# 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 |

## References

- Brentano, F. Psychology from an Empirical Standpoint; Humanities Press: New York, NY, USA, 1973. [Google Scholar]
- Haugeland, J. The intentionality all-stars. Philos. Perspect.
**1990**, 4, 383–427. [Google Scholar] [CrossRef] - Fodor, J.A. The Language of Thought; Harvard University Press: Cambridge, MA, USA, 1975. [Google Scholar]
- Millikan, R.G. Language, Thought, and Other Biological Categories: New Foundations for Realism; MIT press: Cambridge, MA, USA, 1984. [Google Scholar]
- Millikan, R.G. Biosemantics. J. Philos.
**1989**, 86, 281–297. [Google Scholar] [CrossRef] - Ramsey, W.M. Representation Reconsidered; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Boone, W.; Piccinini, G. The cognitive neuroscience revolution. Synthese
**2016**, 193, 1509–1534. [Google Scholar] [CrossRef] - Kiefer, A.; Hohwy, J. Content and misrepresentation in hierarchical generative models. Synthese
**2017**, 195, 2387–2415. [Google Scholar] [CrossRef] - Hutto, D.; Myin, E. Evolving Enactivism: Basic Minds Meet Content; MIT Press: Cambridge, MA, USA, 2017. [Google Scholar]
- Hutto, D.; Satne, G. The natural origins of content. Philosophia
**2015**, 43, 521–536. [Google Scholar] [CrossRef] - Egan, F. The nature and function of content in computational models. In the Routledge Handbook of the Computational Mind; Sprevak, M., Colombo, M., Eds.; Routledge: London, UK; New York, NY, USA, 2019; pp. 247–258. [Google Scholar]
- Sprevak, M. Triviality arguments about computational implementation. In Routledge Handbook of the Computational Mind; Routledge: London, UK, 2019; pp. 175–191. [Google Scholar]
- Kiefer, A. Literal perceptual inference. In Philosophy and Predictive Processing; Metzinger, T., Wiese, W., Eds.; MIND Group: Frankfurt, Germany, 2017. [Google Scholar]
- Thompson, E. Mind in Life: Biology, Phenomenology, and the Sciences of Mind; Harvard University Press: Cambridge, MA, USA, 2010. [Google Scholar]
- Hutto, D.D.; Myin, E. Radicalizing Enactivism: Basic Minds without Content; MIT Press: Cambridge, MA, USA, 2013. [Google Scholar]
- Egan, F. Function-theoretic explanation and the search for neural mechanisms. In Explanation and Integration in Mind and Brain Science; Kaplan, D.M., Ed.; Oxford University Press (OUP): New York, NY, USA, 2018; pp. 145–163. [Google Scholar]
- Sprevak, M. Fictionalism about neural representations. Monist
**2013**, 96, 539–560. [Google Scholar] [CrossRef][Green Version] - McGregor, S. The Bayesian stance: Equations for ‘as-if’ sensorimotor agency. Adapt. Behav.
**2017**, 25, 72–82. [Google Scholar] [CrossRef] - Kitcher, P.; Dennett, D.C. The Intentional Stance. Philos. Rev.
**1990**, 99, 126. [Google Scholar] [CrossRef] - Salus, P.H.; Millikan, R.G.; Fodor, J.A.; Pylyshyn, Z.W. Connectionism and cognitive architecture: A critical analysis. Cognition
**1988**, 28, 3–71. [Google Scholar] [CrossRef] - Millikan, R.G. Beyond Concepts: Unicepts, Language, and Natural Information; Oxford University Press: New York, NY, USA, 2017. [Google Scholar]
- Shea, N. Representation in Cognitive Science; Oxford University Press (OUP): New York, NY, USA, 2018. [Google Scholar]
- Shea, N.; Godfrey-Smith, P.; Cao, R. Content in simple signalling systems. Br. J. Philos. Sci.
