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Graph theoretical analysis of magnetoencephalographic functional connectivity in Alzheimer's disease

C. J. Stam, W. de Haan, A. Daffertshofer, B. F. Jones, I. Manshanden, A. M. van Cappellen van Walsum, T. Montez, J. P. A. Verbunt, J. C. de Munck, B. W. van Dijk, H. W. Berendse, P. Scheltens
DOI: http://dx.doi.org/10.1093/brain/awn262 213-224 First published online: 24 October 2008


In this study we examined changes in the large-scale structure of resting-state brain networks in patients with Alzheimer's disease compared with non-demented controls, using concepts from graph theory. Magneto-encephalograms (MEG) were recorded in 18 Alzheimer's disease patients and 18 non-demented control subjects in a no-task, eyes-closed condition. For the main frequency bands, synchronization between all pairs of MEG channels was assessed using a phase lag index (PLI, a synchronization measure insensitive to volume conduction). PLI-weighted connectivity networks were calculated, and characterized by a mean clustering coefficient and path length. Alzheimer's disease patients showed a decrease of mean PLI in the lower alpha and beta band. In the lower alpha band, the clustering coefficient and path length were both decreased in Alzheimer's disease patients. Network changes in the lower alpha band were better explained by a ‘Targeted Attack’ model than by a ‘Random Failure’ model. Thus, Alzheimer's disease patients display a loss of resting-state functional connectivity in lower alpha and beta bands even when a measure insensitive to volume conduction effects is used. Moreover, the large-scale structure of lower alpha band functional networks in Alzheimer's disease is more random. The modelling results suggest that highly connected neural network ‘hubs’ may be especially at risk in Alzheimer's disease.

  • Alzheimer's disease
  • functional connectivity
  • MEG
  • synchronization
  • small-world networks


A central question in cognitive neuroscience is how cognitive functions depend upon coordinated and integrated activity of specialized, widely distributed brain regions. There is strong support that a network perspective on the brain is required in order to understand higher brain functioning (Varela et al., 2001; Le van Quyen, 2003). How do functional interactions between brain regions take place, and how can this be measured and assessed? For answering these questions an important idea is the so-called functional connectivity that refers to linear or nonlinear statistical interdependencies between time series of physiological signals recorded from different brain regions (Aertsen et al., 1989; Friston, 2001; Lee et al., 2003; Fingelkurts et al., 2005). Functional connectivity is assumed to reflect functional interactions between the underlying brain regions.

The concept of functional connectivity has become very important in the study of brain mechanisms underlying disturbed cognition in Alzheimer's disease, the most frequent cause of dementia in the western population (van der Flier and Scheltens, 2005). Alzheimer's disease is characterized by degeneration of neurons starting in the hippocampus, later spreading to the temporal and parietal cortex, and finally involving most cortical areas. Loss of neurons, involvement of white matter as well as disturbed synaptic transmission, e.g. due to decreased levels of acetylcholine (Osipova et al., 2003), account for abnormal functional interactions between cortical regions. It has even been suggested that Alzheimer's disease can be viewed as a disconnection syndrome (Delbeuck et al., 2003). Support for this concept comes from a number of EEG and magneto-encephalography (MEG) studies using conventional coherence as a measure of functional connectivity (Leuchter et al., 1992; Besthorn et al., 1994; Dunkin et al., 1994; Jelic et al., 1996; Locatelli et al., 1998; Berendse et al., 2000; Knott et al., 2000; Stevens et al., 2001; Adler et al., 2003; Hogan et al., 2003; Jiang 2005; Koenig et al., 2005; Pogarell et al., 2005). In most of these studies a consistent decrease of coherence in the alpha and beta band was reported, whereas results for other bands were more variable. Abnormalities of functional connectivity have also been demonstrated with nonlinear synchronization methods (Jeong et al., 2001; Stam et al., 2002, 2006; Pijnenburg et al., 2004; Babiloni et al., 2004). While these studies in general support the hypothesis of a disconnection syndrome in Alzheimer's disease, two problems need further attention: (i) assessment of functional connectivity with EEG and MEG can be biased by volume conduction, which may yield spurious correlations between nearby sensors and hence render interpretation unreliable; (ii) connectivity studies in Alzheimer's disease are generally very descriptive and lack a more robust framework to discriminate between normal and abnormal networks in the brain.

