Probing feature selectivity of neurons in primary visual cortex with natural stimuli.

Transcription

1 Pobing featue selectiity of neuons in pimay isual cotex with natual stimuli. T. Shapee -3, H. Sugihaa,3, A.V. Kugansky,3, S. Rebik,3, M. P. Styke -3 and K. D. Mille -3. Sloan-Swatz Cente fo Theoetical Neuobiology, Depatment of Physiology, Keck Cente fo Integatie Neuoscience 3, Uniesity of Califonia, San Fancisco, CA ABSTRACT One way to chaacteize neual featue selectiity is to model the esponse pobability as a nonlinea function of the output of a set of linea filtes applied to incoming signals. Taditionally these linea filtes ae measued by pobing neuons with coelated Gaussian noise ensembles and calculating coelation functions between incoming signals and neual esponses. It is also impotant to deie these filtes in esponse to natual stimuli, which hae been shown to hae stongly non-gaussian spatiotempoal coelations. An infomation-theoetic method has been poposed ecently fo econstucting neual filtes using natual stimuli in which one looks fo filtes whose conolution with the stimulus ensemble accounts fo the imal possible pat of the oeall infomation caied the sequence of neual esponses. Hee we gie a fist-time demonstation of this method on eal neual data, and compae esponses of neuons in cat pimay isual cotex dien with natual stimuli, noise ensembles, and moing gatings. We show that the infomationtheoetic method achiees the same quality of filte econstuction fo natual stimuli as that of well-established whitenoise methods. Majo paametes of neual filtes deied fom noise ensembles and natual stimuli, as well as fom moing gatings ae consistent with one anothe. We find that application of the eese coelation method to natual stimuli ensembles leads to significant distotions in filtes fo a majoity of studied cells with non-zeo eesecoelation filte. Keywods: Pimay isual cotex, infomation theoy,, eceptie fields, eese coelation.. INTRODUCTION Neual featue selectiity has taditionally been studied using stimuli that can be descibed by a small numbe of paametes, so that the input/output function can be measued with sufficient detail. The classic example is the use of bas and gatings to study neuons in pimay isual cotex. Such an appoach is contingent on the fact that neuons ae esponsie to the class of stimuli used [,]. Finding an appopiate class of stimuli may equie a seendipitous discoey, such as the one made by Hubel and Wiesel fo neuons in pimay isual cotex []. Howee it is not clea een at the next stage of isual signal pocessing, what is the most appopiate stimuli class to be used (see, howee, Ref. [3,4]. An altenatie appoach is to use a boad ange of stimuli, and, though some sot of statistical aeaging, to find out what filte is being used by a paticula neuon [-4]. Fom the statistical analysis point of iew, it is most adantageous to use stimuli with known statistical popeties, such as (possibly coelated Gaussian noise. While it may suffice fo the fist few stages of isual signal pocessing, neuons at highe leel, whose stuctue we ae most inteested in, ae left unesponsie by noise ensembles. Neuons at all stages of signal pocessing ae esponsie to natual stimuli, i.e. those stimuli that ae deied fom ou suoundings. It tuns out that such stimuli hae specific statistical popeties, which ay supisingly little with a change of enionment fom which they ae deied [-]. The most obust featues ae stong spatiotempoal coelations with chaacteistic / f spectum, whee f is a spatial fequency, scaling of the pobability distibution as a whole, and stong deiation of the pobability distibution fom a Gaussian one. The non-gaussian chaacte of coelations inalidates many methods deeloped to study neual featue selectiity []. Howee, it may be possible to use ideas fom infomation theoy to bypass the poblem of non- Gaussian effects in natual stimuli, thus allowing thei use in lage-scale systematic neuophysiological ecodings []. We decided to test the newly poposed infomation-theoetic method [] on neuons in pimay isual cotex, whose popeties ae well studied. Each of the neuons has been pobed by and natual stimuli ensembles, so that we can compae the filtes obtained with the infomation-theoetic method to the filtes deied fom the neuon s esponses to. In addition, neual esponses whee pobed with moing (sinusoidal o squae gatings to detemine thei optimal oientation and spatial fequency. To analyze neual esponses to eithe o natual stimulus ensembles we use the famewok of the so-called linea-nonlinea model, accoding to which the pobability of Fluctuations and Noise in Biological, Biophysical, and Biomedical Systems II, edited by Deek Abbott, Segey M. Bezuko, Andas De, Angel Sánchez, Poc. of SPIE Vol. 467 (SPIE, Bellingham, WA, X/4/$ doi:.7/.483

