Anomalous Consistency in Mild Cognitive Impairment: a complex networks approach
J. H. Martínez, J. M. Pastor, P. Ariza, M. Zanin, D. Papo, F. Maestú, R. Bajo, S. Boccaletti, J. M. Buldú
aa r X i v : . [ q - b i o . N C ] O c t Chaos, Solitons & Fractals 00 (2018) 1–14
Chaos,Solitons & Fractals
Anomalous Consistency in Mild Cognitive Impairment: A complexnetworks approach.
J. H. Mart´ınez a,b, ∗ , P. Ariza c , M. Zanin d,e , D. Papo f , F. Maest ´u g , Juan. M. Pastor a , R.Bajo g , Stefano Boccaletti h , J. M. Buld ´u i a Complex Systems Group, Technical University of Madrid, Madrid, Spain b Modelling and Simulation Laboratory, Universidad del Rosario de Colombia, Bogot´a, Colombia c Laboratory of Biological Networks, Centre for Biomedical Technology (UPM-URJC), Madrid, Spain d Faculdade de Ciˆencias e Tecnologia, Departamento de Engenharia Electrot´ecnica, Universidade Nova de Lisboa, Lisboa, Portugal e The INNAXIS foundation and Research Institute, Madrid, Spain f Computational Systems Biology Group, Centre for Biomedical Technology (UPM), Madrid, Spain g Laboratory of Cognitive and Computational Neuroscience, Centre for Biomedical Technology (UPM-UCM), Madrid, Spain h CNR-Istituto dei Sistemi Complessi, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Italy i Complex Systems Group, Universidad Rey Juan Carlos, Madrid, Spain
Abstract
Increased variability in performance has been associated with the emergence of several neurological and psychiatric pathologies.However, whether and how consistency of neuronal activity may also be indicative of an underlying pathology is still poorly under-stood. Here we propose a novel method for evaluating consistency from non-invasive brain recordings. We evaluate the consistencyof the cortical activity recorded with magnetoencephalography in a group of subjects diagnosed with Mild Cognitive Impairment(MCI), a condition sometimes prodromal of dementia, during the execution of a memory task. We use metrics coming from non-linear dynamics to evaluate the consistency of cortical regions. A representation known as parenclitic networks is constructed,where atypical features are endowed with a network structure, the topological properties of which can be studied at various scales.Pathological conditions correspond to strongly heterogeneous networks, whereas typical or normative conditions are characterizedby sparsely connected networks with homogeneous nodes. The analysis of this kind of networks allows identifying the extent towhich consistency is a ff ected in the MCI group and the focal points where MCI is especially severe. To the best of our knowledge,these results represent the first attempt at evaluating the consistency of brain functional activity using complex networks theory.c (cid:13) Keywords:
Complex Networks, Functional Brain Networks, Mild Cognitive Impairment, Magnetoencephalography, Synchronization,Consistency
1. Introduction
Excessive variability in performance negatively impacts people’s ability to carry out activities of daily living.Increased short-term fluctuations, particularly in reaction times, that cannot be attributed to systematic e ff ects, such ∗ Corresponding author
Email address: [email protected] (J. H. Mart´ınez)
URL: http://johemart.wix.com/neurobelongings (J. H. Mart´ınez) Chaos, Solitons & Fractals 00 (2018) 1–14 as learning, have been associated with a wide range of cognitive disorders including impaired top-down executive andattentional control processes, and with conditions including healthy ageing, and various neurological and psychiatricdisorders ranging from Parkinson’s disease [1, 2], multiple sclerosis [3], traumatic brain injury [4], schizophrenia [5]and various forms of dementia [6, 7]. A number of studies attest an association between behavioral inconsistency andstructural and functional brain abnormalities. For instance, di ff usion tensor imaging showed a relationship betweenintra-individual variability in reaction times and white matter integrity, with variability increasing with white matterdegradation, pathway connectivity degradation and brain dysfunction [8, 9, 10]. Behavioural inconsistency was alsoassociated with neurotransmitter dysfunction, stress, and fatigue [11, 12, 13, 7, 14].However, whether and how behavioural consistency stems from a corresponding loss of consistency of functionalbrain activity is still unclear. For example, it has been shown that behavioural variability in the response time duringa face recognition task negatively correlates with the brain signal variability, at least when children and adults arecompared [15], the latter having lower response time variability combined with higher signal variability. In the currentwork we will focus on the variability of brain dynamics when carrying the same (memory) task. We study whetherthe dynamics of the recorded signal at dierent cortical regions maintains or not its shape when the same memory taskis carried out, and how the variability of each cortical region is related to that of other regions.In physics, consistency [16] has been studied with a series of di ff erent dynamical models [17, 18, 19]. The emer-gence of a consistent response requires a high synchronization between di ff erent outputs of a nonlinear system (i.e.,di ff erent realizations with di ff erent initial conditions) when the same external input is applied. Nevertheless, consis-tency does not imply the observation of a synchronized state between the external input and the system’s response asit is the case of an entrained or driven system. Figure 1 shows, qualitatively, the di ff erence between a driven systemand a consistent / inconsistent system. Figure 1.
The phenomenon of consistency.
