When is cumulative dose response monotonic? Analysis of incoherent feedforward motifs

Abstract

We study the monotonicity of the cumulative dose response (cDR) for a class of incoherent feedforward motifs (IFFM) systems with linear intermediate dynamics and nonlinear output dynamics. While the instantaneous dose response (DR) may be nonmonotone with respect to the input, the cDR can still be monotone. To analyze this phenomenon, we derive an integral representation of the sensitivity of cDR with respect to the input and establish general sufficient conditions for both monotonicity and non-monotonicity. These results reduce the problem to verifying qualitative sign properties along system trajectories. We apply this framework to four canonical IFFM systems and obtain a complete characterization of their behavior. In particular, IFFM1 and IFFM3 exhibit monotone cDR despite potentially non-monotone DR, while IFFM2 is monotone already at the level of DR, which implies monotonicity of cDR. In contrast, IFFM4 violates these conditions, leading to a loss of monotonicity. Numerical simulations indicate that these properties persist beyond the structured initial conditions used in the analysis. Overall, our results provide a unified framework for understanding how network structure governs monotonicity in cumulative input-output responses.

Figures (16)

• Figure 1: (a) IFFMs with inhibition via $x$ (left) or directly from $u$ (right). (b) An example: suppose that the external input $u$ changes from a baseline value $u=1$ to a new constant value $u=2$ at time $5s$, and $x(0)=1$, $y(0)=1$ (steady state for $u=1$ in system $\dot x = -x+u$, $\dot y = u/x-y$). This acts as a "change detector": $\Delta u(t)$ triggers an activity burst, followed by a return to the adapted value, $y(t) \rightarrow y(0)$.
• Figure 2: Dose response $\mathrm{DR}(u,T)$ for the four IFFM systems under three different initial conditions $x_0,y_0$. The input $u$ is varied over $u \in [10^{-3},10^{3}]$. IFFM1 and IFFM3 exhibit nonmonotone DR, while IFFM2 shows monotone behavior. IFFM4 displays nonmonotonicity.
• Figure 3: Cumulative dose response $\mathrm{cDR}(u,T)$ for the four IFFM systems under the same conditions as Figure \ref{['fig:DR']}. Despite nonmonotone DR, IFFM1 and IFFM3 exhibit monotone cDR, in agreement with the theoretical results. IFFM2 remains monotone, while IFFM4 loses monotonicity due to the sign-indefiniteness of the sensitivity function $G_u$.
• Figure 4: System 1: Dose response $\mathrm{DR}(u,T)$ and cumulative dose response $\mathrm{cDR}(u,T)$ for $u\in[10^{-1},10^{3}]$. (a)--(b) Responses for three fixed initial conditions $x_0$. (c)--(d) Envelope (min--max band) and mean over $x_0\in[0.1,10]$. The results illustrate that $\mathrm{cDR}(u,T)$ is monotone nonincreasing with respect to $u$, consistently with the theoretical analysis.
• Figure 5: System 2: Dose response $\mathrm{DR}(u,T)$ and cumulative dose response $\mathrm{cDR}(u,T)$ for $u\in[10^{-1},10^{3}]$. (a)--(b) Responses for three fixed initial conditions $x_0$. (c)--(d) Envelope (min--max band) and mean over $x_0\in[0.1,10]$. The results show that $\mathrm{cDR}(u,T)$ is monotone nonincreasing with respect to $u$.

Theorems & Definitions (37)

• Remark 1
• Theorem 1
• Proof 1
• Theorem 2
• Proof 2
• Proposition 1
• Proof 3
• Lemma 1
• Proposition 2
• Lemma 2