{"title": "Statistical Reliability of a Blowfly Movement-Sensitive Neuron", "book": "Advances in Neural Information Processing Systems", "page_first": 27, "page_last": 34, "abstract": null, "full_text": "Statistical Reliability of a Blowfly \n\nMovement-Sensitive Neuron \n\nRob de Ruyter van Steveninck .. \n\nBiophysics Group, \n\nRijksuniversiteit Groningen, \nGroningen, The Netherlands \n\nWilliam Bialek \n\nNEe Research Institute \n\n4 Independence Way, \nPrinceton, N J 08540 \n\nAbstract \n\nWe develop a model-independent method for characterizing the reliability \nof neural responses to brief stimuli. This approach allows us to measure \nthe discriminability of similar stimuli, based on the real-time response of a \nsingle neuron. Neurophysiological data were obtained from a movement(cid:173)\nsensitive neuron (HI) in the visual system of the blowfly Calliphom ery(cid:173)\nthrocephala. Furthermore, recordings were made from blowfly photore(cid:173)\nceptor cells to quantify the signal to noise ratios in the peripheral visual \nsystem. As photoreceptors form the input to the visual system, the reli(cid:173)\nability of their signals ultimately determines the reliability of any visual \ndiscrimination task. For the case of movement detection, this limit can \nbe computed, and compared to the HI neuron's reliability. Under favor(cid:173)\nable conditions, the performance of the HI neuron closely approaches the \ntheoretical limit, which means that under these conditions the nervous \nsystem adds little noise in the process of computing movement from the \ncorrelations of signals in the photoreceptor array. \n\n1 \n\nINTRODUCTION \n\nIn the 1940s and 50s, several investigators realized that understanding the reliabil(cid:173)\nity of computation in the nervous system posed significant theoretical challenges. \nAttempts to perform reliable computations with the available electronic computers \n\n\u00b7present address: University Hospital Groningen, Dept. of Audiology, POB 30.001, NL \n\n9700RB Groningen, The Netherlands \n\n27 \n\n\f28 \n\nde Ruyter van Steveninck and Bialek \n\ncertainly posed serious practical problems, and the possibility that the problems of \nnatural and artificial computing are related was explored. Guided by the practical \nproblems of electronic computing, von Neumann (1956) formulated the theoreti(cid:173)\ncal problem of \"reliable computation with unreliable components\". Many authors \nseem to take as self-evident the claim that this is a problem faced by the nervous \nsystem as well, and indeed the possibility that the brain may implement novel solu(cid:173)\ntions to this problem has been at least a partial stimulus for much recent research. \nThe qualitative picture adopted in this approach is of the nervous system as a \nhighly interconnected network of rather noisy cells, in which meaningful signals are \nrepresented only by large numbers of neural firing events averaged over numerous \nredundant neurons. Neurophysiological experiments seem to support this view: If \nthe same stimulus is presented repeatedly to a sensory system, the responses of an \nindividual afferent neuron differ for each presentation. This apparently has led to \na widespread belief that neurons are inherently noisy, and ideas of redundancy and \naveraging pervade much of the literature. Significant objections to this view have \nbeen raised, however (c/. Bullock 1970). \nAs emphasized by Bullock (Ioc.cit), the issue of reliability of the nervous system is a \nquantitative one. Thus, the first problem that should be overcome is to find a way \nfor its measurement. This paper focuses on a restricted, but basic question, namely \nthe reliability of a single neuron, much in the spirit of previous work (cf. Barlow and \nLevick 1969, Levick et al. 1983, Tolhurst at al. 1983, Parker and Hawken 1985). \nHere the methods of analysis used by these authors are extended in an attempt to \ndescribe the neuron's reliability in a way that is as model-independent as possible. \n\nThe second-conceptually more difficult-problem, is summarized cogently in Bul(cid:173)\nlock's words, \"how reliable is reliable?