ABSTRACT
We employ a phenomenological model to simulate the recurrent unprovoked partial seizures of mesial temporal lobe epilepsy. The model is comprised of a nonautonomous, second-order, reciprocal max-type difference equation with incorporated threshold function, https://www.w3.org/1998/Math/MathML"> x n + 1 = max A n + D n · H ϵ ( x n ) x n , B n x n − 1 , n = 0 , 1 , … , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003285380/d401538e-40ed-4f3e-b349-71c9a9560f54/content/mathab_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
where https://www.w3.org/1998/Math/MathML"> H ϵ ( x ) = 0 , 0 < x < ϵ , 1 , x ≥ ϵ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003285380/d401538e-40ed-4f3e-b349-71c9a9560f54/content/mathab_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
with ϵ > 0, and where
https://www.w3.org/1998/Math/MathML"> { A n } n = 0 ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003285380/d401538e-40ed-4f3e-b349-71c9a9560f54/content/mathab_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> { B n } n = 0 ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003285380/d401538e-40ed-4f3e-b349-71c9a9560f54/content/mathab_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> are both positive periodic sequences with minimal period 2.
https://www.w3.org/1998/Math/MathML"> { D n } n = 0 ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003285380/d401538e-40ed-4f3e-b349-71c9a9560f54/content/mathab_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is a positive periodic sequence with minimal period 3.
https://www.w3.org/1998/Math/MathML"> A 0 + D 1 , A 1 + D 1 < B 0 , B 1 < A 0 + D 2 , A 1 + D 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003285380/d401538e-40ed-4f3e-b349-71c9a9560f54/content/mathab_6.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
The state variable xn represents the density of excited neurons at the nth time step in the middle structures of the temporal lobe.
Equation (1) is obtained from the nonautonomous, (k +1)st-order, reciprocal max-type equation https://www.w3.org/1998/Math/MathML"> x n + 1 = max A n ( 0 ) x n , A n ( 1 ) x n − 1 , … , A n ( k ) x n − k , n = 0 , 1 , … , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003285380/d401538e-40ed-4f3e-b349-71c9a9560f54/content/mathab_7.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
by setting k = 1 (i.e., the smallest possible delay in a max-type equation) and inserting a threshold function, Eq.(2). Thus, Eq.(1) as a difference equation suffers from “impaired memory,” thereby mimicking the consequences of hippocampal sclerosis in mesial temporal lobe epilepsy. Equation (3) is unbounded and non-persistent, under certain conditions on the coefficient parameters, and eventually periodic (i.e., oscillatory) under other conditions.
The ictal (i.e., seizure) and interictal (i.e., between seizures) phases are defined as follows:
Let {xn }∞ n=−1 be a solution of Eq.(1). Then the ith ictal phase is the string of terms https://www.w3.org/1998/Math/MathML"> S i = { x ℓ + 1 , x ℓ + 2 , … , x m − 1 , x m } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003285380/d401538e-40ed-4f3e-b349-71c9a9560f54/content/mathab_8.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
such that the following hold:
https://www.w3.org/1998/Math/MathML"> x ℓ + 1 , x m ∈ { x 3 j + 2 } j = 0 ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003285380/d401538e-40ed-4f3e-b349-71c9a9560f54/content/mathab_9.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
https://www.w3.org/1998/Math/MathML"> x l + 1 , x l + 2 , … , x m − 1 ≥ ϵ and x m < ϵ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003285380/d401538e-40ed-4f3e-b349-71c9a9560f54/content/mathab_10.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
https://www.w3.org/1998/Math/MathML"> { x 3 j + 2 } j = 0 ∞ ∩ S i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003285380/d401538e-40ed-4f3e-b349-71c9a9560f54/content/mathab_11.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is strictly decreasing.
https://www.w3.org/1998/Math/MathML"> { x 3 j + 3 } j = 0 ∞ ∩ S i and { x 3 j + 4 } j = 0 ∞ ∩ S i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003285380/d401538e-40ed-4f3e-b349-71c9a9560f54/content/mathab_12.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> are strictly increasing.
The ith interictal phase is the string of terms between the (i – 1)th and ith ictal phases that do not satisfy conditions (i)–(iv), https://www.w3.org/1998/Math/MathML"> S i = { x k + 1 , x x + 2 , … , x l − 1 , x l } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003285380/d401538e-40ed-4f3e-b349-71c9a9560f54/content/mathab_13.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
