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

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

Problem Statement
The paper addresses a critical distinction in neuronal modeling: reproducing a voltage trace or a specific timing metric (such as first-spike latency) does not necessarily imply that a reduced model preserves the underlying dynamical mechanism of the parent system. Specifically, the authors investigate whether a two-dimensional (2D) piecewise-linear planar model can reproduce the logarithmic scaling of first-spike latency observed in its three-dimensional (3D) parent model (the Neurosimilator), despite potentially altering how a stimulus initiates sustained firing. The core challenge is to determine if similar latency scaling implies a shared mechanism of excitability or if it merely reflects a coincidental functional form arising from different dynamical structures.

Methodology
The study employs a comparative analysis between two electronic neuron models:

1. The Parent Model: A 3D Neurosimilator combining smooth regenerative feedback with voltage-controlled switching. It operates in a spiking–rest bistability regime where a stable resting focus and a sustained-spiking attractor coexist.
2. The Planar Model: A reduced 2D switching model introduced in a companion study. It replaces the parent's fast depolarizing state with an instantaneous, threshold-activated term. The dynamics are defined by piecewise-linear ordinary differential equations (ODEs) involving a voltage-like variable ($v$) and a recovery variable ($k$). The phase plane is divided into six linear regions by switching boundaries ($v=v_{th}$, $v=v_{ref}$, $k=v_{th}$).

The authors analyze:

• Waveform Morphology: Comparing normalized voltage traces and recovery timescales.
• Frequency-Current Response: Evaluating event rates over a full input range.
• First-Spike Latency ($T_{FSL}$): Measuring the time from a constant input step to the first threshold crossing.
• Analytical Derivation: Deriving the latency law for the planar model by analyzing the approach to the switching boundary, contrasting it with the parent's basin-access threshold.
• Hardware Validation: Comparing numerical solutions, closed-form analytical trajectories, and signals from a physical electronic circuit implementation of the planar model.

Key Results

1. Divergent Mechanisms for Similar Scaling: Both models exhibit a logarithmic dependence of first-spike latency on the input distance from a reference threshold ($-\ln \mu$). However, the fitted coefficients differ significantly (approx. 99.2 ms for the 3D parent vs. 49.7 ms for the 2D planar model).
2. Distinct Onset Mechanisms:
   • Parent (3D): The firing threshold ($I_{th,3D}$) is a "basin-access threshold" within a bistability regime. An abrupt step accesses the spiking attractor, while a slow ramp to the same input allows accommodation to a stable resting focus. This threshold is distinct from the saddle-node limit point ($I_{LP}$) and does not involve a loss of equilibrium stability.
   • Planar (2D): The firing threshold ($I_{th,2D}$) marks the point where the subthreshold equilibrium reaches a switching boundary and ceases to be admissible. The eigenvalues remain negative; no bifurcation (collision of equilibria) occurs. The latency law is analytically derived from the time required to reach this boundary.
3. Quantitative Differences: While the planar model captures the directional trend of the input-output response, it fails to preserve the quantitative frequency-current gain (approaching 80 Hz vs. <7 Hz for the parent) and the specific waveform morphology (sharper excursions, different recovery timescales).
4. Analytical Tractability: The planar model allows for exact trajectory calculations within its six switching regions, providing analytical access to the stages of spike generation that are obscured in the smooth 3D parent.

Significance and Claims
The paper claims that the observation of similar latency scaling (e.g., logarithmic dependence) is insufficient to validate that a reduced model preserves the mechanism of excitability of its parent. The authors assert that:

• Mechanism vs. Metric: A firing threshold measured via abrupt steps need not mark a bifurcation. In the parent, it is a protocol-dependent selection of an attractor; in the planar model, it is a loss of equilibrium admissibility.
• Validation Requirements: Validating a reduced neuronal model requires examining the stimulus history, attractor access, and equilibrium structure alongside spike timing and local equilibrium stability.
• Role of the Eliminated Variable: The elimination of the third state variable ($w$) in the planar model removes the specific internal dynamical loop responsible for protocol-dependent selection (step vs. ramp) in the parent. While the planar model retains temporal information via the recovery variable $k$, it bypasses the specific history dependence governing the parent's basin selection.
• Limitations: The reduction is heuristic, not an asymptotic reduction of Hodgkin–Huxley kinetics or the parent equations. It sacrifices the true 3D bifurcation topology and the full spectrum of complex behaviors (quasiperiodic, chaotic) for analytical tractability and specific input-output characteristics.

The study concludes that while reduced models are valuable for understanding specific features, they must be scrutinized for what they fail to preserve, particularly regarding how stimuli initiate firing in bistable regimes. Future work is proposed to test whether adding a simple memory term (delayed feedback) to the planar model can recover the protocol-dependent sensitivity lost in the dimensional reduction.