Examining dynamic functional relationships in a pathological brain using evolutionary computation

Abstract

Neuropathological conditions often result in abnormal functional relationship between different regions in the brain and are specific to certain spectral bands that are not known in advance. Typically, these abnormalities are spatially and temporally very localized in nature, and detecting these changes can be clinically very useful. In this article, a novel evolutionary computation-based procedure is introduced to discover such localized changes in a data-driven manner. Given a predefined set of regions of interest (ROIs), the procedure automatically detects a subset of ROIs, a time window, and a frequency band, such that the functional relationship among the ROIs significantly differ between controls and neuropathological cases; the procedure makes no prior assumptions regarding the spectral characteristics of the data. To demonstrate the effectiveness of this procedure, a publicly available EEG dataset of 46 alcoholics and 31 controls is used. In all, 100 cross-validation runs are performed. Using the procedure, many weakened inter-hemispheric functional connections, primarily between the left and the right parietal lobe sensors, are detected in chronic alcoholics. For these functional connections, gamma band (35–50 Hz) activity in 200–400 ms window was found to be significantly different between alcoholics and controls. These results are consistent with the existing literature and helps to validate the procedure. In addition, the procedure is also tested via simulation using a graph generation model with known characteristics, and its general utility to brain imaging literature is discussed.

Appendices

Appendix 1: Graph metrics

• 1. The global clustering coefficient, GCC, of an undirected graph is defined as follows:

  $$\begin{aligned}&GCC = \frac{3\,{*}\,{\textit{Total number of triangles in the network}}}{{\textit{Total connected triples in the network}}}\nonumber \\ \end{aligned}$$

  (7)

  GCC is often used as a measure of network’s local communication efficiency. That is, if there are many triangles in a network then it would mean that the nodes in the network are locally very well connected.

• 2. The local clustering coefficient, LCC, of a node represents how well the neighbors of a node are connected to each other. The local clustering coefficient of a node for an undirected graph can be evaluated as:

  $$\begin{aligned}&{} \textit{LCC} = \frac{{\textit{Number of direct connections between the neighbors of the node}}}{{\textit{Maximum possible direct connections that can exist among the neighbors}}}\nonumber \\ \end{aligned}$$

  (8)

• 3. Average shortest path length, L(G),  of a undirected graph G(V, E) is defined as:

  $$\begin{aligned} L\left( G \right) = \frac{1}{{n{*}\left( {n-1} \right) }}\,\, {*}\,\, {\sum \limits _{i\ne j} Dist\left( {v\_i, v\_j} \right) } \end{aligned}$$

  (9)

  where,

  $$\begin{aligned} V \text { is a set of all nodes in } G \\ E \text { is a set of all edges in }G \end{aligned}$$

  ${{ Dist}(v\_i, v\_j)}$ is the shortest path length between nodes ${v\_i, v\_j}$ such that ${v\_i, v\_j \in V}$.

  The average shortest path length (or its reciprocal value) is often used as a measure of network’s global communication efficiency. That is, smaller the average shortest path length, higher is the network’s efficiency.

• 4. Degree centrality of a node in an undirected graph is defined as the total number of edges that are incident on the node.

• 5. Betweenness centrality of a node in an undirected graph is based on the total number of shortest paths between other nodes in the network that passes through the given node.

Appendix 2: Encoding a small-world network as a function of time and frequency

To encode random and small-world (SW) networks as a function of time and frequency, the rewiring probability, p(t, f), was defined as follows:

$$\begin{aligned}&p\_0 \left( t,f,{ amplitude,support}\_t , { cutoff}\_t , { order}\_t ,\right. \nonumber \\&\quad \left. { support}\_f , { cutoff}\_f , { order}\_f \right) = \nonumber \\&\quad { amplitude}\,{*}\,\frac{1}{1 + \left( \frac{t - { support}\_t}{{ cutoff}\_t} \right) ^{2^{*} order\_t}} \,{*}\, \frac{1}{1 + \left( \frac{f - { support}\_f }{{ cutoff}\_f }\right) ^{2^{*} order\_f}}\nonumber \\ \end{aligned}$$