Nearby EEG electrodes or MEG sensors are likely to pick up activity of identical sources, resulting in strong correlations between recorded signals that reflect simple volume conduction rather than true functional connectivity (Nunez et al., 1997; Srinivasan et al., 2007). Two approaches have been submitted to overcome this problem. First, one may study interdependencies between time series of reconstructed sources rather than signals of recording electrodes or sensors (Gross et al., 2001; David et al., 2002; Tass et al., 2003; Amor et al., 2005; Hadjipapas et al., 2005; Lehmann et al., 2006). While this approach has certainly the added benefit of dealing with interactions between anatomically well-defined brain regions, a major pitfall is the absence of a unique definition of the corresponding source space. Different assumptions may lead to different source models and, hence, different results. However, to date there is no reliable way to decide which model is the proper choice. Second, one may look for time series analysis techniques that extract interdependencies between signals which are not or at least unlikely due to volume conduction. This measure therefore reflects true interactions. An early attempt in this direction was summarized in a study by Nunez and colleagues (1997) who proposed to subtract a baseline random coherence from the measured coherence in order to obtain a reduced, task-related coherence, which is less influenced by volume conduction effects. More recently Nolte and colleagues (2004) proposed to use the imaginary part of the (complex-valued) coherency between two signals. Indeed volume conduction cannot give rise to imaginary coherency, but the magnitude of the imaginary part does not appear to be a proper value to quantify synchronization, since it mixes information on coupling strength and coupling delay. As an alternative, a so-called phase lag index (PLI) was introduced, which reflects the consistency with which one signal is phase leading or lagging with respect to another signal (Stam et al., 2007b). The PLI was shown to be less affected by volume conduction than more traditional measures like coherence, and by the same token, it is rather sensitive to true changes in synchronization. Here we will exploit this capacity to address possible changes in functional connectivity due to Alzheimer's disease, as we see an advantage of PLI to ‘reduced coherency’. Although the latter method might represent an improvement over traditional coherence, it does rely on several a priori assumptions such as stationarity and linearity, and is still sensitive to signal amplitude. PLI is sensitive to non-linear data and can handle non-stationary data, at least to a large degree.

The theoretical framework for understanding large-scale networks is given by ‘modern network theory’ (for a review see Boccaletti et al., 2006), a branch in graph theory, in which networks are represented by a set of nodes (vertices) and connections (edges). See Fig. 1 for an explanation of the basic principles of graph theory used in this study.

Fig. 1

Representation of a network as a graph. Black dots represent the nodes or vertices, and the lines connecting the dots the connections or edges. The left panel shows an unweighted graph. The shortest path length (L) between vertices A and B consists of three edges, indicted by the striped lines. The clustering coefficient (C) of a vertex is the likelihood that its neighbours are connected. For vertex C, with neighbours B and D, the clustering coefficient is 1. When weights are assigned to the edges, the graph is weighted (right panel). Here the weights of the edges are indicated by the thickness of the lines. Figure taken with permission from Stam and Reijneveld. Graph theoretical analysis of complex networks in the brain. Non-linear Biomedical Physics 2007c; 1: 3.

In recent years, graph theory has been introduced to the study of anatomical and functional networks in the central nervous system (Bassett and Bullmore, 2006; Stam and Reijneveld, 2007c). Graph theory provides models of complex networks in the brain, and allows one to better understand the relations between network structure and the processes taking place on those networks. It can also provide a concept of an ‘optimal’ network (for example in terms of balancing segregation and integration, performance and cost), and offers scenarios of how complex networks might develop, and how they might respond to different types of damage. Watts and Strogatz (1998) introduced so-called ‘small-world’ networks, which have a balance between local specialization and global integration that is optimal for information processing, and they showed that several real-life networks possess small-world features. Small-world networks have a relatively high amount of so-called ‘local clustering’, meaning that nodes are often connected to their neighbours, combined with relatively short ‘path lengths’, which means that from any node it takes just a few steps to reach any other node in the network. There is now accumulating evidence that different types of structural brain networks display a ‘small-world’ type network organization characterized by a combination of high local clustering as well as short path lengths (Watts and Strogatz, 1998; Hilgetag et al., 2000; He et al., 2007; Iturria-Medina et al., 2008). A similar approach has also been used to study networks of functional connectivity. In several fMRI studies of healthy subjects, small-world patterns were found (Salvador et al., 2005; Achard et al., 2006; Supekar et al., 2008). The presence of small-world type functional networks in healthy subjects was also confirmed in numerous EEG and MEG studies (Stam, 2004; Bassett et al., 2006; Smit et al., 2007; Stam et al., 2007a). However, only a few studies have yet shown that brain pathology may interfere with the normal small-world architecture. According to Bartolomei et al. (2006) brain networks in patients with low-grade glioma's are more random compared with healthy controls. A similar change in network structure was reported in patients with schizophrenia and in patients with epilepsy during the interictal state (Micheloyannis et al., 2006; Ponten et al., 2007; Rubinov et al., 2007). In a recent pilot study on Alzheimer's disease a loss of the normal small-world architecture was reported (Stam et al., 2006). In view of these findings one might speculate that brain disease in general gives rise to a deviation from the normal, optimal small-world configuration of brain networks. It is not clear, however, how such network changes come about.

