# Equal Space

This project is in preparation for publication to PNAS.

Previous Work in our lab uses nonlinear dimensionality reduction techniques to embed the states of a high-dimensional dynamical system in a submanifold of lower dimension, providing insight on the true coarse dynamics of the system.

The technique treats each state in a simulation or measured timeseries
(or corpus of such timseries)
as a high-dimensional point. The resulting cloud of points is then amenable
to various dimensionality reduction techniques--we choose to use
*diffusion maps* for its ability to uncover nonlinear embedded manifolds.

This new project uses datasets of a similar form--trajectory matrices. However, instead of using rows of the matrix as high-dimensional points, we use columns or patches of the matrix. This technique is applicable to simulatios of many coupled agents (such as typical neural simulations), or method-of-lines discretized partial differential equations. Rows of the trajectory matrices are indexed by timestep, whlie columns are indexed by agent ID or PDE discretization cell.

In the simpler case where columns are used, this method for processing a trajectory uncovers thie underlying continuum of types of dynamics. For PDEs in space and time, this continuum is often isomorphic to the underlying physical space. For ODEs describing the behavior of a population of agents (such as neurons), we uncover an effective space for the agents, which may include (1) physical space, (2) heterogeneous parameters, (3) initial conditions, or (4) steady states.

This project was conceived in support of my primary thesis topic, in which heterogeneous parameters for a population of agents are used as abscissae for describing system state as a sum of smooth basis functions. (This is analogous to the use of Fourier expansions in physical space for the solution of e.g. underground reservoir simulation.) However, when simulations involve many heterogeneous parameters, or when trajectory matrices represent real-world experimental data, it is not always clear what the underlying heterogneous parameters are. Hence, the equal space technique aims to extract the relevant heterogeneities directly from sample trajectories.