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Unformatted text preview: Banburismus and the Brain: Decoding Review the Relationship between Sensory Stimuli, Decisions, and Reward Joshua I. Gold and Michael N. Shadlen Presenter: He Huang 04/22/2010 Outline • • • Turing’s theoretical framework and “Banburismus” Application to 2AFC decisions on RT perceptual tasks Experimental evidence Background: Banburismus at Bletchley Park • Problem: Decrypt the ‘Enigma machine’ Solution: Compare if two messages received are from • the same encoding scheme. Background: Banburismus at Bletchley Park Formalize the problem • Hypothesis testing (given the evidence from the messages): • h0: the encoding scheme for the two messages was different • h1: the encoding system was the same Background: Banburismus at Bletchley Park Formalize the problem h1: the encoding systems that generates (m1, m2) is different; h0: the encoding systems that generates (m1,m2) is the same; • • Pr(m|h0): the likelihood of getting a match given h0. Pr(m|h1): the likelihood of getting a match given h1; What is the decision rule? When is the stopping time? of letters they encoded also negative number describing the weight of ,evidence crossed eithereach other. decision matched barrier, the There- one pair of c tions w te givendecide h1 thePr(m|h1a was greater than the ) h1, denoted other ), on he fore, themade. evidence formulation for statistical When the was expected “match rate” fromdecide h . applicaA similar needed to the encoded mesthe war nt Second, it pre elihoodweight of a match given h0, denoted Pr(m|h0). of getting 0 e- sages woulddeveloped crossed eitherby Wald shortly after tions was evidence independently barrier, the decision be the same as that evidence unencodedPark As Tu from the proBackground: Banburismus at Bletchleyfollows. By Bay of weight of ccordingly, Turing defined the weight of s- messages, (Wald, is equal to 1/13 in ordinary German the war which 1947). advanta e be related to s ded byFormalizeA similaraformulation for statistical applicaawas made.similarly, non-match) in favor of h1 match (or, the problem ed text. In other words, the likelihood of getting a match As Turing was aware, this formulation had two distinct instruct ethe logarithm of the independently by Wald shortly after ver h0 astions was developed ratio of these likelihoods: an (m )Evidence 1favoring h1 Pr(mahdefinite stoppingthan the advantages. First, it had | 1), was greater time that Pr(on m) h1| one given h , denoted / evidence favoring h0 sthe war (Wald, 1947). log ch likelihood of getting a match given hwhen to stop working Pr(Second instructed the codebreaking team , denoted Pr(m|h ). h0|m) 0 0 d As Turing was aware, thisPr(m|h1) formulation had two distinct he Accordingly, Turing defined the weight of evidence proon one of evidence 1) Weight pair of ciphers and turn their attention (to another. log follows n advantages. First, it had aPr(m|h0) stopping time that definite Second, it predicted the accuracy of the decision,hwhere be h0|m as Pr( rela 0). h vided by a match (or, similarly, athe weight ofto stop working instructed the codebreaking non-match) in favor ofcan team when evidence 1 ofollows. By Bayes’ theorem, probabilities a over h0 of the logarithm of the ratio of attention to another. Bayes probabilities, and probabilities: ea logarithmasrule:to several this quantity isthese likelihoods: on one pair Pr(h|m) =otherturn their measured of ciphers Pr(m|h)Pr(h)/Pr(m) s1 be related h pothesis after Pr( log ). Second, it predicted of accuracy of the decision, as ) and Pr(h 0s: units that depend on the basethethe logarithm (Good, Pr(h0 Pr( Pr(m of Pr( 1|m) Pr(hof o- In Banburismus, logarithms 1) base weight|hused follows.hBy Bayes’ theorem, the 10 were 1) evidence describe the p can 979). Weight oflog evidence lweight of evidence, (1) og (2) log Pr(m| actuPr(h0|m) be related to (the Banburismus processh other h1 were called “bans” severalPr(h0) probabilities:0) nd where evidence is sa (1) s: computed weights of evidence in units of 1/10th of lly probab prior probabili wherer(Pr(m|m) and Pr(h11|)m) are called the posterior P h1| h0of probabilities, this quantity is measured ) Pr(h As log logarithm ban—a adeciban—that the codebreakers considered log weight of evidence, hy(2) pothesi assumption m probabilities) and describe) the probability of each Pr(h0smalleston the baseevidence that is (Good, |m Pr(h0of of the logarithm be in units that depend the evidence has been sampled, and Pr(h a ed “about the after all weight evidence is0)eq pothesis 1) 1979). In Banburismus, Pr(h |m) of called were used irectly perceptible htom) ) andlogarithmsare base 10 the posterior human called the prior probabilities and intuition”; Good, 1979). describ d, posterior prob where and