How do neurons operate on sparse distributed representations? A mathematical theory of sparsity, neurons and active dendrites

TL;DR

This paper introduces a mathematical model for sparse neural representations and active dendrites, explaining how neurons recognize patterns accurately and robustly using small synaptic clusters, aligning well with experimental data.

Contribution

The paper presents a novel formal framework linking dendritic processing with sparse coding, including scaling laws and the union property for pattern recognition in neurons.

Findings

Neurons with active dendrites can recognize large patterns with high accuracy using few synapses.

The model predicts NMDA spiking thresholds that match experimental measurements.

Sparse representations enable robust pattern recognition even under adverse conditions.

Abstract

We propose a formal mathematical model for sparse representations and active dendrites in neocortex. Our model is inspired by recent experimental findings on active dendritic processing and NMDA spikes in pyramidal neurons. These experimental and modeling studies suggest that the basic unit of pattern memory in the neocortex is instantiated by small clusters of synapses operated on by localized non-linear dendritic processes. We derive a number of scaling laws that characterize the accuracy of such dendrites in detecting activation patterns in a neuronal population under adverse conditions. We introduce the union property which shows that synapses for multiple patterns can be randomly mixed together within a segment and still lead to highly accurate recognition. We describe simulation results that provide further insight into sparse representations as well as two primary results. First we show that pattern recognition by a neuron with active dendrites can be extremely accurate and robust with high dimensional sparse inputs even when using a tiny number of synapses to recognize large patterns. Second, equations representing recognition accuracy of a dendrite predict optimal NMDA spiking thresholds under a generous set of assumptions. The prediction tightly matches NMDA spiking thresholds measured in the literature. Our model matches many of the known properties of pyramidal neurons. As such the theory provides a mathematical framework for understanding the benefits and limits of sparse representations in cortical networks.