Integrating GNN and Neural ODEs for Estimating Non-Reciprocal Two-Body Interactions in Mixed-Species Collective Motion

TL;DR

A novel deep learning framework for estimating the underlying equations of motion from observed trajectories, integrating graph neural networks with neural differential equations, enabling effective prediction of two-body interactions based on the states of the interacting entities.

Abstract

Analyzing the motion of multiple biological agents, be it cells or individual animals, is pivotal for the understanding of complex collective behaviors. With the advent of advanced microscopy, detailed images of complex tissue formations involving multiple cell types have become more accessible in recent years. However, deciphering the underlying rules that govern cell movements is far from trivial. Here, we present a novel deep learning framework for estimating the underlying equations of motion from observed trajectories, a pivotal step in decoding such complex dynamics. Our framework integrates graph neural networks with neural differential equations, enabling effective prediction of two-body interactions based on the states of the interacting entities. We demonstrate the efficacy of our approach through two numerical experiments. First, we used simulated data from a toy model to tune the hyperparameters. Based on the obtained hyperparameters, we then applied this approach to a more complex model with non-reciprocal forces that mimic the collective dynamics of the cells of slime molds. Our results show that the proposed method can accurately estimate the functional forms of two-body interactions -- even when they are nonreciprocal -- thereby precisely replicating both individual and collective behaviors within these systems.

Figures (5)

• Figure 1: Schematic figure of the integration of GNN and neuralODE. For details, see \ref{['sec:simulation_details']}.
• Figure 1: Snapshots of the simulation results in the mixed-species model and the estimated model. Panels (A, C, E, G) depict results from the mixed-species model, and panels (B, D, F, H) from the estimated model. The upper rows (A-D) represent simulations with parameter set (i), and the lower rows (E-H) with parameter set (ii). Panels (A, B, E, F) include individuals with $c^i=0$ and panels (C, D, G, H) with $c^i=1$. Colors correspond to those used in Figure \ref{['fig:2']}.
• Figure 2: Snapshots of the simulation results in the harmonic interaction model with friction constants. (A) The case with a friction constant of $\gamma = 1 \times 10^{-2}$ and (B) with $\gamma = 1 \times 10^{-1}$. Positions of individuals are indicated by blue circles, and velocities by black arrows. (C-H) The functions estimated from the data with harmonic interaction plotted against the true values. The upper panel displays $F^{(1)}$ and the lower panel $F^{(2)}$ for different cases: (C-D) with $\gamma = 1 \times 10^{-2}$ and $N_\text{tra}=270$, (E-F) with $\gamma = 1 \times 10^{-2}$ and $N_\text{tra}=3$, and (G-H) with $\gamma = 1 \times 10^{-1}$ and $N_\text{tra}=3$. Blue and orange indicates x- or y- element $F$. A black line serves as a guide indicating where the estimated values equal the true values. (I-K) Snapshots of the simulation results in the model estimated from the data with harmonic interaction. The panels represent different scenarios: (I) with $\gamma = 1 \times 10^{-2}$ and $N_\text{tra}=270$, (J) with $\gamma = 1 \times 10^{-2}$ and $N_\text{tra}=3$, and (K) with $\gamma = 1 \times 10^{-1}$ and $N_\text{tra}=3$.
• Figure 2: The functions estimated from data with the mixed-species model, without dependency on species type, plotted against the true values. The rows display different cases: (A-D) with parameter set (iii), (E-H) with parameter set (iv), and (I-L) with parameter set (v). A gray line serves as a guide to indicate where the estimated values equal the true values.
• Figure 3: Snapshots of the simulation results in the mixed-species model. Panel (A) displays a representative snapshot of the training data for the case with parameter set (i), and panel (B) with parameter set (ii) (see Supplemental Table S\ref{['tab:S2']}). Each individual is represented by an arrow located at their position and directed toward their polarity. Black arrows indicate individuals with $c^i=0$, while red arrows represent those with $c^i=1$. (C-J) The functions estimated from the data with the mixed-species model plotted against the true values. The rows display different cases: (C-F) with parameter set (i), and (G-J) with parameter set (ii). A gray line serves as a guide indicating where the estimated values equal the true values. (K-L) Snapshots of the simulation results in the model estimated from data with the mixed-species model. (K) The case with parameter set (i), and (L) with parameter set (ii).