Bayesian Estimation of Stimulus Responses in Poisson Spike Trains

Transcription

1 NOTE Communicated by Kechen Zhang Bayesian Estimation of Stimulus Responses in Poisson Spike Trains Sidney R. Lehky Cognitive Brain Mapping Laboratory, RIKEN Brain Science Institute, Saitama , Japan, and Laboratory of Brain and Cognition, National Institute of Mental Health, Bethesda, M 20892, U.S.A. A Bayesian method is developed for estimating neural responses to stimuli, using likelihood functions incorporating the assumption that spike trains follow either pure Poisson statistics or Poisson statistics with a refractory period. The Bayesian and standard estimates of the mean and variance of responses are similar and asymptotically converge as the size of the data sample increases. However, the Bayesian estimate of the variance of the variance is much lower. This allows the Bayesian method to provide more precise interval estimates of responses. Sensitivity of the Bayesian method to the Poisson assumption was tested by conducting simulations perturbing the Poisson spike trains with noise. This did not affect Bayesian estimates of mean and variance to a signi cant degree, indicating that the Bayesian method is robust. The Bayesian estimates were less affected by the presence of noise than estimates provided by the standard method. 1 Introduction The goal here is to use Bayesian methods to improve estimates of neural responses. The Bayesian analysis accomplishes this by incorporating additional information about spike train statistics into the analysis relative to standard methods, which simply calculate moments (e.g., mean, variance) of the data. In this analysis, we incorporate the assumption that spike counts are Poisson distributed. There have been previous applications of Bayesian methods to neural data (Brown et al., 1998; Martignon et al., 2000; Sanger, 1996; Zhang, Ginzburg, McNaughton, & Sejnowski, 1998). However, the focus of those studies was determining the most likely stimulus given a set of responses within a neural population. Here we are not concerned with determining the most likely stimulus, but rather move the analysis to a lower level and infer the probability distribution of the response ring rate to one particular stimulus, given observations of spike counts within xed time periods. The emphasis is on practical data analysis methods rather than theoretical issues in neural coding. Neural Computation 16, (2004) c 2004 Massachusetts Institute of Technology

2 1326 S. Lehky The Bayes formulation for the case at hand is prob. j n; T/ prob.n j ; T/ prob. ; T/ ; (1.1) prob.n; T/ where n is the observed spike count during time interval T and is the estimated spike rate. Omitting the normalization factor prob.n; T/ in the denominator leaves the proportionality: prob. j n; T/ / prob.n j ; T/ prob. ; T/ posterior / likelihood prior: (1.2) 2 The Likelihood Function The experimental situation we envisage is that a particular stimulus is presented during multiple trials. Within each trial, there is a prestimulus period, when activity of the neuron is at spontaneous level, and a stimulus period. Spike counts from the two periods are used to form likelihood functions for spontaneous and stimulus ring rates, spont and stim. From these, the likelihood function of the response stim spont is derived. The Poisson probability density function (pdf) we assume gives the distribution of spike count n during period T, given mean spike count T: prob.n j ; T/. T/n e T : (2.1) n! What we actually need for a Bayesian estimate, however, is the likelihood of, which involves keeping the same equation but now making n a constant and a variable, opposite to the situation in equation 2.1. This change in perspective converts equation 2.1 into gamma-distribution-shaped likelihood function of, with shape parameter n C 1 and rate parameter 1=T: prob.n j ; T/ likelihood. j n; T/. T/n 0.n C 1/ e T (2.2a) C ne T : (2.2b) (Multiplying this likelihood function by a constant factor T would normalize it to a gamma pdf.) In equation 2.2b, all the multiplicative constants are collected into C. The precise value of C is unimportant for likelihood calculations as we are interested in relative rather than absolute values of the function. Assuming prestimulus and stimulus spike rates are independent, the joint likelihood of spont and stim is formed by multiplying their individual likelihoods: likelihood i. stim ; spont j n i stim ; ni spont ; T stim; T spont / C i ni stim stim ni spont spont e. stimtstimc sponttspont/ (2.3)

