Pole-Zero Geometry, Model Reduction, and Identifiability in Sensory Adaptation
Abstract
Sensory adaptation provides a concrete setting in which low-order system identification can fail qualitatively. We show that one fixed higher-order adaptive system composed entirely of real first-order relaxation modes can be reduced to opposite sides of the second-order pole boundary: low-frequency moment matching gives ρmoment=4.50, whereas finite-window fitting gives ρwindow=3.31, and the inferred pole class changes further with sampling protocol. Thus the real-versus-complex classification of a reduced model is not itself reduction invariant. We then use the general two-state spectrum to connect stochastic identifiability to adaptation: for nontrivial coupling and one-state observation, cross diffusion drops out of the scalar spectrum when the hidden state has no self-relaxation. In the adaptive model, this condition is precisely the integral-memory limit that produces exact adaptation, while leaky memory restores spectral sensitivity. For the exact-adaptation model, the Gaussian path-space irreversibility nevertheless depends on the hidden cross-diffusion channel. Hence {H,Sx} does not determine the irreversibility rate. Independently, for a specified all-even reduced two-state drift with ρ<4, the drift-only lower bound is σ≥ τx−1(4/ρ−1). Published \textit{E.~coli} and \textit{C.~elegans} responses provide biological examples of these limits. The distinction established here between transfer-function invariants, reduction-dependent properties, and hidden-state quantities provides a concrete framework for evaluating the limitations of low-dimensional models of adaptive biological dynamics.
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BibTeX
@misc{kim2026polezero,
title = {{Pole-Zero Geometry, Model Reduction, and Identifiability in Sensory Adaptation}},
author = {Gunn Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01329},
eprint = {2609.01329},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01329},
}
RIS
TY - UNPB
TI - Pole-Zero Geometry, Model Reduction, and Identifiability in Sensory Adaptation
AU - Gunn Kim
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01329
UR - https://arxiv.org/abs/2609.01329
AB - Sensory adaptation provides a concrete setting in which low-order system identification can fail qualitatively. We show that one fixed higher-order adaptive system composed entirely of real first-order relaxation modes can be reduced to opposite sides of the second-order pole boundary: low-frequency moment matching gives ρ_moment=4.50, whereas finite-window fitting gives ρ_window=3.31, and the inferred pole class changes further with sampling protocol. Thus the real-versus-complex classification of a reduced model is not itself reduction invariant. We then use the general two-state spectrum to connect stochastic identifiability to adaptation: for nontrivial coupling and one-state observation, cross diffusion drops out of the scalar spectrum when the hidden state has no self-relaxation. In the adaptive model, this condition is precisely the integral-memory limit that produces exact adaptation, while leaky memory restores spectral sensitivity. For the exact-adaptation model, the Gaussian path-space irreversibility nevertheless depends on the hidden cross-diffusion channel. Hence {H,S_x} does not determine the irreversibility rate. Independently, for a specified all-even reduced two-state drift with ρ<4, the drift-only lower bound is σ≥ τ_x^−1(4/ρ−1). Published \textit{E.~coli} and \textit{C.~elegans} responses provide biological examples of these limits. The distinction established here between transfer-function invariants, reduction-dependent properties, and hidden-state quantities provides a concrete framework for evaluating the limitations of low-dimensional models of adaptive biological dynamics.
ER -