The brain is characterized by heterogeneous patterns of structural connections supporting unparalleled feats of cognition and a wide range of behaviours. New non-invasive imaging techniques now allow comprehensive mapping of these patterns. However, a fundamental challenge remains to understand how the brain’s structural wiring supports cognitive processes, with major implications for personalized mental health treatments. Here, we review recent efforts to meet this challenge, drawing on physics intuitions, models and theories, spanning the domains of statistical mechanics, information theory, dynamical systems and control. We first describe the organizing principles of brain network architecture instantiated in structural wiring under constraints of spatial embedding and energy minimization. We then survey models of brain network function that stipulate how neural activity propagates along structural connections. Finally, we discuss perturbative experiments and models for brain network control; these use the physics of signal transmission along structural connections to infer intrinsic control processes that support goal-directed behaviour and to inform stimulation-based therapies for neurological and psychiatric disease. Throughout, we highlight open questions that invite the creative efforts of pioneering physicists.
From the first measurement of the nerve impulse by Hermann von Helmholtz in 1849 to the cutting-edge superconducting devices used in magnetoencephalography, physics and neuroscience have always been inextricably linked.
Today, network neuroscience — the study of the brain as a complex web of interacting components — draws intuitions and techniques from nearly every branch of physics.
The architecture of structural connections between neurons or brain regions is constrained by requirements of energy minimization and efficient information transfer.
The materialization of long-range correlations and synchronization from the collective firing of individual neurons conjures notions of emergence and criticality from statistical mechanics.
Together, these investigations of brain network structure and function guide targeted treatments for cognitive disorders using theories of network control.
Now more than ever, understanding the complexities of the mind lies at the feet of curious and pioneering physicists.
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Scott, A. Neurophysics (Wiley, 1977).
Koch, C. & Poggio, T. A theoretical analysis of electrical properties of spines. Proc. R. Soc. Lond. B Biol. Sci. 218, 455–477 (1983).
Tyler, W. J. The mechanobiology of brain function. Nat. Rev. Neurosci. 13, 867–878 (2012).
Friston, K., Kilner, J. & Harrison, L. A free energy principle for the brain. J. Physiol. Paris 100, 70–87 (2006).
Plewes, D. B. & Kucharczyk, W. Physics of MRI: a primer. J. Magn. Reson Imaging 35, 1038–1054 (2012).
Hari, R. & Salmelin, R. Magnetoencephalography: from SQUIDs to neuroscience. Neuroimage 20th anniversary special edition. Neuroimage 61, 386–396 (2012).
Boto, E. et al. Moving magnetoencephalography towards real-world applications with a wearable system. Nature 555, 657–661 (2018).
Alivisatos, A. P. et al. Nanotools for neuroscience and brain activity mapping. ACS Nano 7, 1850–1866 (2013).
Piazza, S., Bianchini, P., Sheppard, C., Diaspro, A. & Deisseroth, K. Enhanced volumetric imaging in 2-photon microscopy via acoustic lens beam shaping. J. Biophotonics 11, e201700050 (2018).
Boyden, E. S., Zhang, F., Bamberg, E., Nagel, G. & Deisseroth, K. Millisecond-timescale, genetically targeted optical control of neural activity. Nat. Neurosci. 8, 1263–1268 (2005).
McCulloch, W. S. & Pitts, W. A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biol. 5, 115–133 (1943).
Fries, P. Rhythms for cognition: communication through coherence. Neuron 88, 220–235 (2015).
Betzel, R. F. & Bassett, D. S. Specificity and robustness of long-distance connections in weighted, interareal connectomes. Proc. Natl Acad. Sci. USA 115, E4880–E4889 (2018).
Van Essen, D. C. et al. The WU-Minn human connectome project: an overview. Neuroimage 80, 62–79 (2013).
Markram, H. et al. Reconstruction and simulation of neocortical microcircuitry. Cell 163, 456–492 (2015).
Poo, M. M. et al. China brain project: basic neuroscience, brain diseases, and brain-inspired computing. Neuron 92, 591–596 (2016).
Okano, H., Miyawaki, A. & Kasai, K. Brain/MINDS: brain-mapping project in Japan. Philos. Trans. R. Soc. Lond. B Biol. Sci. 370, 20140310 (2015).
Bassett, D. S. & Gazzaniga, M. S. Understanding complexity in the human brain. Trends Cogn. Sci. 15, 200–209 (2011).
Sethna, J. P. Statistical Mechanics: Entropy, Order Parameters and Complexity (Oxford University Press, 2006).
Bassett, D. S. & Bullmore, E. T. Small-world brain networks revisited. Neuroscientist 23, 499–516 (2016).
Albert, E. & Barabasi, A.-L. Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47 (2002).
Butts, C. T. Revisiting the foundations of network analysis. Science 325, 414–416 (2009).
Costa, Ld. F., Rodrigues, F. A., Travieso, G. & Villas Boas, P. R. Characterization of complex networks: a survey of measurements. Adv. Phys. 56, 167–242 (2006).
Gross, T. & Blasius, B. Adaptive coevolutionary networks: a review. J. R. Soc. Interface 5, 259–271 (2008).
Zhang, X., Moore, C. & Newman, M. E. J. Random graph models for dynamic networks. Eur. Phys. J. B 90, 200 (2017).
Hackett, A., Melnik, s & Gleeson, J. P. Cascades on a class of clustered random networks. Phys. Rev. E 83, 056107 (2011).
Newman, M. E. J. The structure and function of complex networks. Siam Rev. 45, 167–256 (2003).
Motter, A. E. Networkcontrology. Chaos 25, 097621 (2015).
Bassett, D. S., Zurn, P. & Gold, J. I. On the nature and use of models in network neuroscience. Nat. Rev. Neurosci. 19, 566–578 (2018).
Pereda, A. E. Electrical synapses and their functional interactions with chemical synapses. Nat. Rev. Neurosci. 15, 250–263 (2014).
Avena-Koenigsberger, A., Misic, B. & Sporns, O. Communication dynamics in complex brain networks. Nat. Rev. Neurosci. 19, 17–33 (2017).
Ising, E. Beitrag zur theorie des ferromagnetismus [German]. Z. Für Phys. 31, 253–258 (1925).
