CC BY
17
UDC 517.988.5, 51-76
DOI: 10.20310/1810-0198-2017-22-1-7-12
VOLTERRA OPERATOR INCLUSIONS IN THE THEORY OF GENERALIZED NEURAL FIELD MODELS
WITH CONTROL. II
© E. O. Burlakov
Norwegian University of Life Sciences 1432, Norway, As, 3 Universitetetstunet E-mail: eb @bk.ru
We obtained conditions for solvability of Volterra operator inclusions and continuous dependence of the solutions on a parameter. These results were implemented to investigation of generalized neural field equations involving control.
Key words: Volterra operator inclusions; neural field equations; control; existence of solutions; continuous dependence on parameters
3. Integro-differential inclusion with parameter.
In the first part of the present paper (see [1]), we introduced a controllable integro-differential equation and posed the corresponding well-posedness problem. The form of the integro-differential equation introduced describes a broad class of equations used in modeling of electrical activity in the brain cortex (see, e.g. [2], [3]). The particular type of solutions, namely, spatially localized solutions, which we have focused on in the first part of the paper, characterizes the electrical activity of the brain prevalent during it's normal functioning (see, e.g. [4], [5]). Electrical stimulation of the brain using e.g. in therapy of Epilepsy and Parkinson's disease, following the ideas of [6]-[10], was considered as control. The control problem obtained was represented in the form of integro-differential inclusion in the following way
t
w(t,x) € J J f(t,s,x,y,w(s,y),U,X)dyds, t € [a, oo) , x € Rm, A € A, (3.1)
a Rm
lim w(t,x) = 0, t € [a, oo). (1.3)
|ie|—>oo
Using the results from Section 2 of the previous part of the paper (see, [1]), we investigate here the solvability of the problem above and continuous dependence of the solutions obtained on the parameter A from some metric space A. Here U is some compact subset of Rk .
Throughout the section, we will use the following notation. Let L(Q,/x, Rn) be the space of all measurable and integrable functions >Rn with the norm \\x\\l(ü,ij,,r™)= /lx(s)Ms; Co(f2, Rn)
Q
be the space of all continuous functions >Rn satisfying the additional condition lim i9(x) = 0
|ik|—>oo
in the case if fi is unbounded, with the norm ||$||c0(n Rn) = max I $0*01 > and C([a, b], Co(i2, Rn))
' ikSH
be the space of all continuous functions v : [a, b] —> Co(Q, Rn), with the norm \\v\\c(\a,b\,c0(Q.,Rn)) = = max IKi)llco(iW) •
td\a,b\
We assume that for some Ao € A and ro > 0 , the following conditions are fulfilled:
(i) For any i G [a, oo), w € Rn , x € Rm and any u € U, A € A, the function f(t, -,x, -,w, u, A): [a, oo) x Rm —>• Rn is measurable.
(ii) For almost all (s, y) € [a, oo) x Rm and any AeA, the function /(•, s, •, y, •, •, A) is continuous.
(iii) For any b € (a, oo) and any r > 0 , it holds true that
t
' 0
lim sup
i€ [a,b] ,x€Rm \BRm (0,t)
J J f(t,s,x,y,w,u,\0 + A)dyds
a Rm
for any w € -Br«( 0, r) , uniformly for all uGU and A € B\(0, ro) ■
(iv) For any b € (a, oo) and any r > 0 , there exists such £ L([a, b] x Rm, /x, R) that
\f(t,s,x,y,w,u, A)| < g(b,r)(s>y)
for all x € Rm , w € Bru (0, r) , t € [a, b] , ueU , A € £>a(Ao, ro) and almost all (s, y) € [a, b] x Rm .
Definition 3.1. Choose an arbitrary iteSM^o^o)- We define a local solution to the problem (3.1) - (1.3), defined on [a,a+7], 7 € (0, 00), to be a function w1 € € C([a, a+7], Co(Rm, R"')), that satisfies the equation (3.1) on [a,a+7] . We define a maximally extended solution to the problem (3.1) - (1.3), defined on [a,a+rj), rj € (0, 00) , to be a map wv : [a, a+rj) —> Co(Rm, Rn), whose restriction w~( on [a, a+7] with any 7<r? is its local solution and lim ^ ||w7||c([a,a+7],c0(-Rm,-Rn)) =oo- We define an oo -solution to the problem (3.1) - (1.3)
to be a map w : [a, 00) —> Co(Rm, Rn), whose restriction w1 on [a, a+7] for any 7 € (0, 00) is its local solution.