2 a esponse a in ou case gien a stimulus s is some nonlinea function (whose shape is not a pioi specified of pojection of the stimulus s onto a set of filtes ê i, i= k, : P( s = P(f(s ê,..., ˆ s ek ( The most impotant simplification made in Eq. ( is that the numbe of filtes k used by the neuon is much less than the dimensionality of the stimulus itself. Een though neuons may hae many filtes, in this pape we will pesent the esults of simulations fo neuons with up to two filtes. In analyzing data fom neuons in pimay isual cotex, we cay out the analysis as if they had just one filte. We concentate on cells fo which the white-noise filte is non-zeo. This biases ou subset of cells towads simple cells, which ae thought to be well descibed by one filte. Howee ou pimay goal in this wok is to demonstate the use of the infomation-theoetic method on neual data and to eify that the neual filtes it poides ae consistent with othe independent measuements. Unde these cicumstances the use of mostly simple cells is desiable.. METHODS Expeimental ecodings. All expeimental pocedues whee conducted unde a potocol appoed by the Uniesity of Califonia, San Fancisco Committee on Animal Reseach and hae been descibed in detail elsewhee [,3]. Biefly, tains wee ecoded using tetode electodes fom pimay isual cotex of anesthetized adult cats. The wee pesented as two-dimensional spatiotempoal pattens of light intensities on a ideo monito with a efesh ate of Hz. The fame update ate was 6 Hz in the case stimulus ensemble and 3 Hz in the case of a natual stimulus ensemble. In addition cells wee pobed with full-field moing peiodic pattens (gatings. The optimal oientation was detemined fom cell s esponses to a set of spaced oientations and spatial fequency of. cycles/degee and tempoal fequency of Hz. The optimal spatial fequency fo a gien cell was deied fom its esponses to a set of moing gatings of optimal oientation and aiable spatial fequencies (linealy spaced between. and 4 cycles/degee. A cell was classified as simple o complex depending on the atio of the fist Fouie component F of its esponse to a moing gating of optimal oientation and spatial fequency to its mean F : a cell was consideed simple if F /F > and an complex, othewise. To estimate the infomation caied by the aial of one I, an sec long segment of the natual moie was epeated fom - times. isual stimuli s, Filte econstuction fom neual esponses to a white-noise ensemble eese coelation method. Geneally, the poblem of econstucting neual esponses consists of two pats, one of which is to find the undelying linea filtes and the othe one is to find the nonlinea function f( s ê,..., ˆ s ek in Eq. ( that detemines the pobability of a gien stimulus. If stimuli ae dawn fom a Gaussian distibution, then it is possible to use Wick s theoem to show that the poblem of finding the linea filtes sepaates fom that of finding the nonlineaity associated with them. In paticula, in the case of a ensemble, one may obtain one of the linea filtes by computing a coelation function between s and stimuli: e = sp( s P( s, ( ˆSTA which as may be seen fom Eq. ( is computed by taking the aeage stimulus weighted by the numbe of s it elicits and subtacting the aeage stimulus multiplied by the oeall numbe of s, the pocedue also known as the -tiggeed aeage (STA. The magnitude of the filte is ieleant, since escaling of the ecto can be accounted fo by the appopiate escaling of the input/output function ( which conets pojections on the eleant filte into pobability. Theefoe, we will nomalize all of the deied filtes to unit length. Once the filte has been detemined, the nonlinea input/output function can be mapped diectly fom expeimental data, using the Bayes ule: P( s P( s (s ˆ STA P esta f ( s STA = = = (3 P( P( P(s As might be expected such an appoximation is bette fo some classes of cells than othes. Fo neuons in pimay isual cotex, esponses of simple cells ae thought to be descibed by one filte, while esponses of complex cells equie at least two filtes fo thei desciption. Accodingly we find that the amount of infomation that can be accounted fo by pojections onto just one linea filte is significantly highe fo simple cells as opposed to complex. The detailed compaison of simple s. complex cells esponses will be gien in subsequent wok. STA Poc. of SPIE Vol

