Calibration of stochastic, agent-based neuron growth models with Approximate Bayesian Computation

TL;DR

This work tackles the stochastic inverse problem in mechanistic, agent-based neuron growth models by embedding them in a Bayesian framework and applying Approximate Bayesian Computation (ABC) with a Wasserstein-based distance in an SMC scheme. Morphometrics quantify neuron morphology, enabling direct comparison between simulated and real data without hand-crafted summaries, and the framework is implemented through a scalable coupling of BioDynaMo and ABCpy. On synthetic data, the method recovers data-generating parameters and demonstrates sensitivity to morphometrics and distance choices, with Wasserstein distance yielding reliable, efficient calibration; on experimental CA1 pyramidal neurons, the approach yields sharp posteriors and plausible matches in means, though some breadth of variability remains under model-driven QoIs. Overall, the proposed framework offers a robust, scalable path for Bayesian calibration and model adequacy assessment of neuronal growth models, with potential extensions to richer morphometrics and more complex neuron types. All mathematics are expressed with proper notation, including $p(\Theta\mid y_{\mathit{obs}})$, $\mathcal{W}$, and $\epsilon$.

Abstract

Understanding how genetically encoded rules drive and guide complex neuronal growth processes is essential to comprehending the brain's architecture, and agent-based models (ABMs) offer a powerful simulation approach to further develop this understanding. However, accurately calibrating these models remains a challenge. Here, we present a novel application of Approximate Bayesian Computation (ABC) to address this issue. ABMs are based on parametrized stochastic rules that describe the time evolution of small components -- the so-called agents -- discretizing the system, leading to stochastic simulations that require appropriate treatment. Mathematically, the calibration defines a stochastic inverse problem. We propose to address it in a Bayesian setting using ABC. We facilitate the repeated comparison between data and simulations by quantifying the morphological information of single neurons with so-called morphometrics and resort to statistical distances to measure discrepancies between populations thereof. We conduct experiments on synthetic as well as experimental data. We find that ABC utilizing Sequential Monte Carlo sampling and the Wasserstein distance finds accurate posterior parameter distributions for representative ABMs. We further demonstrate that these ABMs capture specific features of pyramidal cells of the hippocampus (CA1). Overall, this work establishes a robust framework for calibrating agent-based neuronal growth models and opens the door for future investigations using Bayesian techniques for model building, verification, and adequacy assessment.

Figures (13)

• Figure 1: Conceptual overview of the calibration methodology. We first define the quantities of interest (QoI $1,\dots,J$) for the neuronal growth simulation. The data is then obtained from either of two data pipes: A) experimental data or B) synthetic data ( [outer color=olive, fill color=olive, inner color=white]1 to [outer color=olive, fill color=olive, inner color=white]4 ). After processing and structuring the data ($J$ QoIs for $K$ neurons), we provide an initial guess of the model parameters -- the prior parameter distribution -- to the SMCABC algorithm DelMoral2012Bernton2019. The algorithm DelMoral2012 describes the parameter distribution with a set of $N$ particles ( [outer color=black, fill color=black, inner color=white]0 ) and enters into a loop ( [outer color=black, fill color=black, inner color=white]1 to [outer color=black, fill color=black, inner color=white]3 ) iteratively moving particles closer to the posterior. Adaptively lowering the tolerance level $\epsilon_i$ ( [outer color=black, fill color=black, inner color=white]1 ) ensures efficient positional updates of the particles in ( [outer color=black, fill color=black, inner color=white]3 ) DelMoral2012. For such updates, the algorithm executes the steps [outer color=gray, fill color=gray, inner color=white]1 to [outer color=gray, fill color=gray, inner color=white]5 simulating ${M^\prime}$ neurons under the model and computing the statistical distance to the data.
• Figure 2: 3D Mechanistic ABM for neuronal growth. Panel (a) shows the agent-based discretization and the early stage of a neuron simulation (pyramidal cell). The center is the spherical soma (cell body). The dendrites are discretized with small, cylindrical agents. Typically, the tips drive the growth Shree2022; hence, we differentiate between the general and tip agents. The agents define a tree-like structure with different segments and branches. (b) Pyramidal cell at the end of the simulation. (c) Experimental pyramidal cell in the mouse hippocampus observed by DeFelipePyramidal2019 (https://neuromorpho.org/neuron_info.jsp?neuron_name=24-mouse-c57-hcca1-id6-sec53-cel3-soma-corr1-6z).
• Figure 3: Error of the Wasserstein distance: We compare the numerical estimation of the Wasserstein distance between two data sets of cardinality N and M. The datasets are sampled from multivariate normal distributions of different dimensionality. They differ in their mean, i.e., a zero vector and a vector filled with ones, but share the covariance matrix $C_{ij} = \delta_{ij} + 0.2 \cdot (1 - \delta_{ij})$. For each combination (N, M), we sample 1000 datasets. The confidence interval in (a) highlights the area between the first and third quartiles. Numerical computation follows Flamary2021potDutta2021Bonneel2011, the analytic solution was discovered by Dowson1982.
• Figure 4: Software implementation and interfaces for the SMCABC sampling step [outer color=black, fill color=black, inner color=white]3 in Figure \ref{['fig:project-overview']}. Persistent BioDynaMo processes (green) reside on all server cores during calibration. The simulation objects that ABCpy (blue) dynamically generates on the MPI ranks are connected to the respective BioDynaMo process via stdin-stdout. Results are written to and retrieved from a disk allocated in RAM to enhance performance. [outer color=gray, fill color=gray, inner color=white]1 parameter proposal, [outer color=gray, fill color=gray, inner color=white]2 simulate data under the model, [outer color=gray, fill color=gray, inner color=white]3 compute morphometrics, and [outer color=gray, fill color=gray, inner color=white]4 evaluate acceptance criterion.
• Figure 5: Simulated data sets simulated under Model 1 (blue) and Model 2 (orange). The histograms are computed on $10^4$ simulated neurons.