For simplicity, take continuous degrees of freedom (dofs) \(x_1,x_2\)
The interaction remains \(X_1\to X_2\) and the derivation is the same for discrete systems

\[ \partial_t p_{12}=\left[\frac{\mathcal L_1}{\tau_1}+\frac{\mathcal L_2(x_1)}{\tau_2}\right]p_{12} \]

\[ \mathcal L_i p=-\partial_{x_i}\!\left[b_i p\right]+\partial_{x_i}^{,2}\!\left[D_i p\right] \]

\(b_i\): drift   \(D_i\): diffusion
\(\mathcal L_i\) acts only \(x_i\), while other states enter as parametric dependencies

For the two-state example, use transition operators and sums in the same calculation

Direct interaction: \(X_1\to X_2\), with \(\tau_1\ll\tau_2\)

\[ s=\frac{t}{\tau_2},\quad\epsilon=\frac{\tau_1}{\tau_2}\ll1,\qquad\partial_s p_{12}=\left[\epsilon^{-1}\mathcal L_1+\mathcal L_2(x_1)\right]p_{12} \]

\[ p_{12}=p_{12}^{\rm eff}+\epsilon p_{12}^{(1)}+\cdots\quad\Rightarrow\quad\mathcal L_1p_{12}^{\rm eff}=0 \]

\[ p_{12}^{\rm eff}=p_1^{\rm st}(x_1)\,p_2(x_2,s),\qquad \mathcal L_1p_1^{\rm st}=0 \]

The autonomous fast regulator reaches its stationary distribution: \(\int dx_1\,p_1^{\rm st}=1\)

Integrate the next order over \(x_1\); probability conservation removes the fast operator

\[ \int dx_1\,\mathcal L_1p_{12}^{(1)}=0 \]

\[ \partial_s p_2=\underbrace{\int dx_1\,\mathcal L_2(x_1)\!\left[p_1^{\rm st}(x_1)p_2\right]}_{\displaystyle \mathcal L_2^{\rm eff}p_2} \]

The effective operator averages over the full distribution of the fast input

\[ p_{12}^{\rm eff}=p_1^{\rm st}(x_1)\,p_2^{\rm eff}(x_2,s)\qquad\Rightarrow\qquad I_{12}=0 \]

Feedback interaction: \(X_1\to X_2\), with \(\tau_2\ll\tau_1\)

\[ s=\frac{t}{\tau_1},\quad\epsilon=\frac{\tau_2}{\tau_1}\ll1,\qquad\partial_s p_{12}=\left[\mathcal L_1+\epsilon^{-1}\mathcal L_2(x_1)\right]p_{12} \]

Hold \(x_1\) fixed while \(x_2\) relaxes under the original operator \(\mathcal L_2(x_1)\)

\[ \mathcal L_2(x_1)\,p_{2|1}^{\rm st}(x_2|x_1)=0,\qquad\int dx_2\,p_{2|1}^{\rm st}(x_2|x_1)=1 \]

Each frozen value of \(x_1\) can give a different stationary response

\[ p_{12}(x_1,x_2,s)=p_{2|1}^{\rm st}(x_2|x_1)\,p_1(x_1,s) \]

\[ \partial_s p_1=\int dx_2\,\mathcal L_1\!\left[p_{2|1}^{\rm st}p_1\right]=\mathcal L_1p_1 \]

The slow operator is also unchanged; the fast conditional density integrates to one

\[ \begin{aligned}p_2(x_2,s)&=\int dx_1\,p_1(x_1,s)p_{2|1}^{\rm st}(x_2|x_1)\\[4pt]I_{12}&=\int dx_1\,p_1(x_1,s)D_{\rm KL}\!\left[p_{2|1}^{\rm st}(\cdot|x_1)\,\Vert\,p_2\right]\end{aligned} \]

Information is positive when the conditional response varies across the populated input states