3 Bayesian Estimation of Neural Responses 1327 (for the ith stimulus presentation). All multiplicative constants are grouped into C i, whose value again is not important. The joint likelihood for N stimulus trials is then found by multiplying the likelihood functions for the individual trials: likelihood. stim ; spont j En stim ; En spont ; T stim ; T spont / NY i1 3 Prior Probability istribution [C i ni stim stim ni spont spont e. stim Tstim C spontt spont/ ]: (2.4) We de ne two prior distributions as examples. One possibility, the agnostic assumption, is that all response magnitudes are equally likely. This is represented by a two-dimensional uniform distribution, priorprob Ag. stim ; spont / 1 a 2 0 spont ; stim a; 0 otherwise (3.1) with a de ning a biophysically plausible range of ring rates. Another possibility, the skeptical assumption, is that there is no stimulus response. That is, spont and stim are on average identical. Given the longterm mean spontaneous spike rate, Ņ spont, this distribution, analogous to equation 2.3, is: priorprob Sk. stim ; spont j Ņ spont ; T stim ; T spont / C. stim / Ņ spontt stim. spont / Ņ spontt spont e. stimtstimc sponttspont/ ; (3.2) where C is the normalizing constant. Besides these example prior distributions, others may be suggested by the particular aspects of an experiment. 4 Posterior Probability istribution The posterior probability of the ring rates is proportional to the prior probability (see equation 3.1 or 3.2), and the joint likelihood function (see equation 2.4): postprob. stim ; spont j En stim ; En spont ; T stim ; T spont / C[priorProb. stim ; spont / likelihood. stim ; spont j En stim ; En spont ; T stim ; T spont /]; (4.1) in which C normalizes the area under postprob. stim ; spont /.

4 1328 S. Lehky What we require, however, rather than the joint pdf of spont and stim, is the distribution of the response resp stim spont. This is accomplished by rst changing variables from prob. stim ; spont / to prob. stim spont ; stim C spont /, which involves a rotation and dilation of the coordinate axes. Then prob. stim spont ; stim C spont / is integrated along the stim C spont axis to give the marginal distribution prob. stim spont /. For convenience, we relabel the transformed variables as follows: sum stim C spont dif stim spont : (4.2) Under the transformed variables, the likelihood function for a single trial (analogous to equation 2.3) becomes: likelihood i. sum ; dif j n i stim ; ni spont ; T stim; T spont / C i. sum C dif / ni stim. sum dif / ni spont e 1 2 [. sumc dif /TstimC. sum dif /Tspont] : (4.3) The joint likelihood over N stimulus trials is then found by taking the product of individual trials, analogous to equation 2.4. The agnostic prior, equation 3.1, remains essentially unchanged under transformed coordinates because it is a constant. The transformed equation for the skeptical prior, equation 3.2, becomes: priorprob Sk. sum ; dif j Ņ spont ; T stim ; T spont / C. sum C dif / Ņ spontt stim. sum dif / Ņ spontt spont e 1 2 [. sumc dif /TstimC. sum dif /Tspont] : (4.4) Given the transformed likelihood and prior probability functions, the posterior probability is postprob. sum ; dif j En stim ; En spont ; T stim ; T spont / C[priorProb. sum ; dif / likelihood. sum ; dif j En stim ; En spont ; T stim ; T spont /]; (4.5) where again C normalizes the distribution. Integrating equation 4.5 with respect to sum takes us from postprob. sum ; dif / to postprob. dif /: postprob. dif j En stim ; En spont ; T stim ; T spont / C Z 1 1 priorprob. sum ; dif / likelihood. sum ; dif j En stim ; En spont ; T stim ; T spont /d sum : (4.6)

