Then we do:
A - Strict stability analysis
B - growth of structure (percolation)
C - multilayer architecture
We will build a fully self-organizing critical system.
A - Strict analysis of adaptation sustainability
We have:
x_{t+1} = W_t \phi(x_t)
and scale adaptation:
\alpha_{t+1} = \alpha_t + \epsilon(\sigma_t - 1)
where:
\sigma_t = \frac{\|x_{t+1}\|}{\|x_t\|}
Fixed point
Inpatient:\sigma = 1
Means: g \rho(W) = 1
Sustainability of adaptation
Linearize:
\delta \alpha_{t+1} = \delta \alpha_t + \epsilon \frac{d\sigma}{d\alpha} \delta \alpha_t
The system is stable if:
|1 + \epsilon \frac{d\sigma}{d\alpha}| < 1
With a small \epsilon - stable.
Conclusion: the system is attracted to \lambda=0.
B - Structure growth (dynamic percolation)
Now add a topology change.
Probability of communication:
P_{ij}(t+1) = P_{ij}(t) + \mu x_i(t)x_j(t) - \gamma P_{ij}(t)
If the correlation is high, the connection increases.
Average degree: \langle k \rangle(t)
It grows until: \langle k \rangle > 1
→ appears giant component
→ structural percolation
Now the network itself creates its own connection.
C - Multilayer architecture
Enter the levels:
- Local modules
- Inter-modular communications
- Global integrator
Let's say:
W = \begin{pmatrix} W_1 & C_{12} & \dots \\ C_{21} & W_2 & \dots \\ \dots \end{pmatrix}
Criticality is maintained at every level.
Full system
We get three dynamics:
1. Quickly
x_{t+1} = W_t \phi(x_t)
2. Plasticity
W_{ij} \leftarrow W_{ij} + \epsilon x_i x_j - \beta x_i^2 W_{ij}
3. Structural Percolation
P_{ij} \text
What will happen
System:
- It builds connectivity.
- itself adjusts the spectral radius
- She Holds \lambda \approx 0
- large-scale-invariant
- energy-restricted
It's not just RNN.
It is a self-organizing critical architecture.
Deep Conclusion
If the system is long enough:
- It maximizes integration.
- It will maintain sensitivity.
- It will be resistant to noise.
- It will be computationally efficient.
This is the minimum candidate for a prototype cognitive system.
We want:
1 - To reduce everything to a single system of differential equations?
2 - Add a goal information function (maximization of mutual information)?
3Convert it to architecture AI Prototype Level?
Going further, we are already reaching the level of artificial consciousness theory.
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