Deep Neural Cellular Potts Models
Authors: Koen Minartz
Tim D'Hondt
Leon Hillmann
Jörn Starruß
Lutz Brusch
Vlado Menkovski
Venue: ICML 2025 poster (2025)
Keywords: Cellular Potts Models
Multi-Cell Dynamics
Cell Migration
Energy-Based Models
Simulation
Primary area: applications->chemistry_physics_and_earth_sciences
TL;DR: We introduce Neural cellular Potts models, a method for data-driven simulation of multicellular systems.

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.

Deep Neural Cellular Potts Models
Koen Minartz'
Tim d' d'Hondt'
Leon Hillmann 213
Jörn Starruß
Lutz Brusch
Vlado Menkovski
Abstract
Laboratoon observation
The cellular Potts model (CPM) is a power- ful computational method for simulating collec- tive spatiotemporal dynamics biological cells. To drive the dynamics. CPMs rely on physics- inspired Hamiltonians. However, ciples remain elusive in biology, these Hamilto- nians only approximate the full complexity of real multicellular systems. To address this limi- tation, we propose NeuralCPM, a more expres- sive cellular Potts model that can be trained di- rectly on observational data. At the core of Neu- raICPM lies the Neural Hamiltonian, neural network architecture that respects universal sym- metries in collective cellular dynamics. More- over, this approach enables seamless integration of domain knowledge by combining known bi- ological mechanisms and the expressive Neural Hamiltonian into hybrid model. Our evalua- tion 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.
CPM
first prin-
NeuraiCPM
Figure I. NeuralCPM is more expressive than traditional cellular Potts models, allowing for accurate simulation of real-world collective cell dynamics, e.g. bi-polar self-organization of biologi- cal cells over hours the top row (Toda et al., 2018).
biology, but also needed for the development of medical treatments. As experimental data alone do not reveal the regulatory logic and self-organization principles of com- plex cell-cell interactions, computational models have to be combined with biological experiments (Marée & Hogeweg, 2001; Hester et al., 201 1; Boutillon al., 2022). One of the most powerful and widely-used numerical methods the cellular Potts model (CPM), which captures the stochastic movement and shape of cells, interactions in multicellular systems, and multiscale dynamics (Graner & Glazier, 1992; Balter et al., 2007).
1. Introduction
Cell migration and multicellular self-organizattio are cru- cial biological processes that drive many phenomena of life, such as embryo growth and the spread of cancer (Friedl Gilmour, 2009; Gottheil et al., 2023). Understanding these cellular dynamics is not only a fundamental goal of
CPMs are based on a Hamiltonian or energy function which maps each possible state the discrete state space of multicellular system
scalar (the energy). To model the evolution of the system over time, CPM-based simulator stochastically perturbs the current state towards states with a lower energy, following the principles of statistical mechan- ics. Consequently, the Hamiltonian directly drives the simu- lated dynamics of cells, and designing the Hamiltonian is the core challenge to arrive at realistic CPM simulations. So far, domain experts need engineer failor-made Hamiltonian for each new problem setting. This Hamiltonian generally consists of weighted sum of symbolic, physics-inspired features of the system, but (1) it is labor-intensive to develop and (2) arguably only partially captures the full complexity of cellular systems due to simplifying assumptions in the structure of the Hamiltonian.
'Department of Mathematics and Computer Science, Eind- hoven University of Technology, Eindhoven, the Netherlands Depaarment of Applied Physics and Science Education, Eind- hoven University of Technology, Eindhoven, the Netherlands "Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, the Netherlands "Center for Information Services and High Performance Computing. TUD Dresden Univer- Koen
sity of Technology, Dresden, Germany. Correspondence Minartz <k.minartz@ @ tue.nl>, Lutz Brusch <lutz, brusch @tu- dresden.de>, Vlado Menkovski <ww.enkovski@nn @l
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Deep Neural Cellular Potts Models
In this work, we tackle these two weaknesses by presenting NeuralCPM: method for learning cellular Potts models with expressive neural network-based Hamiltonians, In con- trast to current practice in the field, NeuralCPM enables fitting a Hamiltonian directly on observational data, without requiring any assumptions on the structure of the Hamilto- nian or problem-specific feature engineering. NeuralCPM also facilitates the seamless integration of biological domain knowledge by using the neural network closure term, complementary to an analytical Hamiltonian based on do- main knowledge. This allows us to constrain the learned model to cellular configurations with guaranteed biologi- cal realism (e.g. compact cells of given number even for unseen tasks, as opposed to potential hallucinations of frag- mented or supernumerous cells), while leveraging the neural network find structure in the observed data that is too complex model with analytical Hamiltonian.
unknown normalization constant as follows:
e-Ho(x)
po(x) = Zo
(1)
where Hg X - R defines scalar- energy func- tion (also called Hamiltonian) parameterized by 9. Zg = Jg e e-Ho(x) dr defines the typically intractable normaliza- tion constant (LeCun et al., 2006; Song & Kingma, 2021). As Ze does not need to be computed, Hş(x) can be any nonlinear regression function as long as it could in principle be normalized (Song& & Kingma, 2021), making EBMs highly flexible class of models. If Ho (a:) is a neural network, we call the model a Deep EBM.
Typically, Deep EBMs are fitted to a target distribution p'(x) by minimizing the negative log-likelihood:
C()) = Ex (s) log pe (x))].
(2)
0
Our main contributions are summarized as follows:
for which gradient descent on the de-facto optimiza- tion algorithm. The gradient of L(0) then looks follows
We propose the neural cellular Potts model, CPM in which the Hamiltonian is parameterized with a novel neural network architecture that respects the symme- tries that are universal in cellular dynamies model- ing. We exploit the strong connection between CPMs and deep energy-based models, generative modeling framework developed in the machine learning com- munity, to directly train the Neural Hamiltonian on observational data.
(Hinton, 2002; Song & Kingma, 2021):
VoL(O) = Epr (m) [oHo(r)] - Eps (a)) Vo (3)
where the expectations can be estimated with Monte Carlo sampling. The main challenge lies in sampling x from the intractable distribution po(r) to estimate which is typically achieved by a Markov Chain Monte Carlo (MCMC) algorithm.
We show how known biological mechanisms can straightforwardly be integrated in the NeuralCPM framework by using the Neural Hamiltonian as a clo- sure model. We find that such moology-informed Neural Hamiltonians not only improve biological consistency of the simulations, but also act as a regularizer that effectively stabilizes the training process, which can be notoriously challenging for deep energy-based models.
2.2. Cellular Potts Model
The CPM is stochastic numerical method used for the sim- ulation of individual and collective cell dynamics (Graner & Glazier, 1992: Savill & Hogeweg, 1997; Balter et al., 2007). In the CPM, cells are modeled as discrete entities with explicit two- or dimee-dimeniional shapes. Because of its capabilities in modeling collective cell behavior, stochas- tic cellular dynamics, and multiscale phenomena, the CPM has become one of the most effective frameworks for sim- ulating multicellular dynamics (Rens & Edelstein-Keshet, 2019; Hirashima al., 2017).
We validate the effectiveness of our proposed method
on three experimental scenarios: (1) parameter fitting of a known analytical cellular Potts model (validation of the learning algorithm); (2) fitting Hamiltonians that are difficult or impossible to attain with analytical functions (validation of the increased expressiveness); and (3) fitting a Hamiltonian on real-world biological data (demonstration of an application to real-world problems).
The CPM works as follows: given a lattice L and a set of cells C, I' E c/4 denotes the time-varying state of a multicellular system, where of = c if cell e E C occu- pies the lattice site i E L. The state x' is evolved over time by an MCMC algorithm that has a stationary distribu- tion characterized by a Hamiltonian // O + R. The MCMC dynamics mimic the protruding dynamics of bio- logical cells: random lattice site i is chosen and its state 2; is attempted to alter to the state of a neighboring site j. The proposed state transition I + x' is accepted with probability min{1, e (-m/TT) where AH = H(x') - -x() is the difference in energy between the proposed state x'
2. Background and Related Work
2.1. Energy-Based Models
Energy-barc based models (EBMs) specify a probability distri- bution over a random variable x defined on a space X' up to
2

