= Solution
Use the source's explicit graded production rules. X is already present at a constant <concentration>; the input at zero activates it rather than starting accumulation of X. Assume its active level exceeds both relevant X thresholds, put $\alpha_x=\alpha_y=\alpha_z=\alpha$, let $Y_s=\beta_y/\alpha$, and take $0<h=K_{yz}/Y_s<1$. Before the input Z is not produced, so use the initial steady value $Z(0)=0$.
For <I1 incoherent feed-forward loop>, activation of X starts Y production; for <I4 incoherent feed-forward loop> it stops Y production. Consequently
$$
\boxed{Y_1(t)=Y_s(1-e^{-\alpha t}),\quad d_1=-\alpha^{-1}\ln(1-h);\qquad
Y_4(t)=Y_se^{-\alpha t},\quad d_4=-\alpha^{-1}\ln h.}
$$
The two delays are the times for Y to cross its output-regulation threshold. In <I1 incoherent feed-forward loop>, initially absent Y permits the high output rate, then accumulated Y represses it. In <I4 incoherent feed-forward loop>, initially abundant Y activates the high output rate, then Y's decay removes that activation. Thus \i[both] have production $\beta_{\rm high}$ before their respective delay and $\beta_{\rm low}$ afterwards.
With $Z_s=\beta_{\rm low}/\alpha$ and $F=\beta_{\rm high}/\beta_{\rm low}$, their common piecewise response, with $d=d_1$ or $d_4$, is
$$
\boxed{\frac{Z(t)}{Z_s}=\begin{cases}
F(1-e^{-\alpha t}),&0\le t\le d,\\
1+\left[F(1-e^{-\alpha d})-1\right]e^{-\alpha(t-d)},&t\ge d.
\end{cases}}
$$
The output is continuous at the production switch. There is an overshoot precisely when $F(1-e^{-\alpha d})>1$; otherwise it rises monotonically to $Z_s$. For a threshold not extremely close to its limiting values and the supplied $F\gg1$, a rapid pulse followed by relaxation to a lower nonzero level is expected. This can provide fast initial response or a transient strong signal, but it is not perfect adaptation to the original zero output.
The original plots below use the illustrative values $h=1/4$ and $F=10$. At $h=1/2$ the two Y curves are complementary and their output delays and Z curves are identical. These threshold values are examples, not extra source data. “AND” here must be read with the stipulated graded rates: replacing them by an ordinary all-or-nothing AND rule would give a different response.
\Image[/past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2018-gene-circuit-and-switch.png]
{title=Graded incoherent feed-forward pulses and the change from one to two attracting toggle states}
{description=Original model trajectories and nullclines. Pulse panels use illustrative $h=1/4$, $F=10$ and dimensionless time $\alpha t$; switch panels use $n=2$ with the explicitly labelled reduced production strengths.}
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