# Mathematics and the Brain: A Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

“There is no certainty in sciences where mathematics cannot be applied”(Leonardo da Vinci)

## 2. Category and Consciousness

#### 2.1. Definition of Category

**Definition**

**1.**

**Definition**

**2.**

#### 2.2. Category and Consciousness

“A neural correlate of a phenomenal family S is a neural system N such that the state of N directly correlates with the subject’s phenomenal property in S.”

#### 2.3. Categories in IIT and TTC

#### 2.3.1. Categories in IIT

#### 2.3.2. Categories in TTC

## 3. Functor, Natural Transformation, and Consciousness

#### 3.1. Definition of Functor and Natural Transformation

**Definition**

**3.**

- 1.
- It maps f: X→Y in C to F(f): F(X)→F(Y) in D;
- 2.
- F(f ∘ g) = F(f) ∘ F(g) for any (composable) pair of f and g in C;
- 3.
- For each X in C, F(1X) = 1F(X).

**Definition**

**4.**

- 1.
- t maps each object X in C to corresponding arrow tX: F(X)→G(X) in D;
- 2.
- For any f: X→Y in C, tY ∘ F(f) = G(f) ∘ tX.

#### 3.2. Functor, Natural Transformation, and Consciousness

#### 3.3. Functors and Natural Transformations in IIT and TTC

#### 3.3.1. Functors and Natural Transformations in IIT

#### 3.3.2. Functor and Natural Transformations in TTC

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Levine, J. Materialism and qualia: The explanatory gap. Pac. Philos. Q.
**1983**, 64, 354–361. [Google Scholar] [CrossRef] - Chalmers, D.J. What Is a Neural Correlate of Consciousness? Neural Correlates of Consciousness: Empirical and Conceptual Questions; MIT Press: Cambridge, MA, USA, 2000. [Google Scholar]
- Northoff, G. Unlocking the Brain: Volume II: Consciousness; Oxford University Press: Oxford, UK, 2014. [Google Scholar]
- Churchland, P. Brain-Wise; MIT Press: Cambridge, MA, USA, 2002. [Google Scholar]
- Northoff, G. Neuro-Philosophy and the Healthy Mind: Learning from the Unwell Brain; Norton Publisher: New York, NY, USA, 2016. [Google Scholar]
- Northoff, G. The Spontaneous Brain. From Mind-Body Problem to World-Brain Problem; MIT Press: Cambridge, MA, USA, 2018. [Google Scholar]
- Searle, J.R. Mind: A Brief Introduction; Oxford University Press: Oxford, UK, 2004. [Google Scholar]
- Arzi-Gonczarowski, Z. Perceive this as that—Analogies, artificial perception, and category theory. Ann. Math. Artif. Intell.
**1999**, 26, 215–252. [Google Scholar] [CrossRef] - Crick, F.; Koch, C. A framework for consciousness. Nat. Neurosci.
**2003**, 6, 119–126. [Google Scholar] [CrossRef] [PubMed] - De Graaf, T.A.; Hsieh, P.J.; Sack, A.T. The ‘correlates’ in neural correlates of consciousness. Neurosci. Biobehav. Rev.
**2012**, 36, 191–197. [Google Scholar] [CrossRef] [PubMed] - Koch, C. The Quest for Consciousness; Oxford University Press: Oxford, UK, 2004. [Google Scholar]
- Northoff, G. Unlocking the Brain: Volume I: Coding; Oxford University Press: Oxford, UK, 2014. [Google Scholar]
- Koch, C.; Massimini, M.; Boly, M.; Tononi, G. Neural correlates of consciousness: Progress and problems. Rev. Neurosci.
