  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.
