Mathematical jeopardy with differential equations

Question

^{Jeopardy! is an American television game show [...] in which contestants are presented with general knowledge clues in the form of answers, and must phrase their responses in the form of questions.}

What kind of mathematical problem is it, to ask for systems of differential equations which have a given function $x(t)$ as an (approximate) solution?

Specifically, I am looking for a simple (the most simple?) pair of differential equations

$\dot{x} = f(x,y)$

$\dot{y} = g(x,y)$

that give rise to something like an action potential:

To make a solution of this problem feasible, one should not fix the target function $x(t)$ too strictly, but only qualitatively:

1. monotonously and slowly growing – like $\alpha t$ or $\alpha(1-e^{-\beta t})$ – for $t \leq t_0 = 1 - \epsilon$ for some $\epsilon \ll 1$ and $\alpha \ll 1$, $\beta \approx 4$

2. having approximately the shape of a narrow Gauss curve $e^{-(x - x_0)^2/\lambda}$ for $t \geq t_1 = 1 + \epsilon$ with $\lambda\ll 1$ and $x_0 \approx 1 + \gamma\epsilon$, $2 < \gamma < 4$

3. any smooth continuation from 1. to 2. (in the very narrow range between $t_0$ and $t_1$), which is also monotonously growing.

Note, that one system of differential equations giving rise to another (and biologically more realistic) kind of action potentials $x(t)$ is given by the FitzHugh-Nagumo model, but I'm looking for significantly simpler ones:

$\dot{x} = x - \frac{x^3}{3} - y + a$

$\dot{y} = bx - cy + d$

with arbitrary $a$ and appropriately chosen $b, c, d$.

Answer 1

1. Very often such solutions appear as a result of search for traveling wave solutions of some spatial systems. Since the form you are looking for is "impulse", it means that you are looking for a system, whose orbit has the properties $x(t)\to 0$ when $t\pm \infty$, which also implies that one particular way is to look at systems with homoclinic trajectories.

2. As I mentioned in the comments, one simple example is the so-called Keller-Segel model, which for the traveling wave solutions takes the form: $$ \dot U=cU-\delta k \frac{U^2}{cV}\,,\\ \dot V=-k U/c.\\ $$ This system can be integrated exactly (see, e.g., Model for chemotaxis by Evelyn F.Keller and Lee A.Segel), and for $U$ you will find something which resembles your drawing.