Analytic modeling of neural tissue: II. Nonlinear membrane dynamics

Area of Science:

• Computational neuroscience
• Neural engineering
• Applied mathematics

Background:

• Analytic solutions fail to capture nonlinear neuroactivity essential for understanding brain dynamics.
• Current methods are often limited to linearized models, hindering advancements in neural engineering.
• Accurate computational modeling of neuroactivity is vital for developing technologies like deep brain stimulation and neuroprosthetics.

Purpose of the Study:

• To establish a robust and straightforward process for modeling neurodynamic phenomena that preserves nonlinear features.
• To apply decomposition methods from homotopy analysis to solve nonlinear ordinary differential equations governing neural conduction.
• To validate the decomposition method against established techniques like B-spline collocation.

Main Methods:

• Utilized George Adomian's decomposition method to solve nonlinear ordinary differential equations for the Ermentrout-Kopell, FitzHugh-Nagumo, and Hindmarsh-Rose models.
• Constructed power series solutions for each variable, recursively determined from initial conditions.
• Employed one-step analytic continuation to extend the region of convergence, creating decomposition splines.

Main Results:

• Achieved rapid convergence with a maximal error of $$ using only eight terms.
• Demonstrated the method's ability to yield solutions for single- and multi-variable models, characterizing action potentials and complex bursting patterns.
• Showcased comparable accuracy to B-spline collocation while preserving the original model equations.

Conclusions:

• The decomposition method provides a stable, computationally efficient, and accurate approach for modeling nonlinear neurodynamic phenomena.
• This technique is a viable tool for advanced neural engineering studies, offering solutions to the original problems without simplification.
• The method's ability to handle nonlinearities makes it suitable for complex neural simulations and the development of advanced neural technologies.