Field Theory
4.6 The entire spectrum in mean field theory
(4.5.21)
= r1
k − 1
N −1 . (4.5.22)
Note that λwsc2 (N, k,1) in eq. (4.5.20) acquires a positive value for too small k-values and a sufficiently large network size N (cf. [FDBV01]). However, for the k-values we investigated (Fig. 4.10, (a) and (b)), the second largest eigenvalues are well predicted by both eqs. (4.5.20) and (4.5.22).
4.6 The entire spectrum in mean field theory
To gain further insights into the entire spectrum we study the density of statesρ(λ) (cf. e.g. [FDBV01]) as defined in its discrete form, i.e. for finite network sizeN, by
ρ(λ) = 1 N
N
X
j=1
δ(λ−λj) , (4.6.1)
whereδ is the Dirac delta function. The evaluation of (4.6.1) for the analytic mean field predictions and for the numerically obtained eigenvalues of undirected and directed networks are shows good qualitative agreement, cf. Fig. 4.11 for all but large topological randomness q → 1. Spectra for networks with parameters other thanN = 1000andk = 50yield qualitatively the same structure. Thus, the largest and smallest eigenvalues, the location and form of bulk peak as well as the entire structure of eigenvalues are well approximated up toq of order 1by the mean field predictions derived analytically.
4.7 Summary and discussion
In this Chapter we introduced a simple two-stage mean-field rewiring scheme which we used to derive analytical predictions for the spectra of graph Laplacians. Sys-tematic numerical checks confirm that this prediction is accurate for the second largest eigenvalue for all but very small degrees or very large topological random-ness. For very small k, our analytic prediction still serves as a valuable guide for the overall dependence all topological parameters. For q close to unity, our mean field prediction can be complemented by existing results from random matrix the-ory. Besides the second largest and smallest eigenvalues, that already give valuable information about initial and asymptotic relaxation dynamics, the bulk spectrum as well as the fine structure of the spectrum is well approximated by our analyt-ical prediction. In particular, the spectral prediction include regular rings, small worlds, and substantially more randomly rewired networks and undirected as well as directed ones.
4.7. Summary and discussion
0
-0.5
-1
ReΛ
0
-0.5
-1
ReΛ
10-5 10-4 10-3 10-2 10-1 1 0
-0.5
-1
topological randomness q
ReΛ
0 5 10 15 >16
a
c b
Figure 4.11: Analytics predicts structure of entire spectrum. Densities of states (eq. (4.6.1) for (a) undirected, (b) directed and (c) mean field networks (N = 1000, k = 50). Dashed white lines show the ex-treme eigenvalues obtained numerically for undirected and directed networks. The solid black lines show the mean field prediction for the extreme eigenvalues. Densities of states for directed and undi-rected networks are averaged over 100realizations for a fixed q-value, while the mean field density is analytically determined by eq. (4.2.29).
In particular, our theoretical predictions agree well with the eigenvalues obtained numerically over almost the entire range of topological randomnessq, thereby com-pleting previous attempts based on perturbation theory forq ≪1[Mon99, BP02].
Although the mean field rewiring is undirected, eigenvalues for directed networks are approximated more accurately and in a wider range of q-values, which is in particular related to the fact that the predictions for the undirected second-largest eigenvalues at q = 1 are larger in real part than the directed ones, while all the mean field eigenvalues converge to −1at q= 1. For ‘small’ k-values the mean field approximation becomes less accurate, which may be due to the fact that the ring structure is destroyed more easily while rewiring. Additionally, the bulk spectra spread much more drastically with q than for largerk-values.
The simple mean field approach presented here definitely leads to an essential reduction of computational efforts when studying randomized (regular or small-world) network models. It may be extended to rewiring approaches starting from other than ring-like structures, for instance to two or three dimensions, as for in-stance, relevant for neural network modeling [SB09]. Checking with appropriate models, it may thus serve as a powerful tool to predict or deduce the relations between structural and dynamical properties of randomized networks.
Furthermore, the analysis of the mean field spectrum presented here could be extended to the Laplacian eigenvectors. Studies of the Laplacian eigenvectors are rare although there are fascinating results as well. For instance, the discrete analogs of solutions of the Schrödinger equation on manifolds can be investigated on graphs (cf. e.g. [BL07]).
In this Chapter we present possible mechanisms determining how synchronous population activity in developing neuronal networks may be suppressed by tar-geted stimulation of so-called functional hubs based on recent experimental find-ings in neuroscience [BGP+09]. We ask the reader to understand that we do not provide an introduction to theoretical neuroscience here. Instead, see e.g. [Tuc88, KJJ96, DA01, GK02, Buz06]. The work we present in this Chapter – unpublished of the time of writing – originates from a collaboration with Birgit Kriener (Institute of Mathematical Sciences and Technology, Norwegian University of Life Sciences, As, Norway) and Marc Timme.
Besides the small-world architecture, there is another network structure beyond the two extremes of totally regular and random ones, known as the scale-free topol-ogy [AB00, DM01, AB02, Cal07] since it is characterized by a heavy-tailed dis-tribution of degree per neuron with no characteristic scale. While most neurons display local connectivity, there is a small number of hub neurons – characterized by the large numbers of cells they connect to – that have long-range connections.
However, whether hubs are in fact present within neural assemblies has just been ex-perimentally examined recently [BGP+09]. Here, a morphological analysis revealed scale-free features in a functional topology of developing hippocampal networks.
The high-connectivity (HC) neurons are a sub-population ofγ -amino-butyric-acid-releasing (GABAergic) interneurons with widespread axonal arbors.
Intriguingly, stimulating a single HC or hub neuron completely suppresses global synchronous activity. When the stimulation is switched off, the synchronous activity returns. If a low-connectivity (LC) or non-hub neuron is driven in the same way, synchronous activity remains almost identically as without stimulation.
Such synchronous oscillations constitute one of the most dominant collec-tive dynamics of complex networks. They occur not only in circuits neurons [BH99, Buz06, OLPT10], but in a large range of systems: ranging from metabolic and gene regulatory networks within cells [WF01, McM02, GdBLC03, TYHC03] to food webs of cross-feeding species [MHH98, WM00] or even to oscillations in the global climate system [SR94, SvRE98]. Thus, to understand the functional role of hubs – not only in neuronal circuits – is a question of paramount importance and has recently attracted attention [Per10, ZLZK10, SHK07, MS08, ASW+06, SR07].
However, the mechanisms underlying the suppression of global oscillations in a neu-robiological system have not yet been understood. Understanding those mechanisms based on the phenomena observed in [BGP+09] is the main aspect we address in this Chapter.