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

We present the first closed-form analytical characterization of local oscillatory dynamics in the adaptive exponential integrate-and-fire (AdEx) model, a key framework for understanding neural excitability and adaptation. By combining standard rescaling with rigorously bounded polynomial approximations of the exponential nonlinearity, we derive three unprecedented analytical results: (1) explicit Hopf bifurcation loci (trace-zero conditions) and stability criteria; (2) closed-form expressions for the first Lyapunov coefficient determining bifurcation type (subcritical vs supercritical) and neural excitability class (type-I vs type-II); and (3) leading-order period coefficients (T2, T3) characterizing how oscillation frequency depends on amplitude near bifurcation. For cubic approximations, we additionally characterize transitions between monoequilibria and triequilibria regimes, with implications for multistability and working memory. We provide rigorous local validity guarantees (|v-v∗|<0.6 ensures <1% and <5% errors for cubic and quadratic approximations, respectively) and quantify Taylor remainders. These closed-form results enable direct parameter-to-behavior mappings without numerical integration. We validate predictions against the full exponential model and demonstrate practical utility through genetic-algorithm-based parameter fitting to experimental AgRP neuron recordings. This work connects analytical tractability with empirical accuracy, offering both mechanistic insights into how adaptation shapes neural oscillations and computational efficiency for fitting models to data. While inherently local by construction, these results complement existing global reduction approaches and provide explicit coefficients unavailable from previous methods, opening new avenues for understanding adaptation-dependent dynamics in spiking neural networks.