Hidden Dynamics in the FitzHugh–Nagumo Neural System Revealed: How Third-Order Analysis Uncovers Hidden Behaviors Better Than Computers

A faculty member at the College of Science, University of Tehran, has succeeded, by presenting precise formulas, in determining the conditions for stability and the existence of invariant tori in the well-known FitzHugh–Nagumo neural excitation model—behaviors that had not previously been detectable in numerical simulations.

According to the Public Relations Office of the University of Tehran, the new analytical method, developed through an international collaboration with researchers from the University of Campinas (UNICAMP) in Brazil, reveals complex and hidden behaviors in neural systems that are difficult to uncover through conventional computer simulations. It can help engineers and neuroscientists more accurately predict the function of neuronal excitability and neurons.

Dr. Nasrin Sadri, a faculty member at the College of Science, explaining her latest findings in neuroscience and mathematical biology, announced the derivation of general and explicit conditions for the existence and stability of limit cycles and invariant tori in the FitzHugh–Nagumo neural system.

The University of Tehran researcher added: “The analytical methods used in this study are capable of identifying complex dynamics that cannot be detected by conventional numerical approaches and require third-order mathematical analyses—an approach that, due to its inherent complexities, had received less attention until now.”

She also emphasized: “The FitzHugh–Nagumo system is a four-parameter family of three-dimensional vector fields that is widely used in modeling neuronal excitability and neurons.”

The faculty member of the Faculty of Mathematical Sciences stated: “The results of this study not only generalize previous studies by providing general and precise conditions for the existence and stability of invariant tori, but also introduce new conditions for the existence of limit cycles for the first time.”

Referring to the Lyapunov–Schmidt reduction method, Dr. Sadri said: “Under certain conditions, first-order methods do not lead to a definite answer, but in this study we have also succeeded in deriving explicit conditions for the bifurcation of limit cycles and stable tori under these conditions.”

Another notable achievement of this research is the explicit statement of the stability condition for bifurcated limit cycles—a subject that had been overlooked in previous studies.

In conclusion, Dr. Sadri announced: “During these investigations, complex dynamics have been identified that require third-order analysis; behaviors that are usually difficult to reveal with common numerical methods due to inherent limitations, and their analytical identification is considered an important step toward a deeper understanding of neural systems and nonlinear dynamics.”

Source: Public Relations Office, University of Tehran
Article: https://www.sciencedirect.com/science/article/pii/S1468121826000763