Gell-Mann-Low Criticality in Neural Networks

Area of Science:

• Computational neuroscience
• Theoretical physics
• Complex systems

Background:

• Criticality is linked to optimal computational capacity in biological systems.
• Current theories often rely on mean-field approaches, neglecting multi-scale interactions crucial for complex nonlinear computation.
• A lack of a renormalized theory for critical brain dynamics limits understanding of biological information processing.

Purpose of the Study:

• To present a renormalized theory for a prototypical neural field theory, the stochastic Wilson-Cowan equation.
• To compute the flow of couplings, parametrizing interactions across increasing length scales.
• To elucidate how critical interactions balance linearity and nonlinearity for optimal computation.

Main Methods:

• Developed a renormalized theory for the stochastic Wilson-Cowan equation.
• Computed the flow of couplings to analyze interactions across different length scales.
• Compared the theory's structure to established models like the Kardar-Parisi-Zhang and Gell-Mann-Low types.

Main Results:

• The theory is of a Gell-Mann-Low type, a characteristic of renormalizable quantum field theories.
• Nonlinear couplings vanish logarithmically slowly towards the Gaussian fixed point, remaining effective across most scales.
• Demonstrated that this critical interaction structure achieves a balance between linearity (for information storage) and nonlinearity (for computation).

Conclusions:

• The developed renormalized theory provides a framework for understanding critical brain dynamics beyond mean-field approximations.
• The findings highlight the importance of multi-scale interactions in achieving optimal computational capacity in neural systems.
• This work offers insights into the fundamental principles governing information processing in biological systems at criticality.