Figure 3. Recurrent neural network and learning drift diffusion model (DDM).

(a) Roll out in time of recurrent neural network (RNN) for one trial. (b) The decision variable for the recurrent neural network (dark gray), and other trajectories of the equivalent DDM for different diffusion noise samples (light gray). (c, d, e) Changes in $ER$, $DT$, and $iRR$ over a long period of task engagement in the RNN (light gray, pixel simulation individual traces; black, pixel simulation mean; pink, Gaussian simulation mean) compared to the theoretical predictions from the learning DDM (blue). (f) Visualization of traces in c and d in speed-accuracy space along with the optimal performance curve (OPC) in green. The threshold policy was set to be $iRR$-sensitive for c–f.

Figure 3—figure supplement 1. Analytical reduction of linear drift-diffusion model (LDDM) matches error-corrective learning neural network dynamics during learning.

(a) The recurrent linear neural network can be analytically reduced. In the reduction, the decision variable draws an observation from one of two randomly chosen Gaussian ‘stimuli’. The observations are scaled by a perceptual weight. After the addition of some irreducible noise, the value of the decision variable at previous time step is added to the current time step. A trial ends once the decision variable hits a predetermined threshold. The dynamics of the perceptual weight capture the mean effect of gradient descent learning in the recurrent linear neural network. (b) Weight w of neural network across task engagement time for multiple simulations of the network (gray), the mean of the simulations (black), and the analytical reduction of the network (blue). (c) Same as in b but for the threshold z. (d) Same as in b but for the error rate. (e) Same as in b but for the decision time. (f) Same as in b but for the instantaneous reward rate (correct trials per second). (g) Learning trajectory in speed-accuracy space for simulations, simulation mean, and analytical reduction (theory). Optimal performance curve (OPC) is shown in red.