{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:11:33Z","timestamp":1760242293971,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2017,3,8]],"date-time":"2017-03-08T00:00:00Z","timestamp":1488931200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NIH Common Fund, the Extracellular RNA Communication Consortium","award":["1U01HL126496-01"],"award-info":[{"award-number":["1U01HL126496-01"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We apply a network complexity measure to the gap junction network of the somatic nervous system of C. elegans and find that it possesses a much higher complexity than we might expect from its degree distribution alone. This \u201cexcess\u201d complexity is seen to be caused by a relatively small set of connections involving command interneurons. We describe a method which progressively deletes these \u201ccomplexity-causing\u201d connections, and find that when these are eliminated, the network becomes significantly less complex than a random network. Furthermore, this result implicates the previously-identified set of neurons from the synaptic network\u2019s \u201crich club\u201d as the structural components encoding the network\u2019s excess complexity. This study and our method thus support a view of the gap junction Connectome as consisting of a rather low-complexity network component whose symmetry is broken by the unique connectivities of singularly important rich club neurons, sharply increasing the complexity of the network.<\/jats:p>","DOI":"10.3390\/e19030104","type":"journal-article","created":{"date-parts":[[2017,3,8]],"date-time":"2017-03-08T07:42:58Z","timestamp":1488958978000},"page":"104","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Complexity and Vulnerability Analysis of the C. Elegans Gap Junction Connectome"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3004-7593","authenticated-orcid":false,"given":"James","family":"Kunert-Graf","sequence":"first","affiliation":[{"name":"Pacific Northwest Diabetes Research Institute, Seattle, WA 98122, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nikita","family":"Sakhanenko","sequence":"additional","affiliation":[{"name":"Pacific Northwest Diabetes Research Institute, Seattle, WA 98122, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Galas","sequence":"additional","affiliation":[{"name":"Pacific Northwest Diabetes Research Institute, Seattle, WA 98122, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,3,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Sporns, O. (2011). The Non-Random Brain: Efficiency, Economy, and Complex Dynamics. Front. Comput. Neurosci., 5.","DOI":"10.3389\/fncom.2011.00005"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Gollo, L.L., Zalesky, A., Hutchison, R.M., van den Heuvel, M., and Breakspear, M. (2015). Dwelling quietly in the rich club: Brain network determinants of slow cortical fluctuations. Philos. Trans. R. Soc. Lond. B Biol. Sci., 370.","DOI":"10.1098\/rstb.2014.0165"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Dehmer, M., Barbarini, N., Varmuza, K., and Graber, A. (2009). A Large Scale Analysis of Information-Theoretic Network Complexity Measures Using Chemical Structures. PLoS ONE, 4.","DOI":"10.1371\/journal.pone.0008057"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Emmert-Streib, F., and Dehmer, M. (2012). Exploring Statistical and Population Aspects of Network Complexity. PLoS ONE, 7.","DOI":"10.1371\/journal.pone.0034523"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1002\/cplx.20382","article-title":"Complexity of Networks I: The SetComplexity of Binary Graphs","volume":"17","author":"Sakhanenko","year":"2011","journal-title":"Complexity"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1002\/cplx.21383","article-title":"Complexity of Networks II: The Set Complexity of Edge-colored Graphs","volume":"17","author":"Ignac","year":"2012","journal-title":"Complexity"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Varshney, L.R., Chen, B.L., Paniagua, E., Hall, D.H., and Chklovskii, D.B. (2011). Structural Properties of the Caenorhabditis elegans Neuronal Network. PLoS Comput. Biol., 7.","DOI":"10.1371\/journal.pcbi.1001066"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"956","DOI":"10.1523\/JNEUROSCI.05-04-00956.1985","article-title":"The neural circuit for touch sensitivity in Caenorhabditis elegans","volume":"5","author":"Chalfie","year":"1985","journal-title":"J. Neurosci."