- #1
Pythagorean
Gold Member
- 4,400
- 312
I have n elements. Say n = 3.
Suppose I have an association matrix that gives the relationship between each element
[tex]
\begin{array}{cc}
0 & 0 & D3\\
D1 & 0 & 0\\
0 & D2 & 0
\end{array} [/tex]
I have a function in mind now, I want to operate and the physical variables representing my three elements, [tex]\vec{y} = [y_1 y_2 y_3] [/tex]. My functional matrix would look like:
[tex]
\begin{array}{cc}
-D1 & D1 & 0\\
0 & -D2 & D2\\
D3 & 0 & -D3
\end{array} [/tex]
so that I get [tex] \vec{y} = [D1(y_2 - y_1) D2(y_3-y_2) D3(y_1-y_3) [/tex]
If you're interested in the physical/biological motivation, we basically have a unidirectional diffusion coupling between electrophysiological neurons here so you're seeing a numerical second derivative. Now, using MATLAB commands (circshift and transpose) I do circular shifts and transposes on the association matrix to nudge it into my functional shape.But I'm having trouble with the more general case. What if I have two-way diffusion, but the diffusion is stronger in one direciton than in the other? Now the association matrix is:
[tex]
\begin{array}{cc}
0 & D4 & D3\\
D1 & 0 & D5\\
D6 & D2 & 0
\end{array} [/tex]
and what we for functional is:[tex]
\begin{array}{cc}
-(D1+D4) & D1 & D4\\
D5 & -(D2+D5) & D2\\
D3 & D6 & -(D3 + D6)
\end{array} [/tex]
so that [tex]\vec{y} = [D1(y_3-y_1) + D4(y2_y1); D2(y_1- y_2) + D5(y_3-y_2) + ...][/tex]
Now, I can design another series of circshifts and tranposes, but it won't work for the case above. I can't find a general set of operations that works for both
This should work in general, for an nxn matrix.
Thank you for your help.
Suppose I have an association matrix that gives the relationship between each element
[tex]
\begin{array}{cc}
0 & 0 & D3\\
D1 & 0 & 0\\
0 & D2 & 0
\end{array} [/tex]
I have a function in mind now, I want to operate and the physical variables representing my three elements, [tex]\vec{y} = [y_1 y_2 y_3] [/tex]. My functional matrix would look like:
[tex]
\begin{array}{cc}
-D1 & D1 & 0\\
0 & -D2 & D2\\
D3 & 0 & -D3
\end{array} [/tex]
so that I get [tex] \vec{y} = [D1(y_2 - y_1) D2(y_3-y_2) D3(y_1-y_3) [/tex]
If you're interested in the physical/biological motivation, we basically have a unidirectional diffusion coupling between electrophysiological neurons here so you're seeing a numerical second derivative. Now, using MATLAB commands (circshift and transpose) I do circular shifts and transposes on the association matrix to nudge it into my functional shape.But I'm having trouble with the more general case. What if I have two-way diffusion, but the diffusion is stronger in one direciton than in the other? Now the association matrix is:
[tex]
\begin{array}{cc}
0 & D4 & D3\\
D1 & 0 & D5\\
D6 & D2 & 0
\end{array} [/tex]
and what we for functional is:[tex]
\begin{array}{cc}
-(D1+D4) & D1 & D4\\
D5 & -(D2+D5) & D2\\
D3 & D6 & -(D3 + D6)
\end{array} [/tex]
so that [tex]\vec{y} = [D1(y_3-y_1) + D4(y2_y1); D2(y_1- y_2) + D5(y_3-y_2) + ...][/tex]
Now, I can design another series of circshifts and tranposes, but it won't work for the case above. I can't find a general set of operations that works for both
This should work in general, for an nxn matrix.
Thank you for your help.