In our dynamic world, decisions about noisy stimuli can require temporal accumulation of evidence to identify steady signals; differentiation to detect unpredictable changes in those signals; or both. Normative models can account for learning in these environments but have not yet been applied to faster decision processes. We present a novel, normative formulation of adaptive learning models that forms decisions by acting as a leaky accumulator with non-absorbing bounds. These dynamics, derived for both discrete and continuous cases, depend on the expected rate of change of the statistics of the evidence and balance signal identification and change detection. We found that, for two different tasks, human subjects learned these expectations, albeit imperfectly, then used them to make decisions in accordance with the normative model. The results represent a unified, empirically supported account of decision-making in unpredictable environments that provides new insights into the expectation-driven dynamics of the underlying neural signals.
The authors suggest a model of sequential information processing that is aware of possible switches in the underlying source of information. They further show that the model fits responses of people in two perceptual decision making tasks and consequently argue that behaviour, which was previously considered to be suboptimal, may follow the normative, i.e., optimal, mechanism of the model. This mechanism postulates that typical evidence accumulation mechanisms in perceptual decision making are altered by the expected switch rate of the stimulus. Specifically, evidence accumulation becomes more leaky and a non-absorbing bound becomes lower when the expected switch rate increases. The paper is generally well-written (although there are some convoluted bits in the results section) and convincing. I was a bit surprised, though, that only choices, but not their timing is considered in the analysis with the model. In the following I’ll go through some more details of the model and discuss limitations of the presented models and their relation to other models in the field, but first I describe the experiments reported in the paper.
The paper reports two experiments. In the first (triangles task) people saw two triangles on the screen and had to judge whether a single dot was more likely to originate from the one triangle or the other. There was one dot and corresponding response per trial. In each trial the position of the dot was redrawn from a Gaussian distribution centred around one of the two triangles. There were also change point trials in which the triangle from which the dot was drawn switched (and then remained the same until the next change point). The authors analysed the proportion correct in relation to whether a trial was a change point. Trials were grouped into blocks which were defined by constant rate of switches (hazard rate) in the true originating triangle. In the second experiment (dots-reversal task), a random dot stimulus repeatedly switched (reversed) direction within a trial. In each trial people had to tell in which direction the dots moved before they vanished. The authors analysed the proportion correct in relation to the time between the last switch and the end of stimulus presentation. There were no blocks. Each trial had one of two hazard rates and one of two difficulty levels. The two difficulty levels were determined for each subject individually such that the more difficult one lead to correct identification of motion direction of a 500ms long stimulus in 65% of cases.
The authors present two normative models, one discrete and one continuous, which they apply across and within trial in the triangles and dots-reversal tasks, respectively. The discrete model is a simple hidden Markov model in which the hidden state can take one of two values and there is a common transition probability between these two values which they call hazard ‘rate’ (H). Observations were implicitly assumed Gaussian. They only enter during fitting as log-likelihood ratios in the form \(\beta*x_n\) where beta is a scaling relating to the internal / sensory uncertainty associated with the generative model of observations and \(x_n\) is the observed dot position (x-coordinate) in the triangles task. In methods, the authors derive the update equation for the log posterior odds (\(L_n\)) of the hidden state values given in Eqs. (1) and (2).
The continuous model is based on a Markov jump process with two states which is the continuous equivalent of the hidden Markov model above. Using Ito-calculus the authors again derive an update equation for the log posterior odds of the two states (Eq. 4), but during fitting they actually approximate Eq. (4) with the discrete Eq. (1), because it is supposedly the most efficient discrete-time approximation of Eq. (4) (no explanation for why this is the case was given). They just replace the log-likelihood ratio placeholder (LLR) with a coherence-dependent term applicable to the random dot motion stimulus. Notably, in contrast to standard drift-diffusion modelling of random dot motion tasks, the authors used coherence-dependent noise. I’d be interested in the reason for this choice.
