Bifurcation of spiking oscillations from a center in resonate-and-fire neurons

The theta rhythm is important for many cognitive functions including spatial processing, memory encoding, and memory recall. The information processing underlying these functions is thought to rely on consistent, phase-specific spiking throughout a theta oscillation that may fluctuate significantly in baseline (center of oscillations), frequency, or amplitude. Experimental evidence shows that spikes can occur at specific phases even when the baseline membrane potential varies significantly, such that the integrity of phase-locking persists across a large variability in spike threshold. The mechanism of this precise spike timing during the theta rhythm is not yet known and previous mathematical models have not reflected the large variability in threshold potential seen experimentally. Here we introduce a straightforward mathematical neural model capable of demonstrating a phase-locked spiking in the face of significant baseline membrane potential fluctuation during theta rhythm. This novel approach incorporates a degenerate grazing bifurcation of an asymptotically stable oscillation. This model suggests a potential mechanism for how biological neurons can consistently produce spikes near the peak of a variable membrane potential oscillation.