{"title": "Recurrent Cortical Amplification Produces Complex Cell Responses", "book": "Advances in Neural Information Processing Systems", "page_first": 90, "page_last": 96, "abstract": null, "full_text": "Recurrent Cortical Amplification Produces \n\nComplex Cell Responses \n\nFrances S. Chance~ Sacha B. Nelson~ and L. F. Abbott \n\nVolen Center and Department of Biology \n\nBrandeis University \nWaltham, MA 02454 \n\nAbstract \n\nCortical amplification has been proposed as a mechanism for enhancing \nthe selectivity of neurons in the primary visual  cortex.  Less appreciated \nis the fact that the same form of amplification can also be used to de-tune \nor  broaden selectivity.  Using  a  network  model  with  recurrent  cortical \ncircuitry,  we  propose  that the  spatial  phase  invariance  of complex  cell \nresponses  arises  through  recurrent  amplification  of feedforward  input. \nNeurons in  the network respond  like simple cells at low  gain and com(cid:173)\nplex ceUs at high  gain.  Similar recurrent  mechanisms  may  playa role \nin generating invariant representations of feedforward input elsewhere in \nthe visual processing pathway. \n\n1 \n\nINTRODUCTION \n\nSynaptic input to neurons in  the primary visual cortex is primarily recurrent, arising from \nother cortical cells. The dominance of this type of connection suggests that it may play an \nimportant role in cortical information processing.  Previous studies proposed that recurrent \nconnections amplify weak feedforward input to the cortex (Douglas et aI., 1995) and selec(cid:173)\ntively amplify tuning for specific stimulus characteristics, such as orientation or direction \nof movement (Douglas et aI., 1995; Ben-Yishai et aI., 1995; Somers et aI., 1995; Sompolin(cid:173)\nsky and Shapley, 1997).  Cortical cooling and shocking experiments provide evidence that \nthere is cortical amplification through recurrent connections, but they do not show increases \nin orientation or direction selectivity as a result of this amplification (Ferster et aI.,  19%; \nChung and  Ferster,  1998).  Recurrent connections can also decrease  neuronal  selectivity \nthrough the same form of amplification, generating responses that are insensitive to certain \nstimulus features.  Although the ability to sharpen tuning may be an  important feature  in \ncortical processing, the capacity to broaden tuning for particular stimulus attributes is also \ndesirable. \n\nNeurons  in  the  primary  visual  cortex  can  be  divided  into  two  classes  based on  their re-\n\n\fRecurrent Cortical Amplification Produces Complex Cell Responses \n\n91 \n\nsponses  to  visual  stimuli  such  as  counterphase  and  drifting  sinusoidal  gratings.  Sim(cid:173)\nple  cells  show  tuning  for  orientation,  spatial  frequency,  and  spatial  phase  of a  grating \n(Movshon  et  aI.,  1978a).  Complex  cells  exhibit orientation  and  spatial  frequency  tun(cid:173)\ning, but are insensitive to spatial  phase (Movshon et aI.,  1978b).  A counterphase grating, \ns(x, t)  =  cos(Kx - cp) cos(wt), is one in which the spatial phase, CP,and spatial frequency, \nK, are held constant but the contrast, s(x, t), varies sinusoidally in time at some frequency \nw.  In response to a counterphase grating, the activity of a simple cell oscillates at the same \nfrequency as the stimulus, w. A complex cell response is modulated at twice the frequency, \n2w.  To create a drifting grating of frequency 1/, s( x, t) = cos( K x - I/t), the  spatial phase \nand spatial frequency are held constant but the grating is moved at velocity 1// K. A simple \ncell  response  to a  drifting  grating  is  highly  modulated at frequency  1/,  while  a  complex \ncell  response to a drifting grating is elevated but relatively unmodulated.  The differences \nbetween complex and simple cell  responses are a direct consequence of the complex cell \nspatial phase insensitivity. \n\nPrevious models of complex cells generate spatial-phase invariant responses through con(cid:173)\nverging sets of feedforward inputs with a wide range of spatial phase preferences but similar \norientation and spatial frequency selectivities (Hubel  and Wiesel, 1962; Mel  et aI.,  1998). \nThese models do not incorporate recurrent connections between complex cells, which are \nknown to be particularly strong (Toyama et al.,  1981).  