CC BY
33
во сколько раз изменится амплитуда соответствующей волны при прохождении ею через вершину графа из г -го ребра в ] -ое, а коэффициент характеризует изменение амплитуды волны, распространяющейся по г -му ребру, после ее отражения от вершины графа.
Подставляя полученные функции из (12) - (13) в формулы (4) на каждом ребре, окончательно получим решение смешанной задачи на графе Г2 в следующем виде:
u(x, t) = <
7i(x - X+t) + gi(x - X-t)
+ 2 '
fi(x - X+t) + Gi(x - X-t) /2(x - X+t) + g2(x - X-t) F2(x - X+t) + g2(x - X-t)
x < X-t, x< 0 , x > X-t, x< 0 ,
x
x
> X+t, x > 0, <X+t, x > 0,
(14)
p(x,t) = <
pc
pc
pc
pc
fi(x - X+t) - gi(x - X-1) + 2 , fi(x - X+t) - Gi(x - X-t) + 2 ~: f2(x - X++t) -g2(x - X2t)
+ 2 ,
F2(x - X+t) - g2(x - X-t)
x< X- t, x< 0, x> X- t, x< 0,
x
x
> X+t, x > 0, <X+t, x > 0.
Таким образом, нами доказана
Теорема 3. Решение смешанной задачи (11) имеет вид (14), (15).
(15)
2
СПИСОК ЛИТЕРАТУРЫ
1. Кошелев В. Б. и др. Математические модели квази-одномерной гемодинамики. М.: МАКС Пресс, 2010.
2. Абакумов Н. В. и др. Методика математического моделирования сердечно-сосудистой системы // Математическое моделирование. 2000. Т. 12. № 2. С. 106-117.
3. Буничева А. Я. и др. Исследование влияния гравитационных перегрузок на параметры кровотока в сосудах большого круга кровообращения // Математическое моделирование. 2012. Т. 24. № 7. С. 67-82.
4. Покорный Ю. В. и др. Дифференциальные уравнения на геометрических графах. М.: Физматлит, 2004.
5. Тихонов А. Н., Самарский А. А. Уравнения математической физики. М.: Изд. МГУ, 2004.
6. Рождественский Б. Л., Яненко Н.Н. Системы квазилинейных и их приложения к газовой динамике. Уравнения математической физики. М.: Наука, 1988.
Поступила в редакцию 5 октября 2016 г.
Безяев Владимир Иванович, Российский университет дружбы народов, г. Москва, Российская Федерация, кандидат физико-математических наук, доцент кафедры прикладной математики, e-mail: [email protected]
Садеков Наиль Халимович, Российский университет дружбы народов, г. Москва, Российская Федерация, студент магистратуры, кафедра прикладной математики, e-mail: [email protected]
1948
UDC 517.925
DOI: 10.20310/1810-0198-2016-21-6-1944-1949
A WAVE PROPAGATION METHOD IN LINEAR HEMODYNAMICS
© V. I. Bezyaev, N. Kh. Sadekov
Peoples Friendship University of Russia 6 Miklukho-Maklay St., Moscow, Russian Federation, 117198 E-mail: [email protected]
In the work examines some problems for the linearized equations of hemodynamics on simple graphs. By the method of propagating waves and by the continuation method obtained exact solutions of these problems.
Key words: hemodynamics; graph; hyperbolic system; a wave propagation method; continuation method
REFERENCES
1. Koshelev V. B. i dr. Matematicheskie modeli kvazi-odnomernoj gemodinamiki. M.: MAKS Press, 2010.
2. Abakumov N. V. i dr. Metodika matematicheskogo modelirovaniya serdechno-sosudistoj sistemy // Matematicheskoe modelirovanie. 2000. T. 12. № 2. S. 106-117.