As mentioned above two questions were addressed in the present study: (i) is it possible to confirm previous EEG and MEG reports of decreased resting state functional connectivity in Alzheimer's disease using a method that is less affected by volume conduction? (ii) can graph analysis reveal abnormalities in the large-scale topology of functional connectivity networks in Alzheimer's disease, and can such network changes be explained by modelling?

Materials and Methods

Patients and controls

Subjects and recordings were identical to Stam et al. (2006). The study involved 18 patients (mean age 72.1 years, SD 5.6; 11 males; mean MMSE 19.2, range: 13–25) with a diagnosis of probable Alzheimer's disease according to the NINCDS-ADRDA criteria (McKhann et al., 1984) and 18 healthy control subjects (mean age 69.1 years, SD 6.8; seven males; mean MMSE 29, range: 27–30), mostly spouses of the patients. Patients and controls were recruited from the Alzheimer Centre of the VU University Medical Centre. Subjects were assessed according to a clinical protocol, which involved history taking, physical and neurological examination, blood tests, MMSE (Folstein et al., 1975) neuropsychological work up (administration of a battery of neuropsychological tests), MRI of the brain according to a standard protocol and routine EEG. The final diagnosis was based upon a consensus meeting in which all the available clinical data and results of the ancillary investigations were considered. As reported in Stam et al. (2006), six patients used cholinesterase inhibitors, which was found to have no influence on functional connectivity. In the control and patient group both benzodiazepine and anti-depressive drug use was reported by one person. The study was approved by the Local Research Ethics Committee and all patients or their caregivers had given written informed consent. Since subjects were included years ago, medical files were checked again recently to verify initial diagnosis; no notable changes (besides disease progression) were discovered.

MEG recording

Magnetic fields were recorded while subjects were seated inside a magnetically shielded room (Vacuumschmelze GmbH, Hanau, Germany) using a 151-channel whole-head MEG system (CTF Systems Inc., Port Coquitlam, BC, Canada). Average distance between neighbouring sensors in this system was 3.1 cm. A third-order software gradient (Vrba et al., 1999) was used after online band-pass filtering between 0.25 and 125 Hz. Sample frequency was 625 Hz. For technical reasons two channels had to be omitted yielding 149 channels or sensors for analyses. Fields were measured during a no-task, eyes-closed condition. At the beginning and at the ending of the recording the head position relative to the coordinate system of the helmet was recorded by leading small alternating currents through three head position coils attached to the left and right pre-auricular points and the nasion on the subject's head. Head position changes during the recording up to ∼1.5 cm were accepted. During the MEG recording, persons were instructed to sit comfortably, close their eyes and reduce eye movements, but remain awake as much as possible. During the recordings, the investigator and MEG technician checked the signal on-line for visual signs of drowsiness (e.g. slow eye movement activity) and observed the patient using a video monitor.

As a filtering process, offline frequency analysis is performed on the raw data, using a Fourier transformation. In the obtained frequency spectrum all frequencies outside the studied bands are set to zero, and using an inverse Fourier transformation the filtered signal is then obtained, with preservation of all phase information of the original data. For the subsequent off-line processing the recordings were converted to ASCII files and down-sampled to 312.5 Hz. For each subject care was taken to find and select exactly three artifact-free epochs of 4096 samples (13.083 s) by two of the investigators (BFJ and IM). MEG registrations were converted to datafiles with a coded filename before epoch selection, so the investigators were blind to the subjects’ diagnosis during this process. Typical artifacts were due to (eye) movements, drowsiness or technical issues. Visual inspection and selection of epochs was realized with the DIGEEGXP software (CS). Epochs were band-pass filtered for the commonly used frequency bands: delta (0.5–4 Hz), theta (4–8 Hz), lower alpha (8–10 Hz), upper alpha (10–13 Hz), beta (13–30 Hz) and gamma (30–45 Hz), and all further analyses were performed for these bands separately.

Phase lag index

The PLI is a measure of the asymmetry of the distribution of phase differences between two signals. It reflects the consistency with which one signal is phase leading or lagging with respect to another signal (Stam et al., 2007b). The PLI performs at least as well as the synchronization likelihood (SL) (Montez et al., 2006) in detecting true changes in synchronization but it is much less affected by the influence of common sources. A more detailed explanation is offered in the Supplementary material to this article.

Beside a global, mean PLI calculation a more regional approach was used. For this analysis MEG sensors were grouped into five regions (frontal, temporal, central, parietal and occipital) for each hemisphere, and average PLI for all sensors within a region (local) or between two regions (long distance) were computed following the procedure described in Stam et al. (2006).