Onsager, L. Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117 (1944).
Brush, S. G. History of the lenz-ising model. Rev. Mod. Phys. 39, 883 (1967).
Sporns, O., Chialvo, D. R., Kaiser, M. & Hilgetag, C. C. Organization, development and function of complex brain networks. Trends Cogn. Sci. 8, 418–425 (2004).
Medaglia, J. D., Lynall, M. E. & Bassett, D. S. Cognitive network neuroscience. J. Cogn. Neurosci. 27, 1471–1491 (2015).
Sporns, O. Contributions and challenges for network models in cognitive neuroscience. Nat. Neurosci. 17, 652–660 (2014).
Petersen, S. E. & Sporns, O. Brain networks and cognitive architectures. Neuron 88, 207–219 (2015).
Misic, B. & Sporns, O. From regions to connections and networks: new bridges between brain and behavior. Curr. Opin. Neurobiol. 40, 1–7 (2016).
Wallace, E., Maei, H. R. & Latham, P. E. Randomly connected networks have short temporal memory. Neural Comput. 25, 1408–1439 (2013).
Rajan, K., Harvey, C. D. & Tank, D. W. Recurrent network models of sequence generation and memory. Neuron 90, 128–142 (2016).
Chaudhuri, R. & Fiete, I. Computational principles of memory. Nat. Neurosci. 19, 394–403 (2016).
Hermundstad, A. M., Brown, K. S., Bassett, D. S. & Carlson, J. M. Learning, memory, and the role of neural network architecture. PLoS Comput. Biol. 7, e1002063 (2011).
Teşileanu, T., Olveczky, B. & Balasubramanian, V. Rules and mechanisms for efficient two-stage learning in neural circuits. Elife 6, e20944 (2017).
Takemura, S. Y. et al. A visual motion detection circuit suggested by drosophila connectomics. Nature 500, 175–181 (2013).
Zhen, M. & Samuel, A. D. C. elegans locomotion: small circuits, complex functions. Curr. Opin. Neurobiol. 33, 117–126 (2015).
Shepherd, G. M. Foundations of the Neuron Doctrine (Oxford University Press, 2015).
White, J. G., Southgate, E., Thomson, J. N. & Brenner, S. The structure of the nervous system of the nematode Caenorhabditis elegans. Philos. Trans. R. Soc. Lond. B 314, 1–340 (1986).
Helmstaedter, M. et al. Connectomic reconstruction of the inner plexiform layer in the mouse retina. Nature 500, 168–174 (2013).
Sporns, O., Tononi, G. & Kötter, R. The human connectome: a structural description of the human brain. PLoS Comput. Biol. 1, e42 (2005).
Hsieh, J. et al. Computed Tomography: Principles, Design, Artifacts, and Recent Advances. (SPIE Bellingham, 2009).
Pierpaoli, C., Jezzard, P., Basser, P. J., Barnett, A. & Di Chiro, G. Diffusion tensor MR imaging of the human brain. Radiology 201, 637–648 (1996).
Basser, P. J., Pajevic, S., Pierpaoli, C., Duda, J. & Aldroubi, A. In vivo fiber tractography using DT-MRI data. Magn. Reson Med 44, 625–632 (2000).
Behrens, T. E. & Johansen-Berg, H. Relating connectional architecture to grey matter function using diffusion imaging. Philos. Trans. R. Soc. Lond. B Biol. Sci. 360, 903–911 (2005).
Stephan, K. E. et al. Advanced database methodology for the collation of connectivity data on the Macaque brain (CoCoMac). Philos. Trans. R. Soc. Lond. B Biol. Sci. 356, 1159–1186 (2001).
Markov, N. T. et al. A weighted and directed interareal connectivity matrix for macaque cerebral cortex. Cereb. Cortex 24, 17–36 (2014).
Young, M. P., Scannell, J. W., Burns, G. A. & Blakemore, C. Analysis of connectivity: neural systems in the cerebral cortex. Rev. Neurosci. 5, 227–250 (1994).
Oh, S. W. et al. A mesoscale connectome of the mouse brain. Nature 508, 207–214 (2014).
Shih, C. T. et al. Connectomics-based analysis of information flow in the Drosophila brain. Curr. Biol. 25, 1249–1258 (2015).
Bullmore, E. & Sporns, O. The economy of brain network organization. Nat. Rev. Neurosci. 13, 336–349 (2012).
Betzel, R. F. & Bassett, D. S. Generative models for network neuroscience: prospects and promise. J. R. Soc. Interface 14, 20170623 (2017).
Bassett, D. S. & Sporns, O. Network neuroscience. Nat. Neurosci. 20, 353–364 (2017).
Thompson, P. M. et al. Genetic influences on brain structure. Nat. Neurosci. 4, 1253 (2001).
Raz, N. et al. Regional brain changes in aging healthy adults: general trends, individual differences and modifiers. Cereb. Cortex 15, 1676–1689 (2005).
Gong, G. et al. Age-and gender-related differences in the cortical anatomical network. J. Neurosci. 29, 15684–15693 (2009).
Kanai, R. & Rees, G. The structural basis of inter-individual differences in human behaviour and cognition. Nat. Rev. Neurosci. 12, 231 (2011).
Banissy, M. J., Kanai, R., Walsh, V. & Rees, G. Inter-individual differences in empathy are reflected in human brain structure. Neuroimage 62, 2034–2039 (2012).
Fleming, S. M., Weil, R. S., Nagy, Z., Dolan, R. J. & Rees, G. Relating introspective accuracy to individual differences in brain structure. Science 329, 1541–1543 (2010).
Hartley, C. A., Fischl, B. & Phelps, E. A. Brain structure correlates of individual differences in the acquisition and inhibition of conditioned fear. Cereb. Cortex 21, 1954–1962 (2011).
Kanai, R., Feilden, T., Firth, C. & Rees, G. Political orientations are correlated with brain structure in young adults. Curr. Biol. 21, 677–680 (2011).
Erdös, P. & Rényi, A. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–60 (1960).
Sherrington, C. S. The Integrative Action of the Nervous System (Yale University Press, 1906).
