{"title": "The Brain Uses Reliability of Stimulus Information when Making Perceptual Decisions", "book": "Advances in Neural Information Processing Systems", "page_first": 1045, "page_last": 1053, "abstract": "In simple perceptual decisions the brain has to identify a stimulus based on noisy sensory samples from the stimulus. Basic statistical considerations state that the reliability of the stimulus information, i.e., the amount of noise in the samples, should be taken into account when the decision is made. However, for perceptual decision making experiments it has been questioned whether the brain indeed uses the reliability for making decisions when confronted with unpredictable changes in stimulus reliability. We here show that even the basic drift diffusion model, which has frequently been used to explain experimental findings in perceptual decision making, implicitly relies on estimates of stimulus reliability. We then show that only those variants of the drift diffusion model which allow stimulus-specific reliabilities are consistent with neurophysiological findings. Our analysis suggests that the brain estimates the reliability of the stimulus on a short time scale of at most a few hundred milliseconds.", "full_text": "The Brain Uses Reliability of Stimulus Information\n\nwhen Making Perceptual Decisions\n\nSebastian Bitzer1\n\nsebastian.bitzer@tu-dresden.de\n\nStefan J. Kiebel1\n\nstefan.kiebel@tu-dresden.de\n\n1Department of Psychology, Technische Universit\u00a8at Dresden, 01062 Dresden, Germany\n\nAbstract\n\nIn simple perceptual decisions the brain has to identify a stimulus based on noisy\nsensory samples from the stimulus. Basic statistical considerations state that the\nreliability of the stimulus information, i.e., the amount of noise in the samples,\nshould be taken into account when the decision is made. However, for perceptual\ndecision making experiments it has been questioned whether the brain indeed uses\nthe reliability for making decisions when confronted with unpredictable changes\nin stimulus reliability. We here show that even the basic drift diffusion model,\nwhich has frequently been used to explain experimental \ufb01ndings in perceptual\ndecision making, implicitly relies on estimates of stimulus reliability. We then\nshow that only those variants of the drift diffusion model which allow stimulus-\nspeci\ufb01c reliabilities are consistent with neurophysiological \ufb01ndings. Our analysis\nsuggests that the brain estimates the reliability of the stimulus on a short time scale\nof at most a few hundred milliseconds.\n\n1\n\nIntroduction\n\nIn perceptual decision making participants have to identify a noisy stimulus. In typical experiments,\nonly two possibilities are considered [1]. The amount of noise on the stimulus is usually varied to\nmanipulate task dif\ufb01culty. With higher noise, participants\u2019 decisions are slower and less accurate.\nEarly psychology research established that biased random walk models explain the response distri-\nbutions (choice and reaction time) of perceptual decision making experiments [2]. These models\ndescribe decision making as an accumulation of noisy evidence until a bound is reached and cor-\nrespond, in discrete time, to sequential analysis [3] as developed in statistics [4]. More recently,\nelectrophysiological experiments provided additional support for such bounded accumulation mod-\nels, see [1] for a review.\nThere appears to be a general consensus that the brain implements the mechanisms required for\nbounded accumulation, although different models were proposed for how exactly this accumulation\nis employed by the brain [5, 6, 1, 7, 8]. An important assumption of all these models is that the\nbrain provides the input to the accumulation, the so-called evidence, but the most established models\nactually do not de\ufb01ne how this evidence is computed by the brain [3, 5, 9, 1]. In this contribution, we\nwill show that addressing this question offers a new perspective on how exactly perceptual decision\nmaking may be performed by the brain.