Statistical analysis and modeling
of neural data
Lecture 5
Bijan Pesaran
19 Sept, 2007
Goals
• Recap last lecture – review Poisson
process
• Give some point process examples to
illustrate concepts.
• Characterize measures of association
between observed sequences of events.
Poisson process
n k
nk
lim P( X  k )  lim   p 1  p 
n 
n  k
 
k
nk
n!      
 lim
  1  
n  n  k ! k! n
   n

t  e
P N (t )  k  
k
k!
 t
Renewal process
• Independent intervals
Xi
• Completely specified by interspike interval
density
f x 
• Convolution to get spike counts
Characterization of renewal
process
• Parametric: Model ISI density.
– Choose density function, Gamma distribution:
f x  , k  

k
k 
k 1 x
x e
– Maximize likelihood of data
log L , k    log  f  X i  , k 
n
i 1
No closed form. Use numerical procedure.
Characterization of renewal
process
• Non-parametric: Estimate ISI density
– Select density estimator
– Select smoothing parameter
Non-stationary Poisson process –
Intensity function
PdN t , t     1 past    t 
 t   f zt 
Conditional intensity function
 t H t   lim
PN t     N t   1 H t 

pt   t Ht 
 0
P N1:n     p
n
dNt
t
t 1
n
1  pt 
1 dNt
log LN1:n     log  pt   dN t   pt
t 1
n
t 1
Measures of association
• Conditional probability
• Auto-correlation and cross correlation
• Spectrum and coherency
• Joint peri-stimulus time histogram
Cross intensity function
M t , N t 
PN   PdN t , t     1
PNM   '  PdN t   , t       1  dM t , t   '  1
mNM    PdN t   , t       1 | dM t , t   '  1
PNM  
mNM   
PM
Cross-correlation function
lim PNM    PN PM
 
CovNM    PNM    PN PM  '

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  '
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 
Questions
• Is association result of direct connection or
common input
• Is strength of association dependent on
other inputs