We  propose that  the  spatial  phase \ninvariance of complex cell  responses can arise from  a broadening of spatial  phase tuning \nby cortical amplification (Chance et aI.,  1998).  The model  neurons exhibit simple cell  be(cid:173)\nhavior when weakly coupled and complex cell behavior when strongly coupled, suggesting \nthat the two classes of neurons in the primary visual cortex may arise from the same basic \ncortical circuit. \n\n2  THE MODEL \n\nThe  activity  of neuron  i  in  the  model  network  is  characterized by a  firing  rate  rio  Each \nneuron  sums feedforward and  recurrent input and  responds as described by  the  standard \nrate-model equation \n\ndri \n\nTrdi = Ii + L.J Wijrj - rio \n\n~ \n\nIi  represents the feedforward input to cell  i, Wij is the weight of the synapse from neuron \nj  to neuron i, and Tr  is a time constant.  Previous studies have suggested that, for a neuron \nreceiving many  inputs,  Tr  is  small, closer to a synaptic time  constant than  the  membrane \ntime constant (Ben-Yishai et al., 1995; Treves, 1993). Thus we choose Tr  = 1 ms. \nThe feedforward input describes the  response of a simple cell with a Gabor receptive field \n\nIi  =  [/ dxGi(x) 100 dt' H(t')s(x, t - t')] + ' \n\nwhere s(x, t)  represents the contrast function of the  visual  stimulus and the  notation  [ 1+ \nindicates rectification. The temporal response function is (Adelson and Bergen, 1985) \n\nH(t')  =  exp(-at')  - - - - -\n\n(  (at')5 \n\n5! \n\n(at')7) \n\n7 ! '  \n\nwhere we use a  = l/ms. The spatial filter is a Gabor function, \nG =  exp ( - 2:; ) cos(kix - <Pi)' \n\nwhere  Ui  determines  the  spatial  extent  of the  receptive  field,  ki  is  the  preferred spatial \nfrequency,  and <Pi  is  the  preferred spatial  phase.  The values of <Pi  are equally distributed \n\n\f92 \n\nF  S.  Chance, S.  B. Nelson and L. F  Abbott \n\nover the  interval  [-180 0 ,  1800 ) .  To give  the  neurons a  realistic  bandwidth, (j i  is  chosen \nsuch that ki(ji  = 2.5 . Initially we consider a simplified case in which  k i  = 1 for all  cells. \nLater we consider the spatial frequency selectivity of neurons in the network and allow the \nvalue of ki  to range from 0 to 3.5 cycles/deg. \n\nIn  this  paper we assume that  the  model  network describes one  orientation column of the \nprimary  visual  cortex, and thus all  neurons have  the same orientation tuning.  All  stimuli \nare of the optimal orientation for the network. \n\nSpatial phase tuning is selectively broadened in the model  because the strength of a recur(cid:173)\nrent connection  between two neurons  is  independent of the  spatial  phase  selectivities of \ntheir feedforward inputs. In the model with all ki  = 1, the recurrent input is determined by \n\n9 \n\nWij =  (N -1) ' \n\nfor all  i  =I  j. N  is the number of cells in the network, and 0  ~ 9  < gmax, where gmax  is \nthe largest value of 9 for which the network remains stable.  In this case gmax  =  1. \n\n3  RESULTS \nThe steady-state solution of the  rate-model  equation is  given  by Ti  =  Ii + L  WijTj .  To \nsolve this equation, we express the  rates and feedfoward inputs in  terms of a complete set \nof eigenvectors ~r of the recurrent weight matrix, L  Wij~r = ~/.l~r for I-L  = 1,2, ... ,N, \nwhere ~/.l are the eigenvalues. The solution is then \n\nThis  equation  displays  the  phenomenon  of cortical  amplification  if one  or more  of the \neigenvalues is near one. If we assume only one eigenvalue, ~1 , is close to one, the factor 1-\n~1 in the denominator causes the I-L  = 1 term to dominate and we find Ti  ~ ~t L  Ij~J (1  -\n~1) -1.  The  input combination  L  Ij~J dominates  the  response,  determining  selectivity, \nand this mode is amplified by a factor 1/(1 - ~1)' We  refer to this amplification factor as \nthe cortical  gain. \nIn  the case where W ij  =  g/(N - 1) for i  =I  j, the largest eigenvalue is  ~1 =  9 and the \ncorresponding eigenvector has all components equal to each other.  For 9  near one, the  re(cid:173)\ncurrent input to neuron i is then proportional to Lj [cos(~-1>j)]+ which, for large numbers \nof cells with uniformly placed preferred spatial phases 1>i, is approximately independent of \n~, the  spatial  phase of the stimulus.  