{"title": "Refractoriness and Neural Precision", "book": "Advances in Neural Information Processing Systems", "page_first": 110, "page_last": 116, "abstract": null, "full_text": "Refractoriness and Neural Precision \n\nMichael J. Berry n and Markus Meister \nMolecular and Cellular Biology Department \n\nHarvard University \n\nCambridge, MA 02138 \n\nAbstract \n\nThe relationship between a neuron's refractory period and the precision of \nits response to identical stimuli was investigated. We constructed a model of \na spiking neuron that combines probabilistic firing with a refractory period. \nFor realistic refractoriness, the model closely reproduced both the average \nfiring rate and the response precision of a retinal ganglion cell. The model is \nbased on a \"free\" firing rate, which exists in the absence of refractoriness. \nThis function may be a better description of a spiking neuron's response \nthan the peri-stimulus time histogram. \n\n1 INTRODUCTION \n\nThe response of neurons to repeated stimuli is intrinsically noisy. In order to take this \ntrial-to-trial variability into account, the response of a spiking neuron is often described \nby an instantaneous probability for generating an action potential. The response \nvariability of such a model is determined by Poisson counting statistics; in particular, the \nvariance in the spike count is equal to the mean spike count for any time bin (Rieke, \n1997). However, recent experiments have found far greater precision in the vertebrate \nretina (Berry, 1997) and the HI interneuron in the fly visual system (de Ruyter, 1997). In \nboth cases, the neurons exhibited sharp transitions between silence and nearly maximal \nfiring. When a neuron is firing near its maximum rate, refractoriness causes spikes to \nbecome more regularly spaced than for a Poisson process with the same firing rate. Thus, \nwe asked the question: does the refractory period play an important role in a neuron's \nresponse precision under these stimulus conditions? \n\n2 FIRING EVENTS IN RETINAL GANGLION CELLS \n\nWe addressed the role of refractoriness in the precision of light responses for retinal \nganglion cells. \n\n2.1 RECORDING AND STIMULATION \n\nExperiments were performed on the larval tiger salamander. The retina was isolated from \nthe eye and superfused with oxygenated Ringer's solution. Action potentials from retinal \n\n\fRefractoriness and Neural Precision \n\n111 \n\nganglion cells were recorded extracellularly with a multi-electrode array, and their spike \ntimes measured relative to the beginning of each stimulus repeat (Meister, 1994). \nSpatially uniform white light was projected from a computer monitor onto the \nphotoreceptor layer. The intensity was flickered by choosing a new value at random from \na Gaussian distribution (mean J, standard deviation oJ) every 30 ms. The mean light level \n(J= 4'10-3 W/m2) corresponded to photopic (daylight) vision. Contrast C is defined here \nas the temporal standard deviation of the light intensity divided by the mean, C = 01/ I. \nRecordings extended over 60 repeats of a 60-sec segment of random flicker. \n\nThe qualitative features of ganglion cell responses to random flicker stimulation at 35 % \ncontrast are seen in Fig. 1. First, spike trains had extensive periods in which no spikes \nwere seen in 60 repeated trials. Many spike trains were sparse, in that the silent periods \ncovered a large fraction of the total stimulus time. Second, during periods of firing, the \nperi-stimulus time histogram (PSTH) rose from zero to the maximum firing rate \n(-200 Hz) on a time scale comparable to the time interval between spikes (-10 ms). We \nhave argued that these responses are better viewed as a set of discrete firing \"events\" than \nas a continuously varying firing rate (Berry, 1997). In general, the firing events were \nbursts containing more than one spike (Fig. IB). Identifiable firing events were seen \nacross cell types; similar results were also found in the rabbit retina (Berry, 1997). \n\nA \n\n8 \n\nC \n\nW\u00b7 \n~' \n~~ \n~~ ,{ \nIf: \n>f. \n\n) .. 