In a typical perceptual evidence-integration experiment (Figure 1A), an observer is presented in each trial with a time-varying stimulus and must report which of two possible stimulus categories it belongs to. Typical examples include judging whether a dynamic visual stimulus is moving leftwards or rightwards (Yates et al., 2017; Katz et al., 2015); whether the orientation of a set of gratings is more aligned with cardinal or diagonal directions (Wyart et al., 2012) whether a combination of tones is dominated by high or low frequencies (Morillon et al., 2014; Hermoso-Mendizabal et al., 2020; Znamenskiy and Zador, 2013); which of two acoustic streams is more intense or dense (Brunton et al., 2013; Pardo-Vazquez et al., 2019; Cisek et al., 2009). Such paradigms have been used extensively in humans, nonhuman primates, and rodents. Here, we focus on experiments in which observers report their choice at the end of a period whose duration is controlled by the experimenter (Kiani and Shadlen, 2009; Wyart et al., 2012; Brunton et al., 2013; Raposo et al., 2012), in contrast to so-called ‘reaction time’ tasks, in which the observer can respond after viewing as brief a portion of the stimulus as they wish (Roitman and Shadlen, 2002; Znamenskiy and Zador, 2013; Pardo-Vazquez et al., 2019; Hermoso-Mendizabal et al., 2020).

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Moreover, we focus on experimental paradigms in which the sensory evidence in favor of each category arrives in a sequence of discrete samples. Samples can correspond to motion pulses (Yates et al., 2017), individual gratings (Wyart et al., 2012), acoustic tones (Morillon et al., 2014; Hermoso-Mendizabal et al., 2020; Znamenskiy and Zador, 2013), numbers (Bronfman et al., 2015; Cisek et al., 2009), or symbols representing category probabilities (Yang and Shadlen, 2007). We refer to this configuration as the DSS paradigm. In this paradigm, the perceptual evidence provided by each sample can be controlled independently, allowing for detailed analyses of how different samples contribute to the behavioral response. The DSS framework can be contrasted with experiments in which the experimenter specifies only the mean stimulus strength on each trial, and variations in sensory evidence over time are not finely controlled or are not easily determined from the raw spatio-temporal stimulus.

Tasks using the DSS paradigm are classically thought to rely on sequential accumulation of the stimulus evidence (Bogacz et al., 2006), which we refer to here as temporal integration. Figure 1A shows an example stimulus sequence composed of n samples that provide differing amounts of evidence in favor of one alternative vs. another (‘A’ vs. ‘B’). In the temporal integration model, the accumulated evidence fluctuates as new samples are integrated and finishes at a positive value indicating overall evidence for stimulus category A (Figure 1B). This integration process can be formalized by defining the decision variable or accumulated evidence $x_i$ and its updating dynamics across stimulus samples: $x_i=x_{i-1}+m_i$ where ${m_i=S}_i+{\epsilon }_i$ represents a noisy version of the true stimulus evidence $S_i$ in the ith sample corrupted by sensory noise ${\epsilon }_i$ . The binary decision r is simply based on the sign of the accumulated evidence $x_n$ at the end of the sample sequence (composed of n samples): $r=A$ if ${\displaystyle x_n>0}$, and $r=B$ if ${\displaystyle x_n<0}$. This procedure corresponds to the normative strategy with uniform weighting that maximizes accuracy. For such perfect integration, $x_n={\Sigma }_i{S}_i+{\Sigma }_i{\epsilon }_i$ , so that the probability of response A is $p(r=A)=\Phi \left({\Sigma }_i\beta S_i\right)$ where $\Phi$ is the cumulative normal distribution function (the normative weight for the stimuli β depends on the noise variance $Var\left(\epsilon \right)$ and the number of samples through $\beta =1/\sqrt{nVar\left(\epsilon \right)}$). Departures from optimality in the accumulation process such as accumulation leak, categorization dynamics, divisive normalization, sensory adaptation, or sticky boundaries may however yield unequal weighting of the different samples (Yates et al., 2017; Brunton et al., 2013; Prat-Ortega et al., 2021; Bronfman et al., 2016; Keung et al., 2019; Keung et al., 2020). To accommodate for these, we allowed the integration model to take any arbitrary weighting of the samples: $p(r=A)=\Phi \left({{\beta }_0+\Sigma }_i{\beta }_i{S}_i\right)$ (see Methods for details). The mapping from final accumulated evidence to choice was probabilistic, to account for the effects of noise from different sources in the decision-making process (Drugowitsch et al., 2016). Thus, this model represented an approximate statistical description for any generative model relying on temporal integration of the stimulus evidence.

Although it has been commonly assumed that observers use evidence-integration strategies to perform these psychophysical tasks, recent work has suggested that observers may employ non-integration strategies instead (Stine et al., 2020). Here, we consider two specific alternative models. The first non-integration model corresponds to an extrema-detection model (Waskom and Kiani, 2018; Stine et al., 2020; Ditterich, 2006). In this model, observers do not integrate evidence across samples but instead base their decision on extreme or salient bits of evidence. More specifically, the observer commits to a decision based on the first sample i in the stimulus sequence that exceeds one of the two symmetrical thresholds, that is such that $\left|m_i\right|\ge \theta$. In our example stimulus, the first sample that reaches this threshold in evidence space is the fifth sample, which points toward stimulus category B, so response B is selected (Figure 1C). This policy can be viewed as a memoryless decision process with sticky bounds. If the stimulus sequence contains no extreme samples, so that neither threshold is reached, the observer selects a response at random. We also explored an alternative mechanism where in such cases the response is based on the last sample in the sequence, following Stine et al., 2020; and a variant of the model where the decision threshold is different on every sample position.

The second non-integration model corresponds to the snapshot model (Stine et al., 2020; Pinto et al., 2018). In this model, the observer attends to only one sample i within the stimulus sequence, and makes a decision based solely on the evidence from the attended sample: $r=A$ if ${\displaystyle m_i>0}$, and $r=B$ if ${\displaystyle m_i<0}$. The position in the sequence of the attended sample is randomly selected on each trial. In our example, the fourth sample is randomly selected, and since it contains evidence toward stimulus category A, response A is selected (Figure 1D). We considered variants of this model that gave it additional flexibility, including: allowing the prior probability over the attended sample to depend on its position in the sequence using a non-parametric probability mass function estimated from the data; allowing for deterministic vs. probabilistic decision-making rule based on the attended evidence; including attentional lapses that were either fixed to 0.02 (split equally between leftward and rightward responses) or estimated from behavioral data. We finally considered a variant of the snapshot model where the decision was made based on a subsequence of K consecutive samples within the main stimulus sequence (${\displaystyle 1\le K<n}$), rather than based on a single sample.

