If we want to really understand the boundary of "life-chaos", without Lyapunov's indicator anywhere. This is a strict tool for determining where the system is attenuated, where it is stable, and where it goes into exponential instability.
1 - Definition
The maximum Lyapunov exponent \lambda_{\max} shows how a small perturbation \delta x behaves: \delta x(t) \sim e^{\lambda t}
- \lambda < 0 →
- \lambda = 0 → critical boundary
- \lambda > 0 → chaos
Criticism = \lambda_{\max} \approx 0
2 - For recurrent neural network
Model No.: x(t+1) = W \phi(x(t))
Linearization:
\delta x(t+1) = W D_\phi \delta x(t), where: D_\phi = \text{diag}(\phi'(x))
Then the Lyapunov indicator:
\lambda_{\max} = \lim_{T\to\infty} \frac{1}{T} \sum_{t=1}^{T} \ln \| W D_\phi(t) \|
3 - Approximate estimate
The average gain of g = \langle \phi'(x) \rangle, Toto:
\lambda_{\max} \approx \ln (g \rho(W))
This is the key formula.
4 - Criticality relationship
\lambda_{\max} = 0 \Rightarrow g \rho(W) = 1
Here's where they converge:
- spectral radius
- branching ratio
- Percolation
- Phase transition
This is a strict transition criterion.
5 - Geometric meaning
The Lyapunov indicator measures:
- trajectories divergence rate
- Attractor stability
- effective phase volume
In the brain:
- \lambda < 0 → Sleep, Suppression
- \lambda \approx 0 → Conscious Dynamics
- \lambda > 0 → pathological instability
6 - Communication with information
Information productivity is associated with the sum of the positive indicators of Lyapunov (entropy of Kolmogorov-Blue):
h_{KS} = \sum_{\lambda_i > 0} \lambda_i
At the critical point:
- h_{KS}
- The system does not lose its stability.
This is the optimal calculation.
7 - Connection with percolation
Percolation gives a structural threshold: \langle k \rangle > 1
Lyapunov gives a dynamic threshold: \lambda_ZQ\max-= 0
Only if you do both:
- There is global connectivity
- Dynamics at the Border of Chaos
8 - Universal Brain Condition
A conscious state is possible if:
\langle k \rangle > 1 \quad \text{and} \quad \lambda_{\max} \approx 0
This brings together:
- Topology
- dynamics
- Computational Efficiency
9 - Deep Conclusion
The Lyapunov index is a thermometer of complexity.
It shows how much the system:
- Sensitive
- Stable
- able to store and process information
Consciousness is not just complexity.
This is the state where \lambda_QQQ\max is close to zero.
1 - Display the Lyapunov spectrum for a large random network (dynamic mean field method)
2 - Linking the Lyapunov score to energy consumption of the brain
3Build a self-organizing network that sets itself up \lambda \to 0
We will be working on a publication level theory.