The evolution of these sparsely coupled states can be expressed as a Langevin or stochastic differential equation; namely, a high dimensional, nonlinear, state-dependent flow plus independent random (Wiener) fluctuations, , with a variance of 2Γ

1

The flow of Equation 1 can be expressed using the Helmholtz decomposition (expressing the flow as a mixture of a dissipative, gradient flow and a conservative, solenoidal flow)

2

the history or path of a time varying state

the paths of a state-dependent flow are determined by state-dependent flow

Parameters of the state-dependent flow

Initial states of parameters of the state-dependent flow

Partition of states comprising the external, sensory, active and internal states of a phenotype

Sensory and active states constitute blanket states

phenotypic states comprise internal and blanket states

The autonomous states of a phenotype are not influenced by external states:

independent random (Wiener) fluctuations

Variance of independent random (Wiener) fluctuations is 2Γ.

gradient flow (depends upon the amplitude of random fluctuations)

solenoidal flow (circulates on the isocontours of potential function called self-information)

the nonequilibrium steady-state density (NESS density)

Figure 1: schematic (i.e., influence diagram) illustrating the sparse coupling among states that constitute a particular partition at two scales