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 form followed by a point nonlinearity and exponential-family noise. The quadratic form 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 (Marmarelis and Marmarelis 1978, Koh and Powers 1985) and the elliptical Linear-Nonlinear-Poisson model (Park and Pillow 2011). Here we show that for canonical form" GQMs, spectral decomposition of the first two response-weighted moments yields approximate maximum-likelihood 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. 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.

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.

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 form followed by a point nonlinearity and exponential-family noise. The quadratic form 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 (Marmarelis and Marmarelis 1978, Koh and Powers 1985) and the elliptical Linear-Nonlinear-Poisson model (Park and Pillow 2011). Here we show that for canonical form" GQMs, spectral decomposition of the first two response-weighted moments yields approximate maximum-likelihood 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."

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 form followed by a point nonlinearity and exponential-family noise. The quadratic form 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 (Marmarelis and Marmarelis 1978, Koh and Powers 1985) and the elliptical Linear-Nonlinear-Poisson model (Park and Pillow 2011). Here we show that for canonical form" GQMs, spectral decomposition of the first two response-weighted moments yields approximate maximum-likelihood 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."

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 form followed by a point nonlinearity and exponential-family noise. The quadratic form 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 (Marmarelis and Marmarelis 1978, Koh and Powers 1985) and the elliptical Linear-Nonlinear-Poisson model (Park and Pillow 2011). Here we show that for canonical form" GQMs, spectral decomposition of the first two response-weighted moments yields approximate maximum-likelihood 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."