Discovering a low-dimensional temperature control architecture across animals

Hibernation is an adaptation to extreme environmental seasonality that has been studied for almost 200 years, but our understanding of the underlying physiological system remains lacking due to the partially observed nature of the system. During hibernation, small mammals, such as the Arctic ground squirrel, exhibit dramatic oscillations in body temperature, typically one of the only physiological states measured, of up to 40 $^{\circ}$C. These spikes are known as interbout arousals and typically occur 10-20 times throughout hibernation. The physiological process that drives interbout arousals is unknown, but two distinct macro-scale mechanisms have been hypothesized. Using model selection for partially observed systems and classical dynamical systems theory, we are able to differentiate between these two hypotheses using only body temperature data recorded from a free-ranging Arctic ground squirrel, and show that our model can capture the broad features of the observed seasonal physiological transitions. We then modify our discovered physiological model of Arctic ground squirrel to include internally-encoded environmental information and find that we can qualitatively match body temperature data recorded from a wide range of species, including a bird, a shrew, and a bear, which also dynamically modulate body temperature. Our results suggest that a low-dimensional, environmentally sensitive core regulator could control body temperature across a diverse range of species – a new understanding of the physiological organization across species. While the findings presented here are applicable to thermophysiology, the general modeling procedure is applicable to time series data collected from partially observed biological, chemical, physical, mechanical, and cosmic systems for which the goal is to elucidate the underlying mechanism or control structure.

💡 Research Summary

The authors address a long‑standing puzzle in thermophysiology: what internal mechanism drives the dramatic temperature oscillations—inter‑bout arousals—observed during hibernation in small mammals such as the Arctic ground squirrel? Two competing macro‑scale hypotheses have been proposed. The “Tempur Arousal Clock” hypothesis posits a non‑temperature‑compensated circadian clock that drives a hidden physiological driver whose period varies with body temperature. The “Hourglass and Threshold” hypothesis, by contrast, assumes a metabolite that decays at a roughly constant rate and triggers an arousal when it falls below a fixed threshold, leading to a hidden driver with a near‑constant period. Distinguishing between these alternatives requires inference of the hidden driver’s dynamics from the only readily observable variable—body temperature (Tb).

To tackle this, the authors treat the system as a partially observed dynamical system and first determine its intrinsic dimensionality using time‑delay embedding of the Tb time series. The analysis reveals a two‑dimensional attractor, suggesting that a minimal model should involve the measured temperature x and a single hidden state y. They then construct a library of candidate monomial terms for the right‑hand sides of the ODEs governing x and y, and apply a recent sparse model selection algorithm designed for partially observed data. The algorithm iteratively estimates coefficients via variational annealing, imposes a sparsity penalty, and thresholds small coefficients to zero, thereby identifying parsimonious models that balance fit quality against complexity.

Eight‑, nine‑, and ten‑term models emerge on the Pareto front. All terms have positive coefficients, and the generic form can be written as

 ẋ = θ₁x² – θ₂x³ – θ₃xy + θ₄x²y – θ_A y³ + θ_B xy²
 ẏ = θ₅x² – θ₆y² – θ_C x³ + θ_D xy

with additional terms appearing in the larger models. When forward‑simulated with the coefficients obtained directly from the selection procedure, these models either produce a single spike or settle to a steady state, failing to generate the sustained limit‑cycle behavior characteristic of inter‑bout arousals. The authors recognize that many biological oscillators (e.g., FitzHugh‑Nagumo, Izhikevich) rely on a separation of timescales and on very small parameter values that are pruned by the sparsity threshold.

To restore realistic oscillations, they augment the temperature equation with a tiny basal heat‑production term θ₀, reflecting the fact that even in torpor a small metabolic heat flux prevents the animal from freezing. They also re‑introduce low‑order terms with small coefficients that were previously eliminated. With these adjustments the system exhibits a stable limit cycle whose period is largely insensitive to temperature, matching the “Hourglass and Threshold” scenario. Bifurcation analysis further shows how gradual changes in environmental cues (photoperiod, ambient temperature) can drive the system through transitions between homeothermy, torpor, and the arousal cycle, providing a mechanistic link between neuro‑biological control and observable physiology.

Having identified a model that supports a constant‑period hidden driver, the authors test its generality across taxa. By encoding environmental inputs (e.g., day length) into the model parameters, they qualitatively reproduce long‑term body‑temperature recordings from a bird, a shrew, and a bear—species that span the spectrum from daily heterothermy to deep hibernation and even an intermediate phenotype. Parameter tuning for each species preserves the core two‑dimensional structure, suggesting that a conserved low‑dimensional controller, modulated by external cues, underlies diverse thermoregulatory strategies.

The paper makes two major contributions. First, it demonstrates that sparse model discovery combined with dynamical‑systems theory can infer hidden physiological drivers from a single observable, overcoming the challenges of partially observed biological systems. Second, it proposes a unifying, low‑dimensional control architecture that can be flexibly adapted to explain temperature dynamics across a wide range of heterothermic endotherms. The work also highlights the importance of retaining small‑magnitude terms in mechanistic models, as they can be essential for generating biologically realistic oscillations.

In conclusion, the study provides a parsimonious yet mechanistically interpretable model of hibernation thermoregulation, validates the “Hourglass and Threshold” hypothesis over the clock‑based alternative, and extends the framework to suggest a conserved thermoregulatory core across species. The methodology is broadly applicable to any partially observed system where the goal is to uncover underlying control structures from limited time‑series data.