  Although temporal coding through spike-time patterns has long been of
interest in neuroscience, the specific structures that could be useful for
spike-time codes remain highly unclear. Here, we introduce a new analytical
approach, using techniques from discrete mathematics, to study spike-time
codes. We focus on the phenomenon of ``phase precession'' in the rodent
hippocampus. During navigation and learning on a physical track, specific cells
in a rodent's brain form a highly structured pattern relative to the
oscillation of local population activity. Studies of phase precession largely
focus on its well established role in synaptic plasticity and memory formation.
Comparatively less attention has been paid to the fact that phase precession
represents one of the best candidates for a spike-time neural code. The precise
nature of this code remains an open question. Here, we derive an analytical
expression for an operator mapping points in physical space, through individual
spike times, to complex numbers. The properties of this operator highlight a
specific relationship between past and future in hippocampal spike patterns.
Importantly, this approach generalizes beyond the specific phenomenon studied
here, providing a new technique to study the neural codes within spike-time
sequences found during sensory coding and motor behavior. We then introduce a
novel spike-based decoding algorithm, based on this operator, that successfully
decodes a simulated animal's trajectory using only the animal's initial
position and a pattern of spike times. This decoder is robust to noise in spike
times and works on a timescale almost an order of magnitude shorter than
typically used with decoders that work on average firing rate. These results
illustrate the utility of a discrete approach, based on the symmetries in spike
patterns, to provide insight into the structure and function of neural systems.
