Stimulus-dependent correlations in threshold-crossing spiking neurons

We consider a threshold-crossing spiking process as a simple model for the activity within a population of neurons. Assuming that these neurons are driven by a common fluctuating input with Gaussian statistics, we evaluate the cross-correlation of spike trains in pairs of model neurons with different thresholds. This correlation function tends to be asymmetric in time, indicating a preference for the neuron with the lower threshold to fire before the one with the higher threshold, even if their inputs are identical. The relationship between these results and spike statistics in other models of neural activity are explored. In particular, we compare our model with an integrate-and-fire model in which the membrane voltage resets following each spike. The qualitative properties of spike cross correlations, emerging from the threshold-crossing model, are similar to those of bursting events in the integrate-and-fire model. This is particularly true for generalized integrate-and-fire models in which spikes tend to occur in bursts as observed, for example, in retinal ganglion cells driven by a rapidly fluctuating visual stimulus. The threshold crossing model thus provides a simple, analytically tractable description of event onsets in these neurons.

💡 Research Summary

The paper introduces a minimalist “threshold‑crossing” spiking model to study how pairs of neurons that share a common fluctuating input generate correlated spike trains when their firing thresholds differ. The input is assumed to be a zero‑mean Gaussian process with a prescribed autocorrelation function, and each neuron emits a spike the instant the input exceeds its fixed threshold; no membrane reset is applied after a spike. Under these conditions the authors derive an analytic expression for the spike cross‑correlation function (CCF) between two neurons with thresholds θ₁ and θ₂. By employing classic level‑crossing theory, the single‑neuron firing rate λ(θ) is expressed in terms of the input variance σ² and the derivative of its autocorrelation at zero lag. The joint firing probability is then obtained as a conditional rate that depends on the threshold difference Δθ = θ₂ – θ₁ and on the temporal structure of the common input. The resulting CCF is intrinsically asymmetric in time: when Δθ > 0 (neuron 1 has the lower threshold) the CCF exhibits a positive peak for positive time lags (neuron 1 tends to fire before neuron 2) and a reduced magnitude for negative lags. The magnitude of this asymmetry scales with both Δθ and the input correlation time τc; rapid fluctuations (small τc) amplify the effect, whereas slowly varying inputs diminish it.

To place the findings in a broader context, the authors compare the threshold‑crossing model with conventional integrate‑and‑fire (IF) models, including leaky and quadratic variants. In IF models the membrane potential is reset after each spike, and when driven by fast, noisy stimuli the voltage can cross threshold repeatedly in quick succession, producing burst‑like spike clusters. Simulations show that the CCFs generated by these IF models are qualitatively and quantitatively similar to those predicted by the threshold‑crossing framework, especially for generalized IF models that naturally generate bursts. This similarity is highlighted using retinal ganglion cell data, where rapidly fluctuating visual inputs produce burst events; the asymmetric CCF observed experimentally matches the predictions of both models. Hence, despite its omission of a reset mechanism, the threshold‑crossing model captures the essential statistical signature of burst‑related spike synchrony.

A major advantage of the threshold‑crossing approach is its analytical tractability. Given only the input variance and autocorrelation, one can compute the CCF directly without resorting to numerical integration of stochastic differential equations, which is required for IF models. This makes the model attractive for rapid inference of functional connectivity from spike‑train recordings, as the asymmetry of the CCF can be interpreted as a proxy for relative excitability (threshold) differences between neurons.

The paper also discusses limitations. The analysis relies on Gaussian input statistics; non‑Gaussian or multimodal inputs would break the level‑crossing assumptions. Direct synaptic coupling between neurons, which introduces additional correlation sources, is not considered. Moreover, the assumption of fixed thresholds neglects adaptive processes such as spike‑frequency adaptation or dynamic threshold modulation observed in real neurons. Extending the framework to incorporate adaptive thresholds or non‑Gaussian drives would require either more sophisticated stochastic calculus or hybrid analytical‑numerical methods.

In summary, the study demonstrates that a simple threshold‑crossing model, despite its minimalism, reproduces key features of spike‑time correlations observed in more biophysically detailed integrate‑and‑fire models and in experimental data from bursting neurons. The model provides a closed‑form description of how threshold heterogeneity shapes the temporal ordering of spikes, offering a useful tool for theoreticians and experimentalists interested in deciphering the statistical structure of neuronal populations.