \nThe motivation for highpass filtering, on the other hand, is questionable as there is no spe(cid:173)\ncific spatial frequency separating local neural activity patterns and the global signal  [4]). \n\nA  different  approach,  Blind  Source  Separation  (BSS),  models  the  components  of the \nrecorded  image  frames  as  independent  sources,  and  the  observations  (recorded  image \nframes)  as  noisy  linear mixtures of the  unknown sources.  After performing the BSS  the \nmapping  component  should  ideally  be  concentrated  in  one estimated  source,  the  global \nsignal  in  another,  and  blood vessel  artifacts, etc. in  further ones.  Previous work ([5])  has \nshown that BSS  algorithms, which are based  on  higher order statistics ([6,  7, 8]), fail  for \noptical imaging data,  because of the high signal to noise ratio. \n\nIn this work we suggest and  investigate versions of the M&S algorithm [9, 10], which are \nrobust against sensor noise,  and  we  analyze their performance on artificial  as  well  as  real \noptical  recording  data.  In  section  2  we describe  an  improved  algorithm,  which  we  later \ncompare to other methods in section 3.  There an  artificial data set is used for the analysis \nof noise robustness, and  benchmark results are presented.  Then,  in section  4, it is  shown \nthat the newly developed algorithm is very  well  able to separate the different components \nof the optical  imaging  data, for  ocular dominance  as  well  as  orientation preference  data \nfrom  monkey  striate cortex.  Finally,  section  5 provides conclusions and  perspectives  for \nfuture work. \n\n2  Second order blind source separation \n\nLet m be the number of mixtures and r  the sample index, i.e. a vector specifying a pixel in \nthe recorded  images.  The observation vectors y(r)  =  (Y1(r) , ... ,Ym\u00a5')f  are assumed \nto be linear mixtures of m unknown sources s(r) =  (Sl (r) , . . . ,Sm  (r))  with A being the \nm  x  m  mixing matrix and n  describing the sensor noise: \n\ny(r) =  As(r) + n \n\n(1) \n\nThe goal  of BSS  is to obtain optimal source estimates s(r)  under the assumption  that the \noriginal sources are independent.  In the  noiseless case W  =  A -1  would be  the optimal \ndemixing  matrix.  In  the  noisy  case,  however,  W  also  has  to  compensate for  the  added \nnoise:  s(r) ==  Wy(r) = W  . A . s(r) + W  . n. BSS algorithms are generally only able to \nrecover the original sources up to a permutation and scaling. \n\nExtended  Spatial  Decorrelation  (ESD)  uses  the  second  order  statistics  of the  observa(cid:173)\ntions to  find  the  source estimates.  If sources  are  statistical  independent all  source cross(cid:173)\ncorrelations \n\nCi(,~) (D.r) = (si(r)Sj(r+ D.r))r = ~ LSi(r)Sj(r+ D.r) \n\n, i =F  j \n\n(2) \n\nr \n\n\fApplication of BSS to  Optical Recording of Brain Activity \n\n951 \n\nmust vanish for all  shifts ~r, while the autocorrelations (i =  j) of the sources remain (the \nvariances).  Note that this implies that the sources must be spatially smooth. \n\nMotivated  by  [to]  we  propose  to  optimize  the  cost  function,  which  is  the  sum  of the \nsquared cross-correlations of the estimated sources over a set of shifts {~r}, \n\nE(W) = L L ((WC(~r)WT)i,jr \n\n(3) \n\n6r  i~j \n\n= L L\\Si(r)Sj(r + ~r))~ , \n\n6r  itj \n\nwith respect to  the demixing matrix  W.  The matrix Ci,j(~r) =  (Yi (r)Yj(r + ~r))r de(cid:173)\nnotes the mixture cross-correlations for a shift ~r. This cost function  is minimized using \nthe Polak Ribiere Conjugate Gradient technique,  where the line search  is  substituted by  a \ndynamic step width adaptation ([11]).  To keep the demixing matrix W  from converging to \nthe zero matrix, we introduce a constraint which keeps the diagonal elements of T  =  W-l \n(in the noiseless case and for non-sphered data T  is an  estimate of the mixing matrix, with \npossible permutations) at a value of 1.0. Convergence properties are improved by sphering \nthe data (transforming their correlation matrix for shift zero to an  identity matrix) prior to \ndecorrelating the mixtures. \n\nNote that  use  of multiple shifts  ~r allows  to  use  more  information about the  auto- and \ncross-correlation structure of the mixtures for the separation process.  