ESL-SNNs: An Evolutionary Structure Learning Strategy for Spiking Neural Networks

Jiangrong ShenQi XuJian K. LiuYueming WangGang PanHuajin Tang

article2023AAAI63 citations

Proposes a biologically inspired evolutionary structure learning framework that enables training spiking neural networks from scratch with dynamic synaptic pruning and regeneration, achieving high accuracy with only a fraction of full network connectivity to minimize memory and power consumption.

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Spiking neural networks are brain-inspired artificial intelligence models designed for high energy efficiency, but expanding their depth to match modern AI performance causes substantial parameter redundancy. Traditional techniques rely on post-training pruning, which removes unnecessary connections only after dense training is complete. Consequently, these models still suffer from excessive computational cost and memory overhead throughout the training phase, hindering their deployment on resource-constrained embedded and neuromorphic hardware.

The article evaluates an evolutionary structure learning framework designed to train sparse spiking neural networks from scratch. Inspired by the dynamic rewiring and structural plasticity observed in biological brains, the framework aims to maintain a fixed, low connection density throughout both training and inference without inheriting weights from pre-trained dense models.

The researchers assessed the approach through empirical evaluations on multi-layer feedforward and deep convolutional spiking architectures. They initialized network connectivity as sparse random graphs and applied periodic connection pruning alongside dynamic regeneration rules—including momentum-based and gradient-based strategies—to explore the parameter space during training. Testing covered standard vision benchmarks, including static image datasets and neuromorphic dynamic vision sensor data, while measuring classification accuracy, parameter counts, and estimated energy consumption across GPU and TrueNorth neuromorphic platforms.

The key findings demonstrate that dynamic sparse training achieves high structural efficiency with minimal accuracy loss. On neuromorphic vision data, the sparse network reached 78.30% accuracy with only 10% connection density, exhibiting an accuracy drop of just 0.28% relative to dense models while outperforming several state-of-the-art architectures. In feedforward network tests, the model maintained 96.58% accuracy with roughly six times fewer connections than the fully connected baseline, while energy estimations showed that running the resulting sparse models on neuromorphic hardware provided approximately one order of magnitude higher energy efficiency compared to GPU implementations. Furthermore, pairing magnitude-based pruning with momentum-based growth proved most effective at discovering optimal network topologies.

These results show that resource-efficient training from scratch is viable for spiking neural networks, significantly lowering memory footprints and compute costs during training as well as inference. This capability facilitates on-chip training on embedded neuromorphic systems, reducing development time and operational energy expenditures without requiring costly dense pre-training.

Organizations developing edge AI and neuromorphic systems should consider adopting dynamic sparse learning frameworks to streamline model training pipelines and lower hardware power requirements. Before committing to large-scale operational deployment, engineering teams should conduct pilot implementations on physical neuromorphic embedded systems to validate real-world hardware latency and energy gains across domain-specific workloads.

Confidence in these findings is supported by consistent results across multiple architectures and standard benchmarks. However, readers should note that energy metrics were derived from theoretical estimations rather than direct physical hardware measurements, and evaluations focused primarily on visual classification tasks. Broader operational verification across diverse real-time tasks will help confirm generalizability.

Dalian University of Technology University of Leeds Zhejiang Lab Zhejiang University inference low-power consumption sparse approximations evolutionary structure learning strategy deep spiking neural networks ESL-SNNs

arXiv: 2306.03693

Paper: Rigging the Lottery: Making All Tickets Winners, Utku Evci et al. (2020). Introduces the RigL dynamic sparse training paradigm of periodic magnitude-based pruning and gradient-based connection regrowth during training from scratch, which ESL-SNNs adapts to spiking architectures.
Paper: Surrogate Gradient Learning in Spiking Neural Networks: Bringing the Power of Gradient-based optimization to spiking neural networks, Emre O. Neftci et al. (2019). Establishes the foundational surrogate gradient framework necessary to differentiate through non-smooth spiking thresholds and compute the parameter gradients used for network rewiring.
Paper: Spatio-Temporal Backpropagation for Training High-Performance Spiking Neural Networks, Yujie Wu et al. (2017). Formulates spatio-temporal backpropagation through time for deep spiking neural networks, providing the core gradient optimization mechanism ESL-SNNs relies on during sparse training.
Paper: Deep Learning in Spiking Neural Networks, Amirhossein Tavanaei et al. (2018). Surveys foundational training methodologies and architectural design principles for deep spiking neural networks.
Paper: Rethinking the Value of Network Pruning, Zhuang Liu et al. (2019). Demonstrates that training sparse architectures from scratch can match or exceed fine-tuning dense pre-trained models, providing key conceptual motivation for ESL-SNNs.
Paper: SNIP: Single-shot Network Pruning based on Connection Sensitivity, Namhoon Lee et al. (2018). Provides foundational insights into single-shot connection pruning at initialization to avoid computationally expensive dense pre-training cycles.
Paper: The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks., Jonathan Frankle et al. (2019). Establishes the existence of sparse, trainable sub-networks at random initialization, inspiring dynamic rewiring methods that seek optimal sparse topologies.
Paper: Learning both Weights and Connections for Efficient Neural Networks, Song Han et al. (2015). Introduces classical magnitude-based iterative pruning for neural networks, which ESL-SNNs integrates as its primary connection removal criterion.