\". Just quantifying reliability is not enough, \nand the qualitative question of whether redundancy, averaging, multiplexing, or yet \nmore exotic solutions to von Neumann's problem are relevant to the operation of \nthe nervous system hinges on a quantitative comparison of reliability at the level \nof single cells with the reliability for the whole system. Broadly speaking, there \nare two ways to make such a comparison: one can compare the performance of \nthe single cell either with the output or with the input of the whole system. As \nto the first possibility, if a single cell responds to a certain stimulus as reliably as \nthe animal does in a behavioral experiment, it is difficult to imagine why multiple \nredundant neurons should be used to encode the same stimulus. Alternatively, if \nthe reliability of a single neuron were to approach the limits set by the sensory \nperiphery, there seems to be little purpose for the nervous system to use functional \nduplicates of such a cell, and the key theoretical problem would be to understand \nhow such optimal processing is implemented. Here we will use the latter approach. \n\nWe first quantify the reliability of response of HI, a wide-field movement-sensitive \nneuron in the blowfly visual system. The method consists essentially of a direct \napplication of signal detection theory to trains of neural impulses generated by \nbrief stimuli, using methods familiar from psychophysics to quantify discriminabil(cid:173)\nity. Next we characterize signal transfer and noise in the sensory periphery-the \nphotoreceptor cells of the compound eye-and we compare the reliability of infor(cid:173)\nmation coded in HI with the total amount of sensory information available at the \ninput. \n\n\fStatistical Reliability of a Blowfly Movement-Sensitive Neuron \n\n29 \n\n2 PREPARATION, STIMULATION AND RECORDING \n\nExperiments were performed on female wild-type blowfly Calliphora erythrocephala. \nSpikes from HI were recorded extracellularly with a tungsten microelectrode, their \narrival times being digitized with 50 {ts resolution. The fly watched a binary \nrandom-bar pattern (bar width 0.029\u00b0 visual angle, total size (30.5\u00b0)2) displayed on \na CRT. Movement steps of 16 different sizes (integer multiples of 0.12\u00b0) were gener(cid:173)\nated by custom-built electronics, and presented at 200 ms intervals in the neuron's \npreferred direction. The effective duration of the experiment was 11 hours, during \nwhich time about 106 spikes were recorded over 12552 presentations of the 16-step \nstimulus sequence. \n\nPhotoreceptor cells were recorded intracellularly while stimulated by a spatially \nhomogeneous field, generated on the same CRT that was used for the HI experi(cid:173)\nments. The CRT's intensity was modulated by a binary pseudo-random waveform, \ntime sampled at 1 ms. The responses to 100 stimulus periods were averaged, and \nthe cell's transfer function was obtained by computing the ratio of the Fourier trans(cid:173)\nform of the averaged response to that of the stimulus signal. The cell's noise power \nspectrum was obtained by averaging the power spectra of the 100 traces of the \nindividual responses with the average response subtracted. \n\n3 DATA ANALYSIS \n\n3.1 REPRESENTATION OF STIMULUS AND RESPONSE \n\nA single movement stimulus consisted of a sudden small displacement of a wide-field \npattern. Steps of varying sizes were presented at regular time-intervals, long enough \nto ensure that responses to successive stimuli were independent. In the analysis we \nconsider the stimulus to be a point event in time, parametrized by its step size Ct. \n\nThe neuron's signal is treated as a stochastic point process, the parameters of which \ndepend on the stimulus. Its statistical behavior is described by the conditional \nprobability P(rICt) of finding a response r, given that a step of size Ct was presented. \nFrom the experimental data we estimate P(rICt) for each step size separately. To \nrepresent a single response r, time is divided in discrete bins of width ~t = 2 ms. \nThen r is described by a firing pattern, which is just a vector q = [qO, ql, .. ] of binary \ndigits qk(k = 0, n - 1), where qk = 1 and qk = 0 respectively signify the presence \nor the absence of a spike in time bin k (cf. Eckhorn and Popel 1974). No response \nis found within a latency time t'at=15 ms after stimulus presentation; spikes fired \nwithin this interval are due to spontaneous