Spectral methods for neural characterization using generalized quadratic models
We describe a set of fast, tractable methods for characterizing neural responses to high-dimensional sensory stimuli using a model we refer to as the generalized quadratic model (GQM). The GQM consists of a low-rank quadratic function followed by a point nonlinearity and exponential-family noise. The quadratic function characterizes the neuron's stimulus selectivity in terms of a set linear receptive fields followed by a quadratic combination rule, and the invertible nonlinearity maps this output to the desired response range. Special cases of the GQM include the 2nd-order Volterra model [1,2] and the elliptical Linear-Nonlinear-Poisson model [3]. Here we show that for "canonical form" GQMs, spectral decomposition of the first two response-weighted moments yields approximate maximumlikelihood estimators via a quantity called the expected log-likelihood. The resulting theory generalizes moment-based estimators such as the spike-triggered covariance, and, in the Gaussian noise case, provides closed-form estimators under a large class of non-Gaussian stimulus distributions. We show that these estimators are fast and provide highly accurate estimates with far lower computational cost than full maximum likelihood. Moreover, the GQM provides a natural framework for combining multi-dimensional stimulus sensitivity and spike-history dependencies within a single model. We show applications to both analog and spiking data using intracellular recordings of V1 membrane potential and extracellular recordings of retinal spike trains.
This paper makes two contributions. First, this paper 
      continues a trend of recent work by Park and Pillow (2011) and Ramirez 
      & Paninski (2012) to provide a firm grounding of the spike-triggered 
      methods STA and STC within forward, likelihood-based modelling. STA and 
      STC are calculated as moments from the stimulus/response ensemble, and are 
      popular for constructing stimulus/response models due to their speed and 
      simplicity. However, as explored extensively in previous work (e.g. 
      Paninski 2003, Sharpee et al 2004, Samengo & Gollisch 2013), the 
      validity and utility of these moments depends on particular restrictive 
      conditions on the stimulus set. The prior work of Park & Pillow (2011) 
      was able to shed light on this, by relating these moments to the 
      parameters of a Generalised Linear Model (GLM), a popular forward model 
      for neural responses. While GLM parameter estimation typically requires 
      log-likelihood maximisation over a full dataset (a potentially slow 
      operation), if one instead maximises the expected log-likelihood (EL) 
      under the stimulus distribution, the STA/STC moments provide a fast and 
      asymptotically consistent means to estimate the GLM. Here, the authors 
      extend this work, by asking how moments such as STA and STC can assist in 
      the estimation of a more complex forward model -- the Generalised 
      Quadratic Model (GQM) -- via the EL framework. In doing so, the paper 
      makes its second contribution, in that it introduces the GQM as a new, 
      general class of models for characterising neural responses. The GQM 
      extends the popular Generalised Linear Model (GLM) by allowing for 
      quadratic (rather than just linear) relationships between stimulus and 
      response (together with a point nonlinearity and exponential-family 
      noise). For Gaussian likelihoods, the authors derive moment-based 
      estimators for the GQM under the conditions of Gaussian-distributed 
      stimuli, and axis-symmetric stimuli. A similar derivation is provided for 
      Poisson likelihoods under a set of assumptions for the stimulus 
      distribution. Low-rank estimates of the parameters are also shown to be 
      asymptotically consistent. Finally, the authors demonstrate an application 
      of these results to real neural data (intracellular voltage and spiking 
      cells), demonstrating a performance gain of the GQM over the GLM. They 
      also show a synthetic example where the vast speed improvements from using 
      moments, as opposed to maximising the likelihood over a full dataset, are 
      evident. The paper is technically sound, giving thorough 
      derivations of the relationships between the GQM and stimulus/response 
      moments under the set of conditions studied. While this aspect of the work 
      is technical in nature, its significance -- as for that of recent work on 
      this topic raised above -- is in establishing the theoretical groundwork 
      that underlies the workings of a common set of analytical techniques. 
      These results also build incrementally upon these previous papers, 
      although the relationship between the exponentiated-quadratic model 
      presented in Park & Pillow (2011) section 2.2, and that given here in 
      section 3.2, should be made clearer. The GQM may generally be a promising 
      tool for neural characterisation, although this paper does not really 
      provide a thorough enough evaluation of the realistic data requirements to 
      properly assess this. This paper, in its current form, does fall a 
      little short in its clarity. Part of my confusion in reading it stems from 
      the fact that half of the time the paper is selling the GQM, and half of 
      the time the paper is demonstrating the deeper relationships between 
      moment-based estimators and the GQM parameters. In addition, the title 
      seems to emphasise spectral methods (I assume this refers to the 
      dimensionality reduction in Section 4), but this only figures as a small 
      part of the paper. Because of the jumps in focus between the EL approach 
      in section 3, dimensionality reduction in section 4, and the GQM demo in 
      section 5, it took me a number of reads just to get a handle on the 
      narrative of the paper. Some bracketing commentary at the start and the 
      end of each section to ground the reader would really be helpful. Also, 
      section 3.3 should really be a part of section 4. Some minor 
      comments: - line 207: E[x_i . x_j^3] is zero. This should be E[x_i 
      . x_j^2] or E[x_j^3] or something like that in order to be a third moment. 
      - line 209: what happens when the stimulus is not white? Also, an 
      explanation of how the general fourth-order moments at the end of the 
      expression for Lambda (line 204), reduce to the matrix M = E[x_i^2 . 
      x_j^2] would be helpful (I admit I didn't follow this logic). - 
      lines 283-292: Typesetting -- mu_x has the subscript in math italics on 
      line 283, but bold in (17) and on line 291. - Section 4. I'm not 
      sure why the notation here was changed from vector (bold) x to matrix 
      (bold) X, and scalar (italic non-bold) y to matrix (bold) Y. It makes this 
      section needlessly confusing (X and Y are matrices now?) and harder to 
      connect with the work in previous sections. Surely the authors could stick 
      with vector x and scalar y. - Fig 4: r^2 = 0.55 in the figure 
      legend, but r^2 = 0.50 in the panel. - line 122: unfinished 
      sentence