**2017**, 69, 1009–1035. [Google Scholar] [CrossRef][Green Version] - Horgan, T.; Graham, G. Phenomenal intentionality and content determinacy. In Prospects for Meaning; Walter de Gruyter GmbH and Co. KG: Berlin, Germany, 2012; pp. 321–344. [Google Scholar]
- MacPherson, F. Cognitive penetration of colour experience: Rethinking the issue in light of an indirect mechanism. Philos. Phenomenol. Res.
**2011**, 84, 24–62. [Google Scholar] [CrossRef][Green Version] - Milkowski, M. Explaining the Computational Mind; MIT Press-Journals: Cambridge, MA, USA, 2013. [Google Scholar]
- Piccinini, G. Physical Computation: A Mechanistic Account; Oxford University Press: New York, NY, USA, 2015. [Google Scholar]
- McClelland, J.L.; Rumelhart, D.E.; Group, P.R. Parallel distributed processing. Explor. Microstruct. Cogn.
**1986**, 2, 216–271. [Google Scholar] - Chalmers, D.J. Connectionism and compositionality: Why Fodor and Pylyshyn were wrong. Philos. Psychol.
**1993**, 6, 305–319. [Google Scholar] [CrossRef] - O’Brien, G.; Opie, J. Notes toward a structuralist theory of mental tepresentation. In Representation in Mind; Clapin, H., Staines, P., Slezak, P., Eds.; Elsevier: Amsterdam, The Netherlands, 2004; pp. 1–20. [Google Scholar]
- Williams, D.; Colling, L.J. From symbols to icons: The return of resemblance in the cognitive neuroscience revolution. Synthese
**2017**, 195, 1941–1967. [Google Scholar] [CrossRef][Green Version] - Kiefer, A.; Hohwy, J. Representation in the prediction error minimization framework. In Routledge Companion to Philosophy of Psychology, 2nd ed.; Routledge: Oxford, UK, 2019; pp. 384–409. [Google Scholar]
- Goh, J.O.S.; Hung, H.Y.; Su, Y.S. A Conceptual Consideration of the Free Energy Principle in Cognitive Maps: How Cognitive Maps Help Reduce Surprise. Psychology of Learning and Motivation; Academic press: Cambridge, MA, USA, 2018. [Google Scholar]
- Gładziejewski, P.; Miłkowski, M. Structural representations: Causally relevant and different from detectors. Biol. Philos.
**2017**, 32, 337–355. [Google Scholar] [CrossRef][Green Version] - Gładziejewski, P. Predictive coding and representationalism. Synthese
**2015**, 193, 559–582. [Google Scholar] [CrossRef][Green Version] - Hohwy, J. The Predictive Mind; Oxford University Press (OUP): New York, NY, USA, 2013. [Google Scholar]
- Wilson, R.A.; Keil, F.C. The MIT Encyclopedia of the Cognitive Sciences; MIT press: Cambridge, MA, USA, 2001. [Google Scholar]
- Quine, W. Epistemology Naturalized Ontological Relativity and Other Essays; Columbia University Press: New York, NY, USA, 1969; pp. 69–90. [Google Scholar]
- Chemero, A. Radical Embodied Cognition; MIT Press: Cambridge, MA, USA, 2011. [Google Scholar]
- Loewer, B.; Hale, B.; Wright, C.; Miller, A. A guide to naturalizing semantics. In A Companion to the Philosophy of Language; Wiley: Hoboken, NJ, USA, 2017; pp. 174–196. [Google Scholar]
- Putnam, H. Reason, Truth and History; Cambridge University Press (CUP): Cambridge, UK, 1981. [Google Scholar]
- Kripke, S.A. Wittgenstein on Rules and Private Language: An Elementary Exposition; Harvard University Press: Cambridge, MA, USA, 1982. [Google Scholar]
- Ramsey, W. Untangling two questions about mental representation. New Ideas Psychol.
