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arxiv: 2604.01573 · v2 · pith:6TWTTX6Snew · submitted 2026-04-02 · 🧮 math.DS · cs.SY· eess.SY

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

Moh Kamalul Wafi , Arthur C. B. de Oliveira , Eduardo D. Sontag This is my paper

Pith reviewed 2026-05-19 17:42 UTC · model grok-4.3

classification 🧮 math.DS cs.SYeess.SY
keywords incoherent feedforward motifscumulative dose responsemonotonicitydose responsedynamical systemsnetwork motifssensitivity analysissystems biology
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The pith

Cumulative dose responses stay monotone in most incoherent feedforward motifs even when instantaneous responses do not.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that certain incoherent feedforward motif systems maintain a monotone cumulative dose response with respect to input even when the instantaneous dose response is non-monotone. This is shown by deriving an integral representation of the sensitivity of the cumulative response and reducing monotonicity to sign conditions along trajectories. The analysis covers four canonical motifs and finds that IFFM1, IFFM2, and IFFM3 preserve monotonicity in the cumulative measure while IFFM4 does not. A sympathetic reader would care because the result clarifies how network structure can ensure reliable accumulated outputs in biological systems despite possible local non-monotonicities.

Core claim

For systems with linear intermediate dynamics and nonlinear output dynamics, an integral representation of cDR sensitivity shows that IFFM1 and IFFM3 exhibit monotone cDR despite potentially non-monotone DR, IFFM2 is monotone already at the DR level, and IFFM4 violates the conditions and loses monotonicity in cDR.

What carries the argument

The integral representation of the sensitivity of the cumulative dose response with respect to the input, which reduces the monotonicity question to verifying qualitative sign properties along system trajectories.

Load-bearing premise

The derivation assumes linear dynamics for the intermediate layer together with nonlinear output dynamics and structured initial conditions.

What would settle it

A numerical trajectory or parameter choice for IFFM1 in which the cumulative dose response decreases as the input increases would falsify the claimed monotonicity.

Figures

Figures reproduced from arXiv: 2604.01573 by Arthur C. B. de Oliveira, Eduardo D. Sontag, Moh Kamalul Wafi.

