Fig. 2

Zebrafish navigation is led by long-lived modes with a hierarchy of timescales prioritizing change of orientation, speed, and egocentric direction bias. (A) Implied timescales of the -th mode of the reversibilized , estimated as , where are the eigenvalues of with , SI Appendix, Fig. S2A. We find three long-lived modes that are well separated from the bulk spectrum, while faster modes are closer to each other. Error bars are bootstrapped 95% CI obtained by sampling 7,500 bouts from each 14 sensory context (11) over 100 seeds (SI Appendix). (A, Inset) We show the averaged transition matrix . (B) Microstates projected along and color coded by the mean absolute change in heading for the bouts belonging to each of the microstates. Increasing is associated with higher amplitude reorientations. (C) Same as (B), but color coded by the mean speed in each microstate . We find that higher is generally associated with higher speeds. (D) Probability density of the mean change in heading for each of the microstates along the third mode . Low values of correspond to leftward bouts, while high values of correspond to rightward bouts. (E) Cumulative distribution function (CDF) of the absolute mean change in heading direction for all bouts belonging to the cruising (red) and wandering (blue) strategies, identified by partitioning along (SI Appendix). (E, Inset) Reorganized , where each microstate in cruising or wandering is placed together, revealing a block diagonal structure. (F) Mean sequence length (MSL) of fish in wandering and cruising strategies. We infer a coarse-grained transition matrix among cruising and wandering , and simulate sequences by sampling the next state according to the transition matrix (SI Appendix). We find that the coarse-grained Markov models inferred for each fish accurately predict the mean sequence lengths in cruising and wandering. Note also the wide variability in mean sequence lengths among fish, from a few bouts to a few hundreds of bouts.