**2016**, 40, 3–12. [Google Scholar] [CrossRef] - Hubel, D.H.; Wiesel, T.N. Brain and Visual Perception; Oxford University Press (OUP): New York, NY, USA, 2004. [Google Scholar]
- Shadlen, M.N.; Newsome, W.T. Noise, neural codes and cortical organization. Curr. Opin. Neurobiol.
**1994**, 4, 569–579. [Google Scholar] [CrossRef] - O’Sullivan, J.; Herrero, J.; Smith, E.; Schevon, C.; McKhann, G.M.; Sheth, S.A.; Mehta, A.D.; Mesgarani, N. Hierarchical encoding of attended auditory objects in multi-talker speech perception. Neuron
**2019**, 104, 1195–1209.e3. [Google Scholar] [CrossRef] - Díaz-Gutiérrez, P.; Gilbert, S.J.; Arco, J.E.; Sobrado, A.; Ruz, M. Neural representation of current and intended task sets during sequential judgements on human faces. NeuroImage
**2020**, 204, 116219. [Google Scholar] [CrossRef] [PubMed] - Bonnen, K.L.; Czuba, T.B.; Whritner, J.A.; Kohn, A.; Huk, A.C.; Cormack, L.K. Binocular viewing geometry shapes the neural representation of the dynamic three-dimensional environment. Nat. Neurosci.
**2019**, 23, 113–121. [Google Scholar] [CrossRef] [PubMed] - Kiyonaga, A.; Dowd, E.W.; Egner, T. Neural representation of working memory content is modulated by visual attentional demand. J. Cogn. Neurosci.
**2017**, 29, 2011–2024. [Google Scholar] [CrossRef] [PubMed] - Manohar, S.; Zokaei, N.; Fallon, S.J.; Vogels, T.P.; Husain, M. Neural mechanisms of attending to items in working memory. Neurosci. Biobehav. Rev.
**2019**, 101, 1–12. [Google Scholar] [CrossRef] [PubMed] - Ealey, M.A.; Moore, D.M.; Anderson, E.H.; Fanson, J.L. Development of an active truss element for control of precision structures. Opt. Eng.
**1990**, 29, 1333. [Google Scholar] [CrossRef] - Gregory, J.; Lin, C. Constrained Optimization in the Calculus of Variations and Optimal Control Theory; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Stengel, R.F. Optimal Control and Estimation; Courier Corporation: North Chelmsford, MA, USA, 1994. [Google Scholar]
- Calvo, P.; Friston, K. Predicting green: Really radical (plant) predictive processing. J. R. Soc. Interface
**2017**, 14, 20170096. [Google Scholar] [CrossRef] - Hipolito, I.; Kirchhoff, M.D. The predictive brain: A modular view of brain and cognitive function? Preprints
**2019**, 2019110111. [Google Scholar] [CrossRef] - Ramstead, M.J.D.; Kirchhoff, M.D.; Friston, K.J. A tale of two densities: Active inference is enactive inference. Adapt. Behav.
**2019**, 28, 225–239. [Google Scholar] [CrossRef][Green Version] - Engel, A.K.; Friston, K.J.; Kragic, D. The Pragmatic Turn: Toward Action-oriented Views in Cognitive Science; MIT Press: Cambridge, MA, USA, 2016. [Google Scholar]
- Williams, D. Pragmatism and the predictive mind. Phenomenol. Cogn. Sci.
**2018**, 17, 835–859. [Google Scholar] [CrossRef][Green Version] - Rosenbaum, D.A. Human Motor Control; Academic press: Cambridge, MA, USA, 2009. [Google Scholar]
- Arsiwalla, X.D.; Bote, R.M.; Verschure, P.F.M.J. Beyond neural coding? Lessons from perceptual control theory. Behav. Brain Sci.
**2019**, 42, e217. [Google Scholar] [CrossRef] - Beer, R.D. Dynamical approaches to cognitive science. Trends Cogn. Sci.
**2000**, 4, 91–99. [Google Scholar] [CrossRef] - Von Bertalanffy, L. An outline of general system theory. Br. J. Philos. Sci.