There is an apparent fundamental difference between the discrete and continuous models which can be seen in Fig. 1 B vs C. In the discrete model, for H>0.5, the log posterior odds may actually switch sign from one observation to the next whereas this cannot happen in the continuous model. Conceptually, this means that the log posterior odds in the discrete model, when the LLR is 0, i.e., when there is no evidence in either direction, would oscillate between decreasing positive and increasing negative values until converging to 0. This oscillation can be seen in Fig. 2G, red line for |LLR|>0. In the continuous model such an oscillation cannot happen, because the infinitely many, tiny time steps allow the model to converge to 0 before switching the sign. Another way to see this is through the discrete hazard ‘rate’ H which is the probability of a sign reversal within one time step of size dt. When you want to decrease dt in the model, but want to maintain a given rate of sign reversals in, e.g., 1 second, H would also have to decrease. Consequently, when dt approaches 0, the probability of a sign reversal approaches 0, too, which means that H is a useless parameter in continuous time which, in turn, is the reason why it is replaced by a real rate parameter (\(\lambda\)) representing the expected number of reversals per second. In conclusion, the fundamental difference between discrete and continuous models is only an apparent one. They are very similar models, just expressed in different resolutions of time. In that sense it would have perhaps been better to present results in the paper consistently in terms of a real hazard rate (\(\lambda\)) which could be obtained in the triangles task by dividing H by the average duration of a trial in seconds. Notice that the discrete model represents all hazard rates \(\lambda>1/dt\) as H=1, i.e., it cannot represent hazard rates which would lead to more than 1 expected sign reversal per \(dt\). There may be more subtle differences between the models when the exact distributions of sign reversals are considered instead of only the expected rates.
Using first order approximations of the two models the authors identify two components in the dynamics of the log posterior odds L: a leak and a bias. [Side remark: there is a small sign mistake in the definition of leak k of the continuous model in the Methods section.] Both depend on hazard rate and the authors show that the leak dominates the dynamics for small L whereas the bias dominates for large L. I find this denomination a bit misleading, because both, leak and bias, effectively result in a leak of log-posterior odds L by reducing L in every time step (cf. Fig. 1B,C). The change from a multiplicative leak to one based on a bias just means that the effective amount of leak in L increases nonlinearly with L as the bias takes over.
To test whether this special form of leak underlies decision making the authors compared the full model to two versions which only had a multiplicative leak, or one based on bias. In the former the leak stayed constant for increasing L, i.e., \(L’ = \gamma*L\). In the latter there was perfect accumulation without leak up to the bias and then a bias-based leak which corresponds to a multiplicative leak where the leak rate increased with L such that \(L’ = \gamma(L)*L\) with \(\gamma(L) = bias / L\). The authors report evidence that in both tasks both alternative models do not describe choice behaviour as well as the full, normative model. In Fig. 9 they provide a reason by estimating the effective leak rate in the data and the models in dependence on the strength of sensory evidence (coherence in the dots reversal task). They do this by fitting the model with multiplicative leak separately to trials with low and high coherence (fitting to choices in the data or predicted by the different fitted models). In both data and normative model the effective leak rates depended on coherence. This dependence arises, because high sensory evidence leads to large values of L and I have argued above that larger L has larger effective leak rate due to the bias. It is, therefore, not surprising that the alternative model with multiplicative leak shows no dependence of effective leak on coherence. But it is also not surprising that the alternative model with bias-based leak has a larger dependence of effective leak on coherence than the data, because this model jumps from no leak to very large leak when coherence jumps from low to high. The full, normative model lies in between, because it smoothly transitions between the two alternative models.
Why is there a leak in the first place? Other people have found no evidence for a leak in evidence accumulation (eg. Brunton et al., 2013). The leak results from the possibility of a switch of the source of the observations, i.e., a switch of the underlying true stimulus. Without any information, i.e., without observations the possibility of a switch means that you should become more uncertain about the stimulus as time passes. The larger the hazard rate, i.e., the larger the probability of a switch within some time window, the faster you should become uncertain about the current stimulus. For a log posterior odds of L=0 uncertainty is at its maximum (both stimuli have equal posterior probability). This is another reason why discrete hazard ‘rates’ H>0.5 which lead to sign reversals in L do not make much sense. The absence of evidence for one stimulus should not lead to evidence for the other stimulus. Anyway, as the hazard rate goes to 0 the leak will go to 0 such that in experiments where usually no switches in stimulus occur subjects should not exhibit a leak which explains why we often find no evidence for leaks in typical perceptual decision making experiments. This does not mean that there is no leak, though. Especially, the authors report here that hazard rates estimated from behaviour of subjects (subjective) tended to be a bit higher than the ones used to generate the stimuli (objective), when the objective hazard rates were very low and the other way around for high objective hazard rates. This indicates that people have some prior expectations towards intermediate hazard rates that biased their estimates of hazard rates in the experiment.