To compare the three decision-making models defined above (i.e., temporal integration, extrema-detection, snapshots), we first examined behavioral data from two monkeys performing a fixed-duration motion integration task (Yates et al., 2017). In this experiment, each stimulus was composed of a sequence of 7 motion samples of 150 ms each where the motion strength toward left or right was manipulated independently for each sample. At the end of the stimulus sequence, monkeys reported with a saccade whether the overall sequence contained more motion toward the left or right direction. The animals performed 72,137 and 33,416 trials for monkey N and monkey P, respectively, allowing for in-depth dissection of their response patterns.

We fit the three models (and their variants) to the responses for each animal individually (see Figure 2—figure supplement 1 for estimated parameters for the different models). We then simulated the fitted model and computed, for simulated and experimental data, the psychophysical kernels capturing the weights of the different sensory samples based on their position in the stimulus sequence (Figure 2B). Psychophysical kernels were non-monotonic and differed in shape between the two animals, probably reflecting the complex contributions of various dynamics and suboptimalities along the sensory and decision pathways (Yates et al., 2017; Levi and Huk, 2020).

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The temporal profile of the kernel was perfectly matched by the integration model, almost by design, as we gave full flexibility to the model to adjust the sample weights. The snapshot model was provided with similar flexibility, as the prior probability of attending each sample could be fully adjusted to the monkey decisions. However, the snapshot model could not match the experimental psychophysical kernel as accurately. It consistently underestimated the magnitude of weighting in monkey P (Figure 2B, bottom row). The extrema-detection model was not endowed with such flexibility of sensory weighting. On the contrary, since the decision was based on the first sample in the sequence reaching a certain criterion, this inevitably generates a primacy effect in the psychophysical kernels – or at best a flat weighting (Stine et al., 2020). The model thus failed to capture the non-monotonic psychophysical kernels from animal data.

Next, we looked at the psychometric curves and choice accuracy predictions of each fitted model (Figure 2C, D). Stine et al., 2020 have argued that integration and non-integration models can capture the psychometric curves equally well. For both animals, the accuracy and psychometric curves were accurately captured by the integration model. In line with Stine et al., we also found that both non-integration models could reproduce the shape of the psychometric curve in monkey N, although the quantitative fit was always better for the integration than non-integration models. By contrast both non-integration models failed to capture the psychometric curve for monkey P (Figure 2B, bottom row). More systematically, the overall accuracy, which is an aggregate measure of the psychometric curve, clearly differed between models, as the accuracy of the non-integration models systematically deviated from animal data for both animals (Figure 2C). In other words, all models produce the same type of psychometric curves up to a scaling factor, and this scaling factor (directly linked to the model accuracy) is key to differentiate model fits. For the snapshot model in monkey P, this discrepancy was explained because the model, limited to using one stimulus sample, could not reach the performance of the animal (compare the maximum accuracy of the model indicated by the blue mark with the accuracy of the animal). (This also explains why the psychophysical kernel of the snapshot model underestimated the true kernel in monkey P.) For the extrema-detection model in monkey P and for both non-integration models in the other animal (monkey N), the model accuracy is not bounded below the subject’s accuracy. In such cases, the model can produce better-than-observed accuracy for certain parameter ranges, but these are not the parameters found by the maximum likelihood procedure, probably because they produce a pattern of errors that is inconsistent with the observed pattern of errors. This indicates an inability of the models to match the pattern of errors of the animal (see Discussion).

Finally, we assessed quantitatively which model provided the best fit, while correcting for model complexity using the Akaike information criterion (AIC, Figure 2A). In both monkeys, AIC favored the integration model over the two non-integration models by a very large margin. We also explored whether (previously unpublished) elaborations of the extrema-detection and snapshot models could provide a better match to the behavioral metrics considered above (Figure 2—figure supplements 2 and 3). We found using the AIC metric that the integration model was preferred over all variants of both non-integration models, for both monkeys. Moreover, these model variants could not replicate the psychophysical kernels as well as the integration model did (Figure 2—figure supplements 2 and 3).

In conclusion, while psychometric curves may not always discriminate between integration and non-integration strategies, other metrics including psychophysical kernels, predicted accuracy and quality of fit (AIC) support temporal integration in monkey perceptual decisions. For one model in one monkey (the snapshot model in monkey P), even the simple metric of overall accuracy compellingly supported temporal integration (Figure 2C). For the other monkey and/or model, where the distinction was less clear, our model-based approach allowed us to leverage these other metrics to reveal strong support for the temporal integration model (Figure 2A–C). While these data rely only on two experimental subjects, we show below further evidence supporting the integration model in humans and rats.

While formal model comparison leads us to reject the non-integration models in favor of the integration models, it is informative to examine qualitative features of the animal strategies and identify how non-integration models failed to capture them. We started by designing two analyses aimed at testing whether choices were consistent with the extrema-detection model, namely by testing whether choices were strongly correlated with the largest evidence samples. In the first analysis, we looked at the subset of trials where the evidence provided by the largest evidence sample in the sequence was at odds with the total evidence in the sequence: we show one example in Figure 3B, where the largest evidence sample points toward response B, while the overall evidence points toward response A. These ‘disagree trials’ represent a substantial minority of the whole dataset: 1865 trials (2.6%) in monkey N and 1831 trials (5.5%) in monkey P. If integration is present, the response of the animal should in general be aligned with the total evidence from the sequence (Figure 3A, red bars). By contrast, if it followed the extrema-detection model (Figure 1C), it should in general follow the largest evidence sample (Figure 3A, green bars). In both monkeys, animal choices were more often than not aligned with the integrated evidence (Figure 3A, black bars), as predicted by the integration model. The responses generated from the extrema-detection model tended to align more with the largest evidence sample, although that behavior was somehow erratic (for monkey N) due to the large estimated decision noise in the model. This rules out that monkey decisions rely on a memoryless strategy of simply detecting large evidence samples, discarding all information provided by lower evidence samples. Our results complement a previous analysis on disagree trials in this task (Levi et al., 2018), by explicitly comparing monkey behavior to model predictions.