The consistency of a dynamical system relies on its ability to respond in the same way when the thesame external input is applied. In all panels (A, B, C), the same external perturbation is applied and the response of the system with two di ff erentinitial conditions ( ~ x and ~ x ′ ) is (qualitatively) shown. In A, we show an example of a driven system, where the output always behaves in the sameway (as the external perturbation) due to a strong forcing of the input. In B, the external perturbation is not able to drive the output which respondsin a di ff erent way when the same input is applied (due to di ff erent initial conditions). In C, we show an example of consistent dynamics. When thesame input is applied, the system’s output always behaves in the same way, despite not mimicking the driving dynamics. Here we propose a new methodology for quantifying neuronal consistency that can be applied to noninvasivediagnostic procedures. This method involves constructing a parenclitic network representation [20, 21] based onanomalous behaviour of a certain set of features. In our case, each node of the parenclitic network, each of them2
Chaos, Solitons & Fractals 00 (2018) 1–14 corresponding to a certain cortical region, is associated with a feature: its dynamical consistency during a memorytask. We construct a network where nodes (i.e., cortical regions) have a link between them with a weight that dependson how their consistency diverges from an expected value. The topological characteristics of the parenclitic networkscan be used to extract information about how consistency is lost / maintained across the whole functional network.In such a representation, atypical or pathological conditions lead to the existence of a high number of links withlarge weights. Previous applications of this methodology to biological data have shown that parenclitic networksobtained from datasets associated with pathological conditions lead to strongly heterogeneous networks, i.e., networkswhere some nodes have a degree that is much higher than expected for links created at random, whereas typical ornormative conditions are characterized by sparsely connected networks with homogeneous nodes [20, 21]. In essence,a parenclitic representation equips the set of abnormalities of a system with a network characterization with topologicalproperties at various scales.Specifically, we study the consistency of functional brain activity in a group of patients diagnosed with mildcognitive impairment (MCI). MCI is a clinical condition in which subjects experience memory loss to a greater extentthan would be expected for age, who while not meeting the criteria for clinically probable Alzheimer’s Disease (AD)are nonetheless at increased risk of developing it. Behavioural evidence shows that, compared to cognitively healthyageing, MCI has been associated with increased response time variability and particularly in those subjects laterdeveloping AD [11, 12, 22, 13, 8, 9]. Abnormal consistency in MCI may therefore represent a measure of functionalintegrity that may help identifying those patients who ultimately lapse into fully-edged dementia.On the other hand, recent studies have shown how the functional networks of MCI patients obtained during amemory test show di ff erences in their topologies when compared to healthy subjects [23, 24]. This fact indicatesthat a global reorganization of brain activity is occurring, raising the question of how the consistency of a brainnetwork is a ff ected by this reorganization. With the aim of studying the consistency of the whole brain network, weused magnetoencephalography (MEG) to record the brain activity of a group of patients su ff ering from MCI and ahealthy control group as they carried out the Sternberg short-term memory task. We then computed the consistencyof each brain site for each individual of each group and finally constructed the parenclitic network for the di ff erencesin consistency between MCIs and controls (see Materials and Methods for details). The structure of the parencliticnetworks highlights those regions whose consistency is most a ff ected by the disease and suggests ways in which thee ff ects of MCI may propagate through the functional brain network.
2. Materials and Methods
All subjects or legal representatives provided written consent to participate in the study, which was approved bythe local ethics committee of the Hospital Cl´ınico Universitario San Carlos (Madrid, Spain). Fourteen right handedpatients with MCI were recruited from the Geriatric Unit of the “Hospital Cl´ınico Universitario San Carlos Madrid”.In addition, fourteen age-matched elderly participants without MCI were included as the control group. In additionto age, years of education were matched to the MCI group (for details see [23]): 10 years for the MCI group and11 years for controls. To confirm the absence of memory complaints, a score of 0 was required in a 4-questionquestionnaire [25]. None of the participants had a history of neurological or psychiatric condition. The diagnosisof MCI was made according to the criteria proposed by Petersen [26, 27]. None of the subjects with MCI hadevidence of depression as measured using the geriatric depression scale (score lower than 9) [28]. MCI subjectsand healthy participants underwent a neuropsychological assessment in order to establish their cognitive status withrespect to multiple cognitive functions. Specifically, memory impairment was assessed using the Logical Memoryimmediate (LM1) and delayed (LM2) subtests of the Wechsler Memory Scale-III-Revised. Two scales of cognitiveand functional status were applied as well: the Spanish version of the Mini Mental State Exam (MMSE) [29], and theGlobal Deterioration Scale / Functional Assessment Staging (GDS / FAST). A summarized demographical informationis stored in Tab. 1.
Magnetoencephalography (MEG) scans were obtained in the context of a modified version of the Sternberg test[30], which is a well-known task for studying working memory. Specifically, it consisted of a letter-probe test [31, 32]3
Chaos, Solitons & Fractals 00 (2018) 1–14 in which a set of five letters was presented and subjects were asked to keep the letters in mind. After the presentationof the five-letter set, a series of single letters (1000 ms in duration) was introduced one at a time, and participants wereasked to press a button with their right hand when a letter of the previous set was detected. The list consisted of 250letters in which half were targets (previously presented letters), and half distracters (not previously presented letters).Subjects undertook a training series before the actual test and the task did not start until the participant demonstratedthat he / she could remember the five letter set. Subjects’ responses were classified into four di ff erent categories: hits,false alarms, correct rejections and omissions. Only hits were considered for further analysis because the authors wereinterested in evaluating the functional connectivity patterns which support recognition success. The percentage of hits(80% control group and 84% MCI group) and correct rejections (92% control group and 89% MCI group) was highenough in both groups, indicating that participants actively engaged in the task. No significant di ff erence betweenthe two groups during the memory task was revealed. Note that despite patients su ff ering from MCI have deficitsin episodic memory, previous results have shown di ff erences in the functional networks obtained during a short-termmemory tests [23, 24] and even at resting state [33]. Therefore, due to the unpredictability of the episodic memorylags, we have studied the e ff ects of the disease during a short-term memory test. MEG signals were recorded with a 254 Hz sampling frequency and a band pass of 0.5 to 50 Hz, using a 147-channel whole-head magnetometer (MAGNES 2500 WH, 4-D Neuroimaging) confined in a magnetically shieldedroom. An environmental noise reduction algorithm using reference channels at a distance from the MEG sensors wasapplied to the data. Letters of the Sternberg test were projected through a LCD videoprojector (SONY VPLX600E),situated outside of a magnetically-shielded room, onto a series of in-room mirrors, the last of which was suspendedapproximately 1 meter above the participant’s face. The letters subtended 1.8 and 3 degrees of horizontal and verticalvisual angle respectively. Prior to functional connectivity analysis, all recordings were visually inspected by anexperienced investigator, and all of them containing visible blinks, eye movements or muscular artifacts were excludedfrom further analysis. Thirty-five epochs (1 second each one) were finally used. This lower bound was determinedby the participant with least epochs, in order to have an equal number of epochs across participants. This way, werandomly selected thirty-five epochs from those individuals with a higher number of valid epochs.
