Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Operator-Theoretic Framework to Simulate Neuromorphic Circuits

Amir Shahhosseini, Thomas Chaffey, Rodolphe Sepulchre

TL;DR

This work addresses scalable simulation of nonlinear neuromorphic circuits by formulating circuit dynamics as a zero-inclusion problem and applying an energy-based splitting into a lossless LTI subnetwork and a nonlinear resistive subnetwork. The resistive part is modeled as a difference of two cyclic monotone operators, enabling efficient fixed-point iterations via a Difference-of-Monotone Douglas–Rachford (DMDR) method, while the lossless subproblem is solved in the frequency domain to exploit diagonalizable dynamics with FFT. The authors demonstrate substantial computational gains on a FitzHugh–Nagumo neuron and a network of 100 heterogeneously coupled neurons, achieving faster convergence than traditional numerical integration and enabling behavior-informed initialization for further speedups. This framework provides a physically interpretable, scalable approach to simulating large-scale neuromorphic circuits and offers a path toward more robust analysis and design of spiking networks. Practical impact includes reduced simulation time and improved capability to incorporate prior behavioral knowledge into solvers, with code publicly available for reproduction.

Abstract

Splitting algorithms are well-established in convex optimization and are designed to solve large-scale problems. Using such algorithms to simulate the behavior of nonlinear circuit networks provides scalable methods for the simulation and design of neuromorphic systems. For circuits made of linear capacitors and inductors with nonlinear resistive elements, we propose a splitting that breaks the network into its LTI lossless component and its static resistive component. This splitting has both physical and algorithmic advantages and allows for separate calculations in the time domain and in the frequency domain. To demonstrate the scalability of this approach, a network made from one hundred neurons modeled by the well-known FitzHugh-Nagumo circuit with all-to-all diffusive coupling is simulated.

An Operator-Theoretic Framework to Simulate Neuromorphic Circuits

TL;DR

This work addresses scalable simulation of nonlinear neuromorphic circuits by formulating circuit dynamics as a zero-inclusion problem and applying an energy-based splitting into a lossless LTI subnetwork and a nonlinear resistive subnetwork. The resistive part is modeled as a difference of two cyclic monotone operators, enabling efficient fixed-point iterations via a Difference-of-Monotone Douglas–Rachford (DMDR) method, while the lossless subproblem is solved in the frequency domain to exploit diagonalizable dynamics with FFT. The authors demonstrate substantial computational gains on a FitzHugh–Nagumo neuron and a network of 100 heterogeneously coupled neurons, achieving faster convergence than traditional numerical integration and enabling behavior-informed initialization for further speedups. This framework provides a physically interpretable, scalable approach to simulating large-scale neuromorphic circuits and offers a path toward more robust analysis and design of spiking networks. Practical impact includes reduced simulation time and improved capability to incorporate prior behavioral knowledge into solvers, with code publicly available for reproduction.

Abstract

Splitting algorithms are well-established in convex optimization and are designed to solve large-scale problems. Using such algorithms to simulate the behavior of nonlinear circuit networks provides scalable methods for the simulation and design of neuromorphic systems. For circuits made of linear capacitors and inductors with nonlinear resistive elements, we propose a splitting that breaks the network into its LTI lossless component and its static resistive component. This splitting has both physical and algorithmic advantages and allows for separate calculations in the time domain and in the frequency domain. To demonstrate the scalability of this approach, a network made from one hundred neurons modeled by the well-known FitzHugh-Nagumo circuit with all-to-all diffusive coupling is simulated.
Paper Structure (17 sections, 21 equations, 3 figures, 1 algorithm)

This paper contains 17 sections, 21 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Energy-based Splitting of Nonlinear Circuits
  3. Preliminaries
  4. Neuromorphic Circuits and Energy-based Splitting
  5. From Neural Networks to Electrical Circuits
  6. Energy-based Splitting of Nonlinear Electrical Circuits
  7. Mixed Monotone Description of Nonlinear Resistors
  8. Connection with Energy-based Representations
  9. Time-Frequency Approach in Mixed Monotone Circuits
  10. Difference of Monotone Splitting Algorithms
  11. Time-Frequency Approach
  12. Results: From a Single Neuron to a Heterogeneous Network
  13. FitzHugh-Nagumo Model
  14. A Spiking Neural Network
  15. Conclusions
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Decomposition of the circuit network $\mathscr{N}$ into a lossless part and a resistive part
  • Figure 2: The FitzHugh-Nagumo model circuit is made of three branches; a linear capacitor, a linear inductor in series with a linear resistor, and finally, a tunnel diode that is a static nonlinear resistor
  • Figure 3: Simulation of the FitzHugh-Nagumo model; The result of the proposed method at iterations 1, 20, and 200 is contrasted with the steady-state results of numerical integration. The solid red line indicates the result of the settled numerical integration method (only steady-state) whereas the black dashed line is the result of the FPI method.