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Deep Neural Cellular Potts Models
and current state x; and T is the so-called lemperature pa- rameter.
this work, most methods consider cells point masses with- out explicit shape (LaChance et al., 2022; Yang al., 2024). Although some machine learning methods for model- ing CPM-like dynamics of single-cell (Minartz et al., 2022) and multicellular systems (Minartz et al., 2023) have been proposed, these are autoregressive models that require se- quence data covering full trajectories of cellular dynamics for training, which can be costly to acquire. Moreover, these methods are black-box surrogates, and cannot exploit any biological knowledge about the system. In contrast, Neural- CPM requires only observations of self-organized states as training data, akin to Neural Cellular Automata (Mordvint- sev et al., 2020), and relies on the powerful CPM framework to reconstruct the dynamics of cells.
As the Hamiltonian H determines the stationary distribution of the Markov chain, the design of И is the key challenge to achieve realistic simulations of cellular dynamics with the CPM. Generally, contains contact energy and vol- ume constraint terms, as originally proposed in (Graner & Glazier, 1992), and additional application compo- nents:
H(x) = Σ (LEN(L)
J (21, (1 )0,
contact energy
Σ EEC
((V(c)- H caase-spocieee (x).
3. Neural Cellular Potts Models
volume constraint
3.1. Neural Hamiltonian Architecture
In Equation 4, (L) denotes the set of all pairs of neighbor- ing lattice sites in I, my) is a contact energy defining adhesion strength between cells H, at site i and x, at site i. and 52. is the Kronecker delta. Furthermore, V(c) is the volume of cell c, (c( is c's target volume, and A is Lagrange multiplier. Since the first introduction of the CPM, many extensions for Hasse-specific have been pro- posed for varying biological applications, taking into ac- count external forcing, active non-equilibrium processes like chemotaxis, and many other biological concepts (Hi- rashima et al., 2017). However, as opposed to physical systems like evolving foams, where the Hamiltonian can be derived from first principles and which have been modeled with the CPM (Graner et al., 2000). an equivalent of first principles remains elusive for living systems. Therefore, the task of designing a suitable Hamiltonian requires significant domain expertise and needs to be repeated for each new case. Moreover, even well-designed Hamiltonians will only partially account for the observed cell dynamics.
Our goal is to learn Neural Hamiltonian He that parame- terizes a CPM of which the stochastic dynamics align with the complex distribution over real cellular behavior. A chal- lenge in modeling physical systems is that they often admit multiple equivalent representations, a property formalized through mathematical symmetries. Incorporating these sym- metries in neural network architectures can greatly enhance model performance (Bronstein et al., 2021). In the context of CPMs, symmetries arise in Hamiltonians that are invari- and to permutations of the set of cells C' and translations of the lattice L. As such, for any transformation that per- mutes C and translates I., the Neural Hamiltonian should satisfy Hx(x)) = Ho(gz).
Generally, GNNs (Gilmer et al., 2017; Kipf & Weelling 2017), Deep Set models (Zaheer et al., 2017). transform- ers (Vaswani et al., 2017) would be the designated building blocks for architectures that respect permutation symme- try. However, in the CPM context, such architectures do not apply, as they operate on node features represented by vectors, whereas in our case x represents regular lattice. Instead, we propose a Neural Hamiltonian architecture that is invariant to both translations and permutations, illustrated in Figure 2. The global structure of this architecture is as follows: first, x is one-hot encoded, such that for each cell e E C. we now have a separate grid Ag representing that cell. (h2); equals
2.3. Neural Networks for Cellular Dynamies Simulation
Neural networks have gained traction as simulation models of complex dynamical systems. The primary objective of most works in this field has been to improve computational efficiency over physics-based numerical simulators (Aziz- zadenesheli et al., 2024; Li et al., ; Kochkov et al., 2021; Gupta & Brandstetter, 2023). However, models of multicellular systems including vertex-based models, phase- field models, and the CPM generally only partially explain the dynamics observed in laboratory experiments (Alert & Trepat, 2020; Brückner & Broedersz, 2024). Here the value of neural simulators lies primarily in improved accuracy and new discoveries, rather than accelerated simulation.
e occupies lattice site i, and O otherwise. [C] Then, we iteratively process the embeddings (Ne) c=0 with the l'th NH layer to produce deep equivariant represen- tation of the system. Each NH layer processes their input cell embeddings by passing each cell's embedding indepen- dently through a neural network の¹ d''s outputs h. are then aggregated using a permutationniivariin our case summation, to yield global context lattice A of the entire system. Finally, all hy are processed in tandem with A by the cell-interaction network vil, after which local
function D. in
Still, little research has been done in machine learning- driven modeling of cellular dynamics, and as opposed to
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Deep Neural Cellular Potts Models
Global pooling over space and cells
layer
NH layer L
MLP
Layer
Me & (ht)
A D h/6
0, v (hL A)
Pool2D
Cell-wise CNN
Pemutttion- invariant aggregation
Optional local
Cell-interaction CNN
spatial pooling
Figure Architecture of Neural Hamiltonian (NH). First, the discrete CPM input undergoes pixel-wise one-hot encoding. Then, iterations of NH layers applied extract deep representation of the system that is equivariant 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 multi-layer perceptron compute the Hamiltonian value.
spatial pooling, for example max-pooling, can be applied compress the representation of the system. To respect the translation symmetry of the problem and promote local- ized pattern recognition, we parameterize şl and val with convolutional neural networks (CNNs). After processing by the NH layers, we pool the embeddings to obtain an invari- ant representation before processing them with a shallow multi-layer perceptron compute the Hamiltonian Ho (x)).
accounted for by Has (1:). this work, we consider the volume constraint and the contact energies of Equation 4 as symbolic components; depending the available knowl- edge of the biological mechanisms in the system at hand, more sophisticated expressions can be included.
3.2. Training
Our training strategy is minimize the negative log- likelihood objective (Equation 2) using gradient descent (Equation 3). Given a dataset D of observed cellular sys- tems, we estimate the expectation over p" (x) in Equation with a batch of datapoints {a, }B /b=1 sampled uniformly from D. The expectation over pe (a) in Equation is esti- mated with a batch of samples {-, /b=1 obtained using B independent MCMC chains. Inspired by (Du & Mordatch, 2019), we also add regularization term weighted by small scalar À to the objective to regularize energy magnitudes of Ho(x) and improve numerical stability. This gives the loss estimate:
Biology-infoo -ndlome- Hamiltonians. A key advantage of NeuralCPM is that it closely follows the cellular Potts mod- cling paradigm, which enables us to seamlessly integrate biological domain knowledge. Specifically, even though symbolic Hamiltonians are approximations, they may still account for partially known mechanisms underlying the observed dynamics. In this case, wish to exploit this domain knowledge to expedite the model fitting task and to yield more interpretable model. We achieve this by using the Neural Hamiltonian as a closure model on top of an interpretable symbolic Hamiltonian:
B Σ Holat) Ho(x5)
He ws (x) WNN HONN (x),
(5)
((())
where Hgs (x) is the symbolic component HaNs (x) is the Neural Hamiltonian component, and ws and WNN are learned weights balance the contributions of both com- ponents. If the parameters @ of the symbolic component are not known, Ho(x) can be trained end-to-end as long as Hgs (x) is differentiable with respect to Ø8h Through its biology-informed structure, Hos (z) expresses a strong prior on the overall Hamiltonian Hg. while HANN (x) is re- sponsible for expressing higher-order terms that cannot
+ + +Hoo(at)) (2 (6)
The computational complexity of training is dominated by running the MCMC sampler obtain samples from po (z). Each MCMC step requires forward pass of Ha(x) and typ- ically many thousands of MCMC steps are required before the chain has converged to an equilibrium sample of pa (z). Therefore, we introduce an approximate sampler that ac- celerates the original CPM sampler by performing multiple

