Spatial Interaction Image of Electroencephalography Signal during Epileptic Seizure on Flat Electroencephalography

Corresoonding Author: Tahir Ahmad CSNano, Ibnu Sina Institute for Scientific and Industrial Research, Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia Email: tahir@ibnusina.utm.my Abstract: Electroencephalogram (EEG) is widely used for recording the brain electrical activity. It plays an important role in the detection and classification of epilepsy seizures. In addition, it is believed to possess the ability to detect the location of epileptic focus to some degree. In this study, an EEG data during seizure obtained is analysed and transformed into an image form, namely Flat EEG (fEEG). The fEEG is a method for mapping high dimensional signal into a low dimensional space. The EEG of the obtained signal is clustered and the interactions of the cluster centers are studied and presented in this study.


Introduction
Epilepsy is a common disorder disease in the nervous system, regardless of geographical location, races or social class. It describes the condition of a patient having recurring "spontaneous" seizures due to the sudden development of synchronous firing in the cerebral cortex caused by lasting cerebral abnormality. Even though surgical treatment is possible for patients, the success of the surgery is highly dependent on the detection of exact localization of epileptic focus.
Fortunately, EEG was invented and widely used for recording the brain's electrical activity. Generally EEG technique can be categorized into two: Intracranial and scalp. Basically intracranial EEG provides several advantages over scalp EEG (Salanova et al., 1993), however the disadvantages are accompanied by significant discomfort and risk of a major complication such as haemorrhage infection are not negligible (Fisch, 2010). On the other hand, scalp EEG may be less sensitive, but it provides the best overview for detecting the localization of the epileptogenic zone (Noachtar and Rémi, 2009).
Though scalp EEG is imperfect in terms of sensitivity, it is considered as the best method used in epilepsy analysis which focuses on the detection and classification of epilepsy seizures. Nonetheless, the aid from a highly skilled electroencephalographer or neurophysiologist are still needed to give best visual inspection of EEG signal (Sanei and Chambers, 2007).
Due to this restriction, we are driven to study the spatial analysis of EEG signals during seizure.

Related Works
In this section, the works done by previous researchers of our research group are reported, namely flat EEG and Fuzzy Neighbourhood Image. In addition, the prototype model of the Spatial Interaction Image are reviewed as follows. Zakaria (2007)

Flat EEG
where, MC = {((x,y)e p ): x,y, e p ∈ ℝ }. Both C EEG and MC were designed and proven as 2-manifolds (Zakaria, 2007). The mapping S t was designed to be a one to one function as well as being conformal. After the mapping, the data obtained can be clusterised by Fuzzy c-Means (FCM) clustering algorithm, followed by cluster validity test, namely Partition Coefficient (PC) and Compactness Separability (CS), in order to enable EEG signal during seizure as in Fig. 2 to be compressed (Fig. 3a) and analysed second by second (Fig. 3b) (Zakaria, 2007;Ahmad et al., 2006).

Fuzzy Neighbourhood Image
As further study on fEEG, Ahmad and his coresearchers constructed fEEG as a digital space (Fig. 4) Ahmad, 2009). They proved that the digital space of the fEEG is an Alexsandr off space and applied relational topology in order to incorporate topological space of real time recorded EEG signal with digital fEEG. This construction is a key foundation of fEEG which has been linked with the idea of fuzzy information granulation. Furthermore, Abdy and Ahmad (2013) introduced a membership function j c µ at time t: is the electrical potential c j at time t; and: Is the distance between pixel p i and cluster center c j at time t for fEEG.

Spatial Interaction Model
According to Kingsley and Fotheringham, a spatial interaction is a broad term encompassing any movement over space that results from a human process (Haynes and Fotheringham, 1984). For example, in our work the interaction will be the flow of potential differences between cluster centers. The original concept of spatial interaction comes from the universal law of gravity, namely Isaac Newton's law of gravity. Between interaction models, the gravity model is the most widely used type of interaction model (Haynes and Fotheringham, 1984). It is a mathematical formulation which utilizes the gravitational force concept. The model is widely used to analyse and forecast spatial interaction patterns (Philbrick, 1971). Since first introduced by Carey (1858), types of gravity models keep on increasing along with the development of applications. Example include gravity model to determine the market area boundary, doubly constrained gravity models for commodity flows analysis, gravity models for studying the migration flows and so on.
The basic assumption concerning many spatial interaction models is that flows are a function of the attributes of the locations of origin, the attributes of the locations of destination and the friction of distance between the concerned origins and the destinations. The general formulation of the spatial interaction model is as follows (Haynes and Fotheringham, 1984): Where: T ij = Interaction between location i (origin) and location j (destination) V i = Attribute of the location of origin i W j = Attribute of the location of destination j S ij = Attribute of separation between the location of origin i and the location of destination j

Image Form of Spatial Interaction on fEEG
Our aim in this section is to view the interaction relationship between cluster centers in the image form at fEEG. There are some definitions which need to be reviewed.