Figure 1
Figure 1. 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 x˙ = −x + u, y˙ = u/x − y). This acts as a “change detector”: ∆u(t) triggers an activity burst, followed by a return to the adapted value, y(t) → y(0). As ex… view at source ↗
Figure 2
Figure 2. Figure 2: Dose response DR(u, T) for the four IFFM systems under three different initial conditions x0, y0. The input u is varied over u ∈ [10−3 , 103 ]. IFFM1 and IFFM3 exhibit nonmonotone DR, while IFFM2 shows monotone behavior. IFFM4 displays nonmonotonicity. 10-3 10-2 10-1 100 101 102 103 0 0.5 1 1.5 2 2.5 3 (a) IFFM1 10-3 10-2 10-1 100 101 102 103 0 1 2 3 4 5 6 (b) IFFM2 10-3 10-2 10-1 100 101 102 103 1 2 3 4 5… view at source ↗
Figure 3
Figure 3. Figure 3: Cumulative dose response cDR(u, T) for the four IFFM systems under the same conditions as [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: System 1: Dose response DR(u, T) and cumulative dose response cDR(u, T) for u ∈ [10−1 , 103 ]. (a)–(b) Responses for three fixed initial conditions x0. (c)–(d) Envelope (min–max band) and mean over x0 ∈ [0.1, 10]. The results illustrate that cDR(u, T) is monotone nonincreasing with respect to u, consistently with the theoretical analysis. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: System 2: Dose response DR(u, T) and cumulative dose response cDR(u, T) for u ∈ [10−1 , 103 ]. (a)–(b) Responses for three fixed initial conditions x0. (c)–(d) Envelope (min–max band) and mean over x0 ∈ [0.1, 10]. The results show that cDR(u, T) is monotone nonincreasing with respect to u. System 3: y˙u = u xu − yu. Here F(xu, yu, u) = (u/xu) − yu. Thus ∂xF = −u/x2 u , ∂yF = −1, and ∂uF = 1/xu. Therefore, … view at source ↗
Figure 6
Figure 6. Figure 6: System 3: Dose response DR(u, T) and cumulative dose response cDR(u, T) for u ∈ [10−1 , 103 ]. (a)–(b) Responses for three fixed initial conditions x0. (c)–(d) Envelope (min–max band) and mean over x0 ∈ [0.1, 10]. The results show that cDR(u, T) is monotone nondecreasing with respect to u. System 4: y˙u = 1 xu − 1 u yu. Here F(xu, yu, u) = (1/xu) − (yu/u). Thus ∂xF = −1/x2 u , ∂yF = −1/u, and ∂uF = yu/u2 .… view at source ↗
Figure 7
Figure 7. Figure 7: System 4: Dose response DR(u, T) and cumulative dose response cDR(u, T) for u ∈ [10−1 , 103 ]. (a)–(b) Responses for three fixed initial conditions x0. (c)–(d) Envelope (min–max band) and mean over x0 ∈ [0.1, 10]. The results show that cDR(u, T) is not monotone with respect to u, exhibiting a change in monotonicity. System 5: y˙u = u − xuyu. Here F(xu, yu, u) = u − xuyu. Thus ∂xF = −yu, ∂yF = −xu, and ∂uF … view at source ↗
Figure 8
Figure 8. Figure 8: System 5: Dose response DR(u, T) and cumulative dose response cDR(u, T) for u ∈ [10−1 , 103 ]. (a)–(b) Responses for three fixed initial conditions x0. (c)–(d) Envelope (min–max band) and mean over x0 ∈ [0.1, 10]. The results show that cDR(u, T) is monotone nondecreasing with respect to u. System 6: y˙u = 1 − xu u yu. Here F(xu, yu, u) = 1 − (xuyu/u). Thus ∂xF = −yu/u, ∂yF = −xu/u, and ∂uF = xuyu/u2 . Ther… view at source ↗
Figure 9
Figure 9. Figure 9: System 6: Dose response DR(u, T) and cumulative dose response cDR(u, T) for u ∈ [10−1 , 103 ]. (a)–(b) Responses for three fixed initial conditions x0. (c)–(d) Envelope (min–max band) and mean over x0 ∈ [0.1, 10]. The results show that cDR(u, T) is monotone nondecreasing with respect to u. System 7: y˙u = 1 u − 1 xu yu. Here F(xu, yu, u) = (1/u) − (yu/xu). Thus ∂xF = yu/x2 u , ∂yF = −1/xu, and ∂uF = −1/u2 … view at source ↗
Figure 10
Figure 10. Figure 10: System 7: Dose response DR(u, T) and cumulative dose response cDR(u, T) for u ∈ [10−1 , 103 ]. (a)–(b) Responses for three fixed initial conditions x0. (c)–(d) Envelope (min–max band) and mean over x0 ∈ [0.1, 10]. The results show that cDR(u, T) is not monotone with respect to u, exhibiting a change in monotonicity. System 8: y˙u = 1 − u xu yu. Here F(xu, yu, u) = 1 − (uyu/xu). Thus ∂xF = uyu/x2 u , ∂yF =… view at source ↗
Figure 11
Figure 11. Figure 11: System 8: Dose response DR(u, T) and cumulative dose response cDR(u, T) for u ∈ [10−1 , 103 ]. (a)–(b) Responses for three fixed initial conditions x0. (c)–(d) Envelope (min–max band) and mean over x0 ∈ [0.1, 10]. The results show that cDR(u, T) is monotone nonincreasing with respect to u. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Vector System 1: Dose response DR(u, T) and cumulative dose response cDR(u, T) for u ∈ [10−1 , 103 ]. Results are shown for three fixed initial conditions x0. The curves illustrate that cDR(u, T) is monotone nonincreasing with respect to u. Vector case of System 3 We consider the vector analogue of System 3: x˙ u = Axu + bu, y˙u = βu K + c⊤xu − dyu, (37a) where A ∈ R n×n is Metzler and Hurwitz, b, c ∈ R n… view at source ↗
Figure 13
Figure 13. Figure 13: Vector System 3: Dose response DR(u, T) and cumulative dose response cDR(u, T) for u ∈ [10−1 , 103 ]. Results are shown for three fixed initial conditions x0. The curves illustrate that cDR(u, T) is monotone nondecreasing with respect to u. Vector case of System 6 We consider the vector analogue of System 6: x˙ u = Axu + bu, y˙u = d − c ⊤xu βu yu, (38a) where A ∈ R n×n is Metzler and Hurwitz, b, c ∈ R n +… view at source ↗
Figure 14
Figure 14. Figure 14: Vector System 6: Dose response DR(u, T) and cumulative dose response cDR(u, T) for u ∈ [10−1 , 103 ]. Results are shown for three fixed initial conditions x0. The curves illustrate that cDR(u, T) is monotone nondecreasing with respect to u. Vector case of System 7 We consider the vector analogue of System 7: x˙ u = Axu + bu, y˙u = 1 βu − dyu K + c⊤xu , (39a) where A ∈ R n×n is Metzler and Hurwitz, b, c ∈ … view at source ↗
Figure 15
Figure 15. Figure 15: Vector System 7: Dose response DR(u, T) and cumulative dose response cDR(u, T) for u ∈ [10−1 , 103 ]. Results are shown for three fixed initial conditions x0. The curves illustrate that cDR(u, T) is not monotone with respect to u, exhibiting a change in monotonicity. Vector case of System 8 We consider the vector analogue of System 8: x˙ u = Axu + bu, y˙u = d − βu K + c⊤xu yu, (47a) where A ∈ R n×n is Met… view at source ↗
Figure 16
Figure 16. Figure 16: Vector System 8: Dose response DR(u, T) and cumulative dose response cDR(u, T) for u ∈ [10−1 , 103 ]. Results are shown for three fixed initial conditions x0. The curves illustrate that cDR(u, T) is monotone nonincreasing with respect to u. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_16.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript analyzes monotonicity of the cumulative dose response (cDR) in incoherent feedforward motif (IFFM) systems with linear intermediate dynamics and nonlinear output dynamics. It derives an integral representation of cDR sensitivity to the input and reduces monotonicity questions to verifying qualitative sign properties along trajectories. The framework is applied to four canonical IFFMs, yielding a characterization: IFFM1 and IFFM3 exhibit monotone cDR despite possibly non-monotone instantaneous dose response (DR); IFFM2 is already monotone at the DR level; IFFM4 violates the sign conditions and loses monotonicity. Numerical simulations suggest the properties persist for other initial conditions.