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Deep Neural Cellular Potts Models
spin-flip attempts in parallel. In principle, this could lead to synchronization artifacts parallel spin-flips might affect each other (Sultan al., 2023); however, we stress that this approach should be regarded as a surrogate sampling proce- dure primarily intended to accelerate training, analogous to how well-established EBM training techniques also success- fully trade-off unbiased sampling for faster training (Tiele- man, 2008). More details about the training algorithm, the
Cell sorting
Type A
Type B
Type D
Type F
Callular MNIST
approximate sampler and implementation can be found in Appendix B.
Open structures
Enclosed structures
Bipelar sorting
4. Experiments
4.1. Experiment Setup
Lab observations
Synthetic training data
Objectives and scope. Our experiments aim to: (I) vali- date the effectiveness of the NeuralCPM training algorithm for fitting Hamiltonians data, and (2) evaluate the ca- pability of using Neural Hamiltonian to model cellular dynamics and self-organization in synthetic and real-world scenarios. To the best of our knowledge, no prior meth- ods have been developed for learning an energy function to model cellular dynamics. Hence, we compare our approach to alternative neural network architectures for the Hamilto- nian well as analytical CPMs. Details on experiments and datasets are in Appendix A and in our code.'
Figure 3. Data used in the experiments. Top row: example data- points of different cell sorting regimes, as illustrated Edelstein- Keshet & Xiao (2023). Middle row: example datapoints from the Cellular MNIST dataset. Bottom row: example datapoints of Toda et al. (2018) (leftmost two images) and synthetic terparts (rightmost two images). The synthetic counterparts are used for training, after which validate the model against the real-world data of Toda et al. (2018).
Experimental scenarios and datasets. To address objec- tive (1), we generate synthetic datasets using the cell-sorting Hamiltonian of Equation 4 (Graner & Glazier, 1992), and measure how well the training algorithm can learn the pa- rameters in this analytical function from data. Different parameterizations lead to different dynamics, and we gen- erate data following the type A, B, D and F regimes as illustrated in (Edelstein-Keshel & Xiao, 2023) and Figure 3.
axial organization. This behavior is surprising because the cells of two different types, expressing (after induction) different type-specific P-cadherin or N-cadherin adhesion molecules, were expected to sort into a concentric or uni- polar configuration (Graner Glazier, 1992).
As Toda al. (2018) performed only six repetitions of this experiment, construct synthetic counterparts of the final configurations for training using Morpheus (Starruß et al., 2014). Specifically, we ensure that cells from the 'red' type move into a central band from random initial conditions, and that the other cells move towards two opposing clusters along the horizontal axis. To achieve this, we prescribe a preferred target coordinate for each cell independently by adding an additional term to the Hamiltonian that penalizes distance from the target coordinate. Additionally, to make sure the training data are isotropic. we randomly rotate the generated configuration. In contrast to this synthetic data generation procedure, biological cells (and NeuralCPM) do not rely on any such preseribed target locations, but update cell positions result of cell-cell interactions in the current configuration. Of course, such synthetic training data approach should be taken with care. Consequently, we validate the cellular dynamics predicted by the trained NeuralCPM model against the real biological dynamics reported in the Supplemental Figure S6B of Toda et al. (2018) to assess model quality.
To address objective (2), we consider two experimental sce- narios. For the first scenario, we introduce the Cellular MNIST dataset: synthetic dataset in which cells form digit- like structures, also illustrated in Figure 3. The goal of this scenario is not to model cellular structures that are biologi- cally relevant, but rather to investigate well-controlled case that is too complex to model with an analytical Hamiltonian and simultaneously allows for relatively straightforward visual verification of simulation quality.
The second scenario for objective (2) concerns biological experiment by Toda et al. (2018). A hallmark during embryo development is the self-organization of the principal body axis from an unstructured group of cells, which can be recapitulated with in-vitro experiments and quantified using tame-lapse microscopy (Toda et al., 2018). Here, we choose the observation of bipolar axis formation cell aggregates as shown in Figure 1, and we refer to this scenario as bipolar
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Deep Neural Cellular Potts Models
A
(0.
]01, ]02, 2)
Hamiloonin
10 1)
J0.
Calisort
Parmameter value
Collarr Hamiltonian *Exteraal Potental
3
CNN
100
Training iteration (x100)
1 NHH layer +CNN
Figure 4. Convergence of the parameters for Type B cell sort- ing. Dashed lines indicate the true values, solid lines indicate the learned values (T = T") over the course of training.
Neural Hamiltonian
Table 1. Error of fitted coefficients for different parameterizations
the cell sorting Hamiltonian.
Noural Hamiltonian +closure
Cell sorting regime
A
B
D
F
(Edelstein Keshet& & Xiao, 2023)
1ogı0 (RMSE) T
-1.26
-0.01
-1.08
-0.99
logio (RMSE) T = 1
(RMSE
-1.67
-0.75
-1.18
-1.20
200
400
600
800
(MCS)
4.2. Fitting Analytical Hamiltonians
Figure 5. Qualitative results for dynamics simulated by CPMs with varying Hamiltonian models trained on Cellular MNIST data.
Metrics and baselines. To assess how well our learning algorithm can fit analytical Hamiltonians to data, we mea- sure the Root Mean Squared Error (RMSE) of the learned parameters of the Hamiltonian. Since the temperature pa- rameter is mainly related to fluctuations in the system over time and so much the statistical properties of the system at equilibrium, it is poorly identifiable from static snapshots. Hence, we report the RMSE for both tempera- tures T' = 1 and 7' = T", where T* is the temperature that minimizes the RMSE between the learned coefficients and the ground truth.
Punfragmented- the proportion of states in which there are at most three fragmented cells. We allow for three fragmented cells as CPMs allow for small temporary fragmentations. At the collective scale, our goal to assess whether cells successfully organize into digit-like structures. Following the Inception Score (Salimans al., 2016), metric used to quantify the quality of generative models for natural im- ages, we calculate Classifier Score (CS) using classifier Po (y((z) that we trained on the cellular MNIST dataset:
CS = exp (E (Ex)po()) [KL),
Results. As shown in Table 1, the final learned parameters approximate the ground-truth very well. In addition, as can be seen in Figure 4, the parameters converge rapidly to the true values, highlighting the efficiency of the learning algo- rithm. Additional results illustrating the learning process and the robustness of the parameter fitting can be found in Appendix C.I.
KL = DK (po (e)()l [Po())2')
Although CS is calculated in the same way the estab- lished Inception Score, the classifier Pe is not well- validated as the Inception network. Consequently, the metric should be interpreted as rough indication of quality com- plementary to visual inspection, where high CS indicates distinct and diverse cellular structures.
4.3. Cellular MNIST
As basclines, we compare against two analytical models as well neural network-based Hamiltonians. The analytical models the prototypical cell sorting Hamiltonian (Graner & Glazier, 1992) (Fquation 4) and the cell sorting Hamil- tonian plus a learnable external potential. The goal of the neural network baselines to evaluate the design choices in the Neural Hamiltonian. To this end, we consider (1) a Convolutional Neural Network; (2) a 1-layer Neural Hamil-
Metrics and baselines. To assess the simulated dynamics, consider both the cell and collective perspectives. From an individual cell point of view, we require cells to have realistic volume and to be contiguous. Consequently, we measure Pvolume the proportion of states in which all cells exceed the lowest and highest measured cell volumes in the training data by at most 10% of the mean cell volume, and
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Deep Neural Cellular Potts Models
Toda pt al..
(real datal
Table Results Cellular MNIST data. Prolume and Puntragmenied assess the validity of the dynamics at the cell level from biological perspective, while CS assesses to what extent cells successfully assemble in distinct digit-like structures.
2018
Callsort Hamiltonian
Model
Pvalume i Punfragmented 1 CS 1
Cellsort Hamiltonian
1.00
1.00
2,47
Hamiltonian
Cellsort External Potential
1.00
1.00
3.11
Cellsort
*Extemal Potental
CNN
0.00
0.05
3.70
1 NH layer + CNN
0.06
0.87
3.53
Neural Hamiltonian
0.11
0,99
4.91
Neural Hamiltonian + closure
1.00
1.00
4.35
CNN
tonian, followed by invariant pooling over representations of cells and a CNN; (3) the vanilla Neural Hamiltonian illustrated in Figure 2; and (4) Neural Hamiltonian closure on top of a cell sorting Hamiltonian with learnable parameters. Models (1) and (2) serve to investigate the im- portance of deep equivariant embeddings over architectures relying on representations without permutation symmetry (1) or on invariant representations (2), while comparing (3) and (4) gives insight on the relevance of including biological knowledge in the Hamiltonian. Details on the model archi-
1 NH layer +CWN
Neural Hamiltonian ian
Neural Homilloonia tan +closure
tectures can be found in Appendix B.2. Additional results concerning further Hamiltonian baselines and ablations can be found in Appendix C.2.
0.0
3.1
6.2
9.3
12.4
Time (hours)
Figure 6. Biological dynamics observed in Toda et al. (2018), and qualitative results for dynamics simulated by CPMs with varying Hamiltonian models trained bipolar axial organization data.
Results. Figure 5 shows qualitative results of simulated trajectories starting from mixed cell cluster configuration; more visualizations can be found in Appendix C.3. As expected, the analytical models are not able to capture com- plex non-linear relationships, reflected in the failure of cells to form a digit-like structure. The CNN Hamiltonian pro- duces dynamics that are clearly unrealistic, because it lacks the inductive bias of permutation symmetry. In contrast, the architectures based on Neural Hamiltonians respect the symmetries of the system and lead to cells organizing in digit-like structures.
4.4. Bipolar Axial Organization
Metrics and baselines. As in Section 4.3, we evaluate simulations for biological consistency and collective behav- ior. For the biological metrics, we use the same indicators Pvetune and Punfragmented- For the collective dynamics, we quantify the bipolar axial organization as follows: first, we rotate the image along the principal axis of the polar cell clusters (green in Figure 3). We then measure the variance of the spatial configuration of each cell type along this axis as well as along the orthogonal axis. We consider the same baselines as in Section 4.3, and provide additional results in Appendix C.2.
In line with these qualitative results, Table 2 shows that the analytical models excel in the biological metrics Pvalume and Punfragmented due to the biology-informed design of their Hamiltonians, but achieve low CS values as they are not expressive enough to model digit-like structures. From the Neural Hamiltonian models, NH achieves the highest CS score, which substantially higher than the NH layer + CNN model, stressing the relevance of deep equivariant representations. However, these models are subject to un- satisfactory biological metries. In contrast, using the NH as closure term yields high scores on the biological and CS metrics, as it enjoys the strong biological structure of the analytical component to constrain the dynamics to be biologically realistic, while using the more expressive NH architecture to guide cells towards digit-like formations.
Results. Figure shows microscopy time-lapse by Toda et al. (2018), well as CPM-simulated trajectories for dif- ferent Hamiltonians; more simulated trajectories can be found in Appendix C.3. As in Section 4.3, the analyti- cal baselines fail to capture bipolar structures due to their insufficient expressiveness, and the CNN-Hamiltonia pro- duces distorted dynamics. However, in this case the Neural Hamiltonian based models without closure term fail pro-
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Deep Neural Cellular Potts Models
type 1 NeuraICPM
type 2 NeuralCPM
type 1 real data (Toda 2018)
type 2 real data (Toda 2018)
Variance fraction along polar axis
Toda et al., 2018 (real data) Synthetic configurations
Variance fraction along polar axis
NeuraICPM (simulated)
0.6
0.4
0.4
0.2
0.2
0.0
100
200
300
400
500
600
Type
Type 2
Time (minutes)
(a) Bipolar cellular organization at equilibrium.
(h) Development of hipolar cellular organization over time.
Figure 7. Bipolar cellular organization as quantified through the fraction of variance along the polar axis for each cell type. A high value indicates strong alignment with the bipolar axis, which is expected for type 2 cells in equilibrium, whereas low value indicates alignment in the orthogonal direction. Almost all observations of Toda et al. (2018) lic within standard deviation of the mean of the simulations (error bars (a) and shaded in (b)), indicating that NeuralCPM reproduces the observed bipolar organization behavior.
data do not contain as much of the noise that is inherent to real observations, which is expressed the slightly more or- ganized configurations. Our NeuralCPM simulations yield self-organization patterns that largely overlap with the obser- vations of Toda et al. (2018). Moreover, Figure 7(b) shows that the self-organizing dynamics in our simulations align with the time-lapse of Toda et al. (2018). Crucially, while the cells in the synthetic training data could only be prepared in a predefined final bipolar configuration, NeuralCPM suc- cessfully predicts the temporal dynamics and spontaneous symmetry breaking observed in the experiments by Toda et al. (2018). A Neural Hamiltonian trained on multicellular structures paired with the well-established CPM dynamics can therefore not only be used to study equilibrium configu- rations, but also elucidates the dynamic pathways of cellular self-assembly towards such states.
Table Results on bipolar axial organization for CPMs with dif- ferent Hamiltonians. We the same biological consistency indi-
cators Prolume and Panfragmented as in Table 2, as well as the RMSE of the variance along the polar and orthogonal axes of the two coll types to quantify how well bipolar axial organization is captured.
Axial
Model
Pvolume 1 Punfingmented
alignment RMSE
Cellsort Hamiltonian
1.00
1.00
147.2
Cellsort External Potential
1.00
1.00
154.4
CNN
0.00
0.07
153.5
1 NH layer CNN
0.00
0.11
329.1
Neural Hamiltonian
0.00
0.17
254.2
Neural Hamiltonian closure
0.77
1.00
37.3
duce reasonable dynamics, due to fast divergence of these models during training, a common issue of EBMs. In our Cellular MNIST experiment, we observed a similar ten- dency, but we were able to mitigate divergence by careful hyperparameter tuning, which we were not able to achieve for the bipolar axial sorting. In contrast, the NH+closure model trained stably out of the box, with minimal adapta- tions compared to the Cellular MNIST design. As such, we empirically observe that the biologically informed analyt- ical term in NH+closure not only improves the biological realism of the simulations, as evident from Table 3, but also acts as an effective regularizer that stabilizes training.
5. Conclusion
This work introduced NeuralCPM, method for simulating cellular dynamics with neural networks. Whereas the cur- rent practice in CPM research is for domain experts to hand- craft an approximate symbolic Hamiltonian for each appli- cation, NeuralCPM parameterizes the Hamiltonian with an expressive neural network. These Neural Hamiltonians can model more complex dynamics than analytical Hamiltoni- ans and can be trained directly observational data. Our results demonstrated that incorporating the symmetries of multicellular systems is crucial to train an effective Neural Hamiltonian. Moreover, we found that using the Neural Hamiltonian as a closure term on top of a biology-informed symbolic model improves biological realism and training stability. Finally, case study on real-world complex self- organizing dynamics showed that NeuralCPM's simulations successfully predict laboratory observations. We conclude
To compare the NH + closure model with the laboratory the six recorded
observations of Toda al. (2018), self-organized states as well as time-lapse of the self- organizing process, which the authors kindly shared with us. Figure 7(a) shows the degree of bipolar organization in the observations, our synthetic training data, and the Neural- CPM simulations. Due to their synthetic nature, our training
8

