1. Neural Firing

2. 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,

3. Fact
We have that:

4. Likelihood
The likelihood over the k’th interval is:

5. Evolution Prior
The prior takes the form,

6. 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.

7. Posterior
We have that,

8. Result 1 The integral is This gives an update of
This means that we can take the integral to be:

9. Result 2
We differentiate the expression in theta setting the result to 0:

10. Result 3
We have:

11. Result for the mean parameters
In other words for the parameters, this becomes:

12. Result for variance parameters
Viewing the whole distribution as a gaussian and taylor expanding

13. Variances II
This gives

14. For alpha and mu
We have, for our parameters,

15. For sigma-squared
We have for sigma,

16. Alternative
Take the approach of auxiliary particle filters. For a given value of we calculate:

17. Alternative II

18. Correlated neural firing processes
Suppose we have many processes indexed by 1,…,J: We model the correlation between them by assuming a multivariate gaussian.

19. We have,

20. Correlated neural firing processes: estimation
We estimate the correlation between parameters by estimating the covariance matrices: ∑α, ∑μ, ∑σ