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ISSN 1810-0198. Вестник Тамбовского университета. Серия Естественные и технические науки Том 23, № 122 2018
DOI: 10.20310/1810-0198-2018-23-122-131-135
EXISTENCE AND STABILITY OF BUMPS IN A NEURAL FIELD MODEL
e K. Kolodina1), A. Oleynik^, J. Wyller1'
Norwegian University of Life Sciences 3 Universitetstunet. As 1433, Norway E-mail: [email protected], [email protected] 2' University of Bergen 41 Allegaten. Realfagbygget, Bergen 5020, Norway El-mail: [email protected]
Abstract. We investigate existence and stability of bumps (localized stationary solutions) in a homogenized 2-popnlation neural field model, when the firing rate functions are given by the unit step function.
Keywords: homogenization theory; existence and stability of stationary solutions of nonlocal neural field models
1. The main results
The set of coupled integro-differential equations
— Ue = Ue + Uee COPe(Ue ®e) ^ie OCTi{Ui 0,) at
r — Ui = Ui+Uei OCPe{ue de) bJu OOPiK 9i)
models the neural activity in the cortical tissue. Here / OQ? is is defined as
(1)
[/CO](z,y)C f(x x\y y')g(x',y')dydx' 0. Y
where x / 0 < A y / A i>0. ue and Ui are the membrane potentials of excitatory and inhibitory neurons, respectively, at the spatial point x, y and time t. The region 0 is the spatial region occupied by the neurons. The functions ujmn ( m, n = e, i) model the coupling strengths (referred to as the connectivity functions) in the network. The functions P.m, m = e,i (referred to as the firing rate functions) are monotonieally increasing and assume values in the interval Y = [0,1]. The connectivity functions are assumed to be 1-periodic in the variable y. The parameter r is the relative inhibition time i.e. r = Tj/re
The work is partially supported by the Norwegian University of Life Sciences and The Research Council of Norway (project № 239070).
where re ( Tt ) is the excitatory (inhibitory) time constant, while 6e and 9{ are the threshold values for firing of the excitatory and the inhibitory neurons, respectively The system (1) is derived from a 2-population neural field model with periodic microstructure incorporated in the connectivity functions by using the 2-scale convergence technique of Nguctseng [1].
The connectivity kernels umn are expressed in terms of the scaling function <& and the footprint functions crmn, m, n = e,i:
V; amn) = ) am^rnamn) (
&mn(y^ (^mn) = Smn (1 + amn cos(27Ty)), > 0, a > 1 (2)
m = n o, <$>/BC\A).
»
The parameters amn, m,n = e,i are referred to as the heterogeneity parameters. We denote the bump solutions by U = [i/e, Ï7J. The components Ue and [/,; can formally be expressed
Ue) — Wee(o,e Xj Ctee) "h W'eei^e 7 ^7 ^te) I )}
Ui(x] ati) = Wei(ae aei) + W^a-e + x;aei) Wa(at
where ae and at are the vectors ae = (aee, a,;e]) and at = (aei, a,;,;) and
Î 1
k*mn
(:x,y)dy(dx.
0 0
Here the pulse width coordinates am > 0 are defined by means of the condition Um(®am) = 0m ( m = e, i ). Necessary conditions for the existence of the bumps read
/e(a; ae) = 9e, /¿(a; a*) = Oi, (3)
where fe and ft are given as
fe(a- ae) Ç Wee(2ae] aee) Wie(ae + af, aie) + Wie(ae
^¿7 Qije))
fi(a]Q>i) Ç \Vei(ae + af, aei) Wei(ai ae,aei) W^af, an).
Here we have introduced the pulse width vector a = (ae, a,;). We obtain the following result:
Theorem 1. Let S and I be the sets E = }(ae, Oj); ae, a^ > 0| and I = 0 < 9m > 1, m = e, and be the 4 -parameter family of vector field Fa = ( fe,fi) :
fi G Â2 where a = (ate, a,;) / D Ç [0, l)4. Then the following holds true:
1. The set Fa(E) is bounded for all a / H.
2. The vectorfield Fa : E e A2 is smooth for all a / H.
3. If the Jacobian DaF0(a0) is non-singular where a0 is a solution of (3) when a = 0, then by the implicit function theorem the intersection between Fa(E) and I is nonempty i.e. there is a k / [0,1) such that
Fa(E) {/ @= m
for a / H k where
n* Ç }a / n;0 > amn < k\ -^n.
EXISTENCE AND STABILITY OF BUMPS IN A NEURAL FIELD MODEL
133
Based on this result, one can prove the following result:
Theorem 2. For a / and DaF0(a0) being non-singular, the generic picture consists of two solutions of the system (3) for each (6e, 9} ) / J C Fa(T,) { I.
This result is obtained by interpreting solutions of the system (3) as a transversal intersection between two level curves and the one-to-one correspondence between the solutions of (3) and the bumps. This result means that the typical situation consists of two bumps for each (0e, 0}) / J.
We next study stability of the bumps Up, = (Ue,Ui). We write the system (1) on the compact form
|U = T{ U + F(U)), where F is the integral operator on the RHS of this system and
T =
V 0 Í
Jo 1/r' ■
Then by imposing a perturbation on U0 i.e. by assuming
U(x, y, t) = U0(x) + V(x, y) exp[Ai], V = (Ve, Vt) and linearizing the resulting equation for V we end up with the eigenvalue problem
A V = G(V), G(V) = T V+F{jo(V)^
Here the Frechét derivative is given by
Lee^e ^ie
F{j0(V) =
LeiYe LijVj
■ V \
ie r i I
I
{LmnVm)(x,y) = ——j—- (umn(am x, y' y)Vm(amiy') +
Y
umn(am+x,y' y)Vm(
l/))dj/.
