Discontinuous phase transition of feature detection in lateral predictive coding

The brain may adopt the strategy of lateral predictive coding (LPC) to construct optimal internal representations for salient features in input sensory signals, reducing the energetic cost of information transmission. Here we first consider the task of detecting one non-Gaussian signal by LPC from Gaussian background signals of the same magnitude, which is intractable by principal component decomposition. We study the emergence of feature detection function from the perspective of tradeoff between energetic cost $E$ and information robustness, and implement this tradeoff by a thermodynamic free energy. We define $E$ as the mean $L_1$-norm of the internal state vectors, and quantify the level of information robustness by an entropy measure $S$. There are at least three types of optimal LPC matrices, one type with very weak synaptic weights and $S \approx 0$, and two functional types either with low energy $E$ or with high entropy $S$ in which one single unit selectively responds to the non-Gaussian input feature. Energy--information tradeoff induce two discontinuous phase transitions between these three types of optimal LPC networks. We then extend the discussion to detecting and distinguishing between two non-Gaussian input features and observe similar discontinuous phase transitions.