The first step is to quantify how consistent the output of each channel is. This can be done by computing howcoherent the outputs of the same channel are when the same task is carried out. With this aim, for each individual,we calculate the Synchronization Likelihood (SL) [34, 23] between each pair of MEG time series within the samechannel, considering only successful identifications of the target letters. SL is a nonlinear measure of correlationindicating synchronization for values close to one and the absence of any correlation when approaching zero. While
Table 1. Demographic and clinical information of the Control and MCI groups [23].
MMSE : Mini Mental State Exam (maximum score is 30);
GDS : Global Deterioration Scale;
LM1 : Logical Memory immediate recall;
LM2 : Logical Memory delayed recall
GDS LM1 LM2
Controls MCIs Controls MCIs Controls MCIs1 3 42 . ± . ± . ± . ± Sample(Sex) Age MMSE
Controls MCIs Controls MCIs Controls MCIs14(9 Female) 14(9 Female) 70 . ± . . ± . . ± . ± Chaos, Solitons & Fractals 00 (2018) 1–14 other measures to evaluate linear or nonlinear correlation between time series could have been used [35], SL hasproven an adequate measure for capturing the interdependencies between MEG time series obtained during a short-term memory tests [24].This way, we evaluate if the cortical activity measured at each channel during a positive identification of a letteris consistent, e.g., has similar temporal evolution when repeating the same task, despite the initial conditions areintrinsically di ff erent. We calculate the average of the pairwise SL of all non-repeated permutations of the 35 timeseries recorded within each magnetometer in order to get the Channel Consistency (CC). Note that the 35 time seriesof each channel are not combined but compared between them to extract the CC.At this stage, we have a dataset based on 147 CC values for the 14 subjects of the two di ff erent groups (MCIs andcontrols), which were used to build the corresponding parenclitic networks [21]. Node 61 N o d e .
08 0 .
09 0 . . . . . . . . . . ControlsMCIsLinear fitting
138 85 124 86 8829272523 123 571405958 1225643142413914337125 353331
Patient 13 e n61, 84 Link
Figure 2.
Creating a parenclitic network.
The channel consistency (CC) of channels 61 and 84 (each one corresponding to a node of theparenclitic network) is plotted for the 14 controls (black circles) and the 14 patients su ff ering from MCI (red squares). A linear fitting of the CCpairs of the control group is calculated (blue line) and it will be taken as the reference for a normal behaviour. e n , accounts for the deviation ofCC from the reference value for the individual n . Each individual, will have its own parenclitic networks, where the weight of the link betweennodes 61 and 84 is calculated as Z n , = | e n , | /σ , , being σ , the standard deviation of the CC values from the reference line. This way,the larger the weight of the link the higher the deviation from the reference given by the control group. This procedure is repeated for each pair ofchannels, leading to a parenclitic network for each individual, whose links quantify how far is the consistency of a pair of channels from the normalbehaviour. The bottom plot shows part of the parenclitic network of patient 13, specifically the neighborhood of node 61 once all pairs of channelsunderwent the same procedure. Chaos, Solitons & Fractals 00 (2018) 1–14 The method for the network construction is explained in Fig. 2, which shows a specific example with channels i =
61 and j =
84. Black dots represent the CC of these two magnetometers for all control subjects and the blueline is their corresponding linear fitting. Note that there is no previous evidence that a linear correlation between theconsistency of two independent channels exists, since, to the best of our knowledge, this is the first study obtainingnetworks from intra-channel consistency. Nevertheless, as we will see, assuming a linear correlation leads to a cleardistinction between groups, based on statistical significant di ff erences between the network parameters of the controland MCI groups. Interestingly, assuming a correlation described by a second-order polynomial leads to the samequalitative results (not shown here). This is due to the fact that the number of points used to estimate the correlationfunction (fourteen in the absence of outliers) is low, since it is di ffi cult to have large datasets of individuals. Acombination of both kinds of correlations together with other non-linear methods to evaluate functional dependenciesbetween nodes deserves its own study and it will probably depend on the kind of data and problem (disease) underinvestigation.The errors of the control group are adjusted to a normal distribution in order to obtain its standard deviation σ i , j . Once the standard deviation is obtained, we recalculate the correlation function excluding the outlier points ofthe control group. Individuals of the control group with an error higher than 2 σ i , j are assumed to be outliers andare not considered for the definition of the correlation function. This way we exclude 6030 outliers from all the N ( N − / e i , j of the joint consistency of channels i and j .The z-score of each pair of channels associated to a subject n will measure how far the consistency of both channelsis from the expected value and it is obtained as Z ni , j = | e ni , j | /σ i , j . The next step is to project the z-score dataset into aparenclitic network. In general, the links of a parenclitic network are created with a weight that is proportional to thedeviations of a certain feature from an expected value [21]. These networks are weighted and non-directed, and theyunveil important topological di ff erences between a reference group and a group with a certain anomaly [20]. In ourcase, the nodes of the network will be the channels measuring the activity of a certain cortical region and the linksbetween a pair of nodes i and j will be the z-score Z ni , j measuring their deviation from its expected value.The procedure followed to obtain the links and weights of nodes i =
61 and j =
84 is repeated for the CC of the147 × / L links with higher weight, leading to a sparse matrix whose topology will be further analyzed. Nevertheless, it is adelicate step, since a very low threshold will maintain spurious data that may hide the observation of the real networkstructure, while a very high threshold could dismiss valuable information. In order to adequately set the thresholdvalue, we repeated the analysis for di ff erent values of L and calculated the corresponding network parameters. Next,we identify the threshold that showed more di ff erences between the network parameters of the control and the MCIgroups [36]. Specifically, we focused on the di ff erences in the local ¯ E l and global ¯ E g e ffi ciency, since the dependenceof these parameters on the deletion of links is smoother than the clustering coe ffi cient C of the shortest path length d[37], and, in turn, showed a maximum di ff erence around the same value of L.After following this procedure, we finally consider only the L =
400 links with the largest Z ni , j of each parencliticnetwork. This way, we obtain a set of weighted sparse networks for the control and MCI subjects with the samenumber of nodes N and links L . By computing a set of network metrics [38], we are able to compare whether thenetworks di ff er in their topological organization and what are the kind of network structures associated to each group.All network metrics were statistically analyzed on the basis of the mean di ff erences between both populations and5000 di ff erent permutations were performed in order to obtain the corresponding p value of each network metric.
