March  2015, 10(1): 223-232. doi: 10.3934/nhm.2015.10.223

EEG-based functional brain networks: Hemispheric differences in males and females

1. 

Department of Computer Engineering, Sharif University of Technology, Azadi Av., Tehran, Iran

Received  June 2014 Revised  December 2014 Published  February 2015

Functional connectivity in human brain can be represented as a network using electroencephalography (EEG) signals. Network representation of EEG time series can be an efficient vehicle to understand the underlying mechanisms of brain function. Brain functional networks whose nodes are brain regions and edges correspond to functional links between them are characterized by neurobiologically meaningful graph theory metrics. This study investigates the degree to which graph theory metrics are sex dependent. To this end, EEGs from 24 healthy female subjects and 21 healthy male subjects were recorded in eyes-closed resting state conditions. The connectivity matrices were extracted using correlation analysis and were further binarized to obtain binary functional networks. Global and local efficiency measures as graph theory metrics were computed for the extracted networks. We found that male brains have significantly greater global efficiency (i.e., global communicability of the network) across all frequency bands for a wide range of cost values in both hemispheres. Furthermore, for a range of cost values, female brains showed significantly greater right-hemispheric local efficiency (i.e., local connectivity) than male brains.
Citation: Mahdi Jalili. EEG-based functional brain networks: Hemispheric differences in males and females. Networks & Heterogeneous Media, 2015, 10 (1) : 223-232. doi: 10.3934/nhm.2015.10.223
References:
[1]

S. Achard and E. Bullmore, Efficiency and cost of economical brain functional networks,, PLoS Computational Biology, 3 (2007). Google Scholar

[2]

S. Achard, R. Salvador, B. Whitcher, J. Suckling and E. Bullmore, A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs,, The Journal of Neuroscience, 26 (2006), 63. doi: 10.1523/JNEUROSCI.3874-05.2006. Google Scholar

[3]

A. F. Alexander-Bloch, N. Gogtay, D. Meunier, R. Birn, L. Clasen, F. Lalonde, R. Lenroot, J. Giedd and E. T. Bullmore, Disrupted modularity and local connectivity of brain functional networks in childhood-onset schizophrenia,, Frontiers in Systems Neuroscience, 4 (2010). Google Scholar

[4]

A.-L. Barabási, Network science,, Philosophical Transactions of the Royal Society A: Mathematical, 371 (2013). doi: 10.1098/rsta.2012.0375. Google Scholar

[5]

A.-L. Barabási et al., Scale-free networks: A decade and beyond,, Science, 325 (2009), 412. doi: 10.1126/science.1173299. Google Scholar

[6]

E. Barzegaran, A. Joudaki, M. Jalili, A. O. Rossetti, R. S. Frackowiak and M. G. Knyazeva, Properties of functional brain networks correlate with frequency of psychogenic non-epileptic seizures,, Frontiers in Human Neuroscience, (2012). doi: 10.3389/fnhum.2012.00335. Google Scholar

[7]

D. S. Bassett, E. Bullmore, B. A. Verchinski, V. S. Mattay, D. R. Weinberger and A. Meyer-Lindenberg, Hierarchical organization of human cortical networks in health and schizophrenia,, The Journal of Neuroscience, 28 (2008), 9239. doi: 10.1523/JNEUROSCI.1929-08.2008. Google Scholar

[8]

D. S. Bassett and E. T. Bullmore, Human brain networks in health and disease,, Current Opinion in Neurology, 22 (2009), 340. doi: 10.1097/WCO.0b013e32832d93dd. Google Scholar

[9]

S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics,, Physics Reports, 424 (2006), 175. doi: 10.1016/j.physrep.2005.10.009. Google Scholar

[10]

E. Bullmore and O. Sporns, Complex brain networks: Graph theoretical analysis of structural and functional systems,, Nature Reviews Neuroscience, 10 (2009), 186. doi: 10.1038/nrn2575. Google Scholar

[11]

E. Bullmore and O. Sporns, The economy of brain network organization,, Nature Reviews Neuroscience, 13 (2012), 336. doi: 10.1038/nrn3214. Google Scholar

