A theoretical framework for brain-computer interfaces: decodability, performance limits, and closed-loop adaptation.

Although brain-computer interfaces (BCIs) have made significant advances in brain-signal acquisition and decoding algorithms, the theoretical foundations of BCI remain fragmented, with limited characterization of the decodability, performance bounds, and dynamic constraints of BCI systems. To address this issue, this study proposes a theoretical framework for BCIs from the perspectives of information theory and statistical decision theory. The BCI system is formalized as a closed-loop stochastic process comprising intention generation, neural encoding, signal observation, statistical decoding, and feedback regulation. On this basis, a hierarchical theoretical framework is established that integrates scientific hypotheses, neuroscientific principles, fundamental theorems, and fundamental laws. The fundamental theorems characterize the conditions for intention decodability, the upper bound of observable information, the lower bound of optimal decoding error, and the convergence of closed-loop learning. The fundamental laws reveal the constraints imposed by signal-to-noise ratio, low-dimensional neural representations, irreversible loss of observational information, class-distribution separability, and non-stationary co-adaptation. Building on this, the paper takes a four-class SSVEP-BCI as a theoretical case study, parameterizes and instantiates the aforementioned theorems and laws, verifies the operability and explanatory power of the proposed framework within a concrete paradigm, and translates the theory into actionable design guidelines for BCI systems. To the best of our knowledge, this paper is the first attempt to propose and systematically elaborate a theoretical framework for BCI from a unified theoretical perspective, and it is expected that this work will provide a theoretical foundation for the analysis, design, and optimization of BCI systems.