Combining Convolution and Delay Learning in Recurrent Spiking Neural Networks

Abstract: Spiking neural networks (SNNs) are rapidly gaining momentum as an alternative to conventional artificial neural networks in resource constrained edge systems. In this work, we continue a recent research line on recurrent SNNs where axonal delays are learned at runtime along with the other network parameters. The first proposed approach, dubbed DelRec, demonstrated the benefit of recurrent delay learning in SNNs. Here, we extend it by advocating the use of convolutional recurrent connections in conjunction with the DelRec delay learning mechanism. According to our tests on an audio classification task, this leads to a streamlined architecture with smaller memory footprint (around 99% savings in terms of number of recurrent parameters) and a much faster (52x) inference time, while retaining DelRec's accuracy. Our code is available at: https://github.com/luciozebendo/delrec_snn/tree/conv_delays

• The paper introduces a CRSNN architecture that replaces dense recurrent connections with a 1D convolutional kernel and integrates adaptive delay learning.
• It achieves over 99.9% parameter reduction and a 52x inference speedup while closely matching performance benchmarks on audio datasets.
• The study demonstrates that learned axonal delays are critical for robust temporal processing, aligning with biological principles to enable efficient neuromorphic design.

Convolutional Delay Learning in Recurrent Spiking Neural Networks: Architecture and Implications

Introduction

This paper addresses the integration of convolutional connectivity with axonal delay learning in recurrent spiking neural networks (RSNNs), targeting memory and computational constraints inherent to dense recurrence for temporally-structured data. Building on the DelRec framework—where recurrent delays are learned alongside weights—the authors propose a hybrid model (CRSNN) that employs lightweight 1D convolutional recurrence while preserving the delay learning mechanism. The primary motivation is the redundancy of dense recurrent connections for data modalities such as speech and cochlear spectrograms, where significant spatial locality exists.

Methodological Advancements

Convolutional Recurrence for RSNNs

The key architectural change is the substitution of the quadratic recurrent weight matrix $W_{\text{rec}} \in \mathbb{R}^{N \times N}$ with a 1D convolutional kernel of size $k$, typically $k = 3$. This confines each neuron's recurrent receptive field to itself and its immediate neighbors, leveraging the local statistical regularity in time-frequency audio representations.

DelRec’s delay learning mechanism is retained with per-neuron, real-valued axonal delays, enabling differentiable gradient-based delay adaptation via a triangular spread function. The spread width is annealed, ensuring smooth optimization and discrete delays at inference time. Convolution is implemented as cross-correlation along the neuronal dimension, compatible with modern deep learning libraries, yielding efficient parameterization and fast execution.

Implementation Considerations

The recurrent pathway composes a spike scheduling buffer managed as a circular matrix indexed by learned delays. The convolution is applied along the neuron dimension, efficiently mapped to a 2D convolution for batched operation. Initialization follows Kaiming (for recurrent weights) and uniform or half-normal distributions (for delays), maintaining compatibility with standard training protocols.

Numerical Results

Parameter and Computational Efficiency

The convolutional formulation reduces recurrent parameter count from $N^2$ to $k$, which for $N=256$, equates to a reduction from 65,536 to 3 per layer—a savings of over 99.9%. The architecture achieves a 52x reduction in inference time compared to the dense DelRec reference on the Spiking Heidelberg Digits (SHD) audio dataset. This efficiency enables deeper/wider networks suitable for resource-limited neuromorphic deployments.

Performance Analysis

On SHD, the convolutional approach attains 91.51% ± 0.70% accuracy, closely tracking the dense DelRec (91.72% ± 0.84%) despite the severe parameter pruning. Performance variance is reduced, indicating more stable and robust convergence properties. On the Spiking Speech Commands (SSC) dataset, the model achieves 78.59% ± 0.39%, with a moderate but notable reduction relative to the original DelRec, attributed to potential sensitivity in hyperparameter selection.

The ablation study establishes the criticality of delay learning: a fixed delay assignment results in a drop of more than 5pp on SHD, directly quantifying the importance of adaptive delay mechanisms for sequence processing.

Delay Statistics and Temporal Modeling

Learned delay distributions are more heterogeneous under the convolutional regime, with higher standard deviations and wider ranges across neurons. This suggests that local connectivity allows for richer, more diverse temporal receptive fields and may promote more distributed representations of temporal dependencies.

Theoretical and Practical Implications

The results demonstrate that the dense recurrent matrix in standard RSNNs is superfluous for domains with strong local spatial correlations, as in cochlear-encoded speech signals. Local connectivity, combined with trainable synaptic delays, suffices for expressive modeling of long-range temporal dependencies without the penalty of redundant parameters. This aligns temporal credit assignment with the biological principle of polychronization, exploiting precise spike-timing and delay diversity.

Practically, these findings unlock scalable, resource-efficient SNN designs for streaming and edge inference, with deployment directly feasible on neuromorphic substrates where memory and energy are severely constrained.

Future Directions

Potential lines of research include:

• Extending convolutional-delay models to higher-dimensional data or other sensor modalities.
• Investigating adaptive kernel sizes or dynamic local/topological connectivity, potentially guided by input statistics.
• Exploring the interplay of learned delays with more biologically-plausible neuron models or intrinsic adaptive processes.
• Examining the effect of delay learning and parameter reduction on adversarial robustness and domain transfer in SNNs.
• Applying the methodology to online continual learning and event-driven sensory signal processing.

Conclusion

This work establishes that convolutional recurrence augmented with learned axonal delays supplies RSNNs with the means for efficient and robust temporal modeling. Substantial memory and inference time reductions are achieved with negligible loss in accuracy for audio-based sequential benchmarks. Delay learning remains essential for high-performing temporal neural architectures. The approach enables practical large-scale SNN deployment, especially for edge and neuromorphic hardware targeting real-time sequential tasks.

Reference: "Combining Convolution and Delay Learning in Recurrent Spiking Neural Networks" (2604.15997)