An algebraic approach to spike-time neural codes in the hippocampus

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.