Mathematical Background
Definition 1 (Cluster Center) (Abdy and Ahmad, 2013) Let ℂ be the set of all cluster centers at time t, i.e.: ℂ t = {c 1 ,c 2 ,c 3 ,…,c m }, where m = number of cluster centers at time t. Each cluster center carries its position with electrical potential on fEEG, i.e.: (c j ) t = ((x j , y j ), v j ) t , with v j = electrical potential of the cluster center jth. Let ℙ be the set of entire pixels of fEEG, Definition 2 (Domain) (Yong et al., 2013) The set of pair-wise cluster centers interaction on fEEG is defined by: Definition 3 (Interaction model) (Yong et al., 2013) The interaction model, :ℂ×ℂ→ℝ is defined by: Where: P k ij T = Pair-wise cluster centers interaction between ith and jth cluster centers with respect to P k d ij = Euclidean distance between cluster center ith and jth v i ,v j = Electrical potential of cluster center ith and jth respectively λ, α = The fuzzy values (in crisp form) of respective pixel on cluster ith and jth β = The distance decay parameter By considering each cluster center c j at time forms a fuzzy region of the potential difference on the fEEG, each pixel will carry a fuzzy membership value, with respect to cluster center ith (or with respect to cluster center jth) which can be determined from

Definition 4 (Fuzzy Membership Function)
The fuzzy membership function of P k with respect to c j is defined by: Where: The maximum distance between all the entire pixels on fEEG at time t d(P k , c j ) t = The distance between pixel and cluster center at time t It has been shown that Equation 4 satisfies some properties as follows.

Theorem 1
is a fuzzy membership function of c j on P k , then: Theorem 1 shows that if a pixel is very far from a cluster center, the fuzzy membership of the pixel with respect to the cluster center will go to 0.

Theorem 2
is fuzzy membership function of c j on P k , then: Theorem 2 indicates that when a pixel is near to the location of a cluster center, the fuzzy membership of the pixel with respect to the cluster center will move to 1.

Theorem 3
Let c∈ℂ t and P i , Theorem 3 shows that the membership value of a pixel which is closer to cluster center will always be greater than the other.
Since each pixel carries different fuzzy membership values which correspond to every single cluster center respectively, the interaction relationship is the total of the pair-wise cluster centers interaction index, which is defined as follows.
Definition 5 (Yong et al., 2013) Total interaction of all cluster centers on a pixel, P i is given as: Furthermore, the transformation of total interaction index of each pixel P i total T into an image index is given by the following definition.

Definition 6 (Image Index)
The image index, i P I is given as: However, the level of grey-scale is given as:

Spatial Interaction Image Model
Based on the definitions in Section 3.1, the spatial interaction image model can be explained by the following algorithm: Step 1: Choose the values for constant, and distance decay parameter, in Equation 3.
Step 2 Step 6: Transform the total interaction index, Step 7: Refine the image index, i P I to image data, Step 8: Plot the image by using Matlab.
In order to implement the spatial interaction image algorithm as described above, an example (3×3 pixels) is given in Table 1.
Given ℂ = {c 1 , c 2 , c 3 }, ℙ = {P 1 , P 2 , P 3 ,…,P n } ( Table  2). First, the assumption for constant k is 1 and is 2. By applying Equation 4, the fuzzy membership values for all P i ∈ℙ corresponding to all c j ∈ℂ t were obtained as shown in Table 2.
From Equation 2, the ℂ 2 was determined with ℂ 2 = {(c 1 , c 2 ), (c 1 , c 3 ), (c 2 , c 3 )}. Next, the The obtained results are all shown in Table 3. Finally the i P I ′ was used to plot the image by Matlab (Fig. 5).

Implementation
In order to implement the spatial interaction image model, a set of 10s fEEG data which contains the information of location of cluster centers and respective potential difference for every second is obtained from Zakaria (2007) as displayed in Table 4. The value for is assumed to be 2 (Haynes and Fotheringham, 1984) and the constant k is assumed to be 1. The images of interaction relationship are obtained for each second as shown in Fig. 6, with brightness level of pixel representing the levels of interaction force. The computing time of this work is 45.6559 sec by using Matlab version R2010b, with a 32-bit-Win7 OS based computer.
The result shows that the region with darker colour (greater interaction) is slowly expanding from time = 1 s to 6 s. After that, it narrows down from time = 6 s to 9 s and retains a similar size at time = 10 s. However, the size of region colour that is closer to black (the strongest interaction force) is large whenever the cluster centers that carry greater electrical potential (refer to Table 4) are close to each other.

Conclusion
In this study, the modelling of spatial interaction model, namely gravity model is performed on the platform of EEG for 10 sec' data. The results were shown in grey-scale image form where the levels of interaction force were represented by brightness level of pixel. From there, it can be concluded that the interaction forces are mostly greater when the cluster center are close to each other, even though it may not happen to the "isolated" cluster center. In other words, the region of great interaction forces is highly reliant on the closeness of cluster centers.
In reality, the bio-signals in the brain is generated by positive charge ions (+ K and + Na), which are supposed to repel each other. In this implementation, the interaction forces between cluster centers in EEG represent the repellent forces between the ions. The greater the repellent forces happening in the brain, the more disturbance is occurring in the functioning of the patient's brain, which lead to epileptic seizure. The identification of these regions of great interaction may be one of the keys for the detection of exact localization of epileptic focus.