**2016**, 17, 307–321. [Google Scholar] [CrossRef] [PubMed] - Tononi, G. An information integration theory of consciousness. BMC Neurosci.
**2004**, 5, 42. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tononi, G.; Boly, M.; Massimini, M.; Koch, C. Integrated information theory: From consciousness to its physical substrate. Nat. Rev. Neurosci.
**2016**, 17, 450–461. [Google Scholar] [CrossRef] - Dehaene, S.; Charles, L.; King, J.R.; Marti, S. Toward a computational theory of conscious processing. Curr. Opin. Neurobiol.
**2014**, 25, 76–84. [Google Scholar] [CrossRef] [Green Version] - Dehaene, S.; Changeux, J.P. Experimental and theoretical approaches to conscious processing. Neuron
**2011**, 70, 200–227. [Google Scholar] [CrossRef] [Green Version] - Dehaene, S.; Naccache, L. Towards a cognitive neuroscience of consciousness: Basic evidence and a workspace framework. Cognition
**2001**, 79, 1–37. [Google Scholar] [CrossRef] - Baars, B.J. Global workspace theory of consciousness: Toward a cognitive neuroscience of human experience. Prog. Brain Res.
**2005**, 150, 45–53. [Google Scholar] [PubMed] - Northoff, G. What the brain’s intrinsic activity can tell us about consciousness? A tri-dimensional view. Neurosci. Biobehav. Rev.
**2013**, 37, 726–738. [Google Scholar] [CrossRef] [PubMed] - Northoff, G.; Huang, Z. How do the brain’s time and space mediate consciousness and its different dimensions? Temporospatial theory of consciousness (TTC). Neurosci. Biobehav. Rev.
**2017**, 80, 630–645. [Google Scholar] [CrossRef] - Lau, H.; Rosenthal, D. Empirical support for higher-order theories of conscious awareness. Trends Cogn. Sci.
**2011**, 15, 365–373. [Google Scholar] [CrossRef] [PubMed] - Rosenthal, D.M. Metacognition and higher-order thoughts. Conscious. Cogn.
**2000**, 9, 231–242. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lamme, V.A.; Roelfsema, P.R. The distinct modes of vision offered by feedforward and recurrent processing. Trends Neurosci.
**2000**, 23, 571–579. [Google Scholar] [CrossRef] - Fingelkurts, A.A.; Fingelkurts, A.A.; Neves, C.F. Natural world physical, brain operational, and mind phenomenal space-time. Phys. Life Rev.
**2010**, 7, 195–249. [Google Scholar] [CrossRef] [Green Version] - Engel, A.K.; Singer, W. Temporal binding and the neural correlates of sensory awareness. Trends Cogn. Sci.
**2001**, 5, 16–25. [Google Scholar] [CrossRef] - Graziano, M.S.; Kastner, S. Human consciousness and its relationship to social neuroscience: A novel hypothesis. Cogn. Neurosci.
**2011**, 2, 98–113. [Google Scholar] [CrossRef] - Tsuchiya, N.; Taguchi, S.; Saigo, H. Using category theory to assess the relationship between consciousness and integrated information theory. Neurosci. Res.
**2016**, 107, 1–7. [Google Scholar] [CrossRef] [Green Version] - Stanley, R.P. Qualia space. J. Conscious. Stud.
**1999**, 6, 49–60. [Google Scholar] - Yoshimi, J. Phenomenology and connectionism. Front. Psychol.
**2011**, 2, 288. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hoffman, W.C. Subjective geometry and geometric psychology. Math. Model.
**1980**, 1, 349–367. [Google Scholar] [CrossRef] [Green Version] - Hoffman, W.C. The Lie algebra of visual perception. J. Math. Psychol.
**1966**, 3, 65–98. [Google Scholar] [CrossRef] - Palmer, S.E. Color, consciousness, and the isomorphism constraint. Behav. Brain Sci.
**1999**, 22, 923–943. [Google Scholar] [CrossRef] [Green Version] - Prentner, R. Consciousness and topologically structured phenomenal spaces. Conscious. Cogn.
**2019**, 70, 25–38. [Google Scholar] [CrossRef] [Green Version] - Fekete, T.; Edelman, S. Towards a computational theory of experience. Conscious. Cogn.
