Active inference, morphogenesis, and computational psychiatry

Under a Laplace assumption, the variational density becomes a normal distribution:

where ∇i~i~L(s~,ã,i~,i~)) denotes the curvature of the Lagrangian with respect to internal states. With this generative model and assumed form for the variational density, we can now evaluate the variational free energy for any given sensory state and perform a gradient descent according to Equation (12).

An agent minimizing variational free energy is essentially updating beliefs of its environment and itself, under the generative model, so that the evolution of the system will inevitably lead to a non-equilibrium steady state of minimal free energy. Again, this appears rather intuitive in a developmental or regenerative biological context: cells function to remodel tissues and organs, in order to minimize the global difference between the current configuration and a species-specific anatomical goal state (Pezzulo and Levin, 2015; Pezzulo and Cisek, 2016). Such cellular behavior is based on perceived signals from their environment, and act with respect to expectations that are genetically encoded, and shaped by cellular learning (Baluška and Levin, 2016).

As free energy corresponds to (an upper bound on) Bayesian model evidence -ln p(s~,ã,i~|m) as introduced in Equation (11)—the ensuing self-organization is also self-evidencing. Self-evidencing in neuroscience refers to the idea that the brain is separated from its environment by a “statistical boundary,” which can be formally described by the statistical construct of a Markov blanket (Kirchhoff et al., 2018). We note that such a description of a system's dynamics in Bayesian terms [such as Bayesian beliefs q(ẽ)] and self-evidencing is a purely technical formulation based on the underlying mathematical definitions within the framework, which can be ascribed to simple systems like macromolecules and cells, as opposed to just higher cognitive organisms.

The simulations in this work encompass a small set of cells that are each equipped with the same generative model—such that they collectively self-organize to minimize variational free energy in an interdependent way, which has all the characteristics of morphogenesis. This example is appropriate to models such as the highly-regenerative planaria (Durant et al., 2016; Levin et al., 2019), which we will use multiple times in illustrating the principles in this work.

All the cells in our simulations are initialized with random signaling profiles near the center of their environment. The morphogenetic task of self-organization to a target configuration thus corresponds, in this framework, to each cell inferring its own location within the ensemble by forming and testing its beliefs (or predictions) q(e) about the hidden causes of the signaling concentrations it senses (i.e., the signaling molecule secretion profiles and hence cell identities of the other cells (Figure 2).

The active states a cell has control are the type of signals it can secrete, and its movement in any direction in 2D. Furthermore, each cell possesses a place-encoded model of a target configuration common to all cells based on signaling concentrations that would be sensed under that configuration. Therefore, for each of the four possible cell types corresponding to specific positions in the cell cluster, cells expect to sense specific concentrations of signaling molecules.

Sensory states s are therefore chemotactic concentrations of intracellular, exogenous and extracellular signals:

where e are the external states of concentrations c and positions x of other cells. The signal concentration sλ at each position of the i-th cell is given through the secretion and diffusion of signaling molecules of each other cell j and of itself, determined by the coefficient:

Here, ecj is the combination of the four signals expressed at each position j, depicted in Figure 2B as color coded around the target positions e*, that are defined in Figure 2C, and

is the distance between the i-th cell and the remaining cells, to which the secreted signal diffuses with diffusion coefficient k.

Analogous to stem cell-like behavior, we specify the same generative model g for each cell:

is the softmax function (or normalized exponential). This is a commonly utilized function in neural networks to enforce a sum to one constraint, which allows an interpretation as a categorical distribution over mutually exclusive outcomes.

Using these expressions—and the equations of motion from the previous section—we can express the flow of internal and active (i.a. autonomous) states from Equation (12) as

while suppressing higher order terms. Here, ϵ = s−g(i) is the prediction error associated with sensory states—the state of chemotactic signal receptors—and can hence be expressed as:

D is the matrix derivative operator on generalized states and the signal precision Π(1) is set to 1. We assumed Gaussian priors (with a mean of 0) over the hidden states with a small precision Π(2) (i.e., high variance) with a log precision of minus two.

With this generative model in place, the internal states organize themselves to minimize (precision-weighted) prediction error based upon predictions of sensed signaling states from neighboring cells. In neuroscience, this scheme is also referred to as predictive coding and can be viewed as a generalized form of Bayesian (variational) filtering as described earlier. Predictive coding describes the dynamics of the system in terms of prediction errors ϵ through accumulation of model evidence lnp(s~,ã,i~|m), which maximizes likelihood p(s~,ã,i~|ẽ1) (Friston et al., 2009). This is exactly the type of process underlying the formulation of variational free energy above.

Matlab software running these simulations, under different conditions can be downloaded as part of the academic SPM software from https://www.fil.ion.ucl.ac.uk/spm/software/ (accessed via a graphical user interface invoked with the command > > DEM). The code of the simulations can be found at https://github.com/LPioL/active_inference_morphopsy/.