activity and are excluded from analysis, \nso k = 0 corresponds to 15 ms after stimulus presentation. \nThe probability distribution of firing patterns, P(qICt), is estimated by counting the \nnumber of occurrences of each realization of q for a large number of presentations of \nCt. This distribution is described by a tree which results from ordering all recorded \nfiring patterns according to their binary representation, earlier times corresponding \nto more-significant bits. Graphical representations of two such trees are shown ill \nFig. 1. In constructing a tree we thus perform two operations on the raw spike \ndata: first, individual response patterns are represented in discrete time bins ~t, \nand second, a permutation is performed on the set of discretized patterns to order \n\n\f30 \n\nde Ruyter van Steveninck and Bialek \n\nthem according to their binary representation. No additional assumptions are made \nabout the way the signal is encoded by the neuron. This approach should therefore \nbe quite powerful in revealing any subtle\" hidden code\" that the neuron might use. \nAs the number of branches in the tree grows exponentially with the number of time \nbins n, many presentations are needed to describe the tree over a reasonable time \ninterval, and here we use n = 13. \n\n. \n\n3.2 COMPUTATION OF DISCRIMINABILITY \n\nTo quantify the performance of the neuron, we compute the discriminability of two \nnearly equal stimuli al and a2, based on the difference in neural response statistics \ndescribed by P{rlaI) and P{rl(2). The probability of correct decisions is maximized \nif one uses a maximum likelihood decision rule, so that in the case of equal prior \nprobabilities the outcome is al if P{robslal) > P{robsl(2), and vice versa. On \naverage, the probability of correctly identifying step al is then: \nPc{aJ) = L P{rlat} . H[P(rlat) - P{rl(2)], \n\n(1) \n\n{r} \n\nwhere H{.) is the Heaviside step function and the summation is over the set of all \npossible responses {r}. An interchange of indices 1 and 2 in this expression yields \nthe formula for correct identification of a2. The probability of making correct \njudgements over an entire experiment in which al and a2 are equiprobable is then \nsimply Pc(al, (2) = [Pc{at) + Pc(a2)]/2, which from now on will be referred to as \nPc. \nThis analysis is essentially that for a \"two-alternative forced-choice\" psychophysical \nexperiment. For convenience we convert Pc into the discriminability parameter d', \nfamiliar from psychophysics (Green and Swets 1966), which is the signal-to-noise \nratio (difference in mean divided by the standard deviation) in the equivalent equal(cid:173)\nvariance Gaussian decision problem. \nUsing the firing-pattern representation, r = q, and computing d' for successive \nsubvectors of q with elements m = 0, .. , k and k = 0, .. , n - 1, we compute Pc for \ndifferent values of k and from that obtain d'{k), the discrimillability as a function \nof time. \n\n3.3 THEORETICAL LIMITS TO DISCRIMINATION \n\nFor the simple stimuli used here it is relatively easy to determine the theoretical limit \nto discrimination based on the photoreceptor signal quality. For the computation \nof this limit we use Reichardt's (1957) correlation model of movement detection. \nThis model has been very successful in describing a wide variety of phenomena \nin biological movement detection, both in fly (Reichardt and Poggio 1976), and \nin humans (van Santen and Sperling 1984). Also, correlation-like operations can \nbe proved to be optimal for the extraction of movement information at low signal \nto noise ratio (Bialek 1990). The measured signal transfer of the photoreceptors, \ncombined with the known geometry of the stimulus and the optics of the visual \nsystem determine the signal input to the model. The noise input is taken directly \n\n\fStatistical Reliability of a Blowfly Movement-Sensitive Neuron \n\n31 \n\n1.0 \n\n0.8 \n\n0.2G\u00b7 \n\n~ 0.6 \n:s \n\nIV \n,Q \n0 \nb. 0.4 \n\n0.2 \n\n0.0 \n\n20 \n\n30 \n\n(ms) \n\ntime \n\n40 \n\n1.0 \n\n0.8 \n\n0.36\u00b7 \n\n0.6 \n\n~ \n~ \n,Q b. 0.4 \n\n0.2 \n\n0.0 \n\n----\n----\n--~ \n\n20 \n\n30 \n\ntime (ms) \n\n40 \n\nFigure 1: Representation of the firing pattern distributions for steps of 0.24\u00b0 and \n0.36\u00b0. Here only 11 time bins are