Structural Dynamics: Memristive Excitable Automata

Abstract

A memristive automaton is a structurally-dynamic excitable cellular automaton where a link connecting two cells is removed or added if one of the cells is in excited state and another cell is in refractory state.

A cellular automaton $\mathcal{A}$ is an orthogonal array of uniform finite-state machines, or cells. Each cell takes finite number of states and updates its states in discrete time depending on states of its closest neighbours. All cells update their states simultaneously by the same rule. We consider eight-cell neighbourhood and three cell-states: resting ∘, excited +, and refractory −. Let u(x) = { y: |x − y| _L ∞ = 1} be a neighbourhood of cell x. A cell x has a set of incoming links { l _xy : y ∈ u(x)} which take states 0 and 1. A link l _xy is a link of excitation transfer from cell y to cell x. A link in state 0 is considered to be high-resistant, or non-conductive, and link in state 1 low resistant, or conductive. A link-state $l^t_{xy}$ is updated depending on states of cells x and y at time step t: $l^t_{xy} = f(x^t, y^t)$. Resting state gives little indication of cell’s previous history, therefore we will consider only non-resting cells contributing to a link state updates. When cells x and y are in the same state (bother cells are in state + or both are in state −) no ’current’ can flow between the cells, therefore scenarios x ^t = y ^t are not taken into account.