A Determinantal Point Process Latent Variable Model for Inhibition in Neural Spiking Data
Point processes are popular models of neural spiking behavior as they provide a statistical distribution over temporal sequences of spikes and help to reveal the complexities underlying a series of recorded action potentials. However, the most common neural point process models, the Poisson process and the gamma renewal process, do not capture interactions and correlations that are critical to modeling populations of neurons. We develop a novel model based on a determinantal point process over latent embeddings of neurons that effectively captures and helps visualize complex inhibitory and competitive interaction. We show that this model is a natural extension of the popular generalized linear model to sets of interacting neurons. The model is extended to incorporate gain control or divisive normalization, and the modulation of neural spiking based on periodic phenomena. Applied to neural spike recordings from the rat hippocampus, we see that the model captures inhibitory relationships, a dichotomy of classes of neurons, and a periodic modulation by the theta rhythm known to be present in the data.
Summary: -------- The authors apply 
      Determinantal Point Processes (DPPs) to the spiking activity of 
      simultaneously recorded neurons. In addition to stimulus dependence, the 
      resulting model captures pairwise competitive interactions of neurons. 
      The authors apply the model to artificial data and hippocampal 
      recordings. Comments: --------- 1) The application 
      of DPPs to neural recordings is novel to best of my knowledge. The 
      incorporation of instantaneous interactions into GLMs for neurons 
      (even coupled ones) such that the resulting model remains tractable is 
      an important open problem. Hence, I think the paper is timely and of 
      interest to the NIPS neuroscience audience. 2) The paper is 
      clearly written and DPPs are introduced quite gently, resulting in a 
      very readable paper (exceptions below). 3) My main criticism is 
      that the authors did not fully convince me that their model is 
      actually an appropriate one for multi-cell recordings. As far as I 
      know, most noise correlations that have been experimentally measured 
      seem to be positive (at least in cortex) and could therefore not be 
      captured by this DPP approach; please correct me if I'm wrong here. In 
      any case, the authors should have argued more thoroughly and given 
      appropriate citations that the scenario of exclusively competitive 
      interactions is an important one in multi-cell recordings. 4) 
      Section 4.3, application to Hippocampus data: Unfortunately, this 
      paragraph does not fully convince me that the DPP based model is a 
      good model for the data. Are the noise correlations (computed from the 
      data) between pairs of excitatory / pairs of inhibitory neurons / 
      pairs of exc-inhi really mostly negative (emphasize here is on noise 
      correlations as GLMs can capture stimulus induced correlations)? The 
      authors should describe the main result figures FIG4 and FIG3(a) in 
      greater detail: why is this latent embedding sensible / what does this 
      kernel matrix tell us about the data? (The additional space required 
      could be obtained by scaling back the experiments on artificial data.) 
      The fact that the method uncovers the theta oscillations is not very 
      surprising (as this is just the GLM part of the model) and could be 
      described more briefly. Table 1: The authors could have made a 
      stronger point for the model if the table included more pairs of 
      models of the type "GLM_component_1+...+GLM_component_n" and 
      "GLM_component_1+...+GLM_component_n+Latent", as this allows for a 
      direct comparison to figure out if adding the DPP part helps. 
      5) Section 3.3: In the model, the stimulus dependence of each 
      neuron is already captured by the weight vector $w_n$. Isn't the 
      introduction of the of $w_\nu$ redundant? 6) The manuscript is 
      missing a more in depth comparison between coupled GLMs and the DPP 
      approach, eg: coupled GLMs are lacking an instantaneous coupling 
      between the neurons, but this could be compensated for by binning the 
      data on finer time scales etc. Minor Comments: 
      --------------- 1) Either I don't fully understand the 
      notation around eqn. 4 and 5 or it is is rather sloppy: I guess 
      $\Pi^{(t)}$ should be defined as 
      $diag(\sqrt{\lambda_1^{(t)}},\ldots)$, instead of the definition given 
      in l157. Furthermore in eqn 5, $K^{(t)}_{S_t}$ probably only contains 
      the term $k_\theta$ and not the GLM part $\lambda_n^{(t)}$, in 
      contrast to eqn 4. If this is correct, then $K^{(t)}_{S_t}$ should be 
      $K_{S_t}$ as the only time dependence would be via ${S_t}$. 
      Furthermore, shouldn't the normalizer in eqn 5 be independent of 
      $S_t$, i.e. contain eg $K_{\mathcal S}$ instead of $K_{S_t}$ (similar 
      for $\Pi_{S_t}$)? 2) In the definition of $K_S$ in l131, it might 
      be worthwhile to again emphasize that $n,n' \in 
      S$.