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We reasoned that the extrema-detection would also leave a clear signature in the ‘subjective weight’ of the samples, defined as the impact of each sample on the decision as a function of absolute sample evidence (Yang and Shadlen, 2007; Waskom and Kiani, 2018; Nienborg and Cumming, 2007). The extrema-detection model predicts that, in principle, samples whose evidence is below the threshold have little impact on the decision, while samples whose evidence is above the threshold have full impact on the decision. By contrast, the integration model predicts that subjective weight should grow linearly with sample evidence. We estimated subjective weights from monkey choices using a regression method similar in spirit to previous methods (Yang and Shadlen, 2007; Waskom and Kiani, 2018), taking the form ${\displaystyle p\left(r_t=A\right)=\sigma \left({\beta }_0+{\mathrm{\Sigma }}_{i\in \left[\mathrm{1...}n\right]}{\beta }_{i}f\left(S_{t,i}\right)\right)}$, where σ is a sigmoidal function. Here f is a function that captures the subjective weight of the sample as a function of its associated evidence. Whereas previous methods estimated subjective weights assuming a uniform psychophysical kernel, our method estimated simultaneously subjective weights $f\left(S\right)$ and the psychophysical kernel $\beta$, thus removing potential estimation biases due to unequal weighting of sample evidence (see Methods). In both monkeys, we indeed found that the subjective weight depends linearly on sample evidence for low to median values of sample evidence (motion pulse lower than 6), in agreement with the integration model (Figure 3—figure supplement 1). Counter to our predictions, simulated data of the extrema-detection model displayed the same linear pattern for low to median values of sample evidence. We realized this was due to the very high estimated sensory noise (Figure 2—figure supplement 1), such that, according to the model, even samples with minimal sample evidence were likely to reach the extrema-detection threshold. In other words, unlike the previous analyses, inferring the subjective weights used by animals was inconclusive as to whether animals deployed the extrema-detection strategy. This somewhat surprising dependency reinforces the importance of validating intuitions by fitting and simulating models (Wilson and Collins, 2019).

Following model comparisons favoring integration over both snapshot and extrema-detection models, the immediately previous analysis relied on a special subset of trials to provide an additional, and perhaps more intuitive, signature of integration, which ruled out extrema-detection as a possible strategy of either monkey. We next employed another novel analysis specifically designed to tease apart unique signatures of the integration and snapshot models. More specifically, we tested whether decisions were based on the information from only one part of the sequence, as predicted by the snapshot model, or from the full sequence, as predicted by the integration model. To facilitate the analysis, we defined early evidence E_t by grouping evidence from the first three samples in the sequence, and late evidence L_t, as the grouped evidence from the last four samples. We then displayed the proportion of rightward responses as a function of both early and late evidence in a graphical representation that we call integration map (Figure 4A). A pure integration strategy corresponds to summing early and late evidence equally, which can be formalized as $p\left(r\right)=\sigma (E_t+L_t)$, where $\sigma$ is a sigmoidal function. Because this only depends on the sum $E_t+L_t$ , the probability of response is invariant to changes in the $(E_t,L_t)$ space along the diagonals, which leave the sum unchanged. These diagonals correspond to isolines of the integration map (Figure 4A, left; Figure 4—figure supplement 2A). In other words, straight diagonal isolines in the integration map reflect the fact that the decision only depends on the sum of evidence $E_t+L_t$ . Straight isolines thus constitute a specific signature of evidence integration.

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We contrasted this integration map with the one obtained from a non-integration strategy (Figure 4A, middle panel; Figure 4—figure supplement 2B). There we assumed that the decision depends either on the early evidence or on the late evidence, as in the snapshot model, with equal probability. This can be formalized as ${\displaystyle p\left(r\right)=0.5\sigma \left(E_t\right)+0.5\sigma \left(L_t\right)}$. In this case, if late evidence is null ($\sigma \left(L_t\right)=0.5$) and early evidence is very strong toward the right ($\sigma \left(E_t\right)\simeq 1$) the overall probability for rightward response is $p\left(r\right)=0.75$. This probability contrasts with that obtained in the integration case where the early evidence would dominate and lead to an overwhelming proportion of rightward responses, that is $p\left(r\right)\simeq 1$. The 25% of leftwards responses yielded by the non-integration model correspond to trials where only the late (uninformative) part of the stimulus is attended and a random response to the left is drawn. More generally, in regions of the space in which either early or late evidence take large absolute values, their corresponding probability of choice saturates to 0 or 1, when that evidence is attended, so the overall response probability becomes only sensitive to the other evidence. As a result, the equiprobable lines bend toward the horizontal and vertical axes (Figure 4A, middle). Finally, to compare predictions from both integration and non-integration models to monkey behavior, we plotted the integration maps for both monkeys (Figure 4A, right; Figure 4—figure supplement 1A). The isolines were almost straight diagonal lines and showed no consistent curvature toward the horizontal and vertical axes. This provides direct evidence that monkey responses predominantly depend on the sum of early and late evidence – a clear signature of temporal integration.