Sporns, O., Tononi, G. & Edelman, G. M. Theoretical neuroanatomy: relating anatomical and functional connectivity in graphs and cortical connection matrices. Cereb. cortex 10, 127–141 (2000).
Hilgetag, C.-C., Burns, G. A., O’Neill, M. A., Scannell, J. W. & Young, M. P. Anatomical connectivity defines the organization of clusters of cortical areas in the macaque and the cat. Philos. Trans. R. Soc. Lon. B 355, 91–110 (2000).
Sporns, O. & Zwi, J. D. The small world of the cerebral cortex. Neuroinformatics 2, 145–162 (2004).
Sporns, O. & Betzel, R. F. Modular brain networks. Annu Rev. Psychol. 67, 613–640 (2016).
Bassett, D. S. et al. Efficient physical embedding of topologically complex information processing networks in brains and computer circuits. PLoS Comput. Biol. 6, e1000748 (2010).
Taylor, P. N., Wang, Y. & Kaiser, M. Within brain area tractography suggests local modularity using high resolution connectomics. Sci. Rep. 7, 39859 (2017).
Lesicko, A. M., Hristova, T. S., Maigler, K. C. & Llano, D. A. Connectional modularity of top-down and bottom-up multimodal inputs to the lateral cortex of the mouse inferior colliculus. J. Neurosci. 36, 11037–11050 (2016).
Sohn, Y., Choi, M. K., Ahn, Y. Y., Lee, J. & Jeong, J. Topological cluster analysis reveals the systemic organization of the Caenorhabditis elegans connectome. PLoS Comput. Biol. 7, e1001139 (2011).
Azulay, A., Itskovits, E. & Zaslaver, A. The C. elegans connectome consists of homogenous circuits with defined functional roles. PLoS Comput. Biol. 12, e1005021 (2016).
Betzel, R. F. & Bassett, D. S. Multi-scale brain networks. Neuroimage 160, 73–83 (2017).
Khambhati, A. N., Sizemore, A. E., Betzel, R. F. & Bassett, D. S. Modeling and interpreting mesoscale network dynamics. Neuroimage 180, 337–349 (2017).
Aicher, C., Jacobs, A. Z. & Clauset, A. Learning latent block structure in weighted networks. J. Complex Netw. 3, 221–248 (2015).
Betzel, R. F., Medaglia, J. D. & Bassett, D. S. Diversity of meso-scale architecture in human and non-human connectomes. Nat. Commun. 9, 346 (2018).
van den Heuvel, M. P. & Sporns, O. Network hubs in the human brain. Trends Cogn. Sci. 17, 683–696 (2013).
Liao, X., Vasilakos, A. V. & He, Y. Small-world human brain networks: perspectives and challenges. Neurosci. Biobehav Rev. 77, 286–300 (2017).
Deco, G., Tononi, G., Boly, M. & Kringelbach, M. L. Rethinking segregation and integration: contributions of whole-brain modelling. Nat. Rev. Neurosci. 16, 430 (2015).
Latora, V. & Marchiori, M. Efficient behavior of small-world networks. Phys. Rev. Lett. 87, 198701 (2001).
Kaiser, M. & Hilgetag, C. C. Nonoptimal component placement, but short processing paths, due to long-distance projections in neural systems. PLOS Comput. Biol. 2, e95 (2006).
Travers, J. & Milgram, S. The small world problem. Phychology Today 1, 61–67 (1967).
Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998).
Gong, G. et al. Mapping anatomical connectivity patterns of human cerebral cortex using in vivo diffusion tensor imaging tractography. Cereb. cortex 19, 524–536 (2008).
Wedeen, V. J., Hagmann, P., Tseng, W.-Y. I., Reese, T. G. & Weisskoff, R. M. Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging. Magn. Reson. Med. 54, 1377–1386 (2005).
de Solla Price, D. J. Networks of scientific papers. Science 149, 510–515 (1965).
Barabasi, A. L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999).
Dall, J. & Christensen, M. Random geometric graphs. Phys. Rev. E 66, 016121 (2002).
Vertes, P. E. et al. Simple models of human brain functional networks. Proc. Natl Acad. Sci. USA 109, 5868–5873 (2012).
Rubinov, M., Ypma, R., Watson, C. & Bullmore, E. Wiring cost and topological participation of the mouse brain connectome. Proc. Natl Acad. Sci. USA 112, 10032–7 (2015).
Kaiser, M. Mechanisms of connectome development. Trends Cogn. Sci. 21, 703–717 (2017).
Stam, C. J. Modern network science of neurological disorders. Nat. Rev. Neurosci. 15, 683–695 (2014).
Scholtens, L. H., Schmidt, R., de Reus, M. A. & van den Heuvel, M. P. Linking macroscale graph analytical organization to microscale neuroarchitectonics in the macaque connectome. J. Neurosci. 34, 12192–12205 (2014).
Chaudhuri, R., Knoblauch, K., Gariel, M. A., Kennedy, H. & Wang, X. J. A large-scale circuit mechanism for hierarchical dynamical processing in the primate cortex. Neuron 88, 419–431 (2015).
Breakspear, M. Dynamic models of large-scale brain activity. Nat. Neurosci. 20, 340–352 (2017).
Bentley, B. et al. The multilayer connectome of Caenorhabditis elegans. PLoS Comput. Biol. 12, e1005283 (2016).
Mejias, J. F., Murray, J. D., Kennedy, H. & Wang, X. J. Feedforward and feedback frequency-dependent interactions in a large-scale laminar network of the primate cortex. Sci. Adv. 2, e1601335 (2016).
Seung, H. S. & Sumbul, U. Neuronal cell types and connectivity: lessons from the retina. Neuron 83, 1262–1272 (2014).
Arnatkeviciute, A., Fulcher, B. D., Pocock, R. & Fornito, A. Hub connectivity, neuronal diversity, and gene expression in the Caenorhabditis elegans connectome. PLoS Comput. Biol. 14, e1005989 (2018).
Nicosia, V., Vértes, P. E., Schafer, W. R., Latora, V. & Bullmore, E. T. Phase transition in the economically modeled growth of a cellular nervous system. Proc. Natl Acad. Sci. USA 110, 7880–7885 (2013).