Graph analysis

In principle, networks can be represented by graphs, which are sets of vertices and corresponding sets of edges (Boccaletti et al., 2006; Stam and Reijneveld, 2007c). One may say that an edge or connection either exists or not but one may also assign a certain weight to an edge that reflects the importance or strength of the relation between two vertices. While the first one yields unweighted graphs in that edges have values of either 0 or 1, the latter produces so-called weighted graphs. To define the corresponding weights a matrix of correlations between signals recorded at different electrodes is generally suitable. We denote the matrix’ coefficients as wij, i.e. they connect vertex i with vertex j and specified their values using the afore-explained PLI. That is we defined a network of 149 vertices (matching the 149 available MEG channels) and used the matrix of PLI values between all pairs of MEG channels as edge weights.

Graphs can be characterized by various measures. Two fundamental ones are the clustering coefficient, which denotes the likelihood that neighbours of a vertex will also be connected to each other, and the average path length, i.e. the average number of edges of the shortest path between pairs of vertices (Fig. 1).

Well ordered networks are strongly clustered and show large path lengths. In contrast, random networks are weakly clustered with small path lengths. Neither ordered nor random networks are good candidates for real networks like the human brain. Hence, Watts and Strogatz (1998) suggested a new type of networks, so-called small-world networks, which have both large clustering coefficients as well as small path lengths. Interestingly, these networks can be designed to be scale-free by having very short path lengths and a power law degree distribution (Barabási and Albert, 1999). Both small-world and scale-free networks are optimal in the sense that they allow efficient information processing with a minimal number of connections. By now it has been shown that many types of network ranging from metabolic and genetic to social are either small-world or scale-free (Amaral and Ottino, 2004; Boccaletti et al., 2006).

The clustering index Ci of a vertex i generally represents the likelihood that other vertices j that are connected to the vertex i will also be connected to each other. This notion can be adopted for use with weighted graphs in various ways (Boccaletti et al., 2006). Here we propose a simple definition, closely related to the proposal of Onnela et al. (2005), which only requires symmetry (wij = wji) and that 0 ≤ wij ≤ 1 holds. Indeed, both conditions are readily fulfilled when using PLI as weight definition. The (weighted) clustering index of vertex i is then defined as Embedded Image(1) Notice that in all sums in (1) terms with k = i, l = i or k = l are skipped. In the special case in which wij equals either 0 or 1, this definition is equivalent to the classical definition for unweighted graphs (Watts and Strogatz, 1998). For isolated vertices, i.e. vertices that do not have any connections, all weights wij vanish, and the clustering index is defined as Ci = 0 (Newman, 2003). The mean clustering coefficient of the entire network can be determined via (1) as Embedded Image(2) Watts and Strogatz (1998) also defined the path length of unweighted graphs. We extend this definition to weighted graphs building on the approach of Latora and Marchiori (2001). In detail, we define the length of an edge as the inverse of the aforementioned edge weight, i.e. Lij = 1/wij if wij ≠ 0, and Lij = +∞ if wij = 0; recall that wij is positive because we use the PLI as edge weight. The length of a weighted path between two vertices is then defined as the sum of the lengths of the edges of this path. The shortest path lij between two vertices i and j is the path between i and j with the shortest length. Analogously to definition (2) the average weighted path length of the entire graph is computed as Embedded Image(3) Notice that instead of the arithmetic mean we here employed the harmonic mean (Newman, 2003), so that we can handle infinite path lengths between disconnected edges, i.e. 1/∞ → 0.

By definition, both values of Cw and Lw depend on edge weights and network structure but also on network size. In order to obtain measures that are independent of network size, the mean edge weight Graphic and the mean path length Graphic were computed, in which 〈C(surrogate)w〉 and 〈L(surrogate)w〉 denote weighted clustering coefficient and path length averaged over an ensemble of 50 surrogate random networks that were derived from the original networks by randomly reshuffling the edge weights. The steps involved in weighted graph analysis of the MEG data are illustrated schematically in Fig. 2.

Fig. 2

Schematic illustration of the steps involved in weighted graph analysis of MEG recordings. At each of the MEG sensors, illustrated in (A), MEG signals are recorded. Epochs of MEG data are filtered, as shown in (B), and correlations between all pairs of channels are determined with the phase lag index. This results in a weighted graph, with the strength of the synchronization between pairs of sensors indicated in colour (blue low, red high PLI), as shown in (C). From each graph the weighted clustering coefficient Cw and the weighted path length Lw are computed. Also, from each graph, and ensemble of random graphs is generated by randomly shuffling the connection weights (D). The Cw and Lw of each of the random graphs is determined and the mean values for the ensemble, 〈C(surrogate)w〉 and 〈L(surrogate)w〉, are determined. Finally, the ratios Graphic and Graphic are computed (E).