We derived subsequent tests based on the integration map. We computed conditional psychometric curves as the probability for rightward responses as a function of early evidence $E_t$ , conditioned on late evidence value $L_t$ (Figure 4B; Figure 4—figure supplement 1B). From the integration formula $p\left(r\right)=\sigma (E_t+L_t)$, we see that a change in late evidence value corresponds to a horizontal shift of the conditional psychometric curves. By contrast, according to the non-integration formula $p\left(r\right)=0.5\sigma \left(E_t\right)+0.5\sigma \left(L_t\right)$, conditioning on different values of late evidence adds a fixed value to the response probability irrespective of early evidence, a vertical shift akin to that introduced by lapse responses (Figure 4B, middle panel). The conditional psychometric curves for monkeys (Figure 4B, right panel; Figure 4—figure supplement 1 and Figure 4—figure supplement 2) displayed horizontal shifts as late evidence was changed, consistently with the integration hypothesis. We sought to quantify these shifts in better detail. To this purpose, we fitted each conditional psychometric curve with the formula $p\left(r\right)=(1-{\pi }_L-{\pi }_R)\sigma (\alpha E_t+\beta )+{\pi }_R$ , where ${\pi }_L$ , ${\pi }_R$ , $\alpha$, and $\beta$ correspond to the left lapse, right lapse, sensitivity, and lateral bias parameters, respectively (Figure 4C, Figure 4—figure supplement 1 and Figure 4—figure supplement 2). The integration model predicts that the bias parameter $\beta$ should vary linearly with L_t, while lapse parameters should remain null (Figure 4D, left panel). By contrast, the non-integration model predicts that the horizontal shift parameter $\beta$ should remain constant while left and right lapse parameters $({\pi }_L,{\pi }_R)$ should vary (middle panel), as these lapse parameters correspond to the trials where early evidence is not attended and the response depends simply on late evidence. Both monkeys showed a very strong linear dependence between late evidence and the horizontal shift $\beta$ (Figure 4D, right panel; see also Figure 4—figure supplement 1), further supporting that late evidence is summed to early evidence. By contrast, the lapse parameters showed no consistent relationship with late evidence $L_t$ (Figure 4E, right panel). Finally, we directly assessed the similarities between the integration maps from monkey responses and from simulated responses for the three models (integration, snapshot, and extrema-detection). The model-data correlation was larger in the integration model than in the non-integration strategies for both monkeys (Figure 4E; unpaired t-test on bootstrapped r values: p < 0.001 for each animal and comparison against extrema-detection and against snapshot model). Overall, integration maps allow to dissect how early and late parts of the stimulus sequence are combined to produce a behavioral response. In both monkeys, these maps carried signatures of temporal integration. For monkey N, the integration model and the data look very similar (Figure 4—figure supplement 2). For monkey P, there is still a qualitative dependency that deviates from non-integration, but which is not as uniquely matched to the integration strategy (although the imperfect coverage of the two-dimensional space impedes further investigations; Figure 4—figure supplement 1). Thus, complementing the statistical model tests favoring integration, this richer visualization allows the data to show us that some degree of integration is occurring, albeit not perfect.

Overall, all our analyses converged to support the idea that monkey decisions in a fixed-duration motion discrimination task relied on temporal integration. We explored whether the same results would hold for two other species and perceptual paradigms. We first analyzed the behavioral responses from nine human subjects performing a variable-duration orientation discrimination task (Cheadle et al., 2014). In each trial, a sequence of 5–10 gratings with a certain orientation were shown to the subject, and the subject had to report whether they thought the gratings were overall mostly aligned to the left or to the right diagonal. In this task, the experimenter can control the evidence provided by each sample by adjusting the orientation of the grating. We performed the same analyses on the participant responses as on monkey data. As for monkeys, we found that the integration model nicely captured psychometric curves, participant accuracy and psychophysical kernels (Figure 5A–C, red curves and symbols). By contrast, both non-integration models failed to capture these patterns (Figure 5A–C, blue and green curves and symbols). The accuracy from both models consistently underestimated participant performance: eight and six out of nine subjects outperformed the maximum performance for the snapshot and extrema-detection models, respectively (Figure 5—figure supplement 1). This suggests that human participants achieved such accuracy by integrating sensory evidence over successive samples. Moreover, subjects overall weighted more later samples (Figure 5C), which is inconsistent with the extrema-detection mechanism. A formal model comparison confirmed that in each participant, the integration model provided a far better account of subject responses than either of the non-integration models did (Figure 5D). We then assessed how subjects combined information from weak and strong evidence samples into their decisions, using the same analyses as for monkeys. As predicted by the integration model, but not by the extrema-detection model, human choices consistently aligned with the total stimulus evidence and not simply with the strongest evidence sample (Figure 5E). Finally, the average integration map for early and late evidence within the stimulus sequence displayed nearly linear diagonal isolines, showing that both were integrated into the response (Figure 5F). Integration maps from participants correlated better with maps predicted by the integration model than with maps predicted by either of the alternative non-integration strategies (Figure 5G; two-tailed t-test on bootstrapped r values: p < 0.001 for six out of nine participants in the integration vs. snapshot comparison; in all nine participants for the integration vs. extrema-detection comparison). Overall, these analyses provide converging evidence that human decisions in an orientation discrimination task rely on temporal integration.

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Finally, we analyzed data from five rats performing a fixed-duration auditory task where the animals had to discriminate the side with larger acoustic intensity (Pardo-Vazquez et al., 2019). The relative intensity of the left and right acoustic signals was modulated in sensory samples of 50ms, so that the stimulus sequence provided time-varying evidence for the rewarded port. The stimulus sequence was composed of either 10 or 20 acoustic samples of 50 ms each, for a total duration of 500 or 1000 ms. We applied the same analysis pipeline as for monkey and human data. The integration model provided a much better account of rat choices than non-integration strategies, based on psychometric curves (Figure 6A), predicted accuracy (Figure 6B), psychophysical kernel (Figure 6C), and model comparison using AIC (Figure 6D). Similar to humans and monkeys, rats tended to select the side corresponding to the total stimulus evidence and not the largest sample evidence in ‘disagree’ trials, as predicted by the integration model (Figure 6E). Finally, the integration map was largely consistent with an integration strategy (Figure 6F), and correlated more strongly with simulated maps from the integration model (unpaired t-test on bootstrapped r values: p < 0.001 for each animal and comparison against extrema-detection and against snapshot model).

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We present here the most relevant features of the behavioral protocol – see Yates et al., 2017 for further experimental details. Two adult rhesus monkeys performed a motion discrimination task. On each trial, a stimulus consisting of a hexagonal grid (5–7 degrees, scaled by eccentricity) of Gabor patches (0.9 cycle per degree; temporal frequency 5 Hz for Monkey P; 7 Hz for Monkey N) was presented. Monkeys were trained to report the net direction of motion in a field of drifting and flickering Gabor elements with an eye movement to one of two targets. Each trial motion stimulus consisted of seven consecutive motion pulses, each lasting 9 or 10 video samples (150 or 166 ms; pulse duration did not vary within a session), with no interruptions or gaps between the pulses. The strength and direction of each pulse $S_{ti}$ for trial t and sample i were set by a draw from a Gaussian rounded to the nearest integer value. The difficulty of each trial was modulated by manipulating the mean and variance of the Gaussian distribution. Monkeys were rewarded based on the empirical stimulus and not on the stimulus distribution. We analyzed a total of 112 sessions for monkey N and 60 sessions for monkey P, with a total of 72,137 and 33,416 valid trials, respectively. These sessions correspond to sessions with electrophysiological recordings reported in Yates et al., 2017 and purely behavioral sessions.

