PWM tasks involve the sequential comparison of two graded stimuli that differ along a physical dimension and are separated by a delay interval of a few seconds (Figure 1A and B; Romo and Salinas, 2003; Akrami et al., 2018; Ashourian and Loewenstein, 2011). A key feature emerging from these studies is contraction bias, where the averaged performance is as if the memory of the first stimulus progressively shifts toward the center of a prior distribution built from past sensory history (Figure 1C). Additionally, biases toward the most recent sensory stimuli (immediately preceding trials) have also been documented (Akrami et al., 2018; Raviv et al., 2012).

Figure 1

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In order to investigate the circuit mechanisms by which such biases may occur, we use two identical one-dimensional continuous attractor networks to model WM and PPC modules. Neurons are arranged according to their preferential firing locations in a continuous stimulus space, representing the amplitude of auditory stimuli. Excitatory recurrent connections between neurons are symmetric and a monotonically decreasing function of the distance between the preferential firing fields of neurons, allowing neurons to mutually excite one another; inhibition, instead, is uniform. Together, both allow a localized bump of activity to form and be sustained (Figure 1D and E; Sebastian Seung, 1998; Wang, 2001; Zhong et al., 2020; Spalla et al., 2021; Wu and Amari, 2005; Wu et al., 2016; Fung et al., 2008; Fung et al., 2010; Trappenberg, 2005). Both networks have free boundary conditions. Neurons in the WM network receive inputs from neurons in the PPC coding for the same stimulus amplitude (Figure 1D). Building on experimental findings (Murray et al., 2014; Siegle et al., 2021; Gao et al., 2020; Wang et al., 2023; Mejías and Wang, 2022; Ding et al., 2022), we designed the PPC network such that it integrates activity over a longer timescale compared to the WM network section ‘The model’. Moreover, neurons in the PPC are equipped with neural adaptation that can be thought of as a threshold that dynamically follows the activation of a neuron over a longer timescale.

To simulate the PWM task, at the beginning of each trial, the network is provided with a stimulus ${\displaystyle s_1}$ for a short time via an external current ${\displaystyle I_{\mathrm{e}\mathrm{x}\mathrm{t}}}$ as input to a set of neurons (see Appendix 1—table 1). Following ${\displaystyle s_1}$, after a delay interval, a second stimulus ${\displaystyle s_2}$ is presented (Figure 1E). The pair ${\displaystyle (s_1,s_2)}$ is drawn from the stimulus set shown in Figure 1B, where they are all equally distant from the diagonal ${\displaystyle s_1=s_2}$, and are therefore of equal nominal discrimination, or difficulty. The stimuli ${\displaystyle (s_1,s_2)}$ are co-varied in each trial so that the task cannot be solved by relying on only one of the stimuli (Hernández et al., 1997). As in the study in Akrami et al., 2018 using an interleaved design, across consecutive trials, the interstimulus delay intervals are randomized and sampled uniformly between 2, 6, and 10 s. The intertrial interval (ITI), instead, is fixed at 5 s.

We additionally include psychometric pairs (indicated in the box in Figure 1B) where the distance to the diagonal, hence the discrimination difficulty, is varied. The task is a binary comparison task that aims at classifying whether ${\displaystyle s_1<s_2}$ or vice versa. In order to solve the task, we record the activity of the WM network at two time points: slightly before and after the onset of ${\displaystyle s_2}$ (Figure 1E). We repeat this procedure across many different trials and use the recorded activity to assess performance (see section ‘Simulation’) for details. Importantly, at the end of each trial, the activity of both networks is not re-initialized, and the state of the network at the end of the trial serves as the initial network configuration for the next trial.

Probing the WM network performance of psychometric stimuli (Figure 1B, purple box, 10% of all trials) shows that the comparison behavior is not error-free and that the psychometric curves (different colors) differ from the optimal step function (Figure 2A, green dashed line). The performance of pyschometric trials is also better for shorter interstimulus delay intervals, as has been shown in previous work (Sinclair and Burton, 1996; Akrami et al., 2018). In our model, errors are caused by the displacement of the activity bump in the WM network due to the inputs from the PPC network. These displacements in the WM activity bump can result in different outcomes: by displacing it away from the second stimulus, they either do not affect the performance or improve it (Figure 2B, right panel, ‘Bias+’), if noise is present. Conversely, the performance can suffer, if the displacement of the activity bump is toward the second stimulus (Figure 2B, left panel, ‘Bias-’). Note, however, that in these two specific trials the activity bump in PPC is strong and it displaces the activity bump in the WM network, but this is not the only kind of dynamics present in the network (see section ‘Multiple timescales at the core of short-term sensory history effects’ for a more detailed analysis of the network dynamics).

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Performance of stimulus pairs that are equally distant from the ${\displaystyle s_1=s_2}$ diagonal can be similarly impacted and the network produces a pattern of errors that are consistent with contraction bias: performance is at its minimum for stimulus pairs in which ${\displaystyle s_1}$ is either largest or smallest, and at its maximum for stimulus pairs in which ${\displaystyle s_2}$ is largest or smallest (Figure 2C, left panel; Ashourian and Loewenstein, 2011; Fassihi et al., 2014; Akrami et al., 2018; Fassihi et al., 2017; Esmaeili and Diamond, 2019). These results are consistent with the performance of humans and rats on the auditory task, as previously reported (Figure 2C, middle and right panels, data from Akrami et al., 2018).

Can the same circuit also give rise to short-term sensory history biases (Akrami et al., 2018; Loewenstein et al., 2021)? We analyzed the fraction of trials the network response was ‘${\displaystyle s_1<s_2}$’ in the current trial conditioned on stimulus pairs presented in the previous trial and found that the network behavior is indeed modulated by the preceding trial’s stimulus pairs (Figure 2D, panel 1). We quantified these history effects as well as how many trials back they extend to. We computed the bias by plotting, for each particular pair (of stimuli) presented at the current trial, the fraction of trials the network response was ‘${\displaystyle s_1<s_2}$’ as a function of the pair presented in the previous trial minus the mean performance over all previous trial pairs (Figure 2D, panel 2; Akrami et al., 2018). Independent of the current trial, the previous trial exerts an ‘attractive’ effect, expressed by the negative slope of the line: when the previous pair of stimuli is small, ${\displaystyle s_1}$ in the current trial is, on average, misclassified as smaller than it actually is, giving rise to the attractive bias in the comparison performance; the converse holds true when the previous pair of stimuli happens to be large. These effects extend to two trials back and are consistent with the performance of humans and rats on the auditory task (Figure 2D, panels 3–6, data from Akrami et al., 2018).

