Degree Correlations Optimize Neuronal Network Sensitivity to Sub-Threshold Stimuli

Schmeltzer, Christian; Kihara, Alexandre Hiroaki; Sokolov, Igor M.; Rüdiger, Stephan

doi:10.1371/journal.pone.0121794

Cited by 30 publications

(34 citation statements)

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“…The characteristic function of stable distribution can be classified using four parameters α, β, σ and µ, see [26,Definition 1.1.6], and the corresponding random variable is denote by S α (σ, β, µ). The parameter α is called the stability index and is the most important parameter for our purposes, since it relates to the exponent γ of the regularly-varying distribution.…”

Section: Stable Distributionsmentioning

confidence: 99%

“…For instance, it was shown that disassortative networks are easier to immunize and a disease takes longer to spread in assortative networks [11]. In the field of neuroscience, it was shown that assortative brain networks are better suited for signal processing [26], while assortative neural networks are more robust to random noise [12]. Under attacks, when edges or vertices are removed, assortative networks appear to be more resilient than disassortive networks [23,27].…”

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confidence: 99%

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Average nearest neighbor degrees in scale-free networks

Yao

Hoorn²,

Litvak

2018

Internet Mathematics

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The average nearest neighbor degree (ANND) of a node of degree k is widely used to measure dependencies between degrees of neighbor nodes in a network. We formally analyze ANND in undirected random graphs when the graph size tends to infinity. The limiting behavior of ANND depends on the variance of the degree distribution. When the variance is finite, the ANND has a deterministic limit. When the variance is infinite, the ANND scales with the size of the graph, and we prove a corresponding central limit theorem in the configuration model (CM, a network with random connections). As ANND proved uninformative in the infinite variance scenario, we propose an alternative measure, the average nearest neighbor rank (ANNR). We prove that ANNR converges to a deterministic function whenever the degree distribution has finite mean. We then consider the erased configuration model (ECM), where self-loops and multiple edges are removed, and investigate the well-known 'structural negative correlations', or 'finite-size effects', that arise in simple graphs, such as ECM, because large nodes can only have a limited number of large neighbors. Interestingly, we prove that for any fixed k, ANNR in ECM converges to the same limit as in CM. However, numerical experiments show that finite-size effects occur when k scales with n. IntroductionThe goal of this paper is to analytically derive the limiting properties of the average nearest neighbor degree (ANND) in a general class of random graphs. The motivation for this analysis is that the ANND is one of the commonly accepted measures for dependencies between degrees of neighbor nodes. Such dependencies are called degree-degree correlations or network assortativity. A network is said to be assortative, if the correlation between degrees of neighbor nodes is positive and disassortative when it is negative. In assortative networks, nodes of high degree have a preference to connect to nodes of high degree. When the network is disassortative, nodes of high degree have a connection preference for nodes of low degree [23]. If there is no connection preference, the network is said to have neutral mixing.Currently, degree-degree correlations are part of the standard set of properties used to characterize the structure of networks. See [24] for a survey of the work on network assortativity. The effect of degree-degree correlations on disease spreading in networks has been extensively addressed in the literature, cf. [2,4,5,6]. For instance, it was shown that disassortative networks are easier to immunize and a disease takes longer to spread in assortative networks [11]. In the field of neuroscience, it was shown that assortative brain networks are better suited for signal processing [26], while assortative neural networks are more robust to random noise [12]. Under attacks, when edges or vertices are removed, assortative networks appear to be more resilient than disassortive networks [23,27]. On the other hand, when different networks interact, assortativity actually decreases the robust...

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Section: Stable Distributionsmentioning

confidence: 99%

mentioning

confidence: 99%

Average nearest neighbor degrees in scale-free networks

Yao

Hoorn²,

Litvak

2018

Internet Mathematics

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“…4). Work by Roxin [18], Schmeltzer et al [19], and unpublished work by Landau and Sompolinsky [33] has shown that the broadness and correlation of the joint degree distribution can lead to qualitative changes in the behavior of a spiking network. Further work is required to investigate whether and why these changes can be explained by the spectrum of the connectivity matrix derived here.…”

Section: Matrices With Heterogeneous Degree Distributionsmentioning

confidence: 99%

“…Note that by requiring that g is bounded and differentiable on the unit square outside of S 0 we allow the synaptic gain function to be a combination of discrete modulation (e.g., cell-type dependent connectivity for distinct cell types, as in [13]) and of continuous modulation (e.g., networks with heterogeneous and possibly correlated in- and out-degree distributions, as in [18, 19]).…”

Section: Introductionmentioning

confidence: 99%

Low-dimensional dynamics of structured random networks

et al. 2016

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Using a generalized random recurrent neural network model, and by extending our recently developed mean-field approach, we study the relationship between the network connectivity structure and its low dimensional dynamics. Each connection in the network is a random number with mean 0 and variance that depends on pre- and post-synaptic neurons through a sufficiently smooth function g of their identities. We find that these networks undergo a phase transition from a silent to a chaotic state at a critical point we derive as a function of g. Above the critical point, although unit activation levels are chaotic, their autocorrelation functions are restricted to a low-dimensional subspace. This provides a direct link between the network's structure and some of its functional characteristics. We discuss example applications of the general results to neuroscience where we derive the support of the spectrum of connectivity matrices with heterogeneous and possibly correlated degree distributions, and to ecology where we study the stability of the cascade model for food web structure.