5 Bayesian Estimation of Neural Responses 1329 This is the nal result we seek the posterior probability distribution of the stimulus response. 5 Refractory Periods Thus far, the analysis has been carried out for a pure Poisson process. A signi cant perturbation away from this ideal is caused by the existence of a refractory period. The refractory period acts to regularize a spike train, reducing the variance of spike counts within a xed period, and therefore the variance of observed ring rates. Given mean interspike interval Nt ISI and refractory period t ref, the fractional variance relative to a pure Poisson process is º Var. / ref Var. / pure Á! t N 2 ISI t ref : (5.1) Nt ISI (By de nition of t ref, Nt ISI < t ref is impossible.) Since Nt ISI T=n: º ± 1 n T t ref ² 2 : (5.2) To take into account the refractory period, therefore, the variance of the likelihood function (as well as nonuniform priors) should be reduced by the refractory period correction factor º. A modi cation to the likelihood function equation 2.2b that approximates that transform is: likelihood. j n; T; º/ C nc1 1 º e T º 0 < º 1 ± n C 1 º 0; (5.3) T where ± is the irac delta function. This approximation leads to small biases in the estimates of spike statistics. Because the purpose of this study is to quantify properties of the Bayesian method, an exact refractory period correction was carried out numerically in the simulations described below. Using a likelihood function and prior corrected for refractory periods, the rest of the analysis proceeds as before. 6 Mean, Variance, and Variance of Variance of Response Estimates The equations that follow are derived for the situations of a pure Poisson process and a pure Poisson process plus refractory period. A uniform prior is assumed. In a later section, we present simulations that examine what happens when the pure Poisson process plus refractory period is further perturbed by adding noise.

6 1330 S. Lehky Response statistics can be presented as either a function of observed spike counts n or the spike density parameter r (which is known in the case of simulations). The relation between these two variables is given by the formula for average spike count per trial: i1 ni N 6.1 Mean rt: (6.1) Bayesian. Under the Bayesian analysis, if spike counts are Poisson distributed, then spike rates have a gamma-distribution-shaped likelihood function (see equation 2.2a). Given a set of N spike trains with spike count n i, i 1; : : : ; N, and duration T, the posterior probability formed by multiplying together a set of such likelihood functions with a uniform prior and the appropriate normalization factor gives a gamma distribution with shape parameter i1 ni C 1 and scale parameter 1=.NT/. Given this posterior probability distribution, the optimal location estimate of can be de ned as either the mode or mean of the distribution, depending on the optimality criterion (Kay, 1993). Using mean gives the minimum mean square error (MMSE) estimate, while using the mode gives the maximum a posteriori (MAP) estimate, which for a uniform prior is also the maximum likelihood (ML) estimate. The MMSE estimate leads to an expected spike rate of i1 E. / bay ni C 1 NT r C 1 NT ; (6.2) while the MAP estimate is i1 ni mode. / bay NT r: (6.3) The trade-off between the two is lower bias or lower mean square error. The estimated value of the response resp stim spont is then either i1 E. resp / bay ni stim C 1 i1 ni spont C 1 NT stim NT spont r stim r resp C N T stim T spont (6.4)

7 Bayesian Estimation of Neural Responses 1331 or i1 ni spont i1 mode. resp / bay ni stim NT stim NT spont r stim r resp : (6.5) In the discussion below, we shall assume the MMSE estimate Standard. Under the standard analysis, the mean ring rate is i1 ni E. / std NT r: (6.6) The expected value of resp then becomes i1 ni spont i1 E. resp / std ni stim NT stim NT spont r stim r spont : (6.7) Thus, the Bayesian MMSE estimate and the standard estimates of the response differ (except for the special case T stim T spont ), with the two estimates asymptotically converging as the sample size NT increases. The Bayesian MAP estimate is identical with the standard estimate. 6.2 Variance Bayesian. Given a posterior probability that is a gamma distribution with shape parameter i1 ni C1 and scale parameter 1=.NT/, expected variance of the ring rate is i1 E.Var. // bay ni C 1.NT/ 2 r NT C 1.NT/ 2 : (6.8) The variance of the response resp is then i1 E.Var. resp // bay ni stim C 1 i1.nt stim / 2 C ni spont C 1.NT spont / 2 1 rstim C r Á spont C 1N 1 N T stim T 2 spont T 2 stim! C 1 Tspont 2 : (6.9)