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Deep Neural Cellular Potts Models
that NeuralCPM can effectively model collective cell dy- namics, enabling the study of more complex cell behavior through computer simulations.
Impact Statement
Modeling multicellular dynamics is crucial for biology, and accurate computer simulations can contribute to accelerated research on wide range of biological phenomena. How- ever, it remains imperative validate the simulation results against experiments, especially when relying on neural sim- ulation models. We do not foresee any potential negative societal consequences other than those associated with gen- eral research in machine learning and cell biology.
Limitations and future work. As the aim of this work was to introduce and validate the core concepts of Neural- CPM, our evaluation has focused on systems with up to 100 cells and with limited direct biological applications. To apply NeuralCPM to large-scale biological tissues, for example in cancer research, we identify three directions for future work. The first is accelerating the NeuralCPM metropolis kinetics, which pose computational challenges for large systems. We hypothesize that the use of efficient sampling techniques for discrete EBMs may alleviate this challenge (Grathwohl et al., 2021; Zhang et al., 2022; Sun et al., 2023). The second limitation concerns the global receptive field of the Neural Hamiltonian architecture. For the applications we considered, global receptive field is biologically plausible, but this is not the case for tissue- scale simulations. To this end, we need to develop a Neural Hamiltonian in which each cell can only sense its imme- diate surroundings. The third limitation is the assumption of dynamics towards an equilibrium distribution. While this assumption is well-motivated in scenarios like embryo development, other applications concern actively migrat- ing cells, leading to non-Markovian dynamics. A condi- tional Hamiltonian that also depends on the system's 's history can address this limitation. In addition, another promising research direction to use NeuralCPM to discover bio- logical mechanisms, for example by using explainable AI techniques for neural network models of dynamical sys- tems (Cranmer et al., 2020; Brunton et al., 2016). Finally, to foster adoption of NeuralCPM by biologists and integrate with other phenomena, e.g. cell division, NeuralCPM can be incorporated into widely-used software packages for CPM simulation (Starruß al., 2014; Swat et al., 2012).
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We thank Satoshi Toda for sharing the microscopy data on bipolar axial self-organization. We are also grateful to the organizers of the workshop 'Simulating tissue dynam- ics with cellular Potts models' for facilitating the meeting and discussions that led to this project. We furthermore thank Quirine J. S. Braat from Eindhoven University of Technology for her comments on the cellular Potts related sections of the manuscript. K.M. acknowledges that this work used the Dutch national e-infrastructure with the sup- port of the SURF Cooperative using grant no. EINF-7724. L.B. acknowledges support by the German BMBF (grants 031L0293D, 03 0311503)
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Deep Neural Cellular Potts Models
A. Data Generation
Cell sorting Training data for fitting an analytical cell sorting Hamiltonian was generated by sampling states with the energy function given equation (4), where in this case case specific Inspired by the original proposal of the CPM by Graner Glazier (1992), we consider cells of two distinct cell types that perform cell sorting due differential adhesion. We distinguish the scenarios b, d, and f from Edelstein-Kesher & Xiao (2023), characterized by different contact energies between cells, which are laid out in Tables 4 to 7. The Lagrange multiplier was set to xv = 0.1 for scenario a, xv = 0.5 for scenario b, λy = 0.1 for scenario d, and xv = 0.05 for scenario f. All sets of simulations were performed with target volume of V = 60 and temperature T = 1 on a 100 x 100 grid with 50 cells, 25 of each type, which were initialized as single pixels randomly scattered within a centered circle with a radius of 25 lattice sites. The resulting datasets comprised 128 independent full lattice snapshots each.
Table 4. Contact energies for scenario
Table 5. Contact energies for scenario
Medium
0.0
Medium 0.0
Type I
0.5
0.333333
Type 2.5
1.0
Type 2
0.5
0.2
0.266667
Type 1.0
4.5
1.0
Medium
Type 1
Type 2
Medium
Type 1
Type
Table 6. Contact energies for scenario d
Table 7. Contact energies for scenario
Medium
0.0
Medium
0.0
Type
15
4.0
Type
2.0
7.0
Type
7.5
7.5
4.0
Type 8.0
5.5
3.0
Medium
Type
Type
Medium
Type
Type
Cellular MNIST Ground truth data for the synthetic structural assembly experiment in section 4.3 was generated in similar way to the cell sorting data described above. Notably, Hasse-specitie (z) now took the form of an external potential
H Hease-spocific (x) = => p(zz)o
(8)
iEL
where u(xs)) can be considered the coupling strength to the potential d.. The coupling u(z;;) was then defined so that only cells of type 2 couple to Φ; with constant strength Jī 10, while Φ; was chosen such that cells of type 2 favor arrangements that mimic handwritten digits from the MNIST data set (Deng, 2012). To that end, the MNIST images were binarized by applying threshold at half brightness. Subsequently, a Euclidean distance transform was performed. Finally, the image was scaled with cubic interpolation from the input resolution 28 x 28 to the chosen domain size of 100 100 pixels. The result was used as di in the Hamiltonian. The distance transform yields sloped potential which pushes the type cells into the shape of the desired digit. Differential adhesion similar to the cell sorting case was imposed to better separate cells of different types; see table 8. In addition to these parameters, we set Xv RJ 0.974, V 100, and T and initialized the system with the same procedure as above. The final data set contained 1280 samples.
Table 8 Contact energies for cellular MNIST
Medium
0.0
Type
6.0
3.0
Type
6.0
6.0
3.0
Medium
Type 1
Type 2
Bipolar axial organization The experimental cell aggregates of Toda et al. (2018) consist of 200 to 240 cells in 3D, which amounts to about 8 cells along a diameter and about 40 cells in the cross-section. To generate the synthetic training data, we therefore consider 40 interacting cells, 20 of each type. We use Morpheus (Starruß al., 2014) to randomly initialize a
12

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Deep Neural Cellular Potts Models
cluster of these 40 cells, after which we assign each cell of type preferred motion in the direction of one of the two poles. This results in artificially creating configurations where each cell of type 2 has clustered together in the pole it was assigned to move to. In addition, the standard cell sorting Hamiltonian applied, with V*(c) 150, À 1, and contact energies as shown in Table 9. In addition, we set Morpheus' temperature parameter for this experiment to T = 2.0. Using this procedure, we generated 1000 samples, which we randomly rotate for training.
Table 9, Contact energies for synthetic bipolar axial organization
Medium
0.0
Type 16.0
6.0
Type
16.0
16.0
6.0
Medium
Type 1
Type
B. Implementation Details
Our implementation is built JAX (Bradbury al., 2018) and Equinox (Kidger & Garcia, 2021).
B.1. Training and Sampling
Training loop A pseudocode description of the training loop is given in Algorithm 1. We initialize the MCMC chains from datapoints. We persistent chains instead of reinitializing in each iteration, following the Persistent Contrastive Divergence algorithm from (Tieleman, 2008), but with the regularization term from (Du & Mordatch, 2019). While using an approximation of the gradient of the max likelihood objective, this approach reduces the amount of compute per optimization step since less MCMC steps and thus less forward passes of Hz( have to be performed. Note that the autodifferentiation step does not backpropagate through the sampling chain. Common hyperparameters are given Table 10. We used the Adam
Algorithm NeuralCPMM training procedure
dataset D (I,,, 1.N sampler reset probabilityy p Initialize K sampling chains {2g / b= Initialize model while not converged do
Input:
iiid
p" (xx), learning rate 7), number of sampling steps K, number of parallel flips P.
B
~ D
{ag 1 ~
For each 25 with probability p reinitialize of
(w, b= ApproxPCPM(Ho, K, (25 1mm)
g - autodiff( (eq. 6)
6- Adam(1,, 0.
end while return 0
optimizer in all experiments with learning rate 1) 1 and standard hyperparameters 31 0.9, (2 0.999, le 8. The number of sampling steps per model update K is determined in units of 'Monte Carlo Steps', i.e. the total size of the lattice| [L] (aka the grid size), as is common in the CPM literature. Thus, K = |L| * Monte Carlo steps. Each training run was terminated after 24 hours if it was not yet finished. models diverged during training or started showing clear artifacts, we manually found the last model state that was not problematic and used this model state for the experiments. Finally, for the Cellular MNIST and bipolar axial organization experiments, we additionally assigned random types to each cell when the state of the system was reinitialized (second line within the while loop of Algorithm 1) ensure broader coverage of the state-space during training.
Approximate sampler Estimating the loss function using an MCMC sampler that mixes quickly is imperative for scalable training, since each MCMC step requires forward pass of Hş(r) in the Metropolis-Hastingg (MH) correction step. The original CPM sampler uses a proposal distribution that perturbs only one lattice site E L at a time, resulting in a slow mixing rate and thus a very expensive training loop (Graner & Glazier, 1992). We instead use approximate parallelized CPM dynamics with proposal distribution that samples multiple sites and changes their cell state in parallel.
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Deep Neural Cellular Potts Models
Table 10. Training hyperparameters used in results of their respective section
Hyperparameter
4.2
4.3
4.4
Batch size B
16
Max. num training steps
lc4
1c4
le4
Monte Carlo steps
1.0
0.5
0.7
Lattice size
200x200
100x 100
149x149
EWA a
0.0
0.99
0.99
Regularizer
0.0
0.0005
0.0005
Num parallel flips P
100
50
Sampler reset probability Preset
100%
2.5%
2.5%
Randomize cell types after reset
No
Yes
Yes
Algorithm 2 ApproxPCPM
Input: Hamiltonian He, number of sampling steps т, number of parallel flips P, initial state x.
for t in to do
x' + x--1
B - L1E LEjEENNO of x
S 111pp1 ~ B²
for lattice site E S* in parallel do M(i) = (j E (i())a x3}
2 =))(())
+ Ho(x')- - Ho(x"11)
Pt min(1,e
u U(0,1) 1)
2 + of if ui S Pi
end for end for Return nT'
Pseudocode for the approximate CPM dynamics is given in Algorithm 2. Let (() define the set of all neighboring lattice sites lattice site The sampler uses a proposal distribution where first, P lattice sites are independently and uniformly sampled from the boundary of cells. This boundary subset contains all / that have a neighboring lattice site that belongs to a different cell than the cell at the lattice site in question. Such structure in the proposal causes state updates where the system (and system energy) actually changes, increasing the convergence speed of the sampler towards more meaningful configurations. After sampling set of lattice sites S from the boundary B, we sample for each site E S lattice site that is mapped to a different cell in C, a set we denote M(i) = (j E N())aa 4 x Then, copy that neighbouring cell into the originally sampled lattice site and perform an MH correction step for each site in S in parallel. That is, we perform MH correction steps in parallel on states where only one lattice site has been changed. Then, we keep all the proposed changes that were accepted and combine them together into the next system state.
The resulting transition probabilities do not satisfy detailed balance because we essentially use faulty MH correction step, and thus we cannot guarantee that the system has stationary distribution defined by Hz(x). However, the standard CPM also does not satisfy detailed balance (Durand & Guesnet, 2016), and we found in preliminary experiments that this custom approximate sampler achieved speedups over even state-of-the-art discrete MCMC samplers such as (Grathwohl et al., 2021; Zhang et al., 2022; Sun et al., 2023). We believe this is because the custom proposal is able to leverage structure (the boundary constraint) in its proposal that is specific to the cellular dynamics problem, and that the gradient approximations of these discrete systems as used neural Hamiltonian H (z)
ofee---f-therrr samplers were rather poor, likely due to the very unsmooth nature of the
14

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Deep Neural Cellular Potts Models
B.2. Model Details and Hyperparameters
Here, we discuss details on the (hyper)parameters of the various Hamiltonian models used in our experiments.
B.2.1. ANALYTICAL. HAMILTONIANS
The cell sorting Hamiltonian is exactly as described in Equation 4, where the parameters [(1,, e2) and À are learnable; we set (c) to the average volume of all cells observed in the training data for all E C. The cell sorting model with external potential the same as the cell sorting Hamiltonian, with the addition of an external potential as described in Equation 8. However, rather than setting the potential up-front as done the data generation (Appendix A), we learn through stochastic gradient descent.
B.2.2. NEURAL HAMILTONIANS
Initial embedding layer. We first compress the sparse one-hot encoded representation to dense representation by applying a learned downsampling through a single linear strided convolutional layer that operates each cell independently. Both the stride and kernel size of this layer are 3 x 3 and x 5 for Cellular MNIST and bipolar axial sorting respectively.
NH layers. For simplicity, choose fixed architecture for d' and eld throughout this work: both are two repetitions of (Conv2D - J). where Conv2D is convolution with kernel size of 3 and is the SiLU activation function (Elfwing et al., 2018). We use summation as permutation-in invarraan aggregation function θ The hidden dimension (amount of channels) per cell for each NH layer increases progressively with the depth of the Neural Hamiltonian model, where the first Conv2D layer in both v and ø' maps the input to an output with out channels s equal to this hidden dimension, and where subsequent Conv2D layers preserve the amount channels within each NH layer:
(he E in channelaxhxxr + 6' + [h E Rour chanrelsxhxuy
(9)
[h] RRi. chamellaxxx A E Rout chaanelsxhxu) + v/l for Rout channelsxhxup
(10)
Additionally, we use a residual connection (He et al., 2016) for each NH layer, connecting its input 12 with its output oc before max-pooling.
The specific hidden dimensions and pooling design for the Neural Hamiltonians is then follows:
Cellular MNIST: 4 NH layers
Hidden dimensions: [8, 16, 32, 32]
Max-pooling downsampling rates: [3, 2, I, 1]
bipolar axial sorting: 6 NH layers
Hidden dimensions: [16, 32, 32, 64, 64, 64]
- Max-pooling downsampling rates: [2. 1, 2, 1, 2, 1]
We then apply a linear convolution with 32 output channels, before pooling over all cells and pixels using summation get vector representation of the system that is invariant to both permutations and translations. This is then processed by an MLP with two hidden layers with SiLU nonlinearity and 32 neurons each before mapping to a scalar output with single linear layer. Similar to the NH layers, the MLP layers model the residual with respect to the input.
For the I-layer Neural Hamiltonian baseline, only the first of these NH layer is applied before pooling over the representation of all cells using pixel-wise summation.
NH-based baseline models. The Neural Hamiltoiia++clouure model uses the neural network to model an additive term on top of the cell sorting Hamiltonian (Equation 4), where we take the same approach as in Section B.2.1 for the analytical component.
15

