Linearizing and forecasting: a reservoir computing route to digital twins of the brain

Exploring the dynamics of complex systems such as the human brain is challenging due to inherent uncertainties and the limited availability of high-quality data. Here, we develop a mathematical theory for noisy linear recurrent neural networks (lRNNs) within the reservoir computing framework and demonstrate their effectiveness in constructing autonomous in silico replicas - digital-twins - of brain activity. We show that the Laplace-transform poles of high-dimensional inferred lRNNs directly encode the spectral properties of observed systems and are linked to the kernels of auto-regressive models. Notably, our approach enables accurate recovery of the system's linear spectrum even when observations undergo conventional preprocessing, including band-pass filtering pipelines commonly used in neural recordings and resting-state fMRI. In these regimes, established techniques such as dynamic mode decomposition often produce spurious spectral estimates. Applying our framework to resting-state fMRI, we successfully predict and decompose BOLD activity into spatiotemporal modes in a low-dimensional latent state space confined around a single equilibrium point. The inferred lRNNs provide interpretable signatures that differentiate subjects and brain areas, supporting biologically meaningful clustering. This flexible digital-twin framework opens the door to virtual experiments and computationally efficient real-time adaptive learning, offering a promising avenue for personalized medicine and intervention strategies.