A cortical field theory - dynamics and symmetries

Area of Science:

• Computational Neuroscience
• Theoretical Physics
• Mathematical Biology

Background:

• Cortical dynamics exhibit complex oscillatory patterns observed in electrophysiology.
• Understanding these dynamics is crucial for deciphering brain function and dysfunction.
• Previous models often simplify the intricate interactions within the cortical sheet.

Purpose of the Study:

• To characterize cortical dynamics using partial differential equations (PDEs).
• To explore diverse dynamics, including wave equations and limit cycle activity, under balanced neuronal excitation and inhibition.
• To investigate the role of symmetries in analyzing neural field models.

Main Methods:

• Formulation of partial differential equations (PDEs) to model cortical activity.
• Analysis of wave equations (e.g., Klein-Gordon model) and limit cycle dynamics.
• Application of Lagrangian formalism to study model symmetries (continuous and discontinuous).
• Leveraging concepts from critical state physics for neural field analogues.

Main Results:

• Derived dynamics include wave equations, limit cycle waves, and transitions between states.
• Identified continuous and discontinuous symmetries offering mathematical and analytical advantages.
• Model predictions show close agreement with electrophysiological findings (spectral power, wave speed, seizure generation, pattern formation).

Conclusions:

• The brain's critical state assumption drives oscillatory and semi-conservative neural field dynamics.
• Symmetry-preserving PDE formulations are vital for mechanistic insights into cortical activity.
• The model provides a framework for understanding neural field theories and their comparison with empirical data.