\nProbabilistic models provide a precise de\ufb01nition of evidence: Evidence is the likelihood of a de-\ncision alternative under a noisy measurement where the likelihood is de\ufb01ned through a generative\nmodel of the measurements under the hypothesis that the considered decision alternative is true. In\nparticular, this generative model implements assumptions about the expected distribution of mea-\nsurements. Therefore, the likelihood of a measurement is large when measurements are assumed,\n\n1\n\n\fby the decision maker, to be reliable and small otherwise. For modelling perceptual decision making\nexperiments, the evidence input, which is assumed to be pre-computed by the brain, should simi-\nlarly depend on the reliability of measurements as estimated by the brain. However, this has been\ndisputed before, e.g. [10]. The argument is that typical experimental setups make the reliability of\neach trial unpredictable for the participant. Therefore, it was argued, the brain can have no correct\nestimate of the reliability. This issue has been addressed in a neurally inspired, probabilistic model\nbased on probabilistic population codes (PPCs) [7]. The authors have shown that PPCs can imple-\nment perceptual decision making without having to explicitly represent reliability in the decision\nprocess. This remarkable result has been obtained by making the comprehensible assumption that\nreliability has a multiplicative effect on the tuning curves of the neurons in the PPCs1. Current stim-\nulus reliability, therefore, was implicitly represented in the tuning curves of model neurons and still\naffected decisions.\nIn this paper we will investigate on a conceptual level whether the brain estimates measurement\nreliability even within trials while we will not consider the details of its neural representation. We\nwill show that even a simple, widely used bounded accumulation model, the drift diffusion model, is\nbased on some estimate of measurement reliability. Using this result, we will analyse the results of a\nperceptual decision making experiment [11] and will show that the recorded behaviour together with\nneurophysiological \ufb01ndings strongly favours the hypothesis that the brain weights evidence using\na current estimate of measurement reliability, even when reliability changes unpredictably across\ntrials.\nThis paper is organised as follows: We \ufb01rst introduce the notions of measurement, evidence and\nlikelihood in the context of the experimentally well-established random dot motion (RDM) stimulus.\nWe de\ufb01ne these quantities formally by resorting to a simple probabilistic model which has been\nshown to be equivalent to the drift diffusion model [12, 13]. This, in turn, allows us to formulate\nthree competing variants of the drift diffusion model that either do not use trial-dependent reliability\n(variant CONST), or do use trial-dependent reliability of measurements during decision making\n(variants DDM and DEPC, see below for de\ufb01nitions). Finally, using data of [11], we show that\nonly variants DDM and DEPC, which use trial-dependent reliability, are consistent with previous\n\ufb01ndings about perceptual decision making in the brain.\n\n2 Measurement, evidence and likelihood in the random dot motion stimulus\n\nThe widely used random dot motion (RDM) stimulus consists of a set of randomly located dots\nshown within an invisible circle on a screen [14]. From one video frame to the next some of the\ndots move into one direction which is \ufb01xed within a trial of an experiment, i.e., a subset of the dots\nmoves coherently in one direction. All other dots are randomly replaced within the circle. Although\nthere are many variants of how exactly to present the dots [15], the main idea is that the coherently\nmoving dots indicate a motion direction which participants have to decide upon. By varying the\nproportion of dots which move coherently, also called the \u2019coherence\u2019 of the stimulus, the dif\ufb01culty\nof the task can be varied effectively.