8 1332 S. Lehky The effect of the refractory period on expected variance can be approximated if we de ne, following equation 5.2, a mean refractory period correction factor: Nº Á 1 P! N 2 i1 ni t ref NT.1 rt ref / 2 : Using this, the corrected equation 6.9 becomes (6.10) ± PNi1 ² Nº stim n i stim C Nº stim E.Var. resp // bay.nt stim / ± 2 PN ² Nº spont i1 ni spont C Nº spont C.NT spont / 2 1 Nºstim r stim C Nº spontr spont N T stim T spont C 1 Á Nº stim N 2 Tstim 2 C Nº! spont Tspont 2 : (6.11) Although it can be convenient to use Nº on pooled data rather than doing full trial-by-trial calculations with different values of º for each trial, this introduces biases in calculated refractory-corrected variances, as well as variances of variances Standard. Given spike counts n that are Poisson distributed, Var.n/ i1 ni =N. We also have mean ring rate E. / as a function of n, equation 6.6. Combining these two relations gives the variance of from its mean, leading to i1 ni E.Var. // std.nt/ 2 r NT : (6.12) Response variance is then i1 ni spont i1 E.Var. resp // std ni stim.nt stim / 2 C.NT spont / 2 1 rstim C r spont : (6.13) N T stim T spont

9 Bayesian Estimation of Neural Responses 1333 Bayesian response variances (see equation 6.9) are larger than those calculated using the standard method (see equation 6.13), although they converge for large sample sizes. With correction for refractory period: E.Var. resp // std Nº stim i1 ni stim.nt stim / 2 C Nº spont i1 ni spont.nt spont / 2 1 N Nºstim r stim T stim C Nº spontr spont T spont : (6.14) 6.3 Variance of the Variance. Variance of the variance is important because it determines the precision of an interval estimate of a parameter Bayesian. From equation 6.8, we have ring rate variance as a function of n. Again, since spike counts are Poisson distributed, Var.n/ i1 n i =N. Combining these two relations gives the variance of variance: i1 ni Var.Var. // bay.nt/ 4 r.nt/ 3 : (6.15) Response variance of variance is Var.Var. resp // bay 1 N 4 Á PNi1 n i stim T 4 stim 1 N 3 Á r stim T 3 stim C C r spont T 3 spont i1 ni spont! T 4 spont! ; (6.16) and with correction for refractory period: Var.Var. resp // bay 1 N 4 Á Nº 2 stim i1 ni stim T 4 stim 1 N 3 Á Nº 2 stim r stim T 3 stim C Nº2 spont C Nº2 spont r spont T 3 spont! i1 ni spont T 4 spont! : (6.17) Standard. For mean spike count per trial i1 n i =N not close to zero, the ring rate estimate is approximately normally distributed. Performing the subtraction resp stim spont provides an even better approximation of normality. A normally distributed variable has variances that are

10 1334 S. Lehky chi-square distributed with N 1 degrees of freedom, and the variance of variance is Var.Var. // std 2.E.Var. ///2 : (6.18) N 1 Substituting the value of E.Var. // in equation 6.12 gives ± PNi1 ² 2 2 n i Var.Var. // std.n 1/.NT/ 4 The response variance of variance is: Var.Var. resp // std and corrected for refractory period: Var.Var. resp // std 2r 2.N 1/.NT/ 2 : (6.19) Á P P 2 N N!2 i1 ni stim i1.n 1/N 4 Tstim 2 C ni spont Tspont 2 2 rstim.n 1/N 2 C r 2 spont : (6.20) T stim T spont Á P 2 Nº N stim i1 ni stim.n 1/N 4 Tstim 2 Nºstim r stim 2.N 1/N 2 T stim C Nº spont! 2 i1 ni spont T 2 spont C Nº 2 spontr spont : (6.21) T spont Firing rate statistics calculated from the above equations for a pure Poisson process (without refractory period) are presented in Table 1, for parameters r stim 24 Hz, r spont 8 Hz, and T stim T spont 0:5 sec. Table 1 shows identical values for E. / under both methods, as expected for the special case T stim T spont. Bayesian variances E.Var. // are slightly larger than standard estimates but converge as the number of trials increases. Finally, the Bayesian method gives much smaller values for the variance of variance Var.Var. // than the standard analysis. 7 Estimating Neural Responses for Spike Trains with a Refractory Period The analysis is now applied to simulated data shown in Figure 1. At the bottom is a raster plotofspike trains from 20 trials in a simulated experiment.