Scholz, J., Klein, M. C., Behrens, T. E. & Johansen-Berg, H. Training induces changes in white-matter architecture. Nat. Neurosci. 12, 1370–1371 (2009).
Baum, G. L. et al. Modular segregation of structural brain networks supports the development of executive function in youth. Curr. Biol. 27, 1561–1572 (2017).
Zuo, X. N. et al. Human connectomics across the life span. Trends Cogn. Sci. 21, 32–45 (2017).
Holme, P. & Saramaki, J. Temporal networks. Phys. Rep. 519, 97–125 (2012).
Li, A., Cornelius, S. P., Liu, Y.-Y., Wang, L. & Barabási, A.-L. The fundamental advantages of temporal networks. Science 358, 1042–1046 (2017).
Hebb, D. The Organization of Behavior (Wiley, 1949).
Magee, J. C. & Johnston, D. A synaptically controlled, associative signal for hebbian plasticity in hippocampal neurons. Science 275, 209–213 (1997).
Montague, P. R., Dayan, P. & Sejnowski, T. J. A framework for mesencephalic dopamine systems based on predictive hebbian learning. J. Neurosci. 16, 1936–1947 (1996).
Song, S., Miller, K. D. & Abbott, L. F. Competitive hebbian learning through spike-timing-dependent synaptic plasticity. Nat. Neurosci. 3, 919 (2000).
Chialvo, D. R. Emergent complex neural dynamics. Nat. Phys. 6, 744 (2010).
Tononi, G., Boly, M., Massimini, M. & Koch, C. Integrated information theory: from consciousness to its physical substrate. Nat. Rev. Neurosci. 17, 450–461 (2016).
Abbott, L. F. & Dayan, P. Theoretical Neuroscience (MIT Press, 2001).
Dechery, J. B. & MacLean, J. N. Emergent cortical circuit dynamics contain dense, interwoven ensembles of spike sequences. J. Neurophysiol. 118, 1914–1925 (2017).
Brody, C. D. Correlations without synchrony. Neural Comput. 11, 1537–1551 (1999).
Brody, C. D. Disambiguating different covariation types. Neural Comput. 11, 1527–1535 (1999).
Sporns, O., Tononi, G. & Edelman, G. M. Connectivity and complexity: the relationship between neuroanatomy and brain dynamics. Neural Netw. 13, 909–922 (2000).
Schneidman, E., Berry, M. J. II, Segev, R. & Bialek, W. Weak pairwise correlations imply strongly correlated network states in a neural population. Nature 440, 1007 (2006).
Levina, A., Herrmann, J. M. & Geisel, T. Dynamical synapses causing self-organized criticality in neural networks. Nat. Phys. 3, 857 (2007).
Vuksanovic, V. & Hovel, P. Functional connectivity of distant cortical regions: role of remote synchronization and symmetry in interactions. Neuroimage 97, 1–8 (2014).
Green, D. J. & Gillette, R. Circadian rhythm of firing rate recorded from single cells in the rat suprachiasmatic brain slice. Brain Res. 245, 198–200 (1982).
Edwards, F. A., Konnerth, A., Sakmann, B. & Takahashi, T. A thin slice preparation for patch clamp recordings from neurones of the mammalian central nervous system. Pflüg. Arch. 414, 600–612 (1989).
Stosiek, C., Garaschuk, O., Holthoff, K. & Konnerth, A. In vivo two-photon calcium imaging of neuronal networks. Proc. Natl Acad. Sci. USA 100, 7319–7324 (2003).
Grewe, B. F., Langer, D., Kasper, H., Kampa, B. M. & Helmchen, F. High-speed in vivo calcium imaging reveals neuronal network activity with near-millisecond precision. Nat. Methods 7, 399 (2010).
Penny, W. D., Friston, K. J., Ashburner, J. T., Kiebel, S. J. & Nichols, T. E. Statistical Parametric Mapping: The Analysis of Functional Brain Images (Elsevier, 2011).
Hämäläinen, M., Hari, R., Ilmoniemi, R. J., Knuutila, J. & Lounasmaa, O. V. Magnetoencephalography–theory, instrumentation, and applications to noninvasive studies of the working human brain. Rev. Mod. Phys. 65, 413 (1993).
Bailey, D. L., Maisey, M. N., Townsend, D. W. & Valk, P. E. Positron Emission Tomography (Springer, 2005).
Raichle, M. E. Behind the scenes of functional brain imaging: a historical and physiological perspective. Proc. Natl Acad. Sci. USA 95, 765–772 (1998).
Zarahn, E., Aguirre, G. K. & D’Esposito, M. Empirical analyses of bold fmri statistics. Neuroimage 5, 179–197 (1997).
Van Den Heuvel, M. P. & Pol, H. E. H. Exploring the brain network: a review on resting-state fmri functional connectivity. Eur. Neuropsychopharmacol. 20, 519–534 (2010).
Bullmore, E. & Sporns, O. Complex brain networks: graph theoretical analysis of structural and functional systems. Nat. Rev. Neurosci. 10, 186–198 (2009).
Zalesky, A., Fornito, A. & Bullmore, E. On the use of correlation as a measure of network connectivity. Neuroimage 60, 2096–2106 (2012).
He, Y. et al. Uncovering intrinsic modular organization of spontaneous brain activity in humans. PloS One 4, e5226 (2009).
Salvador, R. et al. Neurophysiological architecture of functional magnetic resonance images of human brain. Cereb. Cortex 15, 1332–1342 (2005).
Achard, S., Salvador, R., Whitcher, B., Suckling, J. & Bullmore, E. A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs. J. Neurosci. 26, 63–72 (2006).
Bettencourt, L. M., Stephens, G. J., Ham, M. I. & Gross, G. W. Functional structure of cortical neuronal networks grown in vitro. Phys. Rev. E 75, 021915 (2007).
Sadovsky, A. J. & MacLean, J. N. Scaling of topologically similar functional modules defines mouse primary auditory and somatosensory microcircuitry. J. Neurosci. 33, 14048–14060 (2013).
Yue, Q. et al. Brain modularity mediates the relation between task complexity and performance. J. Cogn. Neurosci. 29, 1532–1546 (2017).
