*Context: Q3 in [How to understand the largest Lyapunov exponent?][1]*<br>
>[![fig9][9]][9]

*I have also asked [this question in MSE][3].*



We know $g$ is proportional to (square root of) the variance of $J$'s every entry ($J_{ij}\sim \mathcal{N}(0,g^2/N)$).<br>
Why is it also positively related to the variance of $h_i$.<br>
In other words, why stronger coupling results in stronger neuronal signals, from a math perspective, in particular, for large systems?


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*Clarification of notations in the paper [(Crisanti, 2018)][4]*:<br>
<li>$S_i^a=\phi (gh^a_i)=\phi (g h_i(t_a))$ ($a,b,c,d$ are indexes for time), i.e. $S$ is the signal/output of neuron.</li>
<li>$C_{ab}=\sum \limits _iS_aS_b/N $, i.e. $C$ is autocorrelation of neuronal signals at times $t_a,t_b$. (I think $\sum \limits _iS_aS_b=\sum \limits _i(S_a-\langle S_a\rangle )(S_b-\langle S_b\rangle )$, since the averages of $S_a,S_b$ are both $0$, for the neuronal signals are random centered at zero.)</li><br>

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*Noteworthy details (Crisanti, 2018):* 
>If the matrix $J_{ij}$ is symmetric, i.e., $J_{ji}=J_{ij}$, [...] The dynamics hence converges toward stable fixed points.<br>
If the matrix is nonsymmetric, [...] a richer steady-state behavior emerges: besides fixed points, limit cycles and chaotic behavior are also possible.
<br>
<br>
The assumption of zero average implies that there is not a preferred type of synaptic connection... inhibitory ($N\overline J_{ij}<0$) or excitatory ($N\overline J_{ij}>0$). 
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<br>
[![fig0][5]][5]
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<br>
[![fig00][6]][6]
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<br>
[![fig1][7]][7]
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<br>
[![fig2][2]][2]



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*General questions*: The paper (Crisanti, 2018) is not easy for me. While I do not expect others to read the paper for me, how could I understand a paper like this? For example, what step should I follow, what prerequisite knowledge should I familiarize myself with?



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*Brief summary of topics and methods*: The paper (Crisanti, 2018) uses some stochastic diff eq (SDE) (part of SD), functional, Fourier transform, etc.<br>
The research seems to be within the domain of  **stochastic dynamics (SD)** in statistical mechanics. The author here adopts a frequently used path integral method; while I am not familiar with the method, it seems to be about variational methods, which consider a functional (typically an integral; the author mentions 'action') and a small variation $\delta$, introduced to eq6 (Crisanti, 2018)(?) and resulting in a stochastic diff eq.

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*General background*: The following is an excerpt from a paper (Weinan E, 2000) about stochastic PDEs in hydrodynamic chaos, which seems to be relevant to the above problem of **chaos in neural networks**.
>[![fig3][8]][8]


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*Specific questions*: When $J_{ij}$ is symmetric, there is an energy function (eq 3)(Crisanti, 2018), while when $J_{ij}$ is asymmetric, there is not, why? And why is the energy function in the form of eq3, and its relaxation in the form of eq2 (Crisanti, 2018)?



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**References**:<br>
A. Crisanti and H. Sompolinsky, *Path integral approach to random neural networks*, 2018 https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.062120 this is what is cited by the original paper (excerpted in [the previous post][1]) about $g$, $h_i$, and *where most of the excerpts above come from*.

H. Sompolinsky, A. Crisanti, and H. J. Sommers, *Chaos in Random Neural Networks*, 1988 https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.259 this is a paper that gives solution to the nonlinear ODEs. 

Weinan E, *Stochastic PDEs in Turbulence Theory*, 2000

NIPS 2006 Workshop on
*Dynamical Systems, Stochastic Processes and Bayesian Inference* www0.cs.ucl.ac.uk/staff/c.archambeau/dsb.htm <br>
*Path Integral Method for Estimation of Time Series* www0.cs.ucl.ac.uk/staff/c.archambeau/accepts/restrepo.pdf


  [1]: https://math.stackexchange.com/q/4451013/577710
  [2]: https://i.sstatic.net/hB1cR.png
  [3]: https://math.stackexchange.com/q/4451150/577710
  [4]: https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.062120
  [5]: https://i.sstatic.net/oFTPZ.png
  [6]: https://i.sstatic.net/fuKYz.png
  [7]: https://i.sstatic.net/3tWGy.png
  [8]: https://i.sstatic.net/dBfK5.png
[9]:https://i.sstatic.net/xOm0N.png