It has been shown that inactivating the PPC in rats performing the auditory delayed comparison task markedly reduces the magnitude of contraction bias without impacting non-sensory biases (Akrami et al., 2018). We assay the causal role of the PPC in generating the sensory history effects as well as contraction bias by weakening the connections from the PPC to the WM network, mimicking the inactivation of the PPC. In this case, we see that the performance of the psychometric stimuli is greatly improved (yellow curve, Figure 2E, top panel), consistent also with the inactivation of the PPC in rodents (yellow curve, Figure 2E, bottom panel, data from Akrami et al., 2018). Performance is improved also for all pairs of stimuli in the stimulus set (Figure 2—figure supplement 1A). The breakdown of the network response in the current trial conditioned on the specific stimulus pair preceding it reveals that the previous trial no longer exerts a notable modulating effect on the current trial (Figure 2—figure supplement 1B). Quantifying this bias by subtracting the mean performance over all of the previous pairs reveals that the attractive bias is virtually eliminated (yellow curve, Figure 2F, left panel), consistent with findings in rats (Figure 2F, right panel, data from Akrami et al., 2018).

Together, our results suggest a possible circuit through which both contraction bias and short-term history effects in a PWM task may arise. The main features of our model are two continuous attractor networks, both integrating the same external inputs, but operating over different timescales. Crucially, the slower one, a model of the PPC, includes neuronal adaptation and provides input to the faster one, intended as a WM circuit. Note that a block design where the delay interval is kept fixed yields similar results (Figure 2—figure supplement 2). In the next section, we show how the slow integration and firing rate adaptation in the PPC network give rise to the observed effects of sensory history.

The activity bumps in the PPC and WM networks undergo different dynamics due to the different timescales with which they integrate inputs, the presence of adaptation in the PPC, and the presence of global inhibition. The WM network integrates inputs over a shorter timescale, and therefore the activity bump follows the external input with high fidelity (Figure 3A [purple bumps] and Figure 3B [purple line]). The PPC network, instead, has a longer integration timescale, and therefore fails to sufficiently integrate the input to induce a displacement of the bump to the location of a new stimulus, at each single trial. This is mainly due to the competition between the inputs from the recurrent connections sustaining the bump and the external stimuli that are integrated elsewhere: if the former is stronger, the bump is not displaced. If, however, these inputs are weaker, they will not displace it, but may still exert a weakening effect via the global inhibition in the connectivity. The external input, as well as the presence of adaptation (Figure 3—figure supplement 1B and C), induces a small continuous drift of the activity bump that is already present from the previous trials (lower-right panel of Figure 2B, Figure 3A [pink bumps] and Figure 3B [pink line]). The build-up of adaptation in the PPC network, combined with the global inhibition from other neurons receiving external inputs, can extinguish the bump in that location (see also Figure 3—figure supplement 1 for more details). Following this, the PPC network can make a transition to an incoming stimulus position (that may be either ${\displaystyle s_1}$ or ${\displaystyle s_2}$), and a new bump is formed. The resulting dynamics in the PPC are a mixture of slow drift over a few trials, followed by occasional jumps (Figure 3A).

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As a result of such dynamics, relative to the WM network, the activity bump in the PPC represents the stimuli corresponding to the current trial in a smaller fraction of the trials and represents stimuli presented in the previous trial in a larger fraction of the trials (Figure 3C). This yields short-term sensory history effects in our model (Figure 2D and E) as input from the PPC leads to the displacement of the WM bump to other locations (Figure 3D). Given that neurons in the WM network integrate this input, once it has built up sufficiently, it can surpass the self-sustaining inputs from the recurrent connections in the WM network. The WM bump, then, can move to a new location, given by the position of the bump in the PPC (Figure 3D). As the input from the PPC builds up gradually, the probability of bump displacement in WM increases over time. This in return leads to an increased probability of contraction bias (Figure 3E) and short-term history (one-trial back) biases (Figure 3F), as the interstimulus delay interval increases.

Additionally, a non-adapted input from the PPC has a larger likelihood of displacing the WM bump. This is highest immediately following the formation of a new bump in the PPC or, in other words, following a ‘bump jump’ (Figure 3F). As a result, one can reason that those trials immediately following a jump in the PPC are the ones that should yield the maximal bias toward stimuli presented in the previous trial. We therefore separated trials according to whether or not a jump has occurred in the PPC in the preceding trial (we define a jump to have occurred if the bump location across two consecutive trials in the PPC is displaced by an amount larger than the typical width of the bump [section ‘The model’]). In line with this reasoning, only the set that included trials with jumps in the preceding trial yields a one-trial back bias (Figure 3G).

Removing neuronal adaptation entirely from the PPC network further corroborates this result. In this case, the network dynamics show a very different behavior: the activity bump in the PPC undergoes a smooth drift (Figure 3—figure supplement 2A), and the bump distribution is much more peaked around the mean (Figure 3—figure supplement 2B), relative to when adaptation is present (Figure 4A). In this regime, there are no jumps in the PPC (Figure 3—figure supplement 2A), and the activity bump corresponds to the stimuli presented in the previous trial in a fewer fraction of the trials (Figure 3—figure supplement 2C), relative to when adaptation is present (Figure 3B). As a result, no short-term history effects can be observed (Figure 3—figure supplement 2C and D), even though a strong contraction bias persists (Figure 3—figure supplement 2E).

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As in the study in Akrami et al., 2018, we can further study the impact of the PPC on the dynamics of the WM network by weakening the weights from the PPC to the WM network, mimicking the inactivation of PPC (Figure 2E and F, Figure 2—figure supplement 1A and B). Under this manipulation, the trajectory of the activity bump in the WM network immediately before the onset of the second stimulus ${\displaystyle s_2}$ closely follows the external input, consistent with an enhanced WM function (Figure 2—figure supplement 1C and D).

The drift-jump dynamics in our model of the PPC give rise to short-term (notably one- and two-trial back) sensory history effects in the performance of the WM network. In addition, we observe an equally salient contraction bias (bias toward the sensory mean) in the WM network’s performance, increasing with the delay period (Figure 3E). However, we find that the activity bump in both the WM and the PPC network corresponds to the mean over all stimuli in only a small fraction of trials, expected by chance (Figure 3B, see section ‘Computing bump location’ for how it is calculated). Rather, the bump is located more often at the current trial stimulus (${\displaystyle s_1^t}$), and to a lesser extent, at the location of stimuli presented at the previous trial (${\displaystyle s_2^{t-1}}$). As a result, contraction bias in our model cannot be attributed to the representation of the running sensory average in the PPC. In the next section, we show how contraction bias arises as an averaged effect when single-trial errors occur due to short-term sensory history biases.

