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# Neural Firing

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.

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## Neural Firing

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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

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