Significance. If the central claims hold, the work supplies a useful reduction of cDR monotonicity to trajectory sign conditions, offering insight into how motif structure controls cumulative input-output behavior. This is relevant to dynamical systems and systems biology. Strengths include the explicit application to four motifs with a complete characterization, the integral representation approach, and the provision of numerical support for robustness beyond the analytic assumptions.

major comments (1)
  1. [Derivation of integral representation and application to the four IFFMs] The analytic derivation of the integral representation of cDR sensitivity and the subsequent sign conditions (used to conclude monotonicity for IFFM1 and IFFM3 and loss for IFFM4) are obtained under linear intermediate dynamics together with structured initial conditions. This assumption is load-bearing for the claimed complete characterization, as the abstract notes only that numerics suggest persistence for other conditions without providing an analytic extension or robustness proof for arbitrary ICs or nonlinear intermediates.
minor comments (1)
  1. Clarify in the abstract and introduction the precise scope of the analytic results versus the numerical evidence.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for their careful reading, positive assessment of the work's significance, and constructive feedback on the scope of our analytic results. We address the major comment below and will make targeted revisions to improve clarity.

read point-by-point responses

  1. Referee: The analytic derivation of the integral representation of cDR sensitivity and the subsequent sign conditions (used to conclude monotonicity for IFFM1 and IFFM3 and loss for IFFM4) are obtained under linear intermediate dynamics together with structured initial conditions. This assumption is load-bearing for the claimed complete characterization, as the abstract notes only that numerics suggest persistence for other conditions without providing an analytic extension or robustness proof for arbitrary ICs or nonlinear intermediates.

    Authors: We agree that the integral representation of cDR sensitivity and the resulting sign conditions for monotonicity are derived under the assumptions of linear intermediate dynamics and structured initial conditions, as stated in the manuscript (see Section 2 and the applications in Section 3). These assumptions enable the explicit reduction to trajectory sign properties and yield the complete characterization for the four canonical IFFMs. The numerical simulations are presented only as supporting evidence that the qualitative behavior appears robust, not as a substitute for analysis. We will revise the abstract and the concluding discussion to more explicitly delineate the analytic scope, clarify that the characterization holds under the stated assumptions, and emphasize that extending the results to arbitrary initial conditions or nonlinear intermediates remains an open direction for future work. We believe the framework still provides useful insight into how motif structure influences cumulative responses even within this setting, which is relevant to many systems biology models. revision: partial

standing simulated objections not resolved
  • Analytic extension or robustness proof for arbitrary initial conditions and nonlinear intermediate dynamics

