Cognitive Neurodynamics

, Volume 11, Issue 3, pp 283–292 | Cite as

From abstract topology to real thermodynamic brain activity

  • Arturo Tozzi
  • James F. Peters
Research Article


Recent approaches to brain phase spaces reinforce the foremost role of symmetries and energy requirements in the assessment of nervous activity. Changes in thermodynamic parameters and dimensions occur in the brain during symmetry breakings and transitions from one functional state to another. Based on topological results and string-like trajectories into nervous energy landscapes, we provide a novel method for the evaluation of energetic features and constraints in different brain functional activities. We show how abstract approaches, namely the Borsuk–Ulam theorem and its variants, may display real, energetic physical counterparts. When topology meets the physics of the brain, we arrive at a general model of neuronal activity, in terms of multidimensional manifolds and computational geometry, that has the potential to be operationalized.


Topology Borsuk–Ulam theorem Brain Nervous system Energetic landscape Symmetry break 


  1. Afraimovich V, Tristan I, Varona P, Rabinovich M (2013) Transient dynamics in complex systems: heteroclinic sequences with multidimensional unstable manifolds. Discontin Nonlinearity Complex 2(1):21–41CrossRefGoogle Scholar
  2. Attwell D, Laughlin SB (2001) An energy budget for signaling in the grey matter of the brain. J Cereb Blood Flow Metab Off J Int Soc Cereb Blood Flow Metab 21(10):1133–1145. doi: 10.1097/00004647-200110000-00001 CrossRefGoogle Scholar
  3. Benson AR, Gleich DF, Leskovec J (2016) Higher-order organization of complex networks. Science 353(6295):163–166. doi: 10.1126/science.aad9029 CrossRefPubMedPubMedCentralGoogle Scholar
  4. Biswal BB, Mennes M, Zuo XN, Gohel S, Kelly C et al (2010) Toward discovery science of human brain function. Proc Natl Acad Sci USA 107(10):4734–4739. doi: 10.1073/pnas.0911855107 CrossRefPubMedPubMedCentralGoogle Scholar
  5. Borsuk M (1933) Drei satze uber die n-dimensionale euklidische sphare. Fundam Math 20:177–190Google Scholar
  6. Deco G, Jirsa VK (2012) Ongoing cortical activity at rest: criticality, multistability, and ghost attractors. J Neurosci 32(10):3366–3375. doi: 10.1523/JNEUROSCI.2523-11.2012 CrossRefPubMedGoogle Scholar
  7. Di Concilio A (2013) Point-free geometries: proximities and quasi-metrics. Math Comput Sci 7(1):31–42CrossRefGoogle Scholar
  8. Di Concilio A, Gerla G (2006) Quasi-metric spaces and point-free geometry. Math Struct Comput Sci 16(1):115137CrossRefGoogle Scholar
  9. Disalle R (1995) Spacetime theory as physical geometry. Erkenntnis 42(3):317–337CrossRefGoogle Scholar
  10. Dodson CTJ (1997) A user’s guide to algebraic topology. Kluwer Academic Publishers, DordrechtCrossRefGoogle Scholar
  11. Dol’nikov VL (1992) A generalization of the ham sandwich theorem. Math Notes 52:771–779CrossRefGoogle Scholar
  12. Fraiman D, Chialvo DR (2012) What kind of noise is brain noise: anomalous scaling behavior of the resting brain activity fluctuations. Front Physiol 3:1–11. doi: 10.3389/fphys.2012.00307 CrossRefGoogle Scholar
  13. Friston K (2010) The free-energy principle: a unified brain theory? Nat Rev Neurosci 11(2):127–138. doi: 10.1038/nrn2787 CrossRefPubMedGoogle Scholar
