= Signaling feedback
<Signaling feedback> returns the effect of a downstream event to an earlier stage of <signal transduction>, closing a loop analogous to <feedback control>. Trace the causal arrows: if an earlier relay activates a later relay and the later relay changes the earlier relay's activity, the return path closes the loop. <Negative feedback> opposes the initiating change; <positive feedback> reinforces it. <Signaling feedback> is a possible property of <signal transduction>, not a requirement of every pathway.
For a schematic example, let relay $A$ activate relay $B$. If $B$ inhibits $A$, increased $A$ eventually produces an opposing influence: <negative feedback>. If $B$ activates $A$, increased $A$ produces a reinforcing influence: <positive feedback>. These signs describe effects on activity, rather than the direction in which <molecules> travel. A closed loop containing two inhibitions can have <positive feedback>, because inhibiting an inhibitor reinforces the original change. The sign belongs to the whole loop.
\Image[/signaling-feedback-loops.png]
{title=<Negative feedback>, <positive feedback> and an <incoherent feedforward loop>}
{description=Original causal schematics. Arrows activate; flat-ended lines inhibit. The upper loops return from downstream relay $B$ to earlier relay $A$. In the lower <incoherent feedforward loop>, input $s$ drives both a fast activating branch $x$ and a slower inhibitory branch $a$; their contributions meet at readout $r$. There is no return arrow from $r$ to either branch or to $s$. Shapes and connections illustrate the distinction, rather than a measured pathway.}
<Negative feedback> can restrain excessive activation and help preserve a usable range of responses. <Positive feedback> can support a decisive transition: Alberts and colleagues describe an <enzyme> whose product increases its own activity. Such reinforcement can contribute to switchlike behavior, but a <positive feedback> arrow alone does not prove <bistability>. A different structure, an <incoherent feedforward loop>, sends opposing influences forward from the same input; inhibition there is not a downstream return. <Molecular Biology of the Cell, 4th edition by Bruce Alberts and others/General Principles of Cell Communication>{ref} https://cqb.pku.edu.cn/tanglab/pdf/78.pdf[Ma et al., Defining Network Topologies that Can Achieve Biochemical Adaptation, Cell 138 (2009), pp. 760–773: definitions on p. 764]{ref}
A worked <mathematical model> shows why <negative feedback> does not automatically produce complete <signaling adaptation>. Let $s$ be a fixed normalized input, $r$ a normalized response, $g\geq0$ a dimensionless <feedback gain>, and $\tau>0$ a <time constant>. Suppose response removal contributes $-r$ and a return proportional to the response contributes $-gr$. The <ordinary differential equation> is
$$
\tau\frac{dr}{dt}=s-r-gr=s-(1+g)r.
$$
Setting the <derivative> to zero gives the <equilibrium point> $r_*=s/(1+g)$. A displacement $\delta r=r-r_*$ obeys $\tau\,d(\delta r)/dt=-(1+g)\delta r$, so its decaying <exponential function> establishes <asymptotic stability>. For a unit input switched on at $t=0$, <initial condition> $r(0)=0$, $g=2$ and $\tau=1$ second,
$$
r(t)=\frac{1-e^{-3t/\tau}}{3},\qquad r(1\,\mathrm{s})\simeq0.316737,\qquad \lim_{t\to\infty}r(t)=\frac13.
$$
The <negative feedback> reduces the eventual response from $1$ to $1/3$, but this response rises monotonically and never returns to its zero baseline. Attenuation and complete <signaling adaptation> are different properties. This deliberately simple <mathematical model> specifies no molecular pathway; <integral action> provides a contrasting way to remove a lasting offset in a stable <feedback control> system. https://www.fbswiki.org/wiki/index.php/Feedback_Principles[Åström and Murray, Feedback Systems, Chapter 2: proportional and integral feedback]{ref}
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