[12]

R. J. Davidson, G. E. Schwartz, E. Pugash and E. Bromfield, Sex differences in patterns of eeg asymmetry,, Biological Psychology, 4 (1976), 119. doi: 10.1016/0301-0511(76)90012-0. Google Scholar

[13]

F. De Vico Fallani, L. Astolfi, F. Cincotti, D. Mattia, D. la Rocca, E. Maksuti, S. Salinari, F. Babiloni, B. Vegso and G. Kozmann, et al., Evaluation of the brain network organization from eeg signals: A preliminary evidence in stroke patient,, The Anatomical Record, 292 (2009), 2023. Google Scholar

[14]

V. M. Eguiluz, D. R. Chialvo, G. A. Cecchi, M. Baliki and A. V. Apkarian, Scale-free brain functional networks,, Physical Review Letters, 94 (2005). doi: 10.1103/PhysRevLett.94.018102. Google Scholar

[15]

R. Ferri, F. Rundo, O. Bruni, M. G. Terzano and C. J. Stam, Small-world network organization of functional connectivity of eeg slow-wave activity during sleep,, Clinical Neurophysiology, 118 (2007), 449. doi: 10.1016/j.clinph.2006.10.021. Google Scholar

[16]

E. Fornari, P. Maeder, R. Meuli, J. Ghika and M. G. Knyazeva, Demyelination of superficial white matter in early Alzheimer's disease: A magnetization transfer imaging study,, Neurobiology of Aging, 33 (2012), 7. doi: 10.1016/j.neurobiolaging.2010.11.014. Google Scholar

[17]

G. Gong, Y. He and A. C. Evans, Brain connectivity gender makes a difference,, The Neuroscientist, 17 (2011), 575. doi: 10.1177/1073858410386492. Google Scholar

[18]

G. Gong, P. Rosa-Neto, F. Carbonell, Z. J. Chen, Y. He and A. C. Evans, Age-and gender-related differences in the cortical anatomical network,, The Journal of Neuroscience, 29 (2009), 15684. doi: 10.1523/JNEUROSCI.2308-09.2009. Google Scholar

[19]

P. Hagmann, L. Cammoun, X. Gigandet, R. Meuli, C. J. Honey, V. J. Wedeen and O. Sporns, Mapping the structural core of human cerebral cortex,, PLoS Biology, 6 (2008). Google Scholar

[20]

M. Ingalhalikar, A. Smith, D. Parker, T. D. Satterthwaite, M. A. Elliott, K. Ruparel, H. Hakonarson, R. E. Gur, R. C. Gur and R. Verma, Sex differences in the structural connectome of the human brain,, Proceedings of the National Academy of Sciences, 111 (2014), 823. doi: 10.1073/pnas.1316909110. Google Scholar

[21]

M. Jalili and M. G. Knyazeva, Constructing brain functional networks from eeg: Partial and unpartial correlations,, Journal of Integrative Neuroscience, 10 (2011), 213. doi: 10.1142/S0219635211002725. Google Scholar

[22]

M. Jalili and M. G. Knyazeva, Eeg-based functional networks in schizophrenia,, Computers in Biology and Medicine, 41 (2011), 1178. doi: 10.1016/j.compbiomed.2011.05.004. Google Scholar

[23]

M. Jalili, S. Lavoie, P. Deppen, R. Meuli, K. Q. Do, M. Cu{\'e}nod, M. Hasler, O. De Feo and M. G. Knyazeva, Dysconnection topography in schizophrenia revealed with state-space analysis of eeg,, PLoS One, 2 (2007). doi: 10.1371/journal.pone.0001059. Google Scholar

[24]

A. Joudaki, N. Salehi, M. Jalili and M. G. Knyazeva, Eeg-based functional brain networks: Does the network size matter?,, PloS One, 7 (2012). doi: 10.1371/journal.pone.0035673. Google Scholar

[25]

D. Kimura, Sex differences in the brain,, Scientific American, 267 (1992), 118. Google Scholar

[26]