  14. Glasser MF, Smith SM, Marcus DS, Andersson JLR, Auerbach EJ et al (2016) The human connectome project’s neuroimaging approach. Nat Neurosci 19:1175–1187. doi: 10.1038/nn.4361 CrossRefPubMedGoogle Scholar
  15. Goddard P, Olive D (1985) Algebras, lattices and strings. In: Lepowsky J, Mandelstam S, Singer IM (eds) Vertex operators in mathematics and physics. Mathematical Sciences Research Institute Publications, vol 3. Springer, New York, NYGoogle Scholar
  16. Jirsa VK, Friedrich R, Haken H, Kelso JAS (1994) A theoretical model of phase transitions in the human brain. Biol Cybern 71:27. doi: 10.1007/BF00198909 CrossRefPubMedGoogle Scholar
  17. Jirsa VK, Fuchs A, Kelso JAS (1998) Connecting cortical and behavioral dynamics: bimanual coordination. Neural Comput Arch 10(8):2019–2045CrossRefGoogle Scholar
  18. Kalmbach AS, Waters J (2012) Brain surface temperature under a craniotomy. J Neurophysiol 108(11):3138–3146CrossRefPubMedPubMedCentralGoogle Scholar
  19. Kida T, Tanaka E, Kakigi R (2016) Multi-dimensional dynamics of human electromagnetic brain activity. Front Hum Neurosci 9:713. doi: 10.3389/fnhum.2015.00713 CrossRefPubMedPubMedCentralGoogle Scholar
  20. Kim I-S (1997) Extensions of the Borsuk–Ulam theorem. J Korean Math Soc 34(3):599Google Scholar
  21. Kim SY, Lim W (2015) Frequency-domain order parameters for the burst and spike synchronization transitions of bursting neurons. Cogn Neurodyn 9(4):411–421. doi: 10.1007/s11571-015-9334-4 CrossRefPubMedPubMedCentralGoogle Scholar
  22. Kleineberg K-K, Boguñá M, Serrano MA, Papadopoulos F (2016) Hidden geometric correlations in real multiplex networks. Nature Physics, in press. doi: 10.1038/nphys3812 Google Scholar
  23. Lech RK, Güntürkün O, Suchan B (2016) An interplay of fusiform gyrus and hippocampus enables prototype- and exemplar-based category learning. Behav Brain Res. doi: 10.1016/j.bbr.2016.05.049 PubMedGoogle Scholar
  24. Lenzen VF (1939) Physical geometry. Am Math Mon 46:324–334CrossRefGoogle Scholar
  25. Long MA, Fee MS (2008) Using temperature to analyse temporal dynamics in the songbird motor pathway. Nature 456(7219):189–194CrossRefPubMedPubMedCentralGoogle Scholar
  26. Marsaglia G (1972) Choosing a Point from the Surface of a Sphere. Ann Math Stat 43(2):645–646. doi: 10.1214/aoms/1177692644 CrossRefGoogle Scholar
  27. Matoušek J (2003) Using the Borsuk–Ulam theorem. Lectures on topological methods in combinatorics and geometry. Springer, BerlinGoogle Scholar
  28. Mazzucato L, La Fontanini A, Camera G (2016) Stimuli Reduce the Dimensionality of Cortical Activity. Syst. Neurosci, in press, Front. doi: 10.3389/fnsys.2016.00011 Google Scholar
  29. Mitroi-Symeonidis F-C (2015) Convexity and sandwich theorems. Eur J Res Appl Sci 1:9–11Google Scholar
  30. Noether E (1918) Invariante Variationsprobleme. Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen. Math-phys. Klasse 1918:235–257Google Scholar
  31. Olive DI, Landsberg PT (1989) Introduction to string theory: its structure and its uses. Philos Trans R Soc Lond Ser A Math Phys Sci 329:319–328CrossRefGoogle Scholar
  32. Papo D (2014) Functional significance of complex fluctuations in brain activity: from resting state to cognitive neuroscience. Front Syst Neurosci 8:112. doi: 10.3389/fnsys.2014.00112 CrossRefPubMedPubMedCentralGoogle Scholar
