Scale-covariant spiking wavelets

We establish a theoretical connection between wavelet transforms and spiking neural networks through scale-space theory. We rely on the scale-covariant guarantees in the leaky integrate-and-fire neurons to implement discrete mother wavelets that approximate continuous wavelets. A reconstruction experiment demonstrates the feasibility of the approach and warrants further analysis to mitigate current approximation errors. Our work suggests a novel spiking signal representation that could enable more energy-efficient signal processing algorithms.

💡 Research Summary

The paper “Scale‑covariant spiking wavelets” establishes a rigorous theoretical bridge between classical wavelet analysis and spiking neural networks (SNNs) by exploiting the scale‑covariant properties of leaky integrate‑and‑fire (LIF) neurons. The authors begin by reviewing the continuous wavelet transform, emphasizing the mother wavelet ψ(x;a,b)=|a|^{-1/2}ψ((x−b)/a) and the reconstruction formula that guarantees any square‑integrable signal can be perfectly recovered from its wavelet coefficients. They then introduce scale‑space theory, where a signal f(t) is represented at multiple temporal scales τ via convolution with a smoothing kernel g(t;τ). A key requirement is variation‑diminishing: as τ increases, the representation becomes smoother, preserving essential structures while discarding fine‑scale details.

To obtain a scale‑covariant representation, the authors adopt the “time‑causal limit kernel” Ψ, which is constructed as a cascade of truncated exponential kernels g_exp(t;µ) = (1/µ) e^{−t/µ} u(t). By logarithmically spacing the time constants µ_k (controlled by a distribution parameter c>1) and taking a finite number K of cascades, they approximate a kernel whose Laplace transform satisfies a specific product form guaranteeing both variation‑diminishing and scale‑covariance. The n‑th temporal derivative of Ψ, after L2‑normalisation, yields a continuous mother wavelet W(t;τ,c) that satisfies the admissibility condition ∫W(t)dt = 0.

The central contribution is mapping this continuous construction onto the dynamics of LIF neurons using the Spike‑Response Model (SRM). In the SRM, the membrane potential u(t) is expressed as the convolution of the input f(t) with a leaky integrator κ(t;µ) plus a reset term η(t) triggered by spikes z(t) = Σ_i δ(t−t_i). The resulting differential equation µ·du/dt = −u + f(t) − θ_thr·z(t) is shown to be mathematically equivalent to the scale‑space representation L(t;µ). Consequently, a single LIF neuron implements a scale‑covariant filter: scaling time by s and the membrane time constant by the same factor leaves the filtered output invariant.

Because spikes are inherently non‑negative, the authors introduce a two‑channel scheme to preserve sign information. One channel integrates f(t) (positive polarity) while the other integrates −f(t) (negative polarity). Each channel fires when its membrane potential crosses a threshold θ_thr, producing spike trains z⁺(t) and z⁻(t). At each scale µ_k they define a band‑pass kernel κ(t;µ_k) = C