History Dependent GLM Example
History Dependent GLM Example
An Analysis of the Spiking Activity of Retinal Neurons in Culture (Iygengar and Liu, 1997)
Retinal neurons are grown in culture under constant light and environmental conditions. The spontaneous spiking activity of these neurons is recorded. The objective is to develop a statistical model which accurately describes the stochastic structure of the waiting times, or interspike intervals (ISIs), for this data.
Attempt #1: Poisson Model
Fit a homogeneous Poisson model to the data:
Estimate rate parameter l by maximum likelihood.
KS Plot
Waiting Time (ms)
CDF
Model CDF
Empirical CDF
Graphical comparison of empirical vs model CDFs
Attempt #2: Renewal Models
Fit a variety of renewal models to the data:
Candidates:
Gamma
Inverse Gaussian
Estimate parameters by Maximum Likelihood or Method of Moments
KS Plots
Empirical CDF
Model CDF
Attempt #3: GLM History Model
How do we pick a model order?
The ISI distribution models we constructed previously assume that
Now, let the conditional intensity be a function of past spiking activity using GLM
Reflection Coefficients
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Model Order
Reflection Coefficient
Model AIC
GLM Order
AIC
Minimum AIC
Parsimonious
Model
Maximum Likelihood Model Fit
Coefficient Index
Coefficient Value
Maximum Likelihood Model Fit
Coefficient Index
Exp Coefficient Value
Goodness-of-Fit
Time-rescaling theorem:
Let where be the event times of a point process with conditional intensity .
Then will be i.i.d. exponential random variables.
Problem:
Solution:
Distribution of arbitrary statistics of spike times, , are difficult to compute.
Model CDF
Empirical CDF
Kolmogorov-Smirnov Plots
KS Plots for Different Order GLMs
Model CDF
Empirical CDF
ISI Lag Order
Correlation
Correlation Function for Rescaled ISIs
Original ISIs
Rescaled ISIs
Goodness-of-Fit Summary
Order
AIC
KS
KS Statistic
Renewal Models:
GLM
1 14 50
6589 5931 5892
0.2330 0.0657 0.0462
Exp Gamma Inv. Gauss.
0.2525 0.2171 0.1063
ISI Histogram
ISI (msec)
Probability Density
Exponential
Gamma
Inverse Gaussian
Order 50 GLM
Analysis Summary
Low order GLMs effectively capture history dependent structure in this data.
Model order can be selected by AIC.
Goodness-of-fit can be evaluated by time-rescaling, comparison of empirical to model CDFs, and correlation analyses.
Monkeys were trained to saccade to one of four targets, based on displayed images.
Single cell recording in monkey hippocampus.
Case 2: Peristimulus Time GLM
94.unknown
Time (sec)
Trial
Peristimulus Time Histogram
Spiking Data
Model
Parameter vector:
Basis functions:
Indicator Functions:
Splines:
*
Here is the model we proposed and study:
It assumes that the firing activity (in kth trial) follows a Poisson process with this intensity: g are basis functions and thetas are parameters that modulate the firing intensity.
Indicator Function Basis
Spline Function Basis
Uniform CDF
Empirical CDF
Lag
ACF of rescaled Times
KS Plot
Goodness-of-Fit
Adding History
Adding History
Uniform CDF
Empirical CDF
KS Plot
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