Data-driven geometric phase in biological locomotion

Technical Summary: Data-driven Geometric Phase in Biological Locomotion

Problem Statement
Geometric phase, a concept rooted in gauge theory, quantifies net locomotion (displacement and rotation) in dissipative media over one period of shape deformation. While theoretically well-established for systems like microswimmers (governed by Stokes flow) and broader kinematic locomotion (e.g., snakes, worms), applying this theory to real biological data presents significant challenges. Biological shape data is typically noisy, sparse, weakly periodic, and limited in duration (spanning only a few beat cycles). Traditional linear dimensionality reduction methods, such as Principal Component Analysis (PCA), fail to define a proper "isochron phase" (asymptotic phase) necessary for consistent geometric phase extraction, offering only a "protophase." Furthermore, standard theoretical computations require explicit mechanical laws (e.g., Stokes equations, resistive force theory), which are often unknown or difficult to parameterize for specific biological systems.

Methodology
The authors propose a theory-guided, data-driven Koopman Autoencoder (KAE) framework to recover the limit cycle embedded in imperfect biological data and extract geometric phase without assuming specific mechanical laws. The methodology proceeds as follows:

1. Koopman Operator Theory & Autoencoder Architecture:
   The system reframes the nonlinear shape dynamics as an infinite-dimensional linear system. A deep autoencoder is constructed where:

   • The encoder ($\phi$) approximates the principal Koopman eigenfunction, mapping high-dimensional shape data $q \in \mathbb{R}^N$ to a 2D latent space $z \in \mathbb{C}$.
   • The latent dynamics are constrained to follow linear evolution ($\dot{z} = i\omega z$), effectively extracting a stable limit cycle.
   • The decoder ($\psi$) reconstructs the denoised shape orbit from the latent space.
   • A Koopman matrix $K$ is enforced in the latent space to model the time evolution, accommodating stochastic contraction toward the limit cycle via a Floquet parameter $\gamma$.

2. Loss Function:
   The network is trained to minimize a composite cost function $L_{tot} = L_{rec} + L_{pred} + L_{lin} + L_{rad}$, comprising:

   • Reconstruction loss ($L_{rec}$): Ensures the decoder accurately recovers the input shape.
   • Prediction loss ($L_{pred}$): Ensures the predicted next step in latent space maps correctly to the shape space.
   • Linearity constraint ($L_{lin}$): Enforces the linear dynamics in the latent space.
   • Radius constraint ($L_{rad}$): Forces the latent trajectory to converge to a unit circle.

3. Geometric Phase Extraction:
   Once the limit cycle is identified in the latent space, the oscillator phase $\Theta$ is defined. The gauge field $A(\Theta)$ (representing the mechanical connection) is expanded as a Fourier series in terms of the oscillator phase. The geometric phase $W_C$ is then computed by integrating this gauge field over the limit cycle, relying solely on the gauge-theoretic structure rather than fluid dynamic equations.

4. Geometric Phase Sensitivity Function (Geometric PSF):
   The authors introduce a new quantity, the Geometric PSF ($Z_W$), derived from the oscillator phase sensitivity function ($Z_1$) and the gauge field. This function quantifies the shift in the geometric phase (translation and rotation) resulting from a local perturbation in the shape variable. It is calculated purely from the data-driven limit cycle and gauge structure.

Key Results
The method was validated and applied to three distinct datasets:

• Synthetic Data: Using a noisy limit cycle model of a supercritical Hopf bifurcation, the KAE successfully reconstructed fine waveform details and extracted a clean limit cycle in the latent space, outperforming 2D PCA which produced scattered trajectories. The pull-back metric revealed the nonlinear folding of the shape space onto the latent unit circle.
• Zebrafish and Bull Sperm: Applied to flagellar waveform data (sparse in time, varying amplitudes), the KAE extracted stable limit cycles. The resulting geometric phases (trajectories) were smooth and consistent, despite the data sparsity and lack of assumed Stokes flow equations.
• Caenorhabditis elegans (Nematode): Applied to crawling data with large fluctuations and anisotropic drag characteristics. The method successfully extracted the limit cycle and geometric phase without assuming resistive force theory.

Geometric PSF Analysis:
The calculated Geometric PSF revealed distinct mechanical insights for each organism:

• Zebrafish Sperm: Sensitivity was relatively uniform along the flagellum, peaking at the middle.
• Bull Sperm: Maximum sensitivity occurred in the distal region, correlating with larger distal beat amplitudes.
• Nematode: Maximum sensitivity for both progressive and lateral translation was located at the head, reflecting the dominance of normal drag over tangential drag in crawling.
• Phase Dependence: The analysis showed that shape perturbations most effectively modulate forward propulsion when the head points along the swimming direction, whereas steering is most effective when the head points away, highlighting different optimal strategies for progression versus turning.

Significance and Claims
The paper claims to present a robust framework for extracting geometric phase from biological locomotion data under severe constraints (noise, sparsity, limited cycles) without prior knowledge of mechanical laws.

• Theoretical Contribution: It bridges the gap between gauge-theoretic locomotion theory and real-world data by utilizing Koopman operator theory to define a proper isochron phase, overcoming the limitations of linear methods like PCA.
• Methodological Innovation: The introduction of the Geometric PSF allows for the direct characterization of underlying mechanics (e.g., drag anisotropy, control concentration) from sparse observations, serving as a tool to understand steering and maneuverability.
• Generality: While demonstrated on 2D motion, the authors state the method extends naturally to 3D deformations and more complex dynamics (e.g., multi-flagella or multi-legged systems) via the Koopman reduction to stable limit tori.
• Scope: The work demonstrates that geometric structure can be exploited to infer physical structure from imperfect observations when detailed governing laws are inaccessible, offering a significant alternative to data-hungry machine learning models that require large, clean datasets.