In order to illustrate the statistical origin of contraction bias in our network model, we consider a mathematical scheme of its performance (Figure 4B). In this simple formulation, we assume that the first stimulus to be kept in WM, ${\displaystyle s_1^t}$, is volatile. As a result, in a fraction ${\displaystyle ϵ}$ of the trials, it is susceptible to replacement with another stimulus ${\displaystyle \hat{s}}$ (by the input from the PPC, which has a given distribution ${\displaystyle p_m}$; Figure 4A). However, this replacement does not always lead to an error, as evidenced by Bias- and Bias+ trials (i.e., those trials in which the performance is affected negatively and positively, respectively; Figure 2B). For each stimulus pair, the probability to make an error, ${\displaystyle p_e}$, is integral of ${\displaystyle p_m}$ over values lying on the wrong side of the ${\displaystyle s_1=s_2}$ diagonal (Figure 4C). For instance, for stimulus pairs below the diagonal (Figure 4C, blue squares) the trial outcome is erroneous only if ${\displaystyle \hat{s}}$ is displaced above the diagonal (red part of the distribution). As one can see, the area above the diagonal increases as ${\displaystyle s_1}$ increases, giving rise to a gradual increase in error rates (Figure 4C). This mathematical model can capture the performance of the attractor network model, as can be seen through the fit of the network performance, when using the bump distribution in the PPC as ${\displaystyle p_m}$ and ${\displaystyle ϵ}$ as a free parameter (see Equation 9 in section ‘The probability to make errors is proportional to the cumulative distribution of the stimuli, giving rise to contraction bias’, Figure 4D and E).

Can this simple statistical model also capture the behavior of rats and humans (Figure 2C)? We carried out the same analysis for rats and humans by replacing the bump location distribution of PPC with that of the marginal distribution of the stimuli provided in the task based on the observation that the former is well-approximated by the latter (Figure 4A). In this case, we see that the model roughly captures the empirical data (Figure 4F and G), with the addition of another parameter ${\displaystyle \delta }$ that accounts for the lapse rate. Interestingly, such ‘lapse’ also occurs in the network model (as seen by the small amount of errors for pairs of stimuli where ${\displaystyle s_2}$ is smallest and largest; Figure 4E). This occurs because of the drift present in the PPC network that eventually, for long enough delay intervals, causes the bump to arrive at the boundaries of the attractor, which would result in an error.

This simple analysis implies that contraction bias in the WM network in our model is not the result of the representation of the mean stimulus in the PPC, but is an effect that emerges as a result of the PPC network’s sampling dynamics, mostly from recently presented stimuli. Indeed, a ‘contraction to the mean’ hypothesis only provides a general account of which pairs of stimuli should benefit from a better performance and which should suffer, but does not explain the gradual accumulation of errors upon increasing (decreasing) ${\displaystyle s_1}$, for pairs below (above) the ${\displaystyle s_1=s_2}$ diagonal (Fassihi et al., 2014; Fassihi et al., 2017; Akrami et al., 2018). Notably, it cannot explain why the performance in trials with pairs of stimuli where ${\displaystyle s_2}$ is most distant from the mean stand to benefit the most from it. Altogether, our model suggests that contraction bias may be a simple consequence of errors occurring at single trials, driven by inputs from the PPC that follow a distribution similar to that of the external input (Figure 4B).

Can contraction bias also be observed in the activity of the WM network prior to binary decision-making? Many studies have evidenced contraction bias also in delayed estimation (or production) paradigms, where subjects must retain the value of a continuous parameter in WM and reproduce it after a delay (Papadimitriou et al., 2015; Jazayeri and Shadlen, 2010). Given that we observe contraction bias in the behavior of the network, we reasoned that this should also be evident prior to binary decision-making. Similar to delayed estimation tasks, we therefore analyzed the position of the bump ${\displaystyle \hat{s}}$, at the end of the delay interval, for each value of ${\displaystyle s_1}$. Consistent with our reasoning, we observe contraction bias of the value of ${\displaystyle \hat{s}}$, as evidenced by the systematic departure of the curve corresponding to the bump location from that of the nominal value of the stimulus (Figure 5A). We also find that this contraction bias becomes greater as the delay interval increases (Figure 5A, right). We next analyzed the effect of the previous trial on the current trial by computing the displacement of the bump during the WM delay as a function of the distance between the current trial’s stimulus and the previous trial’s stimulus ${\displaystyle s_1(t)-s_2(t-1)}$ (Figure 5B). We found that when this distance is larger, the displacement of the bump during WM is on average also larger (Figure 5B). This displacement is also attractive. Breaking down these effects by delay, we find that longer delays lead to greater attraction (Figure 5B, right).

Figure 5

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These results point to attractive effects of the previous trial, leading in turn to contraction bias in our model. To better understand the dynamics leading to them, we next looked at the distribution of bump displacements conditioned on a specific value of the second stimulus of the previous trial ${\displaystyle s_2(t-1)}$ (Figure 5C). These distributions are characterized by a mode around 0, corresponding to a majority of trials in which the bump is not displaced, and another mode around ${\displaystyle s_1(t)-s_2(t-1)}$, corresponding to the displacement in the direction of the preceding trial’s stimulus described in section ‘Multiple timescales at the core of short-term sensory history effects’ and Figure 2B. However, note that the variance of this second mode can be large, reflecting displacements to locations other than ${\displaystyle s_2(t-1)}$, due to the complex dynamics in both networks that we have described in detail in section ‘Multiple timescales at the core of short-term sensory history effects’.

In a recent study (Lieder et al., 2019), a similar PWM task with auditory stimuli was studied in human neurotypical (NT), autistic spectrum (ASD) and dyslexic (DYS) subjects. Based on an analysis using a generalized linear model (GLM), a double dissociation between different subject groups was suggested: ASD subjects exhibit a stronger bias toward long-term statistics – compared to NT subjects – while for DYS subjects, a higher bias is present toward short-term statistics.