},{"key":"ref_9","unstructured":"Sawin, E. (1996). Genetic and Cellular Analysis of Modulated Behaviors in Caenorhabditis elegans. [Ph.D. Thesis, Massachusetts Institute of Technology]."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1016\/S0896-6273(00)81199-X","article-title":"C. elegans locomotory rate is modulated by the environment through a dopaminergic pathway and by experience through a serotonergic pathway","volume":"26","author":"Sawin","year":"2000","journal-title":"Neuron"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3184","DOI":"10.1073\/pnas.0409009101","article-title":"A circuit for navigation in Caenorhabditis elegans","volume":"102","author":"Gray","year":"2005","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/0896-6273(95)90242-2","article-title":"Sensory regulation of male mating behavior in Caenorhabditis elegans","volume":"14","author":"Liu","year":"1995","journal-title":"Neuron"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1038\/nature07886","article-title":"A hub-and-spoke circuit drives pheromone attraction and social behaviour in C. elegans","volume":"458","author":"Macosko","year":"2009","journal-title":"Nature"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"8060","DOI":"10.1523\/JNEUROSCI.23-22-08060.2003","article-title":"Genetic and cellular basis for acetylcholine inhibition of Caenorhabditis elegans egg-laying behavior","volume":"23","author":"Bany","year":"2003","journal-title":"J. Neurosci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1002\/neu.10014","article-title":"Serotonin modulates locomotory behavior and coordinates egg-laying and movement in Caenorhabditis elegans","volume":"49","author":"Hardaker","year":"2001","journal-title":"J. Neurobiol."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Bassett, D.S., Greenfield, D.L., Meyer-Lindenberg, A., Weinberger, D.R., Moore, S.W., and Bullmore, E.T. (2010). Efficient physical embedding of topologically complex information processing networks in brains and computer circuits. PLoS Comput. Biol., 6.","DOI":"10.1371\/journal.pcbi.1000748"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"15775","DOI":"10.1523\/JNEUROSCI.3539-11.2011","article-title":"Rich-Club Organization of the Human Connectome","volume":"31","author":"Sporns","year":"2011","journal-title":"J. Neurosci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1038\/nrn3214","article-title":"The economy of brain network organization","volume":"13","author":"Bullmore","year":"2012","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Harriger, L., van den Heuvel, M.P., and Sporns, O. (2012). Rich club organization of macaque cerebral cortex and its role in network communication. PLoS ONE, 7.","DOI":"10.1371\/journal.pone.0046497"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"7456","DOI":"10.1073\/pnas.1324118111","article-title":"Rich-club organization of the newborn human brain","volume":"111","author":"Ball","year":"2014","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5459","DOI":"10.1523\/JNEUROSCI.4259-14.2015","article-title":"Emergence of rich-club topology and coordinated dynamics in development of hippocampal functional networks in vitro","volume":"35","author":"Schroeter","year":"2015","journal-title":"J. Neurosci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"670","DOI":"10.1523\/JNEUROSCI.2177-15.2016","article-title":"Rich-club organization in effective connectivity among cortical neurons","volume":"36","author":"Nigam","year":"2016","journal-title":"J. Neurosci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"5675","DOI":"10.1523\/JNEUROSCI.0754-16.2016","article-title":"Further Insight into the Brain\u2019s Rich-Club Architecture","volume":"36","author":"Pedersen","year":"2016","journal-title":"J. Neurosci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6380","DOI":"10.1523\/JNEUROSCI.3784-12.2013","article-title":"The Rich Club of the C. elegans Neuronal Connectome","volume":"33","author":"Towlson","year":"2013","journal-title":"J. Neurosci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"052805","DOI":"10.1103\/PhysRevE.89.052805","article-title":"Low-dimensional functionality of complex network dynamics: Neurosensory integration in the Caenorhabditis elegans connectome","volume":"89","author":"Kunert","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Kim, S., Kim, H., Kralik, J.D., and Jeong, J. (2016). Vulnerability-Based Critical Neurons, Synapses, and Pathways in the Caenorhabditis elegans Connectome. PLoS Comput. Biol., 12.","DOI":"10.1371\/journal.pcbi.1005084"},{"key":"ref_27","unstructured":"Hu, Y., Brunton, S.L., Cain, N., Mihalas, S., Kutz, J.N., and Shea-Brown, E. (arXiv, 2016). Feedback through graph motifs relates structure and function in complex networks, arXiv."