3 We will be using this expession to map the input/output function fo any filte found and any stimulus ensemble used. Filte econstuction fom neual esponses to a coelated Gaussian ensemble. In the pesence of coelations within the stimulus ensemble, the eese coelation function ( is biased towads those Fouie components that occu most often. Theefoe the neual filte is found by nomalizing expession ( by the coaiance matix of the stimulus ensemble used: = C sp( s P( s, Cij = s i s j - si s j, (4 whee notation êgaussian -estimate notes the fact the filte has been obtained assuming Gaussian statistics fo the input signals. Filte econstuction fom neual esponses to a natual stimuli ensemble most infomatie dimensions. In cases whee the pobability distibution of stimuli deiates significantly fom a Gaussian one, assuming an incoect fom of the nonlineaity leads to incoect estimates fo the linea filtes themseles. One possibility is to e-estimate the nonlineaity using the Bayes ule of Eq. (3 fo each of the tial diections in the stimulus space and find the linea filte used by the neuon though an optimization pocedue. This is in effect what is done when diffeent diections in the stimulus space ae ealuated accoding to the Kullback-Lieble distance between the pobability distibutions of pojections P ( x = δ ( x s and P s ( x = δ ( x s in the a pioi and conditional s ensembles, espectiely [,4]: = P ( x I ( dxp ( x log, ( P ( x whee the subscipt in pobability distibutions notes the fact the they depend on ecto along which they wee ealuated. The infomation I( along a paticula diection quantifies the eleance of this diection fo geneating s. The atio of the pobabilities distibutions P ( x / P ( x in ( gies an estimate of the input/output function f ( s of Eq. (3. If no change occus in the pobability distibution P (x upon obseation of a, then I ( = and the coesponding diection is ieleant fo geneation. The linea filte used by a paticula neuon is detemined though an optimization pocedue whee infomation I( is imized as a function of the ecto. The pesence of local ima necessitate the use of Monte Calo methods, and we implement a simulated annealing scheme, whee infomation is imized in diections of successie gadients of infomation (. Vectos with lowe infomation alues ae accepted with Boltzman pobabilities. We fist pesent the esults of the optimization algoithm in econstucting filtes fo model neuons, which hae that adantage that thei filtes ae known. It is essential that the deiations between the econstucted and model filtes decease with the numbe of s, the paamete that contols signal-to-noise atio. Theefoe the obseed deiations in the econstucted filte ae on aeage due to noise and not to a failue of the optimization pocedue, see below. Upon completion of the optimization scheme, we will compae the infomation I( along a paticula diection to the oeall infomation caied by the aial of one I. The atio of the two quantities descibes the quality of econstuction. Fotunately, the oeall infomation can be estimated in a model-independent way [] fom epeated pesentations of a subset of stimuli in ode to sample the pobability distibution P( s : P( s P( s I = log (6 P( P( s Below we illustate the esults of imizing infomation on two examples taken to mimic popeties of cells in pimay isual cotex. A model neuon with two eleant diections descibing a spatiotempoal eceptie field. In pepaation to analyze esponses of neuons in pimay isual cotex, we hae analyzed a model whee the pobability of a depends on stimulus histoy. The model filte ê is a concatenation of 3 successie spatial filtes, each one of them is a spatial Gabo function, cf. Fig., uppe ow. Een though this paticula example has a space-time sepaable eceptie field, we make no such assumptions duing econstuction. The eleant diection we ae about to find has 4 Poc. of SPIE Vol. 467

