Directional excitability in Hilbert spaces

We introduce a generalized excitable system in which spikes can happen in a continuum of directions, therefore drastically enriching the expressivity and control capability of the spiking dynamics. In this generalized excitable system, spiking trajectories happen in a Hilbert space with an excitable resting state at the origin and spike responses that can be triggered in any direction as a function of the system's state and inputs. State-dependence of the spiking direction provide the system with a vanishing spiking memory trace, which enables robust tracking and integration of inputs in the spiking direction history. The model exhibits generalized forms of both Hodgkin's Type I and Type II excitability, capturing their usual bifurcation behaviors in an abstract setting. When used as the controller of a two-dimensional navigation task, this model facilitates both the sparseness of the actuation and its sensitivity to environmental inputs. These results highlight the potential of the proposed generalized excitable model for excitable control in high- and infinite-dimensional spaces.