M. G. Knyazeva, M. Jalili, R. S. Frackowiak and A. O. Rossetti, Psychogenic seizures and frontal disconnection: Eeg synchronisation study,, Journal of Neurology, 82 (2011), 505. doi: 10.1136/jnnp.2010.224873. Google Scholar

[27]

V. Latora and M. Marchiori, Economic small-world behavior in weighted networks,, The European Physical Journal B-Condensed Matter and Complex Systems, 32 (2003), 249. doi: 10.1140/epjb/e2003-00095-5. Google Scholar

[28]

W. Liao, Z. Zhang, Z. Pan, D. Mantini, J. Ding, X. Duan, C. Luo, G. Lu and H. Chen, Altered functional connectivity and small-world in mesial temporal lobe epilepsy,, PloS One, 5 (2010). doi: 10.1371/journal.pone.0008525. Google Scholar

[29]

Y. Liu, C. Yu, M. Liang, J. Li, L. Tian, Y. Zhou, W. Qin, K. Li and T. Jiang, Whole brain functional connectivity in the early blind,, Brain, 130 (2007), 2085. doi: 10.1093/brain/awm121. Google Scholar

[30]

M. Matsuura, K. Yamamoto, H. Fukuzawa, Y. Okubo, H. Uesugi, M. Moriiwa, T. Kojima and Y. Shimazono, Age development and sex differences of various eeg elements in healthy children and adults-quantification by a computerized wave form recognition method,, Electroencephalography and Clinical Neurophysiology, 60 (1985), 394. doi: 10.1016/0013-4694(85)91013-2. Google Scholar

[31]

S. Micheloyannis, E. Pachou, C. J. Stam, M. Breakspear, P. Bitsios, M. Vourkas, S. Erimaki and M. Zervakis, Small-world networks and disturbed functional connectivity in schizophrenia,, Schizophrenia Research, 87 (2006), 60. doi: 10.1016/j.schres.2006.06.028. Google Scholar

[32]

P. L. Nunez and R. Srinivasan, Electric Fields of the Brain: The Neurophysics of EEG,, 2nd edition, (2006). doi: 10.1063/1.2915137. Google Scholar

[33]

O. Sporns, Small-world connectivity, motif composition, and complexity of fractal neuronal connections,, Biosystems, 85 (2006), 55. doi: 10.1016/j.biosystems.2006.02.008. Google Scholar

[34]

O. Sporns and J. D. Zwi, The small world of the cerebral cortex,, Neuroinformatics, 2 (2004), 145. doi: 10.1385/NI:2:2:145. Google Scholar

[35]

C. Stam, W. De Haan, A. Daffertshofer, B. Jones, I. Manshanden, A. V. C. Van Walsum, T. Montez, J. Verbunt, J. De Munck and B. Van Dijk, et al., Graph theoretical analysis of magnetoencephalographic functional connectivity in Alzheimer's disease,, Brain, 132 (2009), 213. doi: 10.1093/brain/awn262. Google Scholar

[36]

C. Stam, B. Jones, G. Nolte, M. Breakspear and P. Scheltens, Small-world networks and functional connectivity in Alzheimer's disease,, Cerebral Cortex, 17 (2007), 92. doi: 10.1093/cercor/bhj127. Google Scholar

[37]

M. S. Tahaei, M. Jalili and M. G. Knyazeva, Synchronizability of eeg-based functional networks in early alzheimer's disease,, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 20 (2012), 636. doi: 10.1109/TNSRE.2012.2202127. Google Scholar

[38]

L. Tian, J. Wang, C. Yan and Y. He, Hemisphere-and gender-related differences in small-world brain networks: A resting-state functional mri study,, Neuroimage, 54 (2011), 191. doi: 10.1016/j.neuroimage.2010.07.066. Google Scholar

[39]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks,, Nature, 393 (1998), 440. Google Scholar

[40]

C. Yan, G. Gong, J. Wang, D. Wang, D. Liu, C. Zhu, Z. J. Chen, A. Evans, Y. Zang and Y. He, Sex-and brain size-related small-world structural cortical networks in young adults: A dti tractography study,, Cerebral cortex, 21 (2011), 449. doi: 10.1093/cercor/bhq111. Google Scholar