(10)

where,

$$\begin{aligned}&\{0<= t,f | t, f\in {\mathbb {R}}\} \\&\{0<= { amplitude}<= 1 | { amplitude}\in {\mathbb {R}}\} \\&\{0<= { support}\_t , { support}\_f | { support}\_t , { support}\_f \in {\mathbb {R}}\} \\&\{0<= { cutoff}\_t , { cutoff}\_f | { cutoff}\_t , { cutoff}\_f \in {\mathbb {R}}\} \\&\{0 < { order}\_t , { order}\_f | { order}\_t , { order}\_f \in {\mathbb {Z}}\} \end{aligned}$$

$$\begin{aligned}&p\_1 \left( {t,f} \right) =10 *p\_0 \left( t, f, 1, t0\_sw, C0\_sw\_a, 3,\right. \left. f0\_sw, C0\_sw\_b, 3 \right) +\nonumber \\&\qquad {\sum }_{i=1\, { to}\, 10} p\_0 \left( t, f, A\_i, t0\_i, C0\_a\_i, 3, f0\_i, C0\_b\_i, 3 \right) \end{aligned}$$

(11)

$$\begin{aligned}&p\left( {t,f} \right) =1 - 0.85^{*} p\_1 \left( {t,f} \right) /max\left( {p\_1 \left( {t,f} \right) } \right) \end{aligned}$$

(12)

where,

$$\begin{aligned}&\begin{array}{l} \{0<= t0\_sw<= 1000 | t0\_sw\in {\mathbb {R}}\} \\ \{0<= f0\_sw<= 128 | f0\_sw\in {\mathbb {R}}\}\} \\ \{50<= C0\_sw\_a<= 200 | C0\_sw\_a\in {\mathbb {R}}\} \\ \{10<= C0\_sw\_b<= 30 | C0\_sw\_b\in {\mathbb {R}}\} \\ \end{array}\\&\begin{array}{l} \{0<= A\_i<= 1 | A\_i\in {\mathbb {R}}\} \\ \{0<= t0\_i<= 1000 | t0\in {\mathbb {R}}\} \\ \{0<= f0\_i<= 128 | f0\in {\mathbb {R}}\} \\ \{10<= C0\_a\_i<= 150 | C0\_a\_i \in {\mathbb {R}}\} \\ \{10<= C0\_b\_i <= 30 | C0\_b\_i \in {\mathbb {R}}\} \\ \end{array} \end{aligned}$$

The function ${ p\_}_{0}$ in Eq. 10 creates a 2D surface with a single peak. The parameter amplitude controls its height, the parameters ${ support\_}_{t }$ and ${ support\_}_{f}$ control the center point of the function in time and frequency axes, the parameters ${ cutoff\_}_{t}$ and ${ cutoff\_}_{f}$ control the width of the function in time and frequency axes, and the parameters ${ order\_}_{t}$ and ${ order\_}_{f}$ control how smoothly the function value declines to 0. That is, if the parameter order is set to a very high value, the function falls sharply, whereas if the order is set to a very low value, the function declines gradually. The function ${ p\_}_{1}(t,f)$ is created by adding ten ${ p\_}_{0}$ functions with amplitude varying from 0 to 1, and one ${ p\_}_{0}$ function with amplitude set to 10. The p(t, f) is then created by first normalizing ${ p\_}_{1}(t,f)$ and then converting all peaks to troughs. As a result of this, p(t, f) varies between 1 and 0.15. The max value of ${ p\_}_{1}$ function (i.e., where function ${ p\_}_{1}(t,f) = 10$ is situated) converts to a value of 0.15 in function p(t, f); the small-world class of networks exist in the range between 0.15 and 0.2 . The variable, order, while creating p(t, f) was set to 3. If the order is set to a very high number, then the 2D surface will contain 11 sharp falling troughs, hence producing a needle in a haystack type situation.