*This follows from my other posts: https://math.stackexchange.com/q/4451013/577710 https://math.stackexchange.com/q/4451150/577710. Since a question there (Q3) is a bit complex, I decided to have a new post for more discussions on it.*
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?
The author says
> 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.
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*Clarification of notations in the paper*:<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:*
>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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[![fig0][1]][1]
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[![fig00][2]][2]
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[![fig1][3]][3]
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[![fig2][4]][4]
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*General questions*: The paper 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 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(?) and resulting in a stochastic diff equ.
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*General background*: The following is an excerpt from a paper about stochastic PDEs in hydrodynamic chaos, which seems to be relevant to the above problem of chaos in neural networks.
>[![fig3][5]][5]
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*Specific questions*: When $J_{ij}$ is symmetric, there is an energy function (eq 3), 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?
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**References**:<br>
Path integral approach to random neural networks https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.062120 this is what is cited by the author about $g$, $h_i$, and **where most of the excerpts above come from**.<br>
Chaos in Random Neural Networks https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.259 this is a paper that gives solution to the nonlinear ODEs.
[1]: https://i.sstatic.net/oFTPZ.png
[2]: https://i.sstatic.net/fuKYz.png
[3]: https://i.sstatic.net/3tWGy.png
[4]: https://i.sstatic.net/hB1cR.png
[5]: https://i.sstatic.net/dBfK5.png