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“…In the field of neuroscience degree-degree correlations are studied in the context of information spread in the brain. For instance, in [73], it is shown that assortative brain networks are better suited for signal processing, while assortative neural networks are shown to be more robust to random noise in [33].…”

Section: Influence On Network Properties and Processesmentioning

confidence: 99%

Asymptotic analysis of network structures: degree-degree correlations and directed paths

Hoorn

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Opgedragen aan mijn neefje Wouter: Iter tibi prosperitate scientiaque obseratur et aetas amore sapientiaque plena sit. VoorwoordGeen enkele uitzonderlijke prestatie is het werk van slechts één persoon. En hoewel ik, met betrekking tot dit proefschrift, geen uitspraak zal doen over het eerste deel van dit statement, is het tweede deel absoluut waar. Ik wil dan ook graag iedereen bedanken die hieraan op een of andere manier heeft bijgedragen. Sommige mensen wil ik in het bijzonder noemen en daarvoor zal ik deze pagina's gebruiken. Toen ik, iets meer dan vier jaar geleden, in de zomer van 2012 besloot om een PhD te doen, was ik niet helemaal zeker van mij zaak, gezien het feit dat ik al een jaar geen wiskunde meer bedreven had. Voor mijn interview in Enschede was ik dan ook erg zenuwachtig. Helemaal omdat het hier zelfs over een onderwerp ging waar ik nog nooit iets mee gedaan had gedurende mijn tijd als student. Richard, dank je dat je mij deze kans hebt gegeven, voor je vertrouwen in mij als onderzoeker en het opnemen van mij in je groep. Ik zal altijd met veel plezier terugdenken aan mijn tijd in Twente.Voor mijn dagelijkse begeleiding kon ik terecht bij Nelly. Ik weet nog goed dat ze mij tijdens ons eerste overleg meteen vertelde dat ze hele hoge verwachtingen van me had. Ik kan alleen maar hopen dat ik daar ergens in de buurt van ben gekomen. Maar Nelly, jij bent mijn verwachtingen als begeleider meer dan overstegen. Het voelt ergens vreemd om dit in het Nederlands te schrijven. Ondanks het feit dat je deze taal goed beheerst communiceren wij namelijk altijd met elkaar in het Engels. De stewardess van de KLM begreep er in ieder geval niets van. Jij was altijd heel duidelijk in wat je van me verwachtte en had altijd een plan klaar, hoewel dit laatste vaak erg flexibel was. Je gaf me de vrijheid om mijn onderzoek in te richten en liet me vaak mijn gang gaan, waar je hulp bood als ik dat nodig had. Maar je was ook streng, direct en ondubbelzinnig op de momenten dat dit moest. Naast een meer dan uitstekende begeleider ben je een ook een hele goede vriendin van me geworden. Ik heb ontzettend genoten van de conferenties die wij samen hebben bezocht en alle avonturen die wij hebben beleefd. Ik hoop dan ook dat wij nog heel lang vrienden zullen blijven. Maar nu zal ik verder gaan, voordat je me weer vertelt dat ik als een meisje schrijf.Gedurende mijn PhD heb ik de mogelijkheid gehad om twee onderzoeksstages in het buitenland te doen en daar ben ik heel dankbaar voor.Mariana, thank you for inviting me to Columbia. It was my first time collaborating with someone other than my supervisor and I more than appreciated the opportunity you gave me. When writing this thesis, especially when dealing with some technical proofs, I realized how much I learned from you in only these vii Asymptotic analysis of network structures: degree-degree correlations and directed paths two weeks. I also want to thank you for being part of my defense committee and reading my thesis. I hope to continue our collaboration in the future and learn ...

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Degree Correlations Optimize Neuronal Network Sensitivity to Sub-Threshold Stimuli

Cited by 30 publications

References 58 publications

Average nearest neighbor degrees in scale-free networks

Average nearest neighbor degrees in scale-free networks

Low-dimensional dynamics of structured random networks

Asymptotic analysis of network structures: degree-degree correlations and directed paths

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