shown. \n\nfrom the measured photoreceptor noise power spectrum. Details of this computation \nare given in de Ruyter van Steveninck (1986). \n\n3.4 ERROR ANALYSIS AND DATA REQUIREMENTS \n\nThe effects of the approximation due to time-discretization can be assessed by vary(cid:173)\ning the binwidth. It turns out that the results do not change appreciably if the bins \nare made smaller than 2 ms. Furthermore, if the analysis is to make sense, station(cid:173)\narity is required, i.e. the probability distribution from which responses to a certain \nstimulus are drawn should be invariant over the course of the experiment. Finally, \nthe distributions, being computed from a finite sample of responses, are subject to \nstatistical error. The statistical error in the final result was estimated by partition(cid:173)\ning the data and working out the values of Pc for these partitions separately. The \nstatistical variations in Pc were of the order of 0.01 in the most interesting region \nof values of Pc, i.e. from 0.6 to 0.9. This results in a typical statistical error of 0.05 \nin d'. In addition, this analysis revealed no significant trends with time, so we may \nassume stationarity of the preparation. \n\n4 RESULTS \n\n4.1 STEP SIZE DISCRIMINATION BY THE HI NEURON \n\nAlthough 16 different step sizes were used, we limit the presentation here to steps \nof 0.24\u00b0 and 0.36\u00b0; binary trees representing the two firing-pattern distributions are \nshown in Fig. 1. The first time bin describes the probabilities of two possible events: \neither a spike was fired (black) or not (white), and these two probabilities add up \nto unity. The second time bin describes the four possible combinations of finding \n\n\f32 \n\nde Ruyter van Steveninck and Bialek \n\n15r-----~----~----~----~ \n\n3. 0 r---..----......----.....-----r-----r--r--, \n\n10 \n\n5 \n\n2.0 \n\n1.0 \n\npredICted \n\n-\n\nI \n\nI \n\n1 \n\n1 \n\n/ \n\n/ \n\n/ \n\n/ \n\n1 \n\n/ \n\nI \n\n/ \n\n/ \n\nI \n\n/ \n\n/ \n\n1 \n\npredicted \n\n1 - - and shifted \n\n50 \nobservation window \n\n100 \n\n150 \n(ms) \n\n200 \n\n20 \n\ntime \n\n(ms) \n\n30 \n\n40 \n\nFigure 2: Left: Discrimination performance of an ideal movement detector. See \ntext for further details. Right: comparison of the theoretical and the measured \nvalues of d'et). Fat line: measured performance of Hl. Thin solid line: predicted \nperformance, taken from the left figure. Dashed line: the same curve shifted by 5 \nms to account for latency time in the pathway from photoreceptor to HI. This time \ninterval was determined independently with powerful movement stimuli. \n\nor not finding a spike in bin 2 combined with finding or not finding a spike in bin \n1, and so on. The figure shows that the probability of firing a spike in time bin 1 \nis slightly higher for the larger step. From above we compute Pc, the probability of \ncorrect identification, in a task where the choice is between step sizes of 0.240 and \n0.360 with equal prior probabilities. The decision rule is simple: if a spike is fired in \nbin 1, choose the larger, otherwise choose the smaller step. In the same fashion we \napply this procedure to the following time bin, with four response categories and so \non. The value of d' computed from Pc for this step size pair as a function of time \nis given by the fat line at the right in Fig. 2. \n\n4.2 LIMITS SET BY PHOTORECEPTOR SIGNALS \n\nFigure 2 (left) shows the limit to movement detection computed for an array of 2650 \nReichardt correlators stimulated with a step size difference of 0.12 0 , conforming to \nthe experimental conditions. Comparing the performance of HI to this result (the \nfat and the dashed lines in Fig. 2, right), we see that the neuron follows the limit \nset by the sensory periphery from about 18 to 28 ms after stimulus presentation. \nSo, for this time window the randomness of HI's response is determined primarily \nby photoreceptor noise. Up to about 20 Hz, the photoreceptor signal-to-noise ratio \nclosely approached the limit set by the random arrival of photons at the photorecep(cid:173)\ntors at a rate of about 104 effective conversions/so Hence most of the randomness \nin the spike train was caused by photon shot noise. \n\n\fStatistical Reliability of a Blowfly Movement-Sensitive Neuron \n\n33 \n\n5 