i guess this is a rather simple question, but given my non-mathematical background, i'm a bit stuck. i'm trying to find the jacobian matrix for the follwing dynamical system (wilson-cowan model). the dynamics of the model are given by something like this:
$\tau_e\frac{dx}{dt} = -x(t) + [1-x(t)]*\phi(w_{ee}x(t) - w_{ei}y(t) + I_e(t))$ (Eq.1)
$\tau_i\frac{dy}{dt} = -y(t) + [1-y(t)]*\phi(w_{ie}x(t))$ (Eq.2)
where $\phi$ is a sigmoidal input-response function:
$\phi(z) = \frac{1}{1 + \exp(-az)}$
I hope this is enough information. I now want to find the fixed points of this system. i believe i need the partial derivatives of the first two equations. i don't know how to write this properly, so let's assume Eq. 1 = f and Eq. 2 = g, so the jacobian is defined as:
$J = \begin{bmatrix} \frac{df}{dx} & \frac{df}{dy} \\ \frac{dg}{dx} & \frac{dg}{dy} \end{bmatrix}$
i guess my problem now is rather simple. i could find the partial derivatives for these equations without the input-response function, but its presence is really confusing me and i don't know how to solve this. i'd really appreciate some help here!
thanks a lot and best,
m
ps: i'm sorry if the notation isn't always correct - i've tried as best as i could :)
Determining the fixed points of a system is independent of calculating the Jacobian. You need to substitute the fixed points into the Jacobian at some point, so you need to calculate the fixed points first. For this system, with general input function $I_e$, there are no fixed points (since $I_e$ could change the state of the system arbitrarily at any point in time).
If you assume $I_e = 0$, then it's still not possible to find an explicit expression for the fixed points. However, you could try to find the number of fixed points by studying the intersections of graphs you obtain by solving the algebraic equations you get while looking for fixed points.