Analysis of neuronal sequences using pairwise biases

TL;DR

This paper introduces the bias matrix, a new probabilistic tool for analyzing neuronal sequences that captures pairwise firing tendencies and generalizes traditional neuronal templates, providing insights into brain functions like memory and navigation.

Contribution

The paper develops the bias matrix framework, establishing its mathematical properties, and demonstrates its application to real hippocampal data for sequence analysis.

Findings

Bias matrices capture the best total ordering of neurons.

Every simple digraph can be realized as a bias network.

The framework enables sequence correlation and similarity measurement.

Abstract

Sequences of neuronal activation have long been implicated in a variety of brain functions. In particular, these sequences have been tied to memory formation and spatial navigation in the hippocampus, a region of mammalian brains. Traditionally, neuronal sequences have been interpreted as noisy manifestations of neuronal templates (i.e., orderings), ignoring much richer structure contained in the sequences. This paper introduces a new tool for understanding neuronal sequences: the bias matrix. The bias matrix captures the probabilistic tendency of each neuron to fire before or after each other neuron. Despite considering only pairs of neurons, the bias matrix captures the best total ordering of neurons for a sequence (Proposition 3.3) and, thus, generalizes the concept of a neuronal template. We establish basic mathematical properties of bias matrices, in particular describing the fundamental polytope in which all biases reside (Theorem 3.25). We show how the underlying simple digraph of a bias matrix, which we term the bias network, serves as a combinatorial generalization of neuronal templates. Surprisingly, every simple digraph is realizable as the bias network of some sequence (Theorem 3.34). The bias-matrix representation leads us to a natural method for sequence correlation, which then leads to a probabilistic framework for determining the similarity of one set of sequences to another. Using data from rat hippocampus, we describe events of interest and also sequence-detection techniques. Finally, the bias matrix and the similarity measure are applied to this real-world data using code developed by us.