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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



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