When 9  is near zero, the network is at low  gain and \nthe response of neuron i  is roughly proportional to its feedforward input,  [cos(~ - 1>j)]+, \nand is sensitive to spatial  phase. \n\nThe response properties of simple and complex cells to drifting and counterphase gratings \nare duplicated by the model neuron, as shown in figure  1.  For low gain (gain = 1, top panels \nof figures  lA and  IB), the  neuron acts as a simple cell and its activity is  modulated at the \nsame frequency  as  the  stimulus (w for counterphase gratings and  v  for drifting gratings). \nAt high  gain  (gain = 20),  the  neuron  responds  like a  complex cell, exhibiting frequency \ndoubling  in  the  response  to  a  counterphase  grating (bottom  panel  of Figure  1A)  and  an \nelevated  DC  response  to a drifting grating  (bottom  panel,  Figure  IB).  Intermediate  gain \n(gain = 5) produces intermediate behavior (middle panels). \nThe  basis  of this  model  is that  the  amplified  mode  is  independent  of spatial  phase.  If \nthe amplified mode depends on spatial  frequency or orientation, neurons at high gain can \nbe selective for these attributes.  To  show  that the  model  can  retain  selectivity for other \n\n\fRecurrent Cortical Amplification Produces Complex Cell Responses \n\n93 \n\n500 \n\n1000 \n\n500 \n\n1000 \n\n500 \n\n1000 \n\ntime (ms) \n\n1~~1VY)!\\< \n\no \n\n500 \n\n1000 \n\ntime (ms) \n\nFigure  1:  The effects of recurrent input on the responses of a neuron in the model network. \nThe responses  of one  neuron  to  a  2  Hz counterphase  grating  (A)  and to a  2 Hz drifting \ngrating (B) are shown for different levels of network gain.  From top to bottom in A and B, \nthe gain of the network is one, five, and twenty. \n\nstimulus characteristics while maintaining spatial phase insensitivity, we allowed the spatial \nfrequency  selectivity  which each  neuron  receives from  its feedforward  input,  ki' to  vary \nfrom  neuron to neuron and also modified  the  recurrent weight matrix so that the  strength \nof the connection between two neurons, i  and j, depends on  ki  - kj .  The dependence is \nmodeled as a difference of Gaussians, so the recurrent weight matrix is now \n((ki  - kj )2)] \n-\n\n((ki  - kj )2) \n-\n\nW ij =  (N -1)  2exp \n\n9 \n\n[ \n\n20'; \n\n. \n\n20'~ \n\n- exp \n\nThus  neurons  that receive feedforward  input  tuned for  similar spatial  frequencies  excite \neach other and neurons that receive  very differently  tuned feedforward input inhibit each \nother. This produces complex cells that are tuned to a variety of spatial frequencies, but are \nstill insensitive to spatial phase (see figure  2).  The spatial frequency tuning curve width is \nprimarily determined by 0' c  =  0.5 cycle/deg and 0' s  =  1 cycle/deg. \n\nCells  within  the  same  network  do  not  have  to  exhibit  the  same  level  of gain.  In  previ(cid:173)\nous figures,  the  gain of the  network was determined  by  a parameter 9  that described the \nstrength of all  the  connections  between neurons.  In  figure  3, the  recurrent input to cell  i \nis  determined by W ij  = gi/(N - 1), where the values of gi  are chosen randomly within \nthe allowed range.  The gain of each neuron depends on the value of gi for that neuron. As \nshown  in  figure  3, a  range of complex and simple cell  behaviors now  coexist within  the \nsame network. \n\n4  DISCUSSION \n\nIn the recurrent model we have presented, as in Hubel and Wiesel's feedforward model, the \nfeedforward input to a complex cell arises from simple cells. Measurements by Alonso and \n\n\f94 \n\nF  S.  Chance,  S.  B. Nelson and L.  F  Abbott \n\nA ~ 100 \n\n(/) \n\nc: o a. \n~  50 \nx ctS \nE \n#. \n\nO-+-~r----r--~----' \n180 \n-180 \n\n-90  o \n90 \nphase (deg) \n\no \n3 \nspatial frequency (cyc/deg) \n\n2 \n\n1 \n\nFigure 2:  Neurons in  a high-gain network can be selective for spatial frequency while re(cid:173)\nmaining insensitive to  spatial  phase.  Both spatial  phase and spatial frequency  tuning are \nincluded in the feedforward input.  A) The spatial phase tuning curves of three representa(cid:173)\ntive neurons from a high-gain network.  B) The spatial frequency tuning curves of the same \nthree neurons as in A. \n\nMartinez (1998) support this circuitry.  However, direct excitatory  input to complex cells \narising from the LGN has also been reported (Hoffman and Stone, 1971; Singer et aI., 1975; \nFerster  and  Lindstrom,  1983).  