a recipe for constructing a peri-stimulus time histogram hideaki shimazaki dept. of physics kyoto...
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A Recipe for Constructing A Recipe for Constructing a Peri-stimulus Time a Peri-stimulus Time
HistogramHistogram
Hideaki ShimazakiDept. of Physics Kyoto University, Japan
March 1, 2007 Johns Hopkins UniversityMarch 2, 2007 Columbia University
Peri-stimulus Time Peri-stimulus Time HistogramHistogram
Rep
eate
d t
rials
a PST Histograma PST Histogram
Events (Spikes)
From Spikes: Exploring the Neural Code, Rieke et al. 1997
George L. Gerstein and Nelson Y.-S. Kiang (1960 )An Approach to the Quantitative Analysis of Electrophysiological Data from Single Neurons, Biophys J. 1(1): 15–28.
Adrian, E. (1928). The basis of sensation: The action of the sense organs.
Prof. Gersteinme
Bin widthBin width@Kyoto Univ.
1. Bin Width Selection1. Bin Width SelectionUnknown
2ˆMISE t t dt Histogram
Underlying Rate
n 2
2k - vC (Δ) =
Δ
EstimateDatDataa
Method for Selecting the Bin Method for Selecting the Bin SizeSize
N
ii=1
1k k ,
N
N
2i
i=1
1v (k - k)
N
n 2
2k - vC (Δ) = ,
(nΔ)
• Compute the cost function
while changing the bin size Δ.
• Divide the data range into N bins of width Δ. Count the number of events ki in the i th bin.
4 2 1 4 3 1
Mean
Variance
• Find Δ* that minimize the cost function.
ik
n 2
2k - vC (Δ) =
(nΔ)n = 4
N = 6
Time-Varying Rate
Spike Sequences
Time Histogram
The spike count in the bin obeys the Poisson distribution*:
A histogram bar-height is an estimator of :
The mean underlying rate in an interval [0, ]:
0
1t dt
( )
k
nnp k n e
k
n̂
k
n
*When the spikes are obtained by repeating an independent trial, the accumulated data obeys the Poisson point process due to a general limit theorem.
Theory for the MethodTheory for the Method
22
0
1ˆMISE ( ) .n tE dt
2 2
0 0
1 1ˆ ˆMISE ( ) ( )T
t t n tE dt dtE
T
Expectation by the Poisson statistics, given the rate.
2 22
0 0
1 1t tdt dt
Variance of the rate Variance of
an ideal histogram
Decomposition of the Systematic Error
Systematic ErrorSampling Error
Average over segmented bins.
Independent of
Theory: Selection of the Bin SizeTheory: Selection of the Bin Size
Systematic Error
2
0
22
1( ) MISE
ˆ( )
T
n t
n
C dtT
E
22ˆ ˆ ˆ2n n n nC E E E
22 ˆ ˆ ˆn n n nC E E E
n
Variance of a Histogram
2 22ˆ ˆ ˆ( ) .n n nE E E
2 1ˆ ˆ( )n nE En
Introduction of the cost function:
The variance decomposition:
Mean of a Histogram
The Poisson statistics obeys:
Unknown: Variance of an ideal histogramSampling error
Variance of a histogram Sampling error
Data : Britten et al. (2004) neural signal archive
Application to an MT neuron Application to an MT neuron datadata
Too few to make
a Histogram !