Bassett, D. S. & Bullmore, E. Small-world brain networks. Neuroscientist 12, 512–523 (2006).
Rosenbaum, R., Smith, M. A., Kohn, A., Rubin, J. E. & Doiron, B. The spatial structure of correlated neuronal variability. Nat. Neurosci. 20, 107–114 (2017).
Goñi, J. et al. Resting-brain functional connectivity predicted by analytic measures of network communication. Proc. Natl Acad. Sci. USA 111, 833–838 (2014).
Honey, C. et al. Predicting human resting-state functional connectivity from structural connectivity. Proc. Natl Acad. Sci. USA 106, 2035–2040 (2009).
Park, H.-J. & Friston, K. Structural and functional brain networks: from connections to cognition. Science 342, 1238411 (2013).
David, O. & Friston, K. J. A neural mass model for meg/eeg:: coupling and neuronal dynamics. NeuroImage 20, 1743–1755 (2003).
David, O., Cosmelli, D. & Friston, K. J. Evaluation of different measures of functional connectivity using a neural mass model. Neuroimage 21, 659–673 (2004).
Cabral, J., Hugues, E., Sporns, O. & Deco, G. Role of local network oscillations in resting-state functional connectivity. Neuroimage 57, 130–139 (2011).
Ganmor, E., Segev, R. & Schneidman, E. Sparse low-order interaction network underlies a highly correlated and learnable neural population code. Proc. Natl Acad. Sci. USA 108, 9679–9684 (2011).
Medaglia, J. D. et al. Functional alignment with anatomical networks is associated with cognitive flexibility. Nat. Human. Behav. 2, 156–164 (2018).
Hodgkin, A. L. & Huxley, A. F. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952).
FitzHugh, R. Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961).
Beurle, R. L. Properties of a mass of cells capable of regenerating pulses. Philos. Trans. R. Soc. Lond. B 240, 55–94 (1956).
Wilson, H. R. & Cowan, J. D. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12, 1–24 (1972).
Hopfield, J. J. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl Acad. Sci. USA 79, 2554–2558 (1982).
Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence Vol. 19 (Springer Science & Business Media, 2012).
Cash, S. & Yuste, R. Linear summation of excitatory inputs by ca1 pyramidal neurons. Neuron 22, 383–394 (1999).
Ferrell, J. E. & Machleder, E. M. The biochemical basis of an all-or-none cell fate switch in xenopus oocytes. Science 280, 895–898 (1998).
Hearst, M. A., Dumais, S. T., Osuna, E., Platt, J. & Scholkopf, B. Support vector machines. IEEE Intell. Syst. 13, 18–28 (1998).
Kleene, S. C. Representation of Events in Nerve Nets and Finite Automata (RAND Corporation,1951).
Schmidhuber, J. Deep learning in neural networks: an overview. Neural Netw. 61, 85–117 (2015).
Egmont-Petersen, M., de Ridder, D. & Handels, H. Image processing with neural networks–a review. Pattern Recognit. 35, 2279–2301 (2002).
Hinton, G. et al. Deep neural networks for acoustic modeling in speech recognition: the shared views of four research groups. IEEE Signal Process. Mag. 29, 82–97 (2012).
Silver, D. et al. Mastering the game of go with deep neural networks and tree search. Nature 529, 484 (2016).
Newman, C. M. Memory capacity in neural network models: rigorous lower bounds. Neural Netw. 1, 223–238 (1988).
Hertz, J., Krogh, A. & Palmer, R. G. Introduction to the Theory of Neural Computation. (Addison-Wesley/Addison Wesley Longman, 1991).
Moosavi, S. A. & Montakhab, A. Structural versus dynamical origins of mean-field behavior in a self-organized critical model of neuronal avalanches. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 92, 052804 (2015).
Woodrow, W. L. et al. Adaptation to sensory input tunes visual cortex to criticality. Nat. Phys. 11, 659–663 (2015).
Haldeman, C. & Beggs, J. M. Critical branching captures activity in living neural networks and maximizes the number of metastable states. Phys. Rev. Lett. 94, 058101 (2005).
Beggs, J. M. & Plenz, D. Neuronal avalanches in neocortical circuits. J. Neurosci. 23, 11167–11177 (2003).
Kinouchi, O. & Copelli, M. Optimal dynamical range of excitable networks at criticality. Nat. Phys. 2, 348–351 (2006).
Shew, W. L., Yang, H., Petermann, T., Roy, R. & Plenz, D. Neuronal avalanches imply maximum dynamic range in cortical networks at criticality. J. Neurosci. 29, 15595–15600 (2009).
Bertschinger, N. & Natschläger, T. Real-time computation at the edge of chaos in recurrent neural networks. Neural Comput. 16, 1413–1436 (2004).
Lee, S.-G., Neiman, A. & Kim, S. Coherence resonance in a hodgkin-huxley neuron. Phys. Rev. E 57, 3292 (1998).
Hille, B. et al. Ion Channels of Excitable Membranes 507 (Sinauer Sunderland, 2001).
Plant, R. & Kim, M. Mathematical description of a bursting pacemaker neuron by a modification of the hodgkin-huxley equations. Biophys. J. 16, 227–244 (1976).
Andersen, S. S., Jackson, A. D. & Heimburg, T. Towards a thermodynamic theory of nerve pulse propagation. Prog. Neurobiol. 88, 104–113 (2009).
Pakdaman, K., Thieullen, M. & Wainrib, G. Fluid limit theorems for stochastic hybrid systems with application to neuron models. Adv. Appl. Probab. 42, 761–794 (2010).
Nagumo, J., Arimoto, S. & Yoshizawa, S. An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1962).
Niebur, E. & Erdös, P. Theory of the locomotion of nematodes: control of the somatic motor neurons by interneurons. Math. Biosci. 118, 51–82 (1993).
Bryden, J. & Cohen, N. In From Animals to Animats 8: Proc. Eighth Int. Conf. Sim. Adapt. Behav. (eds Schaal, S. et al.)183–192 (MIT Press, 2004).
Arena, P., Patané, L. & Termini, P. S. In 2010 Int. Joint Conf. Neurol Networks https://doi.org/10.1109/IJCNN.2010.5596513 (IEEE, 2010).