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Deep Neural Cellular Potts Models
The 1-layer NH CNN model uses the first layer of the respective Neural Hamiltonians as described earlier in this section, before pooling over all cells but not over all pixels. This yields grid representation of the system that is invariant to permutations, which is subsequently processed by CNN architecture consisting of blocks of two {Conv2D - a repetitions, and a residual connection between the imput and output of each block. The first layer of the convolution block maps the input to the specified hidden dimension, which remains the same for the second layer. We again apply max-pooling to the output of each block to get compressed representation the system. The specific designs are as follows:
Cellular MNIST: 3 convolution blocks
Hidden dimensions: [32, 64, 128]
Max Max-pooling downsampling rates: [1, 2, 2]
bipolar axial sorting: convolution blocks
Hidden dimensions: [32, 64, 64, 128]
Max-pooling downsampling rates: [1, 2, 1, 2]
The output then processed the exact same way as the output of the NH layers described in the paragraph above, with the exception that we already pooled over all cells before the CNN, and thus only aggregate over all pixels.
C. Additional Results
C.1. Fitting Analytical Hamiltonians
The convergence plot of the parameters for type A, type D, and type F cell sorting (analogous to Figure 4, see Edelstein- Keshet & Xiao (2023)) can be found in Figure As in Figure 4, the parameters converge rapidly to their target values over the course of training, demonstrating robustness of the training process over different parameterizations the Hamiltonian. Additionally, Table 11 shows the mean and standard deviation of the fitted parameter values calculated over five independent initializations of the model parameters. The results show that the parameters robustly converge to value close the target, regardless of their initialization.
,00
80, X0
J0L.2 2)
No.
211.2)
JtO,11 1)
22. 21
Jco, 11
/00, 11
1).
(12
Parmametos tor value
Pa Parmameert memameetrr value
Pa raamaa mermmeterr value
0.6
8
10
0.4
20
00
00
100
20
10
00
100
20
00
00
100
Training iteration (x100)
Training ileration (x100)
Training ileration (x100)
Figure 8. Convergence of the parameters for cell sorting type A (left), type D (center), and type F (right). Dashed lines indicate the true values, solid lines indicate the learned values (I' = 7T) over the course of training. The parameters converge rapidly to the true values for all parameterizations of the cell sorting Hamiltonian.
C.2. Additional Results for Baselines and Ablations
In addition to the results presented Sections 4.3 and 4.4, we consider three additional baselines and ablations for the architectural design of the Neural Hamiltonian. The first a GNN Hamiltonian, which enjoys permutation symmetry but lacks translation symmetry and inductive biases for lattice data. The second is an ablation of the Neural Hamiltonian where cells cannot interact with each other, achieved by removing the cell-interaction CNN module from each layer and masking out any information from neighboring cells. Finally, we consider an ablation in which we remove the local pooling step in each layer of the Neural Hamiltonian. Qualitative and quantitative results are shown in Figure 9 and Table 12, respectively.
As expected, we see that the Neural Hamiltonian without cell interactions fails to guide cells towards realistic self-organized states. This because each cell needs to know the position of other cells in order to move towards location that contributes
16

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Deep Neural Cellular Potts Models
Table I. Additional results for the parameter fitting experiment. 'Mean' and 'Std'' columns refer the and standard deviation of the fitted values of the respective parameter, calculated over five independent random parameter initializations.
Type A
Type B
Type D
Type
True value (reference)
True value (reference)
True value (reference)
True value (reference)
Parameter
Mean
Std.
Mean
Std.
Mean
Std.
Mean
Std.
0.1
0.103
0.001
0.5
0.616
0.002
0.1
0.101
0.000
0.05
0.051
0.000
J
0.5
0.499
0.003
2.5
2.441
0.008
15.0
15.022
0.005
2.0
1.995
0.009
2(
0.5
0.497
0.006
1.0
1.012
0.007
7.5
7.504
0.020
8.0
8,039
0.014
J(11)
0,333
0.327
0.005
1.0
1.056
0.003
4.0
3.951
0.006
7.0
6,947
0.011
2(1,2)
0.2
0.207
0.003
4.5
4.494
0.003
7.5
7.478
0.021
5.5
5.502
0.015
222
0.267
0.274
0.007
1.0
1.048
0.006
4.0
4.004
0.01
3.0
3.021
0.007
to a realistic cellular structure. The GNN-based Hamiltonian results structures that somewhat resemble the cellular structures the data, but with substantially lower quality than the models based on our Neural Hamiltonian architecture. The Neural Hamiltonian without any pooling yields good results on cellular MNIST, but diverged during training for bipolar identical NH+closure), and hence we do report these data. A
axial sorting (the remaining hyperparameters possible reason for this could be the higher dimensionality of the lattice. Finally, all reported models performed well from an individual cell perspective, suggesting that the mooloy-innformed term is generally effective.
GNN closure
NH + closure GNN + closure GNN + closure
GNN closure
no interactions no interactions
GNN + closure
NH closure
NH + cosure
NH dosuue
no pooling
closure
NH closure no poaling
NH closure no
200
400
600
800
Time (MCS)
Time (hours)
Figure Qualitative results for additional baselines and ablations Cellular MNIST (left) and bipolar axial sorting (right).
To stress the importance of using Neural Hamiltonian for the bipolar axial organization experiment, Figure 10 shows the development of bipolar organization over time, similar to Figure 7(b), but now for CPM with cell sorting Hamiltonian. For type-2 cells, there reasonable overlap, implying that the degree of collinearity along the principal axis of type-2 cells similar the NeuralCPM case. However, the type- cells do not align, meaning that the analytical model fails 'squeeze"
17

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Deep Neural Cellular Potts Models
Table 12. Quantitative results of additional baselines and ablations for Cellular MNIST and bipolar axial organization experiments.
Cellular MNIST
bipolar axial organization
Axial alignment
Model/ablation
Pvolume 1
Punfrogmented
t
CS t
Pvolume 1
Punfragmented
RMSE 4
GNN
1.00
1.00
3.15
1.00
1.00
100.4
+ closure
Neural Hamiltonian
1.00
0.97
3.06
1.00
1.00
119.3
No interactions closure
Neural Hamiltonian
1.00
1.00
4.24
N/A
N/A
N/A
No pooling closure
Neural Hamiltonian closure (cq. Tables 2 and 3)
1.00
1.00
4.35
0.77
1.00
37.3
them along the orthogonal axis, which NeuralCPM successfully learned.
type 1 Cellsort
type 2 Cellsort
type 1 real data (Toda et al., 2018)
type 2 real data (Toda al., 2018)
Variance fraction along polar axis
0.6
0.5
0.4
0.3
0.2
100
200
Time (minutes)
Figure 10. Development of bipolar cellular organization over time for CPM with Cellsort Hamiltonian.
18

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Deep Neural Cellular Potts Models
C.3. Additional Qualitative Results
Additional qualitative simulation results for the Cellular MNIST and bipolar axial organization experiments can be found in Figures 11 and 12 respectively.
19

---

Deep Neural Cellular Potts Models
Cellsort Hamiltoniaa
Hamiltonian
Hamillonian
*External
Patential
Cellort
Hamiltonian
External Potential
CNN
NH layer +CNN
+CNN
Neural Hamiltonian
Hamiltonian
Neural Hamiltonian + cssuuuee
Hamiltonian +closure
200
Time (MCS)
Time (MCS)
Calloort Hamiltonian
Celisort Homiltonian
Hamiltonaan
Extermal Potential
Calisort Hamiltonian 1.Extemal Potential
Celisort
CNN
CNNN
NHllayer +CNN
1 NHyayr +CNN
Neural Hamiltonian
Hamiltonian
Neural llamltonian +closure
Neural Hamiltonaa +closure
200
400
600
800
200
400
600
800
Time (MCS)
Time (MCS)
Figure Additional qualitative results for Cellular MNIST simulations.
20

---

Deep Neural Cellular Potts Models
Cellsort Hamiltonian
Hamiltonian
Hamiltonian *External Patential
Hamiltonian External Potential
Cellort
CNN
Neural Hamitoniaa an +CNN
Hamiltonia an +NN
Neural Hamiltonian
Hamiltonian
Neural Hamiltoniann + csuure
Hamiltonian +closure
Time (hours)
Time (hours)
Callortt Hamiltonian
Celisort Homiltonian
Hamiltonian EExteraal Potential
Callsort Hamiltonian EExternal Potentill
CNN
CNNN
Hamiltonian +CNN
Ha miltoiiaan +CNN
Lyyyer Neural
Neural Hamiltonian
Hamiltonian
Neural Haailooiian +closure
Hamiltonian +closure
0.0
3.1
9.3
12.4
0.0
6.2
9.3
12.4
Time (hours)
Time (hours)
Figure Additional qualitative results for bipolar axial sorting simulations.
21