[41]

A. Zalesky, A. Fornito, I. H. Harding, L. Cocchi, M. Yücel, C. Pantelis and E. T. Bullmore, Whole-brain anatomical networks: Does the choice of nodes matter?,, Neuroimage, 50 (2010), 970. doi: 10.1016/j.neuroimage.2009.12.027. Google Scholar

show all references

References:
[1]

S. Achard and E. Bullmore, Efficiency and cost of economical brain functional networks,, PLoS Computational Biology, 3 (2007). Google Scholar

[2]

S. Achard, R. Salvador, B. Whitcher, J. Suckling and E. Bullmore, A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs,, The Journal of Neuroscience, 26 (2006), 63. doi: 10.1523/JNEUROSCI.3874-05.2006. Google Scholar

[3]

A. F. Alexander-Bloch, N. Gogtay, D. Meunier, R. Birn, L. Clasen, F. Lalonde, R. Lenroot, J. Giedd and E. T. Bullmore, Disrupted modularity and local connectivity of brain functional networks in childhood-onset schizophrenia,, Frontiers in Systems Neuroscience, 4 (2010). Google Scholar

[4]

A.-L. Barabási, Network science,, Philosophical Transactions of the Royal Society A: Mathematical, 371 (2013). doi: 10.1098/rsta.2012.0375. Google Scholar

[5]

A.-L. Barabási et al., Scale-free networks: A decade and beyond,, Science, 325 (2009), 412. doi: 10.1126/science.1173299. Google Scholar

[6]

E. Barzegaran, A. Joudaki, M. Jalili, A. O. Rossetti, R. S. Frackowiak and M. G. Knyazeva, Properties of functional brain networks correlate with frequency of psychogenic non-epileptic seizures,, Frontiers in Human Neuroscience, (2012). doi: 10.3389/fnhum.2012.00335. Google Scholar

[7]

D. S. Bassett, E. Bullmore, B. A. Verchinski, V. S. Mattay, D. R. Weinberger and A. Meyer-Lindenberg, Hierarchical organization of human cortical networks in health and schizophrenia,, The Journal of Neuroscience, 28 (2008), 9239. doi: 10.1523/JNEUROSCI.1929-08.2008. Google Scholar

[8]

D. S. Bassett and E. T. Bullmore, Human brain networks in health and disease,, Current Opinion in Neurology, 22 (2009), 340. doi: 10.1097/WCO.0b013e32832d93dd. Google Scholar

[9]

S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics,, Physics Reports, 424 (2006), 175. doi: 10.1016/j.physrep.2005.10.009. Google Scholar

[10]

E. Bullmore and O. Sporns, Complex brain networks: Graph theoretical analysis of structural and functional systems,, Nature Reviews Neuroscience, 10 (2009), 186. doi: 10.1038/nrn2575. Google Scholar

[11]

E. Bullmore and O. Sporns, The economy of brain network organization,, Nature Reviews Neuroscience, 13 (2012), 336. doi: 10.1038/nrn3214. Google Scholar

[12]

R. J. Davidson, G. E. Schwartz, E. Pugash and E. Bromfield, Sex differences in patterns of eeg asymmetry,, Biological Psychology, 4 (1976), 119. doi: 10.1016/0301-0511(76)90012-0. Google Scholar

[13]

F. De Vico Fallani, L. Astolfi, F. Cincotti, D. Mattia, D. la Rocca, E. Maksuti, S. Salinari, F. Babiloni, B. Vegso and G. Kozmann, et al., Evaluation of the brain network organization from eeg signals: A preliminary evidence in stroke patient,, The Anatomical Record, 292 (2009), 2023. Google Scholar

[14]

V. M. Eguiluz, D. R. Chialvo, G. A. Cecchi, M. Baliki and A. V. Apkarian, Scale-free brain functional networks,, Physical Review Letters, 94 (2005). doi: 10.1103/PhysRevLett.94.018102. Google Scholar