3. Results
We characterize the following properties of the parenclitic networks: a) the degree and strength of the nodes, b)the clustering coe ffi cient, c) the characteristic path length, d) the local and global e ffi ciency and e) the eigenvectorcentrality of the nodes (i.e., the identification of the network hubs).6 Chaos, Solitons & Fractals 00 (2018) 1–14 Our first inspection of the network topology focuses on the local properties of its nodes. Specifically, we computedthe highest degree K max of the network, which gives the highest number of connections a node has. K max is an indicatorof the existence of network hubs, i.e., nodes with a number of links much higher than the expected from the averageconnectivity.If we take into account the weight associated to the links, we can also compute the node strength S ( i ), which isgiven by the sum of all weights of the links attached to node i . The maximum strength S max and the average strength¯ S of the networks are obtained as the maximum / average of S ( i ) over all nodes. Note that Smax better captures theimportance of a node in the network, since apart from depending on the number of links a node has, it takes intoaccount the weights associated with these links.Figure 3 shows a comparison of the highest degree K max , maximum strength S max and the average strength ¯ S ofthe control and MCI groups. We use the box & whisker representation which highlights the main statistical quantitiesof the datasets, i.e., the first, second and third quartile and the mean. When looking at the highest degree K max we cansee how, in the MCI group, the mean, median (or second quartile), third quartile and, in general, all values are around50 % higher than the corresponding values of the control group (Fig. 3A). This finding evidences the presence ofhubs with higher number of links in the parenclitic network associated to the MCI group. Since the number of linksis limited to L =
400 in both groups, the fact that large hubs arise in the MCI networks also reveal the formation ofmore heterogeneous structures. K max max
50 01 0001 5002 0002 500 010203040506070 ¯ SMCIsControlsA CB
Figure 3.
Highest degree, maximum strength and average strength.
From left to right, box & whisker representation showing the first, second,third quartile and the average of: (A) the highest degree K max , (B) maximum strength S max and (C) average strength ¯ S . Patient and control groupsare orange and green, respectively. Red stars account for outlier values. Figure 3B shows the maximum strength S max for each group. Red stars are outliers that show that the maximumstrengths of the MCI group follow not a normal but a skewed distribution. The behaviour of S max remains similar to K max and evidences that the existence of stronger hubs in the MCI group is reinforced when considering the weightof the links. Therefore both measures indicate the existence of certain nodes that accumulate large deviations in theirexpected value of consistency, leading to more heterogeneous networks.In Fig. 3C we plot the average strength ¯ S of the networks to confirm that the MCI patients have higher deviationsfrom the reference value than the control individuals. As expected, the average strength of the MCI group ¯ S MCI = .
59 is much higher than the control group ( ¯ S control = . Chaos, Solitons & Fractals 00 (2018) 1–14 its links (see Materials and Methods). The higher the value of ¯ S , the larger the deviation of the overall consistencyof the functional network. It is important to remark the di ff erence between the maximum and average strength ofboth groups, which is much higher in the MCI group. This fact reveals the more heterogeneous structure of the MCInetworks, where nodes with strong deviations from the expected value arise (i.e. the higher the strength of a node, themore anomalous its consistency is). K (Degree) P ( k ( i ) > K ) − − CDF MCIsCDF Controls
S (Strength) P ( S ( i ) > S ) − − CDF MCIsCDF Controls
MCIs =−1.28 γ Controls =−1.42
A B γ Figure 4.
CDF of the degree and strength distributions. (A) Cumulative Distribution Functions (CDF) of the probability of finding a node witha degree (strength in (B)) higher than k ( S in (B)). Green circles correspond to the control group and orange circles to the MCI group. When thestrength of the nodes is considered (B), we obtain the power law distribution P ( S ( i ) ≥ S ) ∼ S − γ in both cases, as indicated by a straight line in thelog-log scale. Finally, we calculate the cumulative distribution function (CDF) of the degree and strength of the nodes both in thecontrol and MCI groups. For each node i (i.e., cortical region), we compute its corresponding average degree h k ( i ) i .Next, we obtain the CDF by computing the percentage of nodes with a degree h k ( i ) i ≥ k . The same methodologyis followed to calculate the CDF of the node strength P ( h S ( i ) i ≥ S ). Both distributions are plotted in Fig. 4 withregard to the subjects of each group. Permutation t-test for the degree and strength CDFs was performed. Taking intoaccount 5000 randomizations we found significant statistically di ff erences in the strength distributions with a p-value = ff erences in the degree CDFs were not statistically significant (p-value = k c =
40 showing that nodes with a degree higher than k c are more probable in the MCI group than in the control group. This confirms what the values of Kmax and Smaxalready suggested: a number of nodes in the parenclitic networks of the MCI group grossly deviate from their expectedconsistency.The strength distribution spreads over three orders of magnitude and allows the CDF to show a power law decay ∼ k − γ . The exponent of the power law is lower in the MCI group ( γ MCI = . < γ control = .
