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Properties of networks with partially structured and partially random connectivity
Yashar Ahmadian1, Francesco Fumarola2, Kenneth D Miller1
1Center for Theoretical Neuroscience, Department of Neuroscience, and Swartz Program in Theoretical Neuroscience, and Kavli Institute for Brain Science, College of Physicians and Surgeons, Columbia University, New York, New York 10032, USA.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 14, 2015
Summary
This study analyzes complex random matrices common in neuroscience and biology. It develops new formulas to understand network dynamics and stability, even with non-independent random connections.
Area of Science:
- Mathematical Biology
- Computational Neuroscience
- Random Matrix Theory
Background:
- Many scientific networks exhibit stochastic connectivity, not always independent or identically distributed (iid).
- Understanding these complex random matrices is crucial for analyzing network behavior in fields like neuroscience.
- Existing models often assume simpler matrix structures, limiting their applicability to more general network dynamics.
Purpose of the Study:
- To develop a mathematical framework for characterizing large random matrices with non-trivial mean structure and correlations.
- To provide general formulas for eigenvalue density, transient dynamics, and power spectrum of systems described by these matrices.
- To analyze the conditions leading to persistent outlying eigenvalues in such complex network models.
Main Methods:
- Characterization of large random N×N matrices of the form A=M+LJR, where M, L, R are deterministic and J is a random matrix.
- Derivation of general formulas for eigenvalue density and transient evolution of activity magnitude and power spectrum.
- Analytical solutions for specific examples motivated by neurobiological models, examining conditions for outlying eigenvalues.
Main Results:
- General formulas for eigenvalue density and transient dynamics of linear systems with the specified random matrix structure.
- Analytical results for neurobiologically relevant models, demonstrating the application of the derived formulas.
- Identification of conditions related to singular values of L(-1)(z1-M)R(-1) for the persistence of outlying eigenvalues.
Conclusions:
- The developed framework accurately characterizes the behavior of complex stochastic matrices beyond iid assumptions.
- The formulas provide insights into network stability and response dynamics, applicable to neuroscience and mathematical biology.
- Understanding outlying eigenvalues is linked to specific properties of the deterministic components of the random matrix.
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