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Deep Neural Cellular Potts Models

ICML 2025 · Probabilistic and scientific modeling › Dynamical systems · approach: Stochastic population dynamics · 1 citations (OpenAlex)

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Proposes NeuralCPM, which parameterizes the cellular Potts model Hamiltonian with a permutation- and translation-invariant neural network (Neural Hamiltonian), trained via deep energy-based model maximum likelihood on observational cell configurations, optionally combined with a symbolic closure term.

Problem

Cellular Potts models rely on hand-engineered Hamiltonians that only approximate the complexity of real multicellular dynamics, and no prior method learns an energy function for cell dynamics directly from observed data.

Approach

A Neural Hamiltonian built from per-cell CNN embeddings with summation over cells and global pooling is invariant to translations and cell permutations; it is trained as a deep energy-based model with persistent contrastive divergence using an approximate parallel CPM sampler, and can be added as a closure term on a symbolic cell-sorting Hamiltonian.

Key results

• On Cellular MNIST, Neural Hamiltonian + closure achieves p_volume 1.00, p_unfragmented 1.00 and CS 4.35, while the plain Neural Hamiltonian reaches the highest CS of 4.91 but only 0.11 p_volume.
• On bi-polar axial organization, Neural Hamiltonian + closure reaches axial alignment RMSE 37.3 versus 147.2 for the cell sorting Hamiltonian and 254.2 for the plain Neural Hamiltonian.
• Parameter fitting on type A cell sorting yields RMSE 0.021 at T=T* (0.055 at T=1), and the plain CPM-sampler-based fit outperforms Gibbs and Gibbs-with-Gradients baselines.

Measured effects reported in the paper

Surprising findings

• The plain Neural Hamiltonian without the symbolic closure term diverges during training on bi-polar organization, and the closure term acts as a regularizer that stabilizes training.
• Standard discrete EBM samplers (Gibbs, Gibbs-with-Gradients) fail to fit the cell sorting parameters because they do not perturb along cell boundaries, causing contact energies to inflate.

Limitations

• Evaluated only on systems with up to 100 cells, and MCMC sampling cost grows with system size.
• Global receptive field of the Neural Hamiltonian is not biologically plausible for tissue-scale simulations.
• Assumes dynamics converge to an equilibrium distribution, so non-Markovian active migration is not modeled.

Experimental setup

Datasets: synthetic cell sorting data generated with CPM Cellular MNIST (synthetic) synthetic bi-polar axial organization data generated with Morpheus bi-polar axial organization microscopy observations (Toda et al.)

Benchmarks: Cellular MNIST bi-polar axial organization cell sorting parameter fitting

Metrics: RMSE of fitted parameters p_volume p_unfragmented Classifier Score axial alignment RMSE

Baselines: Cellsort Hamiltonian Cellsort + External Potential CNN Hamiltonian 1 NH layer + CNN Neural Hamiltonian Gibbs sampling Gibbs-with-Gradients CPM sampler

Builds on: cellular Potts model deep energy-based models persistent contrastive divergence Gibbs-with-Gradients

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