{"title": "Optimal Filtering in the Salamander Retina", "book": "Advances in Neural Information Processing Systems", "page_first": 377, "page_last": 383, "abstract": null, "full_text": "Optimal Filtering in the Salamander Retina \n\nFred Riekea,l;, W. Geoffrey Owenb and Willialll Bialeka,b,c \n\nDepart.ment.s of Physicsa and Molecular and Cell Biologyb \n\nU niversit.y of California \nBerkeley, California 94720 \n\nand \n\nNEC Research Inst.itute C \n\n4 Independence \\Vay \n\nPrinceton, N e' ... .J ersey 08540 \n\nAbstract \n\nThe dark-adapted visual system can count photons wit h a reliability lim(cid:173)\nited by thermal noise in the rod photoreceptors -\nthe processing circuitry \nbet.ween t.he rod cells and the brain is essentially noiseless and in fact may \nbe close to optimal. Here we design an optimal signal processor which \nestimates the time-varying light intensit.y at the retina based on the rod \nsignals. \\Ve show that. the first stage of optimal signal processing involves \npassing the rod cell out.put. t.hrough a linear filter with characteristics de(cid:173)\ntermined entirely by the rod signal and noise spectra. This filter is very \ngeneral; in fact it. is the first st.age in any visual signal processing task \nat. 10\\\\' photon flux. \n\\Ve iopntify the output of this first-st.age filter wit.h \nthe intracellular voltage response of the bipolar celL the first anatomical \nst.age in retinal signal processing. From recent. data on tiger salamander \nphot.oreceptors we extract t.he relevant. spect.ra and make parameter-free, \nquantit.ative predictions of the bipolar celll'esponse to a dim, diffuse flash. \nAgreement wit.h experiment is essentially perfect. As far as we know this \nis the first successful predicti ve t.heory for neural dynamics. \n\n1 \n\nIntrod uction \n\nA number of hiological sensory cells perform at. a level which can be called optimal \n-\nt.heir performancf' approaches limits set. by t.he laws of physics [1]. In some cases \n\n377 \n\n\f378 \n\nRieke, Owen, and Bialek \n\nthe behavioral performance of an organism, not just the performance of the sensory \ncells, also approaches fundamental limits. Such performance indicates that neural \ncomput.ation can reach a level of precision where the reliability of the computed \nout.put is limited by noise in the sensory input rather than by inefficiencies in the \nprocessing algorithm or noise in the processing hardware [2]. These observations \nsuggest that we study algorithms for optimal signal processing. If we can make the \nnotion of optimal processing precise we will have the elements of a predictive (and \nhence unequivocally testable) theory for what the nervous system should compute. \nThis is in contrast. t.o traditional modeling approaches which involve adjustment of \nfree parameters to fit experimental data. \n\nTo further develop these ideas we consider the vertebrate retina. Since the classic \nexperiments of Hecht, Shlaer and Pirenne we have known that the dark-adapted \nvisual syst.em can count small numbers of photons [3]. Recent experiment.s confirm \nBarlow's suggestion [4,5] t.hat the reliability of behavioral decision making reaches \nlimits imposed by dark noise in the photoreceptors due to thermal isomerizat.ion of \nt.he photopigment [6]. If dark-adapted visual performance is limit.ed by thermal noise \nin t.he sensory cells then the subsequent layers of signal processing circuitry must be \nextremely reliable. Rather than trying to determine precise limits t.o reliability, we \nfollO\\\\I the approach introduced in [7] and use the not.ion of \"optimal computation\" \nt.o design the optimal processor of visual stimuli. These theoret.ical arguments \nresult in parameter-free predictions for the dynamics of