Motivation

Here we summarize a few common probabilistic neural population models. Adapted from reading notes and class presentations from Neuro QC316 taught by Jan Drugowitsch.

LNP, GLM

These are the simplist models of neurons.

z=exp(kTx)yPoisson(z)

To fit these model, we use maximum likelihood (MLE), or maximum a posteriori (MAP) given a prior of k.

argmaxkL(k|y,x)

Remarks

  • This Generalized linear model has convex likelihood function with unique solutions.
  • If the x is distributed isotropically, then this MLE estimate of k will align with spike triggered average iyixi

Latent Variable Models

In the next few sections, the models will all have a similar format xzy or zy , namely there will be some latent variables that are not input and not output – not observed.

For these models, a common strategy is to use Expectation Maximization. It’s a way to perform MLE over the latent states z and the parameter of the model θ jointly. The optimization is performed alternatively, aka coordinate descent or maximization-maximization procedure .

Formal Procedure

Consider model zy, we have a parametric generative model given any z, p(yz) with θ.

Expectation : Then given an observation y, we can estimate the distribution of the latent assuming a fixed paramter θ

p(zy)=pθ(yz)pθ(z)p(y)

Maximization : Using this distribution p(zy), we can estimate the parameters by MLE, this step is as if we know the input data to y then we can use the normal method to estimate θ.

Formally, this is maximizing the “expected” likelihood function.

Ep(zy)[logL(θ;y,z)]

Remark

  • Exact EM algorithm will increase the marginal likelihood of observed data p(y) so it will converge, but not necessarily to global maxima. Thus multiple restart is needed; and a nice initialization could help!
  • Approximate EM algorithm is not guanranteed to converge! If you use gaussian approximation or other approximations for p(zy) etc.

State Space Model

Latent state: a p dimension linear dynamic system

Observation: a linear readout and Generalized linear model with Poisson

GPFA

Latent state: gaussian processes

  • p independent GPs with different time scales

Observation: a linear readout C with a indevidual neuron based variance R

Thus the whole model is a joint multivariate Gaussian.

To learn these models, we use Expectation Maximization .

LFADS

Latent state: Recurrent neural network

Observation: Linear readout

To learn these models, we use gradient descent training on datasets.