New Mathematical Framework Unifies Neuron Spiking Across Irregular Time Scales

Neurons firing across irregular intervals can now be accurately modeled using a unified mathematical framework that separates biological membrane physics from temporal irregularity. Researchers Billur Kaymakçalan and Behiye İlayda Selçuk have formulated a structure-preserving leaky integrate-and-fire neuron model on arbitrary Hilger time scales, capturing how continuous voltage dynamics translate to discrete spike trains without altering underlying biophysical laws.

The leaky integrate-and-fire model is a cornerstone of computational neuroscience because it balances analytical simplicity with the essential hybrid structure of a spiking neuron. It combines continuous subthreshold voltage evolution with threshold-triggered firing and reset mechanisms. In standard continuous time, the passive membrane dynamics are governed by a first-order linear ordinary differential equation. However, experimental recording times and digital computer simulations often operate on irregular temporal domains, clocks, or event-driven updates. This discrepancy creates a structural challenge regarding whether the same membrane equation can hold across continuous, discrete, and irregular time domains.

To address this challenge, the authors utilized Hilger-calculus time scales, a mathematical formalism designed to unify continuous and discrete analysis within a single framework. A time scale is defined as an arbitrary nonempty closed subset of real numbers, where local temporal geometry is encoded by a forward jump operator and graininess function. While time scales have previously been applied to network-level rate models, researchers have largely left the threshold-and-reset membrane dynamics of single spiking neurons unexamined through this specific lens.

As published in arXiv, the study demonstrates that naive transcriptions of the delta-dynamic leaky integrate-and-fire equation fail across coarse time scales. When the temporal graininess exceeds twice the membrane time constant of 20 milliseconds, the naive multiplier produces sign-alternating overshoots and local mathematical amplification that have no counterpart in biological membranes. To overcome this failure mode, the authors applied the inverse cylinder transformation to construct a positively regressive coefficient. This adjustment ensures that the subthreshold trajectory remains an exact restriction of the classical continuous-time solution, regardless of how irregular the time domain becomes.

How Membrane Physics Remain Unaffected by Temporal Irregularity

The core mathematical finding reveals that subthreshold voltage evolution is entirely independent of temporal graininess. For any time scale, the membrane voltage matches the classical continuous-time voltage at the exact same physical physical time. This preservation of the passive membrane semigroup occurs because the constructed coefficient successfully maintains positive regressivity across all evaluated conditions.

Consequently, the model avoids artificial overshoots and maintains strict monotonicity. When initial voltages start below the resting potential equilibrium, the membrane potential stays bounded within expected physiological ranges across every tested time scale. This confirms that changing the temporal domain alters only the admissible observation times rather than the passive biophysical properties of the membrane itself.

Predicting Firing Times Through Forward Projection Geometry

While subthreshold voltages remain unaffected, spike timing depends heavily on the temporal opportunity structure provided by the time domain. Because a threshold-triggered event can only occur at points belonging to the time scale, the realized firing time is determined by a forward projection operator. The authors established that the first firing time is simply the forward projection of the classical continuous threshold-crossing time onto the time scale.

Building on this geometric approach, the team derived a recursive spike-train law valid on every closed time scale unbounded above. The recursion states that subsequent firing times obey a clean relationship based on the classical reset-to-threshold travel time of approximately 18.33 milliseconds. Because the subthreshold ordinary differential equation is autonomous, this travel time remains constant regardless of absolute position in the time domain, successfully preventing finite-time accumulation phenomena such as Zeno behavior.

Quantifying Spike Timing Error and Firing Rates on Uniform Lattices

The study also establishes strict bounds on spike-timing errors by evaluating the gap introduced by time scale projections. On uniform lattices with a constant spacing step $h$, the spike-time error is bounded above by the graininess parameter. For instance, on a uniform lattice where $h$ equals 2 milliseconds, the projection delay is tightly restricted compared to coarser lattices with larger steps.

When analyzing uniform lattices specifically, the inter-spike interval and firing rate become lattice-quantized quantities governed by integer step counts. As the temporal step size approaches zero, both the inter-spike interval and the firing rate converge smoothly to their classical continuous-time limits. These findings provide a robust mathematical foundation for understanding how discrete temporal sampling impacts neural firing patterns without introducing artificial instability.

The study was posted to the arXiv preprint server on November 12, 2026.