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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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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:
Likelihood • The likelihood over the k’th interval is:
Evolution Prior • The prior takes the form,
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.
Posterior • We have that,
Result 1 • The integral is • This gives an update of • This means that we can take the integral to be:
Result 2 • We differentiate the expression in theta setting the result to 0:
Result 3 • We have:
Result for the mean parameters • In other words for the parameters, this becomes:
Result for variance parameters • Viewing the whole distribution as a gaussian and taylor expanding
Variances II • This gives
For alpha and mu • We have, for our parameters,
For sigma-squared • We have for sigma,
Alternative • Take the approach of auxiliary particle filters. For a given value of we calculate:
Correlated neural firing processes • Suppose we have many processes indexed by 1,…,J: We model the correlation between them by assuming a multivariate gaussian.
Correlated neural firing processes: estimation • We estimate the correlation between parameters by estimating the covariance matrices: ∑α, ∑μ, ∑σ