Leveraging network analysis to support experts in their analyses of subjects with MCI and AD

Abstract

In this paper, we propose a network analysis–based approach to help experts in their analyses of subjects with mild cognitive impairment (hereafter, MCI) and Alzheimer’s disease (hereafter, AD) and to investigate the evolution of these subjects over time. The inputs of our approach are the electroencephalograms (hereafter, EEGs) of the patients to analyze, performed at a certain time and, again, 3 months later. Given an EEG of a subject, our approach constructs a network with nodes that represent the electrodes and edges that denote connections between electrodes. Then, it applies several network-based techniques allowing the investigation of subjects with MCI and AD and the analysis of their evolution over time. (i) A connection coefficient, supporting experts to distinguish patients with MCI from patients with AD; (ii) A conversion coefficient, supporting experts to verify if a subject with MCI is converting to AD; (iii) Some network motifs, i.e., network patterns very frequent in one kind of patient and absent, or very rare, in the other. Patients with AD, just by the very nature of their condition, cannot be forced to stay motionless while undergoing examinations for a long time. EEG is a non-invasive examination that can be easily done on them. Since AD and MCI, if prodromal to AD, are associated with a loss of cortical connections, the adoption of network analysis appears suitable to investigate the effects of the progression of the disease on EEG. This paper confirms the suitability of this idea

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Appendix

1.1 A.1 Permutation disalignment index

Given an EEG window under analysis and two EEG channels, two time series x and y are identified. Given a time t, an embedding dimension m and a time lag L [10], starting from the samples x(t) and y(t), two m-dimensional vectors X_t and Y_t can be constructed:

$$ X_{t}=[x(t),x(t+L),...,x(t+(m-1)L)]^{T} $$

and

$$ Y_{t}=[y(t),y(t+L),...,y(t+(m-1)L)]^{T} $$

A visual explanation of the methodology is provided assuming m = 3 (Fig. 6). When m = 3, X_t, and Y_t are 3-dimensional vectors.

Fig. 6

Schema of PDI computation. The method consists of three steps: 1) EEG recording; 2) Sub-bands extraction; 3) Motifs’ detection and PDI computation. Given a recorded EEG window, for every channel, the four sub-band signals (δ, 𝜃, α, β) are extracted. For every pair of channels x and y, given an embedding dimension m, a time t and a lag L, the time series x and y can be projected into the vectors X_t and Y_t, both of them consisting of three elements. If m = 3, 6 ordinal sequences (Motifs) are possible. Given a time t, the algorithm detects which motif occurred in x and y. In the present example, at time t, the same motif π₄ occurred in both x and y. The procedure is reiterated for every time point t so that, once the EEG window has been fully processed, a final occurrence rate p_{X, Y}(π_i) of every motif π_i in x and y can be estimated, and the PDI can be computed in every sub-band

The absolute values of the elements of X_t and Y_t are discarded and only their relative amplitude (low, medium, high) are taken into account. If 3 levels are considered, 6 ordinal sequences (Motifs) are possible [10]. Motifs are denoted with the symbol π_i (where i = 1,..., 6).

At the tth iteration, the algorithm detects which motif occurs in X_t and in Y_t. For instance, in Fig. 6, motif π₄ occurred in both X(t) and Y (t). Therefore, the algorithm increments η_{X, Y}(π₄), that is the number of the simultaneous occurrences of π₄ in time series x and y.

Then, the algorithm moves to the next sample x(t + 1), two new vectors X_t+ 1 and Y_t+ 1 are constructed, and the procedure is reiterated.

Once the two time series have been fully processed, the algorithm estimates the overall probability that every motif π_i (with i = 1,..., 6) simultaneously occurs in x and y, by normalizing the number of occurrences η_{X, Y}(π_i) against the number of iterations:

$$ p_{X,Y}(\pi_{i})=\frac{\eta_{X,Y}(\pi_{i})}{(N-(m-1)L)} $$

The PDI between x and y is, finally, defined as:

$$ PDI(X,Y)=\frac{1}{1-\alpha}\log\left[ \sum\limits_{i=1}^{m!} p_{X,Y}(\pi_{i})^{\alpha} \right] $$

The more coupled x and y are, the lower PDI is expected to be [42]. The parameter α can tune the sensitivity of PDI to sub-Gaussianity or super-Gaussianity of time series. In this work, it was set to 2, to ensure a balanced sensitivity to either sub-Gaussian or super-Gaussian distributions. Details of this algorithm can be found in [42].

1.2 A.2 Pseudo-code for the construction of a clique network

The pseudo-code for the construction of a clique network is reported in Algorithm 1. Function ConstructCliqueNetwork receives a colored network ${\mathcal {N}}_{\pi }$ and returns a clique network ${\mathcal {C}N}$. First, it constructs an empty network ${\mathcal {C}N}$. Then, it computes all the cliques of ${\mathcal {N}}_{\pi }$. After this, for each clique, it identifies the nodes composing it and, for each node, verifies if it is already present in ${\mathcal {C}N}$. In the affirmative case, it increases the corresponding weight. In the negative case, it adds this node to ${\mathcal {C}N}$ with the corresponding weight set to 1. Finally, for each clique of ${\mathcal {N}}_{\pi }$, it considers the corresponding edges. If an edge is already present in ${\mathcal {C}N}$, it increments the corresponding weight; otherwise, it adds this edge to ${\mathcal {C}N}$ with the corresponding weight set to 1.

1.3 A.3 Pseudo-code for the computation of motifs

The pseudo-code for the computation of motifs is reported in Algorithm 2. Function DeriveMotifs receives the list T of the triads of EEGSet and the list N of the networks in which the presence of the triads of T must be verified. First, for each triad t of T, it computes how many times t is present in patients with MCI and in patients with AD. Then, it verifies conditions (1) and (2) and, on the basis of this verification, it possibly adds t to ${\mathcal {M}}_{M}$ or ${\mathcal {M}}_{A}$. At the end of these activities, basic motifs have been computed and the construction of derived motifs can start. For this purpose, DeriveMotifs first considers the motifs of ${\mathcal {M}}_{M}$. For each motif m_i of ${\mathcal {M}}_{M}$, it verifies if m_i has at least one node in common with another motif m_j. In the affirmative case, it constructs a candidate derived motif m_ij by merging m_i and m_j. After this, it evaluates conditions (1) and (2) for m_ij and, if one of them is valid, adds m_ij to ${\mathcal {M}}_{M}$. In an analogous way, DeriveMotifs proceeds for the construction of the derived motifs of ${\mathcal {M}}_{A}$. Finally, it returns ${\mathcal {M}}_{M}$ and ${\mathcal {M}}_{A}$.