Closed-form latent dynamics for nonlinear encoding with linear embedding

Determine whether there exist nonlinear encoding maps φ: R^{N_rec} → R^K and linear embedding parameters (A ∈ R^{N_rec×N_rec}, M ∈ R^{N_rec×K}) such that, for κ(t) = φ(r(t)) and A · ṙ(t) = −r(t) + M κ(t), the induced latent variables κ(t) obey a closed-form, self-contained dynamical system r-independent of the neural state variables (i.e., κ̇(t) = g(κ(t), u(t)) for some g) and characterize the necessary and sufficient conditions for such constructions.

Background

Within the proposed encoding-embedding framework, the authors analyze the case where the latent variables are defined by a nonlinear encoding κ(t) = φ(r(t)) and the neural dynamics are governed by a linear embedding A ṙ(t) = −r(t) + M κ(t). They derive expressions linking r(t) and κ(t) but note that obtaining a closed-form, self-contained equation for κ̇(t) independent of r(t) is unresolved.

This question matters because the framework’s utility hinges on latent variables forming an autonomous dynamical system that can be perturbed and analyzed causally. Establishing conditions under which nonlinear encoding with linear embedding yields closed latent dynamics would expand the class of biologically plausible models encompassed by the theory.