It's a cuttout from Rempe article "A mathematical model of the sleep/wake cycle" about a function which I don't understand how to describe: $h(t)$ decays exponentially while the system is asleep
and increases while awake. We scale the variables so that the system is defined to
be asleep if $x_A < 0$ and awake if $x_A > 0$. Then $h(t)$ satisfies $h' = α_h(h_{max} − h)$
while awake (when $x_A > 0$) and $h' = −β_hh$ while asleep (when $xA < 0$). Here,
$α_h, β_h$ and $h_{max}$ are constants. Also, a period of cycle last 24h, 16h system is awake and sleeps for 8h.
How to describe a function $h(t)$ in one equation? I think it should be somehow related with $H_{\infty}(x)=\frac{1}{1+e^{-x/0.01}}$ as it widely used in this article.
Here are links to original article and some corrections Article, corrections . $h(t)$ mentioned on $620$page.
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1$\begingroup$ Is there no period/ interval for sleep or for being awake in a cycle? $\endgroup$– NarasimhamCommented May 12, 2020 at 18:32
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$\begingroup$ @Narasimham just mentioned that sleeps for 8hours and awake for 16hours, whole cycle period 24h. $\endgroup$– Adolf MiszkaCommented May 12, 2020 at 18:44
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1$\begingroup$ Represents some blood constituent level? You can improve the question. $\endgroup$– NarasimhamCommented May 12, 2020 at 18:48
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$\begingroup$ Nothing told about blood, this sleep-wake is based on neurons. Here is full article, this function described on 620page : researchgate.net/publication/…, and some corrections to it: researchgate.net/publication/…. I'm actually very new in this field. @Narasimhan $\endgroup$– Adolf MiszkaCommented May 12, 2020 at 19:09
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2$\begingroup$ Do you expect a closed form expression to exist? It seems like the function will have discontinuous derivatives and exponential behaviour so a useful closed form seems unlikely. $\endgroup$– DavidCommented May 12, 2020 at 23:35
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