Paper: Resource-Efficient Neural Networks for Embedded Systems, Wolfgang Roth et al. (2024). Provides a comprehensive hardware-centric evaluation of neural network compression and structural sparsity on embedded devices and neuromorphic hardware.
Paper: SpikingResformer: Bridging ResNet and Vision Transformer in Spiking Neural Networks, Xinyu Shi et al. (2024). Extends efficient spiking architectures into vision transformer and residual hybrid models, offering advanced topologies suitable for sparse dynamic learning.
Paper: DepGraph: Towards Any Structural Pruning, Gongfan Fang et al. (2023). Develops dependency graph mapping to automate structural pruning across diverse neural architectures, generalizing beyond layer-by-layer rewiring.

Cover for ESL-SNNs: An Evolutionary Structure Learning Strategy for Spiking Neural Networks

Abstract

Spiking neural networks (SNNs) have manifested remarkable advantages in power consumption and event-driven property during the inference process. To take full advantage of low power consumption and improve the efficiency of these models further, the pruning methods have been explored to find sparse SNNs without redundancy connections after training. However, parameter redundancy still hinders the efficiency of SNNs during training. In the human brain, the rewiring process of neural networks is highly dynamic, while synaptic connections maintain relatively sparse during brain development. Inspired by this, here we propose an efficient evolutionary structure learning (ESL) framework for SNNs, named ESL-SNNs, to implement the sparse SNN training from scratch. The pruning and regeneration of synaptic connections in SNNs evolve dynamically during learning, yet keep the structural sparsity at a certain level. As a result, the ESL-SNNs can search for optimal sparse connectivity by exploring all possible parameters across time. Our experiments show that the proposed ESL-SNNs framework is able to learn SNNs with sparse structures effectively while reducing the limited accuracy. The ESL-SNNs achieve merely 0.2% accuracy loss with 10% connection density on the DVS-Cifar10 dataset. Our work presents a brand-new approach for sparse training of SNNs from scratch with biologically plausible evolutionary mechanisms, closing the gap in the expressibility between sparse training and dense training. Hence, it has great potential for SNN lightweight training and inference with low power consumption and small memory usage.

Topics

inference low-power consumption sparse approximations spiking neural networks structure-learning evolutionary process sparse recovery

Keywords

evolutionary structure learning strategy deep spiking neural networks ESL-SNNs sparse SNNs neuromorphic chip synaptic plasticity connection evolution sparse training synaptic weights

Introduction
Related Work
Method
The ESL-SNNs Framework
Multi-Layer ESL-SNNs
Convolutional ESL-SNNs
Results
The Experiment Settings
Evaluation of Multi-Layer ESL-SNNs
Evaluation of Convolutional ESL-SNNs
Conclusion
Acknowledgements
References

