Extended Data Fig. 10: ANN with binary landmark presence input, and ANN with non-negative rates, recapitulates all main findings from the external map ANN.

(a) ANN with binary landmark presence input. Here, the ANN must simultaneously infer the landmark locations and the location of the animal, in contrast to the previous “external map” configuration. These determinations are inter-related, thus the much higher difficulty of the task. Structure of the recurrent network. Input neurons encoded noisy velocity (10 neurons) and landmark information (1 neuron). In the internal map setup, the input signaled whether a landmark was present at the current position or not. (b) State space trajectories in the internal map network after the second landmark encounter in two different environments. The dark green / dark blue parts of the trajectories correspond to the sections before the third landmark encounter. Left: Predominantly counterclockwise trajectories, right: Predominantly clockwise trajectories. Landmarks and trajectories were sampled with the same parameters as experiment configuration 1, but the duration of test trials was extended from 10 s (100 timesteps) to 50 s (500 timesteps). Only trials with low error after the second landmark encounter are shown, defined as maximum network localization error smaller than 0.5 rad, measured in a time window between 5 timesteps after the second landmark encounter until the end of the trial. Only the state-space trajectory after the second landmark encounter is displayed. (c) State space dimension is approximately 3, same analysis as in Extended Data Fig. 5p. (d) Example tuning curves, same analysis as in Extended Data Fig. 5m. (e) Linear decoding of position, displacement from last landmark and landmark separation from ANN activity, same analysis as in Extended Data Fig. 6c. A multinomial regression decoder was trained on 4000 trials from experiment configuration 1 (the training distribution of the internal map task) to predict from hidden layer activities which of the four possible environments was present. Performance was evaluated on separate 1000 test trials sampled from the training distribution. (f) Example neurons showing transition from egocentric landmark-relative displacement coding to allocentric location encoding, same analysis as in Extended Data Fig. 6a,b. (g) Example neurons showing conjunctive encoding, same analysis as in Extended Data Fig. 2b. Location tuning curves were determined after the second landmark encounter using 1000 trials from experiment configuration 2 using 20 location bins. Velocity and uncertainty from the posterior circular variance of the enhanced particle filter were binned in three equal bins. (h) Distribution of absolute connection strength between and across location-sensitive “place cells” (PCs) and location-insensitive “unselective cells” (UCs), same analysis as in Extended Data Fig. 5n. (i) Hidden unit activations, corresponding to Fig. 2d. (j) Trajectories from example trials, as in Fig. 2e. (k) Same trajectories as in i&j but with full LM2 state. (l) ANN is robust to perturbations, same as in Extended Data Fig. 5w. (m) ANN maintains pairwise correlation structure across states and environments, same as in Fig. 3a and Extended Data Fig. 5o. (n) ANN with non-negative rates recapitulates the main findings from the conventional ANNs. Training an ANN in the external map condition but with non-negative activity replicated all key results from the other NN types: we observed similar results with respect to location and displacement tuning (r), the transition in linear decodability of displacement to location from the population and dynamically varying decodability of landmark separations within trials (p), the presence of heterogeneous and conjunctive tuning (s), lack of modularity in connectivity between cells with high and low amounts of spatial selectivity (t), and the preservation of cell-to-cell correlations across time within trials and across environments (q). The nonlinearity does affect the distribution of recurrent weights: The distribution of non-diagonal elements in the non-negative network is sparse (excess kurtosis k = 7.8), while it is close to Gaussian for the external and internal map networks with tanh-nonlinearity (k = 0.6 and k = 0.9 respectively; panel u); however, the distributions of eigenvalues of the recurrent weights have similar characteristics for all trained networks (panel v). Structure of the recurrent network: Input neurons encoded noisy velocity (10 neurons) and received external map input (70 neuron), same as the regular external map ANN. Recurrent layer rates were constrained to be non-negative. (o) Example tuning curves, same analysis as before. (p)Linear decoding of position, displacement from last landmark and landmark separation from ANN activity, same analysis as before. (q) ANN maintains pairwise correlation structure across states and environments, same as before. (r) Example neurons showing transition from egocentric landmark-relative displacement coding to allocentric location encoding, same analysis as before. (s) Example neurons showing conjunctive encoding, same analysis before. (t) Distribution of absolute connection strength between and across location-sensitive “place cells” (PCs) and location-insensitive “unselective cells” (UCs), same analysis as before. (u) Distribution of non-diagonal recurrent weights for randomly initialized (untrained), external map, internal map, and non-negative network. The k-value measured denotes excess kurtosis, a measure of deviation from Gaussianity (k = 0 for Gaussian distributions). The presence of a nonlinearity constraint on the ANN affects the distribution of recurrent weights: The distribution of non-diagonal elements in the non-negative network is sparse (excess kurtosis k = 7.8), while it is close to Gaussian for the external and internal map networks with tanh-nonlinearity (k = 0.6 and k = 0.9 respectively). (v) Scatterplot of real and imaginary part of complex eigenvalues of recurrent weight matrix for randomly initialized (untrained), external map, internal map, and non-negative network. The distributions of eigenvalues of the recurrent weights have similar characteristics for all trained networks.