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 (here, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow> <mml:mn>2</mml:mn></mml:msup> </mml:math> ) 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 represent the activity of a neuron over time (a "spike train"). This representation has distinctive properties, including a built-in "point of no return" property that may serve as a substrate for decision-making with progressive refinement over time. Moreover, because (in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow> <mml:mn>2</mml:mn></mml:msup> </mml:math> ) 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. Importantly, this encoding is incomplete: quite different sequences of jump times may lead to the same endpoint. The main results of the paper use the tools of hyperbolic geometry to formalize and delineate the these behaviors.