0: \n\nI' :~ : \n\n'~ I'r \n.~' \n'~*': \n/..'1 \n\n, \n\nI \n\n\\ \n\n\u2022 ' .I \n\n.'J: ! ~ . \n\n,.~. \n\n1 \n~ \n\\>.~ \n~-(-,It: \n'I~~ \n\n>. \n\n..-'w c \n..-c \n\nQ) \n\n2 \n\n1 \n\n0 \n60 \n\n~ 40 \n\u00b7c \nI-\n\n20 \n\n0 \n.- 300 \nN \nI \n\n-Q) \nni a: \n\n0 \n\n43.4 \n\n43.5 \n\n43.6 \n\nTime (5) \n\n43.7 \n\n43.8 \n\nFigure 1: Response of a salamander ganglion cell to random flicker stimulation. \n(A) Stimulus intensity in units of the mean for a O.5-s segment, (B) spike rasters \nfrom 60 trials, and (C) the firing rate r(t). \n\n2.2 FIRING EVENT PRECISION \n\nDiscrete episodes of ganglion cell firing were recognized from the PSTH as a contiguous \nperiod of firing bounded by periods of complete silence. To provide a consistent \ndemarcation of firing events, we drew the boundaries of a firing event at minima v in the \nPSTH that were significantly lower than neighboring maxima PI and P2' such that \n\n~ PIP2 Iv ~ \u00a2 with 95 % confidence (Berry, 1997). With these boundaries defined, every \n\nspike in each trial was assigned to exactly one firing event. \n\n\f112 \n\nM. J Berry and M. Meister \n\nMeasurements of both timing and number precision can be obtained if the spike train is \nparsed into such firing events. For each firing event i, we accumulated the distribution of \nspike times across trials and calculated several statistics: the average time Tj of the first \nspike in the event and its standard deviation OTj across trials, which quantified the \ntemporal jitter of the first spike; similarly, the average number N j of spikes in the event \nand its variance ONj 2 across trials, which quantified the precision of spike number. In \ntrials that contained zero spikes for event i, no contribution was made to Tj or OTj , while \na value of zero was included in the calculation of Nj and ONj 2 . \nFor the ganglion cell shown in Fig. 1, the temporal jitter oT of the first spike in an event \nwas very small (1 to 10 ms). Thus, repeated trials of the same stimulus typically elicit \naction potentials with a timing uncertainty of a few milliseconds. The temporal jitter of \nall firing events was distilled into a single number Tby taking the median o\"er all events. \nThe variance ON 2 in the spike count was remarkably low as well: it often approached the \nlower bound imposed by the fact that individual trials necessarily produce integer spike \ncounts. Because ON 2 \u00ab N for all events, ganglion cell spike trains cannot be completely \ncharacterized by their firing rate (Berry, 1997). The spike number precision of a cell was \nassessed by comp.utin,fo. tHe average variance over events and dividing by the average \nspike count: F = (ON- J j(N). This quantity, also known as the Fano factor, has a value \nof one for a Poisson process with no refractoriness. \n\n3 PROBABILISTIC MODELS OF A SPIKE TRAIN \n\nWe start by reviewing one of the simplest probabilistic models of a spike train, the \ninhomogeneous Poisson model. Here, the measured spike times {t j } are used to estimate \nthe instantaneous rate r(t) of spike generation during a time Lit . This can be written \nfonnallyas \n\nwhere M is the number of repeated stimulus trials and e( x) is the Heaviside function \n\n() 1 x~O} \nex= \n\no x-\nu \nc: \nQ) \n:::J \nCT \n.... \nQ) \nu. \n\n1000 \n\n100 \n\nA \n\n10 \n\n1 \n\n0 \n\n0 \n\n0 \n\n0 \n\n0 \n\n2 \n\n0000 \n\n0 \n\n0 \n\n0 \n\n~OOO \n\n0 \n\n000 0 \n\n4 \n\n6 \n\n8 \n\n10 \n\n12 \n\nInter-Spike Interval (ms) \n\n1.0 \n\nc: \n0 \nU \nc: \n:::J \nU. \n~ 0.5 \n> 0 u \nQ) a:: \n\n0.0 \n\nQ) \n\nB \n\n0 \n\n2 \n\n4 \n\n6 \n\n8 \n\n10 \n\n12 \n\nTime (ms) \n\nFigure 3: Determination of the relative refractory period. (A) The inter(cid:173)\nspike interval distribution (diamonds) is fit by an exponential curve (solid), \nresulting in (B) the recovery function. \n\nNot only is the average firing rate well-matched by the model, but the firing rate in each \ntime bin is also very similar. Figure 4A compares the firing rate for the real neuron to that \ngenerated by the model. The mean-squared error between the two is 4 %, while the \ncounting noise, estimated as the variance of the standard error divided by the variance of \nr{t) , is also 4 %. Thus, the agreement is limited by the finite number of repeated trials. \nFigure 4B compares the free firing rate q( t) to the observed rate firing r{ t). q( t) is equal \nto r(t) at the beginning of a firing event, but becomes much larger after several spikes \nhave occurred. In addition, q(t) is generally smoother than r(t), because there is a \ngreater enhancement in q(t) at times following a peak in r(t). \n\nIn summary, the free firing rate q( t) can be calculated from the raw spike train with no \nmore computational difficulty than r{t), and thus can be used for any spiking neuron. \nFurthermore, q(t) has some advantages over r(t): 1) in conjunction with a refractory \nspike-generator, it produces the correct response precision; 2) it does not saturate at high \nfiring rates, so that it can continue to distinguish gradations in the neuron's response. \nThus, q( t) may prove useful for constructing models of the input-output relationship of a \nspiking neuron (Berry, 1998). \n\nAcknowledgments \n\nWe would like to thank Mike DeWeese for many useful conversations. One of us, MJB, \nacknowledges the support of the National Eye Institute. The other, MM, acknowledges \nthe support of the National Science Foundation. \n\n\fFigure 4: Illustration of the free fIring rate. (A) The observed fIring rate r(t) \nfor real data (solid) is compared to that from the model (dotted). (B) The \nfree rate q(t) (thick) is shown on the same scale as r(t) (thin). All rates \nused a time bin of 0.25 ms and boxcar smoothing 'over 9 bins. \n\nReferences \n\nBerry, M. J., D. K. Warland, and M. Meister, The Structure and Precision of Retinal \nSpike Trains. PNAS, USA, 1997.94: pp. 5411-5416. \n\nBerry II, M. J. and Markus Meister, Refractoriness and Neural Precision. 1. Neurosci., \n1998. in press. \n\nDe Ruyter van Steveninck, R. R., G. D. Lewen, S. P. Strong, R. Koberle, and W. Bialek, \nReliability and Variability in Neural Spike Trains. Science, 1997.275: pp. 1805-1808. \n\nJohnson, D. H. and A. Swami, The Transmission of Signals by Auditory-Nerve Fiber \nDischarge Patterns. J. Acoust. Soc. Am., 1983. 74: pp. 493-501. \n\nMeister, M., J. Pine, and D. A. Baylor, Multi-Neuronal Signals from the Retina: \nAcquisition and Analysis. 1. Neurosci. Methods, 1994.51: pp. 95-106. \n\nMiller, M. 1. Algorithms for Removing Recovery-Related Distortion gtom Auditory-Nerve \nDischarge Patterns. J. Acoust. Soc. Am., 1985.77: pp. 1452-1464. \n\nRieke, F., D. K. Warland, R. R. de Ruyter van Steveninck, and W. Bialek, Spikes: \nExploring the Neural Code . 1997, Cambridge, MA: MIT Press. \n\n\f", "award": [], "sourceid": 1461, "authors": [{"given_name": "Michael", "family_name": "Berry", "institution": null}, {"given_name": "Markus", "family_name": "Meister", "institution": null}]}