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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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## Neural Firing

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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

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