Slow passage through a Hopf-like bifurcation in piecewise linear systems: Application to elliptic bursting.

The phenomenon of slow passage through a Hopf bifurcation is ubiquitous in multiple-timescale dynamical systems, where a slowly varying quantity replacing a static parameter induces the solutions of the resulting slow-fast system to feel the effect of the Hopf bifurcation with a delay. This phenomenon is well understood in the context of smooth slow-fast dynamical systems; in the present work, we study it for the first time in piecewise linear (PWL) slow-fast systems. This special class of systems is indeed known to reproduce all features of their smooth counterpart while being more amenable to quantitative analysis and offering some level of simplification, in particular, through the existence of canonical (linear) slow manifolds. We provide conditions for a PWL slow-fast system to exhibit a slow passage through a Hopf-like bifurcation, in link with possible connections between canonical attracting and repelling slow manifolds. In doing so, we fully describe the so-called way-in/way-out function. Finally, we investigate this slow passage effect in the Doi-Kumagai model, a neuronal PWL model exhibiting elliptic bursting oscillations.

Simplified description of dynamics in neuromorphic resonant tunneling diodes.

In this article, the standard theoretical model accounting for a double barrier quantum well resonant tunneling diode (RTD) connected to a direct current source of voltage is simplified by representing its current-voltage characteristic with an analytically approachable, anti-symmetric N-shaped function. The time and variables involved are also transformed to reduce the number of parameters in the model. Responses observed in previous, more physically accurate studies are reproduced, including slow-fast dynamics, excitability, and bistability, relevant for spiking signal processing. A simple expression for the refractory time of the excitable response is derived and shown to be in good agreement with numerical simulations. In particular, the refractory time is found to be directly proportional to the circuit's intrinsic inductance. The presence or absence of bistability in the dependence of the parameters is also discussed thoroughly. The results of this work can serve as a guideline in prospective endeavors to design and fabricate RTD-based neuromorphic circuits for power and time-efficient execution of neural network algorithms.