Nine adult subjects performed an orientation discrimination task whereby on each trial they reported in each trial whether a series of gratings were perceived to be mostly tilted clockwise or counterclockwise (Drugowitsch et al., 2016). Each DSS consisted of 5–10 gratings. Each grating was a high-contrast Gabor patch (color: blue or purple; spatial frequency = 2 cycles per degree; SD of Gaussian envelope = 1 degree) presented within a circular aperture (4 degrees) against a uniform gray background. Each grating was presented during 100 ms, and the interval between gratings was fixed to 300 ms. The angles of the gratings were sampled from a von Mises distribution centered on the reference angle (${\alpha }_0=45$ degrees for clockwise sequences, 135 degrees for anticlockwise sequences) and with a concentration coefficient $\kappa =0.3$. The normative evidence provided by sample i in trial t in favor of the clockwise category corresponds to how well the grating orientation ${\alpha }_{ti}$ aligns with the reference orientation, that is $S_{ti}=2\kappa cos({2(\alpha }_{ti}-{\alpha }_0))$ .

Each sequence was preceded by a rectangle flashed twice during 100 ms (the interval between the flashes and between the second flash and the first grating varied between 300 and 400 ms). The participants indicated their choice with a button press after the onset of a centrally occurring dot that succeeded the rectangle mask and were made with a button press with the right hand. Failure to provide a response within 1000 ms after central dot onset was classified as invalid trial. Auditory feedback was provided 250 ms after participant response (at latest 1100 ms after end of stimulus sequence). It consisted of an ascending tone (400/800 Hz; 83/167 ms) for correct responses; descending tone (400/400 Hz; 83/167 ms) for incorrect responses; a low tone (400 Hz; 250 ms) for invalid trials.

Trials were separated by a blank interstimulus interval of 1200–1600 ms (truncated exponential distribution of mean 1333 ms). Experiments consisted of 480 trials in 10 blocks of 48. It was preceded with two blocks of initiation with 36 trials each. In the first initiation block, there was only one grating in the sequence, and it was perfectly aligned with one of the reference angles. In the second initiation block, sequences of gratings were introduced, and the difficulty was gradually increased (the distribution concentration linearly decreased from $\kappa =1.2$ to $\kappa =0.3$). Invalid trials (mean 6.9 per participant, std 9.4) were excluded from all regression analyses. The study was approved by the local ethics committee (approval 2013/5435/I from CEIm-Parc de Salut MAR).

Rat experiments were approved by the local ethics committee of the University of Barcelona (Comité d’Experimentació Animal, Barcelona, Spain, protocol number Ref 390/14). Five male Long-Evans rats (350–650 g), pair-housed and kept on stable conditions of temperature (23°C) and humidity (60%) with a constant light–dark cycle (12:12 hr, experiments were conducted during the light phase). Rats had free access to food, but water was restricted to behavioral sessions. Free water during a limited period was provided on days with no experimental sessions.

Rats performed a fixed-duration auditory discrimination task where they had to classify noisy stimuli based on the intensity difference between the two lateral speakers (Pardo-Vazquez et al., 2019; Hermoso-Mendizabal et al., 2020). An LED on the center port indicated that the rat could start the trial by poking in that center port. After this poke, rats had to hold their snouts in the central port during 300 ms (i.e., fixation). Following this period, an acoustic DSS was played. Rats had to remain in the central port during the entire presentation of the stimulus. At stimulus offset, the center LED went off and rats could then come out of the center port and head toward one of the two lateral ports. Entering the lateral port associated with the speaker that generated the larger sound intensity led to a reward of 24 µl of water (correct responses), while entering the opposite port lead to a 5-s timeout accompanied with a bright light during the entire period (incorrect responses). If rats broke fixation during the pre-stimulus fixation period or during the stimulus presentation, the sound was interrupted, the center LED remained on, and the rat had to initiate a new trial starting by center fixation followed by a new stimulus. Fixation breaks were not included in any of the analyses. Stimulus duration was 0.5 s (10 samples) or 1 s (20 samples). Two rats performed 0.5-s stimuli only (77,810 and 54,803 valid trials, respectively); one rat performed 1-s stimuli only (42,474 valid trials); the remaining two rats performed a mixture of 0.5 and 1 s stimuli trials randomly interleaved (5016 trials and 65,212 valid trials, respectively, for one animal; 7374 and 38,829 trials for the other animal). In each trial k one stimulus ${\displaystyle S_K^X(t)}$ was played in each speaker (X = R for the Right speaker and X = L for the Left speaker). Each stimulus was an amplitude-modulated (AM) broadband noise defined by $S_k^X\left(t\right)=[1+sin(f_{AM}t+\phi \left)\right]a_k^X\left(t\right){\xi }_X\left(t\right)$ where f_AM = 20 Hz (sensory samples lasted 50 ms), the phase delay $\phi =3\pi /2$ and ${\xi }_X\left(t\right)$ were broadband noise bursts. The amplitudes of each sound in each frame were $a_k^L\left(t\right)=\left(1+S_{k,f}\right)/2$ and $a_k^R\left(t\right)=\left(1-S_{k,f}\right)/2$ with S_k,f(t) being the instantaneous evidence that was drawn independently in each frame f from a transformed Beta distribution with support [−1,1]. With this parametrization of the two sounds the sum of the two envelopes was constant in all frames a^L_k(t) + a^R_k(t) = 1. There were 7 × 5 stimulus conditions, each defined by a Beta distribution, spanning 7 mean values (−1, −0.5, −0.15, 0, 0.15, 0.5, and 1) and 5 different standard deviations (0, 0.11, 0.25, 0.57, and 0.8). In around the first half of the sessions, only sample sequences in which the total stimulus evidence matched the targeted nominal evidence were used. This effectively introduced weak correlations between samples. In the second half of the sessions, this condition was removed and samples in each stimulus were drawn independently from the corresponding Beta distribution.