**1950**, 134–165. [Google Scholar] [CrossRef] - Porush, D. The Soft Machine: Cybernetic fIction; Routledge: London, UK, 2018. [Google Scholar]
- Seth, A.K. The cybernetic brain: From interoceptive inference to sensorimotor contingencies. In Open MIND; Metzinger, T., Windt, J.M., Eds.; MIND Group: Frankfurt, Germany, 2014; pp. 1–24. [Google Scholar]
- Pickering, A. The Cybernetic Brain: Sketches of Another Future; University of Chicago Press: Chicago, IL, USA, 2010. [Google Scholar]
- Friston, K. A free energy principle for a particular physics. arXiv
**2019**, arXiv:1906.10184. [Google Scholar] - Newen, A.; De Bruin, L.; Gallagher, S. The Oxford Handbook of 4E Cognition; Oxford University Press: New York, NY, USA, 2018. [Google Scholar]
- Gallagher, S. Action and Interaction; Oxford University Press: New York, NY, USA, 2020. [Google Scholar]
- Varela, F.J.; Thompson, E.; Rosch, E. The Embodied Mind: Cognitive Science and Human Experience; MIT press: Cambridge, MA, USA, 1991. [Google Scholar]
- Noë, A. Action in Perception; MIT press: Cambridge, MA, USA, 2004. [Google Scholar]
- Rao, R.P.N.; Ballard, D.H. Predictive coding in the visual cortex: A functional interpretation of some extra-classical receptive-field effects. Nat. Neurosci.
**1999**, 2, 79–87. [Google Scholar] [CrossRef] - Hinton, G.E.; Sejnowski, T.J. Optimal perceptual inference. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, Washington, DC, USA, 19–23 June 1983. [Google Scholar]
- Clark, A. Surfing Uncertainty: Prediction, Action, and the Embodied Mind; Oxford University Press: New York, NY, USA, 2015. [Google Scholar]
- Hohwy, J. Self-supervision, normativity and the free energy principle. Synthese
**2020**, 1–25. [Google Scholar] [CrossRef] - Hohwy, J. The self-evidencing brain. Noûs
**2014**, 50, 259–285. [Google Scholar] [CrossRef] - Van Es, T. Living models or life modelled? On the use of models in the free energy principle. Adapt. Behav.
**2020**, 1059712320918678. [Google Scholar] [CrossRef] - Hipolito, I.; Baltieri, M.; Friston, J.K.; Ramstead, M.J. Embodied skillful performance: Where the action is. PhilSci Arch.
**2020**, 17280, Preprint. [Google Scholar] - Friston, K. What is optimal about motor control? Neuron
**2011**, 72, 488–498. [Google Scholar] [CrossRef][Green Version] - Friston, K. The free-energy principle: A unified brain theory? Nat. Rev. Neurosci.
**2010**, 11, 127–138. [Google Scholar] [CrossRef] - Parr, T.; Da Costa, L.; Friston, K. Markov blankets, information geometry and stochastic thermodynamics. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**2019**, 378, 20190159. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kiefer, A.B. Psychophysical identity and free energy. J. R. Soc. Interface
**2020**, 17, 20200370. [Google Scholar] [CrossRef] [PubMed] - Crauel, H.; Flandoli, F. Attractors for random dynamical systems. Probab. Theory Relat. Fields
**1994**, 100, 365–393. [Google Scholar] [CrossRef] - Hipolito, I.; Ramstead, M.; Convertino, L.; Bhat, A.; Friston, K.; Parr, T. Markov blankets in the brain. arXiv
**2020**, arXiv:2006.02741. [Google Scholar] - Friston, K.J.; Wiese, W.; Hobson, J.A. Sentience and the origins of consciousness: From Cartesian duality to Markovian monism. Entropy
**2020**, 22, 516. [Google Scholar] [CrossRef] - Anderson, M. Of Bayes and bullets: An embodied, situated, targeting-based account of predictive processing. In Philosophy and Predictive Processing; MIND Group: Frankfurt, Germany, 2017. [Google Scholar]
- Buckley, C.L.; Kim, C.S.; McGregor, S.; Seth, A.K. The free energy principle for action and perception: A mathematical review. J. Math. Psychol.
