A Theory of Collective Cell Migration and the Design of Stochastic Surveillance Strategies
In nature, complex emergent behavior arises in groups of biological entities often as a result of simple local interactions between neighbors in space or on a network. In such cases, scientific inquiry is typically aimed at inferring these local rules. Conversely, in teams of robots, the goal is to create decentralized control laws which results in efficient global behavior. These behaviors are designed for tasks such as maintaining formation control, performing effective coverage control or persistently monitoring an environment. With this in mind, we consider the following: 1> the emergence of collective cell migration from local contact and mechanical feedback and 2> the design of unpredictable surveillance strategies for teams of robots.
Collective cell migration is an essential part of tissue and organ morphogenesis during embryonic development, as well as of various disease processes, such as cancer. The vast majority of theoretical descriptions of collective cell behavior focus on large numbers of cells, but fail to accurately capture the dynamics of small groups of cells. Here we introduce a low-dimensional theoretical description that successfully describes single cell migration, cell collisions, collective dynamics in small groups of cells, and force propagation during sheet expansion, all within a common theoretical framework. We also explain the counter-intuitive observation that pairs of cells repel each other upon collision while they coordinate their motion in larger clusters.
Conventional monitoring strategies used by teams of robots are deterministic in nature making it possible for intelligent intruders who study the motion of the patrolling agent to compromise the patrol route. This problem can be solved by designing random walkers on graphs which naturally incorporate unpredictability. Within this framework, we study and provide the first analytic expression for the first meeting time of multiple random walkers, in terms of their transition matrices. We also study two problems related to maximizing unpredictability: given graph and visit frequency constraints, 1> maximize the entropy rate generated by a Markov chain, and 2> maximize the return time entropy associated with the Markov chain, where the return time entropy is the weighted average over all graph nodes of the entropy of the first return times of the Markov chain.