Two  shifts provide \njust enough constraints for  a  unique solution ([to]).  Multiple shifts,  and  the redundancy \nthey  introduce, additionally allow to cancel  out part of the noise by approximate simulta(cid:173)\nneous diagonalization of the corresponding cross correlation matrices. \n\nIn  the  presence  of sensor  noise,  added  after  mixing,  the  standard  sphering  technique  is \nproblematic.  When  calculating the  zero-shift cross-correlation matrix  the variance of the \nnoise contaminates the result, and sphering using a shifted cross-correlation matrix, is rec(cid:173)\nommended ([12]).  For spatially white sensor noise and  sources  with reasonable auto cor(cid:173)\nrelations this technique is more appropriate.  In  the following we denote the standard algo(cid:173)\nrithm by dpaO, and the variant using noise robust sphering by dpa1. \n\n3  Benchmarks for artificial data \n\nThe  artificial  data  set  used  here,  whose  sources  are  approximately  uncorrelated  for  all \nshifts,  is  shown in  the left part of figure  1.  The  mixtures were produced  by  generating  a \nrandom mixing matrix (in this case with condition number 3.73), applying it to the sources, \nand finally adding white noise of different variances. \n\nIn order to measure the performance on  the artificial data set we measure a reconstruction \nerror (RE) between the estimated and the correct sources via (see [l3]): \n\nRE(W) =  od(L \u00a7(r)sT(r)) , \n\nr \n\nod(C) = N  ~ N  _  1  L maXk'I~i,kl  - 1 \n) \n\n1  ~  1 \n\nIC\u00b7 \u00b71 \n\n(4) \n\n( \n\nJ \n\nI \n\nThe correlation between the real  and the estimated sources (the argument to \"od\"), should \nbe close to a permutation matrix, if the separation is successful.  If the maxima of two rows \nare in  the same column, the separation is labeled unsuccessful.  Otherwise, the normalized \nabsolute sum of non-permutation (cross-correlation) elements is computed and returned as \nthe reconstruction error. \n\nWe now compare the method based on optimization of (3) by gradient descent with the fol(cid:173)\nlowing variants of second order blind source separation:  (1) standard spatial decorrelation \n\n\f952  H.  Schaner;  M  Stetter; I.  Schiej3l,  J.  E.  Mayhew,  J.  Lund, N.  McLoughlin and K.  Obermayer \n\n. ~. \n\n-. \n\n'. \n\n'-\n\n' \u2022. \n\nopt \nmean \ncor \n\n-\" \n\n'. \n\n'!i ___ \n\n0.5 \n\n<; \nJiO.4 \nc: \n0 \ngO.3 \nb \n~ \n~02 \n0:: \n\n0.1 \n\n0.5 \n\ng \n~0.4 \n\u00a7 \ngOJ \nl:l \n~ \n80.2 \n\" \n0:: \n0.1 \n\no 0 \n\n5 \nSignal to Noise Ratio (dB) \n\n15 \n\n10 \n\n20 \n\n25 \n\n00 \n\n5 \nSignal to Noi se Ratio (dB) \n\n10 \n\n15 \n\n. __  .  jacO \njacl \ndpaO \ndpal \n\n-\n\n-.. .. \n, ... \n\n-~-.--\n\nhI \n''I, \n\n1 \n\n': \n, \n\n\" \n\n% \n\nH, \n~ \n\n\" \n\n, \n\n~ \n\n:. , --\n\n-----\n20 \n\n25 \n\nFigure  1:  The set of three approximately  uncorrelated source images of the artificial data \nset (left). The two plots (middle, right) show the reconstruction error versus signal to noise \nratio  for  different separation  algorithms.  In  the right plot jac1  and  dpa1  are  very  close \ntogether. \n\nusing the optimal single shift yielding the smallest reconstruction error (opt).  (2) Spatial \ndecorrelation using the shift selected by \n\n.6.rcor  =  argmax{.D.r} \n\nnorm (C(.6.r)  - diag (C(.6.r))) \n\nnorm (diag (C(.6.r)))\u00b7' \n\n(5) \n\nwhere  \"diag\"  sets  all  off-diagonal  elements  of its  argument  matrix  to  zero,  and  \"norm\" \ncomputes  the largest  singular  value of its argument  matrix  (cor).  .6.rcor  is  the  shift for \nwhich  the cross correlations are largest,  i.e.  whose signal  to  noise ratio  (SNR)  should be \nbest.  (3) Standard spatial decorrelation using the average  reconstruction error for all  suc(cid:173)\ncessful shifts in a 61  x  61  square around the zero shift (mean).  (4) A multi-shift algorithm \n([12]), using several elementary rotations (Jacobi method) to build an orthogonal demixing \nmatrix,  which  optimizes  the cost function  (3).  The variants using  standard sphering and \nnoise robust sphering are denoted by dacO) and dac1). cor, opt, and mean use two shifts \nfor their computation; but as one of those is always the zero-shift, there is only one shift to \nchoose and they are called single-shift algorithms here. \n\nFigure  1 gives  two plots which  show  the reconstruction error  (4)  versus  the  SNR  (mea(cid:173)\nsured  in  dB)  for  single shift  (middle)  and  multi-shift (right) algorithms.  The  error  bars \nindicate twice the standard error of the  mean  (2x  SEM), for  10 runs with the same mix(cid:173)\ning matrix,  but newly  generated  noise of the given noise level.  In each of these runs, the \nbest result of three  was  selected  for  the gradient descent  method.  This  is  because,  con(cid:173)\ntrary  to the other algorithms, the gradient descent algorithm depends on the initial choice \nof the demixing matrix.  All multi-shift algorithms (all except opt and mean), used 8 shifts \n(\u00b1r, \u00b1r), (\u00b1r, 0), and (0, \u00b1r) for each r  E {I, 3, 5, 10,20, 30}, so 48 all  together. \n\nSeveral  points are noticeable in  the plots.  (i) The cor algorithm is  generally closer to the \noptimum than  to  the average  successful  shift.  (ii)  A  comparison  between  the two  plots \nshows that the multi-shift algorithms (right plot) are able to perform much better than even \nthe optimal single-shift method.  For low to medium noise levels this is even the case when \nusing the standard  sphering method combined  with the gradient descent algorithm.  (iii) \nThe  advantage of the  noise  robust  sphering  method,  compared  to  the standard  sphering, \nis obvious:  the reconstruction error stays  very  low for all evaluated noise levels,  for both \nthe jac1  and dpa1  algoritlnns.  (iv) The gradient descent technique is more robust than the \nJacobi  method  For the standard  sphering its performance  is much  better than  that of the \nJacobi method. \n\nFigure  1 shows results which were produced using a single mixing matrix.  However,  our \nsimulations show that the algorithms compare qualitatively similar when  using mixing ma-\n\n\fApplication of BSS to Optical Recording of Brain Activity \n\n953 \n\nt  = 1 sec. \n\nt  = 2 sec. \n\nt  = 3 sec. \n\nt  = 4 sec. \n\nt  = 5 sec. \n\nt  = 6 sec. \n\nt  = 7 sec. \n\nFigure 2:  Optical  imaging  stacks.  The top  stack is  a single condition  stack  from  ocular \ndominance experiments,  the lower one a difference stack from  orientation preference ex(cid:173)\nperiments  (images  for 90\u00b0  gratings subtracted from  those for 0\u00b0  gratings).  The stimulus \nwas  present during recording images 2-7  in each  row.  Two  large  blood vessels  in  the top \nand left regions of the raw images were masked out prior to the analysis. \n\ntrices  with condition numbers between 2 and  10.  The noise robust versions of the multi(cid:173)\nshift algorithms generally yield the best separation results of all evaluated algorithms. \n\n4  Application to optical imaging \n\nWe  now  apply extended spatial decorrelation to  the analysis of optical imaging data.  The \ndata consists of recordings from the primary visual cortex of macaque monkeys.  Each trial \nlasted 8 seconds, which were  recorded with frame rates of 15 frames  per second.  A visual \nstimulus (a drifting bar grating of varying orientation) was  presented  between  seconds  2 \nand 8.  Trials were separated by  a recovery period of 15 seconds  without stimulation.  The \ncortex was illuminated at a wavelength of 633  nm.  