# A Variational Synthesis of Evolutionary and Developmental Dynamics

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## Abstract

**:**

Dedicated to the Memory of John O. Campbell. |

## 1. Introduction

**From the perspective of the genotype**, we can consider evolution as belief-updating over generations, where the belief in question corresponds to a probability density over extended genotypes (henceforth, genotype). This belief-based model of allelic change is analogous to treatments of evolution in terms of changes in allele frequencies from generation to generation [15]. This belief updating can be described by the probability of a genotype appearing in subsequent generations, in a way that depends lawfully on the marginal likelihood of extended phenotypes (henceforth, phenotype) in the current generation. The basic idea is that the genotype parameterises or encodes a generative model, which the phenotype uses to infer and act on its environment. On this view, evolution can be regarded as testing hypotheses—in the form of generative models—that this kind of phenotype can persist in this environment. These hypotheses are tested by exposing the phenotype to the environment and are rejected if the phenotype ‘strays from the path’ of a persistent phenotype. In this way, the evolutionary process selects models or hypotheses about persistent phenotypes for which it has the greatest evidence. In short, natural selection is just Bayesian model selection [25,26,31,32].

**From the perspective of a phenotype**, each conspecific is equipped with a generative model and initial conditions that underwrite its epigenetic, developmental and ethological trajectories. The states of the phenotype trace out a path through state-space over its lifetime. These phenotypic states encode or parameterise beliefs about environmental states—and the way the phenotype acts. This parameterization leads to active inference and learning, in which the phenotype tries to make sense of its world and—through a process of belief updating—to realise the kind of creature it thinks it is. (We use the term ‘thinks’ in a liberal sense here and do not mean to imply that all living entities have explicit existential thoughts.) More precisely, what we mean is that these entities behave as if they hold a set of beliefs about the sort of entity they are (e.g., the meta-Bayesian stance as considered in [33]). In virtue of its genetic endowment, it thinks it is a persistent phenotype. If endowed with a good generative model of its environment [34], it will persist and supply evidence of its ‘fit’ to the environment (i.e., ‘fitness’); namely, evidence (i.e., marginal likelihood) that has been accumulated by the slow evolutionary process.

## 2. A Variational Formulation

**RG**for Renormalisation Group) that maps the path of a phenotype $\tilde{\pi}$ to relevant variables at the evolutionary scale $\overline{\pi}=R\circ \tilde{\pi}$. On this view, bottom-up causation is simply the application of a reduction operator, $R\circ \tilde{\pi}$, to select variables that change very slowly. Top-down causation entails a specification of fast phenotypic trajectories in terms of slow genotypic variations, which are grouped into populations, $G\circ \overline{\pi}$, according to the influences they exert on each other. The implicit separation into fast and slow variables can be read as an adiabatic approximation [41] or—in the sense of synergetics—into fast (dynamically stable) and slow (dynamically unstable) modes, respectively [42]. This separation can also be seen in terms of vectorial geometric formulations [43]. Please see [21], who deal carefully with the separation of time scales by analogy with temporal dilation in physics. Intuitively, this analogy rests upon the idea that time can be rescaled, depending upon whether we take the perspective of things that move quickly or slowly.

#### 2.1. Particular Partitions

**States**: $x=(\eta ,s,a,\mu )$. States comprise the external, sensory, active and internal states of a phenotype. Sensory and active states constitute blanket states $b=(s,a)$, while phenotypic states comprise internal and blanket states, $\pi =(b,\mu )=(s,\alpha )$. The autonomous states of a phenotype $\alpha =(a,\mu )$ are not influenced by external states:**External states**respond to sensory and active states. These are the states of a phenotype’s external milieu: e.g., econiche, body, or extracellular space, depending upon the scale of analysis.**Sensory states**respond to fluctuations in external and active states: e.g., chemo-reception, proprioception, interception.**Active states**respond to sensory and internal states and mediate action on the environment, either directly or vicariously through sensory states: e.g., actin filaments, motor action, autonomic reflexes.**Internal states**respond to sensory and active states: e.g., transcription, intracellular concentrations, synaptic activity.

#### 2.2. Ensemble Dynamics and Paths of Least Action

#### 2.3. Different Kinds of Things

**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. In other words, we group states into an ensemble of particles, where each particle has its own internal and blanket states. With a slight abuse of the set builder notation:

**G**operator—that creates particles—and a reduction

**R**operator—that picks out certain particular states for the next scale:

**R**typically selects relevant variables whose slow fluctuations contextualise dynamics at the scale below. Here,

**R**simply recovers the states of a particle that are time invariant or that vary slowly with time (i.e., the initial states and flow parameters). This separation of timescales means that the lifetime of a particle (e.g., phenotype) unfolds during an instant from the perspective of the next scale (e.g., evolution). The separation of timescales could have been achieved without the grouping (partitioning) operator. We could simply have projected onto the eigenvectors of a dynamical system’s Jacobian, effectively taking linear (or nonlinear) mixtures of our system to arrive at fast and slow coordinates. However, all we would be left with are fast and slow continuous variables that have nothing of the character of the individuals, phenotypes, or populations in a system. In short, the grouping operator is key in identifying fast and slow ‘things’—as opposed to just fast and slow coordinates of a dynamical system.

