I am trying to implement LtC and CfC models in Jax. However, after going through the code I found some differences between what was discussed in related publications and what was implemented here. I hope someone could clarify this for me here.
First, in the LtC implementation it seems that nominator and denominator depend on a different nonlinear mapping 'f'. In the nominator one uses an adjacency matrix which has both positive and negative connections and in denominator the adjacency matrix is strictly positive. I understand that one requires this for scaling constant to be always positive, but I am wondering why the choice was made to have effectively two different functions. Given the implementation the actual model should be written as
where only $f_1$ is a strictly positive nonlinear mapping.
Second, given this choice for LtC implementation I am not sure how can one use fitted weights from LtC model inside CfC model. to my understanding for the CfC solution to work the equivalent ODE expression should be
This leads me to the final question, that is of the subtle differences in terms of fix points of this two ODEs
$$ \dot{x} = - \left( \frac{1}{\tau} + f(x, I) \right) x + \left( \frac{1}{\tau} + f(x, I) \right) A $$
and
$$ \dot{x} = - \left(\frac{1}{\tau} + f(x, I) \right) x + f(x, I) A $$
In the first ODE $x^* = A$ is a fixed point solution, which one can also see from CfC solution. In the second equation the fix point depends properly on $f(x, I)$ and to me this seems as more flexible model. However, I am then a bit confused with the claim that the CfC solution provides the same flexibility and is equivalent to the second ODE, rather then the first one.
I would highly appreciate some clarification on this issues.
I am trying to implement LtC and CfC models in Jax. However, after going through the code I found some differences between what was discussed in related publications and what was implemented here. I hope someone could clarify this for me here.
First, in the LtC implementation it seems that nominator and denominator depend on a different nonlinear mapping 'f'. In the nominator one uses an adjacency matrix which has both positive and negative connections and in denominator the adjacency matrix is strictly positive. I understand that one requires this for scaling constant to be always positive, but I am wondering why the choice was made to have effectively two different functions. Given the implementation the actual model should be written as
$$ \dot{x} = - \left(\frac{1}{\tau} + f_1(x, I) \right) x + f_2(x, I)A$$
where only $f_1$ is a strictly positive nonlinear mapping.
Second, given this choice for LtC implementation I am not sure how can one use fitted weights from LtC model inside CfC model. to my understanding for the CfC solution to work the equivalent ODE expression should be
$$ \dot{x} = - \left(\frac{1}{\tau} + f(x, I) \right) x + \left( \frac{1}{\tau} + f(x, I) \right)A $$
where $f$ has to be identical in both terms.
This leads me to the final question, that is of the subtle differences in terms of fix points of this two ODEs
$$ \dot{x} = - \left( \frac{1}{\tau} + f(x, I) \right) x + \left( \frac{1}{\tau} + f(x, I) \right) A $$
and
$$ \dot{x} = - \left(\frac{1}{\tau} + f(x, I) \right) x + f(x, I) A $$
In the first ODE $x^* = A$ is a fixed point solution, which one can also see from CfC solution. In the second equation the fix point depends properly on $f(x, I)$ and to me this seems as more flexible model. However, I am then a bit confused with the claim that the CfC solution provides the same flexibility and is equivalent to the second ODE, rather then the first one.
I would highly appreciate some clarification on this issues.