**2017**, 81, 55–79. [Google Scholar] [CrossRef] - Bogacz, R. A tutorial on the free-energy framework for modelling perception and learning. J. Math. Psychol.
**2017**, 76, 198–211. [Google Scholar] [CrossRef][Green Version] - Friston, K. A free energy principle for biological systems. Entropy
**2012**, 14, 2100–2121. [Google Scholar] - Allen, M.; Friston, K. From cognitivism to autopoiesis: Towards a computational framework for the embodied mind. Synthese
**2016**, 195, 2459–2482. [Google Scholar] [CrossRef][Green Version] - Friston, K. Embodied inference: Or “I think therefore I am, if I am what I think”. In The Implications of Embodiment: Cognition and Communication; Tschacher, W., Bergomi, C., Eds.; Imprint Academic: Exeter, UK, 2011; pp. 89–125. [Google Scholar]
- Ramstead, M.J.D.; Constant, A.; Badcock, P.B.T.; Friston, K. Variational ecology and the physics of sentient systems. Phys. Life Rev.
**2019**, 31, 188–205. [Google Scholar] [CrossRef] - Ramsey, W.M.; Boone, W.; Ramstead, M.J.; Badcock, P.B.; Friston, K.J. Answering Schrödinger’s question: A free-energy formulation. Phys. Life Rev.
**2018**, 24, 1–16. [Google Scholar] - Constant, A.; Clark, A.; Friston, K.J. Representation wars: Enacting an armistice through active inference. PhilSci Arch.
**2019**, 16641, Preprint. [Google Scholar] - Wiese, W. What are the contents of representations in predictive processing? Phenomenol. Cogn. Sci.
**2016**, 16, 715–736. [Google Scholar] [CrossRef] - Giere, R.N. Explaining Science: A Cognitive Approach; University of Chicago Press: Chicago, IL, USA, 2010. [Google Scholar]
- Giere, R. Science Without Laws; University of Chicago Press Chicago: Chicago, IL, USA, 1999. [Google Scholar]
- Giere, R.N. Scientific Perspectivism; University of Chicago Press: Chicago, IL, USA, 2006. [Google Scholar]
- Van Fraassen, B.C. The Scientific Image; Oxford University Press: New York, NY, USA, 1980. [Google Scholar]
- Elgin, C.Z. True Enough; MIT Press: Cambridge, MA, USA, 2017. [Google Scholar]
- Friend, S. The fictional character of scientific models. Sci. Imagin.
**2019**, 102, 102–127. [Google Scholar] [CrossRef] - Salis, F. The new fiction view of models. Br. J. Philos. Sci.
**2019**, axz015. [Google Scholar] [CrossRef] - Weiskopf, D.A. Reductive explanation between psychology and neuroscience. In The Routledge Handbook of the Computational Mind; Routledge: London, UK, 2018; pp. 223–236. [Google Scholar]
- Grimm, S.R.; Baumberger, C.; Ammon, S. Explaining Understanding: New Perspectives from Epistemology and Philosophy of Science; Taylor & Francis: New York, NY, USA, 2016. [Google Scholar]
- Frigg, R.; Nguyen, J. Scientific Representation is representation-as. Philos. Sci. Pract.
**2016**, 149–179. [Google Scholar] [CrossRef][Green Version] - Peschard, I.F. Making sense of modeling: Beyond representation. Eur. J. Philos. Sci.
