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On the threshold of excitable systems: An energy-based perspective

Rodolphe Sepulchre, Guanchun Tong

TL;DR

This paper defines an energy-based threshold for excitable systems by identifying the local maximum of the minimal external energy (the required supply) needed to drive a system from rest to an event. Framed in dissipativity terms, the threshold separates passive subthreshold dynamics from regenerative suprathreshold spikes, and is evaluated through analytical and numerical studies of RC, FitzHugh–Nagumo, and Hodgkin–Huxley models. The approach yields concrete threshold locations (e.g., in FHN and HH) and offers a unified view linking energy transport, dissipation, and spiking, with potential implications for neuromorphic engineering and robust excitability theory. The results demonstrate that energy barriers, rather than fixed voltages, govern spike initiation and inhibition, providing a physically grounded criterion for threshold that accommodates history, current dispersion, and internal dissipation.

Abstract

A fundamental characteristic of excitable systems is their ability to exhibit distinct subthreshold and suprathreshold behaviors. Precisely quantifying this distinction requires a proper definition of the threshold, which has remained elusive in neurodynamics. In this paper, we introduce a novel, energy-based threshold definition for excitable circuits grounded in dissipativity theory, specifically using the classical concept of required supply. According to our definition, the threshold corresponds to a local maximum of the required supply, clearly separating subthreshold passive responses from suprathreshold regenerative spikes. We illustrate and validate the proposed definition through analytical and numerical studies of three canonical systems: a simple RC circuit, the FitzHugh--Nagumo model, and the biophysically detailed Hodgkin--Huxley model.

On the threshold of excitable systems: An energy-based perspective

TL;DR

This paper defines an energy-based threshold for excitable systems by identifying the local maximum of the minimal external energy (the required supply) needed to drive a system from rest to an event. Framed in dissipativity terms, the threshold separates passive subthreshold dynamics from regenerative suprathreshold spikes, and is evaluated through analytical and numerical studies of RC, FitzHugh–Nagumo, and Hodgkin–Huxley models. The approach yields concrete threshold locations (e.g., in FHN and HH) and offers a unified view linking energy transport, dissipation, and spiking, with potential implications for neuromorphic engineering and robust excitability theory. The results demonstrate that energy barriers, rather than fixed voltages, govern spike initiation and inhibition, providing a physically grounded criterion for threshold that accommodates history, current dispersion, and internal dissipation.

Abstract

A fundamental characteristic of excitable systems is their ability to exhibit distinct subthreshold and suprathreshold behaviors. Precisely quantifying this distinction requires a proper definition of the threshold, which has remained elusive in neurodynamics. In this paper, we introduce a novel, energy-based threshold definition for excitable circuits grounded in dissipativity theory, specifically using the classical concept of required supply. According to our definition, the threshold corresponds to a local maximum of the required supply, clearly separating subthreshold passive responses from suprathreshold regenerative spikes. We illustrate and validate the proposed definition through analytical and numerical studies of three canonical systems: a simple RC circuit, the FitzHugh--Nagumo model, and the biophysically detailed Hodgkin--Huxley model.

Paper Structure

This paper contains 18 sections, 22 equations, 7 figures.

Table of Contents

  1. INTRODUCTION
  2. An energy-based definition of the threshold in excitable systems
  3. Hodgkin--Huxley model
  4. Subthreshold, Suprathreshold, and Inhibitory Responses
  5. Why thresholds defy a simple definition?
  6. A dissipativity theoretic definition
  7. Illustration example: an RC circuit
  8. A singular optimal control problem
  9. The exponential ansatz
  10. Generalization to a bi-stable RC circuit
  11. Thresholds in the Fitzhugh--Nagumo model
  12. Fitzhugh--Nagumo model
  13. Energy thresholds in FHN
  14. Thresholds in the Hodgkin--Huxley Model
  15. Numerical procedure: dynamic "voltage clamp"
  16. ...and 3 more sections

Figures (7)

  • Figure 1: Subthreshold and suprathreshold behaviors of an excitable circuit. Reproduced from sepulchre2018excitable with permission.
  • Figure 2: The subtlety of defining threshold sepulchre2018excitable.
  • Figure 3: The N-shape i-v curve for the nonlinear resistor.
  • Figure 4: The red cubic curve represents the $v$-nullcline ($\dot{v}=0$), blue line shows the $w$-nullcline ($\dot{w}=0$). Green circle: stable fixed point
  • Figure 5: One threshold of the Fitzhugh--Nagumo circuit.
  • ...and 2 more figures