Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Oleg Makarenkov, Marianne Bezaire, Michael Hasselmo

TL;DR

This work addresses how neurons can produce spikes at a fixed phase of intracellular theta oscillations despite large baseline fluctuations. It introduces a simple resonate-and-fire model with a small perturbation parameter $\varepsilon$ and a degenerate grazing bifurcation to generate an asymptotically stable, one-spike-per-period limit cycle that grazes the firing threshold. The authors derive a rigorous bifurcation framework around a center, relate the fixed point stability to a multiplier $\rho_\varepsilon$ in $(-1,1)$, and show that spike timing remains phase-locked even under stochastic threshold noise. The findings offer a mechanistic account for phase-locked spiking near the theta peak and suggest robustness to biological noise, with implications for theta-phase coding and memory processing.

Abstract

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.

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

TL;DR

This work addresses how neurons can produce spikes at a fixed phase of intracellular theta oscillations despite large baseline fluctuations. It introduces a simple resonate-and-fire model with a small perturbation parameter and a degenerate grazing bifurcation to generate an asymptotically stable, one-spike-per-period limit cycle that grazes the firing threshold. The authors derive a rigorous bifurcation framework around a center, relate the fixed point stability to a multiplier in , and show that spike timing remains phase-locked even under stochastic threshold noise. The findings offer a mechanistic account for phase-locked spiking near the theta peak and suggest robustness to biological noise, with implications for theta-phase coding and memory processing.

Abstract

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.
Paper Structure (11 sections, 3 theorems, 53 equations, 14 figures)

This paper contains 11 sections, 3 theorems, 53 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. An orbitally stable nearly grazing limit cycle
  4. Use of random processes to produce fluctuations in spike threshold
  5. Results
  6. Autonomous case
  7. The case of stable focus
  8. The case of repeated impacts
  9. Non-autonomous case
  10. Discussion
  11. Proofs of lemmas

Key Result

Theorem 1

Assume that system (reduced) is of center type, i.e. (k1k2) holds. Denote by $t\mapsto(v_0(t),h_0(t))$ the cycle of (reduced) that touches the line $v=v_{th}>0$ (bold cycle in Fig. iiff left). If then, for each $\varepsilon>0$ sufficiently small, the system (impulse1)-(impulse2) admits an asymptotically stable $T_\varepsilon$-periodic limit cycle $t\mapsto(v_\varepsilon(t),h_\varepsilon(t))$ of o

Figures (14)

  • Figure 1: Experimental observations of grid cell activity, adapted from Domnisoru2013. The top trace shows the intracellular membrane potential (with truncated spikes) in gray. The red trace below shows the fluctuation in the baseline potential, while the variations in frequency and amplitude of the theta oscillation are apparent in the theta component shown at the bottom (with envelope in green). The sum of the ramp and theta components is shown in black overlay at top. The flexibility in spike threshold to enable spiking that is phase-locked to the peak is clear at the positions marked with black annotations. Figure adapted and reprinted by permission from Nature Publishing Group.
  • Figure 2: Left: Trajectories of system (\ref{['reduced']}). Right: Discontinuous limit cycle of system (\ref{['impulse1']})-(\ref{['impulse2']}) with one impact per period that this paper intends to establish.
  • Figure 3: Poincaré map of system (\ref{['impulse1']})-(\ref{['impulse2']}) induced by cross-section $v=v_{th}+\varepsilon \bar{v}.$
  • Figure 4: Illustration of the proof of the claim.
  • Figure 5: Graphs of $b_0$ and $\dfrac{\bar{h}^2+b_0/a_0^2}{2\bar{h}}+\bar{h}$ from (\ref{['A1']})-(\ref{['condition2']}) as function of $v_{th}$ with other parameters given by (\ref{['param']}).
  • ...and 9 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Remark 1