Modelling network damage

To understand the general mechanisms underlying network changes in Alzheimer patients two models were compared, adopted from Albert and Barabási (2002). The first model (Random Failure) assumes that network changes are due to a random decrease in strength of edges. The second model (Targeted Attack) assumes that edges connecting high degree vertices (‘hubs’) will be more vulnerable to attack than edges connecting low degree vertices. The models were implemented by taking the PLI data of a control subject, selecting an edge at random, and then decrease its weight by a factor 2 with probability 1 (Random Failure model), or a probability that depended on the degree of both vertices connected by the edge (Targeted Attack model). This procedure was repeated until the average PLI of the network was decreased to the average PLI of the Alzheimer group (Figure 3). Data of all control subjects were treated in a similar way. This resulted in two new data sets, one for each model, which were subjected to the same graph analysis as the original control and Alzheimer data sets.

Fig. 3

Damage modelling procedure. The mean PLI of a control subject network is lowered by randomly weakening edges in the network, until it reaches the same value as in a Alzheimer's disease patient network. The effect of this damage is then examined by comparing the network characteristics of the damaged network to the Alzheimer's disease patient network characteristics. RF = Random Failure, TA = Targeted Attack, Cw = mean weighted clustering coefficient, Lw = mean weighted path length.

Statistical analysis

Statistical analysis was done with SPSS for MS-Windows (version 15). Group differences in respectively gender distribution and PLI and were tested with ANOVA (analysis of variance) and two-tailed t-tests for independent samples (not assuming equal variance). Since graph measures showed a non-Gaussian distribution, group differences were tested with Mann–Whitney U-tests for independent samples. The effect of and medication use on PLI and network measures was assessed using Kruskal–Wallis tests. Associations between cognitive status (MMSE) and PLI or network-derived measures were assessed with Spearman's bivariate correlation test. A significance level of α < 0.05 was used.


Subject characteristics

No effect of gender distribution in the groups on PLI values and network measures was found. In the Alzheimer's disease patient group, six persons used cholinesterase inhibitors (rivastigmine or galantamine). However, use of this medication did not produce a significant effect on PLI or network measure outcomes. This was also the case for the use of other psychoactive drugs in both the patient and control group (see Patients and controls section).

Phase lag index

The average networks for Alzheimer's disease patients and controls computed with PLI in six different frequency bands are shown in Fig. 4.

Fig. 4

Average weighted graphs of Alzheimer's disease patients and controls in six frequency bands. The value of the PLI for all individual pairs of MEG sensors is indicated in colour (blue: low PLI; red: high PLI).

Visual inspection already suggested differences between the two groups, especially in the 8–10 Hz and 13–30 Hz bands. Group differences in mean PLI for each frequency band were tested with two-tailed t-tests for independent samples. The results are shown in Fig. 5.

Fig. 5

Mean PLI averaged over all pairs of MEG sensors for Alzheimer's disease patients and controls in six frequency bands. Error bars are SDs. The mean PLI was significantly lower in Alzheimer's disease patients compared to controls in the lower alpha band (two-tailed t-test, P < 0.022) and the beta band (two-tailed t-test, P = 0.036).

The mean PLI was significantly lower in the Alzheimer's disease group in the 8–10 Hz band (P = 0.022) and in the 13–30 Hz band (P = 0.036). A non-significant trend in the same direction was found in the 10–13 Hz band (P = 0.112). No clear differences could be observed in other bands. By way of illustration, for the two frequency bands with a significant mean difference in PLI more detailed, regional results are shown in Fig. 6.

Fig. 6

Schematic illustration of significant differences in long distance (indicated by arrows) and short distance (indicated by filled squares) PLI in the 8–10 Hz and 13–30 Hz band. Alzheimer's disease patients had lower left sided fronto-temporal, fronto-parietal, temporo-occipital and parieto-occipital PLI in the 8–10 Hz band. Local left frontal and temporal, and right parietal PLI were also decreased in Alzheimer's disease patients (A). For the 13–30 Hz band, Alzheimer's disease patients had lower inter hemispheric frontal, right fronto-parietal and bilateral frontal PLI (B).

For the 8–10 Hz band, Alzheimer's disease patients had significantly lower left fronto-parietal (P = 0.026), fronto-temporal (P = 0.007), parieto-occipital (P = 0.025) and temporo-occipital (P = 0.009) PLI. Local left frontal (P = 0.034), temporal (P = 0.011) and right parietal (P = 0.021) PLI were also decreased in the Alzheimer's disease group. For the 13–30 Hz band, Alzheimer's disease patients showed a decrease in interhemispheric frontal (P = 0.032), right fronto-parietal (P = 0.041) and local right (P = 0.020) and left (P = 0.046) frontal PLI.

