Efficient calculation of heterogeneous non-equilibrium statistics in coupled firing-rate models

Table 1 For convenience, we abbreviate the following quantities. When $j=k$ in the double integrals of $\mathcal{M_{F}}$, the bivariate normal distribution $\varrho_{j,k}$ is replaced with the standard normal distribution $\varrho_{1}$. Note that order of the arguments matters in $\mathcal{M_{F}}$: $\mathcal{M_{F}}(j,k)\neq\mathcal{M_{F}}(k,j)$ in general. The quantities in bottom three rows depend on the statistics of the activity $\mu(\cdot)$, $\sigma( \cdot)$

Abbreviation	Definition
$\varrho_{1}(y)$	$\frac{1}{\sqrt{2\pi}} e^{-y^{2}/2}$
$\varrho_{j,k}(y_{1},y_{2})$	$\frac{1}{2 π \sqrt{1 - c_{j k}^{2}}} exp (- \frac{1}{2} {\vec{y}}^{T} {(\begin{array}{c} 1 & c_{j k} \\ c_{j k} & 1 \end{array})}^{- 1} \vec{y})$
$D_{j,k}$	$c_{jk}\frac{\tilde{\sigma}_{j} \tilde{\sigma }_{k}}{\tau_{j}\tau_{k}}$
$\mathcal{E}_{1}(k)$	$\int F_{k}(\sigma_{k}(t) y+\mu _{k}(t))\varrho_{1}(y)\,dy$
$\mathcal{E}_{2}(k)$	$\int F_{k}^{2}(\sigma_{k}(t) y+\mu _{k}(t))\varrho_{1}(y)\,dy$
$\mathcal{M}_{F}(j,k)$	$\iint F_{k}(\sigma_{k}(t) y_{1}+\mu_{k} (t)) y_{2} \varrho_{j,k}(y_{1},y_{2})\,dy_{1}\,dy_{2}$

Abbreviation	Definition
\(\varrho_{1}(y)\)	\(\frac{1}{\sqrt{2\pi}} e^{-y^{2}/2}\)
\(\varrho_{j,k}(y_{1},y_{2})\)	$\frac{1}{2 π \sqrt{1 - c_{j k}^{2}}} exp (- \frac{1}{2} {\vec{y}}^{T} {(\begin{array}{c} 1 & c_{j k} \\ c_{j k} & 1 \end{array})}^{- 1} \vec{y})$
\(D_{j,k}\)	\(c_{jk}\frac{\tilde{\sigma}_{j} \tilde{\sigma }_{k}}{\tau_{j}\tau_{k}}\)
\(\mathcal{E}_{1}(k)\)	\(\int F_{k}(\sigma_{k}(t) y+\mu _{k}(t))\varrho_{1}(y)\,dy\)
\(\mathcal{E}_{2}(k)\)	\(\int F_{k}^{2}(\sigma_{k}(t) y+\mu _{k}(t))\varrho_{1}(y)\,dy\)
\(\mathcal{M}_{F}(j,k)\)	\(\iint F_{k}(\sigma_{k}(t) y_{1}+\mu_{k} (t)) y_{2} \varrho_{j,k}(y_{1},y_{2})\,dy_{1}\,dy_{2}\)