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

An entropy maximization problem related to optical communication

Motivated by a problem in optical communication, we consider the general problem of maximizing the entropy of a stationary random process that is subject to an average transition cost constraint. Using a recent result of Justenson and Hoholdt, we present an exact solution to the problem and suggest a class of finite state encoders that give a good approximation to the exact solution

On the VC-Dimension of Binary Codes

We investigate the asymptotic rates of length-$n$ binary codes with VC-dimension at most $dn$ and minimum distance at least $\delta n$. Two upper bounds are obtained, one as a simple corollary of a result by Haussler and the other via a shortening approach combining Sauer-Shelah lemma and the linear programming bound. Two lower bounds are given using Gilbert-Varshamov type arguments over constant-weight and Markov-type sets

Computable Lower Bounds for Capacities of Input-Driven Finite-State Channels

This paper studies the capacities of input-driven finite-state channels, i.e., channels whose current state is a time-invariant deterministic function of the previous state and the current input. We lower bound the capacity of such a channel using a dynamic programming formulation of a bound on the maximum reverse directed information rate. We show that the dynamic programming-based bounds can be simplified by solving the corresponding Bellman equation explicitly. In particular, we provide analytical lower bounds on the capacities of $(d, k)$-runlength-limited input-constrained binary symmetric and binary erasure channels. Furthermore, we provide a single-letter lower bound based on a class of input distributions with memory.Comment: 9 pages, 8 figures, submitted to International Symposium on Information Theory, 202

On row-by-row coding for 2-D constraints

A constant-rate encoder--decoder pair is presented for a fairly large family of two-dimensional (2-D) constraints. Encoding and decoding is done in a row-by-row manner, and is sliding-block decodable. Essentially, the 2-D constraint is turned into a set of independent and relatively simple one-dimensional (1-D) constraints; this is done by dividing the array into fixed-width vertical strips. Each row in the strip is seen as a symbol, and a graph presentation of the respective 1-D constraint is constructed. The maxentropic stationary Markov chain on this graph is next considered: a perturbed version of the corresponding probability distribution on the edges of the graph is used in order to build an encoder which operates in parallel on the strips. This perturbation is found by means of a network flow, with upper and lower bounds on the flow through the edges. A key part of the encoder is an enumerative coder for constant-weight binary words. A fast realization of this coder is shown, using floating-point arithmetic