We investigated our model to see whether it is able to show similar phenomenology, and if so, what are the relevant parameters controlling the timescale of the biases in behavior? We identified the adaptation timescale in the PPC as the parameter that affects the extent of the short-term bias, consistent with previous literature (Jaffe-Dax et al., 2018; Jaffe-Dax et al., 2017). Calculating the mean bias toward the previous trial stimulus pair (Figure 9A), we find that a shorter-than-NT adaptation timescale yields a larger bias toward the previous trial stimulus. Indeed, a shorter timescale for neuronal adaptation implies a faster process for the extinction of the bump in PPC – and the formation of a new bump that remains stable for a few trials – producing ‘jumpier’ dynamics that lead to a larger number of one-trial-back errors. In contrast, increasing this timescale with respect to NT gives rise to a stable bump for a longer time, ultimately yielding a smaller short-term bias. This can be seen in the detailed breakdown of the network’s behavior on the current trial when conditioned on the stimuli presented at the previous trial (Figure 9B, see also section ‘Multiple timescales at the core of short-term sensory history effects’ for a more detailed explanation of the dynamics). We performed a GLM analysis as in Lieder et al., 2019 to the network behavior, with stimuli from four trials back and the mean stimulus as regressors (see section ‘Generalized linear model’). This analysis shows that a reduction in the PPC adaptation timescale with respect to NT produces behavioral changes qualitatively compatible with data from DYS subjects; on the contrary, an increase of this timescale yields results consistent with ASD data (Figure 9C).

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This GLM analysis suggests that dissociable short- and long-term biases may be present in the network behavior. Having access to the full dynamics of the network, we sought to determine how it translates into such dissociable short- and long-term biases. Given that all the behavior arises from the location of the bump on the attractor, we quantified the fraction of trials in which the bump in the WM network, before the onset of the second stimulus, was present in the vicinity of any of the previous trial’s stimuli (Figure 9—figure supplement 1B, right panel, and Figure 9—figure supplement 1C), as well as the vicinity of the mean over the sensory history (Figure 9—figure supplement 1B, left panel, and Figure 9—figure supplement 1C). While the bump location correlated well with the GLM weights corresponding to the previous trial’s stimuli regressor (comparing the right panels of Figure 9—figure supplement 1A and B), surprisingly, it did not correlate with the GLM weights corresponding to the mean stimulus regressor (comparing the left panels of Figure 9—figure supplement 1A and B). In fact, we found that the bump was in a location given by the stimuli of the past two trials, as well as the mean over the stimulus history, in a smaller fraction of trials, as the adaptation timescale parameter was made larger (Figure 9—figure supplement 1C).

Given that the weights, after four trials in the past, were still non-zero, we extended the GLM regression by including a larger number of past stimuli as regressors. We found that doing this greatly reduced the weight of the mean stimulus regressor (Figure 9C–E, see section ‘Generalized linear mode’ for more details). Therefore, we propose an alternative interpretation of the GLM results given in Lieder et al., 2019. In our model, the increased (reduced) weight for long-term mean in the ASD (DYS) subjects can be explained as an effect of a larger (smaller) window in time of short-term biases without invoking a double dissociation mechanism (Figure 9D and E). In section ‘Generalized linear model’, we provide a mathematical argument for this, which is empirically shown by including a large number of individual stimuli from previous trials in the regression analysis.

Contraction bias is an effect emerging in WM tasks, where in the averaged behavior of a subject the magnitude of the item held in memory appears to be larger than it actually is when it is ‘small’ and, vice versa, it appears to be smaller when it is ‘large’ (Algom, 1992; Berliner et al., 1977; Hellström, 1985; Poulton and Poulton, 1989; Ashourian and Loewenstein, 2011; Preuschhof et al., 2010; Olkkonen et al., 2014). Recently, Akrami et al., 2018 found that contraction bias as well as short-term history-dependent effects occur in an auditory delayed comparison task in rats and humans: the comparison performance in a given trial depends on the stimuli shown in preceding trials (up to three trials back) (Akrami et al., 2018), similar to previous findings in human 2AFC paradigms (Raviv et al., 2012). These findings raise the question: does contraction bias occur independently of short-term history effects, or does it emerge as a result of the latter?

Akrami et al., 2018 have also found the PPC to be a critical node for the generation of such effects as its optogenetic inactivation (specifically during the delay interval) greatly attenuated both effects. WM was found to remain intact, suggesting that its content was perhaps read-out in another region. Electrophysiological recordings as well as optogenetic inactivation results in the same study suggest that while sensory history information is provided by the PPC, its integration with the WM content must happen somewhere downstream to the PPC. Different brain areas can fit the profile. For instance, there are known projections from the PPC to mPFC in rats (Olsen et al., 2019), where neural correlates of PWM have been found (Esmaeili and Diamond, 2019). Building on these findings, we suggest a minimal two-module model aimed at better understanding the interaction between contraction bias and short-term history effects. These two modules capture properties of the PPC (in providing sensory history signals) and a downstream network holding WM content. Our WM and PPC networks, despite having different timescales, are both shown to encode information about the marginal distribution of the stimuli (Figure 4A). Although they have similar activity distributions to that of the external stimuli, they have different memory properties due to the different timescales with which they process incoming stimuli. The putative WM network, from which information to solve the task is read-out, receives additional input from the PPC network. The PPC is modeled as integrating inputs slower relative to the WM network and is also endowed with firing rate adaptation, the dynamics of which yield short-term history biases and, consequently, contraction bias.

It must be noted, however, that short-term history effects (due to firing rate adaptation) do not necessarily need to be invoked in order to recover contraction bias: as long as errors are made following random samples from a distribution in the same range as that of the stimuli, contraction bias should be observed (Tong and Dubé, 2022). Indeed, when we manipulated the parameters of the PPC network in such a way that short-term history effects were eliminated (by removing the firing rate adaptation), contraction bias persisted. As a result, our model suggests that contraction bias may not simply be given by a regression toward the mean of the stimuli during the interstimulus interval (Karim et al., 2013; Kerst and Howard, 1978), but brought about by a richer dynamics occurring at the level of individual trials (Jou et al., 2004), more in line with the idea of random sampling (Rahnev and Denison, 2018).

The model makes predictions as to how the pattern of errors may change when the distribution of stimuli is manipulated either at the level of the presented stimuli or through the network dynamics. When we tested these predictions experimentally by manipulating the skewness of the stimulus distribution such that the median and the mean were dissociated (Figure 6A), the results from our human psychophysics experiments were in agreement with the model predictions. In further support of this, in a recent tactile categorization study (Hachen et al., 2021), where rats were trained to categorize tactile stimuli according to a boundary set by the experimenter, the authors have shown that rats set their decision boundary according to the statistical structure of the stimulus set to which they are exposed. More studies are needed to fully verify the extent to which the statistical structure of the stimuli affects the performance. Finally, we note that in our model the stimulus distribution is not explicitly learned (but see Maes et al., 2023): instead, the PPC dynamics follows the input, and its marginal distribution of activity is similar to that of the external input. This is in agreement with Hachen et al., 2021, where the authors used different stimulus ranges across different sessions and noted that rats initiated each session without any residual influence of the previous session’s range/boundary on the current session, ruling out long-term learning of the input structure.