Pr(h1 are Pr(h0) Pr( 0| 1 robabilities and describe the probability of each hyh instructed the codebreaking team when to stop working othesison oneall the evidence has been sampled, and after pair of ciphers and turn their attention to another. e r( Background: Banburismus of the and ). h0) and Pr(h1) it predictedthe prior probabilitiesdecision, as Second, are called the accuracy at Bletchley Park escribefollows. By Bayes’ problem weight of evidence can the probability of theorem, the each hypothesis before any Formalize the vidence isrelated to severalexample, there is an equal be sampled. If, for other probabilities: 1 rior probability of either hypothesis (which was the : Pr(h1| in Pr(h1) ssumption madem) Banburismus), then theof evidence, (2) weight of log weight log Pr(h0| to Pr(h0) vidence is equalm) the logarithm of the ratio of the ) osterior probabilities. In this case, for a given weight ifwhere knowledge is equal 1|for the two hypothesis prior Pr(h0|m) and Pr(h m) are called the posterior f evidence (e.g., the stopping point in Banburismus): probabilities and describe the probability of each hypothesis |m) all the evidence has been sampled, and after Pr(h1 eight of evidence B (3) log h0) and Pr(h1)ware called the prior probabilities and , Pr( Pr(h |m) 0 describe the probability of each hypothesis before any here Bevidence is sampled. If, for example, there is an equal is a constant that represents the barrier height h1 is1 true two of How much the hypotheses f favor prior .probabilitymutually exclusive evidence favors theover h0 either hypothesis (which was h1 of h For 1 Pr( 1|m) in Banburismus), then 1 r(h0|m)assumptionhmadeor, equivalently, Pr(h0) the weight of s(h )), and assuming the to the logarithm of the ratio of the evidence is equal weight of evidence is exr1 a constant that represents the barrier height f h1. For two mutually exclusive hypotheses 1 Br(h1|m) or, equivalently, Pr(h0) P ackground: Banburismus1at Bletchley d assuming the weight problem-decision rule of evidence is exFormalize the bans, this rearranges to: Pr(h1|m) 1 1 . B 10 Park P(h1|m) (4) indicates that the posterior probability of h1 B nly on the value of the barrier, B, and not on h1 is true How much the evidence favors h1 over h0 lar samples of evidence, m, encountered. In Decision rule: setting B to achieve an expected level of performance (confidence that h1 s, as ilong as the weight of evidence reaches s true). e probability that h1 is correct is a fixed value. Speed-Accuracy trade off: more evidence needs to be accumulated in order to increase Banburismus, setting thethe correctof the barrier height answer. the the probability of getting mined the weight of evidence to accumulate mmitting to a decision was equivalent to setpected level of performance. For example, Implementation: Banburismus Performed by Neurons (in 2 AFC choice) • How does the activity of sensory neurons compute the “weight of evidence”? • How does the brain make use of the sensory evidence? • What is the decision rule based on this evidence? 992;(and assume opponency, giving an ,analogous expres- , Salzman Pr(x y|h1) 2 2 in these areas h0): 2 √1 sion under that the brain Banburismus Performed by Neurons Q where e2 is was most Sensory xSignals-Turing’s Weight of Evidence , (5) Pr( ,y|h1) 2 22 2 Neuron (y 2 (x 1 2 (x √1 1) 0) 1)(y 302 Q 2 2 1 Using the difference (x-y) in spike rates from two where • (5) ) 0 , neurons (one 2 the common variance, other favors Down favors Up motion, the and are the means of 2 is cisions about (x 2 1 ) (y ) 2 (x )(y )0 1 1 0 1 0 motion) x and y, respectively, under h1, and , is the covariance Qlike Tu- 2 tities 2 1 between x and y. Solving for the weight of evidence (in ically related h1: Up motion ( x is larger) 2 is the common variance, 1 and 0 are the versus h0 yields: units of natural bans) for h1 means of 1966). Briefly, x and rate y, Down motion (y h1, and s case, ah2: respectively, under is larger) is the covariance Pr(x,y|h1(in ) between x Solving for the weight log thought of as and y. weight of evidence of evidence units of natural bans) for h1 versus h0 yields:Pr(x,y|h0) us conditions value but can ( 1 constant 1 Relationship between Neural Responses an Figure 1. Theoretical Pr(x,y|h1) 0) · · hypothetical distributions of respon y) (6) The panels on the left depict (x nt orweight of evidence log trial-to2 example, the upper curve could represent the responses of a dire (1 ) Pr(x,y|h0) then represent the responses of the opposing “antineuron” that pr on numerous responses are normally distributed with mean values of and the exponential and Poisson, lead to similar results (Gold and S ) ( 1numerous 1 ese probabiliNote that factors could cause responses plotted as a function of their diff nonzero values 0 computed from these · · (x y) (6) 2 easurements of , including intrinsic factors like attention and arousal (1 ) • • 0 1 measured pothesis after all the by the neuron and antineuron up to that tions of the total number of spikes generated evidence has been sampled, and hm (Good, barrier at 0B, andprocess are called the decision is rendered for h1. e reaches theReview Pr(h ) the Pr(h1) is stopped and a prior probabilities and were used 303 reaches B (not shown), a decision for h0hypothesiserror. Banburismus Performed by before any ight of evidence describe the probability of each is issued inNeurons cess actuevidence is sampled. If, for example, there is an equal Sensory Signals-Turing’s Weight of Evidence th f 1/10 of prior probability of either hypothesis (which was the onsidered eant that 1968; Link and Heath, 1975; Luce, 1986;the weight of assumption made in Banburismus), then Ratcliff and ce that is , implied Rouder, 1998;equal to the logarithm of the ratio of the evidence is Stone, 1960; Usher and McClelland, 2001; od, Howposterior probabilities. In this case, for a given weight er. 1979). Vickers, 1979). df the co“natural of evidence (e.g., is analogous to in Banburismus): This problem the stopping point one-dimensional ponses and Turing’s Weight of Evidence ted quanlar responses from a pair of sensory neurons when h a pairtrue, barriers (Link, 1992; Ratcliff Brownian motion to or h is of as indicated. For of tradePr(h1|mmotion. The lower curve would ) obabilities es of a direction-selective neuron that prefers upward The weight of evidence articular and Rouder, 1998). means of the normal B distribu(3) log ron” that prefers downward motion. Here wePr(h the simplifying assumption that the make |m) surprisal” r can be equal variances ( represent the drift rates, i (where the subscript i tions ). Note that other0 forms of the distribution, including f and and bans but old and Shadlen, 2001). The panelthe particular Turing’s weight of evidenceThe psychometon the f reward reflects B is a right shows motion strength).the barrier height where constant that represents of their difference; note the linear relationship. m of base ric function describing the probability of correctly reachexclusive hypotheses in favor of h1. For two mutually accumulate/integrate over Decision Model tion-time Figure 2. ing eitherh the is“up” r(h difference between equivalently, given) antineuron is: or “down” barrier on a Pr(h trial The weight (Pr(h favor of evidence in |m) over h the accumulated |m) or, responses of a neuron that prefers h and an 1depicted P computed under thetcondition) thatyh( is)true. The thin, wavy line depicts a simulated 1 0 0 ntlities has that prefers weight of7).evidence atare motion ( d (7) total h (see Equation The curves time1t k ihich the trajectory that represents how the weight of evidence might grow on a single trial as a function of time. The dashed line depicts the expectation 0 Pr(h1)), and assuming the weight of evidence is exn motion (mean value) of this trajectory at each time point. Note that changing the constant of proportionality used to relate the accumulated difference 100 years to the weight of evidence (see text for details) simply scales the ordinate. The two insets illustrate the correspondence between the weight to make ofwhere kthe underlying rule:antineuron this ifatk does 1 viewed as hypothetical (normal, equivariate) probability(8) stimulus is a constant. Note that two time points, equal the not Decision in bans, responsesrearranges| to: by the neuron and antineuron up to that , pi evidencepressed neuron and and 2 hesis x i value of that density functions.that h proportionality in Equation 6, thenBgenerated constantThese is true. If the weightthe distributions of thethe 1number Be process| isvalueand a decision is rendered for h . of functions describe of evidence reaches total at of, spikes the stopped sensory time point, given barrier the llthat “two This is the is computed h is true. If the weight of evidence reaches to (not shown), a decision for h is issued in error. value of that expected outcome when is merely proportional B the weight 1 rtex. We of evidence. Regardless of the value of k, however, the (Recall) Equation 4, 1is the logistic function (here using between . (4) Pr(h |m) t, that is neither which,speed. Higher barriers meant that of1968; 1 and10 B1975; Luce, 1986; Ratcliff and like le (x tween accuracy and Link ence algorithm for updating the weight evidenceHeath,the is ght, when more evidence was e ). Unlike in turn, implied Rouder, 1998; Stone, 1960; Usher and McClelland, 2001; base accumulated, which, in Equation 4, however, the probability 0 0 1 2 1 1 0 0 1 1 1 1 1 0 time measured pothesis after all the by the neuron and antineuron up to that tions of the total number of spikes generated