Network analysis

Results of the weighted graph analysis are shown in Table 1.

View this table:
Table 1

Results of weighted graph analysis for Alzheimer's disease patients and controls in six frequency bands

Alzheimer's diseaseControlAlzheimer's diseaseControlAlzheimer's diseaseControlAlzheimer's diseaseControl
0.5–4 Hz0.12 (0.10–0.32)0.12 (0.10–0.17)4.05 (1.69–4.40)3.92 (2.89–4.59)1.04 (1.03–1.12)1.04 (1.02–1.11)1.09 (1.06–1.33)1.08 (1.05–1.34)
4–8 Hz0.11 (0.09–0.20)0.10 (0.09–0.15)4.23 (2.48–4.99)4.44 (3.22–5.01)1.05 (1.03–1.17)1.04 (1.03–1.13)1.14 (1.04–1.41)1.15 (1.05–1.43)
8–10 Hz0.15 (0.120.21)0.17 (0.130.29)3.27 (2.253.76)2.69 (1.803.73)1.04 (1.021.12)1.07 (1.041.13)1.08 (1.051.32)1.19 (1.071.30)
10–13 Hz0.12 (0.11–0.14)0.13 (0.11–0.22)3.83 (3.28–4.36)3.72 (2.36–4.30)1.04 (1.03–1.10)1.04 (1.03–1.21)1.10 (1.05–1.35)1.12 (1.04–1.45)
13–30 Hz0.06 (0.05–0.06)0.06 (0.05–0.08)7.97 (6.44–9.24)7.61 (5.18–9.35)1.04 (1.02–1.07)1.04 (1.03–1.16)1.11 (1.05–1.50)1.12 (1.04–1.50)
30–45 Hz0.05 (0.05–0.09)0.05 (0.05–0.08)8.70 (5.17–9.07)8.54 (6.06–9.14)1.02 (1.02–1.07)1.02 (1.02–1.07)1.09 (1.06–1.33)1.04 (1.02–1.30)
  • Values are medians, with range printed between parentheses. Cw = mean weighted clustering coefficient; Lw = mean weighted path length; Graphic = mean normalized average weighted clustering coefficient (see Materials and Methods section), Graphic = mean normalized average weighted path length. Significant differences between Alzheimer's disease and controls with non-parametric testing (Mann–Whitney U-test, P < 0.05) are given in bold.

The non-parametric Mann–Whitney U-test for independent samples revealed that Cw was lower in Alzheimer's disease subjects in the 8–10 Hz band (U = 89.5; P = 0.022), but not in the 13–30 Hz band (U = 107.0; P = 0.081). Lw was higher in Alzheimer's disease subjects in the 8–10 Hz band (U = 82.0; P = 0.011). In the 8–10 Hz band Alzheimer's disease patients had a lower Graphic (U = 76.0; P = 0.006) and a lower Graphic (U = 86.0; P = 0.016).

Modelling of network changes

Modelling with the Random Failure and the Targeted Attack model was applied to the data of the 8–10 Hz band since this band showed the most consistent differences in graph measures between the two groups. The average PLI graphs for Alzheimer's disease patients, controls and both models are shown in Fig. 7. On visual inspection, both models look quite similar to the average network in the Alzheimer's disease group. Please note that, by definition, the average PLI of both models is the same as the average PLI of the Alzheimer's disease data.

Fig. 7

Comparison of real and modelled networks in the 8–10 Hz band. Top left: average PLI for the Alzheimer patients. Top right: average PLI for the control subjects. Bottom left: average PLI after application of the ‘Targeted Attack’ model to control data. Bottom right: average PLI after application of the ‘Random Failure’ model to control data.

Further analysis of the model data compared with the real data is shown in Fig. 8. For the Random Failure model the Graphic was not different from the control data, and significantly higher than Graphic of the Alzheimer's disease group (Mann–Whitney U-test, U = 76.5; P = 0.007). In contrast, Graphic of the Targeted Attack model was not significantly different from the Alzheimer's disease group, but significantly lower than Graphic of the control group (U = 87.0; P = 0.018). The weighted path length Graphic showed a decreasing trend going from controls to Random Failure, Targeted Failure and controls (Fig. 8, right panel). Graphic of both models did not differ significantly from control data.

Fig. 8

Comparison of normalized weighted clustering coefficient (left panel) and path length (right panel) for Alzheimer patients, Targeted Attack model, Random Failure model and controls in the 8–10 Hz band. Box plots show median, interquartile range and extremes.

