# Is the Free-Energy Principle a Formal Theory of Semantics? From Variational Density Dynamics to Neural and Phenotypic Representations

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## Abstract

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## 1. Introduction: Neural Representations and Their (Dis)Contents

#### 1.1. The Faces of Representationalism: Realism and Non-Realism

#### 1.2. Towards Anti-Realism: Deficiencies of the Realist View

“The roles provided by commonsense psychology are those that distinguish different types of mental representations. What we need and what is not provided by commonsense psychology is, more generally, the sort of physical condition that makes something a representational state, period. In functional terms, we would like to know what different types of representations perhaps have in common, qua representation. Neither commonsense psychology nor computationalism tells us much about the sort causal/physical conditions that bestow upon brain states the functional role of representing (at least not directly).” ([43], p. 6, emphasis added)

#### 1.3. Representations Under the Free-Energy Principle?

## 2. The Free-Energy Principle and Active Inference: From Information Geometry to the Physics of Phenotypes

#### 2.1. State Spaces, Nonequilibrium Dynamics, and Bears (Oh My)

#### 2.2. Markov Blankets and the Dynamics of Living Systems

#### 2.3. Information Geometries and the Physics of Sentient Systems

#### 2.4. Phenotypes: A Tale of Two Densities

#### 2.5. Living Models: A Mechanistic View on Goal-Directed, Probabilistic Inference and Decision-Making Under the Free-Energy Principle

## 3. Deflationary and Fictionalist Accounts of Neural Representation

#### 3.1. A Deflationary Approach to Neural Representation

- 1)
- A mathematical function that is realized by the cognitive system;
- 2)
- Specific algorithms that the system uses to compute the function;
- 3)
- Representational structures that are maintained and updated by the mechanism;
- 4)
- Computational processes that are defined over representational structures.
- 5)
- Ecological component: physical facts about the typical operating conditions in which the computational mechanism typically operates.

#### 3.2. Fictionalism and Models in Scientific Practice

## 4. A Variational Semantics: From Generative Models to Deflated Semantic Content

#### 4.1. A Deflationary Account of Content Under the Free-Energy Principle

#### 4.2. From a Computational Theory Proper to a Formal Semantics

#### 4.3. Phenotypic Representations? Ontologies?

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### A.1. The Langevin Formalism and Density Dynamics

**Figure A1.**Density dynamics and pullback attractors. This figure depicts the density or ensemble dynamics of random dynamical systems that can be described via the Langevin equation. The left panel depicts the time evolution of two states, as a strange attractor. A point in this space assigns to the system a position along each dimension, and so assigns a value to each state. Here, each dimension represents one of the two states, and the trajectory plots the evolution of states over time. The right panel represents an arbitrary random attractor (a pullback attractor). One can think of this pullback attractor in two ways. First, the attractor can be cast as representing the trajectory of systemic states over time (in this case, two states are represented). The crucial feature of this trajectory is that—after sufficient time has passed—it will revisit specific regions of state space, which make up the pullback attractor itself. The second interpretation is probabilistic: it casts the attracting set as a probability density over the states in which the system can be found when it is sampled at random. The Fokker-Planck equation allows us to describe the evolution of this probability density. This, in turn, licences a solution to the Fokker-Planck equation. The consequence of this is that we can establish a lawful relationship between the probability density and the flow of states at any point in the system’s state space. This solution describes the flow of systemic states in terms of gradients of log density or surprisal and in terms of the amplitude of random fluctuations. In turn, the Helmholtz decomposition allows us to express the nonequilibrium steady-state solution in terms of two orthogonal components. One of these is a curl-free gradient flow that depends on the amplitude of random fluctuations Γ. This component rebuilds probability gradients, effectively countering the effect of random fluctuations on states (i.e., countering their dispersion). The other component is a divergence-free (or solenoidal) flow that circulates on isoprobability contours and that depends upon an antisymmetric (skew) matrix Q. The figure depicts the flow around the peak of a probability density that has a Gaussian or normal form. See [66,137,138] for technical details.

#### A.2. Bayesian Mechanics

#### A.3. Information Geometry and Beliefs

## Glossary

Expression | Description | Units |

Variables | ||

$\omega (\tau )$ | Random fluctuations | a.u. (m) |

$x=\{\eta ,s,a,\mu \}\in X$ | Markovian partition into external, sensory, active, and internal states | a.u. (m) |

$\alpha =\{a,\mu \}\in A$ | Autonomous states | a.u. (m) |

$b=\{s,a\}\in B$ | Blanket states | a.u. (m) |

$\pi =\{b,\mu \}\in P$ | Particular states | a.u. (m) |

$\eta \in E$ | External states | a.u. (m) |

$\Gamma ={\mu}_{m}{k}_{B}T$ | Amplitude (i.e., half the variance) of random fluctuations | J·s/kg |

$Q$ | Rate of solenoidal flow | J·s/kg |

$\ell ={\displaystyle \int d\ell}:d{\ell}^{2}={g}_{ij}d{\lambda}^{j}d{\lambda}^{i}$ | Information length | nats |

${g}_{ij}=E\left[\frac{\partial \Im}{\partial {\lambda}^{i}}\frac{\partial \Im}{\partial {\lambda}^{j}}\right]$ | Fisher (information metric) tensor | a.u. |

Functions, functionals and potentials | ||

$E[x]={E}_{p}[x]={\displaystyle \int x{p}_{\lambda}(x)dx}$ | Expectation or average | |

