{"title": "Nonlinear Processing in LGN Neurons", "book": "Advances in Neural Information Processing Systems", "page_first": 1443, "page_last": 1450, "abstract": "", "full_text": "Nonlinear processing in LGN neurons \n\n\u00f7 \n\n \n \n \nSmith-Kettlewell Eye Research Institute \n\nVincent Bonin*\n\n2318 Fillmore Street \n\nSan Francisco, CA 94115, USA \n\n \n\n{vincent,valerio,matteo}@ski.org \n\n \n\n,  Valerio Mante and Matteo Carandini \n\nInstitute of Neuroinformatics \n\nUniversity of Zurich and ETH Zurich \n\nWinterthurerstrasse 190 \n\nCH-8046 Zurich, Switzerland\n\nAbstract \n\nAccording  to  a  widely  held  view,  neurons  in  lateral  geniculate \nnucleus (LGN) operate on visual stimuli in a linear fashion. There \nis  ample  evidence,  however,  that  LGN  responses  are  not  entirely \nlinear.  To  account  for  nonlinearities  we  propose  a  model  that \nsynthesizes  more  than  30  years  of  research  in  the  field.  Model \nneurons  have  a  linear  receptive  field,  and  a  nonlinear,  divisive \nsuppressive field. The suppressive field computes local root-mean-\nsquare  contrast.  To  test  this  model  we  recorded  responses  from \nLGN of anesthetized paralyzed cats. We estimate model parameters \nfrom  a  basic  set  of  measurements  and  show  that  the  model  can \naccurately  predict  responses  to  novel  stimuli.  The  model  might \nserve  as  the  new  standard  model  of  LGN  responses.  It  specifies \nhow  visual  processing  in  LGN  involves  both  linear  filtering  and \ndivisive gain control. \n\n1  Introduction \n\nAccording  to  a  widely  held  view,  neurons  in  lateral  geniculate  nucleus  (LGN) \noperate  linearly  (Cai  et  al.,  1997;  Dan  et  al.,  1996).  Their  response  L(t)  is  the \nconvolution of the map of stimulus contrast S(x,t) with a receptive field F(x,t): \n\n[\n= \u2217\n\nS F\n\n](\n\n)\n\n \n\nt0\n,\n\n( )\nL t\n\nThe receptive field F(x,t) is typically taken to be a difference of Gaussians in space \n(Rodieck, 1965) and a difference of Gamma functions in time (Cai et al., 1997). \nThis linear model accurately predicts the selectivity of responses for spatiotemporal \nfrequency as  measured  with gratings  (Cai et al.,  1997; Enroth-Cugell and  Robson, \n1966).  It  also  predicts  the  main  features  of  responses  to  complex  dynamic  video \nsequences (Dan et al., 1996). \n\n\fs\n/\ns\ne\nk\ni\np\ns\n \n\n0\n5\n1\n\n \n\nData\nModel\n\n \n\nFigure 1. Response of an LGN neuron to a dynamic video sequence along with the \nprediction made by the linear model. Stimuli were sequences from Walt Disney\u2019s \n\n\u201cTarzan\u201d. From Mante et al. (2002). \n\nThe  linear  model,  however,  suffers  from  limitations.  For  example,  consider  the \nresponse of an LGN neuron to a complex dynamic video sequences (Figure 1). The \nresponse is characterized by long periods of relative silence interspersed with brief \nevents of high firing rate (Figure 1, thick traces). The linear model  (Figure 1, thin \ntraces) successfully predicts the timing of these firing events but fails to account for \ntheir magnitude (Mante et al., 2002).  \nThe limitations of the linear model are not surprising since there is ample evidence \nthat  LGN  responses  are  nonlinear.  For  instance,  responses  to  drifting  gratings \nsaturate as contrast is increased (Sclar et al., 1990) and are reduced, or masked, by \nsuperposition  of  a  second  grating  (Bonin  et  al.,  2002).  Moreover,  responses  are \nselective for stimulus size (Cleland et al., 1983; Hubel and Wiesel, 1961; Jones and \nSillito, 1991) in a nonlinear manner (Solomon et al., 2002).  \nWe  propose  that  these  and  other  nonlinearities  can  be  explained  by  a  nonlinear \nmodel  incorporating  a  nonlinear  suppressive  field.  