Markram, H. The blue brain project. Nat. Rev. Neurosci. 7, 153 (2006).
Kishimoto, K. & Amari, S.-i Existence and stability of local excitations in homogeneous neural fields. J. Math. Biol. 7, 303–318 (1979).
Pinto, D. J. & Ermentrout, G. B. Spatially structured activity in synaptically coupled neuronal networks: I. Traveling fronts and pulses. SIAM J. Appl. Math. 62, 206–225 (2001).
Deco, G., Jirsa, V., McIntosh, A. R., Sporns, O. & Kötter, R. Key role of coupling, delay, and noise in resting brain fluctuations. Proc. Natl Acad. Sci. USA 106, 10302–10307 (2009).
Kuramoto, Y. & Araki, H. Lecture notes in physics, international symposium on mathematical problems in theoretical physics (1975).
Ward, L. M. Synchronous neural oscillations and cognitive processes. Trends Cogn. Sci. 7, 553–559 (2003).
Fries, P. A mechanism for cognitive dynamics: neuronal communication through neuronal coherence. Trends Cogn. Sci. 9, 474–480 (2005).
Palmigiano, A., Geisel, T., Wolf, F. & Battaglia, D. Flexible information routing by transient synchrony. Nat. Neurosci. 20, 1014–1022 (2017).
Schnitzler, A. & Gross, J. Normal and pathological oscillatory communication in the brain. Nat. Rev. Neurosci. 6, 285 (2005).
Petersson, K. M., Nichols, T. E., Poline, J.-B. & Holmes, A. P. Statistical limitations in functional neuroimaging. I. Non-inferential methods and statistical models. Philos. Trans. R. Soc. Lond., B, Biol. Sci. 354, 1239–1260 (1999).
Petersson, K. M., Nichols, T. E., Poline, J.-B. & Holmes, A. P. Statistical limitations in functional neuroimaging ii. signal detection and statistical inference. Philos. Trans. R. Soc. Lond., B, Biol. Sci. 354, 1261–1281 (1999).
Bancaud, J. & Talairach, J. Methodology of stereo eeg exploration and surgical intervention in epilepsy. Rev. Otoneuroophtalmol. 45, 315–328 (1973).
Chauvel, P., Vignal, J., Biraben, A., Badier, J. & Scarabin, J. Stereoelectroencephalography, 80–108 (Springer Verlag, 1996).
Todaro, C., Marzetti, L., Valdes Sosa, P. A., Valdes-Hernandez, P. A. & Pizzella, V. Mapping brain activity with electrocorticography: resolution properties and robustness of inverse solutions. Brain Topogr. https://doi.org/10.1007/s10548-018-0623-1 (2018).
Menon, R. S. & Kim, S.-G. Spatial and temporal limits in cognitive neuroimaging with fmri. Trends Cogn. Sci. 3, 207–216 (1999).
Aguirre, G. K. Functional neuroimaging: technical, logical, and social perspectives. Hastings Cent. Rep. 44, S8–S18 (2014).
Ciric, R. et al. Benchmarking of participant-level confound regression strategies for the control of motion artifact in studies of functional connectivity. Neuroimage 154, 174–187 (2017).
Avants, B. B. et al. A reproducible evaluation of ANTs similarity metric performance in brain image registration. Neuroimage 54, 2033–2044 (2011).
Lynall, M. E. et al. Functional connectivity and brain networks in schizophrenia. J. Neurosci. 30, 9477–9487 (2010).
Bassett, D. S. et al. Hierarchical organization of human cortical networks in health and schizophrenia. J. Neurosci. 28, 9239–9248 (2008).
Khazaee, A., Ebrahimzadeh, A. & Babajani-Feremi, A. Identifying patients with alzheimer’s disease using resting-state fMRI and graph theory. Clin. Neurophysiol. 126, 2132–2141 (2015).
Amari, S.-i, Nakahara, H., Wu, S. & Sakai, Y. Synchronous firing and higher-order interactions in neuron pool. Neural Comput. 15, 127–142 (2003).
Sizemore, A. E. et al. Cliques and cavities in the human connectome. J. Comput. Neurosci. 44, 115–145 (2017).
Giusti, C., Ghrist, R. & Bassett, D. S. Two’s company, three (or more) is a simplex: algebraic-topological tools for understanding higher-order structure in neural data. J. Comput. Neurosci. 41, 1–14 (2016).
Giusti, C., Pastalkova, E., Curto, C. & Itskov, V. Clique topology reveals intrinsic geometric structure in neural correlations. Proc. Natl Acad. Sci. USA 112, 13455–13460 (2015).
Reimann, M. W. et al. Cliques of neurons bound into cavities provide a missing link between structure and function. Front Comput. Neurosci. 11, 48 (2017).
Battaglia, D., Witt, A., Wolf, F. & Geisel, T. Dynamic effective connectivity of inter-areal brain circuits. PLoS Comput. Biol. 8, e1002438 (2012).
Zylberberg, J., Pouget, A., Latham, P. E. & Shea-Brown, E. Robust information propagation through noisy neural circuits. PLoS Comput. Biol. 13, e1005497 (2017).
Kirst, C., Timme, M. & Battaglia, D. Dynamic information routing in complex networks. Nat. Commun. 7, 11061 (2016).
McIntyre, C. C., Savasta, M., Kerkerian-Le Goff, L. & Vitek, J. L. Uncovering the mechanism(s) of action of deep brain stimulation: activation, inhibition, or both. Clin. Neurophysiol. 115, 1239–1248 (2004).
Lozano, A. M. & Lipsman, N. Probing and regulating dysfunctional circuits using deep brain stimulation. Neuron 77, 406–424 (2013).
Liu, Y.-Y. & Barabási, A.-L. Control principles of complex systems. Rev. Mod. Phys. 88, 035006 (2016).
Schiff, S. J. Neural Control Engineering: The Emerging Intersection between Control Theory and Neuroscience (MIT Press, 2012).
Kim, J. Z. et al. Role of graph architecture in controlling dynamical networks with applications to neural systems. Nat. Phys. 14, 91–98 (2018).
Gu, S. et al. Controllability of structural brain networks. Nat. Commun. 6, 8414 (2015).
