Neuronal correlations shape the scaling behavior of memory capacity and nonlinear computational capability of recurrent neural networks

Reservoir computing is a powerful framework for real-time information processing, characterized by its high computational ability and quick learning, with applications ranging from machine learning to biological systems. In this paper, we demonstrate that the memory capacity of a reservoir recurrent neural network scales sublinearly with the number of readout neurons. To elucidate this phenomenon, we develop a theoretical framework for analytically deriving memory capacity, attributing the decaying growth of memory capacity to neuronal correlations. In addition, numerical simulations reveal that once memory capacity becomes sublinear, increasing the number of readout neurons successively enables nonlinear processing at progressively higher polynomial orders. Furthermore, our theoretical framework suggests that neuronal correlations govern not only memory capacity but also the sequential growth of nonlinear computational capabilities. Our findings establish a foundation for designing scalable and cost-effective reservoir computing, providing novel insights into the interplay among neuronal correlations, linear memory, and nonlinear processing.