Optimized Histogram
Optimal bin size decreases
1
2
2
1 2 1 22 0 0
1
nCn
t t dt dtn
Theoretical cost function:Theoretical cost function:
Unknown
DatDataa
1 2t t Mean Correlation function
n 2
2k - vC (Δ) =
(nΔ)
Theory
Empirical
If you know the underlying rate, you can computeSee also Koyama and Shinomoto
J. Phys. A, 37(29):7255–7265. 2004
1 2 1 22 0 0
1nC t t dt dt
n
2 31 1( ) 0 0 0 ( )
3 12nC On
(i) Expansion of the cost function by :
1 3
6~
0 n
Scaling of the optimal bin size:
Theoretical cost function:
Number of sequence s, m
100
101
102
103
10-2
10-1
Scaling of the optimal bin size
-1/3
The expansion of the cost function by
When the number of sequences is large, the optimal bin size is very small
Ref. Scott (1979)
(ii) Divergence of the optimal bin size
2
2
1 1~ | |
1 1 1 1
n
c
C t dt t t dtn
un n
cn t dt
The expansion of the cost function by 1/
Critical number of trials:
cn n
cn n
Optimal bin size diverges.
Finite optimal bin size.
The second order phase transition.
Not all the process undergoes the first order phase transition. Others undergo the second order (discontinuous) phase transition.
When the number of sequences is small, the optimal bin size is very large.
n: small
n: large
nc: critical number of trials
Phase transitions of optimal Phase transitions of optimal bin sizebin size
*1
cn
1 5n 10n
15n
20n
25n
1 n
n: small
n: large
Data : Britten et al. (2004) neural signal archive
2. The extrapolation method2. The extrapolation method
Too few to make
a Histogram !
2
1 1( )m n
kC n C
m n n
Extrapolation
Estimation: At least 12 trials are
required.
Optimized Histogram
Verification of the extrapolation Verification of the extrapolation methodmethod
Extrapolated:
Finite optimal bin size
Original: Optimal bin size diverges
2
1 1( )m n
kC n C
m n n
( )nC
Required # of sequences (Estimation)
Optimal bin size v.s. m
Data Size UsedData Size Used # of sequences used
Required # of trials
cn
t dt
30
We have n=30 Sequneces
Data : Britten et al. (2004) neural signal archive
2. The extrapolation method2. The extrapolation method
Too few to make
a Histogram !
2
1 1( )m n
kC n C
m n n
Extrapolation
Estimation: At least 12 trials are
required.
Optimized Histogram
Line-Graph Time Histogram
( )
k
nnp k n e
k
01.t dt
2tL t
ˆ ˆ ˆ ˆˆ
2t tL
0
1t dt
A line-graph is constructed by connecting top-centers of adjacent bar-graphs.
An estimator of a line-graph
The spike count obeys the Poisson distribution
Rate
Spike Sequences
Time Histogram
^
^
t
Line-Graph Model
( )( ) ( ) ( 0) ( )
2
2 2 1( ) 2 2
3 ( ) 3 3nkCn
( ) ( )ik j( ) ( )ik j (0) ( )ik j ( ) ( )ik j
( ) ( )
1
( )n
p pi i
j
k k j
(i) Define the four spike counts,
{ 0 }p
( ) ( )
1
1 Np p
ii
kkN
( ) ( )( ) ( ) ( )
1
1 Np qp q p q
i ii
s k kk kN
( ) ( )( ) ( )( )
1 1
1 1( ) ( )
p qN np qp q i i
i ii j
k kk j k js
N n n n
( ) ( )( )
2 2( )
p q p qp q s s
n n
(ii) Summation of the spike count
Binned-average
Covariations w.r.t. bins
Bin-average of the covariation of spike count w.r.t. sequences,
(iv) Cost function:
(iii)
(v) Repatn i through iv by changing Δ. Find the optimal Δ that minimizes the cost function.
0 1 1 2 0 1
0 2 2 0 0 1
( ) ( )ik j
( ) ( )ik j
(0) ( )ik j
( ) ( ) 2 ( )i ik j t j
i
j
j
j
The covariances of an ideal line-graph model is
A Recipe for an optimal line-graph TH
The optimal Line-Graph The optimal Line-Graph HistogramHistogram
0.05 0.1 0.15 0.2 0.25 0.3
-100
-75
-50
-25
25
50
75
100
Line-Graph
Bar-Graph
The Line-graph histogram performs better if the rate is smooth.
