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Inferring hidden forcing in a biological oscillator using Kolmogorov-Arnold networks

This paper demonstrates that Kolmogorov-Arnold networks can reconstruct hidden muscular forcing in avian respiratory dynamics from partial air-sac pressure observations, revealing a nontrivial two-phase activation pattern that was independently validated by electromyographic recordings.

Original authors: Julian Szereszewski, Facundo Fainstein, Leandro E. Fernandez, Gabriel B. Mindlin

Published 2026-06-09

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CC BY 4.0

Original authors: Julian Szereszewski, Facundo Fainstein, Leandro E. Fernandez, Gabriel B. Mindlin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out how a musician is playing a complex song, but you can only hear the sound coming out of the speaker. You can't see their hands, you can't see the keys they are pressing, and you can't see the sheet music. All you have is the audio waveform.

Usually, if the sound looks like a simple, rhythmic "whoosh," you might assume the musician is just pressing a key and letting it fade out (a simple "relaxation" motion). But what if the musician is actually pressing two distinct keys in a very specific, hidden pattern to create that same sound?

This is exactly what the researchers in this paper did, but instead of a musician, they studied a bird's breathing system.

The Mystery of the Bird's Breath

Birds have a unique way of breathing. They don't have flexible lungs like humans; instead, they have rigid lungs connected to a system of air sacs (like balloons) that act as reservoirs. When the bird breathes, muscles squeeze these air sacs to push air through the lungs.

The scientists could easily measure the pressure inside these air sacs (like listening to the speaker). However, they couldn't easily measure the muscle force squeezing the sacs (the musician's hands). They wanted to know: What is the exact pattern of muscle squeezing that creates the pressure we see?

The Problem with Old Tools

Previously, scientists tried to guess the answer by writing down a math equation with a few "knobs" (parameters) and turning those knobs until the math matched the pressure data.

The flaw: This is like guessing the song by only trying to match the volume. If the real song has a complex melody, your simple guess might still match the volume but miss the melody entirely. You have to assume the answer looks a certain way before you start, which can blind you to the truth.

The New Tool: The "Transparent" AI

The researchers used a new type of Artificial Intelligence called Kolmogorov–Arnold Networks (KANs).

Think of a standard AI (like a deep neural network) as a black box. You put data in, and a result comes out, but you have no idea what happened inside. It's like a magic trick where the magician hides the mechanism.

A KAN, however, is like a clear glass box.

Inside a standard AI, the "rules" are hidden in millions of tiny, confusing numbers.
Inside a KAN, the rules are built from simple, single-variable curves that you can actually see and draw. It's like the magician showing you exactly how the card trick works, step-by-step, in plain sight.

The Discovery: Two Phases, Not One

The scientists fed the bird's pressure data into this "glass box" AI. The AI learned the hidden math that governs the breathing.

What they expected: Based on the pressure signal alone, it looked like a simple, one-time squeeze of the muscles (a relaxation-like oscillation).

What the AI found: The "glass box" revealed that the muscles are actually doing something much more complex. The AI showed that within a single breath, the muscles activate in two distinct phases:

A strong push at the very beginning of the exhale.
Another strong push near the very end of the exhale.

The pressure signal looked smooth and simple, but the force creating it was actually a double-punch pattern. The pressure signal was "hiding" this complexity.

The Proof: Listening to the Muscles

To prove the AI wasn't just making things up, the scientists went back and recorded the actual electrical activity of the bird's breathing muscles (using tiny electrodes, like a doctor listening to a heart).

The result: The muscle recordings perfectly matched the AI's prediction! The muscles were firing twice per breath, exactly as the "glass box" AI had deduced from the pressure data alone.

The Big Takeaway

This paper shows that if you have a complex system where you can only see the "output" (the pressure), you can use a special, transparent type of AI to reconstruct the hidden "rules" of the system.

By doing this, they didn't just predict the future; they uncovered a hidden secret about how the bird's body works. They proved that the "muscle force" driving the bird's breath is far more intricate than the pressure signal suggests, revealing a two-step dance that was invisible to the naked eye but visible through the math.

In short: They used a see-through AI to listen to a bird's breath and figured out that the bird's muscles are doing a complex double-step dance, a fact that was completely hidden in the sound of the breath itself.