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1038\/nphys209","article-title":"Detecting rich-club ordering in complex networks","volume":"2","author":"Colizza","year":"2006","journal-title":"Nat. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"824","DOI":"10.1126\/science.298.5594.824","article-title":"Network Motifs: Simple Building Blocks of Complex Networks","volume":"298","author":"Milo","year":"2002","journal-title":"Science"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Sporns, O., and K\u00f6tter, R. (2004). Motifs in Brain Networks. PLoS Biol., 2.","DOI":"10.1371\/journal.pbio.0020369"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Qian, J., Hintze, A., and Adami, C. (2011). Colored Motifs Reveal Computational Building Blocks in the C. elegans Brain. PLoS ONE, 6.","DOI":"10.1371\/journal.pone.0017013"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1089\/cmb.2013.0039","article-title":"Describing the Complexity of Systems: Multivariable \u201cSet Complexity\u201d and the Information Basis of Systems Biology","volume":"21","author":"Galas","year":"2014","journal-title":"J. Comput. Biol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1089\/cmb.2015.0051","article-title":"Biological Data Analysis as an Information Theory Problem: Multivariable Dependence Measures and the Shadows Algorithm","volume":"22","author":"Sakhanenko","year":"2015","journal-title":"J. Comput. Biol."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Bohland, J.W., Wu, C., Barbas, H., Bokil, H., Bota, M., Breiter, H.C., Cline, H.T., Doyle, J.C., Freed, P.J., and Greenspan, R.J. (2009). A Proposal for a Coordinated Effort for the Determination of Brainwide Neuroanatomical Connectivity in Model Organisms at a Mesoscopic Scale. PLoS Comput. Biol., 5.","DOI":"10.1371\/journal.pcbi.1000334"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cub.2010.11.056","article-title":"Three-Dimensional Reconstruction of Brain-wide Wiring Networks in Drosophila at Single-Cell Resolution","volume":"21","author":"Chiang","year":"2011","journal-title":"Curr. Biol."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1038\/nature13186","article-title":"A mesoscale connectome of the mouse brain","volume":"508","author":"Oh","year":"2014","journal-title":"Nature"},{"key":"ref_37","unstructured":"Peixoto, T.P. (2014). The graph-tool python library. Figshare."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"White, J.G., Southgate, E., Thomson, J.N., and Brenner, S. (1986). The structure of the nervous system of the nematode Caenorhabditis elegans. Philos. Trans. R. Soc. Lond. B Biol. Sci., 314.","DOI":"10.1098\/rstb.1986.0056"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1523\/JNEUROSCI.11-01-00001.1991","article-title":"The posterior nervous system of the nematode Caenorhabditis elegans: Serial reconstruction of identified neurons and complete pattern of synaptic interactions","volume":"11","author":"Hall","year":"1991","journal-title":"J. Neurosci."},{"key":"ref_40","unstructured":"Durbin, R. (1987). Studies on the Development and Organisation of the Nervous System of Caenorhabditis elegans. [Ph.D. Thesis, University of Cambridge]."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1109\/LCOMM.2004.823426","article-title":"The rich-club phenomenon in the internet topology","volume":"8","author":"Zhou","year":"2004","journal-title":"IEEE Commun. Lett."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/3\/104\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:29:58Z","timestamp":1760207398000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/3\/104"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,8]]},"references-count":41,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,3]]}},"alternative-id":["e19030104"],"URL":"https:\/\/doi.org\/10.3390\/e19030104","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2017,3,8]]}}}