**2011**, 20, 807–827. [Google Scholar] [CrossRef] [Green Version] - Eilenberg, S.; MacLane, S. Relations between homology and homotopy groups of spaces. Ann. Math.
**1945**, 46, 480–509. [Google Scholar] [CrossRef] - Baez, J.C.; Stay, M. Physics, Topology, Logic. and Computation: A Rosetta Stone. Available online: https://arxiv.org/abs/0903.0340 (accessed on 10 October 2019).
- Ehresmann, A.C.; Vanbremeersch, J.P. Hierarchical evolutive systems: A mathematical model for complex systems. Bull. Math. Biol.
**1987**, 49, 13–50. [Google Scholar] [CrossRef] - Ehresmann, A.C.; Vanbremeersch, J.P. Information processing and symmetry-breaking in memory evolutive systems. Biosystems
**1997**, 43, 25–40. [Google Scholar] [CrossRef] - Ehresmann, A.C.; Gomez-Ramirez, J. Conciliating neuroscience and phenomenology via category theory. Prog. Biophys. Mol. Biol.
**2015**, 119, 347–359. [Google Scholar] [CrossRef] [PubMed] - Healy, M.J.; Caudell, T.P.; Goldsmith, T.E. A Model of Human Categorization and Similarity Based Upon Category Theory; Electrical & Computer Engineering Technical Reports; University of New Mexico: Albuquerque, NM, USA, 7 January 2008; Report No.: EECE-TR-08-0010; Available online: https://digitalrepository.unm.edu/ece_rpts/28 (accessed on 10 October 2019).
- Phillips, S.; Wilson, W.H. Categorial compositionality: A category theory explanation for the systematicity of human cognition. PLoS Comput. Biol.
**2010**, 6, e1000858. [Google Scholar] [CrossRef] [PubMed] - Phillips, S.; Wilson, W.H. Systematicity and a categorical theory of cognitive architecture: Universal construction in context. Front. Psychol.
**2016**, 7, 1139. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Allison, T.; Ginter, H.; McCarthy, G.; Nobre, A.C.; Puce, A.; Luby, M.; Spencer, D.D. Face recognition in human extrastriate cortex. J. Neurophysiol.
**1994**, 71, 21–25. [Google Scholar] [CrossRef] - Baroni, F.; van Kempen, J.; Kawasaki, H.; Kovach, C.K.; Oya, H.; Howard, M.A.; Adolphs, R.; Tsuchiya, N. Intracranial markers of conscious face perception in humans. Neuroimage
**2017**, 162, 322–343. [Google Scholar] [CrossRef] [Green Version] - Kanwisher, N.; Yovel, G. The fusiform face area: A cortical region specialized for the perception of faces. Philos. Trans. R. Soc. Lond. B Biol. Sci.
**2006**, 361, 2109–2128. [Google Scholar] [CrossRef] [Green Version] - Tong, F.; Nakayama, K.; Vaughan, J.T.; Kanwisher, N. Binocular rivalry and visual awareness in human extrastriate cortex. Neuron
**1998**, 21, 753–759. [Google Scholar] [CrossRef] [Green Version] - Rangarajan, V.; Hermes, D.; Foster, B.L.; Weiner, K.S.; Jacques, C.; Grill-Spector, K.; Parvizi, J. Electrical stimulation of the left and right human fusiform gyrus causes different effects in conscious face perception. J. Neurosci.
**2014**, 34, 12828–12836. [Google Scholar] [CrossRef] - Chialvo, D.R. Emergent complex neural dynamics. Nat. Phys.
**2010**, 6, 744–750. [Google Scholar] [CrossRef] [Green Version] - Rees, G.; Friston, K.; Koch, C. A direct quantitative relationship between the functional properties of human and macaque V5. Nat. Neurosci.
**2000**, 3, 716–723. [Google Scholar] [CrossRef] - Balduzzi, D.; Tononi, G. Qualia: The geometry of integrated information. PLoS Comput. Biol.
**2009**, 5, e1000462. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tononi, G. Information integration: Its relevance to brain function and consciousness. Arch. Ital. Biol.
**2010**, 148, 299–322. [Google Scholar] [PubMed] - Oizumi, M.; Albantakis, L.; Tononi, G. From the phenomenology to the mechanisms of consciousness: Integrated Information Theory 3.0. PLoS Comput. Biol.
**2014**, 10, e1003588. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hidaka, S.; Oizumi, M. Fast and exact search for the partition with minimal information loss. PLoS ONE
**2018**, 13, e0201126. [Google Scholar] [CrossRef] [PubMed] - Toker, D.; Sommer, F.T. Information integration in large brain networks. PLoS Comput. Biol.
**2019**, 15, e1006807. [Google Scholar] [CrossRef] [Green Version] - Tsuchiya, N.; Andrillon, T.; Haun, A. A reply to “the unfolding argument”: Beyond functionalism/behaviorism and towards a truer science of causal structural theories of consciousness. PsyArXiv
**2019**. [Google Scholar] [CrossRef] [Green Version] - Awodey, S. Category Theory; Oxford University Press: Oxford, UK, 2010. [Google Scholar]
- Haun, A.M.; Oizumi, M.; Kovach, C.K.; Kawaski, H.; Oya, H.; Howard, M.A.; Adolphs, R.; Tsuchiya, N. Conscious perception as integrated information patterns in human electrocorticography. eNeuro
**2017**, 4. [Google Scholar] [CrossRef] - Oizumi, M.; Tsuchiya, N.; Amari, S.I. Unified framework for information integration based on information geometry. Proc. Natl. Acad. Sci. USA
**2016**, 113, 14817–14822. [Google Scholar] [CrossRef] [Green Version] - Tegmark, M. Improved measures of integrated information. PLoS Comput. Biol.
**2016**, 12, e1005123. [Google Scholar] [CrossRef] - Northoff, G. Paradox of slow frequencies—Are slow frequencies in upper cortical layers a neural predisposition of the level/state of consciousness (NPC)? Conscious. Cogn.
**2017**, 54, 20–35. [Google Scholar] [CrossRef] - Northoff, G. The anxious brain and its heart—Temporal brain-heart de-synchronization in anxiety disorders. J. Affect. Disord.
**2019**. Forthcoming. [Google Scholar] - He, B.J.; Zempel, J.M. Average is optimal: An inverted-U relationship between trial-to-trial brain activity and behavioral performance. PLoS Comput. Biol.
**2013**, 9, e1003348. [Google Scholar] [CrossRef] [PubMed] - Northoff, G.; Qin, P.; Nakao, T. Rest-stimulus interaction in the brain: A review. Trends Neurosci.
**2010**, 33, 277–284. [Google Scholar] [CrossRef] [PubMed] - Huang, Z.; Zhang, J.; Longtin, A.; Dumont, G.; Duncan, N.W.; Pokorny, J.; Qin, P.; Dai, R.; Ferri, F.; Weng, X.; et al. Is There a Nonadditive Interaction Between Spontaneous and Evoked Activity? Phase-Dependence and Its Relation to the Temporal Structure of Scale-Free Brain Activity. Cereb. Cortex.
**2017**, 27, 1037–1105. [Google Scholar] [CrossRef] [Green Version] - Boly, M.; Phillips, C.; Tshibanda, L.; Vanhaudenhuyse, A.; Schabus, M.; Dang-Vu, T.; Moonen, G.; Hustinx, R.; Maquet, P.; Laureys, S. Intrinsic brain activity in altered states of consciousness: How conscious is the default mode of brain function? Ann. N. Y. Acad. Sci.
**2008**, 1129, 119–129. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hesselmann, G.; Kell, C.A.; Eger, E.; Kleinschmidt, A. Spontaneous local variations in ongoing neural activity bias perceptual decisions. Proc. Natl. Acad. Sci. USA
**2008**, 105, 10984–10989. [Google Scholar] [CrossRef] [Green Version] - Sadaghiani, S.; Hesselmann, G.; Friston, K.J.; Kleinschmidt, A. The relation of ongoing brain activity, evoked neural responses, and cognition. Front. Syst. Neurosci.
