Three methods for measuring perception. Magnitude estimation. Steven s power law. Matching. P = k S n. Detection/discrimination

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1 Three methods for measuring perception. Magnitude estimation 2. Matching 3. Detection/discrimination Magnitude estimation Have subject rate (e.g., -) some aspect of a stimulus (e.g., how bright it appears or how load it sounds).. Steven s power law Matching P = k S n P: perceived magnitude S: stimulus intensity k: constant In a matching experiment, the subject's task is to adjust one of two stimuli so that they look/sound the same in some respect. Relationship between intensity of stimulus perception of magnitude follows the same general equation in all senses Example: brightness matching Detection/discrimination In a detection experiment, the subject's task is to detect small differences in the stimuli. Psychophysical procedures for detection experiments: Method of adjustment. Yes-No/method of constant stimuli. Simple forced choice. Two-alternative force choice

2 Method of adjustment Yes/no method of constant stimuli Ask observer to adjust the intensity of the light until they judge it to be just barely detectable Example: you get fitted for a new eye glasses prescription. Typically the doctor drops in different lenses asks you if this lens is better than that one. Percent yes responses Tone intensity Do these data indicate that Laurie s threshold is lower than Chris s threshold? Forced choice Present signal on some trials, no signal on other trials (catch trials). Subject is forced to respond on every trial either ``Yes the thing was presented'' or ``No it wasn't'. If they're not sure then they must guess. Advantage: We have both types of trials so we can count both the number of hits the number of false alarms to get an estimate of discriminability independent on the criterion. Versions: simple forced choice, 2AFC, 2IFC Simple forced choice: four possible outcomes yes no Information acquistisition yes no present Hit Miss present Hit Miss absent False alarm Correct reject absent False alarm Correct reject

3 shift Information criterion yes no Two components to the decision-making: information criterion. present Hit Miss Information: Acquiring more information is good. The effect of information is to increase the likelihood of getting either a hit or a correct rejection, while reducing the likelihood of an outcome in the two error boxes. absent False alarm Correct reject : Different people may have different bias/ criterion. Some may may choose to err toward "yes" decisions. Others may choose to be more conservative say "no more often. : probability of occurrence curves Discriminability (d ) N d S+N N: noise only (tumor absent) S+N: signal plus noise (tumor present) d = separation spread Discriminability (d or d-prime ) is the distance between the N S+N curves Example applications of SDT Vision Detection (something vs. nothing) Discrimination (lower vs greater level of: intensity, contrast, depth, slant, size, frequency, loudness,... Memory (internal response = trace strength = familiarity) Neurometric function/discrimination by neurons (internal response = spike count) when no tumor when tumor present N S+N

4 Hits: respond yes when tumor present Correct rejects: respond no when tumor absent when no tumor when tumor present when no tumor when tumor present Misses: respond no when present False alarms: respond yes when absent when no tumor when tumor present when no tumor when tumor present shift SDT: Gaussian case z[p(cr)] z[p(h)] N S+N c d x d = z[p(h)] + z[p(cr)] = z[p(h)] z[p(fa)] β = c = z[p(cr)] p(x = c S + N) p(x = c N) = e (c d )2 / 2 e c2 / 2 G(x;µ,σ) = 2πσ e (x µ) 2 / 2σ 2

5 ROC ROC # #2 Hits Hits False Alarms False Alarms ROC ROC #3 #4 Hits Hits False Alarms False Alarms Receiver operating characteristic (ROC) ROC: Gaussian case N S+N z[p(h)] c z[p(fa)]

6 Gaussian unequal variance N c Text S+N Rapid estimation of full ROC: confidence ratings Hits False Alarms SDT review Measuring thresholds Your ability to perform a detection/discrimination task is limited by internal noise. Information (e.g., signal strength) criterion (bias) are the 2 components that affect your decisions, they each have a different kind of effect. Because there are 2 components (information & criterion), we need to make 2 measurements to characterize the difficulty of the task. By measuring both hits & false alarms we get a measure of discriminability (d ) that is independent of criterion. Low intensity d 3 2 Signal intensity High intensity Two-alternative forced choice Staircase Two options presented on each trial: Two stimuli presented simultaneously at two different positions (e.g., one of which has higher contrast). Two stimuli presented sequentially at the same position. One stimulus presented with two possible choices (e.g., moving right or left). Subject is forced to pick one of the two options. If they're not sure then they must guess. Feedback (correct/incorrect or $) provided after each trial. Advantage: Two options with feedback balances criterion so we can measure proportion correct.

