Impedance-based analysis of a neuromorphic oscillator near a Hopf bifurcation

Impedance-based analysis of a neuromorphic oscillator near a Hopf bifurcation Roberto Fenollosa*1, Juan Bisquert1 1Instituto de Tecnología Química (ITQ). Universitat Politècnica de ValènciaConsejo Superior de Investigaciones Científicas (UPV-CSIC). 46022 València, Spain As neuromorphic computing moves toward energy-efficient, event-driven architectures, the use of oscillator-based neurons has gained renewed interest. These systems aim to emulate the core dynamic features of biological spiking neurons while leveraging physical models that allow compact and tunable implementations. In this context, we investigate a neuromorphic system governed by a set of two coupled differential equations, structurally analogous to the Morris–Lecar model [1,2]. This reduced framework captures key excitability and oscillatory features characteristic of spiking neurons, while remaining analytically tractable. By drawing a formal analogy with an elementary electrical circuit composed of a resistor, a capacitor, and an inductor, we derive an analytical expression for the system’s impedance function [3]. This complex function describes the linear frequency response of the system to small periodic perturbations, and provides a natural bridge between the time-domain dynamics of the model and its frequency-domain characteristics. The impedance reveals how the system processes inputs across a range of frequencies, exhibiting features such as resonance and phase lag. Through numerical simulations, we explore how the system responds as a control parameter is varied. We observe a Hopf bifurcation that marks the transition from a stable fixed point to a regime of self-sustained oscillations [4]. Importantly, we find that this bifurcation is accompanied by a qualitative transformation in the impedance spectrum: the emergence of resonance peaks, frequency selectivity, and distinct shifts in phase response signal the onset of oscillatory behavior. These changes reflect a reorganization of the system’s internal time scales and nonlinear feedback structure. Our findings underscore the utility of impedance spectroscopy as a diagnostic and classification tool for neuromorphic oscillators. The method provides insight into critical transitions—such as the onset of spiking—and allows identification of dynamical regimes using experimentally accessible quantities. This approach is rooted in well-established techniques from electrochemistry and neuroscience, where impedance measurements have long been used to probe the behavior of both chemical and biological oscillators. Acknowledgments This work was funded by the European Research Council (ERC) via Horizon Europe Advanced Grant, grant agreement nº 101097688 (“PeroSpiker”), MENEU project (20250002) by Universitat Poltècnica de València, (CEX2021-001230-S grant by MCIN/AEI/10.13039/501100011033) of the Spanish Ministry of Science and Innovation References [1] C. Morris, H. Lecar, “Voltage oscillations in the barnacle giant muscle fiber,” Biophys J. 35, pp. 193-213 (1981). [2] E. M. Izhikevich, Dynamical Systems in Neuroscience, The MIT Press, Cambridge, Massachusetts, London, England, 2007, pp. 89, 117. [3] A. Bou, J. Bisquert, “Impedance spectroscopy dynamics of biological neural elements: from memristors to neurons and synapses,” J. Phys. Chem. B, 125, pp. 9934– 9949, 2021. [4] J. Bisquert, “Hopf bifurcations in electrochemical, neuronal, and semiconductor systems analysis by impedance spectroscopy,” Appl. Phys. Rev. 9, 011318, 2022.