[15]

R. Ferri, F. Rundo, O. Bruni, M. G. Terzano and C. J. Stam, Small-world network organization of functional connectivity of eeg slow-wave activity during sleep,, Clinical Neurophysiology, 118 (2007), 449. doi: 10.1016/j.clinph.2006.10.021. Google Scholar

[16]

E. Fornari, P. Maeder, R. Meuli, J. Ghika and M. G. Knyazeva, Demyelination of superficial white matter in early Alzheimer's disease: A magnetization transfer imaging study,, Neurobiology of Aging, 33 (2012), 7. doi: 10.1016/j.neurobiolaging.2010.11.014. Google Scholar

[17]

G. Gong, Y. He and A. C. Evans, Brain connectivity gender makes a difference,, The Neuroscientist, 17 (2011), 575. doi: 10.1177/1073858410386492. Google Scholar

[18]

G. Gong, P. Rosa-Neto, F. Carbonell, Z. J. Chen, Y. He and A. C. Evans, Age-and gender-related differences in the cortical anatomical network,, The Journal of Neuroscience, 29 (2009), 15684. doi: 10.1523/JNEUROSCI.2308-09.2009. Google Scholar

[19]

P. Hagmann, L. Cammoun, X. Gigandet, R. Meuli, C. J. Honey, V. J. Wedeen and O. Sporns, Mapping the structural core of human cerebral cortex,, PLoS Biology, 6 (2008). Google Scholar

[20]

M. Ingalhalikar, A. Smith, D. Parker, T. D. Satterthwaite, M. A. Elliott, K. Ruparel, H. Hakonarson, R. E. Gur, R. C. Gur and R. Verma, Sex differences in the structural connectome of the human brain,, Proceedings of the National Academy of Sciences, 111 (2014), 823. doi: 10.1073/pnas.1316909110. Google Scholar

[21]

M. Jalili and M. G. Knyazeva, Constructing brain functional networks from eeg: Partial and unpartial correlations,, Journal of Integrative Neuroscience, 10 (2011), 213. doi: 10.1142/S0219635211002725. Google Scholar

[22]

M. Jalili and M. G. Knyazeva, Eeg-based functional networks in schizophrenia,, Computers in Biology and Medicine, 41 (2011), 1178. doi: 10.1016/j.compbiomed.2011.05.004. Google Scholar

[23]

M. Jalili, S. Lavoie, P. Deppen, R. Meuli, K. Q. Do, M. Cu{\'e}nod, M. Hasler, O. De Feo and M. G. Knyazeva, Dysconnection topography in schizophrenia revealed with state-space analysis of eeg,, PLoS One, 2 (2007). doi: 10.1371/journal.pone.0001059. Google Scholar

[24]

A. Joudaki, N. Salehi, M. Jalili and M. G. Knyazeva, Eeg-based functional brain networks: Does the network size matter?,, PloS One, 7 (2012). doi: 10.1371/journal.pone.0035673. Google Scholar

[25]

D. Kimura, Sex differences in the brain,, Scientific American, 267 (1992), 118. Google Scholar

[26]

M. G. Knyazeva, M. Jalili, R. S. Frackowiak and A. O. Rossetti, Psychogenic seizures and frontal disconnection: Eeg synchronisation study,, Journal of Neurology, 82 (2011), 505. doi: 10.1136/jnnp.2010.224873. Google Scholar

[27]

V. Latora and M. Marchiori, Economic small-world behavior in weighted networks,, The European Physical Journal B-Condensed Matter and Complex Systems, 32 (2003), 249. doi: 10.1140/epjb/e2003-00095-5. Google Scholar

[28]

W. Liao, Z. Zhang, Z. Pan, D. Mantini, J. Ding, X. Duan, C. Luo, G. Lu and H. Chen, Altered functional connectivity and small-world in mesial temporal lobe epilepsy,, PloS One, 5 (2010). doi: 10.1371/journal.pone.0008525. Google Scholar

[29]