DISCUSSION \n\nThe approach presented here gives us estimates for the reliability of a single neuron \nin a well-defined, though restricted experimental context. In addition the theoretical \nlimits to the reliability of movement-detection are computed. Comparing these two \nresults we find that HI in these conditions uses essentially all of the movement \ninformation available over a 10 ms time interval. Further analysis shows that this \ninformation is essentially contained in the time of firing of the first spike. The \nplateau in the measured d'(t) between 28 and 34 ms presumably results from effects \nof refractoriness, and the subsequent slight rise is due to firing of a second spike. \n\nThus, a step size difference of 0.12\u00b0 can be discriminated with d' close to unity, \nusing the timing information of just one spike from one neuron. For the blowfly \nvisual system this angular difference is of the order of one-tenth of the photoreceptor \nspacing, well within the hyperacuity regime (cf. Parker and Hawken 1985). \nIt should not be too surprising that the neuron performs well only over a short \ntime interval and does not reach the values for d' computed from the model at large \ndelays (Fig. 2, left): The experimental stimulus is not very natural, and in real-life \nconditions the fly is likely to see movement changing continuously. (Methods for an(cid:173)\nalyzing responses to continuous movement are treated in de Ruyter van Steveninck \nand Bialek 1988, and in Bialek et al. 1991.) In such circumstances it might be \nbetter not to wait very long to get an accurate estimate of the stimulus at one \npoint in time, but rather to update rough estimates as fast as possible. This would \nfavor a coding principle where successive spikes code independent events, which \nmay explain that the plateau in the measured d'{t) starts at about the point where \nthe computed d'{t) has maximal slope. Such a view is supported by behavioral \nevidence: A chasing fly tracks the leading fly with a delay of about 30 ms (Land \nand Collett 1974), corresponding to the time at which the measured d'{t) levels off. \nIn conclusion we can say that in the experiment, for a limited time window the \nneuron effectively uses all information available at the sensory periphery. Periph(cid:173)\neral noise is in turn determined by photon shot noise so that the reliability of HI's \noutput is set by the physics of its inputs. There is no neuro-anatomical or neuro(cid:173)\nphysiological evidence for massive redundancy in arthropod nervous systems. More \nspecifically, for the fly visual system, it is known that HI is unique in its combina(cid:173)\ntion of visual field and preferred direction of movement (Hausen 1982), and from \nthe results presented here we may begin to understand why: It just makes little \nsense to use functional duplicates of any neuron that performs almost perfectly \nwhen compared to the noise levels inherently present in the stimulus. It remains to \nbe seen to what extent this conclusion can be generalized, but one should at least \nbe cautious in interpreting the variability of response of a single neuron in terms of \nnoise generated by the nervous system itself. \n\nReferences \n\nBarlow HB, Levick WR (1969) Three factors limiting the reliable detection of light \nby retinal ganglion cells of the cat. 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Q Rev Biophys 9:311-375. \nde Ruyter van Steveninck RR (1986) Real-time performance of a movement-sensitive \nneuron in the blowfly visual system. Thesis, Rijksuniversiteit Groningen, the \nNetherlands. \n\nde Ruyter van Steveninck RR, Bialek W (1988) Real-time performance of a \nmovement-sensitive neuron in the blowfly visual system: coding and information \ntransfer in short spike sequences. Proc R Soc Lond B 234: 379-414. \n\nvan Santen JPH, Sperling G (1984) Temporal covariance model of human motion \nperception. J Opt Soc Am A1:451-473. \n\nTolhurst DJ, Movshon JA, Dean AF (1983) The statistical reliability of signals in \nsingle neurons in cat and monkey visual cortex. Vision Res 23: 775-785. \n\n\f", "award": [], "sourceid": 469, "authors": [{"given_name": "Rob", "family_name": "de Ruyter van Steveninck", "institution": null}, {"given_name": "William", "family_name": "Bialek", "institution": null}]}