Stability criteria for the contextual emergence of macrostates in neural networks

Abstract

More than thirty years ago, Amari and colleagues proposed a statistical framework for identifying structurally stable macrostates of neural networks from observations of their microstates. We compare their stochastic stability criterion with a deterministic stability criterion based on the ergodic theory of dynamical systems, recently proposed for the scheme of contextual emergence and applied to particular inter-level relations in neuroscience. Stochastic and deterministic stability criteria for macrostates rely on macro-level contexts, which make them sensitive to differences between different macro-levels.

• Network models

Notes

Notes

[1] A non-technical presentation of the detailed argumentation can be found in Atmanspacher (2007). The KMS condition induces a partition into equivalence classes of mechanical states defining statistical states whose mean energy can be assigned a particular temperature.

[2] Another important example are Hebbian auto-associator networks (Hopfield 1982) trained with random patterns that have been investigated by Amari and Maginu (1988).

[3] We present Amari's proposal in our own notation here and in the following.

[4] This situation resembles quantum mechanical complementarity where observables such as the position and momentum of an electron refer to different, mutually excluding measurement contexts. For a treatment of complementary observables in classical systems see beim Graben and Atmanspacher (2006).

[5] An illustrative example is the particle-wave dualism in quantum mechanics: An electron behaves as a particle in one particular measurement context and as a wave in another.