PowerPoint Presentation
CA1 Region of Rat Hippocampus
Spike Time Series
Time (sec)
Position Specific Spiking
Inhomogeneous Rate Model:
Gaussian Place Field
⎧ ⎫′⎨ ⎬
⎩ ⎭
-11( ) = exp – ( ( ) – ) ( ( ) – )
2
λ α μ μt x t W x t
{ }e x p α
Model parameters:
maximum field height
⎡ ⎤
⎢ ⎥
⎢ ⎥⎣ ⎦
2
1 12
2
12 2
=W
σ σ
σ σ
scale matrix
1 2= ( ),μ μ μ
center
Model covariate:
position
( ) = ( ( ) ( ))1 2x t x t , x t
Point Process Data Likelihood
Intensity Model:
Observed Spike Data:
Data Likelihood:
Estimate model parameters by maximum
likelihood
1
(Spike Train | ) exp( log( ) )
T
k k k
k
L t N tθ λ λ
=
= Δ Δ − Δ∑
( ) ( )k k kN N t t N tΔ = + Δ −
( ( ) | )k k kx t Hλ λ=
1 2
ˆˆ( , ; , )= =L W Wμ μ α α
1 2
ˆˆ( , ; , )= =L W Wμ μ α α
1 2
ˆˆ( , ; , )= =L W Wμ μ α α
1 2
ˆˆ( , ; , )= =L W Wμ μ α α
Likelihood Slices
1μ 2μ
1 2 2
ˆˆ ˆ( ; , , )= = =L W Wμ α α μ μ 2 1 1 ˆˆ ˆ( ; , , )= = =L W Wμ α α μ μ
Parameter Estimates
{ }ˆ 6 .8 2 0 .4 8 H z= ±e x p α
0.072 0 0.001 0.004
0 0.111 0.004 0.003
⎡ ⎤ ⎡ ⎤
±⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
=W
ˆ 0.12 0.09= − ±1μ
ˆ 0.32 0.11= − ±2μ
⎧ ⎫′⎨ ⎬
⎩ ⎭
-11( ) = exp – ( ( ) – ) ( ( ) – )
2
λ α μ μt x t W x t
Model Fit
Model Fit
Spiking Rate in Time
Time (sec)
#
S
pi
ke
s/
33
m
se
c
bi
n
0
4
2
F
iri
ng
R
at
e
(H
z)
Goodness-of-Fit Questions
• How well does this model describe the
data?
• How does this model compare to others?
• How can we refine this model?
Three Models
{ }( ) = exp – ( )λ α βt x t
⎧ ⎫′⎨ ⎬
⎩ ⎭
-11( ) = exp – ( ( ) – ) ( ( ) – )
2
λ α μ μt x t W x t
⎧ ⎫′⎨ ⎬
⎩ ⎭
-11( ) = exp – ( ( ) – ) ( ( ) – ) + ( )
2
sλ γt x t W x t tα μ μ
Gaussian model with random signal dependence:
Linear exponential model:
Gaussian shaped model:
Model MLEs
Linear Est Gaus Est G+R Est
1.48 ±.12 -6.82±.48 -6.78±.51
-.49±.08 -.12±.09 -.14±.08
1.21±.08 -.32±.11 -.30±.10
.072±.001 .072±.002
0.00±.004 -0.00±.006
.111±.003 .110±.022
.063±.135
eα eα eα
1β
2β
1μ
2μ
1μ
2μ
11w
12w
22w
11w
12w
22w
γ
Model Fits
Linear exponential Gaussian Gaussian + Random signal
16187 12567 12568
AIC:
16181 12555 12554
-2*Log Likelihood at MLE:
Time Rescaling
Linear exponential Gaussian Gaussian + Random
ISI Histogram
KS Plots
Linear exponential Gaussian Gaussian + Random
0.315 0.113 0.095
KS Statistics:
Model QQ Plots
Linear exponential Gaussian Gaussian + Random
Rescaled ACFs
Linear exponential Gaussian Gaussian + Random
Point Process Residuals
Linear exponential Gaussian
Residuals vs x2
Residuals vs y2
Lag
Residuals vs x2
Residuals vs y2
Lag
Sample Fano Factor
Sample Fano factor distribution for spikes binned at 33 ms
Sample Fano factor
Summary
Linear model:
– Poorest explanatory / predictive quality
– Misses both small and large quantiles
– Correlations at distant lags
– Quadratic component missing from
model
Quadratic model:
– Better explanatory / predictive quality
– Misses large quantiles
– Correlations at small lags
Addition of Random Signal:
– Doesn’t alter spatial field properties
– Decreased predictive quality
All models:
– Incomplete specifications (KS test)
– Cannot describe large quantiles
– Data not inhomogeneous Poisson
CA1 Region of Rat Hippocampus
Spike Time Series
Position Specific Spiking
Inhomogeneous Rate Model: Gaussian Place Field
Point Process Data Likelihood
Likelihood Slices
Parameter Estimates
Model Fit
Model Fit
Spiking Rate in Time
Goodness-of-Fit Questions
Three Models
Model MLEs
Model Fits
Time Rescaling
KS Plots
Model QQ Plots
Rescaled ACFs
Point Process Residuals
Sample Fano Factor
Summary