Importantly, our results are not limited to the delayed ‘comparison’ paradigm, where binary decision-making occurs. We show that by analyzing the location of the WM bump at the end of the delay interval, similar to the continuous recall tasks, we can retrieve the averaged effects of contraction bias, similar to previous reports (Jazayeri and Shadlen, 2010). Such continuous read-out of the memory reveals a rich dynamics of errors at the level of individual trials, similar to the delayed comparison case, but to our knowledge this has not been studied in previous experimental studies. Papadimitriou et al., 2015 have characterized residual error distribution in an orientation recall task when limiting previous trials to orientations in the range of +35 to+ 85° relative to the current trial. This distribution is unimodal, leading the authors to conclude that the current trial shows a small but systematic bias toward the location of the memorandum of the previous trial. It remains to be tested whether the error distribution remains unimodal if conditioned on other values of the current and previous orientations, similar to our analysis in Figure 5C.

Our model assumes that the stimulus is held in WM through the persistent activity of neurons, building on the discovery of persistent selective activity in a number of cortical areas, including the prefrontal cortex (PFC), during the delay interval (Fuster and Alexander, 1971; Miyashita and Chang, 1988; Funahashi et al., 1989; Funahashi et al., 1990; Romo et al., 1999; Salinas et al., 2000; Zhang et al., 2019). To explain this finding, we have used the attractor framework, in which recurrently connected neurons mutually excite one another to form reverberation of activity within populations of neurons coding for a given stimulus (Hopfield, 1982; Amit, 1992; Battaglia and Treves, 1998). However, subsequent work has shown that persistent activity related to the stimulus is not always present during the delay period and that the activity of neurons displays far more heterogeneity than previously thought (Barak et al., 2013). It has been proposed that short-term synaptic facilitation may dynamically operate to bring a WM network across a phase transition from a silent to a persistently active state (Mongillo et al., 2008; Barak and Tsodyks, 2007). Such mechanisms may further contribute to short-term biases (Barbosa et al., 2020), an alternative possibility that we have not specifically considered in this model.

An important model feature that is crucial in giving rise to all of its behavioral effects is its operation over multiple timescales (Figure 3—figure supplement 2F). Such timescales have been found to govern the processing of information in different areas of the cortex (Murray et al., 2014; Siegle et al., 2021; Gao et al., 2020) and may reflect the heterogeneity of connections across different cortical areas (Stern et al., 2021).

In many early studies, groups of neurons whose activity correlates monotonically with the stimulus feature, known as ‘plus’ and ‘minus’ neurons, have been found in the PFC (Romo et al., 1999; Barak et al., 2010). Such neurons have been used as the starting point in the construction of many models (Miller et al., 2003; Machens et al., 2005; Barak et al., 2013; Barak and Tsodyks, 2014). It is important, however, to note that depending on the area the fraction of such neurons can be small (Esmaeili and Diamond, 2019) and that the majority of neurons exhibit firing profiles that vary largely during the delay period (Machens et al., 2010). Such heterogeneity of the PFC neurons’ temporal firing profiles has prompted the successful construction of models that have not included the basic assumption of plus and minus neurons, but these have largely focused on the plausibility of the dynamics of neurons observed, with little connection to behavior (Barak et al., 2013).

A separate line of research has addressed behavior by focusing on normative models to account for contraction bias (Ashourian and Loewenstein, 2011; Raviv et al., 2012; Rahnev and Denison, 2018; Salinas, 2011). The abstract mathematical model that we present (Figure 4) can be compatible with a Bayesian framework (Ashourian and Loewenstein, 2011) in the limit of a very broad likelihood for the first stimulus and a very narrow one for the second stimulus, and where the prior for the first stimulus is replaced by the distribution of ${\displaystyle \hat{s}}$, following the model in Figure 4B (see section ‘The probability to make errors is proportional to the cumulative distribution of the stimuli, giving rise to contraction bias’ for details). However, it is important to note that our model is conceptually different, that is, subjects do not have access to the full prior distribution, but only to samples of the prior. We show that having full knowledge of the underlying sensory distribution is not needed to present contraction bias effects. Instead, a point estimate of past events that is updated trial to trial suffices to show similar results. This suggests a possible mechanism for the brain to approximate Bayesian inference, and it remains open whether similar mechanisms (based on the interaction of networks with different integration timescales) can approximate other Bayesian computations. It is also important to note the differences between the predictions from the two models. As shown in Figure 6A and Figure 4—figure supplement 1, depending on the specific sensory distributions, the two models can have qualitatively different testable predictions. Data from our human psychophysical experiments, utilizing auditory PWM, show better agreement with our model predictions compared to the Bayesian model.

Moreover, an ideal Bayesian observer model alone cannot capture the temporal pattern of short-term attraction and long-term repulsion observed in some tasks, and the model has had to be supplemented with efficient encoding and Bayesian decoding of information in order to capture both effects (Fritsche and Spaak, 2020). In our model, both effects emerge naturally as a result of neuronal adaptation, but their amplitudes crucially depend on the time parameters of the task, perhaps explaining the sometimes contradictory effects reported across different tasks.

Finally, while such attractive and repulsive effects in performance may be suboptimal in the context of a task designed in a laboratory setting, this may not be the case in more natural environments. For example, it has been suggested that integrating information over time serves to preserve perceptual continuity in the presence of noisy and discontinuous inputs (Fischer and Whitney, 2014). This continuity of perception may be necessary to solve more complex tasks or make decisions, particularly in a non-stationary environment, or in a noisy environment.

Our model is composed of two populations of ${\displaystyle N}$ neurons, representing the PPC network and the putative WM network. We consider that each population is organized as a continuous line attractor, with recurrent connectivity described by an interaction matrix ${\displaystyle J_{ij}}$, whose entries represent the strength of the interaction between neurons ${\displaystyle i}$ and ${\displaystyle j}$. The activation function of the neurons is a logistic function, that is, the output ${\displaystyle r_i}$ of neuron ${\displaystyle i}$, given the input ${\displaystyle h_i}$, is

where ${\displaystyle \beta }$ is the neuronal gain. The variables ${\displaystyle r_i}$ take continuous values between 0 and 1 and represent the firing rates of the neurons. The input ${\displaystyle h_i}$ to a neuron is given by

where ${\displaystyle \tau }$ is the timescale for the integration of inputs. In the first term on the right-hand side, ${\displaystyle J_{ij}{r}_j}$ represents the input to neuron ${\displaystyle i}$ from neuron ${\displaystyle j}$, and ${\displaystyle I_i^{\mathrm{e}\mathrm{x}\mathrm{t}}}$ corresponds to the external inputs. The recurrent connections are given by

