Figure 1. Normative model.
(A) Illustration of the belief-updating process. (B) Discrete-time log-prior odds at a given moment as a function of the belief at the prior moment, plotted as Equation 2 for different values of H. (C) Continuous-time version of the model, with log-prior odds plotted as a function of belief, computed by numerically integrating Equation 4 with dx(t) = 0 over a 16 ms interval. Here expected instability (λ) has units of number of changes per s. Thus, λ → ∞ is analogous to discrete-time H → 0.5. (D–F) Examples of how the normative model (solid lines) and perfect accumulation (dashed gray lines) process a time-dependent stimulus (light vs dark grey for the two alternatives, shown at the top) for different hazard rates (H).
Figure 1—figure supplement 1. Dynamics of the continuous-time model.
(A) Example of the temporal derivative of belief magnitude plotted as a function of belief magnitude (Equation 4), given an average sensory evidence of 7 units of LLR per unit time. The point of intersection with the dotted line (i.e., derivative = 0) corresponds to the approximate steady-state value of the belief (the mode of the probability distribution). (B) Histogram of belief values at steady state for hazard rate λ = 0.1 Hz from panel A, with the approximated solution in Equation 14a shown in magenta. Note the heavy tail, which is typical for moderate-to-strong input to the model. (C) Asymmetric effects of transient ‘perturbations’ of evidence of equal magnitude (35 units of LLR per s) but opposite signs. Belief magnitude re-converges to steady state faster after the positive perturbation (i.e., favoring the current belief) than the negative perturbation (i.e., opposing the current belief), which accounts for the heavy tail pointed towards zero in B. (D–I) Simulated temporal evolution of the probability distribution of belief value, given an average sensory evidence of 7 units of LLR per s, with an abrupt sign reversal at t = 5 s. (D–F) Histograms of belief at the time-points indicated by arrows. (G) Pseudocolor plot of the full probability distributions normalized by the peak probability over the entire time series. Hot/cold colors indicate high/low probabilities. (H) Expected value of the belief variable. (I) Standard deviation of the belief variable. Note that after a change-point, leaky integration is re-started and beliefs pass through the ‘low-confidence’ regime of the model, resulting in a transient increase of variability that is approximately Gaussian. All simulations used 10,000 iterations.