Knowls

Knowl 1 — ESL-SNNs Dynamic Structure Learning Algorithm
algorithm
The Evolutionary Structure Learning for Spiking Neural Networks (ESL-SNNs) framework trains sparse SNNs from scratch while preserving a constrained overall synaptic density. At initialization, sparse layers are assigned connectivity masks generated from an Erdős–Rényi random topology. During training, standard forward and backward propagation update the active weights. Every $T_{\text{iter}}$ training iterations, a fraction $\alpha$ of the active synapses is pruned based on lowest magnitude, and an equal number of new connections are regenerated via a growth rule (e.g., momentum-based growth), with newly regenerated synaptic weights initialized to zero. The fraction $\alpha$ decays over iterations following a cosine annealing schedule.
Input: Training data $\{x_i, c_i\}_{i=1}^N$ , initial rewiring fraction $\alpha$ , mask update interval $T_{\text{iter}}$ , final rewiring iteration $T_{\text{end}}$ , network parameter matrices $W$
Output: Trained sparse weight matrices $W$ and final connection masks $M$
for each assigned sparse layer $k$ do
    Initialize binary mask $M^{(k)}$ using Erdős–Rényi random topology
    Apply mask to weights: $W^{(k)} \leftarrow M^{(k)} \odot W^{(k)}$
end for
Initialize training parameters and optimizers
for iteration $i = 1$ to total_iterations do
    Perform standard forward and backward training pass
    Update active weights $W$
    if $i \pmod{T_{\text{iter}}} = 0$ and $i \le T_{\text{end}}$ then
        Compute current update fraction $\alpha_i = \frac{\alpha}{2} (1 + \cos(\frac{i \pi}{T_{\text{end}}}))$
        for each assigned sparse layer $k$ do
            Prune fraction $\alpha_i$ of active connections in $M^{(k)}$ with lowest absolute weight
            Regenerate fraction $\alpha_i$ of inactive connections in $M^{(k)}$ according to the growth rule
            Initialize weights of newly activated connections to 0
            Update masked weight matrix: $W^{(k)} \leftarrow M^{(k)} \odot W^{(k)}$
        end for
    end if
end for
return Sparse network weights $W$
Knowl 2 — Erdős–Rényi Random Graph Sparse Layer Initialization
equation
In ESL-SNNs, the initial connectivity topology of a sparse layer is generated using an Erdős–Rényi random graph. For a sparse layer $H_k$ containing $n^k$ neurons connected to a preceding layer $H_{k-1}$ containing $n^{k-1}$ neurons with weight matrix $W \in \mathbb{R}^{n^{k-1} \times n^k}$ , the probability $p(w_{ij})$ that an initial synaptic connection exists between neuron $h_i^k \in H_k$ and neuron $h_j^{k-1} \in H_{k-1}$ is given by:

$p(w_{ij}) = \frac{\epsilon (n^k + n^{k-1})}{n^k \cdot n^{k-1}}$

where $\epsilon$ is a scalar factor that directly modulates the global sparsity and initial connection density of the layer.
Knowl 3 — Dynamic Synaptic Pruning, Growth, and Annealing Rules
model/method
Synaptic connection rewiring in ESL-SNNs occurs every $T_{\text{iter}}$ iterations and comprises two symmetric operations: pruning and growth.
1. Magnitude-based pruning: For each sparse layer, the fraction $\alpha$ of currently active connections whose absolute weight values $|w_{ij}|$ are closest to zero is removed ( $m_{ij} = 0$ ).
2. Synaptic growth: An equal number of pruned connections are reactivated ( $m_{ij} = 1$ ) to keep layer sparsity constant, with their initial weight values set to zero. Regrowth strategies include:
  - Momentum-based growth: Selects candidate zero-valued connections with the largest parameter momentum.
  - Gradient-based growth: Selects candidate connections with the highest instantaneous gradient magnitude.
  - Random unfired growth: Selects candidate connections uniformly at random, prioritizing synapses that have remained dormant since initialization.
3. Cosine annealing schedule: The update fraction decays over training iteration $t$ according to:
$f_{\text{decay}}(t; \alpha, T_{\text{end}}) = \frac{\alpha}{2} \left(1 + \cos\left(\frac{t\pi}{T_{\text{end}}}\right)\right)$

where $\alpha$ is the baseline rewiring fraction and $T_{\text{end}}$ is the terminal iteration after which structural connectivity is frozen.
Knowl 4 — Multi-Layer Feedforward Single-Spike ESL-SNN Formulation
model/method
For multi-layer feedforward ESL-SNNs employing single-spike temporal coding, non-leaky integrate-and-fire neurons with exponentially decaying synaptic current kernels are used. For a layer $L_I$ with $N_I$ presynaptic neurons, the membrane potential $V_j(t)$ of postsynaptic neuron $j$ is:

$V_j(t) = \sum_{i=1}^{N_I} \Theta(t - t_i) w_{ij} m_{ij} (1 - \exp(-(t - t_i)))$

where $t_i$ is the firing time of presynaptic neuron $i$ , $\Theta(\cdot)$ is the Heaviside step function, $w_{ij}$ is the synaptic weight, and $m_{ij} \in \{0, 1\}$ is the evolutionary binary mask. When $V_j(t)$ reaches the firing threshold $V_{\text{thr}} = 1$ , neuron $j$ fires at time $t_j$ . Defining the causal set $C_j = \{i : t_i < t_j\}$ , the first spike time in the transformed $z$ -domain ( $z = \exp(t)$ ) is analytically expressed as:

$z_j = \frac{\sum_{i \in C_j} w_{ij} m_{ij} z_i}{\sum_{i \in C_j} w_{ij} m_{ij} - 1}$

Given the output spike times $z_o$ in the $z$ -domain and target class label $g$ , the network is trained end-to-end using the $z$ -domain cross-entropy loss:

$L_{z\text{-domain}}(g, z_o) = -\ln \left( \frac{\exp(-z_o[g])}{\sum_k \exp(-z_o[k])} \right)$
Knowl 5 — Convolutional Multi-Spike ESL-SNN Formulation and TET Loss
model/method
Convolutional ESL-SNNs utilize iterative Leaky Integrate-and-Fire (LIF) neurons discretized by the Euler method. Over discrete time steps $t$ , the membrane potential $u(t)$ updates according to:

$u(t) = \tau u(t-1) + I(t)$

$a(t+1) = \Theta(u(t+1) - V_{\text{th}})$

$u(t+1) = u(t+1)(1 - a(t+1))$

where $\tau = 0.5$ is the membrane leak factor, $I(t) = (W \odot M) x(t)$ represents presynaptic input masked by the evolutionary topology mask $M$ , $V_{\text{th}}$ is the firing threshold, and $a(t+1) \in \{0, 1\}$ is the binary spike output. Output layer neurons accumulate presynaptic inputs without leak or thresholding. The network is supervised across $T$ simulation time steps via the Temporal Efficient Training (TET) loss:

$L_{\text{TET}} = \frac{1}{T} \sum_{t=1}^T L_{\text{CE}}[O(t), y]$

where $L_{\text{CE}}$ is the standard cross-entropy loss between instantaneous output $O(t)$ and true class label $y$ .

Knowl 6 — Multi-Layer ESL-SNN Accuracy and Energy Benchmarks on MNIST

data/table

A three-layer feedforward architecture (784-800-10) trained with ESL-SNNs from scratch achieves comparable accuracy to a fully connected dense baseline while requiring over six times fewer parameters. The sparse ESL-SNN also demonstrates superior accuracy over a dense network scaled down to match its parameter count (103K parameters without pruning reaches ~92% accuracy, compared to 96.58% for ESL-SNN).

Model	Test Accuracy (%)	Accuracy Loss (%)	Connection Density	FLOPS	Energy on GPU (J)	Energy on TrueNorth (J)
SNNs	96.70	–	1.00	635K	$1.13 \times 10^{-5}$	$7.95 \times 10^{-6}$
ESL-SNNs	96.58	-0.12	0.16	103K	$1.84 \times 10^{-6}$	$1.29 \times 10^{-6}$

Energy measurements on neuromorphic hardware (TrueNorth) demonstrate approximately one order of magnitude higher energy efficiency compared to GPU execution (Titan V100).

Knowl 7 — Dynamic Layer-Wise Sparsity Redistribution in Convolutional ESL-SNNs
empirical result
When sparse convolutional VGGSNN models are trained from scratch using ESL-SNNs, the connection densities of individual layers dynamically redistribute during training before stabilizing around epoch 200. The evolutionary rewiring automatically allocates higher connection densities to shallow layers with fewer total parameters (for example, the early 128-channel $3 \times 3$ convolutional layer evolves to 47% density) and allocates substantially sparser connectivity to deeper layers with large parameter counts (for example, the 512-channel $3 \times 3$ convolutional layer evolves to 8% density). This demonstrates that parameter redundancy in SNNs resides predominantly in deep layers with large parameter counts.
Knowl 8 — Synaptic Growth Strategy Comparison in ESL-SNN Rewiring
empirical result
Evaluating different synaptic growth rules (momentum-based, gradient-based, random, and random-unfired) combined with pruning rules (magnitude and SET) on DVS-CIFAR10 across global connection densities of 1.0, 0.1, and 0.01 demonstrates that:
1. Combining the SET pruning rule with the momentum-based growth rule yields the highest stability and accuracy retention across extreme sparsity levels, suffering only a 0.28% accuracy drop at 10% connection density and a 14.08% drop at 1% connection density.
2. Gradient-based growth rules perform poorly at high sparsity because instantaneous gradient information causes regenerated connections to collapse repeatedly into similar topologies, restricting topological parameter exploration across time.