Theoretical Cost Function
2
1 2 1 22 0 0
1
nCn
t t dt dtn
2
21 2 1 22 0 2
22 21
3n
tC t t dt dt
n
0
1 2 1 2 1 2 1 22 20 0 0
2 1
3 3t t dt dt t t dt dt
(Bar-Graph)
(Line-Graph)
2. Theories on the optimal 2. Theories on the optimal bin sizebin size
Generalization of Koyama and Shinomoto J. Phys. A, 37(29):7255–7265. 2004
2 31 1( ) 0 0 0 ( )
3 12nC On
1 3
6~
0 n
1 23
~(0 )n
3 4 52 37 181 49( ) 0 0 0 0 ( )
3 144 5760 2880nC On
1 5
1280~
181 0 n
1 296
~37 (0 )n
-0.3 -0.2 -0.1 0.1 0.2 0.3
20
40
60
80
100
-0.3 -0.2 -0.1 0.1 0.2 0.3
20
40
60
80
100
the second order term vanishes.
The expansion of the cost function by :
A smooth process: A correlation function is smooth at origin.
A jagged process: A correlation function has a cusp at origin.
(0 ) 0 (Bar-Graph)
(Line-Graph)
(Bar-Graph)
(Line-Graph)
(Bar-Graph)
(Line-Graph)
(0 ) 0
(i) Scalings of the optimal bin size
Identification of the scaling Identification of the scaling exponentsexponents
1 2~ n 1/ 2~ n
smooth
zig-zag
1 5~ n
smooth
zig-zag
1 3~ n smooth
zig-zag
Data Size UsedData Size Used Data Size UsedData Size Used
Bar-Graph HistogramLine-Graph Histogram
The second and the first order phase The second and the first order phase transitionstransitions
Power spectrum of a rate process
Cost functions
Optimal Bin Size
Second order(Continuous)
First order(Discontinuous)
Summary of Today’s Summary of Today’s TalkTalk
1. Bin width selection1. Bin width selection•The The methodmethod and and theorytheory
•Application to an MT neuronApplication to an MT neuron
2. Theories on the optimal bin width2. Theories on the optimal bin width•ScalingScaling and and Phase transitionsPhase transitions
3. Advice to experimentalists3. Advice to experimentalists•Extrapolation methodExtrapolation method
FAQ
N
2i
i=1
1v (k - k)
N2
2k - vC(Δ) =
Δ
too precise
Q. Can I apply the proposed method to a histogram for a probability distribution? A. Yes.
Q. I want to make a 2-dimensional histogram. Can I use this method? A. Yes.
Q. Can I use unbiased variance for computation of the cost function?A. No.
Q. I obtained a very small bin width, which is likely to be erroneous. Why?A. You probably searched smaller bin size than the sampling resolution.
Round Up
1
N
2i
i=1
1v (k - k)
N
Erroneously small optimal bin size
ReferenceReference
A Method for Selecting the Bin Size of a Time Histogram
Hideaki Shimazaki and Shigeru ShinomotoNeural Computation in Press
Short Summary: Adavances in Neural Information Processing Systems Vol. 19, 2007
• Web Application for the Bin Size Selection
• Matlab / Mathematica / R sample codes
are available at our homepagehttp://www.ton.scphys.kyoto-u.ac.jp/~shino/
/~hideaki/See alsoKoyama, S. and Shinomoto, S. Histogram bin width selection for time-dependent poisson processes. J. Phys. A, 37(29):7255–7265. 2004
AcknowledgementsAcknowledgements
JSPS Research Grant
Prof. Shigeru ShinomotoDr. Shinsuke Koyama
Hideaki Shimazaki
2007 Ph. D Prof. Shigeru Shinomoto
Dept. of Physics Kyoto University, Kyoto, Japan
2003 MA Prof. Ernst NieburDept. of Neuroscience, Johns Hopkins
University Baltimore, Maryland
2006-2008 JSPS Research Fellow
Introducing myself…
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