Table 1.
Key variables and parameters of the BAttM.
These variables are defined mathematically in the models section below.
Fig 1.
Schematic comparing a pure attractor model of decision making with the Bayesian attractor model.
Both plots show illustrative snapshots of the two evolving decision states while in transit towards a fixed point where a decision will be made. (A) In a pure attractor model, on the way to a fixed point, the decision state (violet) is evolving according to attractor dynamics (grey arrows). From an initial, unstable fixed point (empty, black circle) the decision state is driven by noisy evidence into one of two attracting, stable fixed points, each of which correspond to a decision alternative. (B) In the Bayesian attractor model the same attractor dynamics is used as generative model for sensory observations. The decision state effects, in a top-down fashion, both internal predictions and gain. These are in turn used together with sensory observations to compute gain-modulated prediction errors which drive updates of the decision state. The model represents uncertainty over the decision state (shaded, violet ellipse) and allows to define the decision criterion directly in terms of confidence in the decision. We show in Results that this recurrent principle stabilises the location of fixed points of the attractor dynamics while at the same time maintaining the ability to reliably switch decisions after a change in stimulus.
Fig 2.
Example stimulus of single dot task, with a switch of target location.
(A) The plot shows both x- and y- positions of the single dot throughout an example trial of 1600ms length. Every 40ms a new dot position is drawn. For 800ms positions are drawn from the first target (blue), i.e., a Gaussian with mean position [0.71, 0.71] (dark blue horizontal line) and standard deviation s = 2 in both dimensions. For the next 800ms positions are drawn from the second target (orange) around the mean [-0.71, -0.71] (red horizontal line) with the same standard deviation. (B) Same data as in (A), but plotted in 2D coordinates as when presented on a screen. Note that the observer would see only a single dot of neutral colour at any time throughout the trial and would have to decide whether the dot moves around the first (lower left) or second (upper right) target (indicated by lines).
Fig 3.
Illustration of the inference scheme used for decision making in the BAttM.
In the physical environment a stimulus is presented by the experimenter and observed by the subject. Components inside the shaded rectangle model internal processes of the subject. Sensory processes in the subject’s brain translate the stimulus into an abstract feature representation xt. The input model (i, green) of the BAttM approximates this translation by mapping the stimulus identity (decision alternative At at time t) to a value xt drawn from a Gaussian distribution with mean μt and covariance s2 I. The generative model (ii, orange) states that the decision state z is represented by a Gaussian and evolves according to Hopfield dynamics (Eq 2). The generative model further maps the decision state to different Gaussian densities over observations which mirror those in the input process (Eq 3). Consequently, for the next time step, the generative model predicts the distribution of the decision state,
, and the distribution of the observation,
, which critically depend on model parameters q and r, respectively. The cross-covariance between predicted decision state and predicted observation is denominated
. Bayesian inference (iii, red) iteratively compares observations xt with predictions
and updates the estimate of the decision state (Eq 4) via the Kalman gain Kt which processes the uncertainty defined by
and
(Eq 5). The decision criterion (iv, blue) is defined as a bound λ on an explicit measure of confidence (Eq 6).
Fig 4.
Example trial showing evolution of confidence in alternative 1, p(zt = ϕ1∣XΔt:t) (notice log-scale and initial, very low values), for different values of dynamics uncertainty q.
Larger values of q mean that only smaller confidence values can be reached even after the decision state zt eventually settled into the stable fixed point ϕ1 (compare, e.g., confidence for q = 1 and q = 0.5 at 200ms, note log-scale). Horizontal dotted line: confidence value used as bound (λ = 0.02).
Fig 5.
Example trajectories for the Bayesian attractor model on a binary decision task for varying sensory uncertainty r.
Each of the plots shows three example trials. Note that there are two state variables (blue: alternative 1, orange: alternative 2) for each trial. (A-C) Decision state . (D-F) Confidence (log-scale). Grey, dashed line: threshold used in the model. (A,D) r = 1, decisions are inaccurate and shoot over fixed points (located at [10, 0] and [0, 10]). (B,E) r = 2.2, decisions are relatively fast and accurate, (C,F) r = 3.0, decisions are accurate but can be slow. The same sensory input with noise level (standard deviation) s = 4.7 was used in all three cases. Dynamics uncertainty was q = 0.1 and initial state uncertainty was p0 = 5. Note that for clarity we plotted only the mean of the posterior distributions but not the posterior uncertainties (but see below for examples).
Fig 6.
Mapping from sensory uncertainty r and noise level s to behavioural measures.
(A) Log-log plot of the fraction of correct responses, i.e. accuracy. (B) Mean reaction time for correct responses in ms (including a non-decision time of 200ms, see Methods). Light blue areas correspond to parameter settings where more than 50% of trials resulted in time outs (RT >1000ms). Light red lines show approximated contour lines (see Methods of the underlying grey scale map. In A the lines correspond, from right to left, to 0.6, 0.7, 0.8 and 0.9 fraction of correct responses. In B the lines correspond, from bottom to top, to 400, 500, 600 and 700 ms.
Fig 7.
Re-decision behaviour of Bayesian attractor model for switching stimuli.