For any A € A and 7 € (0,1), we denote by 87(A) the set of v(7) -local solutions to (3.1) -(1.3) corresponding to AeA.
Theorem 3.1. Let the assumptions (i) - (iv) hold true. Assume that the following conditions are satisfied:
1) For the given ro > 0 and any r> 0 there exists fr(s, y) € L([a, 00) x Rm, /x, R) such that for which
sup inf \f(t,s,x,y,wl,ul, A) - f(t,s,x,y,w2,u2, A)| < fr(s,y)\wl - w2\
uieUu2eu
for all wl,w2 € Bnn(0,r), 11, A € £>a(Ao, ro); t€[a, 00), x € Bum(0, r) .
2) For any w € Rn , t € [a, 00), x € Rm , and any sequence Aj —> Ao , it holds true that
sup inf |/(i, -,x, ■,w,u, A0) - f(t,-,x, ■,w,u, Aj)| 0
in measure on [a, 00) x Rm .
Then for each (u,X)gU x!?A(Ao,ro), the problem (3.1) - (1.3) has a local solution, and each local solution can be extended to an 00 -solution or maximally extended solution. Moreover, if
at A = Ao the problem (3.1) - (1.3) has a local solution Wq1 defined on [a, a+7] , then for any A sufficiently close to Ao , the problem (3.1) - (1.3) has a local solution w1 = w1(X) defined on [a, a+7] and the set-valued mapping \t—>§7(A) is lower semi-continuous at Ao . Proof.
From the conditions (i) - (iv), one can observe that:
a) for any (u, A) gU x Ba(Xo, ro) and each b>a,
(•)
J J f(;s,;y,w(s,y),u,\)du(y)dsGC([a,b],Co(Rm,Rn)y,
a Rm
b) F(-, A) : C([a, b],C0(Rm, Rn)) ->■ clos(C([a, b],C0(Rm, Rn))),
F(w,X) = J J f(-,s,-,y,w(s,y),U,X)dv(y)ds,
a Rm
for all A € £>a(Ao, ro) due to the theorem 1.2.29 (see ), continuity of F in the variable w and regularity of the space C([a, b],Co(Rm, Rn)).
We are going to check now the conditions of Theorem 2.2 from the previous section. Choose an arbitrary b € (a, 00), (?o<l, r>0. Let 7<E(0 ,b — a) and w\(t, •) = W2(t, •), t€ € [a, a+7] , where wi,w2 € -Bc([a,6],co(-Rm,-Rn))(0)r) ■ Using assumptions (i)-(iv) and condition 1) of Theorem 3.1, we get the following estimates
sup
1
J J f(t,s,x,y,wl(s,y),u,X)dv(y)ds-
a Rr'
<
i
-J J f(t,s,x,y,w2(s,y),u, \)du(y)ds
a Rm
a+j+S
e/2+ sup / / f(t,s,x,y,w1(s,y),u,X)dv(y)ds—
t&\a,a+"i+S],x&BRm (0,r£) J J
a+7 Rm
-f(t,s,x,y,w2(s,y),u,X) dv(y)ds <
a+j+S
£/2+ SUp / / jr{s,y)\\wl - W2\\C({a,b],BC(R™,R"))dv(y)ds < £
ie[a,a+7+<5],a;eBflm(0,r-£) J J
a+7 Rm
for all (u, A) € U x Ba(Xq, ro) ■ Here, r£> 0 , 5 > 0 can be chosen in a such way that e < qo ■ The estimate above yields that
hc([a,a+-f+S},BC(Rm,Rn))(F1+sw}f+s, FJ+sW2+s) < qodc([a,a+-f+S],BC(Rm ,Rn)) (W~f+S> wj+s) •
Thus, we checked that condition q2) is satisfied. The verification of condition qi) is analogous.
From the proof of Theorem 2.1 (see [1]), we can deduce that, if for each b>a there exist at least one local solution defined on [a, b] with bounded norm, we can obtain an 00-solution. Otherwise, we get a maximally extended solution.
