Moment-closure approaches to statistical mechanics and inference in models of neural dynamics

At the upcoming SAND meeting in Pittsburgh, I'll be presenting our recent work on using moment closures to combine theoretical models with statistical inference. This work has already been published, but this poster provides a quick summary. 

In my postdoc at Edinburgh, I worked on methods to combine neural field modelling and statistical inference. Neural field models capture how microscopic actions of single neurons combine to create emergent collective dynamics. Statistical modelling of spiking data commonly uses Poisson point-process models. These projects combined the two in an interesting way. 

In "autoregressive point-processes as latent state-space models" [PDF], we convert a popular statistical model for spike-train data into a neural field model. This neural field model is a bit unusual: it extends over time rather than space, and describes correlations as well as mean firing rates. This may lead to new tricks for inference and coarse-graining on these types of models. 

In "neural field models for latent state inference", we use a microscopic model of retinal waves to specify a second-order neural field model that doubles as a latent state-space model for spiking observations. This advances methods for developing data-driven neural field models.

[download poster PDF]

Autoregressive point-processes as latent state-space models

In 2016 I started a postdoc with the labs of Guido Sanguinetti and Matthias H. Hennig—and this is the first paper to result!

[get PDF]

A central challenge in neuroscience is understanding how the activity of single cells combines to create the collective dynamics that underlie perception, cognition, and behavior. One way to study this is to build detailed models, and "coarse grain" them to see which details are important. 

Our paper develops ways to relate detailed point-process models to coarse-grained quantities, like average neuronal firing rates and correlations. Point-process models are used for statistical modelling of spike train data. They can reveal effective neural dynamics by capturing how neurons in a population inhibit or excite each-other and themselves. 

Preview of figures: 

Figure 2: Moment closure of autoregressive PPGLMs combines aspects of three modeling approaches:

(A) Log-linear autoregressive PPGLM framework (e.g., Weber & Pillow, 2017). Dependence on the history of both extrinsic covariates x(t) and the process itself y(t) are mediated by linear filters, which are combined to predict the instantaneous log intensity of the process. (B) Latent state-space models learn a hidden dynamical system, which can be driven by both extrinsic covariates and spiking outputs. Such models are often fit using expectation- maximization, and the learned dynamics are descriptive. (C) Moment closure recasts autoregressive PPGLMs as state-space models. History dependence of the process is subsumed into the state-space dynamics, but the latent states retain a physical interpretation as moments of the process history (dashed arrow). (D) Compare to neural mass and neural field models, which define dynamics on a state space with a physical interpretation as moments of neural population activity

System-size expansion and Gaussian moment closure for Quiescent-Active-Refractory model

Epilogue: These notes concern the system size expansion and moment closure later published as "Neural field models for latent state inference: Application to large-scale neuronal recordings" [pdf] (more) 

These notes derive the Kramers-Moyal system size expansion and approximate equation for the evolution of means and covariances of a single-compartment neural mass model with Quiescent (Q), Active (A), and Refractory (R) states. The derivations here are identical to the standard ones for the Susceptible (S), Infected (I), and Recovered (R) (SIR) model commonly used in epidemiology. 

[get these notes as PDF]