signal transfer from t.he \nrod photoreceptor to t.he bipolar cell, the first stage in visual signal processing. We \ncompare these predictions directly with measurements on the intact retina of t.he \nt.iger salamander A mbystoma tigrinum [8,9]. \n\n2 Design of the optimal processor \n\nAll of an organism's knowledge of the visual world derives from the currents In (t) \nflowing in the photoreceptor cells (labeled n). Visual signal processing consists of \nestimating various aspects of the visual scene from observat.ion of these current.s. \nFurthermore, t.o be of use to the organism t.hese estimates must be carried out in real \ntime. The general problem then is to formulate an optimal strat.egy for estimating \nsome functional G[R(r, t)] of the time and position dependent photon arrival rate \nR(r, t) from real time observation of the currents InU). \n\n\\Ve can make considerable analytic progress to\\vards solving this general prohlem \nusing probabilistic methods [7,2]. St.art. by writ.ing an expression for the probability \nof t.he functional G[R(r,t)] conditional on the currents InU), P{G[R(r,t)Jlln(t)}. \nExpanding for low signal-to-noise ratio (SNR) we find that the first term in the \nexpa.nsion of P{ GIl} depends only on a filt.ered version of the rod current.s, \n\nP{G[R(r, t)]IIn(t)} = 8 G [F * In] + higher orJer corrections, \n\n(1) \nwhere * denotes convolution; the filter F depends only on t.he signal a.nd noise \ncharacteristics of t.he photorecept.ors, as described below. Thus the estimation t.ask \ndivides nat.urally int.o two stages -\na universal \"pre-processing\" stage and a t.ask(cid:173)\ne1ept>ndellt stage. The univf'rsal stage is independent both of the stimulus R(r, t) anel \nof the particular functiona.l G[R] we wish to estimate. Intuitively this separa.tion \nmakes sense; in conventional signa.l processing systems detector outputs are first \n\n\fOptimal Filtering in the Salamander Retina \n\n379 \n\nreconstruction \nalgorithm \n\nphoton rate R(I) \n\ntime \n\nestimated rate R t (t) \n\n\u2022\u2022 \n\ntime \n\nFigure 1: Schematic view of photon arrival rate estimation problem. \n\n~ .... - - - rod current \n\nprocessed by a filter whose shape is motivated by general SNR considerat.ions. Thus \nthe view of retinal signal processing which emerges from this calculation is a pre(cid:173)\nprocessing or \"cleaning up\" stage followed by more specialized processing stages. \nVve emphasize that this separat.ion is a mathematical fact, not. a model we have \nimposed. \n\nTo fill in some of the details of the calculation we turn to t.he simplest example of \nthe estimat.ion tasks discussed above -\nest.imation of t.he phot.on arrival rat.e itself \n(Fig. 1): Photons from a light source are incident on a small patch of retina at \na time-varying rate R(t), resulting in a current J(t) in a particular rod cell. The \ntheoret.ical problem is t.o determine the opt.imal st.rategy for est.imat.ing R{t) based \non t.he currents 1(t) in a small collect.ion of rod cells. \\Vit.h an appropriat.e defini(cid:173)\ntion of \"optima.l\" we can pose t.he estimation problem ma.themat.ically and look for \nanalytic or numerica.l solutions. One approach is the conditional probability calcu(cid:173)\nlat.ion discussed above [7]. Alternatively we can solve t.his problem using functional \nmet.hods. Here we outline the funct.ional calculation. \n\nStart by writing the estimated rate as a filtered version of t.he rod currents: \n\nRest(t) \n\nJ dTFl(T)J(t - T) \n\n+ J dT J dT' F2(T, T')J(t - T)/(i - T') + .... \n\n(2) \n\nIn t.he low SNR limit. t.he rods respond linearly (t.hey count photons), and we expect. \nthat. t.he linear term dominates the series (2) . \\Ve then solve analyt.ically for t.he \nfilt.er FdT) which