Reviews:
[{"review_id": "kBcPYWX0NW", "reviewer": "Reviewer_FiGo", "summary": "This paper introduces NeuralCPM, a neural network-based approach to cellular Potts modeling (CPM) for simulating collective cell dynamics. Traditional CPMs rely on physics-inspired Hamiltonians that require significant domain expertise to design and may not fully capture complex biological behaviors. NeuralCPM parameterizes the Hamiltonian with a neural network architecture specifically designed to respect symmetries in cellular dynamics, allowing it to be trained directly from observational data. The framework also enables the integration of known biological mechanisms by combining analytical Hamiltonians with the neural network in a hybrid model. The authors validate their approach through experiments on synthetic data and real-world multicellular systems, demonstrating that NeuralCPM can model cellular dynamics that traditional analytical Hamiltonians cannot capture.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the thorough review and valuable suggestions.\nWe are glad to hear that you found our Neural Hamiltonian well-designed and that you appreciated the validation against real biological data.\nWe address all comments and questions of your review below;\nnew tables and figures (referred to as e.g. Table R1) can be found through https://anonymous.4open.science/r/neuralcpm-rebuttal-results-761F.\n\nDue to the character constraint, we refer to responses to other reviewers where indicated.\n\n### General comments\n\n1. **More initial conditions and parameter ranges for parameter recovery experiment:**\nWe now also tested the approach on more parameter regimes, based on Type D and Type F cell sorting of [1].\nFurther, we tested five different random initializations of the parameters of the Hamiltonian to assess the robustness of the method.\nThe results, shown in Table R1 and Figure R1, demonstrate that the parameters robustly converge to values close to the ground-truth for all four scenarios.\n\n2. **Statistical testing for bi-polar axial organization:**\nWe now added a statistical analysis of the NH + closure model compared to all baseline models as follows:\nwe take 5 groups of 10 simulation runs, calculate the axial alignment RMSE as in Table 2 for each group, and then test whether the mean error of the NH + closure model is significantly lower than the respective baseline.\nWe used Welch’s t-test, and applied a Bonferroni correction to control for multiple testing.\nThe results, shown in Table R3, indicate that the mean error is significantly lower than all baseline models.\n\n### Weaknesses\n\n3. **Approximate parallel sampling:**\nPlease see point 3 of our response to reviewer ZLYN.\n\n4. **Architecture design choices, ablation studies, additional baselines:**\nPlease see point 1 of our response to reviewer KQ4z for a discussion on the newly added baselines and ablations.\n\n5. **Going beyond equilibrium dynamics:**\nNeuralCPM can be extended by also conditioning the Hamiltonian on earlier observed states, and training on full trajectory data. \nThis can be interpreted as learning the energy difference, instead of the absolute value.\nWe already conducted preliminary experiments with this approach that gave promising initial results,\nbut they are not yet ready for publication, and we leave this extension for future work.\n\n6. **Scaling to larger systems:**\nPlease see point 9 of our response to reviewer KQ4z.\n\n7. **Training stability:** \nSee point 10 below.\n\n### Questions\n\n8. **Computational requirements:**\nPlease see point 9 of our response to reviewer KQ4z.\n\n9. **Sensitivity to temperature $T$:**\nSince we can always rescale the Hamiltonian in the CPM, without loss of generality, we can assume that the temperature parameter $T$ equals 1, and train the Hamiltonian for that temperature.\nThe typical use of $T$ roughly corresponds to a parameter that determines the strength of the fluctuations in the system over time [2];\nhowever, it does not substantially affect the behavior of the system at equilibrium.\n\n10. **Techniques to stabilize training, and whether this is a fundamental limitation:**\nWe applied established best practices from the literature for mitigating training instabilities when training discrete EBMs:\nWe used Persistent Contrastive Divergence [3], similar to [4], and we regularized the Hamiltonians during training (Section 3.2).\nWe also tried different architectural configurations during preliminary experiments, e.g. network depth.\nAlthough these techniques are relevant for training EBMs in general, adding the symbolic biology-informed term to the Neural Hamiltonian effectively mitigated training stability issues for our use-case. \nThis is due to the biological terms functioning as ‘guardrails’ for the simulation, preventing biologically unrealistic behaviors.\nFinally, discrete EBMs are an active field of research, and recent advancements (e.g. [4,5,6]) are also applicable to NeuralCPM.\n\n11. **Other neural competing methods:** To the best of our knowledge, there are no earlier proposed neural methods that apply to the cellular dynamics data of Toda et al. and rely only on equilibrium data.\n\n### References\n\n[1] Edelstein-Keshet and Xiao (2023). Simplified Cell Sorting — Morpheus contributed examples.\nhttps://morpheus.gitlab.io/model/m2007.\n\n[2] Glazier and Graner (1993). Simulation of the differential adhesion driven rearrangement of biological cells.\n\n[3] Tieleman (2008). Training restricted Boltzmann machines using approximations to the likelihood gradient.\n\n[4] Grathwohl et al. (2021). Oops I Took A Gradient: Scalable Sampling for Discrete Distributions.\n\n[5] Zhang et al. (2022). A Langevin-like sampler for discrete distributions.\n\n[6] Schröder et al. (2023). Energy discrepancies: a score-independent loss for energy-based models.", "claims_and_evidence": "The first experiment convincingly demonstrates that NeuralCPM can accurately recover parameters from cell sorting models, validating the learning algorithm. The second experiment with the Cellular MNIST dataset shows that NeuralCPM can capture complex spatial arrangements that analytical Hamiltonians cannot, with quantitative metrics supporting this claim. The final experiment applying the approach to real biological data demonstrates practical utility.", "methods_and_evaluation_criteria": "The experimental methodology is sound, with appropriate metrics to evaluate both biological realism (cell volume, fragmentation) and the ability to capture complex collective behaviors (classifier score, axial alignment). The authors properly acknowledge limitations and discuss hyperparameter sensitivity.", "theoretical_claims": "The paper contains few formal theoretical claims requiring proof verification. I checked the mathematical formulation of the permutation invariant neural Hamiltonian architecture in Section 3.1, which appears correct in its construction using permutation-invariant aggregation functions.", "experimental_designs_or_analyses": "For the parameter recovery experiment (Section 4.2), the methodology is valid but could be more rigorous. The authors correctly generate synthetic data using known parameters and evaluate RMSE between learned and ground truth values. However, the rapid convergence shown in Figure 4 suggests this might be a simplified test case. The experiment would be more convincing with diverse initial conditions and parameter regimes.\n\nIn the Cellular MNIST experiment (Section 4.3), the classifier score metric is innovative but potentially problematic - success at recreating visually recognizable digits may not indicate biologically meaningful dynamics. The complementary biological metrics (volume and fragmentation) help address this concern, making the overall analysis approach sound.\n\nFor the bi-polar organization experiment with real biological data (Section 4.4), the axial alignment variance is an appropriate domain-relevant metric. However, the experimental design lacks statistical testing across multiple simulation runs to establish the significance of the reported improvements. Additionally, while the training instability issue with pure neural models is honestly reported, more analysis of this limitation would strengthen the paper. Additionally, including neural competing methods could further strengthen the validity of this experiment.", "relation_to_broader_scientific_literature": "The paper positions NeuralCPM within the intersection of energy-based models from machine learning and cellular Potts models from computational biology. The authors correctly identify the gap in the literature - while neural networks have been used to accelerate physical simulations, their application to improving the expressivity of cellular dynamics models is novel.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "**Strengths:**\n\n- The Neural Hamiltonian architecture is thoughtfully designed to respect fundamental symmetries (permutation, translation) in cellular systems\n- The hybrid approach combining analytical and neural components is particularly valuable, improving both biological realism and training stability\n- The method requires only observations of self-organized states for training, not full trajectories\n- The validation against real biological data demonstrates practical utility beyond synthetic examples\n\n**Weaknesses:**\n\n- The approximate sampling compromises theoretical foundations\n- Architecture design choices lack sufficient justification and ablation studies\n- Limited to equilibrium dynamics, excluding many biological processes\n- No clear pathway for scaling to larger cellular systems\n- Training energy-based models can be unstable, as evidenced by the divergence of some models in the bi-polar sorting experiment", "other_comments_or_suggestions": "No", "questions_for_authors": "- How computationally expensive is NeuralCPM compared to traditional CPMs, both for training and inference? This information would be valuable for potential users considering adopting this approach.\n\n- How sensitive is the model to the choice of temperature parameter T in the CPM? Since you noted this is \"poorly identifiable from static snapshots,\" have you explored whether NeuralCPM could learn an appropriate effective temperature automatically?\n    \n- For the bi-polar axial organization experiment, you mention that Neural Hamiltonian models without the closure term failed due to \"fast divergence during training.\" Could you elaborate on specific techniques you tried to stabilize these models before resorting to the hybrid approach?\n    \n- Does the training instability of pure neural Hamiltonians indicate a fundamental limitation of the approach? What modifications might address this?\n\n- What other neural competing methods might be applicable to the microscopy time-lapse data by Toda et al. (2018)?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "uAv4LXydJ3", "reviewer": "Reviewer_KQ4z", "summary": "This paper introduces Neural Hamiltonians, a novel approach for parameterizing the Hamiltonian function in cellular Potts models. The authors propose using neural networks to learn the Hamiltonian function directly from data, while preserving important physical and biological constraints.\n\nThe key innovation is a formulation that respects critical symmetries inherent to cell populations - specifically, invariance to cell permutations and lattice translations. This ensures the learned models maintain biologically plausible behavior regardless of how cells are indexed or positioned.\n\nAdditionally, the authors present a hybrid framework that combines traditional symbolic Hamiltonians (which incorporate known biological principles) with neural network-based Hamiltonians (which can capture complex patterns from data). The approach includes learnable weights that balance the contribution of these two components, allowing the model to leverage both domain knowledge and data-driven insights.\n\n## update after rebuttal\n\nAfter examining the authors' rebuttal, I can confirm that they have adequately addressed my concerns. Therefore, I have upgraded my recommendation to Accept.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your insightful review. We are glad that you found our work well-written and sound, and that you appreciated the innovations of our approach.\nOur response can be found below; new results (referenced as e.g. Figure R1) \ncan be found via https://anonymous.4open.science/r/neuralcpm-rebuttal-results-761F.\n\n### Weaknesses\n\n1. **Baselines and ablations:** \nWe now added three additional baselines and ablations: 1)\na GNN Hamiltonian, which enjoys permutation symmetry but lacks translation symmetry and inductive biases for lattice data; 2)\nan ablation of the Neural Hamiltonian where cells cannot interact by removing the cell-interaction CNN in each layer, masking out any information from neighboring cells; and 3) \nan ablation of the Neural Hamiltonian where we remove the local pooling step in all NH layers.\nWe show the results for experiments 2 and 3 in Table R2 and Figures R2 and R3.\nFor the bi-polar axial organization experiment, the ablation of removing the pooling in the NH layers diverged during training (remaining hyperparameters identical to NH+closure), and hence we do not report these data.\nThe overall results show that the paper's Neural Hamiltonian + closure design consistently outperforms all baselines in terms of the converged multicellular patterns.\n\n2. **Biological relevance of synthetic data:**\nWe agree that the cellular MNIST data is not directly biologically relevant.\nInstead, for this experiment we aimed to design a scenario with cellular structures that are complex enough to be poorly approximated by a symbolic model,\nyet simple enough to facilitate straightforward (visual) validation of model quality.\nIn a broader biological context, the relevance of this experiment is to test NeuralCPM’s ability to model structures beyond the capabilities of traditional CPMs, which indicates our method's potential to bring real biological use cases such as wound healing, cell migration, and embryo formation within reach of CPM simulations.\n\n3. **Scope of biological experiments:**\nWe only evaluated one real biological use-case in this paper, as our goal was to introduce and validate the core concepts of NeuralCPM before moving to more extensive and complex biological case studies.