with

The interaction kernel, ${\displaystyle K}$, is assumed to be the result of a time-averaged Hebbian plasticity rule: neurons with nearby firing fields will fire concurrently and strengthen their connections, while firing fields far apart will produce weak interactions (Dalgleish et al., 2020). Neuron ${\displaystyle i}$ is associated with the firing field ${\displaystyle x_i=i/N}$. The form of ${\displaystyle K}$ expresses a connectivity between neurons ${\displaystyle i}$ and ${\displaystyle j}$ that is exponentially decreasing with the distance between their respective firing fields, proportional to ${\displaystyle |i-j|}$; the exponential rate of decrease is set by the constant ${\displaystyle d_0}$,that is, the typical range of interaction. The amplitude of the kernel is also rescaled by ${\displaystyle d_0}$ in such a way that ${\displaystyle \sum _{i,j}{K}_{ij}}$ is constant. The strength of the excitatory weights is set by ${\displaystyle J_e}$; the normalization of ${\displaystyle K}$, together with the sigmoid activation function saturating to 1, implies that ${\displaystyle J_e}$ is also the maximum possible input received by any neuron due to the recurrent connections. The constant ${\displaystyle J_0}$, instead, contributes to a linear global inhibition term. Its value needs to be chosen depending on ${\displaystyle J_e}$ and ${\displaystyle d_0}$, so that the balance between excitatory and inhibitory inputs ensures that the activity remains localized along the attractor, that is, it does not either vanish or equal 1 everywhere; together, these three constants set the width of the bump of activity.

The two networks in our model are coupled through excitatory connections from the PPC to the WM network. Therefore, we introduce two equations analogous to Equation 2, one for each network. The coupling between the two will enter as a firing rate-dependent input, in addition to ${\displaystyle I^{\mathrm{e}\mathrm{x}\mathrm{t}}}$. The dynamics of the input to a neuron in the WM network writes

where ${\displaystyle j^W}$ indexes neurons in the WM network, and ${\displaystyle {\tau }_h^W}$ is the timescale for the integration of inputs in the WM network. The first term in the right-hand side corresponds to inputs from recurrent connections within the WM network. The second term corresponds to inputs from the PPC network. Finally, the last term corresponds to the external inputs used to give stimuli to the network. Similarly, for the PPC network we have

where ${\displaystyle j^P}$ indexes neurons in the PPC, and ${\displaystyle {\tau }_h^P}$ is the timescale for the integration of inputs in the PPC network; importantly, we set this to be longer than the analogous quantity for the WM network, ${\displaystyle {\tau }_h^W<{\tau }_h^P}$ (see Appendix 1—table 1). The first and third terms in the right-hand side are analogous to the corresponding ones for the WM network: inputs from within the network and from the stimuli. The second term instead corresponds to adaptive thresholds with dynamics specified by

modeling neuronal adaptation, where ${\displaystyle {\tau }_{\theta }^P}$ and ${\displaystyle D^P}$ set its timescale and its amplitude. We are interested in the condition where the timescale of the evolution of the input current is much smaller relative to that of the adaptation (${\displaystyle {\tau }_h^P\ll {\tau }_{\theta }^P}$). For a constant ${\displaystyle {\tau }_{\theta }^P}$, we find that depending on the value of ${\displaystyle D^P}$, the bump of activity shows different behaviors. For low values of ${\displaystyle D^P}$, the bump remains relatively stable (Figure 3—figure supplement 1C; Hollingworth, 1910). Upon increasing ${\displaystyle D^P}$, the bump gradually starts to drift (Figure 3—figure supplement 1C; Jou et al., 2004; Berliner et al., 1977; Romani and Tsodyks, 2015). Upon increasing ${\displaystyle D^P}$ even further, a phase transition leads to an abrupt dissipation of the bump (Figure 3—figure supplement 1C; Hellström, 1985).

Note that, while the transition from bump stability to drift occurs gradually, the transition from drift to dissipation is abrupt. This abruptness in the transition from the drift to the dissipation regime may imply that only one of the two behaviors is possible in our model of the PPC (section ‘Multiple timescales at the core of short-term sensory history effects’). In fact, our network model of the PPC operates in the ‘drift’ regime (${\displaystyle {\tau }_{\theta }^P=7.5}$, ${\displaystyle D^P=0.3}$). However, we also observe dissipation of the bump, which is mainly responsible for the jumps observed in the model. This occurs due to the inputs from incoming external stimuli that affect the bump via the global inhibition in the model (Figure 3—figure supplement 1A). Therefore, external stimuli can allow the network to temporarily cross the sharp drift/dissipation boundary shown in Figure 3—figure supplement 1B. As a result, the combined effects of adaptation, together with external inputs and global inhibition, result in the drift/jump dynamics described in the main text.

Finally, both networks have a linear geometry with free boundary conditions, that is, no condition is imposed on the profile activity at neuron 1 or ${\displaystyle N}$.

We performed all the simulations using custom Python code. Differential equations were numerically integrated with a time step of ${\displaystyle dt=0.001}$ using the forward Euler method. The activity of neurons in both circuits was initialized to ${\displaystyle r=0}$. Each stimulus was presented for 400 ms. A stimulus is introduced as a ‘box’ of unit amplitude and of width ${\displaystyle 2\delta s}$ around ${\displaystyle s}$ in stimulus space: in a network with ${\displaystyle N}$ neurons, the stimulus is given by setting ${\displaystyle I_i^{\mathrm{e}\mathrm{x}\mathrm{t}}=1}$ in Equation 5 for neurons with index ${\displaystyle i}$ within ${\displaystyle (s\pm \delta s)\times N}$, and ${\displaystyle I_i^{\mathrm{e}\mathrm{x}\mathrm{t}}=0}$ for all the others. Only the activity in the WM network was used to assess performance. To do that, the activity vector was recorded at two timepoints: 200 ms before and after the onset of the second stimulus ${\displaystyle s_2}$. Then, the neurons with the maximal activity were identified at both timepoints and compared to make a decision. This procedure was done for 50 different simulations with 1000 consecutive trials in each, with a fixed ITI separating two consecutive trials, fixed to 5 s. The interstimulus intervals were set according to two different experimental designs, as explained below.