evidence has been sampled, and hm (Good, barrier at 0B, andprocess are called the decision is rendered for h1. e reaches theReview Pr(h ) the Pr(h1) is stopped and a prior probabilities and were used 303 reaches B (not shown), a decision for h0hypothesiserror. Banburismus Performed by before any ight of evidence describe the probability of each is issued inNeurons cess actuevidence is sampled. If, for example, there is an equal Sensory Signals-Turing’s Weight of Evidence th f 1/10 of prior probability of either hypothesis (which was the onsidered eant that 1968; Link and Heath, 1975; Luce, 1986;the weight of assumption made in Banburismus), then Ratcliff and ce that is , implied Rouder, 1998;equal to the logarithm of the ratio of the evidence is Stone, 1960; Usher and McClelland, 2001; od, Howposterior probabilities. In this case, for a given weight er. 1979). Vickers, 1979). df the co“natural of evidence (e.g., is analogous to in Banburismus): This problem the stopping point one-dimensional ted quanlar tradeBrownian motion to a pair of barriers (Link, 1992; Ratcliff Pr(h1|m) obabilities articular and Rouder, 1998). The weight of evidence B means of the normal distribu(3) log Pr(h0|mdrift rates, (where the subscript i ) surprisal” r can be tions represent the i bans but f reward reflects B is particular motion strength).the barrier height the a constant that represents The psychometwhere m of base ric function describing the probability of correctly reachin favor of h1. For two mutually exclusive hypotheses Decision Model tion-time Figure 2. ing eitherh the is“up” r(h difference between equivalently, given) antineuron is: or “down” barrier on a Pr(h trial The weight (Pr(h favor of evidence in |m) over h the accumulated |m) or, responses of a neuron that prefers h and an 1 1 0 7). The curves depicted P computed under the condition that h is true. The thin, wavy line depicts a simulated 1 0 ihich the trajectory that represents how the weight of evidence are grow on a single trial as a function of time. The dashed line depicts the expectation lities has that prefers h (see Equation and assuming the of proportionality of relate the accumulated is exPr(h1)), at each time point. Notemightchanging the constantweight used to evidence difference (mean value) of this trajectory 100 years to the weight of evidence (see text for details) simply that the ordinate. The two insets illustrate the correspondence between the weight 1 to make of evidencepressedrule:antineuronscales i rearranges as hypothetical (normal, equivariate) probability(8) Decision neuron and the distributions of theattotal number of spikesBgenerated by the neuron and antineuron up to that , p two time points, viewed to: and the underlying in bans, this responses 2|i hesis that density functions. These functions describe 1 at Bethe process| is stopped and a decision is rendered for h . sensory time point, given that h is true. If the weight of evidence reaches the barrier , that “two This is the expected outcome when h is true. If the weight of evidence reaches B (not shown), a decision for h is issued in error. 1 rtex. We (Recall) Equation 4, 1is the logistic function (here using . (4) Pr(h |m) t, that is neither which,speed. Higher barriers meant that 1968; 1 and10 B1975; Luce, 1986; Ratcliff and like le tween accuracy and Link Heath, ght, when more evidence was e ). Unlike in turn, implied Rouder, 1998; Stone, 1960; Usher and McClelland, 2001; base accumulated, which, in Equation 4, however, the probability 1 0 0 1 1 1 1 1 0 Pr(x,y|h1) 994). Thus, neural activity in these areas Banburismus Performed by Neurons ncoded, noisy evidence that the brain Neuron where 304 Speed-Accuracy Tradeoff o decide which hypothesis was most Neuron 304 1 (x ) 1 2 2 (y √1 2 ) 0 2 2 P(Correct) QPredicted Effect of Barrier Height Accuracy y Signals to Turing’s Figure 3. 2 2 1 on Speed, Accuracy, and Reward Rate in a Figure 3. Predicted Effect of Direction-Discrimination Task Barrier Height nce on Speed, Accuracy, and Reward Rate in The calculations assume that an experiment a 2 Direction-Discrimination (see uses six levels ofthe Task text for dedifficulty is assume common variance, 1 and consider categorical decisions about The (A) Accuracy. that an experiment tails). calculations The probability of a coruses six levels of difficulty all text for derect choice averaged across(see six motion x Accuracy. The function of barrier gnals by formulating quantities like Tutails). (A)and y, respectively, under h1, and strengths is plotted as a probability of a correct choice averaged ensures six deciheight. A higher barrieracross all that motion strengthsbased on aas a xweight of evi- . vidence that are monotonically related sions between function of barrier Solving for the weig are is plotted