${p}_{\lambda}(x):\mathrm{Pr}[X\in A]={\displaystyle {\int}_{A}{p}_{\lambda}(x)dx}$ | Probability density function parameterised by sufficient statistics $\lambda $ | |

${q}_{\mu}(\eta )$ | Variational density – an (approximate posterior) density over external states that is parameterised by internal states | |

$F(b)\ge \Im (b)$ | Variational free energy free energy – an upper bound on the surprisal of particular states | nats |

Operators | ||

${\nabla}_{x}\Im (x)=\frac{\partial \Im}{\partial x}=\left(\frac{\partial \Im}{\partial {x}_{1}},\frac{\partial \Im}{\partial {x}_{2}},\dots \right)$ | Differential or gradient operator (on a scalar field) | |

Entropies and potentials | ||

$\Im (x)=-\mathrm{ln}p(x)$ | Surprisal or self-information | nats |

$D[q(x)||p(x)]={E}_{q}[\mathrm{ln}q(x)-\mathrm{ln}p(x)]$ | Relative entropy or Kullback-Leibler divergence | nats |

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**Figure 1.**The structure of a Markov blanket. A Markov blanket highlights open systems exchanging matter, energy, or information with their surroundings. The figure depicts the Markovian partition of the system or set of states into internal states (denoted μ), blanket states, which are themselves divided into active states (a) and sensory states (s) states, and external states (η). Internal states are conditionally independent from external states, given blanket states. Variables are conditionally independent of each other by virtue of the Markov blanket. If there is no route between two variables, and they share parents, they are conditionally independent. Arrows go from ‘parents’ to ‘children’. From Hipólito et al. [83].

**Figure 2.**Markov blankets of life. This figure depicts a Markov blanket around a system of interest; here, the brain. The figure associates the Markovian partition to internal (μ), blanket (b), and external states (η); where blanket states can be active (a) or sensory (s) states, depending on their statistical relations to internal and external states. Here, the flow of each kind of state (denoted with a dot) is expressed as a function of other states in the partition (plus some random noise, denoted ω), as a function of the independences that are harnessed by the Markov blanket. Autonomous states (α) are those that follow a free-energy gradient. Particular states (here, denoted π) are those identified with the system itself (internal and blanket states).

**Figure 3.**A generative model. Here, sensory states are observed outcomes, denoted s. The generative model represents external states as hidden states, denoted η. Depicted in this schema are the likelihood mapping from hidden states (denoted

**A**), prior beliefs about the probability of state transitions (

**B**), and the prior beliefs about initial (hidden) states (

**D**). The

**G**term is an expected free-energy that drives policy selection (π) in elaborated generative models that entail the consequences of action (not shown here). The form of this generative model assumes discrete states and steps in time shown from the left to the right. This kind of generative model is known as a hidden Markov model or partially observed Markov decision process.

**Figure 4.**The deflationary account of the content of a representation. This figure depicts the main components of the semantic content of neural representations according to the mathematical-deflationary account of content [11]. The computational component of the representational content (the computational theory proper) is interpreted in a realist sense. The computational component of the content comprises (1) a mathematical function, (2) specific algorithms that realize this function in the system, (3) physical structures that bear representational contents; (4) computational processes that secure these contents, and (5) normal ecological conditions under which the system can operate. The cognitive content is taken in an anti-realist sense, as a kind of explanatory gloss that only has an explanatory, instrumentalist role, as the interpretation given to the neural representation by the experimenter.

**Figure 5.**A formal semantics under the free-energy principle. Bayesian cognitive science does not have to commit to the classic notion of representation carrying propositional semantic content. The free-energy principle allows us to formulate a formal semantics. “Representations” under the free-energy principle are, in essence, formalized under the physics of flow (e.g., dynamical systems theory) and information geometry, and they are better understood as internal structures enabling the system to parse its sensory stream (i.e., as an ontology). Here, we relate the main components of the deflationary account of content [11] to the free-energy formulation. The mathematical function that underwrites the free-energy principle is a variational free-energy functional. The specific algorithm is gradient descent (i.e., flow) on this free-energy functional, which defines the gradients on which the system ‘surfs’ until it reaches a nonequilibrium steady state. Representational structures (i.e., the structures that embody or carry out these processes) correspond to the internal states of a system and associated intrinsic information geometry. The computational process itself is active inference, which provides an overarching framework to use the generative model for policy (action) selection. Finally, the ecological component is defined by the implicit semantics that is entailed by the dual (intrinsic and extrinsic) information geometries: via the associated extrinsic information geometry, the system looks as if it behaves as a functional of beliefs about external states.

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**MDPI and ACS Style**

Ramstead, M.J.D.; Friston, K.J.; Hipólito, I.
Is the Free-Energy Principle a Formal Theory of Semantics? From Variational Density Dynamics to Neural and Phenotypic Representations. *Entropy* **2020**, *22*, 889.
https://doi.org/10.3390/e22080889

**AMA Style**

Ramstead MJD, Friston KJ, Hipólito I.
Is the Free-Energy Principle a Formal Theory of Semantics? From Variational Density Dynamics to Neural and Phenotypic Representations. *Entropy*. 2020; 22(8):889.
https://doi.org/10.3390/e22080889

**Chicago/Turabian Style**

Ramstead, Maxwell J. D., Karl J. Friston, and Inês Hipólito.
2020. "Is the Free-Energy Principle a Formal Theory of Semantics? From Variational Density Dynamics to Neural and Phenotypic Representations" *Entropy* 22, no. 8: 889.
https://doi.org/10.3390/e22080889