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Equations
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Equation #
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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
Group of phenotypes, reflecting particular states in the ensemble of particles in population
7
Expression of the dynamics of each particle in terms of its sensory states—that depend upon the blanket states of other particles—and autonomous states—that only depend upon the states of the particle in question
8
Expression of the dynamics of each particle in terms of its sensory states—that depend upon the blanket states of other particles—and autonomous states—that only depend upon the states of the particle in question
8
the states in the particular ensemble have to be the states of some ‘thing’; namely, the states of a particle at a lower scale. This means that states must be the states of particles (e.g., phenotypic states) that constitute the particular states at the next scale (e.g., phylogenetic states) This recursive truism can be expressed in terms of grouping G operator—that creates particles—and a reduction R operator—that picks out certain particular states for the next scale
9
The composition of the reduction and grouping operators can be construed as mapping from the states of particles at one scale to the next or, equivalently, from particular states at one scale to the next. In short, creating particles of particles, namely, populations.
10
grouping operator G that partitions the states at the i-th scale of analysis into N particles on the basis of the sparse coupling implied by a particular partition.
a reduction R operator—that picks out certain particular states for the next scale.
Figure 2: schematic showing the hierarchical relationship between particles at scales i and i+1. For clarity, only sensory and autonomous states are illustrated in blue and pink, respectively. Note that each variable is a (very large) vector state that itself is partitioned into multiple vector states. At scale i+1, each particle represents an ensemble, the elements of which are partitioned into autonomous and sensory subsets. At scale i, each particle represents an element of an ensemble, which is itself partitioned into sensory and autonomous subsets. The slow states of each element (e.g., phenotype) are recovered by the reduction operator R, to furnish the states at the ensemble level (e.g., genotype). A key feature of this construction is that it applies recursively over scales.
At the phylogenetic scale — a population corresponds to a set of particular kinds (i.e., sensory and autonomous kinds)
Natural selection: a variational formulation
11
if, at non-equilibrium evolutionary steady-state, the likelihood of an agent’s genotype is proportional to the likelihood of its phenotypic trajectory (where \ denotes exclusion), then the following holds (Equation 12).
11
An agent’s autonomous dynamics can be cast as a gradient descent on a Lagrangian, whose path integral (i.e., action) corresponds to negative fitness. This Lagrangian depends upon the flow parameters (and initial states) supplied by the genotype. The agent’s genotype can then be cast as a stochastic gradient descent on negative fitness (i.e., genetic drift)
12
The existence of a nonequilibrium evolutionary steady-state solution to the density dynamics (at both scales) allows us to express the fast and slow dynamics of agents and their autonomous states in terms of Helmholtz decompositions.
13
The gradients of surprisal at the slow scale, with respect to any given agent’s ‘kind’ or genotype, are the gradients of action
14
In the limit of small fluctuations, the autonomous paths become the paths of least action; i.e., when the fluctuations take their most likely value of zero.
15
the genotypic state of the n-th agent
the phenotypic state of the n-th agent
the action (i.e., negative fitness) scoring the accumulated evidence. This evidence is also known as a marginal likelihood because it marginalises over external dynamics; i.e., other agents.
a phenotype’s generative model
the extended genotype
the extended genotype covers both the genetic and epigenetic specification of developmental trajectories and the initial conditions necessary to realise those trajectories, including external states conditions necessary for embryogenesis.
The sentient phenotype
5
variational free energy functional of Bayesian beliefs—about external states—encoded by their internal phenotypic states, under a generative model encoded by their genotype.
16
We can consider the conditional density over external paths as being parameterised by internal paths. We will call this a variational density (noting from 6 that internal paths are conditionally independent of external paths)
17
The Lagrangian and variational free energy share the same minima, where their gradients vanish
18
Density over external states parameterised by internal states
a generative model encoded by their genotype
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