Abstract

Recent whole brain imaging experiments in \textit{C. elegans} have revealed that the neural connectomic dynamics live on a low dimensional manifold with stereotyped transitions between behaviors. Typical theoretical efforts use data to produce a set of local linear models characterizing the data, but it is unknown how a single, global neural network model can generate the observed dynamics. We propose instead a control framework to achieve a global model of the dynamics, whereby underlying linear dynamics is actuated by sparse control signals. The method learns the control signals in an unsupervised way from data, then uses {\em Dynamic Mode Decomposition with control} (DMDc) to create the first global, linear dynamical system that can reconstruct whole-brain imaging data. These internally generated control signals are shown to be implicated in transitions between behaviors. In addition, we analyze the time-delay encoding of these control signals, both reproducing known neural encodings and showing that these transitions can be predicted from previously unknown neurons. Moreover, our decomposition method allows one to understand the observed nonlinear global dynamics instead as linear dynamics with control. Taken together, the possibility of decomposing this neural dataset into linear intrinsic dynamics and spikes constrains the need for nonlinearities in future modeling efforts. The proposed mathematical framework is generic and can be generalized to other neurosensory systems, potentially revealing transitions and their encodings in a completely unsupervised way.