Creutzfeldt-Jacob Disease (CJD) is a rapidly progressive, uniformly fatal Transmissible Spongiform Encephalopathy. Sporadic CJD (sCJD) is the most common form of CJD. Electroencephalography (EEG) is one of the main methods to perform clinical diagnosis of CJD, mainly because of Periodic Sharp Wave Complexes (PSWCs). In this paper, we propose a new numerical coefficient and some network motifs, which characterize the presence of PSWCs in an EEG tracing. Furthermore, network motifs are able to detect what the most active and/or connected brain areas are in the tracing segments with PSWCs.
Creutzfeldt-Jacob Disease (CJD) is a rapidly progressive, uniformly fatal
Transmissible Spongiform Encephalopathy (TSE). It is characterized by the
accumulation of a variant of the host encoded cellular prion protein in the brain [
In the past, approaches to investigating PSWCs in the EEGs of patients
with sCJD were mainly based on signal processing [
This paper is organized as follows: in Section 2, we present some support data structures. In Section 3, first we illustrate our connection coefficient, then we introduce our concept of motif and, finally, we present our approach to motif extraction. Finally, in Section 4, we draw some conclusions. 2
The EEGs to perform our investigation were provided by three different Italian centers (i.e., University “Magna Graecia” of Catanzaro, Neurologic Institute “Carlo Besta” of Milano, and Neurologic Institute of the University of Catania). They regard a group of ten patients with sCJD examined in the last 15 years in these three centers6. We segmented each EEG at disposal in such a way as to separate the tracing segments with PSWCs from those without PSWCs. As a consequence, for each EEG, we had several tracing segments, which could be grouped in two distinct sets, namely, those containing PSWCs and those not containing PSWCs.
Formally speaking, let EEGSet be the set of EEGs at our disposal, let eeg
be an EEG of EEGSet. Starting from eeg, it is possible to define a network N
(resp., N ) representing the set of segments of eeg with PSWCs (resp., without
PSWCs). Specifically: N = ⟨V; E⟩ and N = ⟨V; E⟩. Here, V is the set of the
nodes of N and N . Each node vi ∈ V corresponds to an electrode. In our EEGs,
the electrodes were applied by following the 10-20 system. As a consequence,
|V | = 19. E (resp., E) is the set of the edges of N (resp., N ). Each edge eij ∈ E
connects the nodes vi and vj . It can be represented as eij = (vi; vj ; wij ). Here,
wij is a measure of “distance” between vi and vj , which is an indicator of their
disconnection level. Actually, each measure representing this feature could be
adopted in our model. In the experiments described in this paper, we adopted
the Permutation Disalignment Index (PDI) between vi and vj , which is a new
metric of cross-randomness between channels in multivariate electrophysiological
time-series [
In order to make our model more “user-friendly” and “expressive” and, at the same time, more capable of discriminating strong and weak connections between brain areas, we decided to construct two new networks, namely N and N , obtained from N and N by removing the edges with an “excessive” weight and by coloring the other ones on the basis of their weight. More specifically, blue edges denote strong connections (i.e., small weights), red edges represent intermediate ones and, finally, green edges indicate weak connections. Due to space limitations, we cannot report the technical details concerning the construction of N and N .
In Figure 1, we report the colored networks N and N for a patient with
sCJD. The disposal of the nodes in the networks reflects the 10-20 system, even if
they are rotated 90 degrees clockwise. Observe how the filtering of the edges with
the highest distance, along with the coloration of the other ones on the basis of
the closeness of the corresponding nodes, make this model very expressive. The
trends emerging from these figures have been confirmed in all the other EEGs
at our disposal. In particular, we observe that: (i) for a specific EEG, N has
more edges than N ; furthermore, the edges of N are generally stronger than
the ones of N ; (ii) in both networks, the strongest edges can be found in the
occipital area of the skull.
