Element-wise and recursive solutions for the power spectral density of biological stochastic dynamical systems at fixed points

Stochasticity plays a central role in nearly every biological process, and the noise power spectral density (PSD) is a critical tool for understanding variability and information processing in living systems. In steady state, many such processes can be described by stochastic linear time-invariant (LTI) systems driven by Gaussian white noise, whose PSD is a complex rational function of the frequency that can be concisely expressed in terms of their Jacobian, dispersion, and diffusion matrices, fully defining the statistical properties of the system's dynamics at steady state. Here, we arrive at compact, element-wise solutions of the rational function coefficients for the auto- and cross-spectrum that enable the explicit analytical computation of the PSD in dimensions <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>n</a:mi><a:mo>=</a:mo><a:mn>2</a:mn><a:mo>,</a:mo><a:mn>3</a:mn><a:mo>,</a:mo><a:mn>4</a:mn></a:mrow></a:math>. We further present a recursive Leverrier-Faddeev-type algorithm for the exact computation of the rational function coefficients. Crucially, both solutions are free of matrix inverses. We illustrate our element-wise and recursive solutions by considering the stochastic dynamics of neural systems models, namely, FitzHugh-Nagumo (<b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>n</b:mi><b:mo>=</b:mo><b:mn>2</b:mn></b:mrow></b:math>), Hindmarsh-Rose (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>n</c:mi><c:mo>=</c:mo><c:mn>3</c:mn></c:mrow></c:math>), Wilson-Cowan (<d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>n</d:mi><d:mo>=</d:mo><d:mn>4</d:mn></d:mrow></d:math>), and the stabilized supralinear network (<e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>n</e:mi><e:mo>=</e:mo><e:mn>22</e:mn></e:mrow></e:math>), as well as an evolutionary game-theoretic model with mutations (<f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>n</f:mi><f:mo>=</f:mo><f:mn>5</f:mn><f:mo>,</f:mo><f:mn>31</f:mn></f:mrow></f:math>). We extend our approach to derive a recursive method for calculating the coefficients in the power-series expansion of the integrated covariance matrix for interacting spiking neurons modeled as Hawkes processes on arbitrary directed graphs. Published by the American Physical Society 2024