Supporting  these  measurements  is  evidence  that  certain \nstimuli can excite complex cells without strong excitation of simple cells (Hammond and \nMackay, 1975, 1977; Movshon, 1975) and also that complex cells still respond when simple \ncells are  silenced  (Malpeli,  1983;  Malpeli  et ai,  1986;  Mignard and  Malpeli,  1991).  In \naccordance with this, the weak feedforward simple cell input in the recurrent model  could \nprobably be replaced by direct LGN input, as in the feedforward model of Mel et al. (1998). \n\nThe proposed model makes definite predictions about complex cell responses. If the phase(cid:173)\ninvariance  of complex  cell  responses  is  due  to  recurrent interactions,  manipulations  that \nmodify the balance between feedforward and recurrent drive should change the  nature of \nthe  responses  in  a predictable manner.  The model  predicts that blocking local  excitatory \nconnections should turn complex  cells  into simple cells.  Conversely,  manipulations  that \nincrease cortical  gain  should make simple cells act more  like complex cells.  One way to \nincrease cortical  gain may be to block or partially block inhibition since this increases the \ninfluence  of excitatory  recurrent  connections.  Experiments  along  these  lines  have  been \nperformed, and blockade of inhibition does indeed cause simple cells to take  on complex \ncell properties (Sillito, 1975; Shulz et aI., 1993). \n\nIn a previous study, Hawken, Shapley, and Grosof (1996) noted that the temporal frequency \ntuning  curves  for  complex  cells  are  narrower for  counterphase  stimuli  than  for  drifting \nstimuli.  The recurrent model  reproduces  this  result as long as  the  integration of synaptic \ninputs depends  on  temporal  frequency.  Such a dependence  is  provided, for example, by \nshort-term  synaptic  depression  (Chance  et  aI.,  1998).  Hubel  and  Wiesel's  feedforward \nmodel (1962) does not reproduce this effect, even with synaptic depression at the synapses. \n\nWe have presented a model of primary visual cortex in which complex cell  response char(cid:173)\nacteristics arise from  recurrent amplification  of simple cell  responses.  The complex  cell \nresponses  in the  high  gain  regime  arise  because recurrent connections selectively  deam(cid:173)\nplify selectivity for spatial  phase.  Thus recurrent connections can act to generate invariant \nrepresentation of input data. A similar mechanism could be used to produce responses that \nare independent of other stimulus attributes, such as size or orientation. Given the ubiquity \nof invariant representations in the  visual  pathway, this  mechanism  may  have  widespread \nuse. \n\n\fRecurrent Cortical Amplification Produces Complex Cell Responses \n\n95 \n\n1~~~ \n\no \n\n500 \n\n1000 \n\n1~~~ \n\no \n\n500 \n\n1000 \n\ntime (ms) \n\n~1001JV\\M \no -+-------,.--------. \n\neft.  50 \n\no \n\n500 \n\n1000 \n\ntime (ms) \n\nFigure 3:  Responses to a 4 Hz drifting grating of four neurons from  a large  network con(cid:173)\nsisting of a mixture of simple and complex cells. The two traces on the left represent simple \ncells and the two traces on the right represent complex cells. \n\nAcknowledgements \n\nResearch supported by the Sloan Center for Theoretical Neurobiology at Brandeis Univer(cid:173)\nsity, the  National  Science Foundation  (DMS-95-03261),  the  W.M.  Keck  Foundation, the \nNational Eye Institute (EY-11116), and the Alfred P.  Sloan Foundation. \n\nReferences \n\nAdelson, E. H. & Bergen, J. R. Spatiotemporal energy models for the perception of motion. \n1. Opt. Soc. Am. A.  2,284-299 (1985) \n\nAlonso, J-M. &  Martinez, L. M. Functional connectivity between simple cells and complex \ncells in cat striate cortex. Nature Neuroscience 1,395-403 (1998) \n\nBen-Yishai, R., Bar-Or, L. & Sompolinsky, H. Theory of orientation tuning in visual cortex. \nProc. Natl. Acad. Sci.  USA  92,3844-3848 (1995) \n\nChance  F.  S.,  Nelson  S.  B.  &  Abbott  L.  F.  Complex cells as  cortically amplified  simple \ncells. 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Network 4, 259-284 (1993) \n\n\f", "award": [], "sourceid": 1526, "authors": [{"given_name": "Frances", "family_name": "Chance", "institution": null}, {"given_name": "Sacha", "family_name": "Nelson", "institution": null}, {"given_name": "L.", "family_name": "Abbott", "institution": null}]}