tzl_spGLM

A package for fitting Poisson generalized linear models (GLM) to the spike trains of a neuron recorded in the Poisson Clicks task

system requirements

This software has been tested on Julia 1.10.0 and MATLAB R2023a.

installation

```julia-repl

using Pkg Pkg.add(url="https://github.com/Brody-Lab/tzl_spGLM.git") ``` The installation should take no more than 5 minutes.

data

Spike times and the times of task events are loaded from a MATLAB MAT file. Spikes are counted in time bins of $\Delta t$ seconds aligned to a reference event on each trial to give spike train $y$. GLM's are fitted to the spike train $y$.

An example data file is located at here.

model

On time step $t$ of trial $m$, the spike train observation $y_{m,t}$, given the inputs to the model, is modelled as a Poisson random variable whose intensity is given by

$$ \lambda{m,t} = \text{softplus}\left( w\cdot bm + \sumi (k^{(i)} * xm^{(i)})(t) \right) $$

where $bm$ is a scalar trial-varying input (described below). Both $bm$ and its scalar encoding parameter $w$ is learned from the data.

The time series $x_m^{(i)}$ indicates the input related to event $i$ in the trial (nose-fixation, stereoclick, movement, or response). The input is an impulse function that gives the value of 1 on the time step when the event occurred and is otherwise zero (i.e., the integral of a delta function over each time bin).

The input is convolved $(\cdot * \cdot)$ with linear filter $k_i$ to capture the time-varying effect of event $i$ on the neuron's probability of spiking. The filter is learned as a linear combination of basis functions.

$$ ki = \Phii w_i $$

where the columns of matrix $\Phi$ correspond to basis functions and rows time steps. Example basis functions are shown below. Each plot corresponds to a set of basis functions, and each color is a individual basis function of that set.

For the set related to "response" (i.e., "spoke"), $\Phi$ has two columns, and the encoding vector $w$ is two dimensional.

trial-varying baseline

The trial-varying baseline $b_m$ is learned through L2-penalized linear regression as a preliminary step before fitting the GLM. The inputs (regressors) are the spike count of the simultaneously recorded population 2 seconds before fixation, giving a design matrix $X$ of dimensions trials-by-neurons. The response (regressand) $y$ are the mean firing rate between fixation onset and entry into the side port (side poke). The L2 regularization penalty $\lambda$ was selected across ten values from $10$ to $1e10$, and selected using a 5-fold cross-validation scheme. If either the smallest ($\lambda=10$) or the largest $(1e10)$ value gives the lowest out-of-sample mean squared error, the model was not fit. After identify the optimal L2 penalty $\hat{\lambda}$, the baseline is estimated using all trials as

$$ b \equiv X \left[ (X^\top X + \hat{\lambda}I)^{-1} X^\top y \right] $$

example

Julia REPL

The optimization procedure learns both the parameters (i.e. regressor weights) and a hyperparamer (precision of the Gaussian prior on the parameters, analogous to a L2 penalty coefficient): ``` julia> using SPGLM julia> csvpath = "/mnt/cup/people/zhihaol/Documents/tzluo/analyses/analysis20240422atest_SPGLM/models.csv"; julia> options = SPGLM.Options(csvpath); julia> trials = SPGLM.loadtrials(options); julia> model = SPGLM.Model(options, trials); julia> eo = SPGLM.maximizeevidence!(model)

Iter Function value Gradient norm 0 3.553781e+04 1.285527e+03 * time: 6.103515625e-5 1 3.340892e+04 1.124030e+03 * time: 0.05215096473693848 2 3.038869e+04 7.491107e+02 * time: 0.10418510437011719 3 2.782119e+04 3.308262e+02 * time: 0.15614914894104004 4 2.628437e+04 1.192378e+02 * time: 0.21809101104736328 5 2.546613e+04 2.909452e+01 * time: 0.25608205795288086 6 2.535228e+04 3.249330e+00 * time: 0.29502296447753906 7 2.535107e+04 4.872618e-02 * time: 0.33403897285461426 8 2.535107e+04 1.218098e-05 * time: 0.38610410690307617 9 2.535107e+04 1.225686e-12 * time: 0.44210100173950195 Evidence optimization iteration: 1: precision (a) = 0.5908446386657102 Evidence optimization iteration: 1: norm of the residual gradient of the log-posterior (gresidual) = 1.2256862191861728e-12 Evidence optimization iteration: 1: MAP optimization converged Evidence optimization iteration: 1: approximate log-evidence (𝐸) = -25382.187862691735 Iter Function value Gradient norm 0 2.493416e+04 1.770758e+01 * time: 6.198883056640625e-5 1 2.491330e+04 1.655549e+01 * time: 0.05198216438293457 2 2.487574e+04 1.496841e+01 * time: 0.1039891242980957 3 2.481107e+04 1.249717e+01 * time: 0.15512514114379883 4 2.471561e+04 8.400673e+00 * time: 0.20711398124694824 5 2.462951e+04 2.547959e+00 * time: 0.2629849910736084 6 2.461760e+04 4.920018e-01 * time: 0.30104517936706543 7 2.461753e+04 2.734311e-03 * time: 0.3390049934387207 8 2.461753e+04 1.063443e-07 * time: 0.369035005569458 9 2.461753e+04 2.953931e-14 * time: 0.4078941345214844 Evidence optimization iteration: 2: precision (a) = 0.006298120303099656 Evidence optimization iteration: 2: norm of the residual gradient of the log-posterior (gresidual) = 2.9539305029802065e-14 Evidence optimization iteration: 2: MAP optimization converged Evidence optimization iteration: 2: approximate log-evidence (𝐸) = -24707.60295639283 Iter Function value Gradient norm 0 2.461416e+04 8.121148e-02 * time: 6.29425048828125e-5 1 2.461414e+04 1.074387e-03 * time: 0.05279088020324707 2 2.461414e+04 1.031625e-08 * time: 0.10295701026916504 3 2.461414e+04 8.031770e-15 * time: 0.1548769474029541 Evidence optimization iteration: 3: precision (a) = 0.0051224104379705695 Evidence optimization iteration: 3: norm of the residual gradient of the log-posterior (gresidual) = 8.031769693772617e-15 Evidence optimization iteration: 3: MAP optimization converged Evidence optimization iteration: 3: approximate log-evidence (𝐸) = -24707.271175121816 SPGLM.EvidenceOptimization{Vector{Float64}, Vector{Vector{Float64}}} a: Array{Float64}((3,)) [0.5908446386657102, 0.006298120303099656, 0.0051224104379705695] 𝐸: Array{Float64}((3,)) [-25382.187862691735, -24707.60295639283, -24707.271175121816] MAPg_residual: Array{Float64}((3,)) [1.2256862191861728e-12, 2.9539305029802065e-14, 8.031769693772617e-15] 𝐰: Array{Vector{Float64}}((3,))

julia> characterization = SPGLM.Characterization(model) julia> SPGLM.save(model) julia> SPGLM.save(eo, model.options.outputpath) julia> SPGLM.save(characterization, model.options.outputpath) ``` This should take no more than a few mintues

MATLAB command window

Let's check whether the model actually fit the data, using utilities in +SPGLM and +TZL ```

analysispath = 'V:\Documents\tzluo\analyses\analysis20240423atest_SPGLM' models = readtable(fullfile(analysispath, "models.csv"), 'delimiter', ','); models.outputpath = TZL.cup2windows(string(models.outputpath)); load(fullfile(models.outputpath{1}, 'model.mat')) load(fullfile(models.outputpath{1}, 'characterization.mat')) peths = SPGLM.tabulatepeths(peths); kernels = SPGLM.tabulatekernels(kernels); ``` Plotting the peri-event time histograms (PETH), aligned to the rat's movement onset, and conditioned on whether it made a left or a right choice:

```

referenceevent = "movement"; conditions = ["leftchoice", "rightchoice"]; SPGLM.stylizeaxes colors = get(gca, 'colororder'); h = []; k = 0; for i = 1:numel(conditions) index = peths.condition == conditions(i) & peths.referenceevent == referenceevent; k =k + 1; h(k) = plot(peths.timestepss{index}, peths.observed{index}, '-', 'color', colors(i,:)); k =k + 1; h(k) = plot(peths.timestepss{index}, peths.predicted{index}, '-', ... 'linewidth', 1, 'color', colors(i,:)); end xlabel(['time from ' char(referenceevent) ' (s)']) ylabel('spikes/s') ylim(ylim.*[0,1]) legend(h, {'left choice, observed', 'left choice, predicted', ... 'right choice, observed', 'right choice, predicted'}, 'location', 'northwest') ```

The model captures the choice-dependent peri-movement neuronal activity by treating the moment of movement onset as a transient impulse and fits convolution filters aligned to the impulse. Separate convolution kernels are fit for different choices.

Using similar code as above to plot the PETH aligned to the onset of auditory click trains, which is always preceded by a left and right click being played simultaneously (i.e., a so-called "stereoclick"). The meaning of the color and the line thickness are as above

```

reference_event = "stereoclick"; ```

Or aligned to the onset of nose fixation

```

referenceevent = "cpokein"; ```

We can also check whether we capture the response aligned to either a left or a right click (also called the "click-triggered average").