The integration model for human participants corresponds to a logistic regression model, where the probability of selecting the right choice $p\left(r_t\right)$ at trial t depends on the weighted sum of the sample evidence: ${\displaystyle p\left(r_t\right)=\sigma \left({\beta }_0+{\mathrm{\Sigma }}_{i\in \left[\mathrm{1...}n\right]}{\beta }_i{S}_{ti}\right)}$, where ${\beta }_0$ is a lateral bias, $S_{ti}$ is the signed sample evidence at sample i; ${\beta }_i$ is the sensory weight associated with the ith sample in the stimulus sequence; and $\sigma \left(x\right)=({1+e^{-x})}^{-1}$ is the logistic function. The vector ${\beta }_i$ ’s allowed to capture different shapes of psychophysical kernels (e.g., primacy effects, recency effects) which can emerge due to a variety of suboptimalities in the integration process (leak, attractor dynamics, sticky bounds, sensory after-effects, etc.) (Brunton et al., 2013; Yates et al., 2017; Prat-Ortega et al., 2021; Bronfman et al., 2016; Keung et al., 2019; Keung et al., 2020). Technically, our statistical approach corresponds to the first-order expansion of the Volterra series used to approximate any integration model (Neri, 2004) by a simple and fittable logistic model. In other words, because a more complete description of integration models would be computationally challenging to fit and prone to overfitting, we chose a statistical approximation in the form of the logistic model that captures the essence of any generative model that include temporal integration, that is weighted summation of stimulus evidence.

For the monkey and rat data, we included a session-dependent modulation gain ${\gamma }_t$ to capture the large variations in performance in monkeys across the course of sessions (see Figure 2—figure supplement 1A):

This model corresponds to a bilinear logistic regression model which pertains to the larger family of Generalized Unrestricted Models (GUMs) (Adam and Hyafil, 2020). Parameters ($\beta ,\gamma$) were fitted using the Laplace approximation as described in Adam and Hyafil, 2020. The modulation gain was omitted when applied to human data, yielding a classical logistic regression model.

In the snapshot model, decisions are based on each trial based upon a single sample. The model also includes the possibility for left and right lapses. In each trial, the attended sample is drawn from a multinomial distribution of parameters (${\displaystyle {\pi }_1,...{\pi }_n,{\pi }_L,{\pi }_R}$), where the first terms ${\pi }_i(1\le i\le n)$ correspond to the probability of attending sample i, and ${\pi }_L$ and ${\pi }_R$ correspond to the probability of left and right lapses, respectively. Upon selecting sample i, the probability for selecting the right choice is given by the function $H_i\left(S_t\right)$. In the deterministic version of the model, $H_i$ is simply determined by the sign of the ith sample evidence: $H_i\left(S_t\right)=1$ if ${\displaystyle S_{ti}>0}$, $H_i\left(S_t\right)=0$ if ${\displaystyle S_{ti}<0}$, and $H_i\left(S_t\right)=0.5$ if $S_{ti}=0$ (i.e., random guess if the sample has null evidence). We also define similar functions for lapse responses: $H_R\left(S\right)=1$ and $H_L\left(S\right)=0$, irrespective of the stimulus. In the non-deterministic version of the model, the probability $H_i\left(S_{ti}\right)$ is determined by a logistic function of the attended sample evidence $H_i\left(S_t\right)=\sigma \left({{\beta }_{i}S}_{ti}\right)$ where ${\beta }_i$ describes a sensitivity parameter. The deterministic case can be viewed as the limit of the non-deterministic case when all sensitivity parameters ${\beta }_i$ diverge to $+\infty$, that is when sensory and decision noise are negligible.

The overall probability for selecting right choice (marginalizing over the attended sample, which is a hidden variable) can be captured by a mixture model:

The mixture coefficients ${\displaystyle {\pi }_i\left(i=1,...n,L,R\right)}$ are constrained to be non-negative and sum up to 1. In the non-deterministic model, the parameters also include sensitivity parameters ${\beta }_i$ .The model is fitted using Expectation-Maximization (Bishop, 2006). In the Expectation step, we compute the responsibility $z_{ti}$ , that is the posterior probability that the sample i was attended at trial t (for i = L, R, the probability that the trial corresponded to a lapse trial):

In the Maximization step, we update the value of the parameters by maximizing the Expected Complete Log-Likelihood (ECLL): $Q(\pi ,\beta )={\Sigma }_{ti}{z}_{ti}logp(r_t;\pi ,\beta )$. Maximizing over the mixture coefficients with the unity-sum constraint provides the classical update: ${\pi }_i={\Sigma }_{ti}{z}_{ti}/N$, where N is the total number of trials. In the non-deterministic model, maximizing the ECLL over sensitivity parameters is equivalent to fitting a logistic regression model with weighted coefficients $z_{ti}$, which is a convex problem. Best fitting parameters can be found using Newton–Raphson updates on the parameters:

To speed up the computations, in each M step, we only performed one Newton–Raphson update for each sensitivity parameter, rather than iterating the updates fully until convergence. The EM procedure was run until convergence, assessed by an increment in the log-likelihood $L(\pi ,\beta )$ of less than 10⁻⁹ after one EM iteration. The log-likelihood for a given set of parameters is given by $L(\pi ,\beta )={\Sigma }_{t}logp\left(r_t\right)$. The EM iterative procedure was repeated with 10 different initializations of the parameters to avoid local minima.

Note that for monkey and rat data, since we observed large variations in performance across sessions, the model based its choices on session-gain-modulated evidence ${\stackrel{-}{\mathrm{S}}}_{ti}={\gamma }_t{S}_{ti}$ instead of raw evidence $S_{ti}$ (this had no impact for the deterministic variant since ${\stackrel{-}{\mathrm{S}}}_{ti}$ and $S_{ti}$ always have the same sign). We fitted the model from individual subject responses either with lapses ${\pi }_L$ and ${\pi }_R$ as free parameters, or fixed to ${\pi }_L={\pi }_R=0.01$. Figures in the main manuscript correspond to the deterministic snapshot model with fixed lapses. We also studied variants of the snapshot model where decisions in each trial are based on K attended samples, that is depends on $(S_{ti},..S_{t,i+K-1})$ with $1\le K\le n-1$ and $1\le i\le n-K+1$ is the first attended sample. In the deterministic case, the choice is directly determined by the sign of the sum of the signed evidence for the attended samples. In the non-deterministic case, the evidence for the attended samples are weighted and passed through a sigmoid: ${\displaystyle H_i\left(S_t\right)=\sigma \left({\mathrm{\Sigma }}_{k\in \left[\mathrm{1...}K\right]}{\beta }_{ki}{S}_{t,i+k-1}\right)}$. The model with a single attended sample presented above is equivalent to this extended model when using $K=1$. At the other end, using $K=n$ corresponds to the temporal integration model (without the lateral bias).