Next, we check the condition 2) of Theorem 2.2. from [1]. In order to do that, we take arbitrary
e>0, weC([a,b},Co(Rm,Rn)), c C([o, b], C0(Rm, Rn)), {AJcA, \\w-^IbiM.Co^,^)),
ui^uo 00 ), and estimate
hc([a,b],C0(Rm,R"))(F(w,U0) - F(Wi,Ui))
sup inf
u£U,t£[a,b],x£Rm U&U
i
J J f(t,s,x,y,w(s,y),u, Ao)du(y)ds-
a Rm
i
J J f(t,s,x,y,Wi(s,y),u,Xi)dv(y)ds
a Rm
<
< e/3 + sup inf
ueu,te[a,b],xeBRm (o,r£)ueU
i
J J f(t,s,x,y,w(s,y),u, Ao)du(y)ds-
a Rm
= e/3+ sup
u£U,t£[a,b],x£BRm(0,rs) U&U
i
J J f(t,s,x,y,Wi(s,y),u(s,y),Xi)dv(y)ds
a Rm
t
inf J J (\f(t,s,x,y,w(s,y),u, Ao) - f(t,s,x,y,w(s,y),u, Ai)| +
a Rm
+ 1 f(t, s, x, y, w(s, y),u, X i) - f(t, s, x, y, Wi(s, y),u(s, y), Xi)\jdv{y)ds. Estimating the first summand of the integrand, we get
sup inf \f(t,s,x,y,w(s,y),u, Ao) - f(t,s,x,y,w(s,y),u,Xi)\ <
< sup inf \f(t,s,x,y,w(s,y),u, Ao) - f(te,s,xe,y,we,u(s,y), A0)| + ii&uueu
+ sup inf \f(t£,s,x£,y,w£,u(s,y), A0) - f(t£,s,x£,y,w£,u,Xi)\ + ii&uueu
+ sup inf \f(t£,s,x£,y,w£,u, Xi) - f(t,s,x,y,w(s,y),u(s,y),Xi)\. ii&uueu
Here t£, x£ , w£ are approximations of t, x, w(s,y), taking finite number of values (on their compact ranges of definition). Thus, using the condition 2) of Theorem 3.1 and the assumption (ii), the first and the third summands on the right-hand side of the inequality go to 0 uniformly with respect to (s, y) € [a, b] x Rm and the second summand go to 0 in measure on [a, b] x Rm with respect to the variables s and y .
Next, estimation of |/(t, s, x, y, w(s, y),u, X¿) — f(t, s, x, y, Wi(s, y),u, Aj)| using the assumption (ii), gives uniform convergence of this expression to 0 on [a, b] x Rm . Thus, we can find such I that for any i > I, we get
t
sup / / (\f(t,s,x,y,w(s,y),u0) - f(t,s,x,y,w(s,y),Ui)\ +
■■GBnm(O.rr) J J V
t£[a,b],x£BRm(0,rE)
a Rn
+ 1 f(t,s,x,y,w(s,y),Ui) - f(t,s,x,y,Wi(s,y),Ui)\)dv(y)ds < 2e/3, which concludes the verification of Theorem 2.1 conditions and completes the proof.
□
REFERENCES
1. Burlakov E. O. Volterra operator inclusions in the theory of generalized neural field models with control. I. // Tambov University Reports. Series: Natural and Technical Sciences. 2016. V. 21. № 6. P. 1950-1958. DOI: 10.20310/1810-0198-2016-21-6-1950-1958
2. Burlakov E., Zhukovskiy E., Ponosov A., Wyller J. On wellposedness of generalized neural field equations with delay, // Journal of Abstract Differential Equations and Applications 2015. V. 6. P. 51-80.
3. Burlakov E., Zhukovskiy E.S. Existence, uniqueness and continuous dependence on control of solutions to generalized neural field equations // Tambov Univeersity Reports. Series: Natural and Technical Sciences. 2015. V. 20. № 1. P. 9-16.
4. Taube J.S., Bassett J.P. Persistent neural activity in head direction cells // Cereb. Cortex. 2003. V. 13. P. 1162-1172.
5. Wang X-J. Synaptic reverberation underlying mnemonic persistent activity // Trends Neurosci. 2001. V. 24. P. 455-463.
6. Tass P. A. A model of desynchronizing deep brain stimulation with a demand-controlled coordinated reset of neural subpopulations // Biological cybernetics. 2003. V. 89. P. 81-88.
7. Suffczynski P., Kalitzin S., and Lopes Da Silva F.H. Dynamics of non-convulsive epileptic phenomena modeled by a bistable neuronal network // Neuroscience. 2004. V. 126. P. 467-484.
8. Kramer M.A., Lopour B.A., Kirsch H.E., and Szeri A.J. Bifurcation control of a seizing human cortex // Physical Review E. 2006. V. 73. № 4. P. 1-16.