11 Bayesian Estimation of Neural Responses 1335 Table 1: Stimulus Response Statistics for a Pure Poisson Process. E. resp / E.Var. resp // Var.Var. resp // Trials Standard Bayesian Standard Bayesian Standard Bayesian Notes: These were calculated using parameters r stim 24 Hz, r spont 8 Hz, and T stim T spont 0:5 sec. Bayesian estimates are based on a uniform prior and use the MMSE method. Spikes/sec Trial Time (msec) Figure 1: Simulated spike trains used to demonstrate the Bayesian analysis. At bottom is a raster plot of 20 trials; the top shows the peristimulus time histogram. Spike counts are Poisson distributed, but with a 3 ms refractory period included. The timescale has been shifted by the latency so that zero coincides with the onset of stimulus response.

12 1336 S. Lehky The spikes are Poisson distributed but include a 3 ms refractory period. In each trial, there is a 0.5 sec prestimulus period with spontaneous activity of 8 Hz, followed by a 0.5 sec stimulus period in which activity rises to 24 Hz. Activity then returns to spontaneous. The timescale has been translated by the stimulus latency so that the stimulus period starts at zero. The response distribution is found by starting with a prior distribution and successively multiplying in the likelihood function for each trial. For example, the rst trial has 4 spikes in the prestimulus and 20 spikes in the stimulus period. Substituting n 1 spont 4, n1 stim 20, T1 stim 0:500, and Tspont 1 0:500 into the refractory-period corrected analog of equation 4.3 gives the likelihood. sum ; dif / for that trial. Following incorporation of data from the nal trial, the distribution postprob. sum ; dif / is integrated along sum to give the response distribution, postprob. dif /. (For the discussion below, dif shall be relabeled resp.) Figure 2 illustrates the evolution of the response distribution as data from successive trials are included, starting from two prior distributions. Also shown are the conventional means and standard errors of the response. No standard error bar is shown for the Trial = 1 panel, because the conventional method does not allow calculating that statistic from a single trial. The rst panel plots the priors themselves (0th trial). We can obtain a better idea of the relation between the Bayesian and conventional estimates by comparing statistics from multiple replications of the experiment in Figure 1, determined for a uniform prior. Results of the simulations are given in Table 2, based on 10,000 experiment replications with N stimulus trials per experiment, and with r stim 24 Hz, r stim 8 Hz, and T stim T spont 0:5 sec, where r is the spike density parameter used to generate the spike trains. Table 2 shows E. resp /, mean response over 10,000 replications; E.Var. resp //, mean of the 10,000 response variances; and Var.Var. resp //, variance of the 10,000 response variances. Under these simulations, E. resp / is identical under both methods, as expected for the special case T stim T spont, and equal to r stim r spont. Values of resp are approximately normally distributed in both cases. Table 2: Stimulus Response Statistics from Simulations Using Homogeneous Poisson Spike Trains with a 3 Msec Refractory Period Added. E. resp / E.Var. resp // Var.Var. resp // Trials Standard Bayesian Standard Bayesian Standard Bayesian Note: Simulation parameters are as given in Table 1.