  33. Peters JF (2016) Computational Proximity. Excursions in the Topology of Digital Images. Edited by Intelligent Systems Reference Library. Springer, Berlin. doi: 10.1007/978-3-319-30262-1
  34. Peters JF, Naimpally SA (2012) Applications of near sets. Not Am Math Soc 59(4):536–542. doi: 10.1090/noti817 Google Scholar
  35. Peters JF, Tozzi A (2016a) Region-Based Borsuk–Ulam Theorem. arXiv:1605.02987
  36. Peters JF, Tozzi A (2016b) String-Based Borsuk–Ulam Theorem. arXiv:1606.04031v1
  37. Peters JF, Tozzi A, Ramanna S (2016) Brain tissue tessellation shows absence of canonical microcircuits. Neurosci Lett 626:99–105. doi: 10.1016/j.neulet.2016.03.052 CrossRefPubMedGoogle Scholar
  38. Petty CM (1971) Equivalent sets in Minkowsky spaces. Proc Am Math Soc 29(2):369–374CrossRefGoogle Scholar
  39. Roldán É, Martínez I, Parrondo JMR, Petrov D (2014) Universal features in the energetics of symmetry breaking. Nat Phys 10(6):457–461. doi: 10.1038/nphys2940 CrossRefGoogle Scholar
  40. Schleicher D (2007) Hausdorff dimension, its properties, and its surprises. Am Math Mon 114(6):509–528Google Scholar
  41. Schneidman E, Berry MJ, Segev R, Bialek W (2006) Weak pairwise correlations imply strongly correlated network states in a neural population. Nature 440:1007–1012CrossRefPubMedPubMedCentralGoogle Scholar
  42. Scholz JP, Kelso JAS, Schöner G (1987) Nonequilibrium phase transitions in coordinated biological motion: critical slowing down and switching time. Phys Lett A 123(8):390–394. doi: 10.1016/0375-9601(87)90038-7 CrossRefGoogle Scholar
  43. Sengupta B, Stemmler MB, Friston KJ (2013a) Information and efficiency in the nervous system—A synthesis. PLoS Comput Biol. doi: 10.1371/journal.pcbi.1003157 Google Scholar
  44. Sengupta B, Laughlin SB, Niven JE (2013b) Balanced excitatory and inhibitory synaptic currents promote efficient coding and metabolic efficiency. PLoS Comput Biol. doi: 10.1371/journal.pcbi.1003263 Google Scholar
  45. Sengupta B, Tozzi A, Cooray GK, Douglas PK, Friston KJ (2016) Towards a neuronal gauge theory. PLoS Biol 14(3):e1002400. doi: 10.1371/journal.pbio.1002400 CrossRefPubMedPubMedCentralGoogle Scholar
  46. Simas T, Chavez M, Rodriguez PR, Diaz-Guilera A (2015) An algebraic topological method for multimodal brain networks comparisons. Front Psychol. 6(6):904. doi: 10.3389/fpsyg.2015.00904 PubMedPubMedCentralGoogle Scholar
  47. Stemmler M, Mathis A, Herz AVM (2015) Connecting multiple spatial scales to decode the population activity of grid cells. Sci Adv 1:e1500816CrossRefPubMedPubMedCentralGoogle Scholar
  48. Tognoli E, Kelso JS (2013) On the brain’s dynamical complexity: coupling and causal influences across spatiotemporal scales. Adv Cognit Neurodyn (III), pp 259–265. doi: 10.1007/978-94-007-4792-0
  49. Touboul J (2012) Mean-field equations for stochastic firing-rate neural fields with delays: derivation and noise-induced transitions. Phy D Nonlinear Phenom 241(15):1223–1244. doi: 10.1016/j.physd.2012.03.010 CrossRefGoogle Scholar
  50. Tozzi A (2015) Information processing in the CNS: a supramolecular chemistry? Cogn Neurodyn 9(5):463–477. doi: 10.1007/s11571-015-9337-1 (Review)CrossRefPubMedPubMedCentralGoogle Scholar
