A simple mathematical theory for Simple Volatile Memristors and their spiking circuits

In pursuit of neuromorphic (brain-inspired) devices, memristors (memory-resistors) have emerged as effective components for emulating neuronal circuitry. Here we formally define a class of Simple Volatile Memristors (SVMs) based on a simple conductance equation of motion from which we build a simple mathematical theory on the dynamics of isolated SVMs and SVM-based spiking circuits. Notably, SVMs include various fluidic iontronic devices that have recently garnered significant interest due to their unique quality of operating within the same medium as the brain. Specifically we show that symmetric SVMs produce non self-crossing current-voltage hysteresis loops, while asymmetric SVMs produce self-crossing loops. Additionally, we derive a general expression for the enclosed area in a loop, providing a relation between the voltage frequency and the SVM memory timescale. These general results are shown to materialise in physical finite-element calculations of microfluidic memristors. An SVM-based circuit has been proposed that exhibits all-or-none and tonic neuronal spiking. We generalise and analyse this spiking circuit, characterising it as a two-dimensional dynamical system. Moreover, we demonstrate that stochastic effects can induce novel neuronal firing modes absent in the deterministic case. Through our analysis, the circuit dynamics are well understood, while retaining its explicit link with the physically plausible underlying system.