4 appoximately independent paametes. In a model, a gien sequence of 3 fames leads to a if pojection s = s eaches a cetain theshold alue s t in the pesence of noise, i.e. a occus if s > st + n, whee n is a Gaussian andom aiable of aiance σ and zeo mean. Thus, the model is specified by the filte (eceptie field ê, the theshold alue of pojection s t, and noise aiance σ. Hee, the theshold alue was set at s t =. and a noise standad deiation σ=. (both paametes ae measued in units of standad deiation of the pojection alue s onto the model filte. These alues set the pobability of a pe fame at.6, which fo fames shown at 3Hz coesponds to a fiing ate of Hz. The total numbe of s acoss all epetitions of the moie was, (the moie duation is appoximately 9min, and it was epeated times. The STA ecto o eese coelation function ( is shown in the middle ow of fames, Figue. It epesents a boadened esion of the undelying eleant diection, a consequence of spatiotempoal coelations pesent in the stimuli. If stimuli wee dawn fom a Gaussian pobability distibution, the effects of coelations could be emoed by multiplying the STA ecto by the inese the a pioi coaiance matix, cf. Eq. (4. As mentioned aboe this pocedue fails because natual scenes hae highe than second ode coelations. In contast, it is possible to obtain a good estimate of the eleant diection by imizing infomation diectly. Fames of the ecoeed eleant diection found by imizing infomation ae shown in the bottom ow in Figue. The quality of econstuction, as judged by the final alue of the pojection was on aeage: e =.8. 7, with pojections of the stimulus ensemble on it accounting fo ˆ ± Figue. Analysis of a model neuon with a spatiotempoal filte (uppe ow and the coesponding nonlinea function descibing the pobability of a gien pojection on the model filte. The pojection alues ae gien in units of thei standad deiation. The STA ecto is shown in fames of the middle ow, with occuing on the ight-most fame. It is boadened in both space and time due to spatiotempoal coelations pesent in the stimulus moie. The coesponding ecoeed nonlineaity is shown to the ight. The econstucted ecto as the most infomatie diection is shown in the bottom ow. The model filte is nomalized to unit length, while the othe two ectos ae shown elatie to thei noise standad deiations. ê ê STA.4 P s e P ( s. - - s 4 P ( ˆ ( s e ˆSTA - s ê s ê STA - - Discete time befoe a I ( / I =.999 ±.. The quality of econstuction in tems of pojection between the econstucted and model filtes impoes as the numbe of inceases, as is discussed in detail below. We hae set the model filte to be equal to zeo on the fame duing which a occus to descibe time delay. This and othe acausal fames (not shown could be used to eify algoithm accuacy, as well as typical eo alues in the econstucted ecto. Fo example, the STA ecto has emaining stuctue ay the time lag, while the econstucted ecto has ey little stuctue at this time lag, as quantified with espect to the noise leel. A model neuon with two eleant diections descibing a spatiotempoal eceptie field. A sequence of s fom a model cell with two eleant diections was simulated by pojecting each of the stimuli on two ectos e and e which ae taken to mimic popeties of complex cells found in pimay isual cotex. These ectos ae shown in twospatial dimensions and at 3 discete time lags befoe a in Figue (a,b. Thei spatial dependence is descibed by the same Gaussian enelope and with π/ diffeence in thei spatial phase (sin and cos. A paticula fame leads to a Poc. of SPIE Vol. 467

5 accoding to a logical OR, that is if eithe s o s exceeds a theshold alue s t in the pesence of noise, whee s =s e and s =s e. Similaly to the peious example of a model simple cell, a is elicited if eithe s >s t + n o s >s t +n is tue. Hee, n and n ae independent Gaussian aiables with zeo mean and aiance σ modeling the effects of neual noise. The maginal pobabilities P(s and P(s ae also shown next to the coesponding ectos. The theshold fo eliciting a was set at. standad deiations of the pojection alues s and s, and the noise leel was. in the same units. The esulting tain with espect to the natual moie had an aeage fiing ate of 4 Hz, with the oeall numbe of s ~8, in the data set. We note that een though a simple logical opeation has been chosen to specify the pobability of a gien the two pojection alues, we do not assume a paticula shape of the input/output function duing the econstuction pocess. Due to the contast-inesion symmety, the STA ecto calculated accoding to Eq. ( would be zeo in the case of a stimulus ensemble. In the case of natual stimuli ensemble, the ecto of STA is boadened in both space and time due the stong spatiotempoal coelations pesent in the ensemble, and not supisingly, beas little esemblance to the undelying eleant dimensions, cf. Figue (c. Hee we show that imizing