Direct chain: \(1\to2\to3\), with \(\tau_1\ll\tau_2\ll\tau_3\)

\[ \mathcal L_2^{\rm eff}p_2=\int dx_1\,\mathcal L_2(x_1)\!\left[p_1^{\rm st}(x_1)p_2\right] \]

\[ \mathcal L_3^{\rm eff}p_3=\int dx_2\,\mathcal L_3(x_2)\!\left[p_2^{\rm st,eff}(x_2)p_3\right] \]

Each layer receives an average over its faster input

\[ p_{123}^{\rm eff}=p_1^{\rm st}\,p_2^{\rm st,eff}\,p_3^{\rm eff}(t),\qquad I_{12}=I_{13}=I_{23}=0 \]

Feedback chain: \(3\to2\to1\), with \(\tau_1\ll\tau_2\ll\tau_3\)

\[ \mathcal L_1(x_2)p_{1|2}^{\rm st}=0,\qquad \mathcal L_2(x_3)p_{2|3}^{\rm st}=0 \]

Both operators keep their original form, with each slower input held fixed

\[ p_{123}=p_{1|2}^{\rm st}(x_1|x_2)\,p_{2|3}^{\rm st}(x_2|x_3)\,p_3(x_3,t) \]

\[ I(X_1;X_3|X_2)=0 \]

\(2\to3\to1\), with \(\tau_1\ll\tau_2\ll\tau_3\)

Layer 3 averages the input from 2; layer 1 responds to the frozen state of 3

\[ \mathcal L_3^{\rm eff}p_3=\int dx_2\,\mathcal L_3(x_2)\!\left[p_2^{\rm st}(x_2)p_3\right] \]

\[ p_{123}^{\rm eff}=p_{1|3}^{\rm st}(x_1|x_3)\,p_2^{\rm st}(x_2)\,p_3^{\rm eff}(x_3,t) \]

\[ I_{12}=I_{23}=0 \]

\(3\to1\to2\), with \(\tau_1\ll\tau_2\ll\tau_3\)

Average over the fast layer 1 while keeping the slow source \(x_3\) fixed

\[ \mathcal L_2^{\rm eff}(x_3)p_2=\int dx_1\,\mathcal L_2(x_1)\!\left[p_{1|3}^{\rm st}(x_1|x_3)p_2\right] \]

The effective operator for layer 2 still depends on \(x_3\)

\[ p_{123}^{\rm eff}=p_{1|3}^{\rm st}(x_1|x_3)\,p_{2|3}^{\rm st,eff}(x_2|x_3)\,p_3(x_3,t) \]

At fixed \(x_3\), the two faster layers are independent

\[ p_{12|3}^{\rm eff}=p_{1|3}^{\rm st}\,p_{2|3}^{\rm st,eff}\qquad\Rightarrow\qquad I(X_1;X_2|X_3)=0 \]

Averaging over their shared slow source can correlate the two layers

\[ p_{12}^{\rm eff}=\int dx_3\,p_{1|3}^{\rm st}(x_1|x_3)\,p_{2|3}^{\rm st,eff}(x_2|x_3)\,p_3(x_3,t) \]

\(\tau_1\ll\tau_2\ll\cdots\ll\tau_N\): relax the fastest layer, average it, then continue

\[ \mathcal L_1\,p_{1|\rho(1)}^{\rm st}=0,\qquad\int dx_1\,p_{1|\rho(1)}^{\rm st}=1 \]

\[ \mathcal L_\mu^{\rm eff} f=\int dx_1\cdots dx_{\mu-1}\,\mathcal L_\mu\!\left[\left(\prod_{\nu<\mu}p_{\nu|\rho(\nu)}^{\rm st,eff}\right)f\right] \]

\(f\) is a density of the remaining layers; their slow states stay fixed in the average

