Supplementary Figure 8: Summary of results of chaotic data RNNs receiving input pulses.

We extracted averaged inferred inputs, u_t, from the LFADS models used in Supp. Fig. 7. a,b. To see how timing of the inferred input was related to the timing of the actual input pulse, we determined the time at which u_t reached its maximum value (vertical axis), and plotted this against the actual time of the delta pulse (horizontal axis) for all trials. a. Results using γ = 1.5, b. and γ = 2.5. As shown, for the majority of trials, despite the complex internal dynamics, LFADS was able to infer the correct timing of a strong input. However, LFADS did a better job of inferring the inputs in the case of simpler dynamics for two reasons: In the case of γ = 2.5, 1) the complexity of the dynamics reduces the effective magnitude of the perturbation caused by the input and 2) LFADS used the inferred input more actively to account for non-input-driven dynamics. We include this example of a highly chaotic data RNN to highlight the subtlety of interpreting an inferred input. c-d One possibility in using LFADS with inferred inputs (that is dimensionality of u_t ≥ 1) is that the data to be modeled is actually generated by an autonomous system, yet one, not knowing this fact, allows for an inferred input in LFADS. To study this case we utilized the four chaotic data RNNs described above, that is γ = 1.5, and γ = 2.5, with and without delta pulse inputs. We trained an LFADS model for each of the four cases, with an inferred input of dimensionality 1, despite the fact that two of the four data RNNs generated their data autonomously. After training we examined the strength of the average inferred input, u_t, for each LFADS model (where strength is taken as the root-mean-square of the inferred input, averaged over an appropriate time window, sqrt(〈u²_t〉t₁:t₂)). The results are show in panel c for γ = 1.5 and panel d for γ = 2.5. The solid lines show the strength u_t at each time point when the data RNN received no input pulses, averaged across all examples. The ’◦’ and ’x’ show the strength of u_t at time points when the data RNN received delta pulses, averaged in a time window around t, and averaged over all examples. Intuitively, a ’◦’ is the strength of u_t around a delta pulse at time t, and an ’x’ is the strength of u_t if there was no delta pulse around time t.