A Kernel-Spectral Index for Single-Neuron Input-Output Complexity

The computational capacity of individual neurons in the brain is a central problem in computational neuroscience and an increasingly relevant link between neuroscience and AI. Recent work fitting artificial neural networks to detailed cortical pyramidal neurons showed that reproducing their millisecond-precision input-output behavior can require temporal convolutional networks with five to eight layers, suggesting that cortical neurons may function as deep computational units inside brain circuits. Yet current quantitative studies of single-neuron complexity are largely surrogate-based; consequently, the obtained results are difficult to interpret, and the process often demands massive computational resources. We introduce the Spectral Complexity Index (SCI), a white-box measure that represents presynaptic histories with an explicit kernel geometry and quantifies how many ordered spectral modes are needed to explain somatic response energy. Across 35 biophysically detailed human and mouse cortical neuron models, SCI yields a statistically significant separation of the two species, strongly agrees with the DNN-based Functional Complexity Index on these cells (Spearman correlation: ρ=0.797), and remains stable under sampling and kernel perturbations. Additionally, compared to existing methods such as FCI, the computational cost of SCI is reduced to approximately 1/10 or less. SCI turns single-neuron complexity from a black-box surrogate property into an interpretable spectral object, enabling reproducible, fast, and efficient comparisons of neuronal computation across cells and species.