42) which reveals theexistence of hubs with a larger strength. These hubs play a relevant role in the structure of the network, since theyaccumulate a higher percentage of the link weights: they are the core of the divergences with respect to the normal(healthy) values of consistency.
Next, we calculate two local (clustering coe ffi cient ¯ C and local e ffi ciency ¯ E l ) and two global (average path length¯ L and global e ffi ciency ¯ E g ) properties of the network. The clustering coe ffi cient c i of a node i is the number oftriangles around a node (i.e., number of neighbors that, in turn, are neighbors between them) divided by the highestpossible number of triangles (i.e., the number of triangles if all of its neighbors were connected between them) [39]. c i is calculated using a generalization of this metric for weighted networks [40], and averaged over the whole networkto obtain the clustering coe ffi cient ¯ C per individual. ¯ C is an indicator of the local density of connections inside the8 Chaos, Solitons & Fractals 00 (2018) 1–14 network, and it has been related to the local resilience of a network against removal of links (i.e., the highest theclustering, the highest the local resilience) [41]. Figure 5A shows ¯ C for the two groups under study. We can observehow the MCI network has a largest clustering coe ffi cient, indicating a higher density of connections at the local level.Interestingly, the clustering coe ffi cient is also an indicator of the network randomness since random networks have avalue of ¯ C close to zero. Thus, the lowest value of ¯ C of the control group indicates that its network topology is closerto a random structure.
00 . 050 . 10 . 150 . 20 . 250 . 30 . 35 ¯ C MCIs
00 . 511 . 522 . 53 ¯ d 0123456 ¯ E l ¯ E g Controls
A B C D
Figure 5.
Clustering, shortest path length, local e ffi ciency and global e ffi ciency. Box & whisker representation of: (A) the clustering ¯ C , (B)shortest path length ¯ d , (C) local e ffi ciency ¯ E l and (D) global e ffi ciency ¯ E g . Orange and green bars correspond, respectively, to the MCI and controlgroups. Red stars are the outlier values. P-values of the network parameters are given in Table 2. Now let us have a look at a global property of the network: the average shortest path ¯ d , i.e., the average numberof steps to go from a node to any other. To obtain the value of ¯ d we first calculate the distance matrix D for allparenclitic networks. We assign a length to each link as l i , j = / Z i , j , i.e. the higher the weight of the link, the shorterthe “distance” between node i and j . Each d i , j element of the D matrix is the shortest path between nodes i and j (i.e., the lowest combination of links’ lengths to go from i to j ), which is calculated using the Dijkstra’s algorithm[42]. Finally, the average shortest path ¯ d is just the average of all elements of matrix D . Figure 5B shows that theMCI group has a lower value of ¯ d , which is a consequence of having higher weights (i.e., shorter distances) in itsconnectivity matrix (since, as we have seen, ¯ S MCI > ¯ S control ). Parenclitic networks are capturing how alterations ofthe expected consistency are distributed over the whole network and, therefore, the low value of ¯ d reveals that the lossof consistency propagates with a shorter number of steps in the MCI group. That is not good news for the resilienceof the consistency when MCI emerges.Both parameters ¯ C and ¯ d can be reinterpreted in terms of how e ffi cient is the network when transmitting informa-tion from one node to any other in the network. With this aim, Latora et al. [37] introduced the concept of local ¯ E l and global ¯ E g e ffi ciency, which accounts for the harmonic mean of the inverse of the number of steps between anypair of nodes. E ffi ciency can be defined at a local level, i.e., within the community of neighbours of a certain node,or globally, i.e. over the whole network. This way, high values of local / global e ffi ciency indicate a good transmissionof information (at the local / global scale) in terms of the number of steps. Figures 5C-D show the local ¯ E l and global¯ E g e ffi ciency for both groups. We can observe how in both cases the e ffi ciency is higher in the MCI networks, whichindicates that the network of dysfunctions is better organized.9 Chaos, Solitons & Fractals 00 (2018) 1–14 Table 2 summarizes the average of all network metrics. Note that despite the classical definition of C , d , E l and E g constrain the values of these to the interval [0,1] [38], the fact that we are using weighted connectivity matrices, whichcontain more information about the interdependency between nodes, leads to values that can exceed this range [43].A comparison between control and MCIs group, for each network parameter, was developed via a non parametricKruskal Wallis test, where we have computed the p -values (5000 permutations each) that illustrate how significant thestatistical di ff erences are. Table 2. Summary of the network metrics for the control and MCI groups: Highest degree K max , maximum strength S max , average strength ¯S ,clustering ¯C , average shortest path ¯d , local e ffi ciency ¯E l and global e ffi ciency ¯E g . The p-value of each metric is also indicated. K max ( p = . S max ( p = . ¯S ( p = . ¯C ( p = . ¯d ( p = . ¯E l ( p = . ¯E g ( p = . Parenclitic networks allow detecting those nodes whose features (consistency in our case) diverge the most fromthe expected behaviour. This task is carried