Random graph theory and neuropercolation for modeling brain oscillations at criticality

Area of Science:

• Computational Neuroscience
• Mathematical Biology
• Complex Systems

Background:

• Brain imaging reveals intermittent, singular space-time dynamics.
• Understanding these dynamics requires advanced mathematical frameworks.
• Key aspects include nonlinear dynamics, chaos, phase transitions, and criticality.

Purpose of the Study:

• To review mathematical approaches for interpreting complex brain dynamics.
• To present models explaining critical brain oscillations and state transitions.
• To link theoretical models with experimental observations in brain imaging.

Main Methods:

• Review of nonlinear dynamics (chaos), phase transitions, and criticality in brain function.
• Description of probabilistic cellular automata and random graph models.
• Introduction of a modular neuropercolation model for cortical dynamics.

Main Results:

• Probabilistic models offer alternatives to differential equations for macroscopic brain variables.
• The neuropercolation model describes critical brain oscillations (background, narrow-band, broadband).
• Models capture transitions between synchronized and unsynchronized brain states with long-range spatial correlations.

Conclusions:

• Mathematical modeling, particularly the neuropercolation approach, provides a framework for understanding complex brain dynamics.
• These models can explain experimentally observed phenomena like critical brain oscillations and state transitions.
• The study highlights the utility of probabilistic and network-based models in neuroscience.