Energy dynamics and circuit implementation for a neuron with a memcapacitive membrane

Abstract

The output voltages for the capacitive elements of a neural circuit model can be mapped into dimensionless capacitive variables, which present firing patterns similar to the membrane potentials detected in biological neurons. The inclusion of a memcapacitor also enables consideration of membrane deformation effects, enhancing the model’s capacity to simulate neuronal behavior across varying physiological and environmental conditions. In this study, a capacitor and a memcapacitor are connected through a linear resistor in parallel with other electric components in different branch circuits composed of an inductor and a nonlinear resistor. The electrical activities in a neuron with a double-layer membrane and two capacitive variables are discussed in detail after converting the nonlinear equations for the neural circuit into a theoretical neuron model. A dimensionless neuron model and its corresponding energy function are derived. The field energy function for the neural circuit is converted into an equivalent Hamilton energy function and further validated via the Helmholtz theorem. Furthermore, the average value of energy serves as an indicator for predicting stochastic resonance, as supported by analyzing the distribution of the coefficient of variation. The neuronal firing patterns are shown to be energy-dependent. An adaptive control strategy is proposed to regulate mode transitions in electrical activities of the neuron. An analog equivalent circuit is constructed to experimentally verify the numerical results, thereby supporting the reliability of the proposed neuron model.

1. Introduction

The human brain comprises approximately 10¹¹ neurons interconnected by roughly 10¹⁴ synapses (Herculano-Houzel, 2009). Complex network studies show that these networks exhibit small-world and modular structures with highly connected hub regions, which support efficient information processing and robust dynamics (Bullmore and Sporns, 2009). Inspired by these architectures, neuromorphic computing has become a challenge that aims to emulate the brain’s massively parallel, low-power, and fault-tolerant operation (Indiveri et al., 2013; Beilliard and Alibart, 2021; Huang et al., 2021). Fast and effective signal processing in artificial neural networks is worthy of investigation in terms of model approach and algorithm reliability. In contrast, conventional von Neumann architectures, with strictly separated memory and processing units, incur a severe data “traffic jam” (memory bottleneck) and high energy cost when scaled to brain-like tasks. To overcome this, neuromorphic systems collocate memory and computation (for example, via memristive synapses) so that the weights and processing occur in place (Dong et al., 2021; Li Y and Shen, 2022; Li JY et al., 2023).

From a physical perspective, the membrane potentials of neurons can be effectively modeled by using capacitive components such as traditional capacitors and memcapacitors, whose main characteristics are electric field and charge levels associated with field energy. Modeling the electrical behavior of neurons has a long history, with simplified theoretical models capturing key neuronal features such as spiking, excitability, and bistability, while remaining analytically tractable (Cebrián-Lacasa et al., 2024). In particular, the FitzHugh–Nagumo (FHN) model, introduced in the early 1960s, is a canonical two-variable excitable cell model that reproduces relaxation oscillations, bistability, and excitability to external stimuli (FitzHugh, 1961). As one of the foundational reduced models, FHN condenses the complex Hodgkin–Huxley system into a tractable form amenable to analysis and experimental interpretation.

In recent decades, the FHN model and its improved versions have served as a theoretical and experimental approach tested for understanding excitable media and neuronal synchronization, including studies on coupled pairs and pattern formation (Yanagita et al., 2005; Plotnikov and Fradkov, 2019; Scialla et al., 2021). Coupled FHN neurons have been extensively analyzed for their synchronization behavior, revealing multi-stability, phase-repulsive and phase-attractive regimes, and the onset of complex firing patterns, including cyclic and chaotic dynamics. Extensions to diffusive and boundary-coupled net