Degeneracy in neuronal bifurcation landscapes: Equivalent bifurcation sequences across distinct parameter sets

Degeneracy, the capacity of structurally distinct systems to achieve similar functions, is a fundamental property of living organisms, enabling adaptability and resilience. Neurons can maintain stable activity patterns despite substantial variability in ion channel expression, highlighting how different intrinsic configurations can produce equivalent electrophysiological behaviors. Previous studies have shown that neuronal activity emerges from the interaction between intrinsic membrane properties and ionic concentration dynamics. However, how degeneracy is organized within the dynamical landscape of neurons and how different degenerate configurations respond to external perturbations remain poorly understood. Here, we investigate how distinct biophysical parameter sets generate equivalent bifurcation sequences and how their resilience changes under ionic modulation. Using a minimal neuron model with three variable conductances, we systematically explore how variations in extracellular potassium concentration, representing physiological and pathological conditions, reshape neuronal dynamics. We show that distinct conductance configurations occupy structured regions of parameter space that support the same bifurcation pathways, leading to equivalent activity regimes such as bursting, seizure-like activity, and depolarization block. However, these degenerate configurations exhibit different sensitivities to extracellular perturbations, revealing that functional equivalence under fixed conditions does not imply equivalent robustness. To quantify this effect, we introduce resilience as a measure of the threshold at which a given conductance configuration transitions into a specific activity regime under external ionic changes. Our results demonstrate that degeneracy extends beyond the coexistence of multiple solutions: degenerate configurations form structured landscapes with distinct trajectories under environmental modulation. This framework provides a quantitative approach to investigate how intrinsic variability influences neuronal vulnerability and resilience across different dynamical states.