**2010**, 4, 20. [Google Scholar] [CrossRef] [Green Version] - Sadaghiani, S.; Hesselmann, G.; Kleinschmidt, A. Distributed and antagonistic contributions of ongoing activity fluctuations to auditory stimulus detection. J. Neurosci.
**2009**, 29, 13410–13417. [Google Scholar] [CrossRef] - Arazi, A.; Censor, N.; Dinstein, I. Neural Variability Quenching Predicts Individual Perceptual Abilities. J. Neurosci.
**2017**, 37, 97–109. [Google Scholar] [CrossRef] [Green Version] - Bai, Y.; Nakao, T.; Xu, J.; Qin, P.; Chaves, P.; Heinzel, A.; Duncan, N.; Lane, T.; Yen, N.S.; Tsai, S.Y.; et al. Resting state glutamate predicts elevated prestimulus alpha during self-relatedness: A combined EEG-MRS study on “rest-self overlap”. Soc. Neurosci.
**2011**, 11, 249–263. [Google Scholar] [CrossRef] - Baria, A.T.; Maniscalco, B.; He, B.J. Initial-state-dependent, robust, transient neural dynamics encode conscious visual perception. PLoS Comput. Biol.
**2017**, 13, e1005806. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Liu, C.H.; Ma, X.; Song, L.P.; Fan, J.; Wang, W.D.; Lv, X.Y.; Zhang, Y.; Li, F.; Wang, L.; Wang, C.-Y. Abnormal spontaneous neural activity in the anterior insular and anterior cingulate cortices in anxious depression. Behav. Brain Res.
**2015**, 281, 339–347. [Google Scholar] [CrossRef] [PubMed] - Northoff, G.; Wainio-Theberge, S.; Evers, K. Is temporospatial dynamics the “common currency” of brain and mind? In Quest of “Spatiotemporal Neuroscience”. Phys. Life Rev.
**2019**. [Google Scholar] [CrossRef] [PubMed] - Oizumi, M.; Amari, S.; Yanagawa, T.; Fujii, N.; Tsuchiya, N. Measuring integrated information from the decoding perspective. PLoS Comput. Biol.
**2016**, 12, e1004654. [Google Scholar] [CrossRef] [PubMed] - Leung, A.; Cohen, D.; van Swinderen, B.; Tsuchiya, N. General anaesthesia reduces integrated information in flies. Monash Univ.
**2018**. [Google Scholar] [CrossRef] - Fong, B.; Spivak, D.I. Seven Sketches in Compositionality: An Invitation to Applied Category Theory. 2018. Available online: https://arxiv.org/abs/1803.05316 (accessed on 10 October 2019).
- Arieli, A.; Sterkin, A.; Grinvald, A.; Aertsen, A. Dynamics of ongoing activity: Explanation of the large variability in evoked cortical responses. Science
**1996**, 273, 1868–1871. [Google Scholar] [CrossRef] [Green Version] - Azouz, R.; Gray, C.M. Cellular mechanisms contributing to response variability of cortical neurons in vivo. J. Neurosci.
**1999**, 19, 2209–2223. [Google Scholar] [CrossRef] [Green Version] - Fox, M.D.; Snyder, A.Z.; Zacks, J.M.; Raichle, M.E. Coherent spontaneous activity accounts for trial-to-trial variability in human evoked brain responses. Nat. Neurosci.
**2006**, 9, 23–25. [Google Scholar] [CrossRef] - Fox, M.D.; Snyder, A.Z.; Vincent, J.L.; Raichle, M.E. Intrinsic fluctuations within cortical systems account for intertrial variability in human behavior. Neuron
**2007**, 56, 171–184. [Google Scholar] [CrossRef] [Green Version] - Fox, M.D.; Raichle, M.E. Spontaneous fluctuations in brain activity observed with functional magnetic resonance imaging. Nat. Rev. Neurosci.
