Analyzing a linear system of equations from fMRI-data and extracting stimuli information this is my first question in here and I hope I'll do it according to your expectations and rules and just start right now ;). At first I have to say that I'm not a mathematican so maybe I'm not expressing myself like one..just correct me or delete the question, if it's not clear or understandable, I'll give it a chance for now :). 
Firstly a little bit of introduction:
I have a 41x41 system of linear equations (inhomogen) which I derived with Eureqa by describing the timecourse of fMRI haemodynamic data from a brain area as a function of the timecourses of 40 other brain areas (from Brodmann areas fyi). The person who was measured had to do some specific task and had seen some pictures. So the timecourse data is a function of the stimuli and my equations somehow also describe the stimuli.
e.g. 
$$ \text{BrainArea}_1= f(\text{brainarea}_2,\text{brainarea}_3, \dots, \text{brainarea}_{41}) $$
$$ \text{BrainArea}_2= f(\text{brainarea}_1,\text{brainarea}_3, \dots, \text{brainarea}_{41}) $$
$$ \vdots $$
$$ \text{BrainArea}_{41} = f(\text{brainarea}_1,\text{brainarea}_2, \dots, \text{brainarea}_{40})$$
So..now I can solve this linear system, can compute the eigensystem for example..
but my question is..
1)
what would a mathematican do with such a system..What is the straightforward way to analyze such a linear system of equations? Also, which Eigenvalues and Eigenvectors are of interest (because I got a lot)? Should I only consider the biggest Eigenvalues, or the smallest..only the positive or negative? What would they mean is this case?
2)
Is it possible to extract the timings when the person saw some stimuli from the system or somehow extract more information about the stimuli, when I know the exact time a stimuli occured (which I do know)? Since the data of the timecourses is a function of what stimuli the person saw, one can see the frequency of the stimuli in the frequency domain if fouriertransform is applied. 
But is there something else one can do with this system? 
3) What would change if my system is a nonlinear system of equations? Would there be some fundamental constrictions or limits in my possibilities to analyze the system?
I'm not hoping for some specific solutions to that matter, but maybe for some helpful hints and tips how to go deeper into it, maybe what I could read up on and so on. 
Thanks in advance
 A: Here is my take on an answer to your question. First off: I am not a mathematician, I am a physicist who likes math, so the answer below is probably not what a pure math-guy would give.
Now for the answer:

First you should be more specific with your equations. The way I understand you is that you have a time-series,that is: all your functions rely explicitly or implicitly on $t$ e.g $BrainArea_1(t) = f_1(BrainArea_2(t),\dots, BrainArea_{41}(t), t)$
Which in the given case are probably evaluated for discrete timesteps $t_i$ yielding a time-series of a set of equations.
Secondly, it is not obvious from your formulation of the problem, that the set is a set of linear equations, as the $f_i$ (or $f$?) could be arbitrary.
Then:
1). Let us assume that the system is indeed linear and time-dependent and denote $BA:=(BrainArea_1,\dots,BrainArea_{41})^t$ Then the system becomes
$BA = A(t) BA$
Where $[A(t)]_{i,i}=0$ to account for the reliance only on the other areas.  One way to look at such a system would be, of course, to look at the eigensystem. In this case looking at the largest (in magnitude) eigenvalues is probably more appropriate I think, because the smallest ones (again in magnitude) would be probably attributable to noise in the system.
The associated eigenvector will then yield the combination of areas which is the most active at the given moment.
2). Under the same assumptions as above you will get (already in 1). ) the eigenvalues $\lambda_i \equiv \lambda_i(t_j)$ for a given timestep, same goes for eigenvectors. The analysis from here on is your job as somebody who understands what the respective areas do.