Y. Liu, C. Yu, M. Liang, J. Li, L. Tian, Y. Zhou, W. Qin, K. Li and T. Jiang, Whole brain functional connectivity in the early blind,, Brain, 130 (2007), 2085. doi: 10.1093/brain/awm121. Google Scholar

[30]

M. Matsuura, K. Yamamoto, H. Fukuzawa, Y. Okubo, H. Uesugi, M. Moriiwa, T. Kojima and Y. Shimazono, Age development and sex differences of various eeg elements in healthy children and adults-quantification by a computerized wave form recognition method,, Electroencephalography and Clinical Neurophysiology, 60 (1985), 394. doi: 10.1016/0013-4694(85)91013-2. Google Scholar

[31]

S. Micheloyannis, E. Pachou, C. J. Stam, M. Breakspear, P. Bitsios, M. Vourkas, S. Erimaki and M. Zervakis, Small-world networks and disturbed functional connectivity in schizophrenia,, Schizophrenia Research, 87 (2006), 60. doi: 10.1016/j.schres.2006.06.028. Google Scholar

[32]

P. L. Nunez and R. Srinivasan, Electric Fields of the Brain: The Neurophysics of EEG,, 2nd edition, (2006). doi: 10.1063/1.2915137. Google Scholar

[33]

O. Sporns, Small-world connectivity, motif composition, and complexity of fractal neuronal connections,, Biosystems, 85 (2006), 55. doi: 10.1016/j.biosystems.2006.02.008. Google Scholar

[34]

O. Sporns and J. D. Zwi, The small world of the cerebral cortex,, Neuroinformatics, 2 (2004), 145. doi: 10.1385/NI:2:2:145. Google Scholar

[35]

C. Stam, W. De Haan, A. Daffertshofer, B. Jones, I. Manshanden, A. V. C. Van Walsum, T. Montez, J. Verbunt, J. De Munck and B. Van Dijk, et al., Graph theoretical analysis of magnetoencephalographic functional connectivity in Alzheimer's disease,, Brain, 132 (2009), 213. doi: 10.1093/brain/awn262. Google Scholar

[36]

C. Stam, B. Jones, G. Nolte, M. Breakspear and P. Scheltens, Small-world networks and functional connectivity in Alzheimer's disease,, Cerebral Cortex, 17 (2007), 92. doi: 10.1093/cercor/bhj127. Google Scholar

[37]

M. S. Tahaei, M. Jalili and M. G. Knyazeva, Synchronizability of eeg-based functional networks in early alzheimer's disease,, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 20 (2012), 636. doi: 10.1109/TNSRE.2012.2202127. Google Scholar

[38]

L. Tian, J. Wang, C. Yan and Y. He, Hemisphere-and gender-related differences in small-world brain networks: A resting-state functional mri study,, Neuroimage, 54 (2011), 191. doi: 10.1016/j.neuroimage.2010.07.066. Google Scholar

[39]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks,, Nature, 393 (1998), 440. Google Scholar

[40]

C. Yan, G. Gong, J. Wang, D. Wang, D. Liu, C. Zhu, Z. J. Chen, A. Evans, Y. Zang and Y. He, Sex-and brain size-related small-world structural cortical networks in young adults: A dti tractography study,, Cerebral cortex, 21 (2011), 449. doi: 10.1093/cercor/bhq111. Google Scholar

[41]

A. Zalesky, A. Fornito, I. H. Harding, L. Cocchi, M. Yücel, C. Pantelis and E. T. Bullmore, Whole-brain anatomical networks: Does the choice of nodes matter?,, Neuroimage, 50 (2010), 970. doi: 10.1016/j.neuroimage.2009.12.027. Google Scholar

[1]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks & Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295

[2]

M. D. König, Stefano Battiston, M. Napoletano, F. Schweitzer. On algebraic graph theory and the dynamics of innovation networks. Networks & Heterogeneous Media, 2008, 3 (2) : 201-219. doi: 10.3934/nhm.2008.3.201

[3]

Liu Hui, Lin Zhi, Waqas Ahmad. Network(graph) data research in the coordinate system. Mathematical Foundations of Computing, 2018, 1 (1) : 1-10. doi: 10.3934/mfc.2018001