One pixel corresponds to about 15  J.Lm \non the cortex; the image stacks used for further processing, consisting of 256  x  256 pixels, \ncovered an area of cortex of approximately 3.7 mm 2 . \n\nBlocks of 15 consecutive frames  were averaged,  and averaging over 8 trials using the same \nvisual  stimulus further  improved  the  SNR.  First frame  analysis  (subtraction of the  first, \nblank,  frame  from  the  others)  was  then  applied  to  the  resulting  stack  of 8  frames,  fol(cid:173)\nlowed by  lowpass filtering  with  14 cycles/mm.  Figure 2 shows the resulting image stacks \nfor an  ocular dominance and  an  orientation preference experiment.  One observes  strong \nblood vessel  artifacts (particularly in  the top  row  of images),  which  are  superimposed to \nthe patchy mapping component that pops up over time. \n\nFigure 3 shows results obtained by the application of extended spatial decorrelation (using \ndpaO).  Only  those estimated  sources  containing patterns different from  white  noise are \nshown.  Backprojection of the estimated  sources  onto the original image  stack  yields the \namplitude time  series of the estimated sources,  which is very  useful  in  selecting the map(cid:173)\nping component:  it can  be present in  the recordings only after the stimulus onset (starting \nat t  =  2 sec.).  The  middle part shows  four  estimated  sources  for  the ocular dominance \nsingle condition stack.  The mapping component (first image) is separated from the global \ncomponent (second image)  and  blood vessel  artifacts (second  to  fourth)  quite well.  The \ntime course of the  mapping  component is  plausible as  well:  calculation  of a plausibility \nindex (sum of squared differences  between the normalized time series and a step function, \nwhich is 0 before and  1 after the stimulus onset) gives 0.5 for the mapping component and \n2.31  for the  next best one.  Results for the gradient descent  algorithm are  similar for this \ndata set,  regardless  of the sphering technique  used.  The Jacobi  method also  gives  simi(cid:173)\nlar  results,  but  a  small  blood  vessel  artifact is  remaining  in  the  resulting  map.  The  cor \nalgorithm usually gives much worse separation results.  In the right part of figure 3 two es-\n\n\f954  H  Schaner,  M  Stetter,  I  SchieJ3l, 1. E.  Mayhew, 1.  Lund, N  McLoughlin and K.  Obermayer \n\nFigure 3:  Left:  Summation technique for ocular dominance (aD) experiment (upper) and \norientation preference  (OP)  experiment (lower).  Middle,  Right:  dpaO algorithm applied \nto  the same  aD single condition (middle) and OP (right) stacks.  The images show  the 4 \n(aD) and  2 (OP) estimated components, which are visually different from  white noise.  In \nthe bottom row the respective time courses of the estimated sources are given. \n\ntimated sources (those different from white noise) for the orientation preference difference \nstack can be seen.  Here the proposed algorithm (dpaO) again works very  well (plausibility \nindex is 0.56 for mapping component, compared to 3.04 for the best other component).  It \ngenerally has  to  be applied a few  times (usually around 3 times) to  select the best separa(cid:173)\ntion result Uudging by visual quality of the separation and the time courses of the estimated \nsources), because of its dependence on parameter initialization; in  return it yields the best \nresults of all algorithms used, especially when compared to the traditional summation tech(cid:173)\nnique. \n\nThe  similar results  when  using standard  and  noise robust sphering,  and  the  small  differ(cid:173)\nences between the gradient descent and the Jacobi algorithms indicate, that not sensor noise \nis  the limiting factor for the quality of the extracted maps.  Instead it seems  that, assuming \na  linear mixing  model,  no  better results  can  be obtained  from  the  used  image  stacks.  It \nwill  remain  for  further  research  to  analyze,  how  appropriate the  linear mixing model  is, \nand  whether the underlying biophysical components are  sufficiently  uncorrelated.  