Statistical analysis and modeling
of neural data
Lecture 6
Bijan Pesaran
24 Sept, 2007
Goals
• Recap last lecture – review time domain
point process measures of association
• Spectral analysis for point processes
• Examples for illustration
Questions
• Is association result of direct connection or
common input?
• Is strength of association dependent on
other inputs?
Measures of association
• Conditional probability
• Auto-correlation and cross correlation
• Spectrum and coherency
• Joint peri-stimulus time histogram
Cross-correlation function
PNM   '  PdN t   , t       1  dM t , t   '  1
lim PNM    PN PM
 
CovNM    PNM    PN PM  '

PNM    PN PM  '
CorrNM   
PN PM  '
Cross-correlation function
EdN t   nPN
V dN t   nPN 1  PN 

PNM    PN PM  '
CorrNM   
PN PM  '
Limitations of correlation
• It is dimensional so its value depends on
the units of measurement, number of
events, binning.
• It is not bounded, so no value indicates
perfect linear relationship.
• Statistical analysis assumes independent
bins
Scaled correlation
ScaledCorrNM   
CovNM  
 N M
• This has no formal statistical
interpretation!
Corrections to simple correlation
• Covariations from response dynamics
• Covariations from response latency
• Covariations from response amplitude
Response dynamics
• Shuffle corrected or shift predictor
CovNM    PNM    PN t PM t  '
Non-stationarity
• Assume moments of the distribution
constant over time.
• Simplest solution is to assume stationarity
is local in time
• Moving window analysis
Joint PSTH
J NM t1 , t2   PNM t1 , t2   PN t1 PM t2  '
J
(N )
NM
J NM t1 , t 2 
t1 , t2  
 N t1  M t 2 
Spectral analysis for point
processes
• Regression for temporal sequences
• Naturally leads to measures of correlation
• Statistical properties of estimators wellbehaved
Cross-spectral density

1
 2if


S NM  f  
Cov

e
d
NM

2  

PN
1
 2if
S NN  f  
CovNN  e
d 

2  
2
Spectral representation
for point processes

dN (t ) 
2ift
e
 dZ  f 
f  
T
 2if j
 2ift
ˆ
dZ ( f )   e
dN t    e
t 1
j
Spectral quantities
2


ˆ
SdN ( f )  E dZ N ( f )




~
SdNdM ( f )  E dZ M  f dZ N  f 
CdNdM ( f ) 



~
E dZ M  f dZ N  f 

E dZ M ( f ) E dZ N  f 
2
2

Spectral examples
• Refractoriness – Underdispersion
– Fourier transform of Gaussian variable
• Bursting – Overdispersion
– Cosine function
2if j
2ift
2if ( t  )
ˆ
dZ ( f )   e
 e
e
j
t
Coherence as linear association
 at dN t      bs dM s 
 a      b 
j
j
k
k
2


min E   a  j      b k  
b (t )
 j

k
 a     A( f )e
2if j
j
j
j
f
f
 b    B f dZ  f 
Substitute into loss:
k
M
j
f
S dNdM  f 
B( f ) 
A( f )
S dM  f 
Minimize wrt B(f):
Minimum
value is:
  A f dZ N  f 
 A f 
2
1 C
dNdM
f 
2
S
dN
f
Where:
S dNdM  f 
CdNdM ( f ) 
S dN  f S dM  f 
 f df
Time lags in coherency
dN t   dM t   
 j   k 

 i 2f
 2if k 
2rf  k   
  e
  e S dN  f 
S dNdM ( f )  E   e





CdNdM ( f )  e
2if