We have the following result:
Theorem 3. The spectrum Sp(G) of the operator G can differ from f Sp T + H[k)(
k=h,2 \
only by two values, 1 and 1/r. Here the integral operator H^ : BCL'(Y) ± BC1(Y) e BC1(Y)±BC1(Y), ¿ = 1,2 is given as
(H^vk)(y) = TA^(y' y)vk(y')dy', vk / BC\Y)±BC\Y)
A^\y) =
A(y) + B(y) C{y) D(y)y E(y) + F(y) G(y) H(y)f
and
\ A(y) B(y) C(y) + D(y) \
A{2\y)=
J E(y) F(y) G(y) + H(y)J
where
At,,\ _ "ee(0,D) U(.,\ — Uee(2ae,y) n(q,\ — gg e(aj-ae,y) HflA — "«(flt+fte ,V)
A[>y)-]uiM\> ~ PW ' №)j ' ^iyJ - №)[ >
~ |^(ae)| ' tyy>~ |E/>e)| • ^yV) ~ 1^)11 nyy> ~ |i/!(aO| "
The spectrum of Hik\ k = 1,2 which is computed by means of the Fourier-decomposition method, serves as the basis for the stability method. For the scenario with two bumps for each pair of threshold values 9e,9i, we find that one bump is unstable for all relative inhibition times r and the other one is stable for small and moderate values of r. The latter bump becomes unstable when r exceeds a certain threshold.
Remark 1. More details as well as other results can be found in Kolodina et al. [2].
REFERENCES
1. Nguetseng G. A general convergence result for a functional related to the theory of homoge-nization. SIAM Journal on Mathematical Analysis, 1989, vol. 20. no. 3, pp. 608-623.
2. Kolodina K., Oleynik A., Wyller J. Single bumps in a 2-population homogenized neuronal network model. Physica D: Nonlinear Phenomena, 2018, vol. 310, pp. 40-53.
Received 23 March 2018
Reviewed 26 April 2018
Accepted for press 5 June 2018
There is no conflict of interests.
Kolodina Karma. Norwegian University of Life Sciences. As. Norway, PhD-student at Faculty of Science and Technology, e-mail: [email protected]
Oleynik Anna, University of Bergen, Bergen, Norway, PhD, Postdoctoral fellow at Department of Mathematics, e-mail: [email protected]
Wyller John. Norwegian University of Life Sciences, As, Norway, PhD, Professor at Faculty of Science and Technology, e-mail: [email protected]
For citation: Kolodina K., Oleynik A., Wyller J. Existence and. stability of bumps in a neural field model. Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Reports. Series; Natural and Technical Sciences, 2018, vol. 23, no. 122, pp. 131-135. DOI: 10.20310/1810-0198-2018-23-122-131-135 (In Engl., Abstr. in Russian).
EXISTENCE AND STABILITY OF BUMPS IN A NEURAL FIELD MODEL
135
DOI: 10.20310/1810-0198-2018-23-122-131-135 УДК 51-76
СУЩЕСТВОВАНИЕ И УСТОЙЧИВОСТЬ ВАМПОВ В МОДЕЛИ НЕЙРОННОГО ПОЛЯ
К. Колодина1', А. Олейник2), Й. Виллер11
^ Норвежский университет естественных наук 1433, Норвегия, Ос, ул. Университетская, 3 E-mail: [email protected], [email protected] 2i Бергенский университет 5020. Норвегия, Берген, ул. Аллегатен. 41 E-mail: [email protected]
Аннотация. Исследованы существование и устойчивость болтов (локализованных стационарных решений) усредненной двупопуляционной модели нейронного поля в случае, когда функции активации задаются функцией типа Хевисайда. Ключевые слова: теория усреднения; существование и устойчивость стационарных решений нелокальных моделей нейронных полей
СПИСОК ЛИТЕРАТУРЫ
1. Nguetseng G. A general convergence result for a functional related to the theory of homoge-nization // SIAM Journal on Mathematical Analysis. 1989. Vol. 20. № 3. P. 608-623.
2. Kolodina K., Oleynik A., Wyller J. Single bumps in a 2-population homogenized neuronal network model // Physica D: Nonlinear Phenomena. 2018. Vol. 310. P. 40-53.
Поступила в редакцию 23 марта 2018 г. Прошла рецензирование 26 апреля 2018 г. Принята в печать 5 июня 2018 г. Конфликт интересов отсутствует.
Колодина Карина. Норвежский университет естественных наук, г. Ос, Норвегия, аспирант факультета науки и технологий, e-mail: [email protected]
Олейник Анна. Бергенский университет, г. Берген. Норвегия, научный сотрудник факультета математики, e-mail: [email protected]
Виллер Йон. Норвежский университет естественных наук, г. Ос, Норвегия, профессор факультета науки и технологий, e-mail: [email protected]
Для цитирования: Колодина К., Олейник А., Виллер Й. Существование и устойчивость бампов в модели нейронного поля // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2018. Т. 23. № 122. С. 131-135. DOI: 10.20310/1810-0198-2018-23-122-131-135
Работа выполнена при поддержке Норвежского университета естественных наук и Научно-исследовательского совета Норвегии (проект № 239070).
