Point Process Spectrum Examples
Point Process Spectral Analysis
Examples
Example #1: Simple Poisson Process
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0
1
2
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4
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8
9
10
Time (msec)
Poisson ISI Histogram
0 50 100 150
0
100
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300
400
500
600
700
ISI (msec)
Sample Spectrum from Poisson Data
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Frequency (kHz)
50
0
40
30
20
10
Po
w
er
(H
z)
Example #1: Simple Poisson Process
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0
1
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9
10
Time (msec)
Example #2: Hard Refractory Period
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0
1
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9
10
Time (msec)
Poisson ISI Histogram
0 50 100 150
0
100
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400
500
600
700
ISI (msec)
Refractory Process ISI Histogram
0 50 100 150
0
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700
ISI (msec)
Sample Spectrum from Poisson Data
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Frequency (kHz)
50
0
40
30
20
10
Po
w
er
(H
z)
Refractory Process Sample Spectrum
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Frequency (kHz)
50
0
40
30
20
10
Po
w
er
(H
z)
Example #1: Simple Poisson Process
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0
1
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9
10
Time (msec)
Example #3: Duplicate each spike at
10 ms lag
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0
1
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9
10
Time (msec)
10 ms Repetition ISI Histogram
0 50 100 150
0
100
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800
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1000
ISI (msec)
Repeated Spike Sample Spectrum
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0
0.1
0.2
0.3
0.4
0.5
Frequency (kHz)
50
0
40
30
20
10
Po
w
er
(H
z)
Example #4: Realistic Baseline
Spiking Data
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0
1
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3
4
5
6
7
8
9
10
Time (msec)
Baseline Spiking ISI
0 50 100 150
0
100
200
300
400
500
600
700
ISI (msec)
Sample Spectrum for Baseline Spiking
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Frequency (kHz)
50
0
40
30
20
10
Po
w
er
(H
z)
Case 1: 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 this
activity.
0 5 10 15 20 25 30 35 40 45 50
0
20
40
60
80
100
120
140
160
Retinal Baseline Data Spectrum
Frequency (Hz)
Po
w
er
(H
z)
5 10 15 20 25
0
5
10
15
20
25
30
35
40
45
50
Retinal Baseline Data Spectrogram
Time (sec)
Fr
eq
ue
nc
y
(H
z)
Autocorrelation of interspike intervals
-10 -8 -6 -4 -2 0 2 4 6 8 10
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Lag
C
or
re
la
tio
n
co
ef
fi
ci
en
t
Interval Spectrum
Inverse Lag
Po
w
er
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
100
200
300
400
500
600
700
800
10 20 30 40 50 60 70 80 90
0
0.02
0.04
0.06
0.08
0.1
0.12
ISI Histogram
ISI (msec)
Pr
ob
ab
ili
ty
D
en
si
ty
Exponential
Gamma
Inverse Gaussian
Order 50 GLM
Hippocampal Place Field
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Hippocampal Place Field
0 100 200 300 400 500 600 700 800 900 1000
0
5
10
15
20
25
30
35
40
Place Field Spectrogram
Time (msec)
1000 2000 3000 4000 5000 6000 7000 8000 9000
5
10
15
20
25
30
35
40
45
Fr
eq
ue
nc
y
(H
z)
Place Field Spectrogram
Time (msec)
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
5
10
15
20
25
30
35
40
45
50
Fr
eq
ue
nc
y
(H
z)
Place Field Analysis from Problem Set #3
1
( | ) ( ( ), ( ))exp
=
⎧ ⎫
= ⋅⎨ ⎬
⎩ ⎭
∑
p
S
t i
i
t H t tλ λ βxN yN spikes_hist(:,i)
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Empirical CDF
M
od
el
C
D
F
Conclusions
• Point process spectral analysis is useful for
visualizing point process data and
suggesting classes of conditional intensity
models.
• The interpretation of sample point process
spectra differs from that of continuous
valued signals.