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# Equations

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Equations
0
Section
LaTeX - Math
Description
Equation #
LaTeX
Ontology terms
Notes
A variational formulation
4
path of a phenotype
phenotype
path
relevant variables at the evolutionary scale
Evolution
EcoEvoDevo
bottom-up causation is simply the application of a reduction operator to select variables that change very slowly
causality
Top-down causation entails a specification of fast phenotypic trajectories in terms of slow genotypic variations, which are grouped into populations, , according to the influences they exert on each other
causality
Particular partitions
17
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
Evolution
states
flow
Stochastic
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
flow
solenoidal
the history or path of a time varying state
path
time
State
the paths of a state-dependent flow are determined by state-dependent flow
State
Parameters of the state-dependent flow
State
parameter
flow
Initial states of parameters of the state-dependent flow
parameter
State
flow
Partition of states comprising the external, sensory, active and internal states of a phenotype
External State
Active States
Internal State
Sense State
phenotype
Partition
Sensory and active states constitute blanket states
Sense State
Active States
Blanket State
phenotypic states comprise internal and blanket states
phenotype
Internal State
Blanket State
The autonomous states of a phenotype are not influenced by external states:
State
phenotype
External State
independent random (Wiener) fluctuations
random
Variance of independent random (Wiener) fluctuations is 2Γ.
Variance
random
gradient flow (depends upon the amplitude of random fluctuations)
solenoidal flow (circulates on the isocontours of potential function called self-information)
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
Ensemble dynamics and paths of least action
10
self-information or surprisal of a state; namely, the implausibility of a state being occupied. When the state is an allele frequency and evolves according to Wright–Fisher dynamics, this is sometimes referred to as an ‘adaptive landscape’
3
the Lagrangian, which is the surprisal of a generalised state; namely, the instantaneous path associated with the motion from an initial state. In generalised coordinates of motion, the state, velocity, acceleration, etc are treated as separate (generalised) states that are coupled through the flow
3
the surprisal of a path is called action, namely, the path integral of the Lagrangian.
3
Generalised states afford a convenient way of expressing the path of least action in terms of the Lagrangian
4
Denoting paths of least action with boldface, this is sometimes described as convergence to the path of least action, in a frame of reference that moves with the state of generalized motion
5
We can also express the conditional independencies implied by a particular partition using the Lagrangian of generalized states. Because there are no flows that depend on both internal and external states, external and internal paths are independent, when conditioned on blanket paths:
6
states
generalised states
paths
λ is the Lagrange multiplier, which ensures the generalized motion of states corresponds to the state of generalized motion. When λ is suitably small, solutions of the implicit generalized equations of motion converge (almost) instantaneously to the path of least action.
Different kinds of things
9