On a Simple Mathematical Model for Epilepsy Motivated by Networks Volume 2 - Issue 4

Ahmed E*

• Department of Mathematics, Faculty of Science, Egypt

Received: March 04, 2020;   Published: March 12, 2020

*Corresponding author: E Ahmed, Department of Mathematics, Faculty of Science, Mansoura Egypt

DOI: 10.32474/CTBB.2020.02.000141

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Abstract

Simple mathematical models motivated by networks are given for epilepsy. The coexistence solutions and their stability are derived.

Introduction

Epilepsy is one of the most common brain diseases [1]. Several approaches are used to model it e.g. dynamical systems [1] and networks [2]. In [2] a network is given or a 3-node graph one represents low (unexcited) cells(L). The second represents medium (M) and the third represent high (excited) cells (H). Here this system is approximated by the dynamical system:

(1)

where a, b, c, d are positive constants. The coexistence solution is

(2)

It exists if b/a < 1 and bd > c (3) and is locally asymptotically stable if

(4)

The model (1) can be reduced to a 2-species model as follows:

(5)

and coexistence solution is:

(6)

And is stable if 2L > 1 (7) Healthy state exists if H << L . Now we introduce the drug resistant type in epilepsy [3]. The populations are low L, susceptible high Hs and resistant high Hr. The model is

(8)

The coexistence solution is

In the physically acceptable case Hr << L hence (c /d ) << 1, the coexistence solution is locally asymptotically stable.

References

1. Takao Namikia, Ichiro Tsudab, Satoru Tadokoroa, Shunsuke Kajikawac, Takeharu Kuniedad, (2019) Mathematical structures for epilepsy: High-frequency oscillation andinterictal epileptic slow (red slow). Neuroscience Research.
2. Gustavo Henrique Tomanik, Predicting epileptic seizures using complex networks.
3. Patrick Kwan, Steven C Schachter, Martin J Brodie (2011) Drug-Resistant Epilepsy. N Engl J Med 365(10):919-926.