**2007**, 8, 700–711. [Google Scholar] [CrossRef] - Sylvester, C.M.; Shulman, G.L.; Jack, A.I.; Corbetta, M. Anticipatory and stimulus-evoked blood oxygenation level-dependent modulations related to spatial attention reflect a common additive signal. J. Neurosci.
**2009**, 29, 10671–10682. [Google Scholar] [CrossRef] [PubMed] - Ferri, F.; Costantini, M.; Huang, Z.; Perrucci, M.G.; Ferretti, A.; Romani, G.L.; Northoff, G. Intertrial variability in the premotor cortex accounts for individual differences in peripersonal space. J. Neurosci.
**2015**, 35, 16328–16339. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ferri, F.; Nikolova, Y.S.; Perrucci, M.G.; Costantini, M.; Ferretti, A.; Gatta, V.; Huang, Z.; Edden, R.A.E.; Yue, Q.; D’Aurora, M.; et al. A Neural “Tuning Curve” for Multisensory Experience and Cognitive-Perceptual Schizotypy. Schizophr. Bull.
**2017**, 43, 801–813. [Google Scholar] [CrossRef] [PubMed] - Ponce-Alvarez, A.; He, B.J.; Hagmann, P.; Deco, G. Task-driven activity reduces the cortical activity space of the brain: Experiment and whole-brain modeling. PLoS Comput. Biol.
**2015**, 11, e1004445. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Huang, Z.; Zhang, J.; Wu, J.; Liu, X.; Xu, J.; Zhang, J.; Qin, P.; Dai, R.; Yang, Z.; Mao, Y.; et al. Disrupted neural variability during propofol-induced sedation and unconsciousness. Hum. Brain Map.
**2018**, 39, 4533–4544. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Schurger, A.; Sarigiannidis, I.; Naccache, L.; Sitt, J.D.; Dehaene, S. Cortical activity is more stable when sensory stimuli are consciously perceived. Proc. Natl. Acad. Sci. USA
**2015**, 112, E2083–E2092. [Google Scholar] [CrossRef] [Green Version] - Wolff, A.; Di Giovanni, D.A.; Gómez-Pilar, J.; Nakao, T.; Huang, Z.; Longtin, A.; Northoff, G. The temporal signature of self: Temporal measures of resting-state EEG predict self-consciousness. Hum. Brain Map.
**2019**, 40, 789–803. [Google Scholar] [CrossRef] [Green Version] - Wolff, A.; Gómez-Pilar, J.; Nakao, T.; Northoff, G. Interindividual neural difference in moral decision-making are mediated by alpha power and delta/theta phase coherence. Sci. Rep.
**2019**, 9, 4432. [Google Scholar] [CrossRef] - Bayne, T. The Unity of Consciousness; Oxford University Press: Oxford, UK, 2010. [Google Scholar]
- Ebisch, S.J.H.; Gallese, V.; Salone, A.; Martinotti, G.; di Iorio, G.; Mantini, D.; Perrucci, M.G.; Romani, G.L.; Di Giannantonio, M.; Northoff, G. Disrupted relationship between “resting state” connectivity and task-evoked activity during social perception in schizophrenia. Schizophr. Res.
**2018**, 193, 370–376. [Google Scholar] [CrossRef] - Northoff, G.; Duncan, N.W.; Hayes, D.J. The brain and its resting state activity-experimental and methodological implications. Prog. Neurobiol.
**2010**, 92, 593–600. [Google Scholar] [CrossRef] - Martino, D.J.; Samame, C.; Strejilevich, S.A. Stability of facial emotion recognition performance in bipolar disorder. Psych. Res.
**2016**, 243, 182–184. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) Objects, arrows, domain, codomain: Each arrow f is associated with two objects, dom(f) and cod(f), which are called the domain and the codomain of (f). When dom(f) = X and cod(f) = Y, we denote f: X→Y, as shown in Figure 1a. (The direction of the arrow can be in any direction, from left to right or reverse, whichever is convenient.) A system with arrows and objects is called a diagram. (