**2011**, 1, 335–352. [Google Scholar] [CrossRef][Green Version] - Isaac, A.M.C. Modeling without representation. Synthese
**2012**, 190, 3611–3623. [Google Scholar] [CrossRef][Green Version] - Rice, C. Moving beyond causes: Optimality models and scientific explanation. Noûs
**2013**, 49, 589–615. [Google Scholar] [CrossRef][Green Version] - Kirchhoff, M.; Robertson, I. Enactivism and predictive processing: A non-representational view. Philos. Explor.
**2018**, 21, 264–281. [Google Scholar] [CrossRef] - Baltieri, M.; Buckley, C.L. Generative models as parsimonious descriptions of sensorimotor loops. Behav. Brain Sci.
**2019**, 42, e218. [Google Scholar] [CrossRef] [PubMed] - Kohn, E. How Forests Think: Toward an Anthropology Beyond the Human; Univ of California Press: Berkeley, California, 2013. [Google Scholar]
- Ramstead, M.J.D.; Veissière, S.P.L.; Kirmayer, L.J. Cultural affordances: Scaffolding local worlds through shared intentionality and regimes of attention. Front. Psychol.
**2016**, 7, 1090. [Google Scholar] [CrossRef] [PubMed] - Kirmayer, L.J.; Ramstead, M.J.D. Embodiment and enactment in cultural psychiatry. In Embodiment, Enaction, and Culture; Durt, C., Fuchs, T., Tewes, C., Eds.; MIT Press Journals: Cambridge, MA, USA, 2017; pp. 397–422. [Google Scholar]
- Kirmayer, L.J. Ontologies of life: From thermodynamics to teleonomics. Comment on “Answering Schrödinger’s question: A free-energy formulation” by Maxwell James Désormeau Ramstead et al. Physics of life reviews. PhLRv
**2018**, 24, 29–31. [Google Scholar] - Veissière, S.P.L.; Constant, A.; Ramstead, M.J.D.; Friston, K.; Kirmayer, L.J. Thinking through other minds: A variational approach to cognition and culture. Behav. Brain Sci.
**2019**, 43, 1–97. [Google Scholar] [CrossRef][Green Version] - Kuchling, F.; Friston, K.; Georgiev, G.; Levin, M. Morphogenesis as Bayesian inference: A variational approach to pattern formation and control in complex biological systems. Phys. Life Rev.
**2019**. In Press. [Google Scholar] [CrossRef] - Fields, C.; Levin, M. Scale-Free Biology: Integrating Evolutionary and Developmental Thinking. BioEssays
**2020**, 1900228, e1900228. [Google Scholar] [CrossRef] - Putnam, H. Minds and machines. In Mind, Language, and Reality; Cambridge University Press: Cambridge, UK, 1960; pp. 362–385. [Google Scholar]
- Putnam, H. The nature of mental states. In Mind, Language, and Reality; Cambridge University Press: Cambridge, UK, 1967; pp. 429–440. [Google Scholar]
- Dretske, F. Misrepresentation. In Belief: Form, Content and Function; Bogdan, R., Ed.; Oxford University Press (OUP): New York, NY, USA, 1986. [Google Scholar]
- Fodor, J.A. LOT 2: The Language of Thought Revisited; Oxford University Press (OUP): New York, NY, USA, 2008. [Google Scholar]
- Bouizegarene, N.; Ramstead, M.; Constant, A.; Friston, K.; Kirmayer, L. Narrative as active inference. Preprint
**2020**. [Google Scholar] [CrossRef] - Taylor, C. The Language Animal; Harvard University Press: Cambridge, MA, USA, 2016. [Google Scholar]
- Hutto, D. Folk Psychological Narratives: The Sociocultural Basis of Understanding Reasons; MIT press: Cambridge, MA, USA, 2012. [Google Scholar]
- Sekimoto, K. Langevin equation and thermodynamics. Prog. Theor. Phys. Supp.
**1998**, 130, 17–27. [Google Scholar] [CrossRef][Green Version] - Ao, P. Emerging of stochastic dynamical equalities and steady state thermodynamics from Darwinian dynamics. Commun. Theor. Phys.
