Nonlinear neural network dynamics accounts for human confidence in a sequence of perceptual decisions.

Electrophysiological recordings during perceptual decision tasks in monkeys suggest that the degree of confidence in a decision is based on a simple neural signal produced by the neural decision process. Attractor neural networks provide an appropriate biophysical modeling framework, and account for the experimental results very well. However, it remains unclear whether attractor neural networks can account for confidence reports in humans. We present the results from an experiment in which participants are asked to perform an orientation discrimination task, followed by a confidence judgment. Here we show that an attractor neural network model quantitatively reproduces, for each participant, the relations between accuracy, response times and confidence. We show that the attractor neural network also accounts for confidence-specific sequential effects observed in the experiment (participants are faster on trials following high confidence trials). Remarkably, this is obtained as an inevitable outcome of the network dynamics, without any feedback specific to the previous decision (that would result in, e.g., a change in the model parameters before the onset of the next trial). Our results thus suggest that a metacognitive process such as confidence in one’s decision is linked to the intrinsically nonlinear dynamics of the decision-making neural network.

: Calibration of the Usher-McClelland model: Parameters for each participant after fit of the mean accuracy and response times of the confidence block.
We note that the Usher-McClelland model fits the response times with respect to confidence, but only at intermediate levels of confidence. For some participants, we observe a strong divergence at high confidence (Participant 1,4 and 5). This can be understood by the fact that, in this model, 'firing rate' variables can take negative values (in fact, for the steady state in the absence of any input, the firing rate variables take negative values). This leads to extreme value of confidence for long trials. However, the trend in accuracy is not correct for some participants (Participants 1 et 4). Accuracy is an increasing function of confidence (except for participant 5), but the experimental data do not fall within the bootstrapped confidence interval of the simulations, as can be seen in Fig. S2. We represent the mean confidence as a function of stimulus orientation in correct trials (green), and in error trials (red), for the experimental data (points) and the model simulations (lines). With parameters resulting from the fit on the confidence block, the numerical protocol mimic the experimental one (same number of trials, and same angle values). Due to the discrete levels of confidence, and the high performance in the task, to get enough statistics we combined the data of all subjects. The shaded areas (resp. error bars) denote the 95% bootstrapped confidence interval on the mean for the simulation (resp. data) S3

S3 -STATISTICAL ANALYSES
As explained in the paper, we perform a linear mixed effects analysis [Gelman and Hill, 2007] of the history biases of the reaction times, on both the experimental and the numerical simulations data. The linear mixed effects model (LMM) we consider assumes that the logarithm of the response time at trial n, RT n , is a linear combination of factors as follows: with x repetition a binary variable taking the value 1 if the correct choice for the current trial is a repetition of the previous choice (and 0 otherwise), θ the orientation of the Gabor (in degree), RT n−1 the response times of the previous trials, and Conf n−1 the confidence of the previous trial coded as 0 for low and 1 for high. The subscript p in a coefficient (e.g a 0,p ) indicates that for this parameter we allow for a random slope per participant.
We present the results obtained with this LMM, equation (1), in Table S2 for the experimental data, and in Table S3 for the numerical data. See the main text, section Results, for the discussion.
We also compare this LMM to other ones that do not include all the terms, using the ANOVA function (with the lme4 package [Bates et al., 2015] in RStudio [RStudio Team, 2015]) that performs model comparison based on the Akaike and Bayesian Information Criteria (AIC and BIC) [Bates et al., 2014]. To perform the comparison between the experimental data and the results of the numerical simulations we first transform the response times of each participant using the z-score [Kreyszig, 1979]. This normalization of the response times allows us to study the participants altogether.
We present the results of the models comparison in Table S4. We see that the full model, equation (1), is always to be preferred.
We note that, even when the previous confidence is not taken into account by the LMM, the LMM does not find an effect of previous reaction times. This means that the attractor neural network does not reproduce this effect that is observed in the experimental data.  Table S6: Threshold parameter for each participant after fit of the mean accuracy and response times of the pure and confidence blocks. The ranges ∆z correspond to one sigma deviation of the likelihood with respect to the corresponding parameter (see the Material and Methods Section).

S5 -ESTIMATION OF THE NON-DECISION TIME DISTRIBUTION
We consider that the nondecision time distribution, noted ρ N DT , is described by an ex-Gaussian distribution [Verdonck and Tuerlinckx, 2016]: with erfc the complementary error function. For such distribution, the mean non decision time, N DT , is given by As we assume no correlation between response and non decision times, the total (observed) response time distribution, ρ data , can be written as the convolution of the decision time distribution, ρ decision , with the nondecision time distribution, ρ N DT : (with * standing for the convolution operation). If the decision time distribution is Gaussian, the resulting total distribution is an ex-Gaussian distribution [Grushka, 1972].
Using maximum likelihood estimation, for each subject we fit the empirical response time distribution (all orientations together) by an ex-Gaussian distribution with parameters µ data , λ data , σ data (and we thus have RT data = µ data + 1/λ data ). For what concerns the model, we find that the decision time distribution of the attractor neuronal network is well fitted by a Gaussian distribution with parameters RT network , σ network . We thus model the decision time distribution by the one provided by the attractor network whose parameters have been calibrated as explained above. Hence, we identify the mean and variance of the decision time distribution with the ones of the network: Taking the characteristic function of both sides of Equation 5, we get: We can then identify the terms on both sides of this equations, which, given the use for the decision parameters of the ones of the attractor network, gives the equations and from which we can compute the non-decision time distribution parameters, λ N DT , µ N DT , σ N DT .  Figure S3 shows a schematic representation of the IRM with collapsing bounds. The same analysis as the one in the main text for the IRM without collapsing bounds leads to the following conclusions. One can observe two behaviors. Either the decision is fast and it is possible for the model to be in a high-confidence trial, either the decision is slow and the confidence will be low.
If the decision was fast, the model will lead to the opposite sequential effect due to confidence after the relaxation period (similar case as in the main text). However, if the decision was slow the losing race is going to be close to the decision threshold and the confidence will be low for all of these trials. After the relaxation, the losing race will have the same state as in the high-confidence fast trial case. Therefore there will be no sequential effects. This analysis shows that even an IRM with collapsing bounds and a relaxation mechanism will not account for the sequential effects due to confidence that we can model with an attractor neural network.