minimizes \\2 = (J dt IR(t) - Rest (t)12) -\ni.t. t.he filt.er which \nsatisfies 6\\2j6Fdr) = o. The averages ( .. . ) are taken over t.he ensemble of stimuli \n\n\f380 \n\nRieke, Owen, and Bialek \n\nR( t). The result of t.his optimization is* \n\nFd T) = \n\nJ dw \n\n_e-1u.'T \n2;r \n\n. (R(w)i*(w)) \n\n(li(w)F) \n\n. \n\n(3) \n\nIn the photon counting regime the rod currents are described as a sum of impulse \nresponses 10(t - tiJ) occuring at t.he phot.on arrival times t p , plus a noise term 61(t). \nExpanding for low SNR we find \n\nF() Jdw -i .... ,TS \n\n'I r = \n\n-, -e \n2;r \n\nR(W \n\n)io(w) \n-. \n~J(w) \n\n+ '\" \n\n, \n\n(4) \n\nwhere SR(W) is t.he spectral density of fluctuations in the photon arrival rate, io(w) \nis the Fourier transform of IoU), and Sdw) is the spectral density of current noise \nol(t) in the rod. \n\nThe filter (4) naturally separat.es into two distinct stages: A \"first\" stage \n\nFbip(W) = io(W)/SI(W) \n\n(5) \nwhich depends only on t.he signal and noise properties of the rod cell, and a \"sec(cid:173)\nond\" stage SR(W) which contains our a priori knowledge of the stimulus. The first \nstage filter is the matched filter given the rod signal and noise characteristics; each \nfrequency component. in the output of this filt.er is weight.ed according to its input \nSNR. \n\nRecall from the probabilistic argument above that optimal estimation of some arbi(cid:173)\ntrary aspect of the scene, such as motion, also results in a separation into t.wo pro(cid:173)\ncessing stages. Specifically, estimation of any functional of light intensity involves \nonly a filtered version of the rod currents. This filter is precisely t.he universal filter \nFbip( T) defined in (5). This result makes intuitive sense since the first stage of \nfiltering is simply \"cleaning up\" the rod signals prior to subsequent computation. \nIntuitively we expect that this filtering occurs at an early stage of visual processing. \nThe first opportunity to filter the rod signals occurs in the transfer of signals be(cid:173)\nt.ween the rod and bipolar cells; we identify the transfer function between these cells \nwith the first st.age of our optimal filter. More precisely we ident.ify the intracellular \nvoltage response of the bipolar cell with the output of the filter FbiP ( r). In response \nto a dim flash of light at t = 0 the average bipolar cell voltage response should t.hen \nbe \n\n{'bip(t) ()( J dT Fbip(r)Io(t - r). \n\n(6) \n\n1Vowhere in this prediction process do 'we illsert allY information about the bipolar \nreSp01lSE -\nth( shape of Oltr' prediction is go punEd entirely by signal and noise \nproperties of the rod cell and the theordical prillciple of optimality. \n\n3 Extracting the filter parameters and predicting the \n\nbipolar response \n\nTo complet.e our prediction of t.he dim flash bipolar response we extract the rod \nsingle photon current Io(t) and rod current. noise spect.rum .':h(w') from experimen-\n\n\u00b7'Ve definf> the Fourier Transrorm as j(w) = J dte+iu.,t 1(t). \n\n\fOptimal Filtering in the Salamander Retina \n\n381 \n\n(\\,I \n\n0 \n\n0 \n0 \n\npredicted bipolar ~ \nresponse \n\nmeasured biP~ \nresponse \n\n, .. \n\n.j\\;t;' \n\nmeasured rod \nresponses \n\n(\\,I \n\n. \n\n0 \n\nQ) \n\n11) c: \n0 a. ~ \n11) 9 \n... \n\nC1) \n\n'0 \nC1) \nto \n.