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The input is made of two Poisson shotnoise: one excitatory and one inhibitory that are both convoluted with an exponential waveform to produce the synaptic conductances time courses.
We introduce the quantitity relative to the mean synaptic bombardement.
\begin{equation}
\left\{
\begin{split}
& μGe(ν_e, ν_i) = ν_e \, K_e \, τ_e \, Q_e
& μGi(ν_e, ν_i) = ν_i \, K_i \, τ_i \, Q_i \
& μG(ν_e, ν_i) = μGe + μGi + g_L \
& τ_m(ν_e, ν_i) = \frac{C_m}{μG}\
\end{split}
\right.
\end{equation}
The mean membrane potential is taken as the stationary solution to static conductances given by the mean synaptic bombardement:
\begin{equation} μ_V(ν_e, ν_i) = \frac{μGe \, E_e + μGi \, E_i + g_L \, E_L}{μG} \end{equation}
From shotnoise theory, we have:
\begin{equation}
\begin{split}
P_V(f) = & ∑syn νsyn \, \| \hat{\mathrm{PSP}}(f) \|^2
& = 2 νin \, \frac{Q_I^2 \, τ_S^2 / μ_G^2 }{
\big(1+4 π^2 f^2 τ_S^2 \big) \big(1+4 π^2 f^2 (τ_m^\mathrm{eff})^2 \big)}
\end{split}
\end{equation}
Obtained by computing a PSP event, then shotnoise theory:
Equation for \(δ V(t)\) around μ_V:
\begin{equation}
\left\{
\begin{split}
& τ_m \frac{d δ V}{dt} + δ V = Usyn \, \mathcal{H}(t) \, e\frac{-t{τsyn}}
& Usyn = \frac{Qsyn}{μ_G} (Esyn - μ_V)
\end{split}
\right.
\end{equation}
That has the solution:
\begin{equation} δ V(t) = Usyn \, \frac{τsyn}{τ_m - τsyn} \, \big( e\frac{-t{τ_m}} - e\frac{-t{τsyn}} \big) \, \mathcal{H}(t) \end{equation}
We take the fourier transform:
\begin{equation} \hat{δ V}(f) = Usyn \, \frac{τsyn}{τ_m - τsyn} \, \big( \frac{τm}{2 \, i \, π \, f \, τm +1}
- \frac{τsyn}{2 \, i \, π \, f \, τsyn +1} \big)
\end{equation}
\begin{equation} ∫_\mathbb{R} \| \hat{\mathrm{PSP}}(f) \|^2 \, df = \frac{ \| \hat{\mathrm{PSP}}(0) \|^2}{2 \, (τ_m + τsyn)} \end{equation}
because:
\begin{equation} ∫_\mathbb{R} df \, \| \hat{\mathrm{PSP}}(f) \|^2 = \frac{(Usyn ⋅ τsyn)^2}{2 \, (τ_\mathrm{m}^\mathrm{eff} + τsyn ) } \end{equation}
we get:
\begin{equation} σ_V = \sqrt{ ∑syn Ksyn \, νsyn \, \frac{(Usyn ⋅ τsyn)^2}{2 \, (τ_\mathrm{m}^\mathrm{eff} + τsyn ) } } \end{equation}
and
\begin{equation} τ_V = \Big( \frac{ ∑syn \big( Ksyn \, νsyn \, (Usyn ⋅ τsyn)^2\big) }{ ∑syn \big( Ksyn \, νsyn \, (Usyn ⋅ τsyn)^2 /(τ_\mathrm{m}^\mathrm{eff} + τsyn ) \big) } \Big) \end{equation}
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