Subjects received, in each trial, a pair of sounds played from ear-surrounding headphones. The subject self-initiated each trial by pressing the spacebar on the keyboard. The first sound was then presented together with a blue square on the left side of a computer monitor in front of the subject. This was followed by a delay period, indicated by ‘WAIT!’ on the screen, then the second sound was presented together with a red square on the right side of the screen. At the end of the second stimulus, subjects had 2 s to decide which one was louder, then indicate their choice by pressing the ‘s’ key if they thought that the first sound was louder, or the ‘l’ key if they thought that the second sound was louder. Written feedback about the correctness of their response was provided on the screen for each individual trial. Every 10 trials, participants received feedback on their running mean performance calculated up to that trial. Participants then had to press spacebar to go to the next trial (the experiment was hence self-paced).

The two auditory stimuli, ${\displaystyle s_1}$ and ${\displaystyle s_2}$, separated by a variable delay (of 2, 4, and 6 s), were played for 400 ms, with short delay periods of 250 ms inserted before ${\displaystyle s_1}$ and after ${\displaystyle s_2}$. The stimuli consisted of broadband noise 2000–20,000 Hz, generated as a series of sound pressure level (SPL) values sampled from a zero-mean normal distribution. The overall mean intensity of sounds varied from 60 to 92 dB. Participants had to judge which out of the two stimuli, ${\displaystyle s_1}$ and ${\displaystyle s_2}$, was louder (had the greater SPL standard deviation). We recruited 10 subjects for the negatively skewed distribution and 24 subjects for the bimodal distribution. Each participant performed approximately 400 trials for a given distribution. Several participants took part in both distributions.

The study was approved by the University College London (UCL) Research Ethics Committee (16159/001) (London, UK). Before starting the experiment, participants were provided with an information sheet relevant to the experiment they will be performing and asked to sign an informed written consent. By signing this consent form, the participants consent to freely take part in the study and confirm they understand the information they received. The participants additionally confirm they understand that their participation is voluntary and that they are allowed to stop the experiment at any moment and withdraw the data they provided. Participants consent to allow use of their personal data only for the purpose of scientific research, as determined by applicable law. Identifiable personal data are only available to the researchers and securely stored upon need, while the anonymized version of this data is freely available in a public repository. Once published, the participants’ contribution will remain non-identifiable.

In order to check whether the bump is in a target location (Figure 3B, Figure 2—figure supplement 1D, and Figure 3—figure supplement 2B), we check whether the position of the neuron with the maximal firing rate is within a distance of ±5% of the length of the whole line attractor from the target location (Figure 3A, Figure 2—figure supplement 1C, and Figure 3—figure supplement 2A). In these figures, we compare the probability that, in a given trial, the activity of the WM network is localized around one of the previous stimuli (estimated from the simulation of the dynamics, histograms) with the probability of this happening due to chance (horizontal dashed line). Here we detail the calculation of the chance probability. In general, if we have two discrete independent random variables, ${\displaystyle \hat{X}}$ and ${\displaystyle \hat{Y}}$, with probability distributions ${\displaystyle p_X}$ and ${\displaystyle p_Y}$, the probability of them having the same value is

where ${\displaystyle i,j}$ are the indices for different values of the two random variables and ${\displaystyle \mathbb{I}(x_i=y_j)}$ equals 1 where ${\displaystyle x_i=x_j}$ and 0 otherwise. If the two random variables are identically distributed, the above expression writes

In our case, the two identically distributed random variables are ‘bump location at the current trial’ and the ‘target bump location’ (that are ${\displaystyle s_1^{t-2}}$, ${\displaystyle s_2^{t-2}}$, ${\displaystyle s_1^{t-1}}$, ${\displaystyle s_2^{t-1}}$, and ${\displaystyle ⟨s⟩}$). With the exception of the mean stimulus ${\displaystyle ⟨s⟩}$, all the other variables are identically distributed, with probability ${\displaystyle p_m}$ (that is the marginal distribution over ${\displaystyle s_1}$ or ${\displaystyle s_2}$). We note that the bump location in the WM network follows a very similar distribution to ${\displaystyle p_m}$ (Figure 4A). Then, we compute the chance probability with the above relationship, where ${\displaystyle p_X\equiv p_m}$. For the mean stimulus, instead, we have a probability which is simply equal to 1 for ${\displaystyle s=0.5}$ and 0 elsewhere; therefore, the chance probability for the bump location to be at the mean stimulus then is ${\displaystyle p_m(0.5)}$.

The excess probability (with respect to chance) for the bump location to equal one of the previous stimuli gives a measure of the correlation between these two; in other terms, of the amount of information retained by the network about previous stimuli.

In order to illustrate the statistical origin of contraction bias consistent with our network model, we consider a simplified mathematical model of its performance (Figure 4B). By definition of the delayed comparison task, the optimal decision maker produces a label ${\displaystyle y}$ equal to 1 if ${\displaystyle s_1^t<s_2^t}$, and 0 if ${\displaystyle s_1^t>s_2^t}$; the impossible cases ${\displaystyle s_1^t=s_2^t}$ are excluded from the set of stimuli, but would produce a label which is either 0 or 1 with 50% probability. That is,

In this simplified scheme, at each trial ${\displaystyle t}$, the two stimuli ${\displaystyle s_1^t}$ and ${\displaystyle s_2^t}$ are perfectly perceived with a finite probability ${\displaystyle 1-ϵ}$, with ${\displaystyle ϵ<1}$. Under the assumption that the decision maker behaves optimally based on the perceived stimuli, a correct perception would necessarily lead to the correct label. However, with probability ${\displaystyle ϵ}$, the first stimulus is randomly selected from a buffer of stimuli, that is, is replaced by a random variable ${\displaystyle {\hat{s}}_1}$ that has a probability distribution ${\displaystyle p_m^t}$.

The probability distribution ${\displaystyle p_m^t}$ is the statistics of previously shown stimuli. The information about the previous stimulus is given by the activity of the ‘slower’ PPC network. As shown above, after the presentation of the first stimulus of the trial, the bump of activity is seen to jump to the position encoding one of the previously presented stimuli, ${\displaystyle s_2^{t-1}}$, ${\displaystyle s_1^{t-1}}$, ${\displaystyle s_2^{t-2}}$, etc., with decreasing probability (Figure 3C). Therefore, in calculating the performance in the task, we can take ${\displaystyle p_m^t}$ to be the marginal distribution of the stimulus ${\displaystyle s_1}$ or ${\displaystyle s_2}$ across trials, as in the histogram (Figure 4A).

The probability of a misclassification is then given by the probability that, given the pair ${\displaystyle (s_1^t,s_2^t)}$, at trial ${\displaystyle t}$,

1. the first stimulus is replaced by a random value, which happens with probability ${\displaystyle ϵ}$, and

2. the value of ${\displaystyle {\hat{s}}_1}$ replaced is larger than ${\displaystyle s_2^t}$ when ${\displaystyle s_1^t}$ is smaller and vice versa (Figure 4C).

