Resonate-and-Fire Neuron

• Resonate-and-fire neurons are spiking models defined by underdamped oscillatory subthreshold dynamics and a hard threshold, providing sharp bandpass filtering.
• Their linear resonant circuit design supports temporal feature extraction and frequency-domain multiplexing, making them ideal for sparse, event-driven computing.
• They are applied in neuromorphic systems and edge processing, offering scalable, energy-efficient realizations in hardware such as CMOS and photonic platforms.

A resonate-and-fire neuron is a spiking neuron model characterized by intrinsic underdamped oscillatory subthreshold dynamics, conferring frequency selectivity distinct from classical integrate-and-fire models. Resonate-and-fire neurons combine a linear resonant circuit core—most fundamentally a damped harmonic oscillator in continuous or discrete time—with a hard spike threshold and reset mechanism, yielding event-driven architectures well suited for temporal feature extraction, frequency-domain multiplexing, and sparse but expressive coding. This design modality supports both theoretical analysis and hardware implementation, and underpins several recent advances in neuromorphic computing, deep spiking neural networks, and edge signal processing.

1. Mathematical Formalism and Subthreshold Oscillations

The resonate-and-fire neuron’s core is a second-order linear differential equation capturing subthreshold dynamics:

$$\begin{aligned} \dot{x} &= b x - \omega y + I(t) \ \dot{y} &= \omega x + b y \end{aligned}$$

or, equivalently, in complex notation,

$\dot{z} = (b + i\omega) z + I(t)$

where $z = x + i y$; $x$ is the current-like (recovery) variable, $y$ is the voltage-like (membrane) variable, $\omega$ sets the intrinsic angular (resonant) frequency, $b$ is the damping constant ($b<0$ for stable focus), and $I(t)$ is external input current. Discrete-time implementations use forward Euler:

$$z_{t+\tau} = z_t + \tau\left[(b + i\omega)z_t + I_t\right]$$

In the absence of input, the solution is a damped oscillation at frequency $\omega$ and decay rate $|b|$: $z(t) = z(0) \exp\left[(b + i\omega)t\right]$ The oscillatory phase and amplitude enable “subthreshold resonance”: responses peak for input frequencies near $\omega$ and decay elsewhere, conferring sharp bandpass filter characteristics. The quality factor is $Q = \omega / |2b|$; negative $b$ increases resonance sharpness and persistence (Tarasenko, 2015, Higuchi et al., 2024).

2. Spiking Threshold, Reset, and Variants

Spiking is triggered when the voltage variable crosses a threshold ($y \geq V_{\mathrm{th}}$ or, more generally, $\Re(z) \geq \vartheta$). The reset mechanism may take different forms:

• Hard reset: $y \to y_{\mathrm{reset}}$, $x$ unchanged, or in complex form $z \to y_{\mathrm{reset}} + ix$.
• Soft/parameterized reset: State reset or damping/threshold adaptation.

Reset rules impact spiking regularity, phase, and network behavior. In the balanced resonate-and-fire (BRF) neuron, additional mechanisms such as a dynamic refractory threshold and analytically enforced stability boundary prevent runaway oscillations and ensure stable, sparse spiking (Higuchi et al., 2024, Higuchi et al., 2024).

3. Frequency Selectivity and Resonance Coding

The hallmark of resonate-and-fire neurons is frequency selectivity. If driven by a sinusoidal input $I(t) = A \sin(\omega_{\text{in}} t)$, response amplitude is maximized when $\omega_{\text{in}} = \omega$, with selectivity tuned by $b$, as

$$|\Im[z(\omega_{\text{in}})]| \propto \frac{A\,\omega_{\text{in}}}{\sqrt{b^2 + (\omega - \omega_{\text{in}})^2}}$$

This creates a sharply tuned temporal filter, supporting phase-sensitive coding and frequency multiplexing—multiple neurons with distinct $\omega$ values can be co-activated on a shared medium, each responding only to its “addressed” frequency (Tarasenko, 2015, Shaaban et al., 2024). Frequency and phase selectivity are robust to moderate parameter noise and intrinsic noise sources, supporting efficient implementation in noisy hardware (Liu et al., 15 Nov 2025).