3. Bunicheva A. YA. i dr. Issledovanie vliyaniya gravitacionnyh peregruzok na parametry krovotoka v sosudah bol'shogo kruga krovoobrashcheniya // Matematicheskoe modelirovanie. 2012. T. 24. № 7. S. 67-82.
4. Pokornyj YU. V. i dr. Differencial'nye uravneniya na geometricheskih grafah. M.: Fizmatlit, 2004.
5. Tihonov A. N., Samarskij A. A. Uravneniya matematicheskoj fiziki. M.: Izd. MGU, 2004.
6. Rozhdestvenskij B. L., YAnenko N. N. Sistemy kvazilinejnyh i ih prilozheniya k gazovoj dinamike. Uravneniya matematicheskoj fiziki. M.: Nauka, 1988.
Received 5 October 2016
Bezyaev Vladimir Ivanovich, Peoples' Friendship University of Russia, Moscow, the Russian Federation, Candidate of Physics and Mathematics, Associate Professor of the Applied Mathematics Department, e-mail: [email protected]
Sadekov Nail Khalimovich, Peoples' Friendship University of Russia, Moscow, the Russian Federation, Graduate Student, Applied Mathematics Department, e-mail: [email protected]
Информация для цитирования:
Безяев В.И., Садеков Н.Х. О методе распространяющихся волн в линейной гемодинамике // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2016. Т. 21. Вып. 6. С. 1944-1949. DOI: 10.20310/1810-0198-2016-21-6-1944-1949
Bezyaev V.I., Sadekov N.Kh. O metode rasprostranyayushchihsya voln v linejnoj gemodinamike [A wave propagation method in linear hemodynamics]. Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Review. Series: Natural and Technical Sciences, 2016, vol. 21, no. 6, pp. 1944-1949. DOI: 10.20310/1810-0198-2016-21-6-1944-1949 (In Russian)
1949
UDC 517.988.5, 51-76
DOI: 10.20310/1810-0198-2016-21-6-1950-1958
VOLTERRA OPERATOR INCLUSIONS IN THE THEORY OF GENERALIZED NEURAL FIELD MODELS
WITH CONTROL. I
© E. O. Burlakov
Norwegian University of Life Science 3, Universitetstunet, As, Norway, 1432 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
1. Introduction.
The human brain cortex is the top layer of the hemispheres, of 2 — 4 mm thick, involving about 109 neurons having 60 x 1012 connections [1]. The brain cortex is responsible for such higher functions of the human brain as e.g. memory, reasoning, thought, and language [2], [3]. The basic unit of the brain cortex is the neuron. It consists of dendrites, cell body (soma), and axon. The dendrites receive electrical signals from other neurons and propagate them to the soma. If the total sum of the input electrical potential in the soma exceeds a certain threshold value, the neuron produces the burst of the output electrical signal, which then propagates along the axon to other neurons. Thus, a natural way (see e.g [4]) of studying electrical activity in the neocortex is the framework of cortical networks.
The most well-known representative of such models is the Hopfield network model [5]
Here zi is the electrical activity of the i -th neuron in the network, uij is the connection strength between the i -th and j -th neurons, the non-negative function f gives the firing rate f (z) of a neuron with activity Z .
However, since the number of neurons and synapses in even a small piece of cortex is immense, a suitable modeling approach is to take a continuum limit of the neural networks and, thus, consider so-called neural field models of the brain cortex (rigorous justification of this limit procedure using the notion of parameterized measure is given in [6]). The most well-known and simplest model describing the macro-level neural field dynamics is the Amari model [7]
t > 0, x e Q C Rn.
Here u(t, x) denotes the activity of a neural element u at time t and position x. The connectivity function u determines the coupling strength between the elements and the nonnegative function f gives the firing rate f (u) of a neuron with activity u. Neurons at a position
N
(t) = -Zi(t) + ^2 Uij f(zj t), t > 0, i = !,...,N.
(1.1)
j=i
(1.2)
1950
x and time t are said to be active if f (u(t, x)) > 0. Typically f is a smooth function that has sigmoidal shape.
