I have been looking at reducing a neural field model into isostable equations and eventually into a Kuramoto-Sivashinksy type form to study patterns. I am now however confused on a particular part of the derivation, the problem arises when trying to Taylor expand in the spatial convolution term with my kernel $w(x)$ and sigmoidal function $f(u)$. Following https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.123.164101 , I have used the substitution $$u(x,t) = Y(\theta) + \psi \rho(\theta)$$ where $\theta$ is the phase and $\psi$ is our (isostable) coordinates. Here, $\theta$ $\in$ $\mathbb S^1$ and $\psi$ $\in$ $\mathbb R$ gives the distance from the periodic orbit parametrised by $\rho$ where $Y(t)$ is the stable periodic orbit of the system.
The problem I have is that partial time derivatives of $\theta$ and $\psi$ seem to be popping out and I believe that they shouldn't be there. I am unaware of any trick using the spatial convolution to get rid of these time partial derivatives. Should $\theta$ and $\psi$ even be functions of both $x$ and $t$ ? If so can anyone help with this Taylor expansion please.
I then intend to use the ansatz $\theta = t + \epsilon \phi$, $\phi = \epsilon^2r$ to generate a p.d.e in $\phi$.