Working on some $Q$-curvature equation in dimension $4$, I have been faced with singular metric of the form $(\mathbb{B}, e^{-1/\vert x\vert ^2} \vert dx\vert)$. I try to figure out to what those metrics look like. In order to do so, I refer to the well known $2$-dimensional where we have conical metric $(\mathbb{D}, \vert x\vert^{\beta-1} \vert dx\vert)$ which  are isometric to a cone of angle $2\pi \beta$. If $\beta=n$ is an integer, I can think about it as  the neighborhood of $0$ of and $n$-fold branched unit disc. My question is: is there is a similar interpretation for $(\mathbb{D}, e^{-1/\vert x\vert ^2} \vert dx\vert)$ (in dimension 2)? Can we think about this metric as the limit of some more classical degenerate metric?
Any reference will be welcome.
Thx in advance

Edit: Thanks to the post of @ubik, I can reformulate the question as follows.
Let $g=e^{-\frac{d}{r^2}}(dr^2 +r^2 d\theta^2)$, we set $s=\int_0^r e^{-\frac{d}{2 t^2}}\, dt$, then we get
$$ s= \int_0^r \frac{t^3}{d} \frac{d}{dt}\left( e^{-\frac{d}{2t^2}}\right)\, dt\sim \frac{r^3}{d} e^{-\frac{d}{2r^2}}$$ 
Hence
$$re^{-\frac{d}{2r^2}} \sim s\frac{d}{r^2}$$
$$\log(s)\sim -\frac{d}{2r^2}$$ 
which gives
$$g=ds^2 +J(s)^2d\theta^2$$
where
$$J(s)\sim 2s\log\left(\frac{1}{s}\right)$$
My question what: is the interpretation of this metric, is it a cusp? is there is a standard model?