Description of the structure of the hidden Markov model (HMM).

(A) In the limit of no temporal correlations, the probability over all binary spike words is given by a mixture over different emission distributions. (i) In the case of no correlation within an emission distribution, Q^ind has a simple product form; (ii) our full model includes pairwise correlations within an emission distribution, Q^tree, having a tree graphical structure. (B) Left: The full model also includes temporal correlation through a transition probability matrix, P(α(t)|α(t−1)), which describes the probability of finding mode α at time step t given the mode present in the previous time step, t-1. The hidden state (mode identity), α, evolves probabilistically over time according to the transition matrix, and the observed pattern of spiking at time t, {σ_i(t)}, has a mode-dependent emission probability, . Right: Schematic representation of the tree graphical structure of each emission distribution. (C) In order to find the stationary distribution of the HMM, we solved for the set of mode weights, {w_α}, using the detailed balance equation. (D) The total number of modes, M, was determined by finding the maximum 2-fold cross-validated likelihood of the model. Note that the optimum was relatively shallow, so that the model is not sensitive to the precise number of modes, as long as it lies within a certain range (for this dataset, roughly 30–100 modes).