Degeneracy in neuronal bifurcation landscapes: Equivalent bifurcation sequences across distinct parameter sets
- 1Aix Marseille Univ, INSERM, INS, Inst Neurosci Syst, Marseille, France
Abstract
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. Author summary: Biophysical models expressed through differential equations can reproduce the dynamics of biological systems with varying levels of detail. By changing model parameters, simulations can capture the natural variability that allows biological systems to achieve the same behavior through different mechanisms. This property, called degeneracy, underlies the robustness of living systems to internal and external perturbations. In this work, we identify distinct behaviors in single-neuron models and link them to electrophysiological patterns observed under elevated extracellular potassium during epileptic events. These patterns correspond to specific classes of spontaneous bursting activity within the framework of nonlinear dynamics. We show that different combinations of conductance parameters, at a fixed extracellular potassium level, can generate the same class of patterns, defining degeneracy groups. Finally, we assess the robustness of these groups by analyzing how they respond to changes in extracellular potassium. Our findings provide a basis for studying variability and resilience in the dynamics of more complex neuronal systems.
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BibTeX
@article{cafaro2026degeneracy,
title = {{Degeneracy in neuronal bifurcation landscapes: Equivalent bifurcation sequences across distinct parameter sets}},
author = {Gianmarco Cafaro and Christophe Bernard and Meysam Hashemi and Damien Depannemaecker},
journal = {PLOS Computational Biology},
year = {2026},
month = oct,
doi = {10.1371/journal.pcbi.1014893},
url = {https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1014893},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Degeneracy in neuronal bifurcation landscapes: Equivalent bifurcation sequences across distinct parameter sets
AU - Gianmarco Cafaro
AU - Christophe Bernard
AU - Meysam Hashemi
AU - Damien Depannemaecker
PY - 2026
DA - 2026/10/30
JO - PLOS Computational Biology
DO - 10.1371/journal.pcbi.1014893
UR - https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1014893
AB - 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. Author summary: Biophysical models expressed through differential equations can reproduce the dynamics of biological systems with varying levels of detail. By changing model parameters, simulations can capture the natural variability that allows biological systems to achieve the same behavior through different mechanisms. This property, called degeneracy, underlies the robustness of living systems to internal and external perturbations. In this work, we identify distinct behaviors in single-neuron models and link them to electrophysiological patterns observed under elevated extracellular potassium during epileptic events. These patterns correspond to specific classes of spontaneous bursting activity within the framework of nonlinear dynamics. We show that different combinations of conductance parameters, at a fixed extracellular potassium level, can generate the same class of patterns, defining degeneracy groups. Finally, we assess the robustness of these groups by analyzing how they respond to changes in extracellular potassium. Our findings provide a basis for studying variability and resilience in the dynamics of more complex neuronal systems.
SN - 1553-7358
ER -