Here, we use active inference to simulate dysmorphogenesis and disorders of sensory attenuation, similarly to what we discussed above, in relation to schizophrenic and autistic patients. We first present the standard (non-pathological) simulation of morphogenesis by setting appropriate values of sensory and prior precisions. Then, we show and analyze three simulations of aberrant morphogenesis. The first simulation illustrates the effects of high sensory precision in driving aberrant morphogenesis, analogous to what is observed in negative symptoms of schizophrenia and autism. The second simulation illustrates the effects of excessive prior precision in driving aberrant morphogenesis, analogous to so-called positive symptoms of schizophrenia (e.g., hallucinations and delusions, Powers et al., 2017). The third simulation illustrates the effects of low sensory precision in driving dysmorphogenesis, which might be analogous to forms of hyporeactivity.

Setting appropriate values for the precision terms leads to normal morphogenesis and the normal differentiation and migration of cells, which reach the appropriate location in the morphospace. The cells correctly infer their locations by forming predictions about the hidden causes of the signaling concentrations (without any perturbation of their precision parameters) and they reach the target morphology (see Figure 3).

In this simulation, ascending prediction errors are associated with an appropriate sensory precision, and sensory signals are processed in a context-sensitive way leading to an appropriate collective inference.

This simulation illustrates a deficit of sensory attenuation (analogous to what is observed in autism and schizophrenia). For this, we assigned an excessively high precision to the biochemical signals sensed by all cells (see Figure 4), i.e., a value that greatly exceeds the optimal value in this case. An excessively high sensory precision will increase the influence of ascending prediction errors of sensory channels in which the cells will place more confidence. We can observe that during differentiation, all cells become intestinal cells and they migrate without following the body plan of the planaria; in other words, the cells tend to stay around a circle. By assigning a high sensory precision, the cells develop a deficit of sensory attenuation and do not behave properly. They secrete chemicals as if they were already intestinal cells and the collective organization and communication fail to give them the appropriate information necessary to reach the appropriate morphology. Sensory prediction errors stay high during development, and top-down prediction errors are not taken into account by the cells leading to this development defect. Cells only integrate local cues, and high sensory precision leads to their inability to contextualize sensory information. This behavior renders sensory prediction errors too precise and context insensitive as top-down information won't be taken into account.

This example illustrates a possible development of a homogeneous tumor from a deficit of sensory attenuation. The majority of the cells stay close to each other and do not migrate appropriately. We can also observe some cells that migrate very fast at the beginning of the morphogenesis. We can draw a parallel in development with neurocristopathies (Bolande, 1974). Neurocristopathies are a developmental defect that are characterized by the abnormal specification, migration, differentiation, or death of neural crest cells during embryonic development. High sensory precision leads to increased estimates of and larger updates about environmental volatility as is sometimes observed in relation to autism (Lawson et al., 2017). Excessively precise systems (including cells) try to extract too much information from noisy signals (or too close signals), instead of treating noise as irreducible. This implies that such a system will update its generative model frequently, without improving its estimate over time. The system remains with some unaccounted uncertainty and this may be stressful in biological systems looking for homeostasis.

In Figure 5A, from two cells having too high precision, we have a developmental defect where one head (red) cell can't reach the appropriate location, but instead stays close to intestinal cells and doesn't completely differentiate. With the increasing number of cells having too high precision, the development of the tumor composed only of intestinal cells is also increasing. The cells fail to infer their location and identity as a collective, they are no longer able to communicate and infer with the other cells.

This second simulation illustrates the effects of assigning an excessively high prior on the identity of the cells analogous to what is hypothesized for the positive symptoms of schizophrenia (see Figure 6). Too strong of a prior makes the cells not accept contradictory information effectively, sensory attenuation is too high reducing their ability to handle novel circumstances (such as regenerative repair). Here, all cells believe before differentiation and migration that they are intestinal cells. We can observe that at the beginning of morphogenesis (see Figure 6), cells are confused and migrate in all directions and mostly at the center as if they were intestinal cells. The differentiation in intestinal (yellow) cells last 28 steps of the simulation and then the cells undergo a partial differentiation to their appropriate identity trying to recover the target morphology. We can observe that the cells of the head and tail do not migrate perfectly to the target morphology. Their location and their differentiation are incomplete. Biologically, this resembles the remodeling of scrambled “Picasso head” in tadpoles, with eyes, jaws, and other organs in the wrong locations, into normal frog faces (Levin, 2021a).

This example also illustrates the pluripotential capacities of the cells and the ability of self-assembly to recover from drastic changes in precision and sensory attenuation.

Similar to the previous simulation, we can observe an abnormal migration of cells as in neurocristopathies that result from a derangement of neural crest migration, colonization, or differentiation (Bolande, 1974, 1997).

As we can observe in Figure 5B, from four cells with a too high expectation on their identities, we can observe incomplete differentiation of several cells. After a period of time, cells recover the target morphology with respect to the migration to the appropriate locations, but the differentiation is not complete. There is an important developmental noise, which leads to a disorder of inference for the other cells in terms of differentiation.

This simulation illustrates the effects of setting a very low sensory precision for all cells in the collective (see Figure 7). A low sensory precision relative to the precision of prediction errors higher in the hierarchy, will bias perception toward prior beliefs, which in this case are randomly initialized.

We can observe that intestinal cells do not successfully differentiate; this is indicated by the fact that they remain brown throughout the entire simulation. Only one pharynx cell (blue) differentiates properly. The other pharynx cell is between an undifferentiated state (brown) and a tail state (green). Overall the cells didn't migrate to the exact location of the target morphology, but this brown-green incompletely differentiated pharynx cell deviated the most from its target.