Power-Law Scaling in the Classification Performance of Small-Scale Spiking Neural Networks

TL;DR

This study probes how classification accuracy in ultra-small spiking neural networks with Leaky Integrate-and-Fire neurons scales with network size $N$, stimulus nodes $S$, and task complexity $C$. The authors combine detailed network simulations with an AI-assisted discovery workflow, comparing linear, polynomial, symbolic regression, and random forest approaches, and derive an interpretable additive power-law model: $Accuracy = p_1 \cdot C^{p_2} + p_3 \cdot S + p_4 \cdot N + p_5$, with $p_2 \approx -0.054$. A key finding is that the power-law scaling arises from network processing rather than input structure, as direct stimulus classification shows no meaningful $C$-dependent scaling. This work demonstrates the value of human-AI collaboration for formulating concise, mechanistic scaling laws and offers design principles for resource-efficient neuromorphic computing and AI-enabled scientific discovery.

Abstract

This paper investigates the classification capability of small-scale spiking neural networks based on the Leaky Integrate-and-Fire (LIF) neuron model. We analyze the relationship between classification accuracy and three factors: the number of neurons, the number of stimulus nodes, and the number of classification categories. Notably, we employ a large language model (LLM) to assist in discovering the underlying functional relationships among these variables, and compare its performance against traditional methods such as linear and polynomial fitting. Experimental results show that classification accuracy follows a power-law scaling primarily with the number of categories, while the effects of neuron count and stimulus nodes are relatively minor. A key advantage of the LLM-based approach is its ability to propose plausible functional forms beyond pre-defined equation templates, often leading to more concise or accurate mathematical descriptions of the observed scaling laws. This finding has important implications for understanding efficient computation in biological neural systems and for pioneering new paradigms in AI-aided scientific discovery.

Figures (11)

• Figure 1: Comparison of model predictions against actual values. The diagonal red line represents perfect prediction. All four models show varying degrees of accuracy, with Random Forest achieving near-perfect alignment.
• Figure 2: Feature importance analysis for the Random Forest model. (Left) Standard feature importance shows $C$ (number of categories) is the dominant predictor, accounting for 83.78% of the explained variance. (Right) Permutation importance confirms the robustness of this finding. This result is consistent with the LLM's guidance, both emphasizing the critical role of $C$ in determining the functional relationship.
• Figure 3: Power-law fitting of classification accuracy as a function of the number of categories ($C$) under different combinations of stimulus nodes ($S$) and neurons ($N$). Each subplot shows the accuracy scaling for a specific $(S, N)$ combination, demonstrating the consistent power-law relationship across all network configurations.
• Figure 4: Scatter plot of predicted versus actual classification accuracy for the ensemble model. Close alignment along the diagonal demonstrates good predictive accuracy across the full value range.
• Figure 5: Power-law fitting performance for stimulus input classification data under fixed stimulus node counts ($S=1,2,3$) (no neuronal networks involved). Scatter plots show direct classification accuracy data, lines show individual power-law fits from Table \ref{['tab:conditional_power_law']}. Wide point dispersion relative to fit lines visually confirms "Poor" $R^2$ ratings and the absence of valid power-law distribution.