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Simulating neuronal dynamics in fractional adaptive exponential integrate-and-fire models

Alexandru Fikl, Aman Jhinga, Eva Kaslik, Argha Mondal

TL;DR

The paper introduces FrAdEx, a fractional-order adaptive exponential integrate-and-fire model using Caputo derivatives to capture memory effects in neuronal spiking with state-dependent impulsive resets. It develops an implicit, nonuniform L1-type discretization with adaptive time stepping and a non-iterative Lambert W solution to efficiently handle exponential spike dynamics, accompanied by a complete error model and stability analysis. The method is validated on simple PIF/LIF benchmarks and applied to FrAdEx, showing first-order convergence in spike times and robust performance, while reproducing diverse firing patterns (e.g., chattering, tonic spiking, and adaptation) as the fractional order varies. This framework enables accurate, scalable simulations of memory-rich neuronal dynamics and provides a foundation for extending to higher-order schemes and networks, with potential impact on modeling biophysically realistic neuronal populations.

Abstract

We introduce an efficient discretization of a novel fractional-order adaptive exponential (FrAdEx) integrate-and-fire model, which is used to study the fractional-order dynamics of neuronal activities. The discretization is based on extension of L1-type methods that can accurately handle the exponential growth and the spiking mechanism of the model. This new method is implicit and uses adaptive time stepping to robustly handle the stiff system that arises due to the exponential term. The implicit nonlinear system can be solved exactly, without the need for iterative methods, making the scheme efficient while maintaining accuracy. We present a complete error model for the numerical scheme that can be extended to other integrate-and-fire models with minor changes. To show the feasibility of our approach, the numerical method has been rigorously validated and used to investigate several different spiking oscillations of the model. We observed that the fractional-order model is capable of predicting biophysical activities, which are interpreted through phase diagrams describing the transition from one firing type to another. This simple model shows significant promise, as it has sufficient expressive dynamics to reproduce several features qualitatively from a biophysical dynamical perspective.

Simulating neuronal dynamics in fractional adaptive exponential integrate-and-fire models

TL;DR

The paper introduces FrAdEx, a fractional-order adaptive exponential integrate-and-fire model using Caputo derivatives to capture memory effects in neuronal spiking with state-dependent impulsive resets. It develops an implicit, nonuniform L1-type discretization with adaptive time stepping and a non-iterative Lambert W solution to efficiently handle exponential spike dynamics, accompanied by a complete error model and stability analysis. The method is validated on simple PIF/LIF benchmarks and applied to FrAdEx, showing first-order convergence in spike times and robust performance, while reproducing diverse firing patterns (e.g., chattering, tonic spiking, and adaptation) as the fractional order varies. This framework enables accurate, scalable simulations of memory-rich neuronal dynamics and provides a foundation for extending to higher-order schemes and networks, with potential impact on modeling biophysically realistic neuronal populations.

Abstract

We introduce an efficient discretization of a novel fractional-order adaptive exponential (FrAdEx) integrate-and-fire model, which is used to study the fractional-order dynamics of neuronal activities. The discretization is based on extension of L1-type methods that can accurately handle the exponential growth and the spiking mechanism of the model. This new method is implicit and uses adaptive time stepping to robustly handle the stiff system that arises due to the exponential term. The implicit nonlinear system can be solved exactly, without the need for iterative methods, making the scheme efficient while maintaining accuracy. We present a complete error model for the numerical scheme that can be extended to other integrate-and-fire models with minor changes. To show the feasibility of our approach, the numerical method has been rigorously validated and used to investigate several different spiking oscillations of the model. We observed that the fractional-order model is capable of predicting biophysical activities, which are interpreted through phase diagrams describing the transition from one firing type to another. This simple model shows significant promise, as it has sufficient expressive dynamics to reproduce several features qualitatively from a biophysical dynamical perspective.
Paper Structure (18 sections, 3 theorems, 68 equations, 9 figures, 1 algorithm)

This paper contains 18 sections, 3 theorems, 68 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model
  3. Fractional Calculus Background
  4. Non-Dimensional Model
  5. Numerical Methods
  6. Lambert W Solution
  7. Time Step Limit
  8. Adaptive Time Step
  9. Stability and Convergence
  10. Numerical Simulations
  11. Convergence on a simple PIF model
  12. Adaptivity of a simple LIF model
  13. Convergence of the FrAdEx model
  14. Performance on the FrAdEx model
  15. Neural behaviour under the FrAdEx model
  16. ...and 3 more sections

Key Result

Lemma 6.1

Let $0 < \alpha < 1$ and $y \in \mathrm{PAC}^2([0, T])$, with a countable set of discontinuities $\tau_1 < \cdots < \tau_m < \cdots < \tau_M$. Let the numerical grid $0 = t_0 < \cdots < t_n < \cdots t_{N} = T$ be such that for every $m$ there exists an $n$ such that $\tau_m = t_n$. Then, for every $ and

Figures (9)

  • Figure 1: Reconstruction of the solution at a spike. The initial guess $\hat{\mathbold{y}}^-_{n + 1}$ is pulled back to the approximated time $t_{n + 1}$, where the discontinuous spike occurs.
  • Figure 2: Approximate solution based on a linear approximation (black) and the exact solution (gray). The approximate solution has a detected jump at $t_{k + 1}$ and the exact solution has the jump at $\tau_j \in [t_{k - 1}, t_{k + 1}]$.
  • Figure 3: (a) Membrane potential at $\alpha = 0.95$ and $\Delta t = 5 \times 10^{-4}$. The peak potential $V_{\text{peak}}$ and the reset potential $V_r$ are denoted with dashed lines. (b) Convergence of the piecewise L1 method on the PIF model. The error is computed based on the location of the exact and the approximate spike times $\{\tau_m\}$. The dashed lines denote the expected order $\mathcal{O}(\Delta t_{\text{max}})$.
  • Figure 4: Evolution of the LIF model using an adaptive step size algorithm with $\Delta t_0 = 10^{-1}$ (dotted black) and $\Delta t_{\text{min}} = 10^{-5}$ (dashed black). The bounds on the pointwise error are shown: $\chi_{\text{max}} = 2^2$ (full), $\chi_{\text{max}} = 2^{-2}$ (dashed) and $\chi_{\text{max}} = 2^{-6}$ (dotted).
  • Figure 5: (a) First-order self-convergence of the FrAdEx model and (b) Pointwise error of each of the $5$ numerical spike times for every $(\chi_{\text{min}}, \chi_{\text{max}})$ pair.
  • ...and 4 more figures

Theorems & Definitions (14)

  • Remark 3.1
  • Remark 3.2
  • Remark 4.1
  • Remark 5.1
  • Remark 5.2
  • Remark 5.3
  • Lemma 6.1
  • proof
  • Theorem 6.1: Piecewise L1 Truncation Error
  • proof
  • ...and 4 more