An Agent-Based Model for Step Lengths in a Random Walk

Richard C. Windecker, PhD

N/A (retired from Bell Labs)

"An Agent-Based Model for Step Lengths in a Random Walk"

As an animal searching for prey performs a random walk, processes in the animal’s nervous system make decisions that produce a distribution of step lengths. I will describe in detail an Agent-Based Model for how an animal’s nervous system might make these decisions. The “agents,” that I call “Simple Abstract Neurons,” are NON-deterministic generalizations of well-known digital logic gates. I will use a simple version of the model, with carefully-chosen, made-up parameters to illustrate central concepts. I will give a detailed example of how the model parameters can be adjusted to fit a set of empirical data; in this case, from a diving marine predator: an individual blue shark. The SAN model fits the shark data much more closely than the “best fit” of a theoretical, analytical model. Theoretical studies suggest that when prey is plentiful, an exponential distribution of step lengths is effective. Otherwise, a power-law distribution is optimum. Animals follow such distributions only to the degree that evolutionary pressures may have resulted in an approximation that gives an acceptable balance of cost vs. benefit. But theory provides no insight as to how an animal might produce the observed behavior. The SAN model suggests some answers and makes testable predictions. For example, the model easily and naturally explains how an animal can follow an approximate power-law distribution while avoiding the implied infinities at very short and very long step lengths. Because the underlying processes are stochastic, any set of empirical data is a sample from a range of possible sets. Model-generated “synthetic data” can be used to characterize this range. The model can also be used to perform very insightful “What if?” experiments. SANs are compatible with digital logic; the number of possible pure and hybrid networks is huge. The potential is very large for the SAN model to be adapted to other animal behaviors besides random walks.

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