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p:questions1 [2020/04/03 13:11] – [Question 1] cjjp:questions1 [2020/04/04 06:07] (current) – [Question 6] cjj
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 ======Questions from TK 2020-04-01====== ======Questions from TK 2020-04-01======
 +<color darkgreen>★ Replies are in green.</color>
 =====Question 1===== =====Question 1=====
  
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 =====Question 2===== =====Question 2=====
-The pca in your fig. 1 for both cases, show a good conservation for different sizes, although the two cases have different slope. So can these two mice in different state? +The PCA in your fig. 1 for both cases, show a good conservation for different sizes, although the two cases have different slope. So can these two mice in different state? 
 + 
 +<color darkgreen> 
 +It seems the slopes for the scaling regimes of both cases are about $1/2$, which is also the value quoted in Bialek's talk. However, the deviation also is in the large-eigenvalue limit of the plots. Maybe, these largest eigenvalues are related to the functional states of the mice thus differ from each other. While, the scaling regime is related to some fundamental life dynamics of mice thus stays the same. 
 +</color>
 =====Question 3===== =====Question 3=====
 > However, whether the distributions of $m_i$ and $C_{i,j}$ are enough or the network topology also plays an important role does still need to be checked. Shuffling or re-sampling $m_i$ and $C_{i,j}$ from the observed distributions is a way of checking this. However, they are tricky since the result may not be a valid combination that can be generated by a distribution of system configurations. This is mentioned in http://arxiv.org/abs/0912.5409 and they coped with it by checking the consistency of all marginal distributions of spin pairs of the system and redraw from the distributions if any of the marginal distributions is invalid. I have improved their method so that redrawing is not necessary and all marginal distributions of spin pairs are valid. However, this does not guaranty triplets or higher marginal distributions are all valid and the shuffled $m_i$ and $C_{i,j}$ can still be un-physical. This makes me wonder if this is one of the reason why this paper was not published. And, we need to think of different ways of perturbing the system to find the minimal criteria of when the collective properties of the system are preserved. > However, whether the distributions of $m_i$ and $C_{i,j}$ are enough or the network topology also plays an important role does still need to be checked. Shuffling or re-sampling $m_i$ and $C_{i,j}$ from the observed distributions is a way of checking this. However, they are tricky since the result may not be a valid combination that can be generated by a distribution of system configurations. This is mentioned in http://arxiv.org/abs/0912.5409 and they coped with it by checking the consistency of all marginal distributions of spin pairs of the system and redraw from the distributions if any of the marginal distributions is invalid. I have improved their method so that redrawing is not necessary and all marginal distributions of spin pairs are valid. However, this does not guaranty triplets or higher marginal distributions are all valid and the shuffled $m_i$ and $C_{i,j}$ can still be un-physical. This makes me wonder if this is one of the reason why this paper was not published. And, we need to think of different ways of perturbing the system to find the minimal criteria of when the collective properties of the system are preserved.
  
 What do you mean by marginal distribution? I thought you just arbitrarily select 64, 32 neurons but keep the experimental data of $C_{i,j}$ and $m_i$. I thought that is what Kay did by cutting off the first 8 neurons or the last 4 neurons. What do you mean by marginal distribution? I thought you just arbitrarily select 64, 32 neurons but keep the experimental data of $C_{i,j}$ and $m_i$. I thought that is what Kay did by cutting off the first 8 neurons or the last 4 neurons.
 +
 +<color darkgreen>
 +Randomly selecting neurons and keeping their $m_i$ and $C_{i,j}$ is kind of coarse-graining the network but preserving its structure. For selected neurons $i$ and $j$, their $m_i$, $m_j$ and $C_{i,j}$ are all preserved. But, if we are generating new values from the distributions of $m$ and $C$ then the original network structure will be lost.
 +</color>
 +
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 +The "marginal distribution" for a spin pair $\langle i,j\rangle$ is the joint probability distribution of $\sigma_i$ and $\sigma_j$, $P(\sigma_i,\sigma_j)$, which is obtained from the state distribution by integrating out the all other spins.
 +\[
 +P(\sigma_i,\sigma_j) = \prod_{k\neq i,j}\left(\sum_{\sigma_k}\right) P(\sigma_1,\sigma_2,...,\sigma_N)
 +\]
 +With the marginal distribution $P(\sigma_i,\sigma_j)$ ($\geq 0$, since probability cannot be negative), we can get the values $m_i=P(1,0)+P(1,1)$, $m_j=P(0,1)+P(1,1)$, and $C_{i,j}=P(1,1)$. Since these values are not fully independent, random drawing can produce invalid values that no possible $P(\sigma_i,\sigma_j)$ exists. For example, if $m_i=1$ and $m_j=1$, then $C_{i,j}$ cannot be anything other than $1$.
 +</color>
 +
 =====Question 4===== =====Question 4=====
 In the paper you quoted, they enlarge the system to 120 neurons by  a quote in page 10 In the paper you quoted, they enlarge the system to 120 neurons by  a quote in page 10
 "We thus generated several synthetic networks of 120 neuons  by randomly choosing once more out of the distribution "We thus generated several synthetic networks of 120 neuons  by randomly choosing once more out of the distribution
-of hi and Cij observed experimentally."  --  have you done this to study large system?+of $m_i$ and $C_{i,j}$ observed experimentally."  --  have you done this to study large system? 
 + 
 +<color darkgreen> 
 +I have attempted this. However, after the drawing, the Boltzmann learning that is performed to obtain $h_i$ and $J_{i,j}$ does not converge as well as the original system so I am leaving them running on the cluster. I will check the results after I am back in my office next week. 
 +</color> 
 + 
 +<color darkgreen> 
 +However, the fact that they are not converging well may be an indication that the generated $m_i$ and $C_{i,j}$ may not be fully valid since we only ensure its validity with spin pairs and not with higher order of spin combinations. When the values $m_i$ and $C_{i,j}$ are not valid, no solution of $h_i$ and $J_{i,j}$ can exactly reproduce $m_i$ and $C_{i,j}$ and there will be a none-zero minimum for the error. 
 +</color>
 =====Question 5===== =====Question 5=====
-We can still try to follow our previous method, keeping the rank and shuffle only C(ij) within a small interval, similarly for mi.+We can still try to follow our previous method, keeping the rank and shuffle only $C_{i,j}$ within a small interval, similarly for $m_i$. 
 If we increase the size of the interval, then we will get to completely reshuffle. Then  we can check how sensitive  our result to the network. If we increase the size of the interval, then we will get to completely reshuffle. Then  we can check how sensitive  our result to the network.
 +
 +<color darkgreen>
 +This should be OK. But, as above, we need to check the validity of resulting combination of $m_i$ and $C_{i,j}$.
 +</color>
 =====Question 6===== =====Question 6=====
 In this Bialek's paper for 40 neurons, a statement is different from what I saw before:  In this Bialek's paper for 40 neurons, a statement is different from what I saw before: 
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 TK TK
  
 +<color darkgreen> 
 +I couldn't find a corresponding statement in the [[https://doi.org/10.1016/j.neuron.2017.10.027|78-neuron paper]], either. 
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