Table 1.

Summary of Notation.

c	index of spike trains c = 1, 2, …, C
m	index of simulated Markov chains m = 1, 2, …, M
t	continuous-time index t ∈ [0, T]
ti	spike timing of the ith spike in continuous time
Δ	smallest time bin size
k	discrete-time index k = 1, 2, …, K, KΔ = T
yk	number of counts observed from discrete-time Markov chain, yk ∈ {0, ℕ}
Sk	discrete-time first-order Markov state, Sk ∈ {1, …, L}
S0	initial Markov state at time 0
S0:T, S1:k	history of the Markov state from time 0 to T (or 1 to k)
n	number of state jumps within the latent process S0:T
l	index of state jumps l = 1, 2, …, n
{S(t); 0 ≤ t ≤ T}	realization of hidden Markov process
= (n, τ, χ)	triplet that contains all information of continuous-time Markov chain {S(t)}
τ = (τ0, …, τn)	(n + 1)-length vector of the sojourn times of {S(t)}
χ = (χ0, …, χn)	(n + 1)-length vector of visited states in the sojourn times of {S(t)}
(0)	initial state of MCMC sampler
νl	ν0 = 0, $v_l={\sum }_{r=0}^{l-1}{\tau }_r(l=1,2,\dots ,n+1)$
0:T, 1:K	history of point-process observations from time 0 to T (or 1 to k)
N(t), Nk	counting process in continuous and discrete time, N(t), Nk ∈ {0, ℕ}
dN(t), dNk	indicator of point-process observations, 0 or 1
Pij	transition probability from state i to j for a discrete-time Markov chain, Σj Pij = 1
qij	transition rate from state i to j for a continuous-time Markov chain, Σj qij = 0
ri = qii	total transition rate of state i for a continuous-time Markov chain, ri = Σj≠i qij
πi	initial prior probability Pr(S0 = i)
ak(i)	forward message of state i at time k
bk(i)	backward message of state i at time k
γk(i)	marginal conditional probability Pr(Sk = i ∣ 0:T)
ξk(i, j)	joint conditional probability Pr(Sk−1 = i, Sk = j ∣ 0:T)
	log likelihood of the complete data
R( → ′)	proposal transition density from state to ′
= 123	prior ratio × likelihood ratio × proposal probability ratio
	acceptance probability, = min(1, )
	Jacobian
λk	conditional intensity function of the point process at time k
θ	parameter vector that contains all unknown parameters
p(x)	probability density function
F(x)	cumulative distribution function, $F(x)={\int }_{-\infty }^{x}p(z)dz$
Φ(x)	gaussian cumulative distribution function
erf(x)	error function
(·)	indicator function
(a, b)	uniform distribution within the region (a, b)