~ \n\"iij 9 \n... \nE \n0 \nc: \n\nQ) \n\n9 \n~ .... . \n\nC'! \nor; \n\n0.0 \n\n0.2 \n\n0.4 \n\n0.6 \n\n0.8 \n\n1.0 \n\n1.2 \n\n1.4 \n\ntime (sec) \n\nFigure 2: Comparison of predicted dim flash bipolar voltage response (based entirely \non rod signal and noise characteristics) and measured bipolar voltage response. For \nreference we show rod voltage responses from two different cells which show the typical \nvariations from cell to cell and thus indicate the variations we should expect in different \nbipolar cells. The measured responses are averages of many presentations of a diffuse \nflash occurring at t = 0 and resulting in the absorption of an average of about 5 photons \nin the rod cell. The errors bars are one standard deviation. \n\nt.al data. To compare our predict.ion directly wit.h experiment. we must obt.ain the \nrod characteristics under identical recording conditions as the bipolar measurement. \nThis excludes suct.ion pipette measurement.s which measure t.he current.s directly, \nbut effect. t.he rod response dynamics [10.11]. The bipolar voltage response is mea(cid:173)\nsured intracellularly in t.he eyecup preparation [8]; our approach is t.o use int.racel(cid:173)\nlular volt.age recordings t.o characterize the rod network and thus convert. volt.ages \nto current.s, as in [12]. This approach to the problem Illay seem overly complicat.ed \n- why did we formulat.e the theory in t.erms of currents and not. voltages? It is \nimportant. we formulate our theory in t.erms of the i7ldilliriuaJ rod signal and noise \ncharacteristics. The electrical coupling between rod cells in t.he ret.ina causes t.he \nvoltage noise in nearby rods t.o be correlated; each rod, however, independently \ninjects current noise int.o the network. \n\nThe impedances connecting adjacent. rod cells, the impedance of t.he rod cell itself \nand t.he spat.ial la.yout and connect.ions between rods det.ermine t.he relationship \nbet.ween current.s and voltages in t.he net.work. The rods lie nearly on a square \n\n\f382 \n\nRieke, Owen, and Bialek \n\nlattice with lattice constant 2011111. Using this result we extract t.he impedances from \ntwo independent experiments (12]. Once we have t.he impedances we \"dec.orrelate\" \nthe voltage noise to calculate the uncorrelat.ed current noise. We also convert the \nmeasured single photon voltage response to the corresponding current Io(t). It \nis important. to realize that t.he impedance characteristics of the rod network are \nexperimentally determined, and are not. in any sense free parameters! \n\nAfter completing these calculat.ions the elements of our bipolar prediction are ob(cid:173)\ntained under ident.ical conditions to the experimental bipolar response, and we can \nmake a direct comparison between the t.wo; th ere are no free parameters il1 this \nprediction. As shown in the Fig. 2, t.he predicted bipolar response (6) is in excellent \nagreement wit.h the measured response; all deviat.ions are well within the error bars. \n\n4 Concluding remarks \n\n'Ve began by posing a theoretical question: How can we best recover t.he phot.on \narrival rat.e from observations of the rod signals? The answer, in the form of a linear \nfilter which we apply to t.he rod current, divides into two st.ages -\na stage which is \nmatched to the rod signal anel noise charact.eristics, and a stage which depends on \nthe particular characteristics of the phot.on source we are observing. The first-stage \nfilter in fact. is the universal pre-processor for all visual processing tasks at low SNR. \nvVe identified t.his filter wit.h the rod-bipolar transfer function, and based on this \nhypothesis predicted the bipolar response t.o a dim , diffuse flash. Our prediction \n'''Te emphasize once \nagrees ext.remely well with experiment.al bipolar responses. \nmore that this is not. a \"model\" of the bipolar cell; in fact there is nothing in our \ntheory about the physical propert.ies of bipolar cells. Rather our approach results \nin parameter-free predictions of the computation al properties of these cells from t.he \ngeneral theoretical principle of opt.imal computation. As far as we know t.his is the \nfirst. successful quantit.at.ive predict.ion