THE CONSTRUCT OF CRITICALITY
Okay. Build a model that adjusts itself to criticality
That is, it brings the maximum Lyapunov indicator to zero:
\lambda_{\max} \rightarrow 0
This is no longer analysis; it is design.
1 - Basic network dynamics
Take the discrete model: x(t+1) = W(t)\,\phi(x(t)) + \eta(t)
- W(t) - weights that will adapt
- \phi — tanh
- \eta - low noise
2 - Purpose of self-organization, We know:
\lambda_{\max} \approx \ln(g\rho(W))
Critical condition: g\rho(W) = 1
Therefore, it is necessary to dynamically adjust the scale of the weights.
3 - Global adaptation (simple version)
Enter the scaling parameter \alpha(t):
W(t) = \alpha(t)\tilde{W} where \tilde{W} — fixed random structure.
Adaptation rule: \alpha(t+1) = \alpha(t) + \epsilon \big( \sigma(t) - 1 \big)
where: \sigma(t) = \frac{\|x(t+1)\|}{\|x(t)\|}
If activity fades → zoom in.
If explodes → reduce.
This brings the system to: \lambda_{\max} \to 0
4 - Local (biologically plausible) version
Each neuron regulates its excitability:
W_{ij}(t+1) = W_{ij}(t) + \epsilon x_i(t)x_j(t) - \gamma x_i(t)^2 W_{ij}(t)
The combination:
- Hebbian Reinforcement
- Normalization
It stabilizes the spectral radius.
5 - Add energy limit
Enter "metabolic price": E(t) = \sum_i x_i(t)^2
And fine: W_{ij} \leftarrow W_{ij} - \beta E(t) W_{ij}
This prevents hypersynchronization.
6 - Final System
We get three contours:
- Fast neural dynamics
- Medium plasticity
- Slow normalization of energy
Multi-level feedback.
7 - What will happen? If everything is right:
- spectral radius → 1
- \lambda_{\max} \to 0
- Neural avalanches
- The distribution will be graded
The network itself will find a critical point.This is called: Self-Organized Criticality (SOC)
8 - Why is it serious
Such a system:
- as sensitive as possible
- Stable
- large-scale-invariant
- energy-restricted
This is a candidate for a minimal model of a living cognitive system.
key step.
A) Provide a rigorous stability analysis of this adaptive system
B) Add structural percolation (growth of bonds)
C) Expand to a multilayered model close to the cerebral cortex
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