Variable Metric Splitting Methods for Neuromorphic Circuits Simulation

TL;DR

The paper addresses the challenge of scalable simulation of neuromorphic circuits composed of capacitors, batteries, and memristive elements. It develops a variable-metric forward-backward splitting algorithm (VMFBS) that decomposes the zero-finding problem $G(v)-i^*=0$ into a differentiator $c\operatorname{D}$, a diagonal gradient operator defined by the memductance, and a nonlinear residual treated as an offset, enabling efficient, scalable computation. Key contributions include establishing a memristor-based gradient interpretation under a Riemannian metric, formulating a three-operator splitting problem tailored to neuromorphic networks, and validating the approach on an E-I motif and a PING network with results consistent with established NI methods. This framework offers a principled, scalable path for simulating large-scale neuromorphic circuits and informs future analysis and deployment at scale.

Abstract

This paper proposes a variable metric splitting algorithm to solve the electrical behavior of neuromorphic circuits made of capacitors, memristive elements, and batteries. The gradient property of the memristive elements is exploited to split the current to voltage operator as the sum of the derivative operator, a Riemannian gradient operator, and a nonlinear residual operator that is linearized at each step of the algorithm. The diagonal structure of the three operators makes the variable metric forward-backward splitting algorithm scalable and amenable to the simulation of large-scale neuromorphic circuits.

Figures (3)

• Figure 1: The simulation results of the two-neuron E-I motif using VMFBS algorithm and Adams-Bashforth two-step method.
• Figure 2: The network consists of two populations of neurons, with no synaptic connections within each population. The external current excites the first population, and the excitatory synapses propagate this excitation to the second population. The excitation and spiking of the second population inhibits the first population, creating the well-known E-I motif rhythms.
• Figure 3: The raster plot of the two population network is presented here. The black dashes indicate the spike of the excitatory population, and the red dashes indicate the spikes of the inhibitory neurons. The excitatory synaptic connection only comes into play after $t_{\text{effect}}=120 ms$, indicated by the blue dashed line. A strong rhythmic behavior can be observed with the given external excitation.