One of the key objects in the neuroscience community is the so-called bump-solutions, i.e. solutions satisfying the following condition
lim w(t,x) = 0, t € [a, o). (1.3)
This type of solutions corresponds to the electrical brain activity that is prevalent during its normal functioning, encoding visual stimuli [8], representing head direction [9], and maintaining persistent activity states in working memory [10], [11].
The models of the type (1.1) are important in studies of cortical gain control or pharmacological manipulations [12]. The problems of therapy of Epilepsy, Parkinson's disease, and other disorders of the central nervous system has been recently investigated in [1]-[17]. The modeling frameworks in [1]-[17] incorporate brain electrical stimulation, which is considered as control, and the corresponding optimization problems. The unique solvability of such models and continuous dependence of the solutions obtained on the control involved in the modeling equations has been recently examined in [18], [19]. These works employed the theory of abstract Volterra operators in complete metric spaces and Banach spaces in order to establish the main results on controllable neural field equations. The present paper extends the results of [18], [19]. Here, we deal with controllable neural field equations where the whole right-hand side is parameterized. Generalizing the model (1.2) and adding control to it, we get
t
w(t,x) = J j f (t,s, x, y, w(s, y), u(t., s, x, y), \)dyds, (L4)
a Rm
t € [a, oo), x € Rm,
with respect to the unknown continuous function w: [a, o) x (Rm ^ (Rn , which is spatially localized, i.e. satisfies (1.3). The function u: [a, o) x [a, o) x (Rm x (Rm ^ U ( U - compact subset of Rk ) is a control which is assumed to be essentially bounded. Here A is a parameter from some metric space A.
We can derive the following inclusion arising with respect to the control taking it's values in U with parametrization from A :
w(t,x) € (F(w,A))(t,x), (1.5)
t
(F(w,A))(t,x) = J J f (t,s,x,y,w(s,y),U,A)dyds,
, w(
aj
t € [a, oc), x € IRm, A € A.
In order to approach the latter problem, we investigate solvability an parametric dependence properties of Volterra operator inclusions in the next section.
2. Volterra operator inclusions with parameter.
Let Rm be the m -dimensional real vector space with the norm | • | . Let W be a metric space with the distance pW . We denote by BW (w, r) an open ball of the radius r centered at w € W . We denote Q,(W) to be the set of all non-empty closed subsets of W .
Let an equivalence relation ~ be defined on W . For any two equivalence classes w1, w2 , we introduce
d(w1 ,w2) = inf p(w1,w2). (2.1)
w1£w1, w2£w2
1951
If for any e> 0 and any w1,w2 € W/ ~ , w1 € w1 one can find w2 € w2 such that d(w1,w2) > > p(w1,w2) — e, then (2.1) defines metric in W/ ~ .
We put in correspondence to each y € [0,1] the equivalence relation v(y). We assume that the family of equivalence relations u = {v(y),y € [0,1]} satisfy the following conditions:
v0) Y = 0 corresponds to the relation u(0) = W2 (any two elements are u(0) -equivalent); v1) y = 1 corresponds to equality relation (any two distinct elements are not u(1) -equivalent); v) if y1 >y2 , then v(y1) C v(y2) (any v(y1) -equivalent elements are v(y2) -equivalent);
Definition 2.1. A set-valued map ^ : W ^ Q(W) is said to be a Volterra set-valued map on the family v if for any y € [0,1] and any w1, w2 € W the fact that (w1, w2) € v(y) implies (^w1, ^w2) € v(y) , which means that (w1, w2) € v(y) for any wl € ^w1 and w2 € ^w2 .
For any w € W , let us denote wY to be the v(y) -equivalence class of w .
Hereinafter we assume that (W, pW) is a complete metric space with the equivalence relation v satisfying vo), v1), v). Moreover, we assume that for each y € (0,1), the corresponding equivalence class is closed and the quotient set W/v(y) is a complete metric quotient space with the distance dW/v(1)(w1 ,w2)= inf pW(w1,w2).