larger and y height. A higher barrier ensures on decidence and thus are more accurate thataversions are based of larger weight of evion a natural bans) for h age. The dashed vertical line is the barrier units rate maximum on averratio (Green and Swets, 1966). Briefly, 1 versus Reward are more accurate rate of = Accuracy/ Response Timeh dence and thus associated Reward rate Response Time height The dashedwith the line is the barrier age. reward. (B) Reward vertical reward rate derate. The a sensory neuron—in this case, a rate height associated with the maximum rate of pends on the decision time plus fixed times Pr(x,y|h1) reward. (B) Reward plus a “time out” for dein and between trials rate. The reward rate an kes per second—can be thought of as pends weight time occurs times error. on the maximum reward rate evidence log B=A betweendecisionplusofplus fixedwhenan in decision process is sufficiently out” for trials a “time accurate the and Pr(x,y|h0) error. A maximum consuming. We suggest reward rate occurs when but not overly time le: for a given set of stimulus conditions the decision adjusts the barrier height to that the brain process is sufficiently accurate but not overly time rate of reward. suggest achieve the maximumconsuming. We(C) Psytion), it has an expected value but can that the function using the optimal barrier chometric brain adjusts the barrier height to (1 0) achieve the maximum rate of reward. (C) Psyheight. The probability of a correct response · ly from moment-to-moment or trial-toischometric function using the optimaldefiniplotted for each motion strength. By barrier 2 (1 height. performance of the 0% response tion, the The probabilityon a correctcoherent motion response can take at The best fitting cumulative Weibull, a plotted for each motion reasonable approximation to logistic eural stimulus is 8). chance (not shown).usingline is aonbarrier height. The time fromwhich is isonset until behavioral onstrength.isBy defininumerous stimulustion, the performance responsethe coherent the 0% plotted function (Equation (D) Response times the optimal motion of the motion strengths. The response line motion strength 250 is reasonable approximation to at chance (not shown). The a best time cumulative Weibull, each stimulus h aatcertainis probability. time isisdecisionfitting height.Equation 9)fromwhich ms.aonset until behavioral responsethe logistic factors could cau probabiliB Response times usingThesebarrier (from The time plusstimulus Note that numerous function (Equation 8). (D) the optimal is plotted at each of the motion strengths. The response time is decision time (from Equation 9) plus 250 ms. ated bythe average decision time also depends on B that there are three other intervals, each of a fixed durataking repeated measurements of , including intrinsic factors like att Note that of and the particular list(or group of neurons) to non-decision time t onextrinsic factors like the light level a neuron of motion strengths chosen by the tion: the a and each presentation (e.g., fix probabilities. In this case, for a given weight Review ce (e.g., the stopping point in Banburismus): 305 g Experimental Evidence Pr(h1|m) Neurophysiology: Banburismus in the Brain weight of evidence B (3) Pr(and-error, in which the barrier is raised and lowered until h0|m) the maximum rate of reward is achieved. Interestingly, in this case, the constant of proportionality that relates the accumulated difference (x y) to the weight of evidence 1 is not needed to find the barrier height that leads to the maximum rate of reward. That information is not lost, however: once the barrier height is fixed, it corresponds 0 1 to a particular level of overall performance and thus can be expressed in units of the weight of evidence, such as natural bans. Note that this quantity is not the weight of evidence that would be calculated based on knowledge of the stimulus motion strength and the associated sensory response distributions because that information would 1 lead to perfect performance at all motion B strengths. Rather, it is the weight of evidence that corresponds to a fixed level of uncertainty across all stimulus strengths in an experiment (this quantity will tend to overestimate the evidence from weak stimuli and underestimate the evidence from strong stimuli). Accordingly, the evidence that accumulates during a trial can be interpreted as a fraction of this quantity and thus in units of natural bans—even when the scaling between the decision variable and the weight of evidence is not known (e.g., if the brain does not know the shapes of the sensory response distributions). 1 is a constant that represents the barrier height f h . For two mutually exclusive hypotheses 1 Pr(h |m) or, equivalently, Pr(h ) 1 nd assuming the weight of evidence is exin bans, this rearranges to: Review 303 Pr(h |m) 1 1 . 