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