Mapping Connectomic Structure to Function(s) in Cerebellar-like Networks using Kernel Regression

TL;DR

This paper establishes a mathematical link between the structured connectivity in cerebellar-like networks and their learning capabilities, showing how observed patterns influence the ease of learning certain functions.

Contribution

It extends kernel regression models to relate biological connectivity patterns to the network's inductive bias, providing a simple, analytical framework for understanding function learning in cerebellar-like circuits.

Findings

Structured connectivity shapes learning bias towards oversampled inputs.

Functions depending on specific neuron groups are easier to learn.

The model offers a tractable way to analyze biological learning mechanisms.

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.