9. Schiff S.J. Towards model-based control of Parkin- son's disease // Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2010. V. 368. P. 2269-2308.
10. Ruths J., Taylor P., Dauwels J. Optimal Control of an Epileptic Neural Population Model. Proceedings of the International Federation of Automatic Control. Cape Town, 2014.
11. Borisovich Yu. G., Gelman B.D., Myshkis A.D., Obukhovskii V. V. Introduction to the Theory of Multivalued Maps and Differential Inclusions. 2nd ed. Moscow: Librokom, 2011.
ACKNOWLEDGEMENTS: The present research is supported by by the Russian Fund for Basic Research (project № 16-31-50037).
Received 5 December 2016
Burlakov Evgenii Olegovich, Norwegian University of Life Sciences, As, Norway, postgraduate, e-mail: eb @bk.ru
УДК 517.988.5, 51-76
DOI: 10.20310/1810-0198-2017-22-1-7-12
ОПЕРАТОРНЫЕ ВКЛЮЧЕНИЯ ВОЛЬТЕРРЫ В ОБОБЩЕННЫХ МОДЕЛЯХ НЕЙРОПОЛЕЙ С УПРАВЛЕНИЕМ. II
© Е. О. Бурлаков
Норвежский университет естественных наук 1432, Ос, ул. Университетская, 3 E-mail: eb_@bk.ru
Получены условия разрешимости операторных включений Вольтерры и непрерывной зависимости решений от параметра. Результаты применены к исследованию обобщенных моделей нейрополей с управлением.
Ключевые слова: операторные включения Вольтерры; модели нейрополей; управление; существование решений; непрерывная зависимость от параметров
СПИСОК ЛИТЕРАТУРЫ
1. Buriakov Е.О. Volterra operator inclusions in the theory of generalized neural field models with control. I. // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2016. Т. 21. Вып. 6. С. 1950-1958. DOI: 10.20310/1810-0198-2016-21-6-1950-1958
2. Buriakov Е., Zhukovskiy Е., Ponosov A., Wyller J. On wellposedness of generalized neural field equations with delay, // Journal of Abstract Differential Equations and Applications 2015. V. 6. P. 51-80.
3. Buriakov E., Zhukovskiy E.S. Existence, uniqueness and continuous dependence on control of solutions to generalized neural field equations // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2015. Т. 20. Вып. 1. С. 9-16.
4. Taube J.S., Bassett J.P. Persistent neural activity in head direction cells // Cereb. Cortex. 2003. V. 13. P. 1162-1172.
5 . Wang X-J. Synaptic reverberation underlying mnemonic persistent activity // Trends Neurosci. 2001. V. 24. P. 455-463.
6. Tass P. A. A model of desynchronizing deep brain stimulation with a demand-controlled coordinated reset of neural subpopulations // Biological cybernetics. 2003. V. 89. P. 81-88.
7. Suffczynski P., Kalitzin S., and Lopes Da Silva F.H. Dynamics of non-convulsive epileptic phenomena modeled by a bistable neuronal network // Neuroscience. 2004. V. 126. P. 467-484.
8. Kramer M.A., Lopour B.A., Kirsch H.E., and Szeri A.J. Bifurcation control of a seizing human cortex // Physical Review E. 2006. V. 73. № 4. P. 1-16.
9. Schiff S.J. Towards model-based control of Parkin- son's disease // Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2010. V. 368. P. 2269-2308.
10. Ruths J., Taylor P., Dauwels J. Optimal Control of an Epileptic Neural Population Model. Proceedings of the International Federation of Automatic Control. Cape Town, 2014.
11. Borisovich Yu. G., Gelman B.D., Myshkis A.D., Obukhovskii V. V. Introduction to the Theory of Multivalued Maps and Differential Inclusions. 2nd ed. Moscow: Librokom, 2011.
БЛАГОДАРНОСТИ: Работа выполнена при финансовой поддержке Российского фонда фундаментальных исследований (проект № 16-31-50037).
Поступила в редакцию 5 декабря 2016 г.
Бурлаков Евгений Олегович, Норвежский университет естественных наук, Аас, Норвегия, аспирант, e-mail: eb_@bk.ru
Информация для цитирования:
Burlo,kov Е.О. Volterra operator inclusions in the theory of generalized neural field models with control. II. Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Reports. Series: Natural and Technical Sciences, 2017, vol. 22, no. 1, pp. 7-12. DOI: 10.20310/1810-0198-2017-22-1-7-12
