Liquid Capacitance-Extended Neural Circuits: Synaptic Activation and Dual Liquid Dynamics for Interpretable Bio-Inspired Models

Abstract

Previous work investigated saturated Electrical Equivalent Circuits (EECs) using either electrical or chemical synapses in sparse or fully connected wiring architectures. These studies demonstrated that sparse topologies enhance interpretability and that chemical synapses outperform electrical synapses in robustness. Particularly, compared chemical synapse-based Liquid Time Constant (LTC) networks against electrical synapse-based Continuous-Time Recurrent Neural Networks (CT-RNNs), both rooted in neural ordinary differential equations (Neural ODEs) requiring computationally intensive hybrid ODE solvers for precise integration. Recent advances introduced a liquid capacitance term into biologically inspired saturated LTC models, yielding Liquid Resistance Liquid Capacitance (LRC) networks. This modification was motivated by findings showing nonlinear dependencies of the membrane capacitance on input and state variables. Unlike LTCs, LRCs exhibit smoother dynamics due to their input- and state-dependent capacitance-resistance terms, enabling simpler explicit Euler solvers (one-step integration) instead of costly hybrid methods while keeping interpretable dynamics. Building on these foundations, we extend the liquid capacitance concept to electrical synapse models - termed Liquid Capacitance (LC) networks - and systematically analyze mixed architectures. By default, chemical synapses have an activation per synapse, and electrical synapses have an activation per neuron. To have a systematic analysis, we also consider mixed models, both LCs and LRCs.