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Some more information and answers are here: https://math.stackexchange.com/q/4451013/577710 .

It is said that

..the largest Lyapunov exponent, which measures the average exponential rate of divergence or convergence of nearby network states.

Lyapunov exponents (LEs) measure how fast nearby trajectories or flows diverge from each other.
Q1: Why does the largest LE measure the average divergence rate, instead of the mean LE?

My thought is that the LEs are somehow eigenvalues of a matrix involved in solving the ODEs $$\tau\frac{dh_i}{dt} = -h_i + \sum_{j=1}^N J_{ij} \phi(h_j),$$ so the solutions would possibly look like a linear combination of $e^{\lambda_i t}$. Since $e^{a t}\gg e^{b t}$ when $a > b$, as $t\to\infty$, the term with the largest LE will be much larger than other terms despite the coefficients, and therefore dominate.

I have not solved the ODEs so I am not sure whether my thought is correct.
How can I strictly solve the equation and answer my question?


This seems to have verified my guess. But the details of calculation are still unclear. It is a non-linear ODE and the solution is possibly more complex than that of a linear ODEs, for which the monotonicity of $e^{\lambda_i t}$ straightforwardly gives the result.

So perhaps the question can be restated as how solutions of non-linear ODEs differ from those of linear ODEs, and whether we can still use the linear algebra method of eigenvalues and eigenvectors to solve the former.
One idea is to linearize the ODEs near the fixed point, with Jacobian (of which LEs are, roughly speaking, eigenvalues).
Even if we can do so, the conclusion seems to be valid only near the fixed point; while here we need to consider $t\to\infty$ for chaos (unstable), and therefore it is almost certain that we would go away from the fixed point.


Q2: Why do the other LEs matter for characterizing chaos?

Q3: We know $g$ is proportional to (sqrt of) the variance of $J$'s every entry ($J_{ij} \text{~} \mathcal{N}(0, g^2/N)$).
Why is it also positively related to the variance of $h_i$. In other words, why stronger coupling results in stronger neuronal signals, from a math perspective?

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    $\begingroup$ I think this question might be more appropriate for math.stackexchange.com . $\endgroup$ May 15, 2022 at 14:06
  • $\begingroup$ math.stackexchange.com/q/4451013/577710 $\endgroup$ May 15, 2022 at 14:25
  • $\begingroup$ Can you give a reference for what you're referring to in Q3? $\endgroup$
    – stafusa
    May 15, 2022 at 18:19
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    $\begingroup$ Path integral approach to random neural networks journals.aps.org/pre/abstract/10.1103/PhysRevE.98.062120 this is what is cited by the author about $g, h_i$; Chaos in Random Neural Networks journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.259 this is a paper that gives solution to the nonlinear ODEs (also see the post in mathstackexchange which has more information). $\endgroup$ May 15, 2022 at 18:27
  • $\begingroup$ Ok, this is really a separate question, you did well to give it its own question at Math SE. Unfortunately I won't be able to study the paper and don't feel I can answer Q3 from a superficial reading. So my answer here remains partial — or, especially if you get a good response over there, you might remove Q3 from the question here, as long as no one else answered it. $\endgroup$
    – stafusa
    May 15, 2022 at 21:16

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Q1: Why does the largest LE measure the average divergence rate, instead of the mean LE?

Like you write, it's because the largest exponent dominates.

how solutions of non-linear ODEs differ from those of linear ODEs, and whether we can still use the linear algebra method of eigenvalues and eigenvectors to solve the former.

The solutions of course differ as, for instance, typical linear ODEs can't exhibit chaos. Nonetheless you can still use the linear algebra arguments, because Lyapunov exponents describe (infinitesimal) local behavior, which means that at every point of the calculation you can consider the system's local linear approximation.

Q2: Why do the other LEs matter for characterizing chaos?

For one, it can be used to talk about things such as hyperchaos (= 2+ positive LEs). And, even more importantly, you'll usually want to see a negative exponent, to have bounded trajectories, and in a Hamiltonian system, they all sum to zero.

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  • $\begingroup$ > Q3: why stronger coupling results in stronger neuronal signals, from a math perspective? That I can't answer right now. $\endgroup$
    – stafusa
    May 15, 2022 at 18:18
  • $\begingroup$ Q1 The second largest LE can be only slightly less than the largest LE. Q2 The author says that a dynamical system with a symplectic structure (like a Hamiltonian system) has point-symmetric Lyapunov spectrum, as you mentioned. But why? $\endgroup$ May 15, 2022 at 18:21
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    $\begingroup$ @CharlieChang Physically, in a Hamiltonian system no energy is lost or gained, and that means that the sum of its LEs has to be zero, because a sum greater than zero means energy is being in average injected into the system and diverges asymptotically, while a sum lower than zero, means that energy is being lost in average and goes asymptotically zero. A mathematical take is given in this answer: physics.stackexchange.com/q/214007/75633 . $\endgroup$
    – stafusa
    May 15, 2022 at 18:44
  • $\begingroup$ So that is saying that energy is conserved in this system due to the reversal symmetry w.r.t time (similar to time translation invariance physics.stackexchange.com/a/708414/273056, and in contrast to quantum hall that breaks time-reversal (TR) symmetry)? However, it seems energy conservation comes from (continuous) time translation symmetry, instead of (discrete) time reversal symmetry (which 'doesn't lead to conserved quantities.') researchgate.net/post/… $\endgroup$ May 16, 2022 at 3:55
  • $\begingroup$ ..'if H is not an explicit function of time then the same argument shows that you will have time-translation symmetry' 'the essential starting point is to correctly identify the elements of the theory that embodies the time invariance. In the Newtonian description, this is the force...Therefore by differentiating with respect to time the sum of kinetic and potential energy is trivial to check that ...' physics.stackexchange.com/a/614764/273056 // So energy conservation here may be about time-translation. //Your link also says 'continuous-time-translation is a symplectic transformation' $\endgroup$ May 16, 2022 at 4:02

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