infomation as a function of diection in the stimulus space allows us to ecoe one of the eleant dimensions een unde moe natual stimulus conditions. The esult of infomation imization is shown in panels (d. The single most infomatie diection accounts on aeage fo I / I =.. of the oeall infomation caied by aial of single, and has pojection of ( ±.83±.3onto the plane of the two eleant ectos ( e ˆ, e ˆ. Similaly to the case of a model simple cell with one eleant dimension, pojection on the eleant plane inceases with inceasing numbe of s, see below. Een though the single most infomatie diection accounts fo about half of the infomation in the caied by time of aial of one, it gies alues of pefeed oientation and spatial fequency which ae simila to those of the undelying two model ectos. P s (a ê.4.. ( (b ê s ê - P( s s ê (c ê STA 6 4. P s ( STA (d - P( s. - - s - - Discete time befoe a Figue. Analysis of a model neuon with two eleant dimensions descibing a spatiotempoal eceptie field. To descibe the time dependence of the eceptie field the time axis has been discetized; 3 sepaate time lags ae consideed, with a occuing on the ight most fame at time lag. The fist and second eleant dimensions of the model neuon ae shown in the two uppe ows, with maginal pobabilities gien pojections of the two ectos. Row (c descibes the spatiotempoal STA with the coesponding pobability gien pojection alues of stimuli on it. The most infomatie ecto is shown in ow (d and has pojection.83±.3 fo a data set with ~8, s. 6 Poc. of SPIE Vol. 467

6 Signal to noise atios of econstuction filtes. Ou estimate of infomation I( along a paticula diection in the stimulus space is lagely limited by the knowledge of the conditional pobability distibution P( s.the numbe of ecoded s N is the numbe of aailable samples fom this pobability distibution. Theefoe one might expect that with inceasing numbe of s, the most infomatie ecto obtained fo a gien sequence will appoach undelying eleant dimensions used by the neuon. This is indeed the case, and in ê the pojection of the (nomalized most infomatie ecto on the case of a neuon with one eleant dimension ê appoaches one in the limit of lage N [,6]: T'[ A ]/ ln, whee A is the Hessian N of infomation ealuated at and tace T ' is taken in the subspace othogonal to the ecto along which Hessian is ealuated. The main conclusion is that the expected eos when imizing infomation ae well behaed and decease as /N with inceasing numbe of s. We emphasize that the expected deiation between the most infomatie ecto and the undelying eleant dimension ae of the same ode of magnitude as eos of the eese coelation method, whee the neual filte is obtained accoding to Eq. (4 [,6]. Since the eese coelation and infomation imization methods gie simila eos when analyzing neual esponses to Gaussian noise ensembles, the eos of the eese coelation method will be lage when it is applied to a non-gaussian stimulus ensemble. In Figue 3 we show how the pojection alue between the econstucted and model ecto inceases (eo deceases when the numbe of epetitions, and theefoe numbe of s, inceases. Most impotantly the econstuction quality of one filte pe cell is simila in the case of a simple and complex cell. Real neuons ae likely to hae moe than one eleant dimension. Howee, we might be able to descibe some of thei neual chaacteistics by finding just one of the eleant filtes fo each neuon, as demonstated in Fig. aboe. Simulations of Figue 3 show that scaling with N.8.6 simple cell complex cell ( ( + I ( / I I ( / I.. (a (b N N Figue 3. (a Analysis of a model simple cell with one eleant diection. The econstuction quality as quantified by inceases to one with inceasing numbe of s N. Regadless of the numbe of s, the most infomatie dimension accounts fo nealy all of the infomation caied by the aial of one (the oeestimate of the faction of infomation explained I ( / I obseed fo small N is due to oefitting. (b Analysis of a model complex cell with two eleant dimensionsê, ê. Similaly to the case of a model neuon with one eleant dimension, the econstuction quality as quantified by ( ( ˆ + e appoaches one with inceasing numbe of s N. Regadless of the numbe of s, the most infomatie dimension accounts fo appoximately half I, as should be expected gien paametes of the model neuon that weighs contibutions fom the two eleant dimensions equally. Poc. of SPIE Vol