Correlation with MMSE

No significant correlations between MMSE and PLI or network measures were found in the Alzheimer's disease patient group (Fig. 9). When correlation with MMSE was analysed for all subjects (Alzheimer's disease and control) put together in one group, we found significant effects between MMSE and mean PLI in the beta band (Spearman's r = 0.570, P = 0.001) and between MMSE and Graphic in the lower alpha band (Spearman's r = 0.475, P = 0.008).

Fig. 9

In the left panel the correlation of mean PLI in the lower alpha band and MMSE is shown (Spearman's r = 0.570, P = 0.001), in the right panel the correlation of the mean clustering coefficient with MMSE in the beta band (Spearman's r = 0.475, P = 0.008). Alzheimer's disease and controls group were combined for this analysis.


The present study showed that resting-state functional connectivity of MEG is decreased in Alzheimer's disease patients in the lower alpha and beta bands using a recently developed measure, the PLI that appears invariant against volume conduction. This finding supports the concept of Alzheimer's disease as a disconnection syndrome. Moreover, changes in functional connectivity in Alzheimer's disease patients did not involve all brain regions to the same extent, suggesting a heterogeneous disruption of overall network structure. This idea was confirmed by graph analysis of the functional connectivity data, which revealed lower normalized clustering coefficients and path lengths in the Alzheimer's disease group in the lower alpha band. This type of change suggests that brain networks in Alzheimer's disease patients are closer to random networks than those of non-demented control subjects. The modelling results suggest that this change was brought about by a preferential decrease of connections between high degree nodes (‘hubs’), rather than a non-specific decrease of connection strength.

Volume conduction

A decrease of resting state functional connectivity in Alzheimer's disease patients in the alpha and often also in the beta band has been reported in many EEG and MEG studies (Leuchter et al., 1992; Besthorn et al., 1994; Dunkin et al., 1994; Jelic et al., 1996; Locatelli et al., 1998; Knott et al., 2000; Stevens et al., 2001; Adler et al., 2003; Hogan et al., 2003; Jiang 2005; Koenig et al., 2005; Pogarell et al., 2005). However, a major point of criticism is that such studies were done on the raw EEG and MEG time series. It is well known that estimates of statistical interdependencies in EEG and MEG may be biased by the effects of volume conduction and, in the case of EEG, by the influence of the reference electrode (Nunez et al., 1997; Guevara et al., 2005). More specifically, nearby EEG electrodes or MEG sensors are likely to pick up activity of the same source, and thereby to display spuriously high correlations between their time series. This problem can be solved to a large degree by acknowledging that spurious couplings due to volume conduction or active reference electrodes cannot give rise to phase delays between channels. The PLI is only sensitive to phase synchronization between two channels when one is consistently leading or lagging in phase with respect to the other. That is, with PLI any coupling with a phase difference which centres around 0 mod π are discounted. Put differently, our finding of a significant decrease of PLI in the lower alpha and the beta band cannot be explained by volume conduction but strongly supports the idea that resting-state functional connectivity is decreased in Alzheimer's disease. Since the PLI results are largely in line with the previous studies we can conclude that the influence of volume conduction and reference electrode in these studies may have been smaller than has sometimes been suggested. However, a detailed comparison of our study with a previous study, in which the same data were analysed with several linear and nonlinear measures, does display a few differences (Stam et al., 2006). For example, if we compare Fig. 6 of the present study with Figs 3, 4 and 7 of Stam et al. (2006), one finds that the PLI in the beta band only showed decreases in the Alzheimer's disease group, whereas coherence and SL also showed centro-parietal increases. A possible explanation could be that the increases in connectivity reported for SL and coherence might be influenced by volume conduction, while the decreases seems to be confirmed by the PLI and may reflect true loss of connectivity, but this should be subject to further study.

Resting state

Functional connectivity can be determined in relation to tasks as well as during a resting state. More recently there has been a growing interest in resting state functional connectivity because it appears that in particular memory-related brain networks are consistently activated during this state (Gusnard and Raichle, 2001; Laufs et al., 2003; Damoiseaux et al., 2006). Moreover, resting state functional connectivity has a strong genetic component, and shows characteristic changes in various psychiatric and neurological disorders (Posthuma et al., 2005; Stam, 2005, 2006).

Network analysis

In the present study Cw was decreased in the lower alpha and beta band and Lw was increased in the lower alpha band in the Alzheimer's disease group. It should be stressed that these changes in Cw and Lw are likely to be influenced by changes in the PLI. A lower mean level of PLI will decrease the estimate of Cw, irrespective of changes in network structure. Similarly, a lower PLI will give rise to longer weighted path lengths. These results should be compared to Fig. 5 in Stam et al. (2007a). Here Cw and Lw were compared between controls and Alzheimer's disease patients for the same threshold, showing a non-significant trend to a lower Cw and a significant increase of Lw in the Alzheimer's disease group. By using the same threshold for both groups, differences in mean PLI could have influenced the results. Thus changes in Cw and Lw in both studies are consistent, but cannot be taken as ‘pure’ measures of changes in network structure as they are likely to be influenced by the lower mean level of connectivity in the Alzheimer's disease group.

