Recent technical developments have dramatically increased our ability to monitor the neural activity of awake, behaving subjects. Here we focus on multielectrode arrays that provide the means for simultaneously measuring the individual spiking activity of about one hundred neurons, as well as the local field potentials associated with neural activity. The resulting data thus tracks the dynamics of a high dimensional system involved in the neural information processing required to perform the task that the subject is involved in. The challenge to theorists is to provide conceptual frameworks and develop mathematical and numerical tools for the analysis of such data.
As is the case for many other high dimensional dynamical systems, dimensionality reduction techniques provide a powerful tool for arriving at compact representations that reveal informative features and facilitate subsequent analysis of system dynamics. We will review standard techniques for linear dimensionality reduction within the framework of latent variable models, and discuss the extension to nonlinear dimensionality reduction techniques. The analysis of the spiking activity of networks of neurons reveals the need to apply nonlinear techniques, as the existence of curved low dimensional manifolds within which the dynamics evolve emerges as a consequence of network connectivity. In contrast, the decoding problem can be analyzed within the linear paradigm. The application of singular value decomposition techniques to this scenario allows us to identify task relevant subspaces and null subspaces, and to establish that variance is allocated in a manner consistent with optimal control theory.