Analytical Solution of System of FitzHugh-Nagumo Model: A Case Study

This study explores the FitzHugh-Nagumo model, a mathematical system used to simulate the activity of neurons. We apply the Adomian Decomposition Method (ADM) to generate approximate solutions using polynomial series. While these formula-based approximations are highly accurate for capturing short-term changes, the analysis reveals a critical limitation: they eventually fail over longer timeframes. This is because the neuron model is designed to produce stable, repeating cycles (oscillations), whereas polynomial approximations naturally grow to infinity rather than looping back. Consequently, this analytical method cannot accurately reproduce the neuron's long-term, rhythmic behavior. To accurately capture the long-term dynamics and spiking behavior of the neuron, numerical integration approaches are the necessary and most reliable option.