out by finding the network hubs and quantifying their importance. Withthis aim, we calculate the degree k ( i ), strength S ( i ) and eigenvector centrality ec ( i ) of nodes belonging to both pop-ulations, all these metrics commonly used as quantifiers of the network hubs. As explained in the previous section,the degree and strength account respectively for the number of links and total accumulated weight of a node. Bothmetrics rely on the local properties of the nodes, which is not the case of ec ( i ). Eigenvector centrality is a global mea-sure of importance that takes into account not only the number of connections / weights of a node, but the number ofconnections / weights of its neighbors [38]. ec ( i ) is calculated from the eigenvector associated to the largest eigenvalueof the weighted connection matrix W whose elements are, in our case, Z i , j . We proceed as follows: two vectors ~ k MCI and ~ k control of length N =
147 (one element per node) contain the average degree of the nodes of each specified group.The di ff erence of the elements of both vectors ∆ h ~ k MCI , control i = ~ k MCI − ~ k control accounts for the di ff erence of nodedegree between both groups and reflects what nodes increase (or decrease) their importance in the network. Figure6A shows ∆ h ~ k MCI , control i , where two peaks stand out over the rest of the degree variations. Nodes 32 and 61 have amuch higher degree in the MCI parenclitic networks than in the control ones. This fact indicates that these two nodesaccumulate the majority of variations related to the consistent behaviour. In a similar way, we obtain the variationsof the strength ~ S MCI , control and eigenvector centrality ~ ec MCI , control . Figure 6B-C shows the di ff erence between groupsof these two metrics ∆ h ~ S MCI , control i (B), ∆ h ~ ec MCI , control i (C). Independent of the metric, again two peaks appear atnodes 32 and 61, confirming that they are the nodes whose consistency is a ff ected the most by the disease. It is worthnoting that these two nodes are not necessarily those nodes whose consistency increased / decreased the most. Node61 has an 8.14% variation in its consistency ( ff ect the interplay, based on consistency, between nodes.In Fig. 6 we plot a 3D representation of these focal nodes together with their local network of interactions. Thisplot shows the local basin of influence of the network hubs and gives an idea about where the disease is being moresevere, at least when consistency is taken into account. It is valuable to compare the position of the most a ff ectednodes with previous results on how MCI a ff ects functional networks. In Buld et al. [24], it was shown that both the10 Chaos, Solitons & Fractals 00 (2018) 1–14 frontal and the occipital lobes contain those nodes of the functional network whose synchronization with other partsof the network was most impaired. Interestingly, these two lobes also contain the two nodes that the consistency-based parenclitic networks revealed to be most a ff ected by MCI. Comparing both results we observe that, despitebeing in the di ff erent lobes, nodes 32 and 61 are not the hub nodes of the functional network, nor are they the nodeswhose local properties inside the functional networks were most modified by the disease. This fact indicates that, withthe projection of brain dynamics into parenclitic networks, we are evaluating a di ff erent e ff ect of the disease on thefunctioning of the brain network.
4. Discussion
It is worth mentioning that although we assumed a linear correlation between the channel consistencies, we havenot proved that this fitting is the one capturing the real interplay between the consistency of brain regions. Furtherstudies should be devoted to investigating the existence, or co-existence, of higher order correlation functions, al-though we obtained similar results with a second-order polynomial adjustment (not shown here). In any case, wedemonstrated that assuming a linear correlation leads to dierences between groups and allows identifying the nodeswhose topological properties are aected the most by the emergence of the disease.Graph theoretical measures have proven to represent good indicator of the emergence and evolution of a series ofbrain diseases, an aspect that renders them of enormous practical application. The emergence of brain dysfunction canbe quantified using network metrics, which are altered in a disease-specific way [44]. For example, during epilepticseizures, functional brain networks become more regular, modifying their degree distribution and losing part of theirmodular structure [45, 46]. On the contrary, functional networks of schizophrenic patients become more random,with a consequent decrease of both the normalized clustering coe ffi cient and shortest path [47]. Another exampleis Alzheimer’s disease (AD) which consists of a disconnection syndrome leading to an increased shortest path anddecreased network clustering, both leading to a severe impairment of the desirable properties a ff orded by small-worldnetworks [48]. The e ff ect of MCI on the structure of functional brain networks has also been investigated showingan increased network synchronization and a propensity to enhance long-range connections [24, 49]. Nevertheless, the ∆ < k M C I , C o n t r o l >
30 60 90 120 − ∆ < S M C I , C o n t r o l >
30 60 90 120 − ∆ < e c M C I , C o n t r o l > Node Numbers
30 60 90 120 −
32 61ABC
Figure 6.
Localizing focal nodes in the consistency impairment.
