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Oscillatory dynamics in the adaptive exponential integrate-and-fire model.

Ilknur Kusbeyzi Aybar¹, Fatma Kocaman¹, Mert Can Turkmen^2,3

¹Department of Mathematics, Faculty of Engineering and Natural Sciences, Yeditepe University, 34755 Istanbul, Turkiye.

Chaos (Woodbury, N.Y.)

|December 16, 2025

Summary

We derived analytical solutions for neural firing dynamics in the adaptive exponential integrate-and-fire (AdEx) model. These findings offer precise predictions of neural excitability and adaptation without numerical simulation.

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Area of Science:

Computational Neuroscience
Mathematical Biology
Systems Neuroscience

Background:

The adaptive exponential integrate-and-fire (AdEx) model is crucial for studying neural excitability and adaptation.
Analytical characterization of its oscillatory dynamics remains a challenge.

Purpose of the Study:

To derive closed-form analytical expressions for local oscillatory dynamics in the AdEx model.
To provide explicit criteria for bifurcation analysis and parameter-to-behavior mapping.

Main Methods:

Utilized standard rescaling and bounded polynomial approximations of the exponential nonlinearity.
Derived explicit Hopf bifurcation loci, stability criteria, and Lyapunov coefficients.
Quantified Taylor remainders to ensure local validity.

Main Results:

Obtained closed-form expressions for bifurcation loci, stability, and excitability class (Type-I/II).
Derived leading-order period coefficients (T2, T3) for amplitude-dependent frequency.
Characterized transitions between mono- and tri-equilibria regimes for cubic approximations.

Conclusions:

The derived closed-form solutions enable direct parameter-to-behavior mapping, bypassing numerical integration.
Validated against the full AdEx model and demonstrated utility in fitting experimental data.
Offers mechanistic insights into adaptation's role in neural oscillations and computational efficiency.