Clustering of neural codewords revealed by a first-order phase transition

A network of neurons in the central nervous system collectively represents information by its spiking activity states. Typically observed states, i.e., codewords, occupy only a limited portion of the state space due to constraints imposed by network interactions. Geometrical organization of codewords in the state space, critical for neural information processing, is poorly understood due to its high dimensionality. Here, we explore the organization of neural codewords using retinal data by computing the entropy of codewords as a function of Hamming distance from a particular reference codeword. Specifically, we report that the retinal codewords in the state space are divided into multiple distinct clusters separated by entropy-gaps, and that this structure is shared with well-known associative memory networks in a recallable phase. Our analysis also elucidates a special nature of the all-silent state. The all-silent state is surrounded by the densest cluster of codewords and located within a reachable distance from most codewords. This codeword-space structure quantitatively predicts typical deviation of a state-trajectory from its initial state. Altogether, our findings reveal a non-trivial heterogeneous structure of the codeword-space that shapes information representation in a biological network.