7 Absolute relative thresholds Weber s law Difference in intensity Absolute threshold Relative thresholds Background intensity Ernst Weber, c85 Gustav Fechner, c85 Weber s law Fechner s analysis Increase in intensity Background intensity

8 Weber s law: Fechner s derivation Weber s law: Fechner s derivation R x x+dx d = R σ Intensity R = log(x) = log(x + dx) At threshold: d = R = σ R = log(x) = log(x + dx) R = σ σ = log(x + dx) log(x) = log x + dx x = log + dx x e σ = dx x dx x = k Weber s law: contrast ratio derivation R = x x = = x + dx x R = dx x intensity of test flash intensity of background dx x = σ d = R σ At threshold: d = R = σ Weber's law: To perceive a difference between a background level x the background plus some stimulation x+dx the size of the difference must be proportional to the background, that is, dx = k x where k is a constant. Fechner's interpretation: The relationship between the stimulation level x the perceived sensation s(x) is logarithmic, s(x) = log(x). Main difference: Fechner's is an interpretation of Weber's law, a hypothesis. Visual stimulus Behavioral protocol Dots Aperture Receptive field Pref target Neurometric function Psychometric function Two-alternative forced choice Null target deg Fixation Point Fix Pt Neuronal response? Choice probability Behavioral judgement Dots Targets sec

9 Stimulus manipulation: motion coherence Psychometric function % coherence 5% coherence % coherence Britten, Shadlen, Newsome & Movshon, 992 Motion coherence MT neurons Motion coherence MT neurons Motion stimulus no coherence 5% coherence % coherence Responses of MT neurons Preferred direction Neural responses are noisy Each tick is an action potential Perceptual decision Decision rule: Monitor the responses of two neurons on each trial, the one being recorded another selective for the opposite motion direction. Choose pref if pref response > non-pref response. Each line corresponds to a stimulus presentation Average across all trials Firing rate (sp/sec) Trial number Time (msec) f n f p f n (r) f p (r) r Neuronal response f p (r) : response PDF for pref direction f n (r) : response PDF for non-pref direction