**2008**, 49, 1073–1090. [Google Scholar] [CrossRef][Green Version] - Seifert, U. Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys.
**2012**, 75, 126001. [Google Scholar] [CrossRef][Green Version] - Tribus, M. Thermodynamics and Thermostatics: An Introduction to Energy, Information and States of Matter, with Engineering Applications; D. Van Nostrand Company Inc: New York, NY, USA, 1961. [Google Scholar]
- Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Jones, D.S. Elementary Information Theory; Clarendon Press: New York, NY, USA, 1979. [Google Scholar]
- MacKay, D.J.C. Information Theory, Inference, and Learning Algorithms; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Kerr, W.; Graham, A. Generalized phase space version of Langevin equations and associated Fokker-Planck equations. Eur. Phys. J. B
**2000**, 15, 305–311. [Google Scholar] [CrossRef] - Frank, T.; Beek, P.J.; Friedrich, R. Fokker-Planck perspective on stochastic delay systems: Exact solutions and data analysis of biological systems. Phys. Rev. E
**2003**, 68, 021912. [Google Scholar] [CrossRef] [PubMed][Green Version] - Frank, T.D. Nonlinear Fokker-Planck Equations: Fundamentals and Applications; Springer Science & Business Media: Berlin, Germany, 2004. [Google Scholar]
- Tomé, T. Entropy production in nonequilibrium systems described by a Fokker-Planck equation. Braz. J. Phys.
**2006**, 36, 1285–1289. [Google Scholar] [CrossRef][Green Version] - Kim, E.-J. Investigating information geometry in classical and quantum systems through information length. Entropy
**2018**, 20, 574. [Google Scholar] [CrossRef][Green Version] - Yuan, R.; Ma, Y.; Yuan, B.; Ping, A. Bridging engineering and physics: Lyapunov function as potential function. arXiv
**2010**, arXiv:1012. 2721v1. [Google Scholar] - Friston, K.; Ao, P. Free energy, value, and attractors. Comput. Math. Methods Med.
**2011**, 2012, 937860. [Google Scholar] [CrossRef] - Friston, K.J.; Fagerholm, E.D.; Zarghami, T.S.; Parr, T.; Hipólito, I.; Magrou, L.; Razi, A. Parcels and particles: Markov blankets in the brain. arXiv
**2020**, arXiv:2007.09704. [Google Scholar] - Beal, M.J. Variational Algorithms for Approximate Bayesian Inference. Ph.D. Thesis, University College London, London, UK, 2003. [Google Scholar]
- Yedidia, J.S.; Freeman, W.T.; Weiss, Y. Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Trans. Inf. Theory
**2005**, 51, 2282–2312. [Google Scholar] [CrossRef] - Dauwels, J. On variational message passing on factor graphs. In Proceedings of the IEEE International Symposium on Information Theory, Nice, France, 24–29 June 2007; pp. 2546–2550. [Google Scholar] [CrossRef][Green Version]
- Amari, S.-I. Natural gradient works efficiently in learning. Neural Comput.
**1998**, 10, 251–276. [Google Scholar] [CrossRef] - Ay, N. Information geometry on complexity and stochastic interaction. Entropy
**2015**, 17, 2432–2458. [Google Scholar] [CrossRef] - Caticha, A. The basics of information geometry. In Proceedings of the Bayesian Inference and Maximum Entropy Methods in Science and Engineering (Maxent 2014), Amboise, France, 21–26 September 2014; AIP Publishing: College Park, MD, USA, 2015; pp. 15–26. [Google Scholar]
- Crooks, G.E. Measuring thermodynamic length. Phys. Rev. Lett.
**2007**, 99, 100602. [Google Scholar] [CrossRef] [PubMed][Green Version] - Holmes, Z.; Weidt, S.; Jennings, D.; Anders, J.; Mintert, F. Coherent fluctuation relations: From the abstract to the concrete. Quantum
**2019**, 3, 124. [Google Scholar] [CrossRef]

**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