We calculate the di ff erences in the node degree k ( i ) (A), node strength s ( i ) (B)and eigenvector centrality ec ( i ) (C). In all cases, nodes 32 and 61 accumulate the highest variations, indicating that are the nodes whose consistencyis a ff ected the most by the disease. On the right plot, we show the position and the local network of connections of these two nodes, where only the30 links with higher weights have been plot. Chaos, Solitons & Fractals 00 (2018) 1–14 studies focused on the coordinated activity between cortical regions and on how the disease alters the topology ofconnections within the functional networks. In the current work, we are concerned with another type of consequencesof the emergence of MCI: the loss of a consistent response [16], i.e., in our case, the impairment of the abilityof a cortical region to behave in the same way when undergoing the same cognitive task, despite having di ff erentinitial conditions. We have taken advantage of a new kind of network representation, the parenclitic network [21],where a link between two nodes quantifies the deviations of a certain feature of these nodes from an expected (healthy)behaviour. We have measured the consistency of 147 cortical brain regions by means of magnetoencephalography (seeMaterials and Methods) and constructed a parenclitic network capturing dysfunctions of the expected consistency.The analysis of the topological features of a control group of healthy individuals and a group of patients su ff eringfrom MCI shows that the parenclitic networks can provide useful information to evaluate how disease alters theconsistency between cortical regions. First, we report a higher network strength in the MCI group when the samenumber of links are considered in both groups. This fact indicates higher deviations from the expected consistencyperformance in the MCI group. Furthermore, we observe the appearance of strong hubs in the patients group, whichreveals that the disease is specially severe at certain cortical regions. Specifically, nodes 32 (frontal lobe) and node 61(occipital lobe) are detected to be the focal points of the consistency impairment. Nevertheless, the loss of consistencyis not restricted to certain specic regions. This is reflected by the fact that global network parameters such as theaverage path length or global eficiency also capture di ff erences between the control and the MCI group. On thecontrary, the number of steps needed to go from one node to any other in the parenclitic network is much lower in theMCI group, which indicates that the disease alters network consistency in quite a fundamental way.At the local scale, the MCI group shows high values of clustering and local e ffi ciency of the parenclitic networkindicating that inconsistency does not emerge in isolated regions but in groups of densely interconnected nodes.Finally, we have also seen that the networks associated to the control group are more random than those of theMCI group, which has been demonstrated to be a common signature of parenclitic networks in preliminary works [20],[21]. It is important to highlight that the control networks are not purely random in the sense of the definition given byErd¨os-Renyi [38], but their network properties are closer to random networks when compared with the MCI. Similarly,the MCI networks are closer to star-like networks, despite having more than one central hub and connections betweentheir peripheral nodes. To the best of our knowledge, this is the first result concerning the construction of parencliticnetworks to understand brain functioning and specifically the e ff ect of a neurodegenerative disease. We believe thatthis technique could be extremely useful to evaluate how di ff erent brain diseases deteriorate the normal functioning ofthe brain activity. At the same time, we must note that parenclitic networks are obtained from the correlation betweentime series of a brain region, which in turn rely on the physiological properties of the brain. Unfortunately, parencliticnetworks do not give any information about the function of brain regions. In that sense, how parenclitic and, moregenerally, functional networks are related to brain function and to the physiological properties of the brain is stilllargely unknown.
5. acknowledgements
This work was supported by the Spanish Ministry of S&T [FIS2009-07072], the Community of Madrid under theR& D Program of activities MODELICO-CM [S2009ESP-1691], the Spanish Ministry of economy and competitive-ness [PSI2012-38375-C03-01, MTM2012-39101], Fundaci´on Carolina Doctoral Scholarship Program and Colcien-cias Doctoral Program 568, as well to a grant from the CAM [S2010 / BMD-2460].
References [1] R. Camicioli, M. Wieler, C. de Frias, W. Wayne Martin, Early, untreated parkinson’s disease patients show reaction time variability, NeurosciLett 441 (2008) 77–80.[2] C. de Frias, R. Dixon, R. Camicioli, Neurocognitive speed and inconsistency in parkinsons disease with and without incipient dementia: An18-month prospective cohort study., J Int Neuropsychol Soc 18 (2012) 746772.[3] A. Bodling, D. Denney, L. SG., Individual variability in speed of information processing: An index of cognitive impairment in multiplesclerosis., Neuropsychology 26 (2012) 357367.[4] L. Collins, C. Long, Visual reaction time and its relationship to neuropsychological test performance., Arch Clin Neuropsychol 11 (1996)613623.[5] D. Manoach, Prefrontal cortex dysfunction during working memory performance in schizophrenia: Reconciling discrepant findings.,Schizophr Res 60 (2003) 285298. Chaos, Solitons & Fractals 00 (2018) 1–14 ff ect sizes.,PLoS One 6(4) (2011) e16973.[8] A. Fjell, L. Westlye, I. Amlien, K. Walhovd, Reduced white matter integrity is related to cognitive instability., J Neurosci 31 (2011)1806018072.[9] C. Tamnes, A. Fjell, L. Westlye, Y. W. K. stby, Becoming consistent: developmental reductions in intraindividual variability in reaction timeare related to white matter integrity., J Neurosci 32 (2012) 972982.[10] S. Teipel, T. Meindl, M. Wagner, B. Stieltjes, S. Reuter, K.-H. Hauenstein, M. Filippi, U. Ernemann, M. Reiser, H. Hampel, Longitudinalchanges in fiber tract integrity in healthy ageing and mild cognitive impairment: A dti followup study., J Alzheimers Dis 22 (2010) 507522.