6 We are aware that the number of patients under examination is low. However, this is
due to the fact that sCJD is a very rare disease and, consequently, it is very difficult
to collect data about it.
As pointed out in the Introduction, one of the main features to investigate in
neurodegenerative patients is the connection level of brain areas. This feature is also
relevant in the problem we are facing. In fact, in the literature, it was shown that,
in presence of PSWCs, brain areas are more connected than in absence of them
[
In network analysis, one of the most important (and, at the same time, simple and basic) tools for investigating network connection is the concept of clique. We recall that, given a network, a clique of dimension k represents a totally connected subnetwork with k nodes. On the basis of this reasoning, a quantitative coefficient for discriminating the networks corresponding to the tracing segments with PSWCs from the ones associated with the tracing segments without PSWCs could highly benefit from cliques.
In particular, this coefficient should take the following considerations into account: (i) Both the dimension and the number of cliques are important as connectivity indicators. (ii) The concept of clique is intrinsically exponential; in other words, a clique of dimension n + 1 is exponentially more complex than a clique of dimension n.
Due to space limitations, we do not report here the technical description of connection coefficient. However, in Table 1, we report the values ccN and ccN of this coefficient for N and N , along with the percentage of decrease observed when passing from ccN to ccN , for all the patients at our disposal.
From the analysis of these tables we can draw two important results. In fact: – ccN is always higher than ccN except for the patient CJD 19 for whom the two coefficients have the same value. However, N 19 and N 19 are associated ccN
ccN ccN ccN for all the patients at our disposal with a very particular EEG. Indeed, N 19 is totally connected and, therefore, has only a unique clique coinciding with it. N 19 , instead, is totally connected except for only one edge; as a consequence, it has only two cliques, each consisting of 18 nodes.
As a consequence, we can say that connection coefficient is really a
quantitative parameter capable of distinguishing the tracing segments with PSWCs
from the ones without PSWCs.
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other in presence of PSWCs than in absence of them [
Motifs
Motifs were already investigated and exploited in past approaches adopting
network analysis. In those scenarios, motifs are considered as \patterns of
interconnections occurring in complex networks at numbers that are signi cantly higher than
those in randomized networks" [
As will be clear in the following, our approach to deriving motifs exploits the support data structures introduced in Section 2, along with a further support network, strongly based on the clique concept, which we call clique network. The clique network CN (resp., CN ) corresponding to N (resp., N ) and to the set C (resp., C) of the cliques of CN (resp., CN ), is defined as CN = ⟨CV; CE⟩ and CN = ⟨CV ; CE⟩. Here: (i) CV represents the set of the nodes of CN ; there is a node vi ∈ CV for each node vi ∈ V . A weight wi is associated with vi; it represents the number of cliques of C in which vi is involved; (ii) CE represents the set of the edges of CN . There is an edge (vi; vj ; wij ) ∈ CE if the edge (vi; vj ) is present in at least one clique of C. wij denotes the number of cliques of C in which (vi; vj ) is present; (iii) CV and CE are analogous to CV and CE, but
Motifs Cz-Fz-P4 Fz-P4-Pz Fz-O2-P4 F4-O2-T6 C4-F4-T6 F4-P4-T6 Cz-Fz-Pz F7-P3-Pz Cz-Fz-O2 F7-O2-P3 Fz-O2-Pz P4-T3-T4 F7-Pz-T5 F7-Pz-T3 F7-O2-Pz F7-O1-Pz F7-O2-T5 F7-O2-T3 F7-O1-O2 Pz-T3-T4 T3-T4-T5 T3-T4-T6 O2-T3-T4 O1-T3-T4
Table 2. The basic motifs extracted by our approach for C, instead of for C. The edges of CN and CN can be “colored” in a way analogous to the edges of N .