7 of pojection alues of the most infomatie diection onto the eleant plane ( e ˆ, e ˆ of a complex cell is simila to that of pojection alues onto the single eleant dimension ê of a simple cell. We also note that the dependence of the infomation alues accounted fo one most infomatie dimension on N is not as stong as the incease in the pojection alue. 3. RESULTS We hae pobed neuons in cat pimay isual cotex with thee diffeent stimuli ensembles: stimuli, natual stimuli, and moing gatings. Fo each of the neuons in ou dataset, we calculate two estimates of the filte used by the neuon fo poducing s: ecto ê is obtained as a -tiggeed aeage fo a white noise stimulus ensemble using Eq. (; ecto natual stimuli ê is obtained as the most infomatie dimension fo a natual stimulus ensemble. Fom the two estimates fo the neual filte we calculate such majo paametes as pefeed oientation and spatial fequency. In addition, an independent estimate fo pefeed oientation and spatial fequency fo each neuon is obtained fom its esponses to moing gatings. Only neuons with non-zeo ê ecto hae been included in the cuent analysis. Ou fist esult is that all thee estimates of the pefeed oientation and spatial fequency ae consistent with one anothe, cf. Figue 4. The ageement between pefeed oientation alue deied fom natual stimuli filte ê and that obtained fom gating esponses is as good as o bette, than ageement natual stimuli between pefeed oientation alue obtained the filte ê thoough a well established method of eese coelation. Similaly, the alues of the pefeed spatial fequencies obtained fom theses thee independent measuements ae consistent with one anothe, cf. Figue 4(b. Thus poiding suppot fo the alidity of looking fo most infomatie dimensions as a method to ecoe neual filtes fo coelated non-gaussian stimuli ensembles, such those deied fom the enionment. We next compae the two estimates of neual filtes ê and ênatual stimuli diectly with each othe. We also apply the eese coelation method with coections fo powe spectum (4 to natual stimuli ensemble. The obtained ecto êgaussian estimate would hae been a alid estimate fo the neual filte if natual stimuli obeyed Gaussian statistics o if a paticula neuon unde study is stictly linea []. We find that the most infomatie ecto ê natual stimuli is always close to ê (in a diect Euclidean sense in the space of light intensities than the coected fo powe spectum eese coelation ecto ê, cf. Figue. One might ask whethe impoement shown in Figue (a fom filte estimate ê to the infomation-theoetic one ê natual stimuli is significant. In Figue (b, we show that in about ~7% of cells this diffeence is significant. To detemine whethe the two methods gie significantly diffeent esults, we calculate distibutions of pojection alues of estimates deied by eithe one of the two methods fom diffeent data segments, and find the mean and standad deiation of pojection alues ˆ egaussian estimate and ˆ natual stimuli enatual stimuli. The diffeence between ectos ê obtained though eese coelation pocedue (4 and ê natual stimuli obtained by imizing infomation is consideed significant if the aeage pojection alue lies below the aeage natual stimuli by moe than one standad deiation. The pecentage does not change when we distinguish between simple and complex cells, howee only cell with non-zeo filte whee included in the subset. 4. DISCUSSION Neual featue selectiity can be studied with espect to aious stimulus ensembles, be it collection of bas, sequence of gatings, o stimuli. Ultimately, we ae only inteested in neual esponses to atificial stimulus ensembles to the degee to which they poide an appoximation of neual esponses in thei woking egime, which fo sensoy neuons mean pocessing of signals deied fom the enionment. Een though one might ague in each 8 Poc. of SPIE Vol. 467

8 (a 6 cells oientation(natual stimuli (b spatial fequency (natual stimuli χ = 47., Q=.9 oientation(gatings χ = 3.69 Q=.98. spatial fequency (gatings oientation ( χ = 44.78, Q=.9 oientation (gatings cells cells 6 cells.8.8 spatial fequency (.6.4. χ = 47.4 Q=.76. spatial fequency (gatings oientation ( spatial fequency (natual stimuli χ = 44., =.96 - oientation (natual stimuli χ = 9.9 Q=.998. spatial fequency ( Figue 4. (a Paiwise compaison of pefeed oientation alues deied fom the natual stimuli filte, the white noise filte, and moing gatings acoss an ensemble of 6 cells that had non-zeo filte. Standad deiations aeaged acoss the ensemble ae shown as eo bas in a cone of each subplot. Solid lines