In  the \nmeantime  the maps  obtained  by  the ESD  algorithm are  superior to  those  obtained  using \nconventional techniques like summation of the image stack. \n\n5  Conclusion \n\nThe results presented  in  the previous sections show the advantages of the proposed algo(cid:173)\nrithm:  In  the comparison  with other spatial  decorrelation algorithms the  benefit  in  using \nmultiple shifts compared  to  only two  shifts is  demonstrated.  The robustness  against sen(cid:173)\nsor noise is  improved, and  in  addition, the selection of multiple shifts is  less  critical than \nselecting a single shift,  as  the resulting multi-shift system of equations contains more re(cid:173)\ndundancy.  In comparison with the Jacobi  method,  which  is restricted to  find  only orthog(cid:173)\nonal demixing matrices, the greater tolerance of demixing by a gradient descent technique \nconcerning noise and incorrect sphering are demonstrated. The application of second order \nblind separation  of sources to optical  imaging data shows that these techniques  represent \nan  important alternative to the conventional approach, bandpass filtering followed by sum(cid:173)\nmation of the image  stack,  for extraction  of neural  activity  maps.  Vessel  artifacts can  be \nseparated from the mapping component better than using classical approaches.  The spatial \ndecorrelation algorithms are very  well adapted to the optical imaging task, because of their \nuse of spatial smoothness properties of the mapping and other biophysical components. \n\nAn  important field  for future research  concerning BSS  algorithms is  the incorporation of \nprior knowledge  about  sources  and  the  mixing  process,  e.g.  that  the  mixing  has  to  be \ncausal:  the  mapping  signal  cannot occur before the  stimulus is  presented.  Assumptions \n\n\fApplication of BSS to Optical Recording of Brain Activity \n\n955 \n\nabout the  time course  of signals could  also  be helpful,  as  well  as  knowledge about their \nspatial  statistics.  Smearing and  scattering limit the resolution of recordings of biological \ncomponents, and, depending on the wavelength of the light used for illumination, the map(cid:173)\nping component constitutes only  a  certain percentage of the changes  in  total  light reflec(cid:173)\ntions. \n\nAcknowledgments \n\nThis work has been supported by the Wellcome Trust (050080IZJ97). \n\nReferences \n\n[I]  T.  Bonhoeffer and  A.  Grinvald.  Optical imaging based on intrinsic  signals:  The methodology. \nIn  A.  Toga and J.  C.  Maziotta, editors, Brain mapping:  The methods, pages 55-97, San Diego, \nCA,  1996. Academic Press, Inc. \n\n[2]  G.  G.  Blasdel and G.  Salama.  Voltage-sensitive dyes reveal  a modular organization in  monkey \n\nstriate cortex.  Nature, 321 :579-585, 1986. \n\n[3]  G. G.  Blasdel.  Differential imaging of ocular dominance and orientation selectivity  in  monkey \n\nstriate cortex. 1.  Neurosci., 12:3115-3138, 1992. \n\n[4]  M.  Stetter,  T.  Otto,  T.  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Jutten, and P.  Louba(cid:173)\nton, editors, Proceedings of the 1. lCA99 Workshop, Aussois, volume  I, pages 87-92, 1999. \n\n[13]  B.-U.  Koehler and R.  Orglmeister.  Independent component analysis using autoregressive mod(cid:173)\n\nels.  In  1.-F.  Cardoso, C.  Jutten, and P.  Loubaton, editors, Proceedings of the lCA99 workshop, \nvolume  I, pages 359-363, 1999. \n\n\f", "award": [], "sourceid": 1662, "authors": [{"given_name": "Holger", "family_name": "Schoner", "institution": null}, {"given_name": "Martin", "family_name": "Stetter", "institution": null}, {"given_name": "Ingo", "family_name": "Schie\u00dfl", "institution": null}, {"given_name": "John", "family_name": "Mayhew", "institution": null}, {"given_name": "Jennifer", "family_name": "Lund", "institution": null}, {"given_name": "Niall", "family_name": "McLoughlin", "institution": null}, {"given_name": "Klaus", "family_name": "Obermayer", "institution": null}]}