Hierarchical Wilson-Cowan Models and Connection Matrices.

Area of Science:

• Computational neuroscience
• Dynamical systems theory
• Network science

Background:

• The Wilson-Cowan model describes neural population dynamics.
• Connection matrices represent cortical neural wiring.
• Classical models struggle to incorporate complex network structures like small-world properties.

Purpose of the Study:

• To investigate the interplay between the Wilson-Cowan model and connection matrices.
• To develop a generalized Wilson-Cowan framework compatible with empirical neural wiring data.
• To explore the small-world property in neural dynamics.

Main Methods:

• Formulating Wilson-Cowan equations on locally compact Abelian groups.
• Developing a p-adic version of the Wilson-Cowan model.
• Conducting numerical simulations using neural network models and p-adic approximations of cortical connection matrices.

Main Results:

• Demonstrated well-posedness of the Cauchy problem for Wilson-Cowan equations on these groups.
• Showed the classical Wilson-Cowan model is incompatible with the small-world property.
• Validated the p-adic Wilson-Cowan model against classical predictions and experimental data.

Conclusions:

• The p-adic Wilson-Cowan model successfully incorporates connection matrices and neural network properties.
• This generalized model offers a pathway to study complex brain network dynamics, including the small-world property.
• The proposed framework advances computational neuroscience by bridging dynamical models with structural connectivity.