After having introduced clique networks, we define more restrictive colored networks and clique networks by removing green edges from the networks defined previously. Specifically, we define N , N , CN and CN . We also define the sets N Set (resp., N Set , N Set , N Set , CN Set , CN Set ) comprising all the networks N (resp., N , N , N , CN , CN ) associated with the EEGs of EEGSet. Finally, let t be a generic triad. We call nocc (resp., nocc , nocc , nocc , cnocc , cnocc ) the number of occurrences of t in N Set (resp., N Set , N Set , N Set , CN Set , CN Set ).
After having defined all support data structures and parameters, we are able to describe our motif extraction approach. It consists of two main steps, the former devoted to the extraction of basic motifs and the latter conceived to the construction of derived ones. Preliminarily, it is necessary to specify what is a basic motif in our context. Specifically: let t be a totally connected triad of N Set . If: (1) t is present frequently in N Set and is absent in N Set , and (2) this trend is con rmed (also only to a lesser extent) for N Set and N Set and also for CN Set and CN Set , then t is a basic motif. In particular, t is a motif characterizing the tracing segments with PSWCs against the ones without PSWCs.
In order to quantify the concept of “frequently”, we define a threshold thf = f · |N Set |7. In this case, Condition (1) becomes (nocc ≥ thh) ∧ (nocc = 0), whereas Condition (2) becomes (nocc > nocc ) ∧ (cnocc > cnocc ).
In a dual fashion, it is possible to define the basic motifs associated with N Set and characterizing the tracing segments without PSWCs against the ones with PSWCs. In the following, we indicate by M (resp., M ) the set of motifs extracted starting from the triads of N Set (resp., N Set ). In an analogous way, it is possible to derive the basic motifs of the sets M , M , CM and CM , obtained starting from the triads of N Set , N Set , CN Set and CN Set . The basic motifs derived by our approach are reported in Table 2.
Once basic motifs have been extracted and a first version of M , M , M , M , CM and CM has been obtained, it is possible to construct derived (and, possibly, much more complex and significant) motifs starting from them.
Our approach to constructing new derived motifs starts from the already known ones. It uses nodes common to two or more known motifs as “junction points”. Formally speaking, let m1 = ⟨V1; E1⟩ and m2 = ⟨V2; E2⟩ be two motifs of M such that V1 ∩ V2 ̸= ∅. Then, it is possible to construct a candidate 7 We experimentally set the value of f to 0.30. motif as the union of m1 and m2: m12 = ⟨V1 ∪ V2; E1 ∪ E2⟩. Once m12 has been constructed, analogously to what we have seen for basic motifs, it is necessary to evaluate nocc , nocc , nocc , nocc , cnocc and cnocc 8. If, for these parameters, conditions (1) and (2) presented above hold, then m12 can be added to M , i.e., M = M ∪ {m12}. Clearly, the addition of a new motif in M could lead to the possibility that new candidate motifs are constructed. As a consequence, the enrichment process of M is iterative and terminates when, during an iteration, no new motif is added to M . In an analogous fashion, the derived motifs of M , M , M , CM and CM can be extracted.
In Figure 2, we report the most significant derived motifs extracted by our approach. The motif on the left derives from the tracing segments with PSWCs. It indicates that, in presence of PSWCs, the most active areas of the human brain reside in its right part. The other two motifs derive from the tracing segments without PSWCs. They indicate that, in absence of PSWCs, the most active areas of the human brain reside in its left part (motif on the center) and in its occipital part (motif on the right). 4
In this paper, we have proposed a network analysis based approach to characterizing PSWCs in EEGs of patients with sCJD. We have also introduced a new form of network motifs, well suited for this application context.
Clearly, this paper is only a first attempt of applying network analysis to
characterize specific aspects of sCJD. In the future, we plan to investigate this
possibility in more depth. As an example, the component of our approach
performing motif extraction could be enriched in several directions. Furthermore,
our approach does not currently perform analyses in the sub-bands , , and
of an EEG; adding this feature could be extremely useful, in particular for the
sub-band , which was proven to play a relevant role in the analysis of sCJD [