A Variational Synthesis of Evolutionary and Developmental Dynamics

Abstract

This paper introduces a variational formulation of natural selection, paying special attention to the nature of 'things' and the way that different 'kinds' of 'things' are individuated from-and influence-each other. We use the Bayesian mechanics of particular partitions to understand how slow phylogenetic processes constrain-and are constrained by-fast, phenotypic processes. The main result is a formulation of adaptive fitness as a path integral of phenotypic fitness. Paths of least action, at the phenotypic and phylogenetic scales, can then be read as inference and learning processes, respectively. In this view, a phenotype actively infers the state of its econiche under a generative model, whose parameters are learned via natural (Bayesian model) selection. The ensuing variational synthesis features some unexpected aspects. Perhaps the most notable is that it is not possible to describe or model a population of conspecifics per se. Rather, it is necessary to consider populations of distinct natural kinds that influence each other. This paper is limited to a description of the mathematical apparatus and accompanying ideas. Subsequent work will use these methods for simulations and numerical analyses-and identify points of contact with related mathematical formulations of evolution.

Keywords: Bayesian; Markov blanket; evolution; natural selection; nonequilibrium; particular partition; renormalisation group; self-organisation; variational inference.

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.

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.

Figure 4

Phylogeny and ontogeny as bottom-up and top-down causation.