\nWe will now consider what kind of evidence the brain can in principle extract from the RDM stim-\nulus in a short time window, for example, from one video frame to the next, within a trial. For\nsimplicity we call this time window \u2019time point\u2019 from here on, the idea being that evidence is ac-\ncumulated over different time points, as postulated by bounded accumulation models in perceptual\ndecision making [3, 1].\nAt a single time point, the brain can measure motion directions from the dots in the RDM display. By\nconstruction, a proportion of measurable motion directions will be into one speci\ufb01c direction, but,\nthrough the random relocation of other dots, the RDM display will also contain motion in random\ndirections. Therefore, the brain observes a distribution of motion directions at each time point. This\ndistribution can be considered a \u2019measurement\u2019 of the RDM stimulus made by the brain. Due to the\nrandomness of each time frame, this distribution varies across time points and the variation in the\ndistribution reduces for increasing coherences. We have illustrated this using rose histograms in Fig.\n1 for three different coherence levels.\n\n1Note that the precise effect on tuning curves may depend on the particular distribution of measurements\n\nand its encoding by the neural population.\n\n2\n\n\fFigure 1: Illustration of possible motion direction distributions that the brain can measure from\nan RDM stimulus. Rows are different time points, columns are different coherences. The true,\nunderlying motion direction was \u2019left\u2019, i.e., 180\u25e6. For low coherence (e.g., 3.2%) the measured\ndistribution is very variable across time points and may indicate the presence of many different\nmotion directions at any given time point. As coherence increases (from 9% to 25.6%), the true,\nunderlying motion direction will increasingly dominate measured motion directions simultaneously\nleading to decreased variation of the measured distribution across time points.\n\nTo compute the evidence for the decision whether the RDM stimulus contains predominantly motion\nto one of the two considered directions, e.g., left and right, the brain must check how strongly these\ndirections are represented in the measured distribution, e.g., by estimating the proportion of motion\ntowards left and right. We call these proportions evidence for left, eleft, and evidence for right,\neright. As the measured distribution over motion directions may vary strongly across time points, the\ncomputed evidences for each single time point may be unreliable. Probabilistic approaches weight\nevidence by its reliability such that unreliable evidence is not over-interpreted. The question is: Does\nthe brain perform this reliability-based computation as well? More formally, for a given coherence,\nc, does the brain weight evidence by an estimate of reliability that depends on c: l = e \u00b7 r(c)2 and\nwhich we call \u2019likelihood\u2019, or does it ignore changing reliabilities and use a weighting unrelated to\ncoherence: e(cid:48) = e \u00b7 \u00afr?\n\n3 Bounded accumulation models\n\nBounded accumulation models postulate that decisions are made based on a decision variable. In\nparticular, this decision variable is driven towards the correct alternative and is perturbed by noise.\nA decision is made, when the decision variable reaches a speci\ufb01c value. In the drift diffusion model,\nthese three components are represented by drift, diffusion and bound [3]. We will now relate the\ntypical drift diffusion formalism to our notions of measurement, evidence and likelihood by linking\nthe drift diffusion model to probabilistic formulations.\nIn the drift diffusion model, the decision variable evolves according to a simple Wiener process with\ndrift. In discrete time the change in the decision variable y can be written as\n\n\u221a\n\n\u03b4y = yt \u2212 yt\u2212\u03b4t = v\u03b4t +\n\n\u03b4ts\u0001t\n\n(1)\n\n2For convenience, we use imprecise denominations here. As will become clear below, l is in our case a\n\nGaussian log-likelihood, hence, the linear weighting of evidence by reliability.