# Learning collective multicellular dynamics with an interacting mean field neural SDE model

## TL;DR

This paper introduces scIMF, a new model that captures complex interactions between cells in dynamic systems using deep learning and stochastic differential equations.

## Contribution

The novel contribution is scIMF, a deep-generative model that integrates cell-cell interactions using McKean-Vlasov SDEs and cell-wise attention for high-dimensional temporal scRNA-seq data.

## Key findings

- scIMF outperforms existing methods in reconstructing gene expression and inferring cellular velocities.
- The model reveals biologically interpretable non-reciprocal interactions in multicellular systems.
- scIMF captures asymmetric interactions in vivo and symmetric ones in vitro, reflecting nonequilibrium dynamics.

## Abstract

The advent of temporal single-cell RNA sequencing (scRNA-seq) data has enabled in-depth investigation of dynamic processes in heterogeneous multicellular systems. Despite remarkable advancements in computational methods for modeling cellular dynamics, integrating cell-cell interactions (CCIs) into these models remains a major challenge. This is particularly true when dealing with high-dimensional gene expression profiles from large populations of interacting cells, where the intricate interplay between cells can be obscured by data complexity. To address this, we present scIMF, a single-cell deep-generative Interacting Mean Field model that learns collective multicellular dynamics. Leveraging the McKean-Vlasov stochastic differential equation framework, scIMF provides a mathematical foundation for describing interacting multicellular systems, where each cell’s evolution depends on the population’s empirical distribution. By incorporating a cell-wise attention mechanism, the model efficiently captures nonlocal and asymmetric CCIs, enabling realistic reconstruction of complex intercellular relationships in high-dimensional spaces. Benchmarking across diverse temporal scRNA-seq datasets demonstrates that scIMF outperforms state-of-the-art methods in reconstructing gene expression at unobserved time points and in inferring cellular velocities. Furthermore, scIMF uncovers biologically interpretable, non-reciprocal interaction patterns of cells, providing a principled framework for studying complex, particularly non-equilibrium biological systems.

Interacting particle systems (IPSs) are ubiquitous in nature, from physics to biology, giving rise to complex dynamics at the level of individual constituents and in the system as a whole. While the mathematical theory of IPSs has matured considerably, computational methods lag in learning non-reciprocal from high-dimensional population snapshots, particularly for collective multicellular dynamics in temporal scRNA-seq data. We introduce scIMF, a deep generative model that overcomes this by integrating McKean-Vlasov stochastic differential equations with a cell-wise attention mechanism. Our framework efficiently infers nonlocal, non-reciprocal cell-cell interactions directly from data, outperforming state-of-the-art methods. Crucially, scIMF reveals asymmetric interactions signifying nonequilibrium dynamics in vivo while capturing symmetry in vitro, providing a transformative tool to study collective behaviors in complex dynamical systems.