hae slope one. Consistency with a hypothesis that pefeed oientation alues ae the same is ealuated by χ -test, and Q gies the pobability that the obseed discepancies ae chance fluctuations [7]. (b Paiwise compaison of pefeed spatial fequency alues deied fom the natual stimuli filte, the filte, and moing gatings. Again, only cells with non-zeo filte hae been included. paticula case whethe a chosen stimulus ensemble is natual enough, ou pimay pupose in this wok was to demonstate that neual filte popeties can be deied with espect to any, sufficiently diese ensemble, of stimuli that expeimente may think is eleant. Most impotantly, in doing so we do not assume that stimuli ae Gaussian, since ensembles of both isual and auditoy stimuli taken fom ou enionment hae been shown to deiate stongly fom a Gaussian distibution. Ou paticula ensemble of natual stimuli has been obtained by walking with a ideo camea though a pak. What we hae shown is that despite the complicated spatiotempoal coelations pesent in ou ensemble, the neual filtes could be in essence decoelated though the infomation-theoetic pocedue of imizing infomation. To some degee this decoelation can be achieed by such standad methods as eese coelation with powe spectum nomalization Eq. (4. Howee, fo majoity (7% of een simple cells, which ae thought to be well descibed by one filte, these leads to significant filte distotions and could make us oeestimate the diffeence between neual filtes obtained fom neual esponses to and moe natual stimuli ensembles. When compaing filtes deied fom neual esponses to and natual stimulus ensembles, we find that thei majo paametes, such as size, pefeed oientation and spatial fequency ae consistent with one anothe and Poc. of SPIE Vol

9 (a (b.8 natual stimuli * simple complex unclassified.4 * simple complex unclassified natual stimuli ênatual stimuli Figue. (a The filte estimate obtained fo natual stimuli ensemble by imizing infomation is always close to the white-noise filte êwhite noise than the estimate êgaussian estimate obtained fom natual stimuli ensemble by eese coelation pocedue with powe spectum coections (4. (b Diffeences between the two filte estimates êgaussian estimate and ênatual stimuli ae significant fo 69% of simple cells (36 cells total, 69% of complex cells (3 cells total and fo all 8 of unclassified cells, see text fo significance citeion. Only cells with non-zeo filte ê whee included in the analysis. with diect measuements of these paametes obtained with moing gatings. This can be iewed as additional eification of the algoithm since it is unlikely that majo paametes of neual filtes (eceptie fields would dastically change when stimulus ensemble is substituted with anothe. In tems of signal to noise atios of the two filte estimates, we find that the ageement between paametes obtained fom natual stimuli filte and moing gatings is as good as that between paametes obtained fom filte and moing gatings. This suggests that the infomation-theoetic method achiees compaable signal to atios in filte econstuction as those of eese coelation method fo noise ensemble. Analytically, the conegence popeties fo the infomation-theoetic method hae been shown to be compaable those of eese coelation method [] fo the same numbe of ecoded s in a dataset. Aside fom the fact that majo paametes of filtes deied fom and natual stimuli ensemble ae the same, one might notice that the diect dot poduct between the two filtes, cf. Figue a, is often much smalle than could be explained by the obseed noise leel, as judged by how diffeent, on aeage, two filtes could be when calculated fom diffeent stimulus pesentations. This opens a seies of questions as to what paametes of neual filtes do change with the stimulus ensemble, some of which we hope to addess in the subsequent wok. The infomation-theoetic method poides us with means fo a igoous econstuction of neual esponses to natual stimuli. In conclusion, we demonstate fo the fist time on eal neual data that it is possible to econstuct neual filtes fom natual stimuli without assuming the lineaity of neuons unde study. In paticula, the filte estimate obtained by imizing infomation is always close to the filte than that obtained assuming that the cell is completely linea (eese coelation function coected fo non-unifom powe spectum. The method does not equie extaodinay amount of physiological data and pefoms as good as standad methods (eese coelation, -tiggeed aeage gien simila numbe of s. Acoss the population of cells with non-zeo Poc. of SPIE Vol. 467

