Schematic of LOOPER—A method for constructing simple models of nonlinear dynamical systems based on experimental observations.

Overview: A) Activity of a hypothetical system given by three neurons. The state of this system is characterized by the neuronal population activity vector, which records the instantaneous firing of all neurons (Neurons 1, 2 and 3). Note that the activity is confined to a helical manifold. Time is implicit and flows in the direction indicated by arrows in A through F. In this example, neuronal activity is recorded under three different task conditions shown by color in A-F. B) Diffusion mapping is used to unwrap the helical structure in A to reveal a simpler (2D) projection that contains all of the neuronal activity. While the unwrapped space is 2-dimensional, the actual dynamics of the system are largely confined to a set of three one dimensional trajectories each starting from a unique set of initial conditions (red, green and blue circles). So long as trajectories stay separate, the information stored in the state of the system is preserved. At the moment when red and blue trajectories merge, the system is no longer able to distinguish them (blue arrow). The merging of the trajectories can be quantified by the overlap in the distributions of neuronal activity (marginal histograms). On the contrary, the green trajectory is separated from the rest at the end of the trial. Thus, the distribution of the green points is distinct from the other two. C) Shows a diffusion map, normalized as a transition probability matrix, constructed from the observations in A. The three task conditions (three trials of each) are shown by three colored squares. The most likely transitions in each case are along the one off diagonal of the matrix. This corresponds to the actual observations. High transition probabilities away from this main diagonal correspond to similar dynamics observed on different trials. Note that the off diagonal bands appear late in the trials for blue and red conditions. In contrast, the green and red conditions are initially similar but diverge towards the end of the trial. D) Observations in C naturally inform a way to coarse grain the diffusion map. States characterized by similar transition probabilities (rows of the matrix in C) can be readily aggregated. This results in a course grained transition probability matrix where each entry is a cluster of experimental observations aggregated on the basis of similarity of rows in the diffusion map in C. This coarse grained transition probability matrix is shown as a graph. The position of each state (S1, S2, S2…) in the latent space is given by the mean of all experimental observations that comprise the state. Note that in this example transitions from each state (arrows) are predominantly to a single other state. This expresses the assumption that the most salient information can be represented by one dimensional trajectories. This assumption allows further simplification of the system by clustering states (S1, S2, …) into trajectories (ordered sequences of states). E) Finally, states within each trajectory are interpolated to reveal a model that consists entirely of one dimensional trajectoriess (each colored surface is a distinct trajectory). The width of each trajectory shows the variance of the experimental observations projected onto the latent space F) A useful way to describe the data is the “computational scaffold” which is the trajectory ID assigned to the observed data at each time point. Note that the data tends to separate out the initial conditions (red, green and blue) during the middle of the trial based on trajectory ID. Further, the way that the trajectories merge and split can give valuable insight into the types of information used by the system.