Topological Conjugacy Between Seizure and Flat Electroencephalography

Problem statement: Seizure and Flat EEG which are modeled as two dist inct dynamical systems share the same dynamics in the topological viewpoint. Approach: Motions of seizure and Flat EEG of the dynamical systems were written as s et points. A function that maps between these two sets is then built. By using topological conjug acy, they are shown to share the same dynamics. Results: Seizure and Flat EEG were shown to share the same dynamics along with other properties such as order isomorphic, homeomorphic and the uniq ue representation of any event during seizure as an open set. Conclusion: This study shows that the dynamics of seizure can be transported to Flat EEG.


INTRODUCTION
Epilepsy is a general term used for a group of disorders that cause disturbances in electrical signal of the brain. In epilepsy there is a miniature brainstorm of certain groups of brain cells and this is often associated with a sudden and involuntary contraction of a group of muscles and loss of consciousness. It can happen in a small area of the brain or the whole brain. Depending on the part of the brain that is affected, it causes involuntary changes in body movement or function, sensation, awareness, or behavior where these changes are known as epileptic seizure.
Electroencephalography (EEG) is the recording of electrical activity originating from the brain. It plays an important diagnostic role in epilepsy and provides supporting evidence of a seizure disorder as well as assisting with classification of seizures. EEG has been used extensively to record the abnormal brain activity associated with epileptic seizures. It is recorded on the surface of the scalp using electrodes, thus the signal is retrievable non-invasively. The type of activity and the area of the brain that is recorded from EEG will assist the physician in prescribing the correct medication for certain type of epilepsy. Patients with epilepsy that cannot be controlled by medication will often have surgery in order to remove the damaged tissue. Thus the EEG plays an important role in localizing this tissue.  (Ahmad et al., 2008). The model is consists of four elements, Magnetic Contour Plane (MC), Base Magnetic Place (BM), Fuzzy Magnetic Field (TM) and Topographic Magnetic Field (TM) each homeomorphic to each other . The novel model was generalized in (Ahmad et al., 2010). Similar concept of topological mapping was also used in (Nordin and Ali, 2009) to provide navigation and localization for visually impaired people. In (Ahmad et al., 2006), a new method for mapping high dimensional signal, namely EEG into a low dimensional space (MC) has been developed. The whole process of this novel model consists of three main parts. The first part is flattening the EEG where the transformation of three dimensional space into two dimensional space that involved location of sensor in patients head with EEG signal (Fig. 1). This flattening process can preserves magnitude and orientation of the surface (Ahmad et al., 2006). Secondly, the EEG is processed using Fuzzy C-Means (FCM).
Finally, the optimal number of clusters is determined using cluster validity analysis. This new model enables tracking of brainstorm during seizure (Ahmad et al., 2006). Figure 2 are examples of Flat EEG. Red dots represent the electrodes while green dots On the other hand, seizure was modeled as a continuous dynamical system in  by assuming that it is governed by a set of n scalar differential equation with a solution of the form ( ) w t, , α β . For a particular initial state and initial time, the motion, i.e., state space trajectory is written as α β . Besides, the augmented dynamic trajectory, S={(w 1 ,…,w , t): w i , t∈R}(denoted as X t in , but we rename it as S to show that it is the augmented trajectory of seizure) that resulted when the motion is defined over an interval of time was also proven to exhibit linear ordering properties under the relation induced by the motion, f. More research has been carried out on Flat EEG, in (Faisal and Tahir, 2010) for instance, Flat EEG on MC was presented as an algebraic structure. In (Faisal and Tahir, 2010), MC is rewrite as square matrices and transformed into upper triangular matrices using QR-Schur decomposition and finally as a semigroup of upper triangular matrices under matrix multiplication.
However, in this study the transformation of the dynamicity of seizure to visual platform, namely Flat EEG will be discussed and presented.