[11] H. Christensen, K. Deart, K. Anstey, R. Parslow, A. Sachdev, P. Jorm, Within-occasion intraindividual variability and preclinical diagnosticstatus: Is intraindividual variability an indicator of mild cognitive impairment?., Neuropsychology 19 (2005) 309–317.[12] R. Dixon, D. Garret, T. Lentz, S. MacDonald, E. Strauss, D. Hultsch, Neurocognitive markers of cognitive impairment: Exploring the rolesof speed and inconsistency., Neuropsychology 21 (2007) 382–399.[13] J. Duchek, D. Balota, D. Holtzman, C.-S. Fagan, AM. Tse, A. Goate, The utility of intraindividual variability in selective attention tasks asan early marker for alzheimer’s disease., Neuropsychology 23 (2009) 746–758.[14] G. Moy, P. Millet, S. Haller, S. Baudois, F. De Bilbao, K. Weber, K. Lvblad, F. Lazeyras, P. Giannakopoulos, C. Delaloye, Magnetic resonanceimaging determinants of intraindividual variability in the elderly: Combined analysis of grey and white matter., Neuroscience 186 (2011)88–93.[15] A. McIntosh, N. Kovacevic, R. Itier, Increased brain signal variability accompanies lower behavioral variability in development., PLoSComput Biol 4(7) (2008) e1000106.[16] A. Uchida, R. McAllister, Consistency of nonlinear system response to complex drive signals., Phys. Rev. Lett 93 (2004) 244102.[17] D. Goldobin, Coherence versus reliability of stochastic oscillators with delayed feedback., Phys. Rev. E 78 (2008) 060104(R).[18] A. Uchida, K. Yoshimura, P. Davis, S. Yoshimori, R. Roy, Local conditional lyapunov exponent characterization of consistency of dynamicalresponse of the driven lorenz system., Phys. Rev. E 78 (2008) 036203.[19] T. P´erez, A. Uchida, Reliability and synchronization in a delay-coupled neuronal network with synaptic plasticity., Phys. Rev. E 83 (2011)061915.[20] M. Zanin, S. Boccaletti, Complex networks analysis of obstructive nephropathy data., Chaos 21 (2011) 033103.[21] M. Zanin, J. Medina Alcazar, J. Vicente Carbajosa, P. Sousa, D. Papo, E. Menasalvas, B. S., Parenclitic networks’ representation of data sets.,arXiv:1304.1896 [physics.soc-ph].[22] E. Gorus, R. De Raedt, M. Lambert, J.-C. Lemper, T. Mets, Reaction times and performance variability in normal aging, mild cognitiveimpairment, and alzheimer’s disease., J Geriatr Psychiatry Neurol 21 (2008) 204–218.[23] R. Bajo, F. Maest´u, A. Nevado, M. Sancho, R. Guti´errez, et al., Functional connectivity in mild cognitive impairment during a memory task:Implication for the disconnection hypotesis., J Alzheimers Dis 22 (2010) 183–193.[24] J. M. Buld´u, R. Bajo, F. Maest´u, N. Castellanos, I. Leyva, P. Gil, et al., Reorganization of functional networks in mild cognitive impairment.,PLoS One 6 (2011) e19584.[25] A. Mitchell, Is it time to separate subjective cognitive complaints from the diagnosis of mild cognitive impairment?., Age and ageing 37(5)(2008) pp.4979.[26] M. Grundman, et al., Mild cognitive impairment can be distinguished from alzheimer disease and normal aging for clinical trials., Archivesof Neurology 61(1) (2004) 59–66.[27] R. Petersen, Mild cognitive impairment as a diagnostic entity., Journal of internal medicine 256(3) (2004) pp.183–94.[28] J. A. Yesavage, J. O. Brooks, On the importance of longitudinal research in alzheimer’s disease., Journal of the American Geriatrics Society39(9) (1991) pp.942–4.[29] A. Lobo, et al., Cognocitive mini-test (a simple practical test to detect intellectual changes in medical patients)., Actas luso-espa˜nolas deneurologia, psiquiatria y ciencias afines 7(3) (1979) pp.189–202.[30] S. Sternberg, High-speed scanning in human memory., Science 153 (1966) 652–654.[31] L. de Toledo Morrell, S. Evers, T. Hoeppner, F. Morrell, D. Garron, et al., A ’stress’ test for memory dysfunction: Electrophysiologicmanifestations of early alzheimer’s disease., Archives of Neurology 48 (1991) 605–609.[32] F. Maest´u, A. Fernandez, P. Simos, P. Gil-Gregorio, C. Amo, et al., Spatio-temporal patterns of brain magnetic activity during a memory taskin alzheimer’s disease., NeuroReport 12 (2001) 3917–3922.[33] E. Seo, D. Lee, J.-M. Lee, J.-S. Park, B. Sohn, Whole-brain functional networks in cognitively normal, mild cognitive impairment, andalzheimers disease., PLoS ONE 8(1) (2013) e53922.[34] C. Stam, B. Dijk, Synchronization likelihood: an unbiased measure of generalized synchronization in multivariate data sets., Physica D 163(2002) 236–251.[35] E. Pereda, R. Quiroga, J. Bhattacharya, Nonlinear multivariate analysis of neurophysiological signals., Progress in neurobiology 77(1-2)(2005) 1–37.[36] M. Zanin, P. Sousa, D. Papo, R. Bajo, J. Garc´ıa Prieto, F. del Pozo, E. Menasalvas, S. Boccaletti, Optimizing functional network representationof multivariate time series., Sci Rep ; 2. 630 (2012) 1–6.[37] V. Latora, M. Marchiori, E ffi cient behavior of small-world networks., Physical Review Letters 87 (2001) 198701.[38] M. E. J. Newman, Networks: An introduction., Oxford University Press, 2010.[39] D. Watss, S. Strogatz, Collective dynamics of small word networks., Nature 393.[40] J. Onnela, J. ASaramaki, J. Kertesz, K. Kaski, Intensity and coherence of motifs in weighted complex networks., Phy Rev E Stat NonliinearSoft Matter Phys 71.[41] S. Bocaletti, V. Latora, Y. Moreno, M. Chavez, D. Hwang, Complex networks: Structure and dynamics., Physics Reports 424 (2006) 175–308.[42] E. Dijkstra, A note on two problems in connexion with graphs., Numerische Mathematik 1 (1959) 269–271. Chaos, Solitons & Fractals 00 (2018) 1–14 ff ertshofer, B. Jones, I. Manshanden, A. van Cappellen van Walsum, T. Montez, J. Verbunt, J. de Munck, B. vanDijk, H. Berendse, P. Scheltens, Graph theoretical analysis of magnetoencephalographic functional connectivity in alzheimers disease., Brain132 (2009) 213224.[49] A. Navas, D. Papo, S. Boccaletti, F. del Pozo, R. Bajo, F. Maest´u, P. Gil, I. Sendi˜na Nadal, J. M. Buld´u, Functional hubs in mild cognitiveimpairment., arXiv:1307.0969.ertshofer, B. Jones, I. Manshanden, A. van Cappellen van Walsum, T. Montez, J. Verbunt, J. de Munck, B. vanDijk, H. Berendse, P. Scheltens, Graph theoretical analysis of magnetoencephalographic functional connectivity in alzheimers disease., Brain132 (2009) 213224.[49] A. Navas, D. Papo, S. Boccaletti, F. del Pozo, R. Bajo, F. Maest´u, P. Gil, I. Sendi˜na Nadal, J. M. Buld´u, Functional hubs in mild cognitiveimpairment., arXiv:1307.0969.