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

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.