Conditions for closed latent dynamics under nonlinear encoding and nonlinear embedding

Ascertain the conditions under which models with both nonlinear encoding φ: R^{N_rec} → R^K and nonlinear dynamical embedding maps produce latent variables κ(t) that satisfy self-contained closed-form dynamical equations (κ̇(t) = g(κ(t), u(t))) independent of r(t).

Background

The authors discuss extending their framework to models with nonlinear encoding and nonlinear embedding. While such models could offer greater generality and biological realism, it is unresolved when the latent variables admit closed, self-sufficient dynamics independent of the high-dimensional neural state.

Resolving this would clarify the theoretical landscape of latent-variable modeling in neuroscience, bridging static autoencoders and dynamical systems that enable causal predictions under perturbations.