The discussed forms of leak implement a property of the model that the authors called a ‘non-absorbing bound’. I find this wording also a bit misleading, because ‘bound’ was usually used to indicate a threshold in drift diffusion models which, when reached, would trigger a response. The bound here triggers nothing. Rather, it represents an asymptote of the average log posterior odds. Thus, it’s not an absolute bound, but it’s often passed due to variance in the momentary sensory evidence (LLR). I can also not follow the authors when they write: “The stabilizing boundary is also in contrast to the asymptote in leaky accumulation, which increases linearly with the strength of evidence”. Based on the dynamics of L discussed above the ‘bound’ here should exhibit exactly the described behaviour of an asymptote in leaky accumulation. The strength of evidence is reflected in the magnitude of LLR which is added to the intrinsic dynamics of the log posterior odds L. The non-absorbing bound, therefore, should be given by bias + average of LLR for the current stimulus. The bound, thus, should rise linearly with the strength of evidence (LLR).
Fitting of the discrete and continuous models was done by maximising the likelihood of the models (in some fits with many parameters, priors over parameters were used to regularise the optimisation). The likelihood in the discrete models was Gaussian with mean equal to the log posterior odds (\(L_n\)) computed from the actual dot positions \(x_n\). The variance of the Gaussian likelihood was fitted to the data as a free parameter. In the continuous model the likelihood was numerically approximated by simulating the discretised evolution of the probabilities that the log posterior odds take on particular values. This is very similar to the approach used by Brunton2013. The distribution of the log posterior odds \(L_n\) was considered here, because the stream of sensory observations \(x(t)\) was unknown and therefore had to enter as a random variable while in the triangles task \(x(t)=x_n\) was set to the known x-coordinates of the presented dots.
The authors argued that the fits of behaviour were good, but at least for the dots reversal task Fig. 8 suggests otherwise. For example, Fig. 8G shows that 6 out of 12 subjects (there were supposed to be 13, but I can only see 12 in the plots) made 100% errors in trials with the low hazard rate of 0.1Hz and low coherence where the last switch in stimulus was very recent (maximally 300ms before the end of stimulus presentation). The best fitting model, however, predicted error rates of at most 90% in these conditions. Furthermore, there is a significant difference in choice errors between the low and high hazard rate for large times after the last switch in stimulus (Fig. 8A, more errors for high hazard rate) which was not predicted by the fitted normative model. Despite these differences the fitted normative model seems to capture the overall patterns in the data.
The authors present an interesting normative model in discrete and continuous time that extends previous models of evidence accumulation to situations in which switches in the presented stimulus can be expected. In light of this model, a leak in evidence accumulation reflects a tendency to increase uncertainty about the stimulus due to a potentially upcoming switch in the stimulus. The model provides a mathematical relation between the precise type of leak and the expected switch (hazard) rate of the stimulus. In particular, and in contrast to previous models, the leak in the present model depends nonlinearly on the accumulated evidence. As the authors discuss, the presented normative model potentially unifies decision making processes observed in different situations characterised by different stabilities of the underlying stimuli. I had the impression that the authors were very thorough in their analysis. However, some deviations of model and data apparent in Fig. 8 suggest that either the model itself, or the fitting procedure may be improved such that the model better fits people’s behaviour in the dots-reversal task. It was anyway surprising to me that subjects only had to make a single response per trial in that task. This feels like a big waste of potential choice data when I consider that each trial was 5-10s long and contained several stimulus switches (reversals).