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Stochasticity plays a central role in nearly every biological process, andthe noise power spectral density (PSD) is a critical tool for understandingvariability and information processing in living systems. In steady-state, manysuch processes can be described by stochastic linear time-invariant (LTI)systems driven by Gaussian white noise, whose PSD is a complex rationalfunction of the frequency that can be concisely expressed in terms of theirJacobian, dispersion, and diffusion matrices, fully defining the statisticalproperties of the system's dynamics at steady-state. Here, we arrive at compactelement-wise solutions of the rational function coefficients for the auto- andcross-spectrum that enable the explicit analytical computation of the PSD indimensions n=2,3,4. We further present a recursive Leverrier-Faddeev-typealgorithm for the exact computation of the rational function coefficients.Crucially, both solutions are free of matrix inverses. We illustrate ourelement-wise and recursive solutions by considering the stochastic dynamics ofneural systems models, namely Fitzhugh-Nagumo (n=2), Hindmarsh-Rose (n=3),Wilson-Cowan (n=4), and the Stabilized Supralinear Network (n=22), as well asan evolutionary game-theoretic model with mutations (n=5, 31).

Element-wise and Recursive Solutions for the Power Spectral Density of Biological Stochastic Dynamical Systems at Fixed Points