10 (4) LIP neurons? 4 indicates that the posterior probability of h 1 only on the value of the barrier, B, and not on ular samples of evidence, m, encountered. In rds, as long as the weight of evidence reaches he probability that h is correct is a fixed value. Figure 2. Decision Model 1 logarithm of the ratio of the equal to the between x and y. Solving for the weight of evidence (in nically related obabilities. Incommon variance, this case, for a given bans)are the versus h yields: weightfor h means of 2 is the units of natural 1966). Briefly, 1 and 0 1 0 (e.g., the stopping pointExperimentalsEvidence in Banburismus): the covariance Turing’s and y, of x Weight respectively, under h1, and i is case, a rate Evidence Pr(x For Neurophysiology: Banburismus |h ) the Brain s from a pair of sensory neurons whenfor or h1 is true, asof evidence 1(in h0evidence indicated.,y in between x Solving thought of as and y. weight of the weight log Pr(h1|m) neuron that prefers upward motion. The lower curve would tion-selective ofweight of evidence versus h (3) units natural bans) for h1 B 0 yields:Pr(x,y|h0) lushconditions Pr( 0|m) fers downward motion. Here we make the simplifying assumption that the value but can d equal variances ( 2 ). Note that other forms x,y|h distribution, including of the ) 1 (1 Pr( 0) 1 · · (x y) (6) adlen, 2001). thatpanel on the rightlog shows Turing’s weight constant The evidence ent 1. weight ofrepresents the barrier height of evidence(1 or trial-to2 ) Pr(x,y|h0) rence; note the linear relationship. honFor two mutually exclusive hypotheses numerous 1. 1 Pr(h1|m) or, ) ( ese probabili- equivalently,1numerous 1 and lowered until Note that Pr(hthe)barrier is1 factors could cause nonzero values 0 and-error, in which 0 raised · · (x y) (6) t the maximum rate of reward is achieved. Interestingly, in the 2 constant assuming the weight of evidenceof(xproportionality thatofrelates the( ) like7) 2.total weight of evidencethis case,time t iskto ex- factors d ( attention and arousal at difference (1 the weight ) y easurements of , including intrinsic () evidence accumulated y) 0 is not needed to find the barrier height that leads to the ans, this rearrangesand extrinsicoffactors likeisthe light level or other variations to: maximum rate reward. That information not lost, neurons) to a however: could cause nonzero the corresponds Note that numerous factorsoncelevelbarrier height is fixed, itnot equal values if For example, to a particular g thewhere k is a constant. Note that of overalldoesevidence, such forthe motion-discriminadistribuin the stimulus. ofk performance and thus can the be expressed in units the weight of 1 of , including intrinsic factorsNote that this quantity is not the weight value like attention the as natural Equation 6, constant of proportionality inbans. would be calculatedathen and arousal is typically prethat based on knowleightPr(hsuch tion task,themotion in and associated direction 3. of extrinsic factors B. of evidencestimulus motion strength(4)thegivenvariations 1|m) edge of and is computed1is merely response distributions becauseto informa- weight like theproportional other light level that 1 sentedsensory a variety or that the 0 stimulus (e.g., at of strengths (i.e., h0 and h1 each tion would lead to perfect performance at all motion in the stimulus. For example, forthe weight of evidence that correof evidence. Regardlessstrengths. Rather, it is the motion-discriminaof the value of k, however, the sponds to a fixed level of uncertainty across all stimulus ndicates that the posterior strengths weight of quantity willtypically preprobability of algorithm motion in a the in theexperiment (this evidence tion task, for updating overestimate an directionstimuli andtend to is the given evidence from weakh1is underly onsented accumulatebarrier,evidenceevidence fromin spike0 rates h1 each the value a the the thestrengths strong stimuli). Accordingly, B, that accumulates on same: at of variety ofestimate the and not during ah can be over difference (i.e., trial and interpreted as a fraction of this quantity and thus in units time. of evidence, Figure 2, thiswhen the In r samplesAs illustrated inm, ofencountered.scaling between accumunatural bans—even temporally the decision variable and the weight of evidence is not lating evidence can be known (e.g., if the reaches thought of as simply of , as long as the weight of evidencebrain does not know the shapes a single the sensory response distributions). Review 305 ...
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This note was uploaded on 09/15/2011 for the course COGS 1 taught by Professor Lewis during the Spring '08 term at UCSD.

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