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.

\[z= \exp(k^Tx)\\ y\sim Poisson(z)\]

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

\[\arg\max_k\mathcal L(k|y,x)\]

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 $\sum_i y_ix_i$

Latent Variable Models

In the next few sections, the models will all have a similar format $x\to z\to y$ or $z\to y$ , 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 $\theta$ jointly. The optimization is performed alternatively, aka coordinate descent or maximization-maximization procedure .

Formal Procedure

Consider model $z\to y$, we have a parametric generative model given any $z$, $p(y\mid z)$ with $\theta$.

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

\[p(z\mid y)=\frac{p_\theta(y\mid z)p_\theta(z)}{p(y)}\]

Maximization : Using this distribution $p(z\mid y)$, 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 $\theta$.

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

\[\mathbb E_{p(z\mid y)} [\log\mathcal L(\theta;y,z)]\]

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 $p(z\mid y)$ 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.