**b**) Composition: If there are two arrows f and g, such that cod(f) = dom(g), there is a unique arrow, (

**c**) g ∘ f, called the composition of f and g. A diagram is called commutative when any compositions of arrows having the common codomain and domain are equal. (

**d**) Associative law: (h ∘ g) ∘ f = h ∘ (g ∘ f). In other words, the diagram is commutative. (

**e**) Unit law: For any object X there exists an arrow 1X: X→X, such that the diagram is commutative for any f: X→Y. In other words, f ∘ 1X = f = 1Y ∘ f for any f. 1X is called the identity of X.

**Figure 2.**(

**a**) In integrated information theory (IIT), category is defined by an object that is a stochastic network with transition probability matrix (TPM). The exemplar network is composed of a copy gate A and B, which copies the state of the other gate with a time delay of 1. The state of the gate is either on or off. The table on the right describes its TPM. An arrow in category N0 is “decomposition” of the network with TPM (Note that we grossed over various details that are important for IIT3.0 (e.g., distinction between past and future). In particular, how decomposed subnetwork should be embedded with the original network requires careful consideration of so-called “purview” in IIT3.0. Within the IIT’s algorithm, what we call “decomposition” corresponds to a step where one evaluates all potential candidate φ or small phi. For example, for a system ABC, its power set, A, B, C, AB, BC, AC, ABC needs to be evaluated. In some cases, decomposed candidate small phis may not exist, thus it may better be called as “potential decomposition”. However, for simplicity, we prefer to call it as “decomposition”.). Decomposition allows IIT to quantify the causal contribution of a part of the system to the whole. (

**b**) Disconnection arrows find the minimally disconnected network, which captures the concept of the amount of integration in IIT.

**Figure 3.**Schematic depiction of a functor: a structure-preserving mapping from one category to another category.

**Figure 4.**(

**a**) Definition of “inclusion functor”. (

**b**) Subcategory C is included by category D if inclusion functor F: C->D exists. Note that C does not need to be “a part of” D to be “included” (unlike a commonsense definition of “inclusion”).

**Figure 5.**(

**a**) Inclusion Functor i: N0→N1. N0 is included in N1 through Inclusion Functor i. (

**b**) Expansion Functor e: N0→N1. e is a different structure preserving mapping from N0 to N1 (i.e., a functor from N0 to N1), but there is “natural transformation” from i to e.

**Figure 6.**Schematic depiction of a natural transformation: a structure-preserving mapping from one functor to another functor.

**Figure 7.**(

**a**) Inclusion functor, i, expansion functor, e, in the IIT category N0 (actual) and N1 (all possible). Objects in N0 and N1 (e.g., [AB]) are a network with TPM, and arrows in N0 and N1 are manipulation of network/TPM that is allowed in IIT. Within N0, we consider only decomposition arrows. N1 is enriched by additional disconnection arrows that represent an operation that finds a “minimally disconnected” network with TPM within N1. An expansion functor, e, finds the minimally disconnected network (e.g., [AB]’) of the original network (e.g., [AB]), as well e also preserves the structure of N0, and qualifies as a functor. A red arrow within N1 that goes from the actual to the minimally disconnected network corresponds to integrated information, φ. (

**b**) Considering decomposition arrows in N0 allows N0 to consist of a powerset of the network. If natural transformation, t, from the inclusion to the expansion functor exists, t gives us a power set of φ’s, the original and the minimally disconnected network with TPMs. This corresponds to system level integration, Φ.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Northoff, G.; Tsuchiya, N.; Saigo, H.
Mathematics and the Brain: A Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness. *Entropy* **2019**, *21*, 1234.
https://doi.org/10.3390/e21121234

**AMA Style**

Northoff G, Tsuchiya N, Saigo H.
Mathematics and the Brain: A Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness. *Entropy*. 2019; 21(12):1234.
https://doi.org/10.3390/e21121234

**Chicago/Turabian Style**

Northoff, Georg, Naotsugu Tsuchiya, and Hayato Saigo.
2019. "Mathematics and the Brain: A Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness" *Entropy* 21, no. 12: 1234.
https://doi.org/10.3390/e21121234