Walks with jumps: a neurobiologically motivated class of paths in the hyperbolic plane

Abstract

We introduce the notion of a "walk with jumps", which we conceive as an evolving process in which a point moves in a space (for us, typically $\mathbb{H}^2$) over time, in a consistent direction and at a consistent speed except that it is interrupted by a finite set of "jumps" in a fixed direction and distance from the walk direction. Our motivation is biological; specifically, to use walks with jumps to encode the activity of a neuron over time (a ``spike train``). Because (in $\mathbb{H}^2$) the walk is built out of a sequence of transformations that do not commute, the walk's endpoint encodes aspects of the sequence of jump times beyond their total number, but does so incompletely. The main results of the paper use the tools of hyperbolic geometry to give positive and negative answers to the following question: to what extent does the endpoint of a walk with jumps faithfully encode the walk's sequence of jump times?