The  qualitative  notion  of  a \nsuppressive  field  was  proposed  three  decades  ago  by  Levick  and  collaborators \n(1972).  We  propose  that  the  suppressive  field  computes  local  root-mean-square \ncontrast, and operates divisively on the receptive field output.  \n\nBasic elements  of  this  model appeared  in  studies  of contrast  gain  control  in  retina \n(Shapley  and  Victor,  1978)  and  in  primary  visual  cortex  (Cavanaugh  et  al.,  2002; \nHeeger,  1992;  Schwartz  and  Simoncelli,  2001).  Some  of  these  notions  have  been \napplied to  LGN  (Solomon  et  al.,  2002),  to  fit responses  to  a limited  set  of  stimuli \nwith  tailored  parameter  sets.  Here  we  show  that  a  single  model  with  fixed \nparameters predicts responses to a broad range of stimuli.  \n\n2  Model \n\nIn the model (Figure 2), the linear response of the receptive field L(t) is divided by \nthe output of the suppressive field. The latter is a measure of local root-mean-square \ncontrast clocal. The result of the division is a generator potential \n\n( )\nV t\n\n=\n\nV\n\nmax\n\n( )\nL t\n+\nc\n\nlocal\n\nc\n50\n\n, \n\nwhere c50 is a constant.  \n\n\f \n\nF(x,t)\n\nReceptive Field\n\nL(t)\n\nStimulus\n\nS(x,t)\n\nH(x,t)\n\nS*(x,t)\n\nclocal\n\nFilter\n\nSuppressive Field\n\nV0\n\nR(t)\n\nRectification\n\nc50\n\nFiring\nrate\n\n \n\nFigure 2.  Nonlinear model of LGN responses. \n\nThe suppressive field operates on a filtered version of the stimulus, S*=S*H, where \nH is a linear filter and * denotes convolution. The squared output of the suppressive \nfield is the local mean square (the local variance) of the filtered stimulus: \n\nc\n\n2\nlocal\n\n= \u222b\u222b\n\n*S\n\n(\n\n)\nx\n, G\nt\n\n2\n\n( )\nx\n\nx\nd dt\n\n, \n\nwhere G(x) is a 2-dimensional Gaussian.  \nFiring rate is a rectified version of generator potential, with threshold Vthresh:   \n\n( )\nR t\n\n=\n\n\uf8ef\n( )\nV t\n\uf8f0\n\n\u2212\n\nV\nthresh\n\n\uf8fa\n\uf8fb . \n+\n\nTo  test  the  nonlinear  model,  we  recorded  responses  from  neurons  in  the  LGN  of \nanesthetized paralyzed cats. Methods for these recordings were described elsewhere \n(Freeman et al., 2002).  \n\n3  Results \n\nWe proceed in two steps: first we estimate model parameters by fitting the model to \na  large  set  of  canonical  data;  second  we  fix  model  parameters  and  evaluate  the \nmodel by predicting responses to a novel set of stimuli.  \n \n\nA\n\nB\n\n60\n\n40\n\n20\n\n)\ns\n/\ns\ne\nk\ni\np\ns\n(\n \n \ne\ns\nn\no\np\ns\ne\nR\n\n0\n0.01\n\n0.2\n\nSpatial Frequency (cpd)\n\n)\ns\n/\ns\ne\nk\ni\np\ns\n(\n \ne\ns\nn\no\np\ns\ne\nR\n\n4\n\n80\n\n60\n\n40\n\n20\n\n0\n0.5\n\n5\n\n50\n\nTemporal Frequency (Hz)\n\nFigure 3. Estimating the receptive field in an example LGN cell. Stimuli are \ngratings varying in spatial (A) and temporal (B) frequency. Responses are the \n\nharmonic component of spike trains at the grating temporal frequency. Error bars \n\nrepresent standard deviation of responses. Curves indicate model fit.  \n\n \n\n\fA\n\n)\ns\n/\ns\ne\nk\ni\np\ns\n(\n \ne\ns\nn\no\np\ns\ne\nR\n\n100\n\n50\n\n0\n\nC\n\n)\ns\n/\ns\ne\nk\ni\np\ns\n(\n \ne\ns\nn\no\np\ns\ne\nR\n\n100\n80\n60\n40\n20\n0\n0.5\n\n \n\nB\n\n100\n80\n60\n40\n20\n\n)\ns\n/\ns\ne\nk\ni\np\ns\n(\n \ne\ns\nn\no\np\ns\ne\nR\n\n0\n\n1.00\n\n0.25 0.50 0.75 1.00\n\nMask contrast\n\n0.25\n\n0.50\n\n0.75\n\nTest contrast\n\nD\n\n)\ns\n/\ns\ne\nk\ni\np\ns\n(\n \ne\ns\nn\no\np\ns\ne\nR\n\n100\n80\n60\n40\n20\n0\n0.01\n4.00\nMask spatial frequency (cpd)\n\n0.20\n\n \n\n4.0\n\n32.0\n\nMask diameter (deg)\n\nFigure 4. Estimating the suppressive field in the example LGN cell. Stimuli are \n\nsums of a test grating and a mask grating. Responses are the harmonic component of \n\nspike trains at the temporal frequency of test. A: Responses to test alone. B-D: \n\nResponses to test+mask as function of three mask attributes: contrast (B), diameter \n\n(C) and spatial frequency (D). Gray areas indicate baseline response (test alone, \n\n50% contrast). Dashed curves are predictions of linear model. Solid curves indicate \n\nfit of nonlinear model. \n\n3 . 