13 Bayesian Estimation of Neural Responses 1337 Rel Prob ensity Trial=0 (Prior prob) Trial= Rel Prob ensity Rel Prob ensity Trial= Trial= l resp 60 (spikes/sec) Trial= Trial= l resp (spikes/sec) Figure 2: Evolution of the Bayesian probability distribution of responses. resp stim spont / as data from increasing numbers of trials accumulate. The rst panel (Trial 0) shows two prior distributions the uniform distribution (dotted line) and a distribution that assumes stimulus and spontaneous ring rates are identical (dashed line). Solid vertical and horizontal lines in the panels indicate means and standard errors calculated in the standard way. All curves have been normalized to a height of one.

14 1338 S. Lehky Prob ensity Bayesian Standard Var( l resp ) (spikes/sec) 2 Figure 3: istributions of response variances for standard and Bayesian methods, derived from 25,000 replications of an experiment with ve stimulus trials per experiment.n 5/. The probability density functions were estimated from the simulated data by kernel smoothing. Although both Var. resp / distributions have almost identical E.Var. resp // values (see Table 2), the Bayesian method produces a much smaller Var.Var. resp //. The distribution produced by the standard method is chi square with N 1 4 degrees of freedom (see equation 7.1), while the Bayesian method leads to approximately normally distributed variances. Moving to E.Var. resp //, the variances for both methods are smaller than those for a pure Poisson process (comparing Table 2 with Table 1). This re- ects the presence of a 3 msec refractory period in the simulated spike trains that served as data. Bayesian variances are slightly larger than the standard ones, with the values converging as the number of trials N increases. Although both methods produce almost identical values of E.Var. resp //, we see in Table 2 that Var.Var. resp // is substantially smaller for the Bayesian method (as was the case calculated for a pure Poisson process in Table 1). This relationship can also be seen in the plot of the distribution of Var. resp / for N 5 trials (see Figure 3). As resp is normally distributed, under the standard analysis the distribution of Var. resp / is chi-square distributed with m N 1 degrees of freedom: prob.var. / j m; ¾ / E.Var. // m¾ 2 ± ².Var. // m 2 1 exp Var. / 2¾ 2 2. m 2 / 0. m 2 /¾ m Var.Var. // 2m¾ 4 : (7.1)

15 Bayesian Estimation of Neural Responses 1339 Combining expressions for E.Var. // in equations 6.14 and 7.1, a calculated value of ¾ can be derived: s Nº stim r stim ¾ C Nº spontr spont : (7.2).N 1/NT stim.n 1/NT spont For the N 5 simulations shown in Figure 3, it can be determined that the distribution for the standard method does indeed follow a chi-square distribution with a least-squares t of ¾ 1:70, compared to a calculated value of ¾ 1:68. The value of Var.Var. resp // observed in these simulations was 67.9, compared to 64.0 calculated from equation The bias in variance of variance calculations arises because of the use of an average refractory period correction factor Nº rather than trial-by-trial values of º. In contrast, the Bayesian method, despite also having a normally distributed response estimate, produces a Var. resp / that is not chi square but approximately normally distributed. The reason is that under the Bayesian method, Var. resp / is not calculated from the sample of resp estimates, but rather from the width of the likelihood functions. For the N 5 example, the observed Var.Var. resp // in the simulations was 1.10, lower than the value of 1.60 calculated from equation 6.17, again showing a bias in the calculated estimate caused by doing refractory period corrections on pooled data rather than on a trial-by-trial basis. 8 Effects of Noise on Bayesian Estimates The Bayesian analysis assumes that spike trains (either spontaneous or stimulus) form homogeneous Poisson processes, in which ring rates do not vary with time within each epoch. In reality, the data will deviate from this. Here we examine the robustness of the Bayesian statistics by testing how well the method works when inhomogeneous Poisson spike trains serve as inputs. Two cases are considered: fast- uctuation noise, in which spike rates vary rapidly relative to the duration of a single trial, and slow- uctuation noise, in which spike rates are constant within a trial but vary randomly from trial to trial. In both cases, we retain the presence of a 3 msec refractory period. Fast- uctuation noise consisted of Poisson spike trains whose randomly varying instantaneous spike density parameter was described by low-pass ltered gaussian white noise. The white noise perturbed spike density was set with means Nr spont 8 Hz and Nr stim 24 Hz and standard deviations of r.t/ equal to 0.5 Nr. The noise was passed through a Butterworth low-pass lter: A out A in 1 [1 C. f=f c / 2n ] 1 2 n 4; f c 20Hz: (8.1) Since spike rate uctuations on average integrated to zero within each trial, they did not change the spike count but merely rearranged their timing,