\n\n3\n\n0\u00b045\u00b090\u00b0135\u00b0180\u00b0225\u00b0270\u00b0315\u00b01234567890\u00b045\u00b090\u00b0135\u00b0180\u00b0225\u00b0270\u00b0315\u00b024681012140\u00b045\u00b090\u00b0135\u00b0180\u00b0225\u00b0270\u00b0315\u00b0510152025300\u00b045\u00b090\u00b0135\u00b0180\u00b0225\u00b0270\u00b0315\u00b02468100\u00b045\u00b090\u00b0135\u00b0180\u00b0225\u00b0270\u00b0315\u00b02468101214160\u00b045\u00b090\u00b0135\u00b0180\u00b0225\u00b0270\u00b0315\u00b051015202530 3.2%time point 1 9.0%25.6%time point 2\fwhere v is the drift, \u0001t \u223c N (0, 1) is Gaussian noise and s controls the amount of diffusion. This\nequation bears an interesting link to how the brain may compute the evidence. For example, it has\nbeen stated in the context of an experiment with RDM stimuli with two decision alternatives that\nthe change in y, often called \u2019momentary evidence\u2019, \u201dis thought to be a difference in \ufb01ring rates of\ndirection selective neurons with opposite direction preferences.\u201d [11, Supp. Fig. 6] Formally:\n\n\u03b4y = \u03c1left,t \u2212 \u03c1right,t\n\n(2)\nwhere \u03c1left,t is the \ufb01ring rate of the population selective to motion towards left at time point t.\nBecause the \ufb01ring rates \u03c1 depend on the considered decision alternative, they represent a form of\nevidence extracted from the stimulus measurement instead of the stimulus measurement itself (see\nour de\ufb01nitions in the previous section). It is unclear, however, whether the \ufb01ring rates \u03c1 just represent\nthe evidence (\u03c1 = e(cid:48)) or whether they represent the likelihood, \u03c1 = l, i.e., the evidence weighted by\ncoherence-dependent reliability.\nTo clarify the relation between \ufb01ring rates \u03c1, evidence e and likelihood l we consider probabilistic\nmodels of perceptual decision making. Several variants have been suggested and related to other\nforms of decision making [6, 16, 9, 7, 12, 17, 18]. For its simplicity, which is suf\ufb01cient for our\nargument, we here consider the model presented in [13] for which a direct transformation from\nprobabilistic model to the drift diffusion model has already been shown. This model de\ufb01nes two\nGaussian generative models of measurements which are derived from the stimulus:\n\np(xt|left) = N (\u22121, \u03b4t\u02c6\u03c32)\n\n(3)\nwhere \u02c6\u03c3 represents the variability of measurements expected by the brain. Similarly, it is assumed\nthat the measurements xt are sampled from a Gaussian with variance \u03c32 which captures variance\nboth from the stimulus and due to other noise sources in the brain:\n\np(xt|right) = N (1, \u03b4t\u02c6\u03c32)\n\n(4)\nwhere the mean is \u22121 for a \u2019left\u2019 stimulus and 1 for a \u2019right\u2019 stimulus. Evidence for a decision is\ncomputed in this model by calculating the likelihood of a measurement xt under the hypothesised\ngenerative models. To be precise we consider the log-likelihood which is\n\nxt \u223c N (\u00b11, \u03b4t\u03c32)\n\n\u221a\nlleft = \u2212 log(\n\n2\u03c0\u03b4t\u02c6\u03c3) \u2212 1\n2\n\n(xt \u2212 1)2\n\n\u03b4t\u02c6\u03c32\n\n\u221a\nlright = \u2212 log(\n\n2\u03c0\u03b4t\u02c6\u03c3) \u2212 1\n2\n\n(xt + 1)2\n\n\u03b4t\u02c6\u03c32\n\n.\n\n(5)\n\nWe note three important points: 1) The \ufb01rst term on the right hand side means that for decreasing\n\u02c6\u03c3 the likelihood l increases, when the measurement xt is close to the means, i.e., \u22121 and 1. This\ncontribution, however, cancels when the difference between the likelihoods for left and right is\ncomputed. 2) The likelihood is large for a measurement xt, when xt is close to the corresponding\nmean. 3) The contribution of the stimulus is weighted by the assumed reliability r = \u02c6\u03c3\u22122.\nThis model of the RDM stimulus is simple but captures the most important properties of the stim-\nulus. In particular, a high coherence RDM stimulus has a large proportion of motion in the correct\ndirection with very low variability of measurements whereas a low coherence RDM stimulus tends\nto have lower proportions of motion in the correct direction, with high variability (cf. Fig. 1). The\nGaussian model captures these properties by adjusting the noise variance such that a high coherence\ncorresponds to low noise and low coherence to high noise: Under high noise the values xt will vary\nstrongly and tend to be rather distant from \u22121 and 1, whereas for low noise the values xt will be close\nto \u22121 or 1 with low variability. Hence, as expected, the model produces large evidences/likelihoods\nfor low noise and small evidences/likelihoods for high noise.