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Element-wise and Recursive Solutions for the Power Spectral Density of Biological Stochastic Dynamical Systems at
Shivang Rawat1,2, Stefano Martiniani1,2,3
1Courant Institute of Mathematical Sciences, New York University, New York 10003, USA.
Physical Review Research
|April 27, 2026
Summary
This study introduces new analytical methods to compute the noise power spectral density (PSD) for stochastic biological systems. These methods simplify calculations for complex systems, offering insights into biological variability and information processing.
Area of Science:
- Computational Biology
- Systems Neuroscience
- Theoretical Neuroscience
Background:
- Stochasticity is fundamental to biological processes, influencing variability and information processing.
- Noise Power Spectral Density (PSD) is key for analyzing stochastic Linear Time-Invariant (LTI) systems.
- Existing methods for PSD calculation in biological systems can be computationally intensive.
Purpose of the Study:
- To develop novel, efficient analytical methods for computing the PSD of stochastic LTI systems.
- To provide explicit solutions for PSD rational function coefficients, reducing computational complexity.
- To enable precise analysis of steady-state dynamics in diverse biological models.
Main Methods:
- Derived compact, element-wise solutions for auto- and cross-spectrum coefficients.
- Developed a recursive Leverrier-Faddeev-type algorithm for coefficient computation.
- Solutions avoid matrix inversions, enhancing computational efficiency.
Main Results:
- Achieved explicit analytical computation of PSD for dimensions n = 2, 3, 4.
- Validated methods on neural models (Fitzhugh-Nagumo, Hindmarsh-Rose, Wilson-Cowan, Stabilized Supralinear Network) and evolutionary game models.
- Extended the approach to derive a recursive method for integrated covariance matrix calculations in Hawkes process models.
Conclusions:
- The novel analytical and recursive methods offer significant computational advantages for PSD analysis in biological systems.
- These findings facilitate a deeper understanding of stochastic dynamics and information processing in neuroscience and evolutionary biology.
- The approach is broadly applicable to various complex biological systems exhibiting stochastic behavior.
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