A digital map is a simple dynamical system that is related to various digital dynamical systems including cellular automata, dynamic binary neural networks, and digital spiking neurons. Depending on parameters and initial condition, the map can exhibit various periodic orbits and transient phenomena to them. In order to analyze the dynamics, we present two simple feature quantities. The first and second quantities characterize the plentifulness of the periodic phenomena and the deviation of the transient phenomena, respectively. Using the two feature quantities, we construct the steady-versus-transient plot that is useful in the visualization and consideration of various digital dynamical systems. As a first step, we demonstrate analysis results for an example of the digital maps based on analog bifurcating neuron models.

This letter studies a digital return map that is a mapping from a set of lattice points to itself. The digital map can exhibit various periodic orbits. As a typical example, we present the digital logistic map based on the logistic map. Two fundamental results are shown. When the logistic map has a unique periodic orbit, the digital map can have plural periodic orbits. When the logistic map has an unstable period-3 orbit that causes chaos, the digital map can have a stable period-3 orbit with various domain of attractions.

The digital spiking neuron (DSN) consists of digital state cells and behaves like a simplified neuron model. By adjusting wirings among the cells, the DSN can generate spike-trains with various characteristics. In this paper we present a theorem that clarifies basic relations between change of wirings and change of characteristics of the spike-train. Also, in order to explore learning potential of the DSN, we propose a learning algorithm for generating spike-trains that are suited to an application example. We then show significances and basic roles of the presented theorem in the learning dynamics.