Bifurcation analysis of a neural network model

Area of Science:

• Computational neuroscience
• Dynamical systems theory
• Mathematical modeling of neural networks

Background:

• The Wilson and Cowan model is a foundational mathematical framework for understanding neural population dynamics.
• Analyzing parameter dependencies is crucial for predicting the complex behaviors of neural systems.

Purpose of the Study:

• To conduct a detailed analysis of the Wilson and Cowan neural network model.
• To investigate the influence of key parameters on the model's dynamical behavior.
• To map and characterize distinct operational regions within the model's parameter space.

Main Methods:

• Modeling the neural network using a system of two ordinary differential equations.
• Analyzing the evolution of average activities in excitatory and inhibitory neuronal populations.
• Partitioning the parameter plane into regions of equivalent behavior using bifurcation curves.
• Constructing representative phase diagrams for each identified region.

Main Results:

• The parameter plane is systematically divided into distinct regions, each characterized by specific dynamical behaviors.
• Bifurcation curves delineate the boundaries between these regions, predicting qualitative shifts in model dynamics.
• Long-period oscillations were identified for specific parameter values.
• A novel dynamical behavior was discovered where the system converges to either a stationary state or a limit cycle based on initial conditions.

Conclusions:

• The study provides a comprehensive qualitative description of the Wilson and Cowan model's behavior across different parameter regimes.
• The findings enable predictions of dynamic changes as model parameters are varied.
• The identification of new dynamical behaviors, including long-period oscillations and initial-condition-dependent attractors, enhances our understanding of neural network dynamics.