[论文解读] Mapping Connectomic Structure to Function(s) in Cerebellar-like Networks using Kernel Regression

William Dorrell, Peter E. Latham|arXiv (Cornell University)|Jan 14, 2026

Neurobiology and Insect Physiology Research被引用 0

一句话总结

该论文通过核回归分析性地将结构化的类似小脑的连接性与学习性能联系起来，展示了偏置和分组投影如何塑造归纳偏好与泛化能力。

ABSTRACT

Cerebellar-like networks, in which input activity patterns are separated by projection to a much higher-dimensional space before classification, are a recurring neurobiological motif, present in the cerebellum, dentate gyrus, insect olfactory system, and electrosensory system of the electric fish. Their relatively well-understood design presents a promising test-case for probing principles of biological learning. The circuits' expansive projections have long been modelled as random, enabling effective general purpose pattern separation. However, electron-microscopy studies have discovered interesting hints of structure in both the fly mushroom body and mouse cerebellum. Recent numerical work suggested that this non-random connectivity enables the circuit to prioritise learning of some, presumably natural, tasks over others. Here, rather than numerical results, we present a robust mathematical link between the observed connectivity patterns and the cerebellar circuit's learning ability. In particular, we extend a simplified kernel regression model of the system and use recent machine learning theory results to relate connectivity to learning. We find that the reported structure in the projection weights shapes the network's inductive bias in intuitive ways: functions are easier to learn if they depend on inputs that are oversampled, or on collections of neurons that tend to connect to the same hidden layer neurons. Our approach is analytically tractable and pleasingly simple, and we hope it continues to serve as a model for understanding the functional implications of other processing motifs in cerebellar-like networks.

研究动机与目标

• 理解小脑样网络中非随机连接基序如何影响学习性能。
• 提供一个可处理的解析框架，将连接结构通过核回归与归纳偏好联系起来。
• 展示过度连接的输入或输入分组如何影响对函数的可学习性。
• 将见解推广到比 toy 模型更现实的网络配置。

提出的方法

• 将小脑样线路径建模为固定的非线性展开后接线性读出，映射到核回归。
• 通过展开层表示定义核 k(x, x') = φ(Jx) · φ(Jx')。
• 使用高斯为基础、解析可控的协方差模型来描述展开权重 J，其协方差为 Σ。
• 考察两种连接基序：偏置连接（对角矩阵 Σ，方差不相等）和分组连接（输入在块内相关）。
• 推导这些方案的核，并分析特征函数/特征值以表征归纳偏好（可学习性 ≈ λi/(λi+κ)）。
• 通过数值方法和生物学上可行的稀疏性/激活约束，将结果扩展到更现实的模型。

Figure 1: A) Schematic of fly mushroom body circuit. Odorants trigger activity in olfactory receptor neurons (ORNs). ORNs contain a unique receptor protein, signalled by their colour; neurons with the same receptor protein send projections to a shared glormulus. There they synapse onto projection ne

实验结果

研究问题

• RQ1 结构化的（偏置/分组）连接在展开层如何改变核及其特征结构？
• RQ2 偏置与分组连接带来哪些归纳偏好，它们如何影响输入-输出映射的可学习性？
• RQ3 在更现实、稀疏化和生物兼容的模型下，解析结论是否仍成立？
• RQ4 框架是否能够解释由于连接结构，小脑样电路在某些任务上更快学习的情况？

主要发现

• 结构化连接改变核，改变表示相似性和特征结构。
• 偏置连接在沿着过度连接输入轴变化的函数的学习难度上提高了学习的便利性。
• 相关/分组连接使学习偏向处理同一连通组成员的函数。
• 随机连接产生的核主要取决于角度相似性，具有球谣函数和平滑性偏好。
• 特征值谱所预测的归纳偏好解释了观测到的基元在特定任务上的泛化优势。

Figure 2: A) A single linear layer can only classify linearly seperable data. B) However adding a fixed first layer of nonlinear processing can permit a linear readout layer to perform nonlinear classifications. C) Given any finite dataset there are infinitely many possible generalisations to unseen