10 filtes, majo paametes of neual filtes deied fom noise ensembles and natual stimuli wee consistent with one anothe as well as with diect measuements of these paametes obtained with moing gatings. ACKNOWLEDGEMENTS We thank William Bialek fo many helpful suggestions on aious stages of this wok. We ae ey gateful to the UCSF Keck Cente community fo poiding compute time that made this eseach possible. Additional computing esouces wee poided by the National Patneship fo Adanced Computational Infastuctue at the San Diego Supecompute Cente though NSF coopeatie ageement ACI-969. T.S. was suppoted by gants fom the Sloan and Swatz Foundations. We also acknowledge financial suppot fom National Eye Institute though gant R- EY39. REFERENCES. D.H. Hubel and T.N. Wiesel, Receptie fields, binocula inteaction, and functional achitectue in the cat s isual cotex, J. Physiol., 6, D. L. Ringach, G. Sapio, and R. Shapley, A subspace eese-coelation technique fo the study of isual neuons, Vision Res., 37, 4-464, J. L. Gallant, J. Baun, D. C. an Essen, Selectiity fo pola, hypebolic and catesian gatings in macaque isual cotex, Science, 9, -3, J.L. Gallant, C.E. Conno, S. Rakshit, J.W. Lewis, and D.C. an Essen, Neual esponses to pola, hypebolic, and Catesian gatings in aea V4 of the macaque monkey, J. Neuophysiol., 76, , E. de Boe and P. Kuype, Tiggeed coelation, IEEE Tans. Biomed. Eng.,, ( R. R. de Ruyte an Steeninck and W. Bialek, Real-time pefomance of a moement-sensitie neuon in the blowfly isual system: coding and infomation tansfe in shot sequences, Poc. R. Soc. Lond. B, 34, , F. Rieke, D. Waland, R. R. de Ruyte an Steeninck, and W. Bialek, Spikes: Exploing the neual code. MIT pess, Cambidge, N. Benne, W. Bialek, and R. R. de Ruyte an Steeninck, Adaptie escaling optimizes infomation tansmission, Neuon, 6, 69-7,. 9. F. E. Theunissen, K. Sen, and A. J. Doupe, Spectal-tempoal eceptie fields of nonlinea auditoy neuons obtained using natual sounds, J. Neuosci.,, 3-33,.. K. Sen, F. E. Theunissen, and A. J. Doupe, Featue analysis of natual sounds in the songbid auditoy foebain, J. Neuophysiol., 86, 44-48,.. F. E. Theunissen, S.V. Daid, N.C. Singh, A. Hsu, W. E. Vinje, and J. L. Gallant, Estimating spatio-tempoal eceptie fields of auditoy and isual neuons fom thei esponses to natual stimuli, Netwok: Comput. Neual Syst.,, 89-36,.. E. J. Chichilnisky, A simple analysis of neuonal light esponses, Netwok: Comput. Neual Syst (. 3. O. Schwatz, E.J. Chilchilnisky, E.P. Simoncelli, Chaacteizing neual gain contol using -tiggeed coaiance, in Adances in Neual Infomation Pocessing Systems 4, edited by T. G. Dietteich, S. Becke, and Z. Ghahamani, MIT Pess, Cambidge, MA. 4. J. Touyan, B. Liu, and Y. Dan, Isolation of eleant isual featues fom andom stimuli fo cotical complex cells, J. Neuosci.,, 8-88,.. D. J. Field, Relations between the statistics of natual images and the esponse popeties of cotical cells, J. Opt. Soc. Am. 4, ( D. L. Rudeman and W. Bialek, Statistics of natual images: scaling in the woods, Phys. Re. Lett., 73, 84-87, D. L. Rudeman, The statistics of natual images, Netwok: Comput. Neual Syst.,, 7-48, D. J. Field, What is the goal of sensoy coding?, Neual Comp., 6, 9-6, D. W. Dong and J. J. Attick, Statistics of natual time-aying images, Netwok: Comput. Neual Syst., 6, 34-38, 99.. E. Simoncelli and B. A. Olshausen, Natual image statistics and neual epesentation, Annu. Re. Neuosci., 4, 93-6,.. T. Shapee, N.C. Rust, W. Bialek, Analyzing Neual Responses to Natual Signals: Maximally infomatie dimensions, Neual Comp., 6, 3- (4. Poc. of SPIE Vol. 467

11 . R.C. Liu, S. Tzone, S. Rebik and K.D. Mille (, Vaiability and infomation in a neual code of the cat lataeal geniculate nucleus, J. Neuophys., 86, A.A. Emondi, S.P. Rebik, A.V. Kugansky and K.D. Mille, Tacking neuons ecoded fom tetodes acoss time, J. Neuosci. Methods, in pess (4. 4. T. L. Adelman, W. Bialek, and R. M. Olbeg, The infomation content of eceptie fields, Neuon, 4, , 3.. N. Benne, W. Bialek and R. R. de Ruyte an Steeninck, Synegy in a neual code, Neual Computation,, 3- (. 6. L. Paninski, Conegence popeties of thee -tiggeed analysis techniques, Netwok: Comput. Neual Syst. 4, (3. 7. W. H. Pess, S. A. Teukolsky, W. T. Vettenbeg, B. P. Flanney, Numeical Recipes in C: the at of scientific computing, (Cambidge Uniesity Pess, 99. Poc. of SPIE Vol. 467