from a theory of neural computation. \n\nThus far our results are limited t.o t.he dark-adapted regime; however the theoreti(cid:173)\ncal analysis present.ed here depends only on low SNR. This observat.ion suggest.s a \nfollow-up experiment. t.o test t.he role of adaptation in t.he rod-bipolar transfer func(cid:173)\ntion. If the retina is first a.dapt.ed to a constant background illuminat.ion and then \nshown dim flashes on t.op of the background we can use the analysis presented here \nto predict t.he adaptEd bipolar rpsponse from the adapted rod impulse response and \nnoise. Such an experiment.s would answer a number of interesting questions about. \nret.inal processing: (l) Does the processing remain optima.l at. higher light. levels? \n(2) Does t.he bipolar ('ell ~till function as t.he universal pre-processor? (:3) Do the \nroJ anel bipolar ('ells adapt t.oget.her in such a way that the optimal first.-stage filter \nremains IInchanged, or does t.he rod-bipola.r transfer function also adapt.? \n\nCan t.hese iJeas be ext.ended t.o ot.her systems, particularly spiking cells'? A number \nof other signal processing syst.ems exhibit. nearly optimal performance [2]. One \nexample we are currently st.udying is the extraction of movement information from \nthe array of photoreceptor voltages in the insect compound eye \n[13). In related \nwork. A tick and Redlich [l4] have argueJ that t.he receptive field characteristics of \nret.inal ganglion C(-'lIs call be quantitat.ively predicted from a principle of opt.imal \nencoding (see also [15)). A more general quest.ion we are currently pursuing is \nthe efficiency of t.he coding of sensory information in neural spike t.rains. Our \n\n\fOptimal Filtering in the Salamander Retina \n\n383 \n\npreliminary results indicate that the information rate in a spike train ca.n be as high \nas 80% of the maximum information rate possible given the noise characteristics of \nspike generat.ion [16]. From these examples we believe t.hat \"optimal performance\" \nprovides a general theoretical framework which can be used to predict t.he significant \ncomputational dynamics of cells in many neural systems. \n\nAcknowledgnlellts \n\nWe thank R. Miller and W. Hare for sharing their data and ideas, D. 'Varland and \nR. de Ruyter van Steveninck for helping develop many of the methods we have used \nin this analysis, and J. Atick, J. Hopfield and D. Tank for many helpful discussions. \n\\V. B. thanks the Aspen Center for Physics for the environment whic h catalyzed \nt.hese discussions. VVork at. Berkeley was supported by t.he National Inst.itutes of \nHealth through Grant No. EY 03785 to \\VGO, and by the National Science Foun(cid:173)\ndation t.hrough a Presidential Young Investigator Award to 'VB, supplemented by \nfunds from Cray Research, Sun Microsystems, and the NEC Research Institute, and \nthrough a Graduate Fellowship to FR. \n\nReferences \n\n1. \\V. Bialek. Ann. Ret'. 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In \nD. Touretzky. editor. Advances in Neural Information Proctssillg Systems 2, \npages :36-43. Morgan Kaufmann, San Mateo. Ca., 1990. \n\n14. J . .J. Atick and N. Redlich. Neural Comp'utation, 2:308, 1990. \n15. W. Bialek, D. Ruderman, and A. Zee. In D. Touretzky, edit.or, Adl'ances in \nNeural Iuformation Processing Systems .'3. Morgan Kaufma.nn, San Mateo, Ca., \n1991. \n\n16. F. Rieke, \\V. Yamada, K. Moortgat, E. R. Lewis, and \\V. Bialek. Proceedings \n\nof the 9th. International Symposium on Htarillg, 1991. \n\n\f", "award": [], "sourceid": 433, "authors": [{"given_name": "Fred", "family_name": "Rieke", "institution": null}, {"given_name": "W.", "family_name": "Owen", "institution": null}, {"given_name": "William", "family_name": "Bialek", "institution": null}]}