Spatial Birth-Death Point Process Models for Controllable Morphogenesis

Abstract

Controlling morphogenetic processes - where the total mass of agents varies as the system evolves - is a particularly difficult inverse design problem. In this work, we present a model for bio-inspired shape formation using spatial point processes, a popular framework for describing many types of spatio-temporal reproductive processes. We frame the problem as one of controlling spatial birth-death processes, and given a target stationary distribution for the cell-like points, construct a continuous-time Markov chain Monte Carlo (MCMC) procedure by which the points reproduce to achieve and maintain the target shape. Within this framework, we consider several types of point processes, beginning with the spatially uncorrelated Poisson process and extending to the repulsive Strauss process, demonstrating how the combination of global and local information allows the points to determine their relative birth-death rates. To improve biological plausibility, we also extend the procedure to include reversible split-and-merge transitions and spatial Hawkes-based birth-death rates, thereby modeling the locality of cell division. This stochastic, continuous-time MCMC formulation of the problem possesses desirable properties for multi-agent systems, including recovery from partial destruction via ergodicity of the constructed Markov chain, and enables extensions into multi-type and periodic processes.