  Songs of many songbird species consist of variable sequences of a finite
number of syllables. A common approach for characterizing the syntax of these
complex syllable sequences is to use transition probabilities between the
syllables. This is equivalent to the Markov model, in which each syllable is
associated with one state, and the transition probabilities between the states
do not depend on the state transition history. Here we analyze the song syntax
in a Bengalese finch. We show that the Markov model fails to capture the
statistical properties of the syllable sequences. Instead, a state transition
model that accurately describes the statistics of the syllable sequences
includes adaptation of the self-transition probabilities when states are
repeatedly revisited, and allows associations of more than one state to the
same syllable. Such a model does not increase the model complexity
significantly. Mathematically, the model is a partially observable Markov model
with adaptation (POMMA). The success of the POMMA supports the branching chain
network hypothesis of how syntax is controlled within the premotor song nucleus
HVC, and suggests that adaptation and many-to-one mapping from neural
substrates to syllables are important features of the neural control of complex
song syntax.