\nThis intuitive relation between stimulus and probabilistic model is the basis for us to proceed to\nshow that the reliability of the stimulus r, connected to the coherence level c, appears at a prominent\nposition in the drift diffusion model. Crucially, the drift diffusion model can be derived as the sum\nof log-likelihood ratios across time [3, 9, 12, 13]. In particular, a discrete time drift diffusion process\ncan be derived by subtracting the likelihoods of Eq. (5):\n\n\u03b4y = lright \u2212 lleft =\n\n(xt + 1)2 \u2212 (xt \u2212 1)2\n\n2\u03b4t\u02c6\u03c32\n\n=\n\n2rxt\n\u03b4t\n\n.\n\n(6)\n\nConsequently, the change in y within a trial, in which the true stimulus is constant, is Gaussian:\n\u03b4y \u223c N (2r/\u03b4t, 4r2\u03c32/\u03b4t). This replicates the model described in [11, Supp. Fig. 6] where the\nparameterisation of the model, however, more directly followed that of the Gaussian distribution\n\n4\n\n\fand did not explicitly take time into account: \u03b4y \u223c N (Kc, S2), where K and S are free parameters\nand c is coherence of the RDM stimulus. By analogy to the probabilistic model, we, therefore, see\nthat the model in [11] implicitly assumes that reliability r depends on coherence c.\nMore generally, the parameters of the drift diffusion model of Eq. (1) and that of the probabilistic\nmodel can be expressed as functions of each other [13]:\n\nv = \u00b1 2\n\n\u03b4t2 \u02c6\u03c32 = \u00b1r\n\n2\n\u03b4t2\n\ns =\n\n2\u03c3\n\u03b4t\u02c6\u03c32 = r\n\n2\u03c3\n\u03b4t\n\n.\n\n(7)\n\n(8)\n\nThese equations state that both drift v and diffusion s depend on the assumed reliability r of the\nmeasurements x. Does the brain use and necessarily compute this reliability which depends on\ncoherence? In the following section we answer this question by comparing how well three variants\nof the drift diffusion model, that implement different assumptions about r, conform to experimental\n\ufb01ndings.\n\n4 Use of reliability in perceptual decision making: experimental evidence\n\nWe \ufb01rst show that different assumptions about the reliability r translate to variants of the drift dif-\nfusion model. We then \ufb01t all variants to behavioural data (performances and mean reaction times)\nof an experiment for which neurophysiological data has also been reported [11] and demonstrate\nthat only those variants which allow reliability to depend on coherence level lead to accumulation\nmechanisms which are consistent with the neurophysiological \ufb01ndings.\n\n4.1 Drift diffusion model variants\n\nFor the drift diffusion model of Eq. (1) the accuracy A and mean decision time T predicted by the\nmodel can be determined analytically [9]:\n\nA = 1 \u2212\n\n1 + exp( 2vb\ns2 )\n\n1\n\n(cid:18) vb\n\n(cid:19)\n\nT =\n\nb\nv\n\ntanh\n\ns2\n\n(9)\n\n(10)\n\nwhere b is the bound. These equations highlight an important caveat of the drift diffusion model:\nOnly two of the three parameters can be determined uniquely from behavioural data. For \ufb01tting\nthe model one of the parameters needs to be \ufb01xed. In most cases, the diffusion s is set to c = 0.1\narbitrarily [9], or is \ufb01t with a constant value across stimulus strengths [11]. We call this standard\nvariant of the drift diffusion model the DDM.\nIf s is constant across stimulus strengths, the other two parameters of the model must explain dif-\nferences in behaviour, between stimulus strengths, by taking on values that depend on stimulus\nstrength. Indeed, it has been found that primarily drift v explains such differences, see also be-\nlow. Eq. (7) states that drift depends on estimated reliability r. So, if drift varies across stimulus\nstrengths, this strongly suggests that r must vary across stimulus strengths, i.e., that r must depend\non coherence: r(c). However, the drift diffusion formalism allows for two other obvious variants\nof parameterisation. One in which the bound b is constant across stimulus strengths, b = \u00afb, and,\nconversely, one in which drift v is constant across stimulus strengths, v = \u00afv \u221d \u00afr (Eq. 7). We call\nthese variants DEPC and CONST, respectively, for their property to weight evidence by reliability\nthat either depends on coherence, r(c), or not, \u00afr.