Jeganathan, J. et al. Fronto-limbic dysconnectivity leads to impaired brain network controllability in young people with bipolar disorder and those at high genetic risk. Neuroimage Clin. 19, 71–81 (2018).
Muldoon, S. F. et al. Stimulation-based control of dynamic brain networks. PLoS Comput. Biol. 12, e1005076 (2016).
Taylor, P. N. et al. Optimal control based seizure abatement using patient derived connectivity. Front Neurosci. 9, 202 (2015).
Medaglia, J. D. et al. Network controllability in the inferior frontal gyrus relates to controlled language variability and susceptibility to TMS. J. Neurosci. 38, 6399–6410 (2018).
Holt, A. B., Wilson, D., Shinn, M., Moehlis, J. & Netoff, T. I. Phasic burst stimulation: a closed-loop approach to tuning deep brain stimulation parameters for Parkinson’s disease. PLoS Comput. Biol. 12, e1005011 (2016).
Holmes, G. Disturbances of vision by cerebral lesions. Br. J. Ophthalmol. 2, 353 (1918).
Owen, A. M., Downes, J. J., Sahakian, B. J., Polkey, C. E. & Robbins, T. W. Planning and spatial working memory following frontal lobe lesions in man. Neuropsychologia 28, 1021–1034 (1990).
Walsh, V. & Cowey, A. Transcranial magnetic stimulation and cognitive neuroscience. Nat. Rev. Neurosci. 1, 73 (2000).
Amassian, V. E. et al. Measurement of information processing delays in human visual cortex with repetitive magnetic coil stimulation. Brain Res. 605, 317–321 (1993).
Pascual-Leone, A., Grafman, J. & Hallett, M. Modulation of cortical motor output maps during development of implicit and explicit knowledge. Science 263, 1287–1289 (1994).
Pascual-Leone, A., Gates, J. R. & Dhuna, A. Induction of speech arrest and counting errors with rapid-rate transcranial magnetic stimulation. Neurology 41, 697–702 (1991).
Walsh, V., Ellison, A., Battelli, L. & Cowey, A. Task–specific impairments and enhancements induced by magnetic stimulation of human visual area V5. Proc. R. Soc. Lond., B, Biol. Sci. 265, 537–543 (1998).
Kringelbach, M. L., Jenkinson, N., Owen, S. L. & Aziz, T. Z. Translational principles of deep brain stimulation. Nat. Rev. Neurosci. 8, 623 (2007).
George, M. S., Lisanby, S. H. & Sackeim, H. A. Transcranial magnetic stimulation: applications in neuropsychiatry. Arch. General. Psychiatry 56, 300–311 (1999).
Perlmutter, J. S. & Mink, J. W. Deep brain stimulation. Annu. Rev. Neurosci. 29, 229–257 (2006).
Tass, P. et al. Detection of n: m phase locking from noisy data: Application to magnetoencephalography. Phys. Rev. Lett. 81, 3291 (1998).
Santaniello, S. et al. Therapeutic mechanisms of high-frequency stimulation in parkinson’s disease and neural restoration via loop-based reinforcement. Proc. Natl Acad. Sci. USA 112, E586–E595 (2015).
Zeki, S. A Vision of the Brain (Blackwell Scientific Publ., 1993).
Chiken, S. & Nambu, A. Disrupting neuronal transmission: mechanism of dbs? Front. Syst. Neurosci. 8, 33 (2014).
Berényi, A., Belluscio, M., Mao, D. & Buzsáki, G. Closed-loop control of epilepsy by transcranial electrical stimulation. Science 337, 735–737 (2012).
Kedzior, K. K., Gierke, L., Gellersen, H. M. & Berlim, M. T. Cognitive functioning and deep transcranial magnetic stimulation (dtms) in major psychiatric disorders: a systematic review. J. Psychiatr. Res. 75, 107–115 (2016).
Ching, S. et al. Real-time closed-loop control in a rodent model of medically induced coma using burst suppression. Anesthesiology 119, 848–860 (2013).
Holt, A. B. & Netoff, T. I. Origins and suppression of oscillations in a computational model of parkinson’s disease. J. Comput. Neurosci. 37, 505–521 (2014).
Heck, C. N. et al. Two-year seizure reduction in adults with medically intractable partial onset epilepsy treated with responsive neurostimulation: final results of the RNS System Pivotal trial. Epilepsia 55, 432–441 (2014).
Crinion, J. et al. Spatial normalization of lesioned brains: performance evaluation and impact on fmri analyses. Neuroimage 37, 866–875 (2007).
Santaniello, S., Fiengo, G., Glielmo, L. & Grill, W. M. Closed-loop control of deep brain stimulation: a simulation study. IEEE Trans. Neural Syst. Rehabil. Eng. 19, 15–24 (2011).
Iudice, F. L., Garofalo, F. & Sorrentino, F. Structural permeability of complex networks to control signals. Nat. Commun. 6, 8349 (2015).
Posner, M. I., Snyder, C. R. & Solso, R. in Cognitive Psychology: Key Readings (Key Readings in Cognition) 205–223 (eds Balota, D. & Marsh, E.) (Psychology Press, 2004).
Fuster, J. M. & Alexander, G. E. Neuron activity related to short-term memory. Science 173, 652–654 (1971).
Goldman, P. S. & Rosvold, H. E. Localization of function within the dorsolateral prefrontal cortex of the rhesus monkey. Exp. Neurol. 27, 291–304 (1970).
Bechara, A., Damasio, A. R., Damasio, H. & Anderson, S. W. Insensitivity to future consequences following damage to human prefrontal cortex. Cognition 50, 7–15 (1994).
Dias, R., Robbins, T. & Roberts, A. Dissociation in prefrontal cortex of affective and attentional shifts. Nature 380, 69 (1996).
Gu, S. et al. Optimal trajectories of brain state transitions. Neuroimage 148, 305–317 (2017).
Betzel, R. F., Gu, S., Medaglia, J. D., Pasqualetti, F. & Bassett, D. S. Optimally controlling the human connectome: the role of network topology. Sci. Rep. 6, 30770 (2016).
