v****n
发帖数: 4 | 1 This is a unsolved intriguing math question only for real hackers.
The dynamics of a ring network is determined by
d(Xi)/dt = - Xi + max( sum_over_j(Wij * Xj) + bi , 0)
a. neuron i is arranged in a ring network. i = 1,2,...N, and then go back to 1.
N could be very large.
b. max function means
max(x, 0) = x if x > 0,
0 otherwise
c. connection matrix Wij is the connection strength between neuron i and j,
and Wij is only a function the distance between these two neuron | B***y
发帖数: 83 | 2
1) Is Wij some function of t or just some constant depending on (i-j) ?
2) what is the exact meaning of " the moving profile is keeping its waveform"
and " a cost function for this"?
It will be helpful if you can make it clear.
【在 v****n 的大作中提到】
: This is a unsolved intriguing math question only for real hackers.
: The dynamics of a ring network is determined by
: d(Xi)/dt = - Xi + max( sum_over_j(Wij * Xj) + bi , 0)
: a. neuron i is arranged in a ring network. i = 1,2,...N, and then go back to 1.
: N could be very large.
: b. max function means
: max(x, 0) = x if x > 0,
: 0 otherwise
: c. connection matrix Wij is the connection strength between neuron i and j,
: and Wij is only a function the distance between these two neuron
| v****n
发帖数: 4 | 3
** Wij is constant only depending on (i-j)
** the basic idea is that with arbitrary W(i-i) it is unlikely to produce
the travelling wave. The task is how to find the optimal values for W(i-j)
and how to keep updating W(i-j) if noise is added into W(i-j) constantly.
Usually in network research, people would conceive some kind of cost
function for a specific task, and then the problem would be under control
by keeping lowering the cost function. This is just technique term. Not
【在 B***y 的大作中提到】
:
: 1) Is Wij some function of t or just some constant depending on (i-j) ?
: 2) what is the exact meaning of " the moving profile is keeping its waveform"
: and " a cost function for this"?
: It will be helpful if you can make it clear.
| f*******d
发帖数: 339 | 4
Sounds like a soliton. How about this idea: start with a desired wave form,
then set dX_i/dt ==0, this give you the equation
-Xi + max(W_ij X_j) =0, since you now have X_i, you can solve for W_ij. If
you
make some further assumptions on the form of W_ij, you can generate a
solution.
【在 v****n 的大作中提到】
: This is a unsolved intriguing math question only for real hackers.
: The dynamics of a ring network is determined by
: d(Xi)/dt = - Xi + max( sum_over_j(Wij * Xj) + bi , 0)
: a. neuron i is arranged in a ring network. i = 1,2,...N, and then go back to 1.
: N could be very large.
: b. max function means
: max(x, 0) = x if x > 0,
: 0 otherwise
: c. connection matrix Wij is the connection strength between neuron i and j,
: and Wij is only a function the distance between these two neuron
| B***y
发帖数: 83 | 5 still I am not sure about the meaning of "travelling wave" in this case.
Let's think of Xi's as a vector (X1, X2, ..., X_N), then usually for a travelling wave
we mean that : There is a solution of the differential equation X(t) = (X1(t),
X2(t), ... X_N(t)) such that as t approaches to -\infty (minus infinity), there
is a limit for X(t); and as t approaches to +\infty (plus infinity), X(t) also
approaches to another limit. That's why it is called travelling wave: it travels
between two ultimate |
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