\n\n4.2 Experimental data\n\nIn the following we will analyse the data presented in [11]. This data set has two major advantages\nfor our purposes: 1) Reported accuracies and mean reaction times (Fig. 1d,f) are averages based on\n15,937 trials in total. Therefore, noise in this data set is minimal (cf. small error bars in Fig. 1d,f)\nsuch that any potential effects of over\ufb01tting on found parameter values will be small, especially in\n\n5\n\n\frelation to the effect induced by different stimulus strengths. 2) The behavioural data is accompanied\nby recordings of neurons which have been implicated in the decision making process. We can,\ntherefore, compare the accumulation mechanisms resulting from the \ufb01t to behaviour with the actual\nneurophysiological recordings. Furthermore, the structure of the experiments was such that the\nstimulus in subsequent trials had random strength, i.e., the brain could not have estimated stimulus\nstrength of a trial before the trial started.\nIn the experiment of [11], that we consider here, two monkeys performed a two-alternative forced\nchoice task based on the RDM stimulus. Data for eight different coherences were reported. To avoid\nceiling effects, which prevent the unique identi\ufb01cation of parameter values in the drift diffusion\nmodel, we exclude those coherences which lead to an accuracy of 0.5 (random choices) or to an\naccuracy of 1 (perfect choices). The behavioural data of the remaining six coherence levels are\npresented in Table 1.\n\nTable 1: Behavioural data of [11] used in our analysis. RT = reaction time.\n\ncoherence (%):\naccuracy (fraction):\nmean RT (ms):\n\n3.2\n0.63\n613\n\n6.4\n0.76\n590\n\n9\n\n0.79\n580\n\n12\n0.89\n535\n\n25.6\n0.99\n440\n\nThe analysis of [11] revealed a nondecision time, i.e., a component of the reaction time that is\nunrelated to the decision process (cf. [3]) of ca. 200ms. Using this estimate, we determined the\nmean decision time T by subtracting 200ms from the mean reaction times shown in Table 1.\nThe main \ufb01ndings for the neural recordings, which replicated previous \ufb01ndings [19, 1], were that i)\n\ufb01ring rates at the end of decisions were similar and, particularly, showed no signi\ufb01cant relation to\ncoherence [11, Fig. 5] whereas ii) the buildup rate of neural \ufb01ring within a trial had an approximately\nlinear relation to coherence [11, Fig. 4].\n\n4.3 Fits of drift diffusion model variants to behaviour\n\nWe can easily \ufb01t the model variants (DDM, DEPC and CONST) to accuracy A and mean decision\ntime T using Eqs. (9) and (10). In accordance with previous approaches we selected values for the\nrespective redundant parameters. Since the redundant parameter value, or its inverse, simply scales\nthe \ufb01tted parameter values (cf. Eqs. 9 and 10), the exact value is irrelevant and we \ufb01x, in each model\nvariant, the redundant parameter to 1.\n\nFigure 2: Fitting results: values of the free parameters, that replicate the accuracy and mean RT\nrecorded in the experiment (Table 1), in relation to coherence. The remaining, non-free parameter\nwas \ufb01xed to 1 for each variant. Left: the DDM variant with free parameters drift v (green) and\nbound b (purple). Middle: the DEPC variant with free parameters v and diffusion s (orange). Right:\nthe CONST variant with free parameters s and b.\n\nFig. 2 shows the inferred parameter values. In congruence with previous \ufb01ndings, the DDM variant\nexplained variation in behaviour due to an increasing coherence mostly with an increasing drift v\n(green in Fig. 2). Speci\ufb01cally, drift and coherence appear to have a straightforward, linear relation.