At each stage, find the normalized stationary density of the effective operator

\[ \mathcal L_\mu^{\rm eff}\,p_{\mu|\rho(\mu)}^{\rm st,eff}=0,\qquad\int dx_\mu\,p_{\mu|\rho(\mu)}^{\rm st,eff}=1 \]

\[ p_{1,\ldots,N}^{\rm eff}=p_N^{\rm eff}(x_N,t)\prod_{\mu=1}^{N-1}p_{\mu|\rho(\mu)}^{\rm st,eff}\!\left(x_\mu\mid\{x_\nu\}_{\nu\in\rho(\mu)}\right) \]

\[ \partial_t p_N^{\rm eff}=\frac{1}{\tau_N}\mathcal L_N^{\rm eff}p_N^{\rm eff} \]

\(\rho(\mu)\) contains the slower layers retained as conditions

A propagation path starts at a slower source and contains a direct interaction

\[ \nu\to\cdots\to\mu:\quad\nu>\mu,\qquad \exists\ (\alpha\to\beta)\ \text{with }\alpha<\beta \]

It is minimal when every internal layer is faster than the target \(\mu\)

\[ \rho(\mu)=\left\{\nu>\mu:\ \nu\to\mu\text{ is an edge, or an mPP }\nu\to\mu\text{ exists}\right\} \]

These paths determine which slow states enter the effective stationary densities

Freeze storage \(s\) and input \(h\) while the fast variables relax

\[ a(h)=e^{-\beta\Delta E}(1+e^{\beta h}),\qquad b(s)=1+e^{\beta\kappa\sigma s/N_S} \]

\[ \begin{gathered}a(h)(1-\theta)=b(s)\theta\\p_{R|S,H}^{\mathrm{eff,st}}(r=1\mid s,h)=\theta(s,h)=\frac{a(h)}{a(h)+b(s)}\end{gathered} \]

\(\beta\): inverse temperature; \(\Delta E\): receptor barrier; \(\kappa\): inhibition strength
\(\sigma\): cost per storage unit; \(N_S\): storage capacity

\[ \begin{gathered}\mu_r=e^{-\beta(V-gr)},\qquad\mu_rp_{U|R}^{\mathrm{eff,st}}(u\mid r)=(u+1)p_{U|R}^{\mathrm{eff,st}}(u+1\mid r)\\p_{U|R}^{\mathrm{eff,st}}(u\mid r)=e^{-\mu_r}\frac{\mu_r^u}{u!}\end{gathered} \]

Average the Poisson readout laws over receptor activity

\[ \begin{aligned}p_{U|S,H}^{\mathrm{eff,st}}(u\mid s,h)&=[1-\theta(s,h)]\operatorname{Pois}(u;\mu_0)+\theta(s,h)\operatorname{Pois}(u;\mu_1)\\\bar u(s,h)&=[1-\theta(s,h)]\mu_0+\theta(s,h)\mu_1\end{aligned} \]

Equal bare receptor-pathway rates, in units of \(\tau_R^{-1}\); frozen slow variables

\(V\): readout production cost; \(g\): receptor-induced reduction (paper’s \(c\))
Constant birth and linear death; unbounded readout counts; frozen slow variables

Average the storage birth rate over fast readout activity

\[ \begin{gathered}w_{s\to s+1}(u)=\frac{u e^{-\beta\sigma}}{\tau_S}\\b_s(h)=\frac{\bar u(s,h)e^{-\beta\sigma}}{\tau_S},\qquad d_s=\frac{s}{\tau_S}\end{gathered} \]

The conditional mean sets storage production, and each stored unit decays spontaneously

Evolve the joint distribution of storage and input

\[ \begin{aligned}\partial_tp^{\mathrm{eff}}_{S,H}(s,h;t)&=\mathcal L_H p^{\mathrm{eff}}_{S,H}(s,h;t)+b_{s-1}(h)p^{\mathrm{eff}}_{S,H}(s-1,h;t)+d_{s+1}p^{\mathrm{eff}}_{S,H}(s+1,h;t)\\&\quad-[b_s(h)+d_s]p^{\mathrm{eff}}_{S,H}(s,h;t)\end{aligned} \]

\[ \frac{d\langle S\rangle}{dt}=\langle b_S(H)-d_S\rangle \]