10 6.4% correct Neurometric function rp : response to pref direction rn : response to non-pref direction f p (r) : response PDF for pref direction fp fn r fn (r) : response PDF for non-pref direction Fn (r) : response CDF for non-pref direction P (correct) = f p [r]fn [r] Response distributions Response Neurometric function (spikes/trial) 6.4% rn rp Neuronal response P (correct) = P(rp > rn ) = r fn ( r )dr dr fn (r )dr = Fn (r).2 P (correct) = Response f p (r)fn (r)dr.2 Neurometric vs. psychometric functions 4752 Britten et al. l MT Neurons Psychophysical Performance at roughly equal rates as the criterion response increased from to impulses/trial. In general, the curvature of the ROC away from the diagonal indicates the separation of the preferred psychometric null response distributions (Bamber, 975). Green Swets (966) showed formally that the normalized neurometric area under the ROC corresponds to the performance expected The Journal of Neuroscience, December 992, fz(l2) 4755 of an ideal observer in a two-alternative, forced-choice paradigm like the one employed in the present study. Again, one can intuit that this is reasonable. At 2.8% correa lation, 99% of the area of the unit square in Figure 5B falls monkey J --f/ beneath the ROC, corresponding to the near-perfect performance we would expect based on the response distributions for / 2.8% correlation in Figure 5.4. In contrast, only 5% of the / unit square falls beneath the ROC for.8% correlation, corre/ sponding as expected to rom performance. For each correlation level tested, we used this method to compute the probability that the decision rule would yield a monkey E-k correct response, the results are shown in Figure 5C. These / data capture the sensitivity of the neuron to directional signals in the same manner that the psychometric function captures E m,onkey YT,,, perceptual sensitivity to directional signals. As for the psychometric data, we fitted the neurometric data with smooth curves IO 5 2 of the form given by Equation. This function provided an Mean neuronal threshold excellent description of the neurometric data; the fit could be rejected for only 2 of the 2 6 neurometric functions in our data set (x2 test, p <.5). Application of Equation 2 resulted in a significantly improved fit for only one neuron. For the example in Figure 5C, the threshold parameter, (Y,was 4.4% correlation, the slope parameter, & was.3. For each neurometric Neuronal threshold (%) function, these parameters can be compared to the equivalent. A from comparison of absolute neuronal parameters Figure obtained the corresponding psychometric function. threshold for the 2 6 experiments in our sample. Solid circles indicate experiments in which neuronal threshold were sta Comparison of psychometric neurometric functions. Fig- in which tistically indistinguishable; open circles illustrate experiments the two were significantly different. functions The broken obdiagonal is ure 6A shows themeasures psychometric neurometric Correlation (%) Britten, Newsome & threshold Movshon, 992 thethe line experiment on which all Shadlen, points would in fallfigures if neuronal tained from illustrated 4 5. Thepredicted threshold perfectly. The frequency histogram at the uptwo functions areremarkably similar both in their location along Figure 6. Psychometric neurometricfunctionsobtainedin six per right was formed by summing across the scatterplot within diago in theirbins, overall shape.the apparent similarity experiments.the open symbols broken lines depict psychometric the abscissa nally oriented The resulting histogram is a scaled replica of the neurometricdata. data,whilethe solid symbols solid lines represent of the twodistribution functionsofwas reflected a close threshold ratios in depicted in correspondence Figure 7. Thesix examples illustratetherangeof relationships presentin ourdata. betweenthe threshold parameters,o(, the slopeparameters, L I A, Resultsof the experiment illustrated I in Figures I 4 5. PsychoI p, obtained from the Weibull fits (Eq. ) to the two data sets. physical neuronal data were statistically indistinguishable in this agonal line depicts the line of equality on all points The neurometric threshold of 4.4% correlation which compared fa- would experiment. Thresholds slope parameters are given in the captions neuronal threshold perfectly Mean in neuronal slope vorably tofall theifpsychometric threshold of 6.%predicted correlation, for Figures 4 5. B, A second experiment which psychometric threshold. Summing within a diagonally arranged set of bins neurometric data were statistically indistinguishable. Psychometric = theperslopeparameterswere similar aswell (neurometric p =.3; Figure 9. A comparison of average neuronal leads to the frequency histogram in the upper right comer of 7.8% correlation, p =.2; neurometric o( = 23.