Knowing one's place: a free-energy approach to pattern regulation

This figure shows the results of a simulation in terms of the solution or trajectory implied by active inference. This simulation used a local linear approximation to integrate the generalized descent on free energy (equation (4.3)) in 32 time steps, using the target morphology for eight cells in figure 2. Each time step can be thought of as modelling migration and differentiation over several minutes. The upper panels show the time courses of expectations about cell identity (left), the associated active states mediating migration and signal expression (middle) and the resulting trajectories; projected onto the first (vertical) direction—and colour-coded to show differentiation. These trajectories progressively minimize free energy (lower left panel), resulting in expectations that establish a relatively unique differentiation of the ensemble (lower middle panel). This panel shows the softmax function of expectations for each of eight cells, which can be interpreted as the posterior beliefs that each cell (column) occupies a particular place in the ensemble (rows). The columns have been reordered so that the maximum in each row lies along the leading diagonal. The lower right panel shows the ensuing configuration using the same format as figure 2. Here, the trajectory is shown in small circles (for each time step). The insert corresponds to the target configuration. (Online version in colour.)

These results are compelling in the sense that they show cells can resolve ambiguity about their fate, even when that resolution depends upon the context established by other (equally ambiguous) cells. Subsequent work will increase these models’ biological realism by associating various quantities with intracellular signalling and transcription, to enable testing of specific predictions about the outcomes of various experimental interventions. To illustrate the potential of this sort of modelling, we conclude with a brief simulation of regeneration and dysmorphogenesis to illustrate the sorts of behaviour the simple system above can exhibit.

Real biological tissues demonstrate remarkable self-repair and dynamic reconfiguration. For example, planarian regeneration provides a fascinating model of reconfiguration following removal of body parts or complete bisection (reviewed in [4]); early embryos of many species are likewise regulative and fully remodel following bisection. To illustrate this sort of behaviour, we simulated morphogenesis for eight time steps and then cut the partially differentiated embryo into two. We then simulated cell division by replicating the cells in each half (retaining their locations and partial differentiation) and continued integrating the morphogenetic scheme for each part separately.

The left panels of figure 4 shows normal morphogenesis over eight time steps, while the upper row shows the subsequent development of the progenitor tail cells over 32 iterations. This effectively simulates the regeneration of a head. The complementary development of the head cells can be seen in the lower panels. One can see that the cells (that were originally destined to become tail and head cells) undergo a slight dedifferentiation before recovering to produce the target morphology. In this example, the head of the split embryo takes slightly longer to attain the final form. This example illustrates the pluripotential nature of the cells and the ability of self-assembly to recover from fairly drastic interventions. Figure 4.Simulating regeneration. This figure reports simulated interventions that induce dynamic reorganization. The left panels show the normal self-assembly of eight cells or clones over iterations using the format of the previous figures (with a veridical generative model). The right panels show the development of the ensemble after it has been split into two (by the dashed line) and duplicated. The upper panels show the fate of the cells that were destined to be the tail, whereas the lower panels show the corresponding development of cells destined to be the head. Both of these (split embryo) clones dynamically reconfigure themselves to produce the target morphology, although the head cells take slightly longer before it ultimately converges (results not shown). (Online version in colour.)

To illustrate dysmorphogenesis (e.g. induced birth defects), we repeated the above simulations while changing the influence of extracellular signals—without changing the generative model (genetic and epigenetic processes). Figure 5 shows a variety of abnormal forms when changing the levels of (or sensitivity to) exogenous, intracellular and extracellular signals. For example, when the sensitivity to exogenous gradients is suppressed, the cells think they have not migrated sufficiently to differentiate and remain confused about their identity. Conversely, if we increase the sensitivity to the vertical gradient, the cells migrate over smaller vertical distances resulting in a vertical compression of the final form. Doubling the sensitivity to intracellular signals causes a failure of migration and differentiation and generalized atrophy. More selective interventions produce a dysmorphogenesis of various segments of the target morphology: reducing sensitivity to the second chemotactic signal—that is expressed by head cells—reduces the size of the head. Similarly, reducing sensitivity to the third signal induces a selective failure of body cells to differentiate. These simulations are not meant to be exhaustive but illustrate the predictions one could make following experimental manipulations (e.g. pharmacological) of intercellular signalling. Figure 5.Perturbing self-assembly. The upper right and left panels show deviations from the target configuration when the self-assembly is confounded by changing the concentration of (or sensitivity to) exogenous gradients. The first intervention illustrates a failure of migration and differentiation that results from non-specific suppression of exogenous signals, while selectively increasing the vertical gradient produces compression along the corresponding direction.

More exotic forms of dysmorphogenesis result when decreasing the intracellular (intrinsic: lower left panel) and extracellular (extrinsic: lower right panels) receptor sensitivities. (Online version in colour.)

In summary, we have introduced a variational (free energy) formulation of ergodic systems and have established a proof of principle (through simulations) that this formulation can explain a simple form of self-assembly towards a specific pattern. We have framed this in terms of morphogenesis, relating the generative model implicit in any self-organizing system to genetic and epigenetic processes. There are clearly many areas for subsequent expansion of this work; for example, cell division and neoplasia: see [67]—and the vast knowledge accumulated about the molecular biology and biophysics of morphogenesis. Furthermore, we have only considered generative models that have a fixed point attractor. In future work, it would be interesting to include equations of motion in the generative model, such that certain cell types could express fast dynamics and generate dynamic behaviours (like pulsation) following differentiation. Finally, future work should explore whether and how the self-organization and autopoietic dynamics that we demonstrated in the morphogenetic domain generalize to larger-scale phenomena such as brains and societies, as has been variously proposed [45,68].