[4]

Deena Schmidt, Janet Best, Mark S. Blumberg. Random graph and stochastic process contributions to network dynamics. Conference Publications, 2011, 2011 (Special) : 1279-1288. doi: 10.3934/proc.2011.2011.1279

[5]

Barton E. Lee. Consensus and voting on large graphs: An application of graph limit theory. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1719-1744. doi: 10.3934/dcds.2018071

[6]

Robert Carlson. Spectral theory for nonconservative transmission line networks. Networks & Heterogeneous Media, 2011, 6 (2) : 257-277. doi: 10.3934/nhm.2011.6.257

[7]

Maya Mincheva, Gheorghe Craciun. Graph-theoretic conditions for zero-eigenvalue Turing instability in general chemical reaction networks. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1207-1226. doi: 10.3934/mbe.2013.10.1207

[8]

A. C. Eberhard, J-P. Crouzeix. Existence of closed graph, maximal, cyclic pseudo-monotone relations and revealed preference theory. Journal of Industrial & Management Optimization, 2007, 3 (2) : 233-255. doi: 10.3934/jimo.2007.3.233

[9]

Kirill D. Cherednichenko, Alexander V. Kiselev, Luis O. Silva. Functional model for extensions of symmetric operators and applications to scattering theory. Networks & Heterogeneous Media, 2018, 13 (2) : 191-215. doi: 10.3934/nhm.2018009

[10]

Xianmin Geng, Shengli Zhou, Jiashan Tang, Cong Yang. A sufficient condition for classified networks to possess complex network features. Networks & Heterogeneous Media, 2012, 7 (1) : 59-69. doi: 10.3934/nhm.2012.7.59

[11]

Serap Ergün, Bariş Bülent Kırlar, Sırma Zeynep Alparslan Gök, Gerhard-Wilhelm Weber. An application of crypto cloud computing in social networks by cooperative game theory. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-15. doi: 10.3934/jimo.2019036

[12]

Giulia Ajmone Marsan, Nicola Bellomo, Massimo Egidi. Towards a mathematical theory of complex socio-economical systems by functional subsystems representation. Kinetic & Related Models, 2008, 1 (2) : 249-278. doi: 10.3934/krm.2008.1.249

[13]

Zhuwei Qin, Fuxun Yu, Chenchen Liu, Xiang Chen. How convolutional neural networks see the world --- A survey of convolutional neural network visualization methods. Mathematical Foundations of Computing, 2018, 1 (2) : 149-180. doi: 10.3934/mfc.2018008

[14]

D. Warren, K Najarian. Learning theory applied to Sigmoid network classification of protein biological function using primary protein structure. Conference Publications, 2003, 2003 (Special) : 898-904. doi: 10.3934/proc.2003.2003.898

[15]

Martin Lara, Sebastián Ferrer. Computing long-lifetime science orbits around natural satellites. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 293-302. doi: 10.3934/dcdss.2008.1.293

[16]

Jean-Philippe Cointet, David Chavalarias. Multi-level science mapping with asymmetrical paradigmatic proximity. Networks & Heterogeneous Media, 2008, 3 (2) : 267-276. doi: 10.3934/nhm.2008.3.267

[17]

George Dassios, Michalis N. Tsampas. Vector ellipsoidal harmonics and neuronal current decomposition in the brain. Inverse Problems & Imaging, 2009, 3 (2) : 243-257. doi: 10.3934/ipi.2009.3.243

[18]

Carole Guillevin, Rémy Guillevin, Alain Miranville, Angélique Perrillat-Mercerot. Analysis of a mathematical model for brain lactate kinetics. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1225-1242. doi: 10.3934/mbe.2018056

[19]

Stéphane Chrétien, Sébastien Darses, Christophe Guyeux, Paul Clarkson. On the pinning controllability of complex networks using perturbation theory of extreme singular values. application to synchronisation in power grids. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 289-299. doi: 10.3934/naco.2017019

[20]

Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68.

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]