% correlation, (3 = psychometric p =.7). This similarity of psychometric formance across animals. A, The geometric mean was of neuronal Figure ; this is simply a scaledreplica of the distribution of.3. C, An experiment in which threshold sub- threshold plottedneuronal against threshold. the geometric mean of for neurometric data wasa common feature of our data, Figure stantially loweris than Psychometric = 3.7% cor-threshold threshold ratios shown in Figure 7. Despite the symmetrical each of the three animals in the study. The vertical error bars indicate illustrates a secondexample. Although absolute threshold reveals relation, p =.68; = 4.8% while D, bars An show6b distribution of the ratios aboutthe unity, the scatterplot SEM neurometric for neuronal (Y threshold, the horizontal error SEM experiment in for which neuronal threshold than levels higher undercorrelation the conditions of this experiment, the threshold. was The substantially broken line islower the best-fitting regres- were only a modest between the two measures (r =.29), sion threshold. = The 3.% correlation, neurometric psychometric data sets were againfor quite similar through Psychometric the data points.(yb, geometric mean (3of=neurometric most of this correlation is accounted by the interanimal 2.5; neurometric 4.7% correlation, fl=.58. mean E, Anofexperiment slopelyis=plotted against the geometric psychometricin slope.(neuronal: Error CY = 23.%, p =.3 ; : (Y = 7.8%, p differencesillustrated in Figure 9A. Thus, which thresholdsbarswere but slopes similar the regression line were are asdissimilar. described Psychometric in A. =.2). Higher absolute thresholds typically occurred when the LY= 3.9% correlation, B =.36; neurometric = 4.% correlation, p = neuronal thresholds are not strongly correlated on an cell-bypropertiescell of the neuron under required a psychophysi.79. F, An experiment in which threshold slope were dissimilar. basis, although theystudy are closely related on the average. The Psychometric (Y= 3.% correlation, (3 =.9; neurometric LY= 27.% tally nonoptimal of the discrimina (e.g., un- since absencepresentation of a strongcell-to-cell correlation is not surprising of the variance in cx (rz, the amount of variance correlation, =.8. usually small receptive fields,large eccentricities, or high speeds). neuronal sensitivity to motion direction varies widely (even accounted for, =.438). Adding neuronalthreshold to the model remaining in Figurea 6monkey s exemplify the range of variwithinpanels MT) whereas sensitivity is as a coregressorrevealed a significant predictive effect ofthe neurelatively constant for any given set ofthresholds stimulus conditions. ation in our data. Neuronal could to 2 impulses/trial, the proportion preferred re-accounting ronal threshold (p < O.Ol),ofbut the effectdirection wassmall, of integration timetheonneuron neuronal(fig. be strikingly Efect dissimilar, with either 6) or the for only an additional 2% of the variance in ROC sponsesexceeding criterion alsofell toward zero. Thus, the thresholds. The more comparison of The neuronal monkey (Fig. 6CJ ) being sensitive. slope parameters threshold (r*.458). experiment-to-experiment variafor 2.8% correlation fell=along thethus, upper left margins of summarized Figurebut 7 issignificant basedon experiments in tions inin Figure neuronal 5Bthreshold wereinnot highly correlated vari- alsothresholds appear dissimilar (Fig.in6E,F), differthe unit square (triangles). contrast, the ROC with could whichwere the monkey wasless required to view the dot stimulus ations in fell threshold, even though were in slope ences observed frequently thanrom significant for.8% correlation near the diagonal line bisecting thethe unitmeans for 2 full seconds before indicating its judgement of motion strikingly similar (e.g., Fig. 7). differencesin threshold (seebelow). square(open circles); sincethe preferred null responsedisdirection. A potential flaw in the analysis results from the un amplifies thiscorrelation, point by showing the relationshipaofparticularly surprisingaspectof our data wasthe existence tributions werefigure very similar at.8% the proportion known time interval over which monkeytointegrated neuronal to threshold in criterion richer detail. MT neurons that were substantially morethe sensitive motion inforof preferred null direction responses exceeding fell Theofscat mation in reaching its decision. Temporal integration can interplot depicts the actual neuronal thresholds measuredin each of the 2 6 experiments in our sample. 5fluence both neuronal thresholds, the % motion coherence comparison captured in Figure 7 is reasonableonly if the inthe solid circlesindicate experimentsin which the neurometric.9 psychometric functions were statistically indistinguishable 45tegration interval is similar for both setsof data. For the physasdescribedearlier; the open circles showexperiments in which 4iological data presented thus far, the integration interval was always 2 set sinceconstruction of the ROCs was always based the two data setswere demonstrably different. The broken di- Predicting the monkey s decisions Response distributions for pref non-pref decisions at a fixed motion coherence Response (impulses/sec) Proportion correct Neurometric & psychometric functions: accuracy vs motion coherence Neuron Behavior Motion strength (% coherence) P (null.6 > crit) (spikes/trial).4 P (null.6 > crit).8 I 7. I.o. Correlation (%) I,'"I Figure 5. Analysis of physiological data. A, This three-dimensional plot illustrates frequency