  51. Tozzi A (2016) Borsuk–Ulam Theorem Extended to Hyperbolic Spaces. In: Computational Proximity. Excursions in the Topology of Digital Images, edited by JF Peters, pp 169–171. doi: 10.1007/978-3-319-30262-1
  52. Tozzi A, Peters JF (2016a) Towards a fourth spatial dimension of brain activity. Cogn Neurodyn 10(3):189–199. doi: 10.1007/s11571-016-9379-z CrossRefPubMedGoogle Scholar
  53. Tozzi A, Peters JF (2016b) A topological approach unveils system invariances and broken symmetries in the brain. J Neurosci Res 94(5):351–365. doi: 10.1002/jnr.23720 CrossRefPubMedGoogle Scholar
  54. Tozzi A, Fla T, Peters PJ (2016a) Building a minimum frustration framework for brain functions in long timescales. J Neurosci Res. doi: 10.1002/jnr.23748 Google Scholar
  55. Tozzi A, Zare M, Benasich AA (2016b) New perspectives on spontaneous brain activity: dynamic networs and energy matter. Front Hum Neurosci. doi: 10.3389/fnhum.2016.00247 PubMedPubMedCentralGoogle Scholar
  56. Van Essen DC (2005) A population-average, landmark- and surface-based (PALS) atlas of human cerebral cortex. Neuroimage 28:635–666CrossRefPubMedGoogle Scholar
  57. Wang H, Wang B, Normoyle KP, Jackson K, Spitler K (2014a) Brain temperature and its fundamental properties: a review for clinical neuroscientists. Front Neurosci 8(8):307PubMedPubMedCentralGoogle Scholar
  58. Wang Z, Li Y, Childress AR, Detre JA (2014b) Brain entropy mapping using fMRI. PLoS ONE 9(3):1–8. doi: 10.1371/journal.pone.0089948 Google Scholar
  59. Wang Y, Wang R, Zhu Y (2017) Optimal path-finding through mental exploration based on neural energy field gradients. Cogn Neurodyn 11(1):99–111. doi: 10.1007/s11571-016-9412-2 CrossRefPubMedGoogle Scholar
  60. Watanabe T, Hirose S, Wada H, Imai Y, Machida T, Shirouzu I, Masuda N (2013) A pairwise maximum entropy model accurately describes resting-state human brain networks. Nat Commun 4:1370. doi: 10.1038/ncomms2388 CrossRefPubMedPubMedCentralGoogle Scholar
  61. Watanabe T, Kan S, Koike T, Misaki M, Konishi S, Miyauchi S, Masuda N (2014) Network-dependent modulation of brain activity during sleep. NeuroImage 98:1–10. doi: 10.1016/j.neuroimage.2014.04.079 CrossRefPubMedGoogle Scholar
  62. Weeks JR (2002) The shape of space, 2nd edn. Marcel Dekker, inc, New YorkGoogle Scholar
  63. Weyl H (1982) Symmetry. Princeton University Press, Princeton. ISBN 0-691-02374-3Google Scholar
  64. Willard S (1970) General topology. Dover Pub. Inc, MineolaGoogle Scholar
  65. Xu X, Wang R (2014) Neurodynamics of up and down transitions in a single neuron. Cogn Neurodyn 8(6):509–515. doi: 10.1007/s11571-014-9298-9 CrossRefPubMedPubMedCentralGoogle Scholar
  66. Yan H, Zhao L, Hu L, Wang X, Wang E, Wang J (2013) Nonequilibrium landscape theory of neural networks. PNAS 110(45):4185–4194CrossRefGoogle Scholar
  67. Zare M, Grigolini P (2013) Chaos, Solitons & Fractals Criticality and avalanches in neural networks. Chaos, Solitons and Fractals: The Interdisciplinary Journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 55:80–94. doi: 10.1016/j.chaos.2013.05.009 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Center for Nonlinear Science, Department of PhysicsUniversity of North TexasDentonUSA
  2. 2.Department of Electrical and Computer EngineeringUniversity of ManitobaWinnipegCanada

Personalised recommendations