\nStill, we argue that the results on this scenario sufficiently demonstrate NeuralCPM’s potential for\nfuture applications, as the CPM formalism is very versatile (evidenced by hundreds of CPM models of biological processes in public repositories: https://artistoo.net/examples.html, https://morpheus.gitlab.io/#examples, https://compucell3d.org/Simulation%20Movies),\nand can now be derived from data with NeuralCPM.\n\n4. **Ablation study:** \nPlease see point 1 above.\n\n### Questions\n\n5. **MNIST dataset:**\nPlease see point 2 above.\n\n6. **Visualization for Cellsort Hamiltonian:**\nPlease see Figure R4.\nFor type-2 cells, there is reasonable overlap, implying that\nthe degree of collinearity along the principal axis of type-2 cells is similar to the NeuralCPM case.\nHowever, the type-1 cells do not align,\nmeaning that the Cellsort model fails to ‘squeeze’ them along the orthogonal axis, which NeuralCPM successfully learned.\n\n7. **Hamiltonian components:** \nThe goal of the experiment is to investigate if the NeuralCPM training framework can recover unknown parameters in symbolic Hamiltonians. \nTo this end, the Hamiltonian consisted only of symbolic terms and randomly initialized parameters.\nWe now also added additional results to investigate the robustness of this approach;\nplease see point 1 of our response to reviewer FiGo for a discussion.\n\n8. **Ablation study:** \nPlease see point 1 above.\n\n9. **Computational complexity:**\nDue to the Neural Hamiltonian, NeuralCPM requires more computational effort than traditional CPMs with symbolic Hamiltonians. \nThe amortized time for a simulation with the Neural Hamiltonian model was 44 seconds (Figure 5) and 62 seconds (Figure 6), while the Cellsort Hamiltonian + External potential, which exploits locality assumptions for an efficient calculation, required 0.3 and 0.5 seconds respectively. \nSo, although the Neural Hamiltonian is more expensive, the simulation times remain manageable.\nFurther, approximating the energy difference with a first order Taylor expansion [1,2] can further accelerate the computation since this enables approximating the energy difference for multiple spin flips at the cost of a single Hamiltonian evaluation. \nSo, using such a technique to evaluate 10+ spin flips in parallel could give an order of magnitude speedup, offering a promising path for scaling NeuralCPM to tissue-scale simulations.\nFinally, all training runs took at most 24 hours, and the tradeoff in terms of relative time per training step is comparable to inference.\n\n### References\n\n[1] Grathwohl et al. (2021). Oops I Took A Gradient: Scalable Sampling for Discrete Distributions.\n\n[2] Zhang et al. (2022). A Langevin-like sampler for discrete distributions.", "claims_and_evidence": "See the strengths and weaknesses below.", "methods_and_evaluation_criteria": "See the strengths and weaknesses below.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "The experimental methodology and analytical approach used in this work are fundamentally sound and well implemented. The authors have come up with a reasonable framework for evaluating their Neural Hamiltonian method. While the overall experimental design is sound, there are several specific observations and suggestions regarding these experiments that I have detailed in the sections below.", "relation_to_broader_scientific_literature": "See the summary.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "**Strengths**\n\n- The paper is well-written, with clear explanations of complex concepts that make the work accessible despite the interdisciplinary nature of the research.\n\n- The authors' approach of using neural networks to parameterize Hamiltonian functions in cellular Potts models represents an interesting innovation. This technique appears to be both effective and mathematically sound.\n\n- The Neural Hamiltonian framework successfully captures complex non-local cell interactions that traditional analytical methods struggle to model, at least as could be observed in the cellular MNIST experiments. This capability can be a substantial advancement in the field of computational biology and multicellular modeling.\n\n- The combination of symbolic (biologically-informed) and neural network-based Hamiltonians is particularly valuable. This can be an elegant balance between incorporating established biological knowledge and leveraging data-driven insights.\n\n**Weaknesses** \n\n- The baseline models included in the experimental comparisons are somewhat simplistic. For instance, the CNN baseline is predictably ill-suited for the task, making its poor performance unsurprising. \n\n- The connection between the synthetic datasets (particularly cellular MNIST) and real cellular dynamics remains vague. The paper does not sufficiently justify how success on these artificial benchmarks translates to biological validity or relevance.\n\n- The experiments with real biological data are notably limited in scope and depth. These experiments fail to fully demonstrate the potential advantages of Neural Hamiltonians in addressing meaningful biological questions. More extensive validation with diverse real-world cellular data would strengthen the paper's claims and better highlight the practical utility of the method.\n\n- The paper would benefit from more detailed ablation studies to isolate and quantify the contribution of each component of the proposed framework, particularly regarding the hybrid approach and symmetry constraints.", "other_comments_or_suggestions": "The abbreviation CNN is used once in the text before using the full term.", "questions_for_authors": "- How well does the cellular MNIST dataset reflect the complexity and characteristics of real cell population dynamics? Could you elaborate on the specific biological phenomena that this synthetic benchmark captures, and what limitations it might have in representing actual cellular behaviors?\n\n- Could you provide the visualization referenced in Figure 7b for the cellsort Hamiltonian? This would help me better understand the comparative performance of your method in this specific scenario.\n\n- For the first experiment, which components of the Neural Hamiltonian design are necessary for accurate parameter estimation? Have you conducted ablation studies to determine how removing or modifying certain elements affects the overall performance? \n\n- Given the novelty of your architecture, it would be valuable to see a systematic ablation study on the architectural components and their impact on experimental outcomes. Specifically: How crucial is the cell interaction CNN to the overall performance? What is the specific contribution of the pooling layers in capturing cellular dynamics? How does removing symmetry constraints affect model performance? What is the optimal balance between the symbolic and neural components in the hybrid approach?\n\n- How does the computational complexity of training and inference with Neural Hamiltonians compare to traditional approaches? Is there a significant trade-off between model expressivity and computational requirements?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "uZa6mDQZD0", "reviewer": "Reviewer_ZLYN", "summary": "In this thesis, we propose NeuralCPM, a neural network-based cellular Potts model (CPM), which aims to learn the dynamics of multicellular systems through neural Hamiltonian. The core idea of the method is to use Deep Energy-Based Model (EBM) to train Hamiltonian in order to overcome the problem of traditional CPM which relies on hand-designed energy functions. The paper claims that NeuralCPM is able to fit more complex cell behaviors and demonstrates its superiority by comparing it with experimental data.\nThe paper provides several experiments to support these claims, including:\n1. a synthetic data experiment to verify that NeuralCPM can learn the parameterized Hamiltonian and accurately simulate cellular MNIST data.\n2. Cellular MNIST dataset experiments: demonstrate that NeuralCPM can generate morphology-specific cellular arrangements, such as number shapes.\n3. real biological experiments (Toda et al., 2018): to validate the effectiveness of NeuralCPM in the bipolar axial organization task.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the thorough review! We appreciate that you found our methodology clear, innovative, and rigorous.\nWe address your specific comments in our response below; any new tables and figures (referenced as e.g. Figure R1)\ncan be found via https://anonymous.4open.science/r/neuralcpm-rebuttal-results-761F.\n\n### General comments\n\n1. **Temperature sensitivity:** \nSince we can always rescale the Hamiltonian in the CPM, we can assume without loss of generality that the temperature equals 1, and train the Hamiltonian for that temperature. \nThe typical effect of $T$ roughly corresponds to the strength of the fluctuations in the system over time [1]; \nhowever, it does not substantially affect the behavior of the system at equilibrium, and a particular Hamiltonian will result in qualitatively similar self-organized patterns for varying temperatures.\n\n### Questions\n\n2. **Use of synthetic data for bi-polar axial organization experiment:** \nTo generate synthetic training data corresponding to Toda et al., we used cell motion along imposed trajectories such that cells of one type (‘red’) move from their random initial position into a\ncentral band and cells of the other type leave the central band towards opposing cluster borders. \nSo by construction, we generate training data with a bi-polar axis. \nAdditionally, to make sure the training data are isotropic, we randomly rotate the final configuration.\nBiological cells (and NeuralCPM) do not rely on any such prescribed trajectories, but update cell positions as a result of cell-cell interactions in the current configuration.\nStill, we argue that the synthetic final self-organized states are statistically comparable to those observed in Toda et al. (2018) in terms of bi-polar organization, shown in Figure 7a. \nAs the training process only relies on these final states, the simplifications we made in the data generation do not necessarily cause adverse effects to the model quality. \nWe test this in Figure 7, where NeuralCPM’s simulations are shown to align very well with the real data observations of Toda et al. \nOf course, in general one should take such a synthetic approach with care, and in this light this experiment should be seen as a proof-of-principle towards more complex biological applications. \nWe will update our paper to include an explicit discussion about this point.\n\n3. **Approximate parallel CPM sampler and deviations from detailed balance:** \nNotably, the simulation dynamics from the standard CPM already lack detailed balance [2].\nThough the approximate parallel sampler can theoretically suffer from synchronization issues when parallel spin-flips affect each other (see e.g. [3]), we stress that it should be regarded as a surrogate sampler to accelerate training;\nother established training techniques for EBMs, like Persistent Contrastive Divergence [4], also successfully sacrifice unbiased sampling for accelerated training, as unbiased maximum likelihood estimation is generally intractable.\n\n4. **Lack of permutation symmetry in CNN baseline:** \nBased on your suggestion, we now added an additional GNN baseline which enjoys permutation symmetry to further strengthen our experimental evaluation, in addition to other ablations (see point 1 in our response to reviewer KQ4z). \nThe quantitative and qualitative results are shown in Table R2 and Figures R2+R3 respectively. \nThe results indicate that the Neural Hamiltonian outperforms the GNN baseline in terms of self-organization (CS and alignment RMSE metrics), emphasizing the importance of inductive biases for lattice data, e.g. translation symmetry and locality, in addition to permutation symmetry.\n\n### References\n\n[1] Glazier and Graner (1993). Simulation of the differential adhesion driven rearrangement of biological cells.\n\n[2] Durand and Guesnet (2016). An efficient Cellular Potts Model algorithm that forbids cell fragmentation.\n\n[3] Sultan et al. (2023). A parallelized cellular Potts model that enables simulations at tissue scale.\n\n[4] Tieleman (2008). Training restricted Boltzmann machines using approximations to the likelihood gradient.", "claims_and_evidence": "1. The core methodology proposed in the paper, NeuralCPM and its Neural Hamiltonian, is conceptually clear and reasonable, especially the emphasis on the symmetry of the cellular system, the integration of domain knowledge, and the rational use of the deep energy model (EBM) framework, which is supported by sufficient theoretical basis, and the argumentation process is logical, rigorous, and clear, and the proposed theories and methodologies are credible and innovative in their own right. The proposed theories and methods are credible and innovative.\n2. In terms of experiments, the validity and advantages of the method are verified through three different scenarios (parameter fitting, complex structure simulation, and real biological data application). The experiments are well-designed, and the quantitative metrics (e.g., RMSE, Classifier Score, etc.) can clearly reflect the performance of the model. The experimental results clearly show that the Neural Hamiltonian model outperforms the traditional analytical Hamiltonian model and the basic neural network architecture in simulating the complex cellular behaviors. However, the article mentions that “the CNN-based Hamiltonian produces unrealistic dynamics in the absence of substitution symmetry”, which suggests that the limitations of the chosen baseline approach are more obvious and may affect the comprehensiveness of the experimental comparisons.\n3. despite the overall strong evidence, there are still some aspects of the paper that need further refinement. For example, the authors rely on synthetic data as the training set in the real biological data validation experiments, and the sample size of the real experiments is on the low side (only six repetitions of the experiments), and whether the resulting generalization ability is sufficient for generalization to other real systems still needs to be further verified. In addition, the stability of the training process still needs to be verified and illustrated in different scenarios.", "methods_and_evaluation_criteria": "1. The authors have selected reasonable evaluation metrics in their experiments, including biological consistency metrics for cell volume and fragmentation, and Classifier Score for quantifying the structural expressiveness of cell populations, and this evaluation clearly reflects the performance of the model. In addition, the authors use several experimental scenarios with different difficulties, including parameter fitting of theoretical Hamiltonian functions, complex image structures (Cellular MNIST) and real biological experiments (e.g., bipolar cellular organization experiments by Toda et al.), to demonstrate the generality and validity of the model, and in particular, the comparative validation with real experimental data enhances the credibility of the results.