1  Cha r a c te ri zi ng   t he  r ec e pti v e  fi e l d \n\nWe  obtain  the  parameters  of  the  receptive  field  F(x,t)  from  responses  to  large \ndrifting  gratings  (Figure  3).  These  stimuli  elicit  approximately  constant  output  in \nthe suppressive field, so they allow us to characterize the receptive field. Responses \nto different spatial frequencies constrain F(x,t) in space (Figure 3A). Responses to \ndifferent temporal frequencies constrain F(x,t) in time (Figure 3B).   \n\n3 . 2  Cha r a c te ri zi ng   t he   suppre ssi v e  f ie l d \n\nTo characterize the divisive stage, we start by measuring how responses saturate at \nhigh contrast (Figure 4A). A linear model cannot account for this contrast saturation \n(Figure  4A,  dashed curve). The  nonlinear model  (Figure  4A,  solid  curve) captures \nsaturation because increases in receptive field output are attenuated by increases in \nsuppressive  field  output.  At  low  contrast,  no  saturation  is  observed  because  the \noutput of the suppressive field is dominated by the constant c50. From these data we \nestimate the value of c50. \nTo obtain the parameters of the suppressive field, we recorded responses to sums of \ntwo drifting gratings (Figure 4B-D): an optimal test  grating at 50% contrast, which \nelicits  a  large  baseline  response,  and  a  mask  grating  that  modulates  this  response. \nTest  and  mask  temporal  frequencies  are  incommensurate  so  that  they  temporally \nlabel  a  test  response  (at  the  frequency  of  the  test)  and  a  mask  response  (at  the \n\n\f \n\nfrequency of the mask) (Bonds, 1989). We vary mask attributes and study how they \naffect the test responses.  \nIncreasing mask contrast progressively suppresses responses (Figure 4B). The linear \nmodel fails to account for this suppression (Figure 4B, dashed curve). The nonlinear \nmodel  (Figure  4B,  solid  curve)  captures  it  because  increasing  mask  contrast \nincreases  the  suppressive  field  output  while  the  receptive  field  output  (at  the \ntemporal frequency of the test) remains constant. With masks of low contrast there \nis little suppression because the output of the suppressive field is dominated by the \nconstant c50.  \nSimilar effects are seen if we increase mask diameter. Responses decrease until they \nreach a plateau (Figure 4C). A linear model predicts no decrease (Figure 4C, dashed \ncurve). The nonlinear model (Figure 4C, solid curve) captures it because increasing \nmask  diameter  increases  the  suppressive  field  output  while  it  does  not  affect  the \nreceptive  field  output.  A  plateau  is  reached  once  masks  extend  beyond  the \nsuppressive  field.  From  these  data  we  estimate  the  size  of  the  Gaussian  envelope \nG(x) of the suppressive field. \nFinally, the strength of suppression depends on mask spatial frequency (Figure 4D). \nAt high frequencies, no suppression is elicited. Reducing spatial frequency increases \nsuppression. This dependence of suppression on spatial frequency is captured in the \nnonlinear  model  by  the  filter  H(x,t).  From  these  data  we  estimate  the  spatial \ncharacteristics  of  the  filter.  From  similar experiments  involving  different  temporal \nfrequencies (not shown), we estimate the filter\u2019s selectivity for temporal frequency. \n\n3 . 