Information-Theoretic Error Bounds for Source Localization in Neural Sensing

摘要

We formulate a point-source localization problem in d dimensions, where a source inside the ball of radius R emits a signal that is picked up by various sensors located at the surface of the ball. For d=3, this can model problems in neural sensing, where a net of electroencephalography (EEG) or magnetoencephalography (MEG) sensors try to locate the source of a distinct neural event such as a seizure. For a power law decay model with exponent α>0 for the sensors, we obtain a lower bound on the minimax risk for localizing the source that is asymptotically d^2σ^2R^2α+2/nα^2PK under mean-squared error loss, where σ^2 is the noise variance, P is the signal power, K is the number of sensors, and n is the number of independent measurements. In the case d≤ 2(α+1) with uniformly distributed sensor locations, we then give a matching upper bound, including getting the exact constant correct, for the asymptotic minimax rate in a neighborhood of the origin. We show that there is a phase transition at d=2(α+2), above which a certain Fisher information quantity is minimized at the boundary of the ball, and below which it is minimized at the origin. At the critical dimension d=2(α+2), the Fisher information is constant throughout the entire parameter space. For the special case d=3, we supplement and compare this information-theoretic analysis with a simulated forward EEG model that uses a realistic head model derived from population-averaged magnetic resonance imaging data.