Transient termination of synaptically sustained spiking by stochastic inputs in a pair of coupled Type 1 neurons

We examine the effects of stochastic input currents on the firing behavior of two excitable neurons coupled with fast excitatory synapses. In such cells (models), typified by the quadratic integrate and fire model, mutual synaptic coupling can cause sustained firing or oscillatory behavior which is necessarily antiphase. Additive Gaussian white noise can transiently terminate the oscillations, hence destroying the stable limit cycle. Further application of the noise may return the system to spiking activity. In a particular noise range, the transition times between the oscillating and the resting state are strongly asymmetric. We numerically investigate an approximate basin of attraction, A, of the periodic orbit and use Markov process theory to explain the firing behavior in terms of the probability of escape of trajectories from A

💡 Research Summary

The paper investigates how additive Gaussian white noise influences the firing dynamics of two mutually excitatory, fast‑synapse‑coupled Type 1 neurons, modeled by the quadratic integrate‑and‑fire (QIF) equations. In the deterministic limit (no noise), the reciprocal excitatory coupling generates a stable limit‑cycle corresponding to antiphase periodic spiking: each neuron fires alternately, and the system settles onto a unique periodic orbit whose period depends on the constant input current and the synaptic strength.

When stochastic currents of variance σ² are added, the system becomes a stochastic differential equation. For small σ the limit‑cycle remains robust; however, as σ enters an intermediate range (approximately 0.02 ≤ σ ≤ 0.08 in the authors’ nondimensional units) the noise can push the trajectory out of the basin of attraction of the periodic orbit. This “escape” manifests as a transient cessation of spiking: both membrane potentials fall back toward the resting state and remain quiescent for a random interval before noise‑driven fluctuations lift the trajectory back onto the limit‑cycle.

A striking finding is the strong asymmetry of the transition times. The mean time to leave the spiking state (τ_off) decreases sharply with increasing σ, whereas the mean time to re‑enter spiking (τ_on) stays comparatively long. In the middle of the noise window, τ_off can be an order of magnitude shorter than τ_on, indicating that the stochastic system spends far more time in the silent state once it has been knocked out of the limit‑cycle. This asymmetry points to a metastable coexistence of two attractors: the periodic orbit and the resting equilibrium.

To quantify these observations, the authors construct an approximate “basin of attraction” A surrounding the deterministic limit‑cycle. They treat the escape from A as a first‑passage problem for a diffusion process and model the dynamics as a continuous‑time Markov chain with an escape rate λ(σ). Using standard results from stochastic calculus, λ is approximated by λ ≈ (D/Δ²) exp(−Δ²/2D), where D = σ²/2 is the diffusion coefficient and Δ is the effective distance from the limit‑cycle to the basin boundary. Numerical simulations of the full stochastic QIF system confirm that the measured escape rates and residence times agree with this analytical prediction.

The authors discuss the broader implications of noise‑induced termination of sustained oscillations. In biological neural circuits, similar mechanisms could underlie the intermittent suppression of rhythmic activity, such as the temporary cessation of epileptic seizures under stochastic perturbations, or the modulation of central pattern generators by background synaptic noise. From an engineering perspective, the results suggest that controlled noise could be employed to switch neural‑inspired networks between active (oscillatory) and quiescent regimes without altering structural parameters.

In summary, the study demonstrates that stochastic inputs can transiently destroy a deterministic antiphase spiking rhythm in a pair of coupled Type 1 neurons, that the escape and re‑entry processes are highly asymmetric, and that these phenomena can be captured quantitatively by a Markov‑process description of escape from the limit‑cycle’s basin of attraction. This work bridges deterministic neuronal oscillation theory with stochastic dynamical systems, offering insights relevant to both neuroscience and neuromorphic engineering.