[Paper Review] 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 Research0 citations

TL;DR

The paper analytically links structured cerebellar-like connectivity to learning performance via kernel regression, showing how biased and grouped projections shape inductive bias and generalisation.

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.

Motivation & Objective

• Understand how non-random connectomic motifs in cerebellar-like networks affect learning performance.
• Provide a tractable analytic framework connecting connectivity structure to inductive bias via kernel regression.
• Show how overconnected inputs or input groupings influence learnability of functions.
• Generalise insights beyond toy models to more realistic network configurations.

Proposed method

• Model cerebellar-like circuits as a fixed nonlinear expansion followed by linear readout, mapped to kernel regression.
• Define kernel via the expansion layer representation k(x, x') = φ(Jx) · φ(Jx').
• Use a Gaussian-based, analytically tractable covariance model for the expansion weights J with covariance Σ.
• Examine two connectivity motifs: biased connectivity (diagonal Σ with unequal variance) and grouped connectivity (correlated inputs in blocks).
• Derive the kernel for these schemes and analyze eigenfunctions/eigenvalues to characterise inductive bias (learnability of eigenfunctions ≈ λi/(λi+κ)).
• Extend results to more realistic models via numerics and biologically plausible sparsity/activation constraints.

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

Experimental results

Research questions

• RQ1 How does structured (biased/grouped) connectivity in the expansion layer alter the kernel and its eigenstructure?
• RQ2 What inductive biases arise from biased vs. grouped connectivity, and how do they affect learnability of input-output mappings?
• RQ3 Do the analytic conclusions hold under more realistic, sparsified, and biology-compatible models?
• RQ4 Can the framework explain when cerebellar-like circuits learn certain tasks more quickly due to connectivity structure?

Key findings

• Structured connectivity alters the kernel, changing representational similarity and the eigenstructure.
• Biased connectivity increases learning ease for functions varying along the overconnected input axis.
• Correlated/grouped connectivity biases learning toward functions that treat members of a connected group similarly.
• Random connectivity yields a kernel depending mainly on angular similarity, with spherical-harmonic eigenfunctions and smoothness bias.
• Inductive bias shifts predicted by the eigenvalue spectrum explain task-specific generalisation advantages of observed motifs.

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