Robust learning of low-dimensional dynamics from large neural ensembles
Recordings from large populations of neurons make it possible to search for hypothesized low-dimensional dynamics. Finding these dynamics requires models that take into account biophysical constraints and can be fit efficiently and robustly. Here, we present an approach to dimensionality reduction for neural data that is convex, does not make strong assumptions about dynamics, does not require averaging over many trials and is extensible to more complex statistical models that combine local and global influences. The results can be combined with spectral methods to learn dynamical systems models. The basic method extends PCA to the exponential family using nuclear norm minimization. We evaluate the effectiveness of this method using an exact decomposition of the Bregman divergence that is analogous to variance explained for PCA. We show on model data that the parameters of latent linear dynamical systems can be recovered, and that even if the dynamics are not stationary we can still recover the true latent subspace. We also demonstrate an extension of nuclear norm minimization that can separate sparse local connections from global latent dynamics. Finally, we demonstrate improved prediction on real neural data from monkey motor cortex compared to fitting linear dynamical models without nuclear norm smoothing.
The authors present a method for reducing the 
      dimensionality of neural activity data. They take advantage of recent 
      advances in applied mathematics to solve sparse and low-rank matrix 
      approximation problems using convex relaxation, as well as of Bregman 
      divergences for exponential families. They apply the method to synthetic 
      data. I find the work mathematically sound and the presentation 
      generally clear. However, the authors may want to devote more attention to 
      introducing the problem. Indeed, in the beginning of Section 2, they 
      postulate the existence of a low-dimensional subspace of neuronal 
      activity. It would seem to me that the dynamics in this low-dimensional 
      subspace, Eq.(1), should be mediated by neurons and their synaptic 
      connections. However, Eq.(1) does not include the synaptic connectivity 
      matrix D, which instead appears in Eq.2. Because the authors point out 
      that the low-dimensional dynamics would be present even in the absence of 
      inputs (u_t = 0) I don’t understand what physical substrate underlies the 
      dynamics of low-dimensional activity. Additionally, while I admire 
      the authors for their effort to introduce fashionable applied mathematics 
      techniques into neuroscience, it is difficult to evaluate the value of 
      their method until it is applied to actual data and generates insight into 
      the function of a biological system. Minor on line 291: 
      minimizing D -> minimizing over D