16 1340 S. Lehky making the spike train more bursty. Fano factors remained at 1.0 over the timescale of a complete trial, unchanged from homogeneous Poisson trains. For slow- uctuation noise, both spontaneous and stimulus spike rates were constant within a trial, but changed randomly from trial to trial according to a log-normal distribution: 8 < 1 prob.r j ¹; ¾ / r¾ p 2¼ e.ln.r/ ¹/ 2 2¾ 2 r > 0 : 0 r 0 : (8.2) Parameters for the spontaneous ring rate were ¹ 2:02, ¾ 0:35, and for the stimulus rate, ¹ 3:15, ¾ 0:22. Those values satis ed two criteria. First, the mean spike densities were the same as used previously, Nr spont 8 and Nr stim 24, and second, Fano factors were approximately 2.0 in both cases. Using noise-perturbed spike trains, stimulus response statistics were calculated as before. Results for the two noise conditions are given in Tables 3 and 4, to be compared with results for homogeneous Poisson trains in Table 2. Comparison of Tables 2 and 3 shows that fast- uctuation noise has no effect on either Bayesian or standard estimates, not surprising given that on extended timescales, spike count is conserved under this noise. Slow- uctuation noise produces no signi cant change in Bayesian estimates of E.Var. resp //, as a comparison of Tables 2 and 4 shows. On the other hand, standard estimates of E.Var. resp // were increased substantially. Thus, under this noise condition, Bayesian variance estimates become more accurate than standard ones (closer to the noise-free estimates), in addition to being more precise (smaller variance of the variance). In the slow- uctuation noise case, we have spike trains that appear non- Poisson by one criterion, a Fano factor of two, yet the estimates of a Poisson- Table 3: Response Statistics Using Inhomogeneous Poisson Spike Trains with a 3 Msec Refractory Period and Whose Firing Rates are Perturbed with Fast- Fluctuation Noise. E. resp / E.Var. resp // Var.Var. resp // Trials Standard Bayesian Standard Bayesian Standard Bayesian Notes: Fast- uctuation noise varies rapidly relative to the duration of a stimulus trial, as de ned in equation 8.1. Simulation parameters are as given in Table 1.

17 Bayesian Estimation of Neural Responses 1341 Table 4: Response Statistics Using Inhomogeneous Poisson Spike Trains with a 3 Msec Refractory Period and Whose Firing Rates are Perturbed with Slow- Fluctuation Noise. E. resp / E.Var. resp // Var.Var. resp // Trials Standard Bayesian Standard Bayesian Standard Bayesian Notes: Slow- uctuation noise is slow relative to the duration of a stimulus trial, as de ned in equation 8.2. Simulation parameters are as given in Table 1. Table 5: Example of Bayesian Estimation Applied to Actual ata. E. resp / E.Var. resp // Var.Var. resp // Trials Standard Bayesian Standard Bayesian Standard Bayesian Notes: The data consisted of 10 trials recorded from a unit in macaque striate cortex, presented with grating stimuli. The Fano factor was 1.7. Bayesian estimation was applied to 50 replications of a bootstrap resampling of the data, with 5 of the 10 trials chosen at random with replacement for each resampling. based model are little affected as spike generation is still Poisson at a local timescale. The insensitivity of Bayesian estimates to either type of noise perturbation away from a homogeneous Poisson process indicates the robustness of the method. As a nal item on noise-perturbed spike trains, we apply the Bayesian method to actual data recorded from monkey striate cortex (see Table 5). In this case, E. resp / is slightly different for the two methods because T stim 6 T spont (see equation 6.4). The pattern of E.Var. resp // values for the actual data appears to follow those of the simulations with slow- uctuation noise added, in which Bayesian variances are substantially smaller than standard variances. We noted previously that under the slow- uctuation noise condition, the smaller Bayesian variance estimates are more accurate than the standard estimates. 9 iscussion The Bayesian and standard methods provide similar estimates for the mean E. resp / and variance E.Var. resp // of the response ring rate for parameters typical of experimental conditions. The estimates of both methods converge