The premise underlying active inference is based upon a variational free energy that is distinct from thermodynamic free energy in physics. In fact, strictly speaking, the formalism used in this paper ‘sits on top of’ classical physics. This is because we only make one assumption; namely, that the world can be described as a random dynamical system. Everything follows from this, using standard results from probability theory. So how does variational free energy relate to thermodynamic free energy? If we adopt the models assumed by statistical thermodynamics (e.g. canonical ensembles of microstates), then variational free-energy minimization should (and does) explain classical mechanics (e.g. [69,70]). In a similar vein, one might hope that there is a close connection with generalizations of thermodynamic free energy to far-from-equilibrium systems [71] and non-equilibrium steady state (e.g. [52,72]). This might be important in connecting variational and thermodynamic descriptions. For example, can one cast external states as a thermal reservoir and internal states as a driven system? In driven systems—with a continuous absorption of work followed by dissipation—the reliability or probability of such exchange processes might be measured by the variational free-energy, so that stable structures are formed at its minima [73]. However, questions of this sort remain an outstanding challenge.

Our simulations rest on several simplifying assumptions, such as what is coded in the cells, the (place-coded) nature of the target location, and the specifics of cell–cell communication. It is important to note that the framework does not depend explicitly on these assumptions—and extending the model to include more detailed and realistic descriptions of genetic coding and signalling is an open challenge. A potentially exciting application of these schemes is their use as observation models of real data. This would enable the model parameters to be estimated, and indeed the form of the model to be identified using model comparison. There is an established procedure for fitting models of time-series data—of the sort considered in this paper—that has been developed for the analysis of neuronal networks. This is called dynamic causal modelling [74,75] and uses exactly the same free-energy minimization described above. Although the nature of neuronal network and morphogenetic models may appear different, formally they are very similar: neuronal connectivity translates into kinetic rate constants (e.g. as controlled by the precisions above) and neuronal activity translates to the concentration or expression of various intracellular and extracellular signals. In principle, the application of dynamic causal modelling to empirical measurements of morphogenesis should provide a way to test specific hypotheses through Bayesian model comparison [76] and ground the above sort of modelling in a quantitative and empirical manner. It should also be pointed out that is now clear that even non-neuronal cells possess many of the same ion channel- and electrical synapse-based mechanisms as do neurons and use them for pattern formation and repair [77–82].

This suggests the possibility that significant commonalities may exist between the dynamics of neural networks and of bioelectric signalling networks that control shape determination [20,21,83].

The example in this paper uses a fairly arbitrary generative model. For example, the choice of four signals was largely aesthetic—to draw a parallel with DNA codons: the first signal is expressed by all cells and is primarily concerned with the intercellular spacing. The remaining three signals provide for eight combinations or differentiated cell types (although we have only illustrated four). One might also ask about the motivation for using an exogenous gradient. This is not a necessary component of the scheme; however, our simulations suggest that it underwrites a more robust solution for larger ensembles (with more than four clones). Furthermore, it ensures that the orientation of the target morphology is specified in relation to an extrinsic frame of reference. Having said this, introducing a hierarchical aspect to the generative model may be a more graceful way to model real morphogenesis at different spatial scales.

We have hypothesized that the parameters of each cell's model are genetically encoded. Another intriguing possibility is that the cells learn some model parameters during epigenesis and growth through selective expression of a fixed code. In other words, during growth, each cell might acquire (or adjust) the parameters of its generative model such that the target morphology emerges during epigenesis. The organism as a whole may therefore exhibit a self-modelling process—where they are essentially modelling their own growth process. A useful analogy here is self-modelling robots, which learn models of their own structure (e.g. their body morphology) and use those models to predict the sensory consequences of movement—and maintain their integrity or recover from injuries [84]. In computational neuroscience and machine learning, various methods have been proposed for this form of (structure) learning [85]. Technically, the difference between learning and inference pertains to the optimization of parameters and expected states with respect to model evidence (i.e. free energy). We have focused on inferring (hidden) states in this paper, as opposed to parameters (like log precisions). It may be worth concluding that although aspects of the target morphology are genetically encoded (in the parameters of the generative models); epigenetic processes (active inference) are indispensable for patterning. Indeed, it is possible to permanently alter the regenerative target morphology in some species; for example, perpetually two-headed planaria can be created by a brief modification of the electric synapses [44,86].

Finally, we have not considered the modelling of different phases of development or hierarchical extensions of the basic scheme. Having said this, the variational formulation offered here provides an interesting perspective on morphogenesis that allows one to talk about the beliefs and behaviour of cells that have to (collectively) solve the most difficult of inference problems as they navigate in autopoietic frame of reference. It is likely that this is just a first step on an important roadmap to formalize the notion of belief and information-processing in cells towards the efficient, top-down control of pattern formation for regenerative medicine and synthetic bioengineering applications. Perhaps the last word on this spatial inverse problem should go to Helmholtz [35], p. 384Each movement we make by which we alter the appearance of objects should be thought of as an experiment designed to test whether we have understood correctly the invariant relations of the phenomena before us, that is, their existence in definite spatial relations.