In contrast, the normalized coefficients Graphic and Graphic are corrected for differences in mean PLI between subjects, since each network is compared to its own random counterpart. The most important result is thus the decrease of Graphic and Graphic in the Alzheimer's disease group in the lower alpha band. Within the framework of the Watts and Strogatz model this suggests that network architecture in Alzheimer's disease patients is significantly closer to that of random networks. However, Graphic was very close to one in both groups, and much lower than reported in other studies, where Graphic was usually around two (Stam, 2004; Salvador et al., 2005; Achard et al., 2006; Bassett et al., 2006; Stam et al., 2007a). It is possible that correlations between nearby sensors due to volume conduction could have produced spuriously high estimates of Cw in previous EEG and MEG studies.

Damage modelling

Modelling was used to investigate whether the observed network changes in Alzheimer's disease in the 8–10 Hz band could be explained by a general mechanism. In the literature on complex networks generally two types of network damage are considered: Random Failure, where edges and/or vertices are lost randomly, and Targeted Attack, where damage mainly affects high degree, critical vertices and/or edges (section 3 in Boccaletti et al., 2006, for a more practical application see Kaiser et al., 2007). In our study the Targeted Attack model performed better than the random error model in explaining the network changes in Alzheimer's disease, in particular with respect to the clustering coefficient (Fig. 7). While both models lowered the mean PLI to the level observed in the Alzheimer's disease group, only the Targeted Attack model produced a clustering coefficient as low as in the patients, whereas the Random Failure model did not change the clustering coefficient at all. These results suggest that the disease process in Alzheimer's disease may specifically affect association fibres connecting brain areas that are highly connected to the rest of the brain, that is: higher order association areas. The distribution of amyloid plaques in Alzheimer's disease is in agreement with this suggestion (Nordberg, 2007).

Several studies have investigated the nature of network changes in different types of brain pathology. In the case of brain tumours, schizophrenia and interictal recordings of patients with epilepsy pathological networks were characterized by a smaller Cw and a smaller Lw (Bartolomei et al., 2006; Micheloyannis et al., 2006; Ponten et al., 2007; Rubinov et al., 2007). Considering the model of Watts and Strogatz, where the edges of a fully ordered network with degree K (number of edges per vertex) are rewired randomly with a certain probability P, a lower Cw and Lw would correspond with a higher value of the rewiring probability, and a more random network. The findings in the Alzheimer's disease group in the present study seem to fit in the same scheme: decrease of both Cw and Lw, and a more random network in the patient group. Moreover, the values were close to (although significantly different from) 1, which indicates that the difference between real and random networks was very small. The one finding that does not fit in this pattern is the increase in beta band path length for Alzheimer's disease patients reported in the previous pilot study (Stam et al., 2007a). This result was obtained only for some values of degree K, with K identical for both groups (Fig. 5 in Stam et al., 2007a). One explanation could be that in the EEG pilot study disconnected points (which occur already for values of K = 3) were excluded from the computation of the path length, whereas in the present study they were included (see formula in Materials and methods section). This is an essential difference, excluding or including disconnected points may decrease or increase the estimated path length considerably. The lower alpha band, which was the only band to show clear changes in normalized clustering coefficient and path length in the current MEG study, was not investigated in the EEG study. Therefore, the evidence in favour of more random network topology in Alzheimer's disease seems to be stronger, and in line with changes in other disorders. To be able to find a disease-specific ‘network change profile’ probably requires further exploration of this network approach and its relation to clinical features of Alzheimer's disease. Possibly ‘network randomization’ may be a final common pathway for different types of brain damage, resulting from loss of neurons and connections as well a random outgrowth of new connections. A related concept of increased entropy relating to ageing and Alzheimer's disease has recently been formulated by Drachmann: ‘Increasing entropy, manifest through a complex network of interacting age related changes, is seen as the fundamental driving cause of neural and cognitive decline in the elderly, as well as the overriding etiologic principle in further transition to sporadic Alzheimer's disease’ (Drachmann, 2006). It would be of considerable interest to study how different types of treatment will interfere with this process of network randomization, and how the network parameters relate to disease severity and cognitive performance.

Supplementary material

Supplementary material is available at Brain online.


Alzheimer Nederland; Dutch Science Foundation (NWO, grant #452-04-344).


The authors thank Mrs Els van Deventer for continuing support in retrieving relevant literature.


  • Abbreviations:
    mini mental state examination
    phase lag index
    synchronization likelihood


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