histograms of responses obtained from a single MT neuron at five different correlation levels. The horizontal axis shows the amplitude of the neuronal response, the vertical axis indicates the number of trials on which a particular response was obtained. The depth axis shows the correlation of motion signal used to elicit the response distributions. Open bars depict responses obtained for motion in the neuron s preferred direction, while solid bars illustrate responses for null direction motion. For this neuron, each distribution is based on 6 trials. B, ROCs for the five pairs of preferred-null response distributions illustrated in A. Each point on an ROC depicts the proportion of trials on whichbritten, the preferredshadlen, direction response exceeded & a criterion level plotted Newsome Movshon, 992 against the proportion of trials on which the null direction response exceeded criterion. Each ROC was generated by increasing the criterion level from to 2 spikes in one-spike increments. Increased separation of the preferred null response distributions in A leads to an increased deflection of the ROC away from the diagonal. C, A neurometric function that describes the sensitivity of an MT neuron to motion signals of increasing strength. The curve shows the performance of a simple decision model that bases judgements of motion direction on responses like those illustrated in A. The proportion of correct choices made by the model is plotted against the strength of the motion signal. The proportion of correct choices at a particular correlation level is simply the normalized area under the corresponding ROC curve in B. For this neuron, data were obtained at seven correlation levels; response distributions ROC curves are illustrated for five of these levels in A B. The neurometric function was fitted with sigmoidal curves of the form given in Equation. In this experiment, neuronal threshold (a) was 4.4 correlation the unitless siope parameter for the neurometric function (p) was.3..8 I 7. I.o. Correlation (%) I,'"I Figure 5. Analysis of physiological data. A, This three-dimensional plot illustrates frequency histograms of responses obtained from a single MT neuron at five different correlation levels. The horizontal axis shows the amplitude of the neuronal response, the vertical axis indicates the number of trials on which a particular response was obtained. The depth axis shows the correlation of motion signal used to elicit the response distributions. Open bars depict responses obtained for motion in the neuron s preferred direction, while solid bars illustrate responses for null direction motion. For this neuron, each distribution is based on 6 trials. B, ROCs for the five pairs of preferred-null response distributions illustrated in A. Each point on an ROC depicts the proportion of trials on which the preferred direction response exceeded a criterion level plotted against the proportion of trials on which the null direction response exceeded criterion. Each ROC was generated by increasing the criterion level from to 2 spikes in one-spike increments. Increased separation of the preferred null response distributions in A leads to an increased deflection of the ROC away from the diagonal. C, A neurometric function that describes the sensitivity of an MT neuron to motion signals of increasing strength. The curve shows the performance of a simple decision model that bases judgements of motion direction on responses like those illustrated in A. The proportion of correct choices made by the model is plotted against the strength of the motion signal. The proportion of correct choices at a particular correlation level is simply the normalized area under the corresponding ROC curve in B. For this neuron, data were obtained at seven correlation levels; response distributions ROC curves are illustrated for five of these levels in A B. The neurometric function was fitted with sigmoidal curves of the form given in Equation. In this experiment, neuronal threshold (a) was 4.4 correlation the unitless siope parameter for the neurometric function (p) was.3. Visual stimulus Neurometric Neurometricfunction function Neuronal response Psychometricfunction function Psychometric? Choice probability Behavioral judgement Predicting the monkey s decisions Choice probability: Accuracy with which one could predict monkey s decision from the response of the neuron given that you know the distributions. fn (r) fp (r) r f p (r) pref decisions Correct trials non-pref Error trials decisions Neuronal response Trial number Shadlen, Britten, Newsome & Movshon, 996 fp (r) : response PDF when monkey reports pref direction fn (r) : response PDF when monkey reports non-pref direction

11 Choice probability Computational model Example neuron Across all neurons recorded MT neuron pool Pooled "up" signal + Σ Pooling noise Decision + Pooling noise Pooled "down" signal Noise is partially correlated across neurons. Responses are pooled non-optimally over large populations of neurons including those that are not the most selective. Additional noise is added after pooling. Britten, Newsome, Shadlen, Celebrini & Movshon, 996 Shadlen, Britten, Newsome & Movshon, 996