Pasqualetti, F., Zampieri, S. & Bullo, F. Controllability metrics, limitations and algorithms for complex networks. IEEE Trans. Control Netw. Syst. 1, 40–52 (2014).
Yan, G. et al. Network control principles predict neuron function in the Caenorhabditis elegans connectome. Nature 550, 519–523 (2017).
Tang, E. & Bassett, D. S. Control of dynamics in brain networks. Preprint in arXiv https://arxiv.org/abs/1701.01531 (2018).
Tang, E. et al. Developmental increases in white matter network controllability support a growing diversity of brain dynamics. Nat. Commun. 8, 1252 (2017).
Cornblath, E. J. et al. Sex differences in network controllability as a predictor of executive function in youth. NeuroImage 188, 122–134 (2019).
Adamantidis, A. R., Zhang, F., Aravanis, A. M., Deisseroth, K. & De Lecea, L. Neural substrates of awakening probed with optogenetic control of hypocretin neurons. Nature 450, 420 (2007).
Deisseroth, K. Optogenetics. Nat. Methods 8, 26 (2011).
Gunaydin, L. A. et al. Ultrafast optogenetic control. Nat. Neurosci. 13, 387 (2010).
Grosenick, L., Marshel, J. H. & Deisseroth, K. Closed-loop and activity-guided optogenetic control. Neuron 86, 106–139 (2015).
Prakash, R. et al. Two-photon optogenetic toolbox for fast inhibition, excitation and bistable modulation. Nat. Methods 9, 1171 (2012).
Rickgauer, J. P., Deisseroth, K. & Tank, D. W. Simultaneous cellular-resolution optical perturbation and imaging of place cell firing fields. Nat. Neurosci. 17, 1816 (2014).
Becker, C. O., Bassett, D. & Preciado, V. M. Large-scale dynamic modeling of task-fMRI signals via subspace system identification. J. Neural Eng. 15, 066016 (2018).
Coron, J.-M. Control and Nonlinearity 136 (American Mathematical Soc., 2007).
Klickstein, I., Shirin, A. & Sorrentino, F. Locally optimal control of complex networks. Phys. Rev. Let. 119, 268301 (2017).
Haynes, G. & Hermes, H. Nonlinear controllability via lie theory. SIAM J. Control 8, 450–460 (1970).
Sussmann, H. J. & Jurdjevic, V. Controllability of nonlinear systems. Differ. Equ. 12, 95–116 (1972).
Hermann, R. & Krener, A. Nonlinear controllability and observability. IEEE Trans. Autom. Contr. 22, 728–740 (1977).
Cornelius, S. P., Kath, W. L. & Motter, A. E. Realistic control of network dynamics. Nat. Commun. 4, 1942 (2013).
Whalen, A. J., Brennan, S. N., Sauer, T. D. & Schiff, S. J. Observability and controllability of nonlinear networks: the role of symmetry. Phys. Rev. X 5, 011005 (2015).
Isidori, A. Nonlinear Control Systems (Springer Science & Business Media, 2013).
Chopra, N. & Spong, M. W. On exponential synchronization of kuramoto oscillators. IEEE Trans. Autom. Contr. 54, 353–357 (2009).
Lynn, C. W. & Lee, D. D. Statistical mechanics of influence maximization with thermal noise. EPL 117, 66001 (2017).
Lynn, C. W. & Lee, D. D. In Thirty-Second AAAI Conference on Artificial Intelligence 679–686 (AAAI, 2018).
Amunts, K. & Zilles, K. Architectonic mapping of the human brain beyond Brodmann. Neuron 88, 1086–1107 (2015).
Cohen, M. R. & Kohn, A. Measuring and interpreting neuronal correlations. Nat. Neurosci. 14, 811–819 (2011).
van den Heuvel, M. P., Bullmore, E. T. & Sporns, O. Comparative connectomics. Trends Cogn. Sci. 20, 345–361 (2016).
Persichetti, A. S., Aguirre, G. K. & Thompson-Schill, S. L. Value is in the eye of the beholder: early visual cortex codes monetary value of objects during a diverted attention task. J. Cogn. Neurosci. 27, 893–901 (2015).
Dore, B. P. et al. Brain activity tracks population information sharing by capturing consensus judgments of value. Cereb Cortex https://doi.org/10.1093/cercor/bhy176 (2018).
Constantinescu, A. O., O’Reilly, J. X. & Behrens, T. E. J. Organizing conceptual knowledge in humans with a gridlike code. Science 352, 1464–1468 (2016).
Kailath, T. Linear Systems (Prentice-Hall, Inc., 1980).
Liu, Y.-Y., Slotine, J.-J. & Barabási, A.-L. Controllability of complex networks. Nature 473, 167–173 (2011).
Klickstein, I., Shirin, A. & Sorrentino, F. Energy scaling of targeted optimal control of complex networks. Nat. Commun. 8, 15145 (2017).
The authors are grateful to L. Papadopoulos, J. Z. Kim and V. Buch for helpful comments on an earlier version of this manuscript. The authors also thank A. E. Sizemore for artistic inspiration. D.S.B. and C.W.L. acknowledge support from the Penn NSF MRSEC Grant DMR-1720530, the John D. and Catherine T. MacArthur Foundation, the Alfred P. Sloan Foundation, the ISI Foundation, the Paul Allen Foundation, the Army Research Laboratory (W911NF-10-2-0022), the Army Research Office (Bassett-W911NF-14-1-0679, Grafton-W911NF-16-1-0474 and DCIST- W911NF-17-2-0181), the Office of Naval Research, the US National Institute of Mental Health (2-R01-DC-009209-11, R01-MH112847, R01-MH107235 and R21-M MH-106799), the US National Institute of Child Health and Human Development (1R01HD086888-01), the US National Institute of Neurological Disorders and Stroke (R01 NS099348) and the US National Science Foundation (BCS-1441502, BCS-1430087, NSF PHY-1554488 and BCS-1631550).
Nature Reviews Physics thanks Y. He and the other anonymous reviewer(s) for their contribution to the peer review of this work.
The authors declare no competing interests.
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Lynn, C.W., Bassett, D.S. The physics of brain network structure, function and control. Nat Rev Phys 1, 318–332 (2019). https://doi.org/10.1038/s42254-019-0040-8
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