**Sensory kinds**mediate the effects of external kinds on the internal members of the population in question: e.g., nutrients or prey.**Active kinds**mediate the effects of internal kinds on external kinds: e.g., agents who mediate niche construction, from a molecular through to a cultural level, depending upon the scale of analysis.**Internal kinds**influence themselves and respond to changes in sensory and active kinds.

#### 2.4. Natural Selection: A Variational Formulation

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

## 3. The Sentient Phenotype

**Lemma**

**2.**

**Proof.**

**Remark**

**2.**

## 4. Variational Recipes

- First, generate an ensemble of particles (i.e., extended phenotypes) by sampling their flow parameters and initial states (i.e., extended genotypes) from some initial density.
- For each particle, find the path of least action using a generalised Bayesian filter (i.e., active inference).
- After a suitable period of time, evaluate the path integral of variational free energy (i.e., action) to supply a fitness functional.
- Update the flow parameters and initial states, using a stochastic gradient descent on the action (i.e., Darwinian evolution).

#### A Numerical Study of Synaptic Selection

**Figure 3.**Synaptic selection. This figure reports the results of numerical studies using fast free-energy minimising processes to model phenotypic dynamics and slow free-energy minimising processes to select phenotypic configurations or morphologies that, implicitly, have the greatest adaptive fitness or adapt to fit their environment. In this example, we focus on the selection of synapses of a brain cell (i.e., neuron) that samples presynaptic inputs from its neuropil (i.e., environment). The details of the generative model—used to simulate intracellular dynamics as a gradient flow on variational free energy—can be found in [107]. The key thing about these simulations is that—after a period of time—certain synapses were eliminated if Bayesian model selection suggested that their removal increased Bayesian model evidence (i.e., decreased variational free energy). (

**A**): Findings in [111] suggest that neurons are sensitive to the pattern of synaptic input patterns. The image shows a pyramidal cell (blue) sampling potential presynaptic inputs from other cells (yellow) with postsynaptic specialisations (red). (

**B**): In this model, pools of presynaptic neurons fire at specific times, thereby establishing a hidden sequence of inputs. The dendritic branch of the postsynaptic neuron comprises a series of segments, where each segment contains a number of synapses (here: five segments with four synapses each). Each of the 20 synapses connects to an axon of a specific presynaptic pool. These provide presynaptic (sensory) inputs at specific times over the length of a dendrite. If each of the 20 synapses were deployed in an orderly fashion across the five segments—as in the connectivity matrix—an orderly sequence of postsynaptic activations would be detected, and, implicitly. (

**C**): The lower panels show the deployment of synaptic connections over 64 ‘generations’ (i.e., cycles), in which the precision (a.k.a. sensitivity) of synapses was used to eliminate synapses if they did not contribute to model evidence. Each ‘lifetime’ of the cell was 120 (arbitrary) time units, during which time two waves of activation were detectable. The upper panels score the ensuing increase in marginal likelihood or adaptive fitness (negative free energy) over the 64 generations. The left panel shows the accompanying increase in the sensitivity (i.e., log-precision) of the 20 synapses as they find the collective arrangement that maximises adaptive fit or model evidence for this (neuronal) environment.

## 5. Discussion

## 6. Limitations

^{3}, and pbx of the bithorax complex in a single animal [113]. In complementary fashion, the planarian Dugesia japonica reproduces by fission followed by regeneration and has a heterogeneous, mixoploid genome with no known heritable mutants [114]; the phenotype of this animal has, however, remained stable for many thousands of generations in laboratories, and in all likelihood for millions of years in the wild. The phenotype can, moreover, be perturbed in saltatory fashion from one-headed to two-headed by an externally imposed bioelectric change; this altered phenotype is bioelectrically reversible but otherwise apparently permanent [115]. Engineering methods can create even more radically diverse phenotypes without genetic modifications, as demonstrated by the ‘xenobots’ prepared from Xenopus laevis skin cells, which adopt morphologies and behaviours completely unlike those that skin cells manifest when in the frog [116,117].

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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

**Figure 2.**Schematic showing the hierarchical relationship between particles at scales i and i + 1. For clarity, 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 (e.g., ${\pi}_{m}^{(i+1)}$ is population m), the elements of which are partitioned into autonomous and sensory subsets (e.g., ${\alpha}_{{m}_{n}}^{(i+1)}$ is the n-th autonomous genotype from population m). At scale i, each particle represents an element of an ensemble (e.g., ${\pi}_{\mathcal{l}}^{(i)}$ is the $\mathcal{l}$-th phenotype), 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.

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**MDPI and ACS Style**

Friston, K.; Friedman, D.A.; Constant, A.; Knight, V.B.; Fields, C.; Parr, T.; Campbell, J.O.
A Variational Synthesis of Evolutionary and Developmental Dynamics. *Entropy* **2023**, *25*, 964.
https://doi.org/10.3390/e25070964

**AMA Style**

Friston K, Friedman DA, Constant A, Knight VB, Fields C, Parr T, Campbell JO.
A Variational Synthesis of Evolutionary and Developmental Dynamics. *Entropy*. 2023; 25(7):964.
https://doi.org/10.3390/e25070964

**Chicago/Turabian Style**

Friston, Karl, Daniel A. Friedman, Axel Constant, V. Bleu Knight, Chris Fields, Thomas Parr, and John O. Campbell.
2023. "A Variational Synthesis of Evolutionary and Developmental Dynamics" *Entropy* 25, no. 7: 964.
https://doi.org/10.3390/e25070964