Fig. 4.

Schematic for analyzing a dynamic neural code. (A) Graphical model of a POMDP problem with a solution implemented by neurons implicitly encoding beliefs. (B) We find how behaviorally relevant variables (here, beliefs) are encoded in measured neural activity through the function ${\check{b}}_t={\phi }_{\mathrm{e}\mathrm{n}\mathrm{c}}\left({\mathbf{r}}_t\right)$. (C) We then test our hypothesis that the brain recodes its beliefs rationally by testing whether the dynamics of the behaviorally estimated belief $\widehat{b}$ match the dynamics of the neurally estimated beliefs $\check{b}$, as expressed through the update dynamics ${\widehat{f}}_{\mathrm{d}\mathrm{y}\mathrm{n}}\left({\widehat{b}}_t,o_t\right)$ and recoding function ${\check{f}}_{\mathrm{r}\mathrm{e}\mathrm{c}}\left({\check{b}}_t,o_t\right)$. (D) Similarly, we test whether the brain decodes its beliefs rationally by comparing the behaviorally and neurally derived policies ${\widehat{\pi }}_{\mathrm{a}\mathrm{c}\mathrm{t}}$ and ${\check{\pi }}_{\mathrm{d}\mathrm{e}\mathrm{c}}$. Quantities estimated from behavior or from neurons are denoted by up-pointing or down-pointing hats, $\widehat{}$ and $\stackrel{ˇ}{}$, respectively (SI Appendix, Table S1).