\nThe same \ufb01nding holds for the DEPC variant. In contrast to the DDM variant, however, which also\nexhibited a slight increase in the bound b (purple in Fig. 2) with increasing coherence, the DEPC\n\n6\n\n051015202530coherence (%)0510152025bDDM051015202530coherence (%)0.000.010.020.030.040.05sDEPC051015202530coherence (%)01020304050607080sCONST0.000.020.040.060.080.10v0.0000.0010.0020.0030.004v02004006008001000120014001600b\fvariant explained the corresponding differences in behaviour by decreasing diffusion s (orange in\nFig. 2). As the drift v was \ufb01xed in CONST, this variant explained coherence-dependent behaviour\nwith large and almost identical changes in both diffusion s and bound b such that large parameter\nvalues occurred for small coherences and the relation between parameters and coherence appeared\nto be quadratic.\n\nFigure 3: Drift-diffusion properties of \ufb01tted model variants. Top row: 15 example trajectories of y\nfor different model variants with \ufb01tted parameters for 6.4% (blue) and 25.6% (yellow) coherence.\nTrajectories end when they reach the bound for the \ufb01rst time which corresponds to the decision\ntime in that simulated trial. Notice that the same random samples of \u0001 were used across variants\nand coherences. Bottom row: Trajectories of y averaged over trials in which the \ufb01rst alternative (top\nbound) was chosen for the three model variants. Format of the plots follows that of [8, Supp. Fig. 4]:\nLeft panels show the buildup of y from the start of decision making for the 5 different coherences.\nRight panels show the averaged drift diffusion trajectories when aligned to the time that a decision\nwas made.\n\nWe further investigated the properties of the model variants with the \ufb01tted parameter values. The top\nrow of Fig. 3 shows example drift diffusion trajectories (y in Eq. (1)) simulated at a resolution of\n1ms for two coherences. Following [11], we interpret y as the decision variables represented by the\n\ufb01ring rates of neurons in monkey area LIP. These plots exemplify that the DDM and DEPC variants\nlead to qualitatively very similar predictions of neural responses whereas the trajectories produced\nby the CONST variant stand out, because the neural responses to large coherences are predicted to\nbe smaller than those to small coherences.\nWe have summarised predicted neural responses to all coherences in the bottom row of Fig. 3 where\nwe show averages of y across 5000 trials either aligned to the start of decision making (left pan-\nels) or aligned to the decision time (right panels). These plots illustrate that the DDM and DEPC\nvariants replicate the main neurophysiological \ufb01ndings of [11]: Neural responses at the end of the\ndecision were similar and independent of coherence. For the DEPC variant this was built into the\nmodel, because the bound was \ufb01xed. For the DDM variant the bound shows a small dependence\non coherence, but the neural responses aligned to decision time were still very similar across coher-\nences. The DDM and DEPC variants, further, replicate the \ufb01nding that the buildup of neural \ufb01ring\ndepends approximately linear on coherence (normalised mean square error of a corresponding linear\nmodel was 0.04 and 0.03, respectively). In contrast, the CONST variant exhibited an inverse rela-\ntion between coherence and buildup of predicted neural response, i.e., buildup was larger for small\ncoherences. Furthermore, neural responses at decision time strongly depended on coherence. There-\nfore, the CONST variant, as the only variant which does not use coherence-dependent reliability, is\nalso the only variant which is clearly inconsistent with the neurophysiological \ufb01ndings.