In the extrema-detection model, a choice is selected according to the first sample in the sequence whose absolute evidence value reaches a certain threshold $\theta$, that is ${\displaystyle p\left(r_t|\theta \right)=H\left(m_{ti}\right),|m_{ti}|\ge \theta ,|m_{tj}|<\theta }$ for all ${\displaystyle j<i}$ . Here, $m_{ti}$ is the sample evidence corrupted by sensory noise ${\epsilon }_{ti}$ which is distributed normally with variance ${\sigma }^2$ : $m_{ti}=S_{ti}+{\epsilon }_{ti}$ with ${\epsilon }_{ti}\sim N(0,{\sigma }^2)$. H is the step function. If the stimulus sequence ends and no sample has reached the threshold, then the decision is taken at chance. As described in Waskom and Kiani, 2018, the probability for a rightward choice at trial t can be expressed as:

where $\Phi$ is the cumulative normal distribution. We also included the possibility for left and right lapses with probability ${\pi }_L$ and ${\pi }_R$ . Following Stine et al., 2020, we explored an alternative default rule called ‘last sample’ rule: if the stimulus extinguishes and the threshold has not been reached, then the decision is based on the (noisy) last sample rather than simply by chance. This changes the equation describing the probability for rightward choices to:

We also explored a variant of the model where the threshold changes on every sample, that is the equations above are changed by substituting $\theta$ with ${\theta }_i$ . As for the snapshot model, we used the session-gain-modulated evidence ${\stackrel{-}{\mathrm{S}}}_{ti}$ instead of raw evidence $S_{ti}$ for fitting the model to monkey and rat data. The four parameters of the model $(\theta ,\sigma ,{\pi }_L,{\pi }_R)$ were estimated from each subject data by maximizing the log-likelihood with interior-point algorithm (function fmincon in Matlab) and 10 different initializations of the parameters. (In the varying-threshold variant, there are n + 3 parameters which are estimated similarly.)

Psychophysical kernels were obtained from subject and simulated data by running a logistic regression model: $p\left(r_t\right)=\sigma ({\beta }_0+{\Sigma }_i{\beta }_i{S}_{ti})$. Standard errors of the weights ${\beta }_i$ were obtained from the Laplace approximation. For psychometric curves, we first defined the weighted stimulus evidence T_t at trial t as the session-modulated weighted sum of signed sample evidence; with the weights obtained from the logistic regression model above ${T_t={\gamma }_t\Sigma }_i{\beta }_i{S}_{ti}$ . We then divided the total stimulus evidence into 50 quantiles (10 for human subjects) and computed the psychometric curve as the proportion of rightward choices for each quantile.

The boundary performance for the snapshot and extrema-detection models corresponds to the best choice accuracy out of all the parameterizations for each model. In the snapshot model, the boundary performance corresponds to the deterministic version with no-lapse, where the attended sample is always the sample $i^{*}$ whose sign better predicts the stimulus category over all animal trials, that is ${\pi }_{i^{*}}=1$ and ${\pi }_i=0$ if ${i\ne i}^{*}$ . For the extrema-detection model, the boundary performance corresponds to the lapse-free model with no sensory noise ($\sigma =0$) and a certain value for threshold $\theta$ that is identified for each subject by simple parameter search.

Finally, model selection was performed using the AIC $AIC=2p-2L_{ML}$ , where p is the number of model parameters and $L_{ML}$ is the likelihood evaluated at maximum likelihood parameters.

We selected for each animal the subset of trials corresponding to when the largest evidence sample was at odds with the total stimulus evidence (disagree trials), that is where $sign(S_{tj},|S_{tj}|\ge |S_{ti}|\vee i)\ne sign\left({\Sigma }_i{S}_{ti}\right)$. For this subset of trials, we computed the proportion of animal choices that were aligned with the overall stimulus evidence. We repeated the analysis for simulated data from the integration and extrema-detection models. We computed error bars following a parametric bootstrap procedure: for each bootstrap, we simulated the model (integration/extrema-detection) using parameters sampled from their posterior distribution (based on the Laplace approximation). We then applied the analysis on disagree trials for simulated data, and used these bootstrap values to define confidence intervals.

In order to estimate the impact of each sample on the animal choice as a function of sample evidence, we built and estimated the following statistical model:

As can be seen, this model is equivalent to the temporal integration model under the assumption that f is a linear function. Rather, here we wanted to estimate the function f (as well as the session gain ${\gamma }_t$ , lateral bias ${\beta }_0$, and sensory weight ${\beta }_i$). Including the session gain was necessary for estimating f accurately from the monkey and rat behavioral data, since the distribution of pulse strength $S_{ti}$ was varied across sessions and could otherwise induce a confound. We assumed that f is an odd function, that is $f\left({-S}_{ti}\right)=-f\left(S_{ti}\right)$. This equation takes the form of a GUM and was fitted using the Laplace approximation method as described in Adam and Hyafil, 2020. In the monkey experiment, sample evidence could take only a finite number of values, so f was simply estimated over these values. In the human experiment, sample evidence could take continuous values. In this case, we defined a Gaussian Process prior over f with squared exponential kernel with length scale 0.1 and variance 1.

We designed a new analysis tool to characterize the statistical mapping from the multidimensional stimulus space ${\mathit{S}}_t=(S_{t1},...S_{tn})\in {\mathfrak{R}}^n$ onto binary choices $r_t\in \left[\mathrm{0,1}\right]$. We first collapsed the stimulus sequence ${\mathit{S}}_t$ onto the two-dimensional space defined by early evidence ${\mathit{E}}_t$ and late evidence ${\mathit{L}}_t$ defined by ${E_t={\gamma }_t\Sigma }_{1\le i\le [n/2]}{\beta }_i{S}_{ti}$ and ${L_t={\gamma }_t\Sigma }_{[n/2]+1\le i\le n}{\beta }_i{S}_{ti}$ , where the weights ${\beta }_i$ and session gains ${\gamma }_t$ correspond to parameters estimated from the temporal integration model (session gains were omitted for human participants). Next we plotted the integration map which represents the probability for rightward choices as a function of $(E_t,L_t)$. The map was obtained by smoothing data points with a two-dimensional Gaussian kernel. More specifically, for each pair value (E,L), we selected the trials whose early and late evidence values $E_t$ and $L_t$ fell within a certain distance to (E,L), that is ${\displaystyle d_t=dist\left(\left(E,L\right)\left(E_t,L_t\right)\right)<2}$. We then computed the proportion of rightward choices for the selected trials, with a weight for each trial depending on the distance to the pair value $w_t=N\left(\right(E_t,L_t);(E,L),0.1^{2}I)$. Because the space (E,L) was not sampled uniformly during the experiment, we represent the density of trials by brightness. For each subject, we obtained integration maps both from subject data and from model simulations. For each model, we computed the Pearson correlation between the maps obtained from the corresponding simulation and from the subject data. We tested the significance of correlation measures between models by using a bootstrapping procedure: we calculated the correlation measure r from 100 bootstraps for each model and participant, and then performed an unpaired t-test between bootstrapped r.

