neural firing
DESCRIPTION
Neural Firing. Notation I. x(t)=signal vector; N(t)=#spikes fired up to time t; H(k)=[θ 1:k-1 ,x 1:k ,N 1:k ] t[k],t[k]+ ∆t[k] =likelihood over interval t k , t k +∆t k,i ∆t k,i ~ interval: t k +∑ i=1:j-1 ∆t k,j , t k + ∑ i=1:j ∆t k,j ,. Fact. - PowerPoint PPT PresentationTRANSCRIPT
Neural Firing
Notation I x(t)=signal vector;
N(t)=#spikes fired up to time t; H(k)=[θ1:k-1 ,x1:k ,N1:k]
t[k],t[k]+∆t[k]=likelihood over interval tk, tk+∆tk,i
∆tk,i~ interval: tk+∑i=1:j-1 ∆tk,j, tk+ ∑i=1:j ∆tk,j,
Fact We have that:
0
| ( ), ( ), ( )
( ) ( ) 1| ( ), ( ), ( )lim t
t x t t H t
P N t t N t x t t H tt
Likelihood The likelihood over the k’th
interval is:
, ,
,
1
1, , ,
, , , , , ,
, , ,
, , ,
, ,
| , 1 ;
log log (1 )log 1
log ;
exp
exp
k i k i
k k i
i
k i
k k
k k
N Nt t t k k k k k i k k i
t t t k i k k i k i k i k i
k i k k i k k i
Nt t t k k i k k i
t t k k i
N H t t
N t N t
N t t
t t
t
L
L
L
L, 1 , 1
,
,[ , ] [ , ];
1k i k k k i k k
k i
k k it t t t t t
N
t
Evolution Prior The prior takes the form,
1 1
11: 1 1 1
1 1 1 1
(k=1,....,);
1| exp '2
| | | ,
k k k k
k k k k k k k k k
k k k k k k k k
F
F Q F
H H N d
More Notation and assumptions We put
We assume that And assume α,μ,σ are independent
apriori. Letting Θ be any one of the parameters, α,μ,σ.
, ,; N Nk k i k k ii i
t t
, ,k k k k
Posterior We have that,
1,
, , ,
1 1 1
log( | , ) log log |
log
log | |
k kk k k t t k k
k i k i k i ki i
k k k k k
H N H
t N
H d
L
Result 1 The integral is
This gives an update of
This means that we can take the integral to be:
| 1 1 1 1log | log | |k k k k k k k kH H d
| 1 1| 1 1|; ' ;
: F=1; Qk k k k k k k
k
F F F Q
Assume
1| 1 | 1 | | | 1 |log | 'k k k k k k k k k k k k kH
Result 2 We differentiate the expression in theta
setting the result to 0:
|
, , , ,
1| 1 | | | 1 |
log( | , )
log
1 '2
k k k k
k i k i k i k ii
k k k k k k k k k k
H N
t N
Result 3 We have:
|
|
, 1, , | | 1 |
|
,| | 1 | , ,
|
log( | , )0
log;
log
k k k k
k k
k ik i k i k k k k k k k
k k
k ik k k k k k k i k i k
i k k
H N
N t
N t
Result for the mean parameters In other words for the parameters, this
becomes:
| | 1 , ,|
, || | 1 , , ,| 2
| 1
2, |2 2
| | 1 , , ,| 22| 1
;
;
2
k k k k k k i k ik ki
k i k kk k k k k i k i k ik k
i k k
k i k kk k k k k i k i k ik k
ik k
N t
XN t
XN t
Result for variance parameters
Viewing the whole distribution as a gaussian and taylor expanding
1| | |
1| 1| |
2
| , , , , |,
'
'
log
k k k k k k
k k k k k k
k k k i k i k i k i k kik k
W
W
t N
Variances II This gives
2
,1 1| 1| , , ,2
1|
2,
,1|
log
log
k ik k k k k i k i k i
i k k
k ik i
k k
W W N t
t
For alpha and mu We have, for our parameters,
1 1| 1| ,[ ] [ ] ;k k k k k iW W t
1 1| 1| , , ,2
2,
,2
1[ ] [ ]k k k k k i k i k ii k
k i kk i
k
W W N t
Xt
For sigma-squared We have for sigma,
2,1 1
| 1| , , ,22
22
,,22
[ ] [ ] k i kk k k k k i k i k i
ik
k i kk i
k
XW W N t
Xt
Alternative Take the approach of auxiliary
particle filters. For a given value of we calculate:
( )1it
( ) ( )1 1
1, , 1,
1 '2[̂ ] arg max
log
j jk k
k i i k i k ii i
jt N
Alternative II
Correlated neural firing processes Suppose we have many processes
indexed by 1,…,J: We model the correlation between them by assuming a multivariate gaussian.
We have,
, , ,
2,
, 2
2 2 2
[ ] 1 [ ]exp [ ]
[ ][ ] exp [ ]
2 [ ]
[1],..., [ ] ( , )
[1],..., [ ] ( ; )
[1],..., [ ] ( ; )
k i k i k i
i k kk i i
k
i i
i i
i i
P N j j j
X jj j
j
J
J
J
NN
Correlated neural firing processes: estimation We estimate the correlation
between parameters by estimating the covariance matrices: ∑α, ∑μ, ∑σ
2,
, , 2
2 2 2
[ ]( [ ] 1) [ ] exp [ ]
2 [ ]
[1],..., [ ] ( , )
[1],..., [ ] ( ; )
[1],..., [ ] ( ; )
i k kj k i k i i
k
i i
i i
i i
X jP N t j j
j
J
J
J
NN