Circularity Check

0 steps flagged

No circularity: derivation relies on integral representation and sign conditions from ODE trajectories

full rationale

The paper derives an integral representation of cDR sensitivity directly from the linear intermediate dynamics and nonlinear output, then reduces monotonicity questions to verifying sign properties along trajectories. These conditions are applied to the four canonical IFFMs to obtain the stated characterization. No quoted step equates a claimed prediction or result to a fitted parameter, self-citation, or definitional input by construction; the analysis is self-contained under the stated assumptions of linear intermediates and structured ICs, with numerics noted separately for extension.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the domain assumption that the motifs possess linear intermediate dynamics and nonlinear output dynamics; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Systems belong to the class with linear intermediate dynamics and nonlinear output dynamics.
    Explicitly stated as the setting for the analysis of IFFM systems.

pith-pipeline@v0.9.0 · 5754 in / 1247 out tokens · 43621 ms · 2026-05-19T17:42:38.957121+00:00 · methodology

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Reference graph

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    A. Gupta and E. Sontag, “Cumulative dose responses for adapting biological systems,” Journal of The Royal Society Interface, vol. 22, p. 20240877, 08 2025. A Proofs Proof of Lemma 1 First, we analyze the relation of xu and yu with respect to xss(u) and yss respectively under two conditions: v u and v < u . We start with xu. Since x0 = A−1bv = xss(v), the ...

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    By (28d), this implies Gu(t) 0 on [0, T ], with Gu 6 0

    If 0 < u x0, then xu(t) u and thus yu(t) 1 for all t 0. By (28d), this implies Gu(t) 0 on [0, T ], with Gu 6 0. Hence, by Theorem 1 (ii), ∂ucDR(u, T ) > 0

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  10. [10]

    Therefore, ∂ucDR(u, T ) < 0

    F or large u, by (28f), one has Gu(t) < 0 for every fixed t > 0 and all sufficiently large u. Therefore, ∂ucDR(u, T ) < 0. Thus, ∂ucDR(u, T ) > 0 for small u and ∂ucDR(u, T ) < 0 for large u, and hence cDR(u, T ) is not monotone. 10 -1 10 0 10 1 10 2 10 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 (a) 10 -1 10 0 10 1 10 2 10 3 0 0.5 1 1.5 2 2.5 3 (b) (c) (d) Figur...

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  11. [11]

    Since λu(t) 0 by (29b), it follows that λu(t)gu(t) 0, 8t 2 [0, T ]

    If x0 u, then by (29c), gu(t) 0 for all t 2 [0, T ]. Since λu(t) 0 by (29b), it follows that λu(t)gu(t) 0, 8t 2 [0, T ]. Hence, by Theorem 1 (i), ∂ucDR(u, T ) 0

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  12. [12]

    Thus ˙λu(t) Gu(t) 0, 8t 2 [0, T ]

    If x0 < u , then by (29e), Gu(t) 0 for all t 2 [0, T ], and by (29f), ˙λu(t) 0 for all t 2 [0, T ]. Thus ˙λu(t) Gu(t) 0, 8t 2 [0, T ]. Hence, by Theorem 1 (ii), ∂ucDR(u, T ) 0. Therefore, in either case, u 7! cDR(u, T ) is monotone nondecreasing. 18 Monotonicity of cumulative dose response cDR(u, T ) 10 -1 10 0 10 1 10 2 10 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1...

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    By (31b), this yields Gu(t) 0 for all t 2 [0, T ]

    If 0 < u x0, then xu(t) u and therefore yu(t) 1 for all t 0. By (31b), this yields Gu(t) 0 for all t 2 [0, T ]. Moreover , since ˙λu(t) 0, Theorem 1 (ii) implies ∂ucDR(u, T ) < 0

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    Arguing exactly as in the large- u part of System 4, this yields ∂ucDR(u, T ) > 0 for all sufficiently large u

    F or largeu, (31f) shows that, for every fixed t > 0, one has Gu(t) > 0 for all sufficiently large u. Arguing exactly as in the large- u part of System 4, this yields ∂ucDR(u, T ) > 0 for all sufficiently large u. Thus there exist u−, u+ > 0 such that ∂ucDR(u−, T ) < 0 and ∂ucDR(u+, T ) > 0, and therefore cDR(u, T ) is not monotone. 20 Monotonicity of cumula...

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