In summary, the probability of an error at trial ${\displaystyle t}$ is given by

We reproduce here the theoretical result from Loewenstein et al., 2021, which provides a normative model for contraction bias in the Bayesian inference framework, and apply it to the different stimulus distributions described in section ‘The stimulus distribution impacts the pattern of contraction bias through its cumulative’.

A stimulus with value ${\displaystyle s}$ is encoded by the agent through a noisy representation ${\displaystyle \hat{r}\sim \ell (\cdot |s)}$. Before the presentation of the stimulus, the agent has an expectation of its possible values which is described by the probability ${\displaystyle \pi }$. Assuming that it has access to the internal representation ${\displaystyle r}$, as well as the probability distributions ${\displaystyle \ell }$ and ${\displaystyle \pi }$, the agent can infer the perceived stimulus ${\displaystyle \hat{s}}$ through Bayes rule:

where ${\displaystyle p(r)=\int d{s}^{\mathrm{\prime }}\ell (r|s^{\mathrm{\prime }})\pi (s^{\mathrm{\prime }})}$. In this Bayesian setting, the probability distributions for the noisy representation and expected measurement are interpreted as the likelihood and the prior, respectively.

In the delayed comparison task, at the time of the decision, the two stimuli ${\displaystyle s_1}$ and ${\displaystyle s_2}$ are assumed to be encoded independently, although with different uncertainties, due to the different delays leading to the time of decision: ${\displaystyle \ell (r_1,r_2|s_1,s_2)={\ell }_1(r_1|s_1){\ell }_2(r_2|s_2)}$, with ${\displaystyle \mathrm{v}\mathrm{a}\mathrm{r}[{\ell }_1]>\mathrm{v}\mathrm{a}\mathrm{r}[{\ell }_2]}$. Similarly, the expected values of the stimuli are assumed to be independent but also identically distributed: ${\displaystyle \pi (s_1,s_2)=\pi (s_1)\pi (s_2)}$.

The optimal Bayesian decision maker uses the inference of the stimuli through Equation 10 to produce an estimate of the probability that ${\displaystyle s_1<s_2}$, given the internal representations,

where ${\displaystyle \mathrm{\Theta }}$ is the Heaviside function and yields a label ${\displaystyle \hat{y}=1}$ (truth value of ‘${\displaystyle s_1<s_2}$’) when such probability is higher than 1/2, and ${\displaystyle \hat{y}=0}$ otherwise. Therefore, the probability that the Bayesian decision maker yields the response ‘${\displaystyle s_1<s_2}$’ given the true values of the stimuli ${\displaystyle s_1}$ and ${\displaystyle s_2}$ are the average of the label ${\displaystyle \hat{y}}$ over the possible values of their representations, that is, over the likelihood:

In the regression formulas in Equations 14 and 15, it is possible to give an interpretation of the parameter ${\displaystyle w_{mean}}$, that is, the weight of the contribution from the covariate corresponding to the mean of the past stimuli. Let us consider two regression models, one in which, in addition to a regressor corresponding to the mean stimulus, regressors corresponding to the stimulus history are included up to trial ${\displaystyle h}$, and another in which ${\displaystyle h=\mathrm{\infty }}$, that is, infinitely many past stimuli are included as regressors. In this case, Equation 15 rewrites

If we assume that the weights obtained from the regression have roughly an exponential dependence on time (Figure 9C and D), we can write

By equating Equations 15 and 16, we would find that

where

that is, an average over the geometric distribution ${\displaystyle g_j=(1-\gamma ){\gamma }^j}$, from time ${\displaystyle t-(h+1)}$ backward. Since for ${\displaystyle \gamma }$ large enough we have ${\displaystyle ⟨s{⟩}_{\gamma }=⟨s⟩}$, we can identify

This derivation indicates that the magnitude ${\displaystyle w_{mean}}$ in the infinite history model, given by Equation 15, is a function of the discount factor ${\displaystyle \gamma }$ as well as the weight of the first trial left out from the finite history regression (${\displaystyle w_{h+1}}$). A higher ${\displaystyle \gamma }$ value, that is, a longer timescale for damping of the weights extending into the stimulus history, yields a higher ${\displaystyle w_{mean}}$. We can obtain ${\displaystyle \gamma }$ for each condition (NT, ASD, and DYS) by fitting the weights obtained as a function of trials extending into the history (Figure 9C and D). As predicted by Equation 20, a larger window for short-term history effects (as in the ASD case relative to NT) yields a larger weight for the covariate corresponding to the mean stimulus. Finally, Equation 20 also predicts that ${\displaystyle w_{mean}}$ is proportional to ${\displaystyle w_{h+1}}$, the number of trials back we consider in the regression, ${\displaystyle h}$, implying that the number of covariates that we choose to include in the model may greatly affect the results. Both of these predictions are corroborated by plotting directly the value of ${\displaystyle w_{mean}}$ obtained from the regression (Figure 9E).

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The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication. For the purpose of Open Access, the authors have applied a CC BY public copyright license to any Author Accepted Manuscript version arising from this submission.

We are grateful to Loreen Hertäg for helpful comments on our figures and Arash Fassihi for helpful discussions. We also thank Guilhem Ibos for pointing out a typo in our figure legends in a previous version of this manuscript. This work was supported by BBSRC (BB/N013956/1, BB/N019008/1), Wellcome Trust (200790/Z/16/Z), Simons Foundation (564408), EPSRC (EP/R035806/1), Gatsby Charitable Foundation (GAT3755), and Wellcome Trust (219627/Z/19/Z).

The study was approved by the University College London (UCL) Research Ethics Committee [16159/001] (London, UK). Before starting the experiment, participants were provided with an information sheet relevant to the experiment they will be performing and asked to sign an informed written consent. By signing this consent form, the participants consent to freely take part in the study and confirm they understand the information they received. The participants additionally confirm they understand that their participation is voluntary and that they are allowed to stop the experiment at any moment and withdraw the data they provided. Participants consent to allow use of their personal data only for the purpose of scientific research, as determined by applicable law. Identifiable personal data are only available to the researchers and securely stored upon need, while the anonymized version of this data is freely available in a public repository. Once published, the participants' contribution will remain non-identifiable.

1. Preprint posted: January 27, 2023
2. Sent for peer review: March 9, 2023
3. Reviewed Preprint version 1: August 2, 2023
4. Reviewed Preprint version 2: January 5, 2024
5. Version of Record published: April 24, 2024

You can cite all versions using the DOI https://doi.org/10.7554/eLife.86725. This DOI represents all versions, and will always resolve to the latest one.

© 2023, Boboeva et al.

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