\n\n7\n\n02004006008001000time from start (ms)\u221220\u22121001020yDDM 6.425.60100200300time from start (ms)05101520mean of y3.26.49.012.025.6-300-200-100DTtime from end (ms)02004006008001000time from start (ms)\u22121.0\u22120.50.00.51.0DEPC0100200300time from start (ms)0.00.20.40.60.81.0-300-200-100DTtime from end (ms)02004006008001000time from start (ms)\u2212600\u2212400\u22122000200400600CONST0100200300time from start (ms)0200400600800100012001400-300-200-100DTtime from end (ms)\f5 Discussion\n\nWe have investigated whether the brain uses online estimates of stimulus reliability when making\nsimple perceptual decisions. From a probabilistic perspective fundamental considerations suggest\nthat using accurate estimates of stimulus reliability lead to better decisions, but in the \ufb01eld of percep-\ntual decision making it has been questioned that the brain estimates stimulus reliability on the very\nshort time scale of a few hundred milliseconds. By using a probabilistic formulation of the most\nwidely accepted model we were able to show that only those variants of the model which assume\nonline reliability estimation are consistent with reported experimental \ufb01ndings.\nOur argument is based on a strict distinction between measurements, evidence and likelihood which\nmay be brie\ufb02y summarised as follows: Measurements are raw stimulus features that do not relate to\nthe decision, evidence is a transformation of measurements into a decision relevant space re\ufb02ecting\nthe decision alternatives and likelihood is evidence scaled by a current estimate of measurement\nreliabilities. It is easy to overlook this distinction at the level of bounded accumulation models,\nsuch as the drift diffusion model, because these models assume a pre-computed form of evidence as\ninput. However, this evidence has to be computed by the brain, as we have demonstrated based on\nthe example of the RDM stimulus and using behavioural data.\nWe chose one particular, simple probabilistic model, because this model has a direct equivalence\nwith the drift diffusion model which was used to explain the data of [11] before. Other models may\nhave not allowed conclusions about reliability estimates in the brain. In particular, [13] introduced\nan alternative model that also leads to equivalence with the drift diffusion model, but explains dif-\nferences in behaviour by different mean measurements and their representations in the generative\nmodel. Instead of varying reliability across coherences, this model would vary the difference of\nmeans in the second summand of Eq. (5) directly without leading to any difference on the drift\ndiffusion trajectories represented by y of Eq. (1) when compared to those of the probabilistic model\nchosen here. The interpretation of the alternative model of [13], however, is far removed from basic\nassumptions about the RDM stimulus: Whereas the alternative model assumes that the reliability of\nthe stimulus is \ufb01xed across coherences, the noise in the RDM stimulus clearly depends on coherence.\nWe, therefore, discarded the alternative model here.\nAs a slight caveat, the neurophysiological \ufb01ndings, on which we based our conclusion, could have\nbeen the result of a search for neurons that exhibit the properties of the conventional drift diffusion\nmodel (the DDM variant). We cannot exclude this possibility completely, but given the wide range\nand persistence of consistent evidence for the standard bounded accumulation theory of decision\nmaking [1, 20] we \ufb01nd it rather unlikely that the results in [19] and [11] were purely found by\nchance. Even if our conclusion about the rapid estimation of reliability by the brain does not en-\ndure, our formal contribution holds: We clari\ufb01ed that the drift diffusion model in its most common\nvariant (DDM) is consistent with, and even implicitly relies on, coherence-dependent estimates of\nmeasurement reliability.\nIn the experiment of [11] coherences of the RDM stimulus were chosen randomly for each trial.\nConsequently, participants could not predict the reliability of the RDM stimulus for the upcoming\ntrial, i.e., the participants\u2019 brains could not have had a good estimate of stimulus reliability at the\nstart of a trial. 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Front Neurosci, 6:75,\n\n2012.\n\n9\n\n\f", "award": [], "sourceid": 666, "authors": [{"given_name": "Sebastian", "family_name": "Bitzer", "institution": "TU Dresden"}, {"given_name": "Stefan", "family_name": "Kiebel", "institution": "TU Dresden"}]}