Operator-Splitting Methods for Neuromorphic Circuit Simulation

TL;DR

This work develops an operator-theoretic framework to simulate neuromorphic circuits by decomposing the circuit dynamics into a difference of monotone operators and solving the resulting zero-finding problem with a consensus-based DM Douglas–Rachford algorithm. By splitting both neuron-level and network-level dynamics into tractable monotone components and leveraging a lifted product space, the method achieves scalable, event-based simulations that capture multimodal spikes and bursts while enabling parameter continuation and modular analysis. Theoretical results establish convergence for the three-operator case and a lifted, multi-operator extension; computational strategies include shifting to enforce monotonicity, inner resolvent iterations, and time-frequency hopping to exploit LTI structure. Demonstrations on single neurons, bursting behavior, and half-center oscillators illustrate accurate event timing, robustness to resolution changes, and ability to harness prior knowledge for fast, modular simulations. The approach offers a principled alternative to standard NI methods, with potential impact on parametric sensitivity studies, neuromodulation analyses, and scalable neuromorphic circuit design, aided by openly available MATLAB code.

Abstract

A novel splitting algorithm is proposed for the numerical simulation of neuromorphic circuits. The algorithm is grounded in the operator-theoretic concept of monotonicity, which bears both physical and algorithmic significance. The splitting exploits this correspondence to translate the circuit architecture into the algorithmic architecture. The paper illustrates the many advantages of the proposed operator-theoretic framework over conventional numerical integration for the simulation of multiscale hierarchical events that characterize neuromorphic behaviors.

Figures (8)

• Figure 1: Modeling the neuron's membrane with a capacitive element is a classical approach in neurodynamics. The first-order lags filter the voltage and generate voltages at different timescales ($v_f, v_s,$ and $v_{\mu s}$). The interaction of positive and negative conductances at different timescales captures the essence of excitability.
• Figure 2: The spiking network also has the same single-layer architecture. It can be seen that the state (voltage) of all neurons can contribute to the dynamics of the $k^{th}$ neuron through synaptic currents. The effect of these synaptic connections is combined with the internal dynamics (denoted by $int$ superscripts) of the $k^{th}$ neuron (stemming from its own ion channels) and equates the externally injected current to the $k^{th}$ neuron.
• Figure 3: Addition of a positive shift operator to the composition of monotone operators makes the upper block monotone. The same shift is subtracted from the composition of the anti-monotone operator and the monotone operator to make it anti-monotone. This virtual addition does not alter the dynamics of the system.
• Figure 4: The effect of the all the synaptic connections of the network on the $k^{th}$ neuron is demonstrated here. The green link represents the aggregate current that the $k^{th}$ neuron receives from its synaptic connections. Given the active nature of the synaptic connections, they rely upon the voltage of the pre-synaptic neuron. The combination of the externally injected current, synaptic currents and internal dynamics' currents govern the behavior of the neuron.
• Figure 5: Simulation of a spiking neuron subjected to external current. The external injected current cannot excite the system if it is small and only creates small bumps in the potential. In contrast, by having a slightly larger external current, the neuron spikes. As it can be seen, the signal gets closer to the solution as the algorithm progresses and, it converges to it. These results are verified by an NI solver.

Theorems & Definitions (23)

• Definition 1
• Definition 2
• Definition 3
• Remark 1
• Definition 4
• Definition 5
• Theorem 1
• Remark 2
• Theorem 2
• proof