Autoregressive Point Processes as Latent State-Space Models: A Moment-Closure Approach to Fluctuations and Autocorrelations

Abstract

Modeling and interpreting spike train data is a task of central importance in computational neuroscience, with significant translational implications. Two popular classes of data-driven models for this task are autoregressive point-process generalized linear models (PPGLM) and latent state-space models (SSM) with point-process observations. In this letter, we derive a mathematical connection between these two classes of models. By introducing an auxiliary history process, we represent exactly a PPGLM in terms of a latent, infinite-dimensional dynamical system, which can then be mapped onto an SSM by basis function projections and moment closure. This representation provides a new perspective on widely used methods for modeling spike data and also suggests novel algorithmic approaches to fitting such models. We illustrate our results on a phasic bursting neuron model, showing that our proposed approach provides an accurate and efficient way to capture neural dynamics.

Neural Field Models for Latent State Inference: Application to Large-Scale Neuronal Recordings

“…To develop a state-space formalism for inference and data assimilation in neural eld models of retinal waves, we rst use a master-equation approach (Buice and Cowan, 2007a;Ohira and Cowan, 1993;Bresslo , 2009) to de ne a three-state stochastic neural eld model. We then outline a moment-closure approach (Schnoerr et al, 2017;Rule and Sanguinetti, 2018) to close a series expansion of network interactions in terms of higher moments (Buice et al, 2010), and obtain a second-order neural eld model with eld equations for both mean and covariance. We illustrate that a Langevin approximation (Riedler and Buckwar, 2013) of this model recapitulates spatiotemporal wave phenomena when sampled.…”

“…We assume spontaneous, Poisson transitions between neural states, with a single quadratic pairwise interaction wherein active (A) cells excite nearby quiescent (Q) cells. Such a quadratic interaction can be viewed more generally as a locally-quadratic approximation of pairwise nonlinear excitatory interaction (Rule and Sanguinetti, 2018;Ale et al, 2013). Consider the following four state transitions of neurons:…”

“…The Gaussian moment closure approximation sets all cumulants beyond the variance to zero, yielding an expression for the third moment in terms of the rst and second moments of Eq. 2 and giving closed ordinary di erential equations for the means and covariances (Schnoerr et al, 2017;Rule and Sanguinetti, 2018). The evolution of the rst moment (mean concentrations) is as follows:…”

“…We build on recent advances in stochastic spatiotemporal modeling, in particular a recent result by Schnoerr et al (2016) which showed that a spatiotemporal agent-based model of reaction-di usion type, similar to the ones underpinning many neural eld models, can be statistically approximated by a spatiotemporal point process with a de ned intensity evolution equation. Subsequently, Rule and Sanguinetti (2018) illustrated a moment-closure approach for mapping stochastic models of neuronal spiking onto latent statespace models, preserving the essential coarse-timescale dynamics. Here, we demonstrate that a similar approach can yield state-space models for neural elds derived directly from a mechanistic microscopic description.…”

Neural field models for latent state inference: Application to large-scale neuronal recordings

“…density) field that evolves in time. Subsequently, Rule and Sanguinetti [29] illustrated a moment-closure approach for mapping stochastic models of neuronal spiking onto latent state-space models, preserving the essential coarse-timescale dynamics. Here, we demonstrate that a similar approach can yield state-space models for neural fields derived directly from a mechanistic microscopic description.…”