\n\n2. However, some of the experiments in the paper have certain limitations, such as the simulation data constructed simplifies the complexity of real biological systems to a certain extent (e.g., there are discrepancies between the synthesized data and the real data in the bi-polar axial organization experiments), and may not be able to fully reflect the nonlinear and stochastic characteristics of the real cellular dynamics scenarios. In addition, there is a clear lack of selection of CNN baseline models, which makes it possible that some of the resultant comparisons may underestimate the capabilities of other potential architectures.", "theoretical_claims": "This paper does not contain an explicit proof of theory, so there is no need to check the correctness of the proof. The main contribution of the authors is to propose a deep learning method called NeuralCPM and to clarify the soundness of the theoretical framework by means of experiments and a symmetry constrained exposition.", "experimental_designs_or_analyses": "I thoroughly checked the soundness and validity of the experimental designs and analyses presented in the manuscript. The paper primarily conducts three sets of experiments: fitting known analytical Hamiltonians, simulating complex multicellular structures using Cellular MNIST data, and applying the model to a real-world biological dataset (bi-polar axial organization).\n\nThe experimental design for parameter fitting using analytical Hamiltonians is sound and well-structured. The authors employed Root Mean Squared Error (RMSE) to quantitatively assess the accuracy of parameter estimation, clearly demonstrating the model’s capability in learning physically meaningful parameters from data. However, given the central role of the temperature parameter in CPM dynamics, the experimental setup could benefit from further clarification on temperature sensitivity and robustness.\n\nIn the Cellular MNIST and bi-polar axial organization experiments, the authors selected appropriate metrics, such as cell volume, fragmentation, and Classifier Score, to capture both biological realism and structural complexity effectively. These metrics convincingly evaluate both cellular-level and collective-level model performance. However, there are some concerns: firstly, the choice of baseline models is limited—particularly the CNN baseline lacks permutation symmetry, leading to unrealistic dynamics. Including more suitable baselines or justifying these choices explicitly would strengthen the comparative validity of results. Secondly, in the bi-polar axial organization scenario, synthetic training data may oversimplify biological complexity, and the potential gap between synthetic and real-world data could be more explicitly discussed. Addressing these concerns would enhance the robustness and generalizability of the experimental validation.", "relation_to_broader_scientific_literature": "None", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "None", "other_comments_or_suggestions": "None", "questions_for_authors": "Q1. In the bi-polar axial organization experiment, synthetic data generated by Morpheus were used for training. Could you elaborate on how well these synthetic data represent real-world complexity? Specifically, how sensitive is your model to the simplifications made during data generation? Clarifying this would strengthen confidence in your method’s applicability to real biological scenarios.\n\nQ2. You introduced an approximate sampler to accelerate the MCMC dynamics. Could you explain more explicitly how deviations from detailed balance affect your results? Have you empirically or theoretically evaluated how these deviations impact the final distributions obtained by your trained models? Addressing this point would help assess potential biases introduced by the approximation.\n\nQ3. Baselines such as the CNN-based Hamiltonian lacked permutation symmetry, potentially limiting their comparative value. Could you justify the selection of these baseline models? Alternatively, did you consider other relevant baselines that respect the symmetries in cellular systems? This clarification is important to fully understand the relative advantages of your proposed architecture.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "yKfgCzZrNo", "reviewer": "Reviewer_U1hT", "summary": "The paper introduces NeuralCPM, a novel cellular Potts model (CPM) that employs a neural network to parameterize the Hamiltonian, diverging from traditional CPMs that rely on manually-defined, physics-inspired analytical Hamiltonians.\n\nThis Neural Hamiltonian is designed to respect symmetries inherent in cellular dynamics, such as permutation and translation invariance, and can be trained directly on observational data. A key feature is its ability to integrate domain knowledge through a hybrid model, combining known biological mechanisms with the neural network to enhance biological realism and expressiveness.\n\nThe authors evaluate NeuralCPM across three scenarios: synthetic data for parameter fitting of known Hamiltonians, the Cellular MNIST dataset to demonstrate increased expressiveness in modeling complex structures, and real-world bi-polar axial organization data from Toda et al. (2018) to showcase practical applicability.\n\nMain findings include NeuralCPM's superior performance over traditional CPMs in capturing complex cellular dynamics, achieving higher biological consistency and pattern fidelity, as evidenced by metrics like the Classifier Score (CS) and axial alignment RMSE.\n\n## update after rebuttal\n\nThe authors have clarified my concerns. I think this is a strong paper, as a result, I am increasing my score.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the rigorous review. We are happy to learn that you appreciated the originality, significance and clarity of our work.\nWe address your specific comments in our response below; \nany new tables and figures (referred to as e.g. Figure R1) can be found via https://anonymous.4open.science/r/neuralcpm-rebuttal-results-761F.\n\n### General comments\n\n1. **Direct comparison between simulated and observed dynamics:** \nIn Figures 6 and 7b of the paper, we show a qualitative respectively quantitative comparison of the simulated dynamics and the observed real-world biological cell dynamics of Toda et al.\nBoth figures show that the simulations of NeuralCPM align with the observations. \nTo further emphasize this, we now include a new Figure R4, following the same procedure as Figure 7b, but now using simulations based on the Cellsort Hamiltonian. \nComparing Figure 7b and R4, we see that NeuralCPM’s dynamics align substantially better with the observations of Toda et al.\n\n\n### Weaknesses\n\n2. **Computational complexity:** \nIndeed, MCMC sampling limits scalability to some extent, but we would like to stress that our current applications did not suffer from prohibitive computation times.\nTo illustrate, the amortized time for a single simulation with the Neural Hamiltonian model was 44 seconds for cellular MNIST (Figure 5) and 62 seconds for bi-polar axial organization (Figure 6), and all models were trained for at most 24 hours. \nAdditionally, as we briefly mentioned in the conclusion of our paper, we expect that exploiting a first order Taylor expansion to approximate the energy difference [1,2] can accelerate the computation substantially, since this can enable approximating the energy difference for many parallel spin flips at the cost of a single Hamiltonian evaluation. Moreover, recent advances in Deep EBM training (e.g. [3]) can also be applied to NeuralCPM in the future.\n\n3. **Training stability:**\nNeuralCPM relies on EBM training techniques, which can be sensitive to hyperparameters. \nHowever, for our experimental scenarios, we found that adding the biological term to the Hamiltonian greatly stabilized training. \nThis is because the biological term effectively acts as a ‘guardrail’ that constrains the MCMC simulation to a biologically reasonable manifold (e.g. cells have reasonable volumes and are contiguous objects). \nWe further expect that ongoing research on training (discrete) EBMs (e.g. [1,2,3,4] as well as future works) can carry over to NeuralCPM training.\n\n### Questions\n\n4. **Clarification on synthetic data for bi-polar experiment:** \nTo generate synthetic training data corresponding to Toda et al., we used cell motion along imposed trajectories such that cells of one type (‘red’) move from their random initial position into a central band and cells of the other type leave the central band towards opposing cluster borders. \nSo by construction, we generate training data with a bi-polar axis. \nAdditionally, to make sure the training data are isotropic, we randomly rotate the final configuration. \nIn contrast, biological cells (and NeuralCPM) do not rely on any such prescribed trajectories, but update cell positions as a result of cell-cell interactions in the current configuration.\nHowever, the patterns that result from this procedure are comparable to those observed in Toda et al. (2018) in terms of bi-polar axial organization (see Fig. 7). \nFinally, our model needs only these final patterns for training, and a comparison of the model with the real observations of Toda et al. shows that the spontaneous symmetry-breaking dynamics of NeuralCPM align with the dynamics observed in Toda et al. (2018).\n\n### Other comments or suggestions\n\n5. **Typos and formatting:** \nWe thank the reviewer for their detailed examination of the manuscript, and we will make sure to remove these errors in the updated version of the paper.\n\n### References\n\n[1] Grathwohl et al. (2021). Oops I Took A Gradient: Scalable Sampling for Discrete Distributions.\n\n[2] Zhang et al. (2022). A Langevin-like sampler for discrete distributions.\n\n[3] Schröder et al. (2023). Energy discrepancies: a score-independent loss for energy-based models.\n\n[4] Sun et al. (2023). Discrete Langevin samplers via Wasserstein gradient flow.", "claims_and_evidence": "The primary claim is that NeuralCPM is more expressive and better at modeling complex cellular dynamics than traditional CPMs. This is supported by:\n\n- Synthetic Data Experiments: NeuralCPM accurately fits parameters of known analytical Hamiltonians for cell sorting, validated by convergence plots (e.g., Figures 4 and 8), showing rapid alignment with ground-truth values.\n- Cellular MNIST Experiment: The model generates digit-like cellular structures, with a higher CS (Table 1) than analytical baselines, indicating improved expressiveness. Qualitative results (Figure 5) further support this.\n- Real-World Data Experiment: NeuralCPM predicts bi-polar axial organization, aligning with laboratory observations (Figure 7), with a significantly lower axial alignment RMSE (37.3) compared to baselines (Table 2).\n\nThe evidence is generally clear and convincing.", "methods_and_evaluation_criteria": "The overall approach of learning the Hamiltonian that was previously manually defined analytically makes sense.\n\nThe Neural Hamiltonian architecture is thoughtfully designed, using convolutional neural networks (CNNs) and permutation-invariant aggregation to respect cellular dynamics symmetries, which is apt for the problem.\n\nThe hybrid model integrating domain knowledge is a sensible approach, enhancing interpretability and leveraging biological insights.\n\nThe evaluation seems appropriate. However, for real-world data, direct comparison metrics between simulated and observed dynamics could further validate the approach.", "theoretical_claims": "This paper does not introduce new theoretical results.", "experimental_designs_or_analyses": "The authors use three different experiments. Overall, the designs are solid, with minor concerns addressed via questions below.", "relation_to_broader_scientific_literature": "NeuralCPM advances CPM research by integrating neural networks.", "essential_references_not_discussed": "I don't think there is related work that is essential but not discussed in this paper.", "other_strengths_and_weaknesses": "### Strengths\n\n- Originality: Combining neural networks with CPMs is innovative and useful, removing the need for manual Hamiltonian design.\n- Significance: Improved modeling of complex biological systems could impact fields like cancer research.\n- Domain Knowledge Integration: The hybrid model enhances trust and interpretability for biologists.\n- Clarity: The paper is very well written and very clear.\n\n### Weaknesses\n\n- Computational Complexity: MCMC sampling limits scalability, though future work is proposed.\n- Training Stability: Challenges with deep EBM training are noted but not fully resolved, affecting reproducibility.", "other_comments_or_suggestions": "- Typos: \"classs\" (Page 2, line 109) should be \"class\"; \"desings\" (Page 14) should be \"designs\".\n- Formatting: Ensure consistent figure captions (e.g., Figure 7a vs. 7(a)).", "questions_for_authors": "Synthetic Data for Bi-Polar Experiment: Could you elaborate on how the synthetic training data for the bi-polar axial organization experiment was generated to reflect real-world dynamics? Specifically, how were cell motion preferences assigned, and what ensures their biological plausibility?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper introduces Neural Hamiltonians, a novel approach for parameterizing the Hamiltonian function in cellular Potts models. The reviewers agreed that the experiments strongly support the capability of the Neural Hamiltonians model in simulating collective cell dynamics. However, reviewers raised concerns regarding potential limitations in biologically realistic simulation experiments, insufficient ablation studies to justify architectural choices, and limited baseline comparisons. After examining the reviews and rebuttals, I'm convinced that the authors have adequately addressed most concerns. While there remains room for improvement in demonstrating the model's robustness, particularly in real biological applications, I believe this paper offers a promising first step in this direction. Therefore, I recommend acceptance.
Decision: Accept (poster)