Technical Summary: Inferring Hidden Forcing in a Biological Oscillator using Kolmogorov–Arnold Networks

Problem Statement
Inferring the forces that drive a dynamical system from partial observations remains a fundamental challenge in physics and biology. In many systems, such as avian respiration, only a limited set of variables (e.g., air-sac pressure) is experimentally accessible, while the effective driving terms arise from hidden degrees of freedom (muscular activation). A critical difficulty is that qualitatively different underlying mechanisms can produce observables with similar temporal structures. For instance, the pressure signal in birds suggests a simple relaxation-like oscillation, potentially masking a more complex driving force. Traditional approaches face limitations: parametric fitting requires strong prior assumptions about functional forms, while standard machine learning (e.g., deep neural networks) acts as a "black box," encoding dynamics in opaque parameters that lack physical interpretability. Methods like SINDy offer interpretability but require a pre-specified library of candidate functions, which can bias reconstruction if the true structure is unknown.

Methodology
The authors propose a data-driven framework using Kolmogorov–Arnold Networks (KANs) to reconstruct the governing equations of the avian respiratory system directly from experimental pressure measurements.

Theoretical Framework: The study utilizes the Kolmogorov–Arnold representation theorem, which states that any continuous multivariate function can be expressed as a superposition of compositions of one-dimensional functions. Unlike standard Multilayer Perceptrons (MLPs) where nonlinearities are fixed at nodes, KANs associate trainable univariate functions with the network edges. This architecture allows the learned functional dependencies to be inspected and simplified into symbolic expressions after training.
Experimental System: Data was collected from naturally breathing canaries (Serinus canaria). Air-sac pressure ( $p$ ) was measured via a flexible cannula connected to a pressure transducer carried in a backpack. The signal was sampled at 44.15 kHz.
Data Processing: The pressure time series was averaged over four respiratory cycles, smoothed using a Savitzky–Golay filter and a low-pass filter, and differentiated to obtain $p$ , $\dot{p}$ , and $\ddot{p}$ .
Model Reconstruction: The authors modeled the system as a second-order differential equation: $\ddot{p} = F(p, \dot{p})$ . Independent KAN models were trained to approximate the vector field $F$ using $(p, \dot{p}, \ddot{p})$ samples.
Decomposition: To isolate the driving force, the learned vector field was decomposed. Linear terms dependent on $p$ and $\dot{p}$ were constrained to match the known passive mechanical response (elastic and viscous terms). The remaining nonlinear contributions were grouped into an effective forcing term, $G(t)$ , which is proportional to the time derivative of the muscular drive $f(t)$ .

Key Results

Vector Field Reconstruction: The KAN successfully reconstructed the phase-space trajectory $(p, \dot{p})$ , reproducing the observed oscillatory cycles of the respiratory system.
Hidden Structure Discovery: While the raw pressure signal appeared to follow a relaxation-like oscillation, the reconstructed vector field revealed a nontrivial structure in the underlying forcing. The inferred muscular drive $f(t)$ exhibits a two-phase activation pattern within each respiratory cycle.
Independent Validation: This two-phase structure was not apparent from the pressure signal alone. The authors validated this prediction through independent electromyographic (EMG) recordings of the expiratory muscles. The EMG data confirmed strong activation at both the onset and near the end of the expiratory phase, matching the temporal structure inferred by the KAN model.

Key Contributions

Interpretable Dynamical Reconstruction: The paper demonstrates that KANs can serve as a bridge between flexible machine learning and interpretable dynamical descriptions. By learning nonlinearities along edges, the method yields explicit functional forms that can be inspected to reveal physical mechanisms.
Recovery of Latent Variables: The study shows that it is possible to infer hidden driving variables (muscular forcing) from partial observations (pressure) without pre-specifying the functional form of the equations.
Correction of Phenomenological Assumptions: The work challenges the assumption that avian respiratory pressure is driven by simple phenomenological waveforms, revealing a more complex, two-phase neural control structure.

Significance and Claims
The paper claims that data-driven reconstruction of dynamical laws, when combined with interpretable representations like KANs, can uncover hidden physical structures and provide access to unobserved driving variables in partially observed systems. The authors emphasize that this approach offers a general route to infer latent forces where distinct mechanisms might otherwise produce indistinguishable observables. By extracting mechanistic insight from complex biological dynamics beyond what is directly observable, this method establishes a pathway for understanding systems where the effective driving terms arise from hidden degrees of freedom. The results are modest in scope, focusing specifically on the avian respiratory system, but the methodology is presented as a generalizable strategy for inferring latent forces in dynamical systems.