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Non-Hermitian localization in biological networks.

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Area of Science:

  • * Theoretical neuroscience
  • * Complex systems
  • * Mathematical physics

Background:

  • * Sparse neural networks utilize random matrices for modeling connectivity.
  • * Non-Hermitian random matrices are crucial for understanding complex network dynamics.
  • * Eigenvalue spectra and localization properties dictate neural network activity.

Purpose of the Study:

  • * To investigate the spectral and localization properties of N-site banded non-Hermitian random matrices in sparse neural networks.
  • * To understand how eigenvalue distributions control spontaneous and induced neural activity.
  • * To analyze the impact of directional bias on network states.

Main Methods:

  • * Direct numerical diagonalization of large N matrices.
  • * Transfer matrix techniques for analyzing large N limits.
  • * Electrostatic analogy to connect eigenvalue distributions and localization length.
  • * Perturbation theory for large directional bias limits.

Main Results:

  • * Approximately equal excitatory and inhibitory connections lead to localized eigenfunctions and complex eigenvalue spectra.
  • * Eigenvalues condense on real/imaginary axes; spectrum exhibits 90° rotational symmetry for large N.
  • * Directional bias creates a hole in the complex plane's density of states, with extended states on its rim.
  • * Dale's law compliant networks show similar spectral properties.

Conclusions:

  • * The spectral and localization properties of non-Hermitian random matrices are fundamental to sparse neural network function.
  • * Directional bias significantly alters network states by modifying the density of states.
  • * Findings have implications for understanding neural computation and related complex systems like ecological networks.