Noisy exemplars of alternative 1 (blue) and subsequently of alternative 2 (orange) were shown with a switch at 800ms (cf. Fig 2). For varying combinations of sensory uncertainty r and dynamics uncertainty q we plotted the mean (over 1000 trials) percentage of time spent in the correct decision state (grey shading). (A-C) Bottom panels show three example trials for the parameter combinations indicated by the corresponding points in the main panel. Top row: decision state, bottom row: confidence (log-scale) with threshold (grey, dashed line). A: fast, but sometimes fickle re-decisions, B: slower but reliable re-decisions, C: no re-decisions. For point A the mean % time spent in the correct decision is larger, because decision and re-decisions are on average faster. The overall level of confidence reached increases from A to C, as previously shown in Fig 4.
Fig 8.
Example of a decision making trial with evolution of cross-covariance and gain for parameters of point B in Fig 7.
Noisy exemplars of alternative 1 (blue) and subsequently of alternative 2 (orange) were shown with a switch at 800ms (cf. Fig 2). (A) Inferred decision state with mean state variables (lines) and two times their standard deviation (shading) indicating posterior uncertainty over decision state. State variable associated with alternative 1 shown in blue and associated with alternative 2 shown in orange. (B) Absolute cross-covariances between predicted observations and predicted decision state over time. Colours indicate cross-covariances associated with corresponding state variables as in A. Cross-covariances are large during their transition between fixed points. Once a fixed point is reached (i.e. a decision has been made) cross-covariances drop quickly. (C) Absolute gain values (elements of Kt) over time. Colouring as in B. Gain values are scaled cross-covariances, i.e., within-trial changes in gain are mostly driven by changes in cross-covariances.
Fig 9.
Evolution of decision state for pure attractor model (left) and Bayesian attractor model (right) for different input strengths or different uncertainty parameters, respectively.
There are two alternatives indicated by blue (alternative 1) and orange (alternative 2). Thinner lines indicate smaller stimulus strength. For the first 800ms, input reflecting alternative 1 was shown, with a switch to input caused by alternative 2 at 800ms. (A) In the pure attractor model speed and accuracy of initial and re-decisions is controlled by the input which we set to It = [ΔtI+vt,0], if alternative 1 is correct, and It = [0,ΔtI+vt], if alternative 2 is correct (vt ∼ 𝓝(0,0.22)). We varied the value of I as indicated in the plot legend. If I is large, i.e., the task is easy, initial decisions and switches are fast (thick lines). The position of the fixed point, to which the dynamics converges, depends strongly on I. (B, C): In the Bayesian attractor model timing and accuracy of initial decisions and re-decisions depend on the uncertainties in the model, but, critically, the location of the fixed points of the dynamics remain the same for different uncertainties. B and C share the same observations with noise level s = 1. In B: q = 0.5. In C: r = 1.9.
Fig 10.
Example evolution of the posterior density of the decision state and the associated confidence values for one trial with a switch of stimulus at 800ms (vertical, dotted line).
(A) Posterior density of the decision state with mean (coloured lines) and two times standard deviation (shading) of decision state variables as in Fig 8A. Grey, dashed lines in A show the decision times for the initial decision (92ms) and the re-decision after the switch (1160ms). (B) Solid lines indicate confidence values for both alternatives, i.e., the posterior probability density values that the decision state is in one of the stable fixed points of the attractor dynamics. The decision threshold is indicated as grey, dashed line. The parameters of the model were those of Fig 7B (r = 2.4, s = 4, q = 0.5).
Fig 11.
Confidence in relation to stimulus strength as predicted by the BAttM for the experiment of [54].
These confidence values result from continuing accumulation of evidence for 100ms after the internal threshold was crossed but before a corresponding motor response was completed (cf. [35]). Negative coherences: left motion stimulus, positive coherences: right motion stimulus. For each coherence level we simulated 2,500 trials (5,000 for 0% coherence) using the BAttM. Shown are mean confidence values and their standard errors. Parameters were those listed in Table 2 with q = 0.5.
Fig 12.
Model fit to experimental data presented in [54].
Eight different coherence levels ranged from 0% to 75%. (A) Model parameters (red: sensory uncertainty r, green: noise level s) inferred from the behavioural data. For each coherence and parameter we show an approximate posterior distribution estimated from 501 posterior samples (see Methods) where darker colours correspond to larger probability as indicated by the colour bars on the right. Both abscissa and ordinate are in log-scale. Red line: linear fit between sensory variance r2 and coherence that also exposes a linear relation between drift and coherence in the drift diffusion model. (B) Fit of mean RT of all responses. Black dots with light grey outline: behavioural data [54]. Greyscale rectangles: estimated posterior distribution over mean reaction time. (C) Fit of accuracy (fraction of correct responses). Format as in B. Black, horizontal bars for coherences greater than 9% indicate probabilities larger than 0.2 for an accuracy of 1. This means that for high coherences parameter values as indicated in A predicted an accuracy of 1.
Table 2.
Fitted parameter values (best fitting sample for each coherence).
Fig 13.
Network diagram for two-alternative Hopfield network (cf. Eqs 9, 10) with interpolated output that was used as generative model.
The network is driven by constant input g modulated by self and lateral inhibition between state variables z1 and z2. The strength of inhibition between state variables is determined by blat (note that self-inhibition is not linear, but moderated by a sigmoid function σ(z)) while the strength of self-inhibition and the strength of the constant input is controlled by blin. After passing through another sigmoid function σ(z) the state variables interpolate target positions (cf. description of single dot task above) stored in M and consequently produce the (mean) prediction μ.