= Synthetic insulin-glucagon response model
{title2=$\tau_I\dot I=I_\infty(g)-I,\quad\tau_H\dot H=H_\infty(g)-H$}
This deliberately synthetic <differential equation> model illustrates response direction and lag. Every plotted number is chosen for teaching; none is a biological measurement or a fitted human parameter. Let externally prescribed, <dimensionless> $g>0$ represent a <glucose> stimulus relative to a reference. Let $I$ and $H$ be normalized <insulin> and <glucagon> response variables, with independently chosen <time constants> $\tau_I=4$ and $\tau_H=6$ in arbitrary time units:
$$
I_\infty(g)=\frac{g^4}{1+g^4},\qquad H_\infty(g)=\frac{1}{1+g^4},\qquad \tau_I\dot I=I_\infty(g)-I,\qquad \tau_H\dot H=H_\infty(g)-H.
$$
Here $I_\infty$ rises and $H_\infty$ falls with $g$. Their targets add to one by construction; the evolving $I+H$ need not equal one because their <time constants> differ. Start with $g=1$ and $I=H=1/2$. Set $g=1.6$ from $t=10$ to $40$, $g=0.65$ from $t=40$ to $70$, and $g=1$ otherwise. Within a constant-input interval beginning at $t_0$, the <exponential decay> solution is
$$
I(t)=I_\infty+[I(t_0)-I_\infty]e^{-(t-t_0)/\tau_I},\qquad H(t)=H_\infty+[H(t_0)-H_\infty]e^{-(t-t_0)/\tau_H}.
$$
Each response is a <continuous function> of time even when the imposed stimulus jumps. The chosen starting values stay between zero and one, since each solution is a <weighted arithmetic mean> of its interval's starting value and target.
\Image[/synthetic-insulin-glucagon-step-response.png]
{title=Synthetic <insulin> and <glucagon> responses to imposed <glucose> steps}
{description=Original illustration of the <synthetic insulin-glucagon response model>. All curves are constructed examples in arbitrary units. Top: imposed <dimensionless> stimulus. Middle: responses with unequal <time constants>. Bottom: illustrative <liver> release and tissue uptake tendencies, which do not determine the imposed stimulus. No experimental data or clinical ranges are represented.}
For an illustrative downstream comparison, define release tendency $R=0.3+0.6H/(1+I)$ and uptake tendency $U=(0.25+0.5I)g$, in the same arbitrary flux units. At the reference target, $R=U=0.5$. At the high-stimulus targets, $R\approx0.343$ and $U\approx1.094$; at the low-stimulus targets, $R\approx0.742$ and $U\approx0.212$. These chosen functions encode greater uptake with <insulin> and greater release with <glucagon>, moderated by <insulin>; they are not established <enzyme kinetics>. The $g$ trace is imposed externally, so the figure does not demonstrate self-correction of <blood glucose>. A closed model would also need a <conservation of mass> equation, nutrient input, finite <glycogen> stores, and additional tissue interactions. The qualitative restoring loops are shown separately in the <insulin> circuit.
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