3  P re di c ti ng  re spo nse s  t o  no v el   sti m uli  \n\nWe have seen that with a fixed set of parameters the model provides a good fit to a \nlarge set of measurements (Figure 3 and Figure 4). We now test whether the model \npredicts responses to a set of novel stimuli: drifting gratings varying in contrast and \ndiameter. \nResponses  to  high  contrast  stimuli  exhibit  size  tuning  (Figure  5A,  squares):  they \ngrow with size for small diameters, reach a maximum value at intermediate diameter \nand are reduced for large diameters (Jones and Sillito, 1991). Size tuning , however, \nstrongly  depends  on  stimulus  contrast  (Solomon  et  al.,  2002):  no  size  tuning  is \nobserved  at  low  contrast  (Figure  5A,  circles).  The  model  predicts  these  effects \n(Figure  5A,  curves).  For  large,  high  contrast  stimuli  the  output  of  the  suppressive \nfield is  dominated  by clocal,  resulting  in  suppression  of  responses.  At low  contrast,  \nclocal is much smaller than c50, and the suppressive field does not affect responses. \nSimilar considerations can be made by plotting these data as a function of contrast \n(Figure 5B). As predicted by the nonlinear model (Figure 5B, curves), the effect of \nincreasing contrast depends on stimulus size: responses to large stimuli show strong \nsaturation  (Figure  5B,  squares),  whereas  responses  to  small  stimuli  grow  linearly \n(Figure  5B,  circles).  The  model  predicts  these  effects  because  only  large,  high \ncontrast  stimuli  elicit  large  enough  responses  from  the  suppressive  field  to  cause \nsuppression.  For  small,  low  contrast  stimuli,  instead,  the  linear  model  is  a  good \napproximation.  \n\n\fA\n\n100\n\n)\ns\n/\ns\ne\nk\ni\np\ns\n(\n \ne\ns\nn\no\np\ns\ne\nR\n\n80\n\n60\n\n40\n\n20\n\n0\n\nB\n\n0.50\n\n4.00\n\nDiameter (deg)\n\n32.00\n\n0.00 0.25 0.50 0.75 1.00\n\nContrast\n\n \n\n \n\nFigure 5. Predicting responses to novel stimuli in the example LGN cell. Stimuli are \ngratings varying in diameter and contrast, and responses are harmonic component of \nspike trains at grating temporal frequency. Curves show model predictions based on \nparameters as estimated in previous figures, not fitted to these data. A: Responses as \nfunction of diameter for different contrasts. B: Responses as function of contrast for \n\ndifferent diameters.  \n\n3 . 4  M o del   pe r f or m a nc e  \n\nTo  assess  model  performance  across  neurons  we  calculate  the  percentage  of \nvariance  in  the  data  that  is  explained  by  the  model  (see  Freeman  et  al.,  2002  for \nmethods).  \nThe model provides good fits to the data used to characterize the suppressive field \n(Figure  4),  explaining  more  than  90%  of  the  variance  in  the  data  for  9/13  cells \n(Figure 6A).  Model parameters are then held fixed, and the model is used to predict \nresponses  to  gratings  of  different  contrast  and  diameter  (Figure  5).  The  model \nperforms  well,  explaining  in  10/13  neurons  above  90%  of  the  variance  in  these \nnovel data (Figure 6B, shaded histogram). The agreement between the quality of the \nfits  and  the  quality  of  the  predictions  suggests  that  model  parameters  are  well \nconstrained and rules out a role of overfitting in determining the quality of the fits. \nTo further confirm the performance of the model, in an additional 54 cells we ran a \nsubset  of  the  whole  protocol,  involving  only  the  experiment  for  characterizing  the \nreceptive  field  (Figure  3),  and  the  experiment  involving  gratings  of  different \ncontrast  and  diameter  (Figure  5).  For  these  cells  we  estimate the  suppressive  field \nby fitting the model directly to the latter measurements. The model explains above \n90%  of  the  variance  in  these  data  in  20/54  neurons  and  more  than  70%  in  39/54 \nneurons (Figure 6B, white histogram).  \nConsidering the large size of the data set (more than 100 stimuli, requiring several \nhours of recordings per neuron) and the small number of free parameters (only 6 for \nthe  purpose  of  this  work),  the  overall,  quality  of  the  model  predictions  is \nremarkable.  \n\n\fA\n\nB\n\n \n\nEstimating the suppressive field\n\nn=13\n\ns\nl\nl\ne\nc\n \n#\n\n6\n4\n2\n0\n\nSize tuning at different contrasts\n\ns\nl\nl\ne\nc\n \n\n#\n\n15\n10\n5\n0\n\n0\n\nn=54\n\n50\n\nExplained variance (%)\n\n100\n\n \n\nFigure 6. Percentage of variance in data explained by model. A: Experiments to \nestimate the suppressive field. B: Experiments to test the model. Gray histogram \n\nshows quality of predictions. White histogram shows quality of fits. \n\n4  Conclusions \n\nThe  nonlinear  model  provides  a  unified  description  of  visual  processing  in  LGN \nneurons.  Based  on  a  fixed  set  of  parameters,  it  can  predict  both  linear  properties \n(Figure  3),  as  well  as  nonlinear  properties  such  as  contrast  saturation  (Figure  4A) \nand  masking  (Figure  4B-D).  Moreover,  once  the  parameters  are  fixed,  it  predicts \nresponses to novel stimuli (Figure 5).  \nThe model explains why responses are tuned for stimulus size at high contrast but \nnot  at  low  contrast,  and  it  correctly  predicts  that  only  responses  to  large  stimuli \nsaturate with contrast, while responses to small stimuli grow linearly. \nThe model implements a form of contrast gain control. A possible purpose for this \ngain  control  is  to  increase  the  range  of  contrast  that  can  be  transmitted  given  the \nlimited dynamic range of single neurons. Divisive gain control may also play a role \nin population coding: a similar model applied to responses of primary visual cortex \nwas  shown  to  maximize  independence  of  the  responses  across  neurons  (Schwartz \nand Simoncelli, 2001). \nWe  are  working  towards  improving  the  model  in  two  ways.  First,  we  are \ncharacterizing the dynamics of the suppressive field, e.g. to predict how it responds \nto transient stimuli. Second, we are testing the assumption that the suppressive field \ncomputes root-mean-square contrast, a measure that solely depends on the second-\norder moments of the light distribution.   \nOur ultimate goal is to predict responses to complex stimuli such as those shown in \nFigure  1  and  quantify  to  what  degree  the  nonlinear  model  improves  on  the \npredictions of the linear model. Determining the role of visual nonlinearities under \nmore natural stimulation conditions is also critical to understanding their function. \n\nThe  nonlinear  model  synthesizes  more  than  30  years  of  research.  It  is  robust, \ntractable and generalizes to arbitrary stimuli. As a result it might serve as the new \nstandard  model  of  LGN  responses.  Because  the  nonlinearities  we  discussed  are \nalready present in the retina (Shapley and Victor, 1978), and tend to get stronger as \none  ascends  the  visual  hierarchy  (Sclar  et al.,  1990),  it  may  also  be  used  to  study \nhow responses take shape from one stage to another in the visual system.  \n\n\f \n\nAc kno wl e dg me nt s \n\nThis  work  was  supported  by  the  Swiss  National  Science  Foundation  and  by  the \nJames  S  McDonnell  Foundation  21st  Century  Research  Award  in  Bridging  Brain, \nMind & Behavior. \n\nRe f er e nce s \n\nBonds, A. B. (1989). 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Extraclassical  receptive  field \nproperties  of  parvocellular,  magnocellular,  and  koniocellular  cells  in  the  primate  lateral \ngeniculate nucleus. J Neurosci 22, 338-349. \n\n\f", "award": [], "sourceid": 2539, "authors": [{"given_name": "Vincent", "family_name": "Bonin", "institution": null}, {"given_name": "Valerio", "family_name": "Mante", "institution": null}, {"given_name": "Matteo", "family_name": "Carandini", "institution": null}]}