Next, we analyzed the conditional psychometric curves, that is the psychometric curves for the early evidence conditioned on the value of late evidence, which correspond to vertical cuts in the integration map. To do so, we first binned late evidence $L_t$ by bins of width 0.5. Conditional psychometric curve represents the probability of rightward choices as a function of early evidence $E_t$, separately for each late evidence bin. For each late evidence bin, we also estimated the corresponding bias $\beta$, left lapse ${\pi }_L$ and right lapse ${\pi }_R$ by fitting the following function on the corresponding subset of trials:

We analyzed the activity of 82 LIP neurons recorded over 43 sessions of the motion discrimination tasks (Yates et al., 2017). We applied the following procedure to extract the integration map for LIP neurons. For each neuron n, we computed the spike count ${s_t}^{\left(n\right)}$ in a window of 500 ms width following each stimulus offset, which is where LIP neurons were found to have maximal selectivity to motion evidence from the entire pulse sequence (Yates et al., 2017). We then applied a Poisson GLM ${{E(s}_t}^{\left(n\right)})=exp({w_0}^{\left(n\right)}+{\Sigma }_i{w_i}^{\left(n\right)}{S}_{ti})$ for each neuron n to extract the impact of each sample i on the individual neural spike count ${w_i}^{\left(n\right)}$ . For each trial t, we used these weights to compute the neuron-weighted early and late evidence defined by and ${E_t}^{\left(n\right)}={\Sigma }_{1\le i\le 3}{w_i}^{\left(n\right)}{S}_{ti}$ and ${L_t}^{\left(n\right)}={\Sigma }_{4\le i\le 7}{w_i}^{\left(n\right)}{S}_{ti}$ . Note that this weighting converts the evidence onto the space defined by the preferred direction of the neuron, such that positive evidence signals evidence toward the preferred direction and negative evidence signals evidence toward the anti-preferred direction. We then merged the vectors for normalized spike counts ${\stackrel{-}{s}}_t^{\left(n\right)}={s_t}^{\left(n\right)}/exp\left({w_0}^{\left(n\right)}\right)$, early evidence ${E_t}^{\left(n\right)}$ and late evidence ${L_t}^{\left(n\right)}$ across all neurons. The normalized spike counts were binned by values of early and late evidence (bin width: 0.02), and the average over each bin was computed after convolving with a two-dimensional Gaussian kernel of width 0.1. The neural integration map represents the average normalized activity per bin.

Simulations of spiking data for the integration and non-integration models were proceeded as follows. First, the neural integration model corresponds to linear summing with neuron-specific weights which are then passed through an exponential nonlinearity; the spike counts for each trial are generated using a Poisson distribution whose rate is equal to the nonlinear output (Figure 4—figure supplement 3a, top). This corresponds exactly to the generative process of the Poisson GLM described above. For the extrema-detection model (Figure 4—figure supplement 3a, middle), we hypothesized that LIP activity would only be driven by the sample that reaches the threshold (and dictates the animal response). To this end, we first simulated the behavioral extrema-detection model for all trials, using parameters $(\theta ,\sigma ,{\pi }_L,{\pi }_R)$ fitted from the corresponding animal, to identify which sample i reaches the subject-specific threshold. We then assumed that the spiking activity of the neuron would follow the stimulus value at sample i $S_{ti}$ (signed by the preferred direction of the neuron $p^{\left(n\right)}$ through):

Again the spike count were generated from a Poisson distribution with rate ${{E_{ED}(s}_t}^{\left(n\right)})$.

Finally, for the snapshot model (Figure 4—figure supplement 3a, bottom), we assumed that the neuron activity would merely reflect the sensory value of the only sample it would attend. We assumed that the probability mass function to attend each of the seven samples would be neuron specific, so we used the normalized weights of the Poisson GLM for that specific neuron as defining such probability (weights were signed by the neuron preferred direction so that the vast majority of weights were positive; negative weights were ignored). For each trial, we thus randomly sampled the attended sample i using this probability mass function and then simulated the spike count ${s_t}^{\left(n\right)}$ from a Poisson distribution with rate

We simulated spiking activity for each neuron and for each integration and non-integration model, and then used simulated data to compute neural integration maps exactly as described above for the actual LIP neuron activity.

The funders had no role in study design, data collection, and interpretation, or the decision to submit the work for publication.

The authors thank Jake Yates for sharing information regarding the monkey experimental data. The authors are supported by the Spanish State Research Agency (RYC-2017-23231 to AH), Spanish Ministry of Economy and Competitiveness together with the European Regional Development Fund grant SAF2015-70324-R (to JR), European Research Council grant ERC-2015-CoG-683209 (to JR), NIH grant R01EY017366 (to ACH and JWP), NIH BRAIN Initiative grant NS104899 (to JWP), and the Simons Collaboration on the Global Brain (SCGB AWD543027, JWP).

Informed consent was obtained from all participants. The experiment with human participants was approved by the UPF ethics committee (approval 654 2013/5435/I from CEIm-Parc de Salut MAR).

Rat experiments were approved by the local ethics committee of the University of Barcelona 658 (Comitéd'Experimentació Animal, Barcelona, Spain, protocol number Ref 390/14). Monkey experiment: All experimental protocols were approved by The University of Texas Institutional Animal Care and Use Committee (AUP-2012-00085, AUP-2015-00068) and in accordance with National Institutes of Health standards for care and use of laboratory animals.

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