A Planar Switching Neuron Shares Logarithmic First-Spike Latency with Its Three-Dimensional Parent But Not Its Onset Mechanism

Abstract

Reducing an electronic neuron from three variables to two can preserve a familiar voltage trace while changing the underlying dynamics that create it. We examine this distinction in a piecewise-linear planar reformulation of the three-dimensional neurosimilator. The planar model produces repetitive spike-like excursions without an imposed reset and admits an exact linear solution inside each of six switching regions. Direct comparison reveals only partial agreement between the two models. Although both generate large voltage excursions and an event rate that rises with applied drive, their waveforms and frequency-current gains differ substantially. A reanalysis of the three-dimensional, near-threshold data also corrects an earlier interpretation: over the resolved interval, the first-spike latency depends logarithmically on the normalized distance from threshold, matching the behavior of the planar model. However, this common functional form does not imply a shared onset mechanism. The three-dimensional equilibrium branch reaches a saddle-node limit point, whereas the planar equilibrium remains a stable node until it exits its admissible region at a switching boundary. While a stagewise analytical reconstruction captures individual voltage excursions, crossing-time errors accumulate unless Newton corrections are applied. The reduced system is therefore a useful electronic neuron model with transparent switching dynamics, rather than a dynamically equivalent copy of its three-dimensional parent. \