MATERIALS AND METHODS
We start as in  to model our series of Flat EEG as a dynamical system. Assuming that it is a continuous dynamical system and governed by a set of m+1 scalar differential equation with a solution of the form e(t,λ,γ). For a given initial state and initial time we can write the motion as g: T→R m+1 which is e(t,λ 0 ,γ 0 ). Hence, when we define over an interval of time, the motion produces a set of points known as the augmented dynamic trajectory which can be written as: where, for each time, t, (e p (x 1 ,y 1 ,z 1 ),…,e p (x m ,y m ,z m ),k,t) represents one Flat EEG with e p (x i ,y i ,z i ), the electrical potential recorded from sensor (x i ,y i ,z i ) and k as the number of cluster centers. Notice that the motion g induce a temporal ordering, g ≺ on Flat EEG S which can be formalize as: Proof: By using the theorem from  which states that every temporal ordering on an augmented dynamic trajectory is a linear ordering, then We will use the definition of order isomorphism given in (Steve, 2008). We rewrite the definition as (with no changes in meaning), Definition 1: Let (P,≤ p ) and (Q,≤ q ) be two linearly ordered sets and h: P→Q a function, then (P,≤ p ) is order isomorphic to (Q,≤ q ) if h is bijective and for all x,y∈P and h(x),h(y)∈Q, x≤ p y if and only if h(x)≤ Q h(y). We start by introducing lemma 2 and theorem 1 which will serve as our tools to prove theorem 2. Therefore, θ is surjective. Since θ is both an injective and surjective, therefore the function θ is bijective. as desired. ■ Linearly ordered hausdorff topological space: In the following, we will show that the motions of seizure and Flat EEG that are modeled as a dynamical system is linearly ordered Hausdorff topological space. The ordering relation that will be used in constructing the interval topology is the induced strict total order (irreflexive, asymmetric and transitive) which we will denote it as g • ≺ . We start by introducing a result obtained from (Kopperman et al., 1998).

Corollary 1 (Kopperman et al., 1998):
If τ is the topology of a well-formed space X , the statements • X is a GO space • X is Hausdorff are equivalent.
this will then generate the following basis: and eventually, we obtained the interval topology: This generated interval topology makes the pair ( ) S S,τ a linearly ordered topological space. Note that, if the topology of an ordered space X is generated by collection of rays, then it is called a well-formed space (Kopperman et al., 1998). Using this fact, ( ) S S,τ is then a well-formed space. Together with the fact from (Bennett and Lutzer, 1996)  Since an interval topology is generated by subbasis, any open set can be written as a union of finite intersections of elements of the subbasis. Therefore, to prove the continuity of a function, it is suffices to show that the inverse image of each subbasis element is open (James, 2000). an order-open ray clearly: for some a S ∈ since 1 − θ is bijective Proof: Since θ is bijective (theorem 1), continuous (theorem 5) and its inverse is continuous (theorem 6) therefore, θ is a homeomorphism. ■ Topologically conjugating: One usual way to relate two dynamical systems is with the topological notion of conjugacy (Erik and Joseph, 2010 (Erik and Joseph, 2010). Using this concept, we show that the dynamic functions that act on S and Flat EEG S share the same dynamics.
Firstly, we rewrite the dynamic function that act on S and S Flat EEG as f: R n+1 →R n+1 and g: R m+1+1 →R m+1+1 respectively by including the information of system state at that particular time into the domain and also the time, t into the range. This changes will not create any inconsistency as both functions operate only on the time, t. Thus, function f and g can respectively be defined as:  Thus, the two dynamic systems with their respective function acting on them share the same dynamics. ■ Figure 4 portray the framework.

RESULTS
In this study, we have shown that seizure and Flat EEG that are modeled as a dynamical systems share the same dynamics. Besides, their augmented dynamic trajectory is linearly ordered and order isomorphic to each other by the relation induced from their motion. By endowing the interval topology, the LOTS are proven to be Hausdorff and homeomorphic to each other. Additionally, we show that any event of seizure can be characterized uniquely by an open set from its LOTS.

DISCUSSION
We have shown that seizure and Flat EEG share the same dynamics by using the concept of topological conjugacy. Therefore, the dynamics and orders of seizure are embedded in EEG signals.

CONCLUSION
In this study, we linked seizure and Flat EEG from few aspects. By modeling Flat EEG as a continuous dynamical system, we composed it into a set of points, Flat EEG S that exhibit linear ordering properties. A function, θ is then introduced to show that the set of points are order isomorphic to S and homeomorphic when endowed with the interval topology. Finally, we show that seizure and Flat EEG that are modeled as a dynamical systems share the same dynamics by using the same function.

ACKNOWLEDGEMENT
The researchers would like to thank their family members for their continuous support and Ministry of Science, Technology Innovation for granting the National Science Fellowship scholarship during his study.