Knowl 9 — Convolutional ESL-SNN Classification Performance Across Vision Benchmarks

data/table

Convolutional ESL-SNN architectures (Sparse ResNet-19 and Sparse VGGSNN) trained from scratch achieve competitive classification accuracy under significant parameter reductions across static and neuromorphic datasets.

Dataset	Methods	Network Architecture	Test Accuracy (%)	Accuracy Loss (%)	Connection Density
CIFAR-10	ADMM-based	7Conv, 2FC	89.53	-3.85	0.10
CIFAR-10	Grad R	6Conv, 2FC	92.84	-0.34	0.12
CIFAR-10	TET (Dense)	ResNet-19	92.79	–	1.00
CIFAR-10	ESL-SNNs	Sparse ResNet-19	91.09	-1.70	0.50
CIFAR-100	TET (Dense)	ResNet-19	74.47	–	1.00
CIFAR-100	ESL-SNNs	Sparse ResNet-19	73.48	-0.99	0.50
DVS-CIFAR10	Streaming Rollout	DenseNet	66.80	–	1.00
DVS-CIFAR10	Conv3D	LIAF-Net	71.70	–	1.00
DVS-CIFAR10	TET (Dense)	VGGSNN	78.58	–	1.00
DVS-CIFAR10	ESL-SNNs	Sparse VGGSNN	78.30	-0.28	0.10

On the neuromorphic DVS-CIFAR10 benchmark, ESL-SNNs achieve 78.30% accuracy with 10% connection density, outperforming dense baseline models such as DenseNet (66.80%) and LIAF-Net (71.70%) while losing only 0.28% relative to dense VGGSNN.

Coverage note — No substantial contributed material was omitted.

References

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Citation

MLA

Shen, J., et al. “ESL-SNNs: An Evolutionary Structure Learning Strategy for Spiking Neural Networks”. arXiv, 2023, http://arxiv.org/abs/2306.03693v1.

APA

Shen, J., Xu, Q., Liu, J. K., Wang, Y., Pan, G., & Tang, H. (2023). ESL-SNNs: An Evolutionary Structure Learning Strategy for Spiking Neural Networks. arXiv. http://arxiv.org/abs/2306.03693v1

Chicago

Shen, J., Q. Xu, J. K. Liu, Y. Wang, G. Pan, and H. Tang. 2023. “ESL-SNNs: An Evolutionary Structure Learning Strategy for Spiking Neural Networks”. arXiv. http://arxiv.org/abs/2306.03693v1.

Harvard

Shen, J. et al. (2023) “ESL-SNNs: An Evolutionary Structure Learning Strategy for Spiking Neural Networks”, arXiv [Preprint]. Available at: http://arxiv.org/abs/2306.03693v1.

Vancouver

1. Shen J, Xu Q, Liu JK, Wang Y, Pan G, Tang H (2023) ESL-SNNs: An Evolutionary Structure Learning Strategy for Spiking Neural Networks. arXiv

BibTeX

@article{shen2023esl,
  title = {ESL-SNNs: An Evolutionary Structure Learning Strategy for Spiking Neural Networks},
  author = {Shen, Jiangrong and Xu, Qi and Liu, Jian K. and Wang, Yueming and Pan, Gang and Tang, Huajin},
  year = {2023},
  journal = {arXiv},
  url = {http://arxiv.org/abs/2306.03693v1},
  eprint = {2306.03693}
}

Metadata:arXiv

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ESL-SNNs: An Evolutionary Structure Learning Strategy for Spiking Neural Networks

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — ESL-SNNs Dynamic Structure Learning Algorithm

Knowl 2 — Erdős–Rényi Random Graph Sparse Layer Initialization

Knowl 3 — Dynamic Synaptic Pruning, Growth, and Annealing Rules

Knowl 4 — Multi-Layer Feedforward Single-Spike ESL-SNN Formulation

Knowl 5 — Convolutional Multi-Spike ESL-SNN Formulation and TET Loss