18 1342 S. Lehky as the data sample size increases, by increasing either the number of trials or the duration of each trial, so that the Bayesian model is asymptotically unbiased. Given a uniform prior, the Bayesian estimate is a maximum likelihood estimate and therefore asymptotically ef cient (given certain regularity conditions) (Kay, 1993). That is, the Bayesian E.Var. resp // estimate asymptotically approaches the Cramér-Rao bound. As the Bayesian estimate also asymptotically approaches the standard estimate, the implication is that the standard estimate is, at minimum, also asymptotically ef cient. The Bayesian method provides much smaller estimates of the variance of the variance.var.var. resp /// than the standard method, despite having almost identical values of expected variance E.Var. resp //. That is because the two methods compute variances using entirely different procedures. Under the standard method, the variance of the ring rate Var. / is computed directly from the sample of itself by plugging those values into the standard formula for variance, without making assumptions about the distribution of. Under the Bayesian method, statistics are calculated indirectly from the data in a two step process, by rst tting a model (the likelihood model) and then determining statistics from the model curve t rather than directly from the data. We can identify three bene ts to using the Bayesian analysis relative to the standard analysis. The rst is smaller values for the variance of the variance. This allows more precise interval estimates of a parameter, which in turn improves the precision of tests of signi cance. The second is reduced sensitivity of variance estimates to perturbations in the data caused by noise. This can be seen by comparing Tables 2 and 4. The nal bene t is the ability to estimate response variance after only a single trial. Underlying the Bayesian analysis was the assumption that spike trains have Poisson statistics. Sensitivity to this assumption was tested by perturbing spike trains away from Poisson statistics with two kinds of noise, whose uctuations were fast or slow relative to the timescale of a single trial. espite the noise perturbations, Bayesian estimates of response mean and variance were not signi cantly affected. This indicates that the Bayesian method is robust. Given the bene ts of Bayesian analysis listed above and the robustness of the method, it appears to be a promising candidate for practical data analysis. Acknowledgments I thank Keiji Tanaka and Leslie Ungerleider for their support. Portions of this work were completed while I was visiting the Sloan-Schwartz Center for Theoretical Neurobiology at the Salk Institute.

19 Bayesian Estimation of Neural Responses 1343 References Brown, E. N., Frank, L. M., Tang,., Quirk, M. C., & Wilson, M. A. (1998). A statistical paradigm for neural spike train decoding applied to position prediction from ensemble ring patterns of rat hippocampal place cells. J. Neurosci., 18, Kay, S. M. (1993). Fundamentals of statistical signal processing: Estimation theory. Upper Saddle River, NJ: Prentice Hall. Martignon, L., eco, G., Laskey, K., iamond, M., Freiwald, W., & Vaadia, E. (2000). Neural coding: Higher order temporal patterns in the neurostatistics of cell assemblies. Neural Comput., 12, Sanger, T.. (1996). Probability density estimation for the interpretation of neural population codes. J. Neurophysiol., 76, Zhang, K., Ginzburg, I., McNaughton, B. L., & Sejnowski, T. J. (1998). Interpreting neuronal population activity by reconstruction: Uni ed framework with application to hippocampal place cells. J. Neurophysiol., 79, Received April 16, 2003; accepted ecember 3, 2003.