Deep Neural Cellular Potts Models

TL;DR

NeuralCPM addresses the limitation that hand-crafted Hamiltonians in cellular Potts models cannot capture the full complexity of multicellular dynamics. By learning a Neural Hamiltonian that respects translation and permutation symmetries, and optionally coupling it with a biology-informed symbolic term as a closure, the method can model intricate collection dynamics directly from observational data. The key contributions include a symmetry-preserving neural architecture for the Hamiltonian, a training approach leveraging negative log-likelihood with an approximate MCMC sampler, and demonstrations across synthetic cell sorting, Cellular MNIST-like structures, and real-world-inspired bipolar self-organization. The results show improved expressiveness over analytical Hamiltonians while maintaining biological realism and training stability, enabling more accurate simulations of complex tissue behaviors with potential applications in biology and medicine.

Abstract

The cellular Potts model (CPM) is a powerful computational method for simulating collective spatiotemporal dynamics of biological cells. To drive the dynamics, CPMs rely on physics-inspired Hamiltonians. However, as first principles remain elusive in biology, these Hamiltonians only approximate the full complexity of real multicellular systems. To address this limitation, we propose NeuralCPM, a more expressive cellular Potts model that can be trained directly on observational data. At the core of NeuralCPM lies the Neural Hamiltonian, a neural network architecture that respects universal symmetries in collective cellular dynamics. Moreover, this approach enables seamless integration of domain knowledge by combining known biological mechanisms and the expressive Neural Hamiltonian into a hybrid model. Our evaluation with synthetic and real-world multicellular systems demonstrates that NeuralCPM is able to model cellular dynamics that cannot be accounted for by traditional analytical Hamiltonians.

Figures (10)

• Figure 1: NeuralCPM is more expressive than traditional cellular Potts models, allowing for more accurate simulation of real-world collective cell dynamics, e.g. bi-polar self-organization of biological cells over 12 hours in the top row Toda2018Science.
• Figure 2: Architecture of the Neural Hamiltonian (NH). First, the discrete CPM input undergoes a pixel-wise one-hot encoding. Then, $L$ iterations of NH layers are applied to extract a deep representation of the system that is equivariant to translations and permutations of cell indices. Finally, the extracted representation is pooled globally over the spatial and cell axes, yielding an invariant global representation of the system which is processed by a multi-layer perceptron to compute the Hamiltonian value.
• Figure 3: Data used in the experiments. Top row: example datapoints of cell sorting type A (leftmost two images) and B (rightmost two images), as illustrated in edelstein2023simplecellsort. Middle row: example datapoints from the Cellular MNIST dataset. Bottom row: example datapoints of Toda2018Science (leftmost two images) and synthetic counterparts (rightmost two images). The synthetic counterparts are used for training, after which we validate the model against the real-world data of Toda2018Science.
• Figure 4: Convergence of the parameters for Type B cell sorting. Dashed lines indicate the true values, solid lines indicate the learned values ($T=T^*$) over the course of training.
• Figure 5: Qualitative results for dynamics simulated by CPMs with varying Hamiltonian models trained on Cellular MNIST data.