4. Computational Roles and Circuit Realizations

Resonate-and-fire neurons implement a computationally lightweight mechanism for spectral decomposition and temporal pattern detection. In neuromorphic and hardware-oriented settings, state variables are realized as capacitor voltages, and the oscillator is constructed with minimal elements (e.g., MOSFET-based negative differential resistance, LC tanks, or bulk-driven transconductance amplifiers) (Nabil et al., 3 Jun 2025, Liu et al., 15 Nov 2025, Leo et al., 8 Dec 2025). Their event-driven spiking yields sparse, energy-efficient encoding. Photonic and spintronic implementations exploit materials with intrinsic negative differential resistance to realize GHz-class operation and ultralow energy per spike (fJ range) suitable for high-density associative memories and hybrid neural networks (Azam et al., 2018, Adair et al., 16 Oct 2025).

Key performance and design metrics (extracted from empirical reports):

Technology	Area (µm²)	Power (nW)	Freq. range (Hz)	Energy/spike
22nm FDSOI RAF	6524	1.6–132.6	100 – 500k	265 fJ
350nm CMOS (nonlin)	~10,000	>1,000	1 – 50k	>20 pJ
Photonic RTD	sub-mm²	<0.1	~20M (optics-limited)	sub-pJ
Skyrmion–MTJ	0.01	—	1–6G (GHz)	1 fJ

All tabulated values are reported in (Liu et al., 15 Nov 2025, Azam et al., 2018, Leo et al., 8 Dec 2025, Adair et al., 16 Oct 2025).

5. Generalization: Balanced, Dendritic, and Parallel RF Neurons

Advanced architectures extend the standard resonate-and-fire neuron via dendritic composition, parallelization, and adaptive reset:

• Balanced resonate-and-fire (BRF): Introduces adaptive threshold/refractory, smooth reset, and an analytical "divergence boundary" ensuring discrete-time stability. BRF neurons exhibit superior training convergence, gradient stability, and spike efficiency relative to adaptive-leaky-integrate-and-fire (ALIF) neurons in deep recurrent SNNs. The loss landscape is empirically nearly convex, and gradient backpropagation through time is well conditioned (Higuchi et al., 2024, Higuchi et al., 2024).
• Dendritic-RF (D-RF): Aggregates multiple RF branches per neuron, each tuned to different frequency bands. Linear combination and adaptive soma thresholds provide enhanced spectral coverage, sparse activity, and scalable O(L log L) training/inference, with robust performance on long-sequence benchmarks (Zhang et al., 21 Sep 2025).
• Parallel resonate-and-fire (PRF): Leverages complex-state ODEs and a parallelizable reset mechanism to accelerate SNN training to O(L log L) time for long sequences, with accuracy matching state-space models and drastically reduced energy (Huang et al., 2024).

6. Applications: Neuromorphic Processing and Edge Computing

Practical uses of resonate-and-fire architectures span radar hand-gesture recognition (direct range estimation without FFT (Shaaban et al., 2024)), audio and speech classification (spiking keyword spotting (Liu et al., 15 Nov 2025, Huber et al., 1 Apr 2025)), sequence learning (sMNIST, LRA, SHD (Zhang et al., 21 Sep 2025, Huang et al., 2024)), and neuromorphic wireless split computing for low-power IoT (spectrum analysis and compressed spike communication (Wu et al., 24 Jun 2025)). Hardware implementations demonstrate reliable event-driven operation, high frequency tunability (~Hz to GHz), and robust performance despite process, voltage, and temperature variations (Leo et al., 8 Dec 2025, Liu et al., 15 Nov 2025).

7. Synchronization, Phase Locking, and Network Functions

Resonate-and-fire neurons support robust synchrony through both spike-mediated and subthreshold oscillatory coupling. Phase reduction theory shows that when neurons exhibit significant post-spike plateau potentials, subthreshold voltage contributes strongly to synchrony, quantified via the phase response curve (PRC) and “reset-induced shear” effects (Chartrand et al., 2018). Grazing bifurcation theory formalizes the emergence of robust, phase-locked spiking even in the presence of stochastic threshold variability—key for encoding timing in fluctuating oscillatory regimes (e.g., theta rhythm) (Makarenkov et al., 15 Oct 2025). This underpins network-level rhythmogenesis found in models of central pattern generators and oscillatory circuits in biological systems (Tolmachev et al., 2018).

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Resonate-and-fire neurons offer a unified framework connecting biological resonance, event-driven computation, and engineered frequency coding. Modern extensions and silicon-level realizations have established their practical relevance in neuromorphic SNN hardware, temporal learning, and robust, energy-efficient AI systems (Higuchi et al., 2024, Liu et al., 15 Nov 2025, Huang et al., 2024, Zhang et al., 21 Sep 2025).