Responses increase production, while surviving storage changes the next response, establishing a feedback loop

\(p^{\mathrm{eff}}_{S,H}(s,h;t)\); \(\mathcal L_H\) evolves the input
\(0\le s\le N_S\); reflecting boundaries \(d_0=0\), \(b_{N_S}=0\); evolve \(p^{\mathrm{eff}}_{S,H}\) numerically

Sum over internal states to obtain the input–response distribution

\[ p^{\mathrm{eff}}_{U,H}(u,h;t)=\sum_{s=0}^{N_S}\sum_{r=0}^1p_{U|R}^{\mathrm{eff,st}}(u\mid r)p_{R|S,H}^{\mathrm{eff,st}}(r\mid s,h)p^{\mathrm{eff}}_{S,H}(s,h;t) \]

\[ p^{\mathrm{eff}}_U(u;t)=\int_0^\infty p^{\mathrm{eff}}_{U,H}(u,h;t)\,dh,\qquad p_H(h;t)=\sum_{u=0}^\infty p^{\mathrm{eff}}_{U,H}(u,h;t) \]

Readout counts are discrete; input values are continuous

Calculate information, then compare matching stimulus phases

\[ \begin{aligned}I_{U,H}(t)&=\sum_{u=0}^{\infty}\int_0^\infty dh\,p^{\mathrm{eff}}_{U,H}(u,h;t)\log\frac{p^{\mathrm{eff}}_{U,H}(u,h;t)}{p^{\mathrm{eff}}_U(u;t)p_H(h;t)}\\&=\mathcal{H}[U_t]-\int_0^\infty dh\,p_H(h;t)\,\mathcal{H}[U_t\mid H_t=h]\end{aligned} \]

\[ \Delta I_{U,H}=I_{U,H}^{\mathrm{hab}}-I_{U,H}^{\mathrm{in}},\qquad\Delta\langle U\rangle=\langle U\rangle^{\mathrm{hab}}-\langle U\rangle^{\mathrm{in}} \]

\(\Delta I_{U,H}>0\): more information; \(\Delta\langle U\rangle<0\): weaker response

Same input statistics at the compared phases; natural logarithms; timescale-separation approximation

Fix the input statistics: \(p_H^{\mathrm{st}}(h)=H_{\mathrm{st}}^{-1}e^{-h/H_{\mathrm{st}}}\)

\[ I^{\mathrm{st}}_{U,H}(\beta,\sigma),\qquad E_{\mathrm{tot}}^{\mathrm{st}}=\delta Q_R^{\mathrm{st}}+\frac{\tau_S J_{\mathrm{int}}^{\mathrm{st}}}{\sigma} \]

\[ \delta Q_R^{\mathrm{st}}=\beta\left(H_{\mathrm{st}}+\kappa\sigma\frac{\langle S\rangle_{\mathrm{st}}}{N_S}\right) \]

\(\delta Q_R\): receptor-cycle dissipation measure; \(J_{\mathrm{int}}\): net storage energy flux
\(\tau_S J_{\mathrm{int}}/\sigma\) is its normalized contribution

Choose the relative weight of information and energetic cost

\[ \mathcal L(\beta,\sigma)=\gamma I^{\mathrm{st}}_{U,H}(\beta,\sigma)-(1-\gamma)E_{\mathrm{tot}}^{\mathrm{st}}(\beta,\sigma) \]

\[ 0\le\gamma\le1,\qquad(\beta^*,\sigma^*)\in\operatorname*{arg\,max}_{\beta,\sigma}\mathcal L(\beta,\sigma) \]

Vary \(\gamma\) to find the best stationary information at different costs

Test the selected parameters under repeated stimulation

High information gain and intermediate response reduction
Storage cost varies at fixed \(\beta\); vertical axes are rescaled