Below we cite some important properties of single-valued Volterra operators (see [20]) that can be naturally extended to Volterra set-valued maps.
1. Choose an arbitrary set r c [0,1] , {0,1} C r, and for any decreasing (or any increasing) sequence {Yi} , it holds true that lim Yi € r . Let u = {v(y),Y € r} . We define the mapping
i—^^o
n: [0,1] ^ r as n(Y) = inf{{ € r},C > y (n(Y)=inf {C € r},C < y) , and put in correspondence to any Y the equivalence relation u(n(Y)). If the set-valued map ^: W ^ Q(W) is a Volterra mapping on the family v , then it is a (set-valued) Volterra on its subfamily u .
2. If for some y0 € (0,1), w € W it holds true that ^w n wY0 = 0 , then the set wY0 is invariant with respect to the Volterra set-valued mapping ^ : W ^ Q(W) and the relation v(y) can be considered only on the elements of wY0 C W . The set wY0 is a complete metric space with respect to the metric of the whole space W . Thus, the family of the equivalence relations satisfying the conditions v0,v1,v is also defined on wY0 . The quotient set wY0/v(y) , Y < Y0 , consists of the unique element. If y>Y0 , the quotient set wY0/v(y) is a complete metric space. Moreover, the fact that ^ : W ^ Q(W) is a Volterra set-valued map on the family v implies that the restriction ^70: ^ tt(w10) of ^ is a set-valued Volterra map on the family v .
3. For each y € (0,1), we define the canonical projection nY : Q(W) ^ Q(W/v(y)) as nYW =
= U , W € Q(W). For a set-valued Volterra mapping ^ : W ^ Q(W) on the family v , we
wew
define the map : W/v(y) ^ Q(W/v(y)) as = nY^w , where w is an arbitrary element
of wY . Choose an arbitrary y0 € (0,1). The family v(y0) generates the corresponding equivalence relation on W/v(yo) . Let C € (0, y0) , and let the elements w1,w2 € W be u(C) -equivalent. Then any w1 € w\0 , w2 € wYi0 are also u(C) -equivalent, which defines the notion of equivalence of the classes w^0 and wYi0. Namely, the classes wli0 and w'ii0 are vY0 (a) -equivalent (a € (0,1)), if there exist (which, actually, means "any") w1 € w}(0 , w2 € w?(0 satisfying the equivalence relation u(C), C = Y0a. Thus, the family vY0 = {vY0(a)} of equivalence relations is defined on W/v(y0) . The
quotient set [W/u(Y0)j /uY0 with the distance
d(WY0a,WY0a) = inf2 ^/v^^a,w2f0^)= inf9_2 PW(wSw2)
is isometric to W/u(Y0a) and, hence, is a complete metric space as well. If the set-valued map ^: W ^ Q(W) is a Volterra map on the family u , then the operator ^70: W/v(y0) ^ Q(W/v(y0)) is a (set-valued) Volterra operator on the family vY0
1952
Below we introduce the notion of local contraction for set-valued maps, which allows to investigate the solvability and parametric dependence properties of operator inclusions.
We consider the following inclusion
w £ Vw, (2.2)
where V : W ^ Q,(W) is a Volterra set-valued map on the family v of equivalence relations.
Definition 2.2. We define a v(y) -local solution of the inclusion (2.2), 7 £ (0,1), to be an equivalence class wY £ W/v(j), that satisfies the inclusion wY £ VYwY . Identifying the element w, satisfying (2.2), with it's u(1) -equivalence class w , we consider it a global solution to the inclusion (2.2) . We define a v(y) -maximally extended solution to (2.2) , 7 £ (0,1) , to be a map putting in correspondence to each £ £ (0,j) a v(£) -local solution w^ , and satisfying the following two conditions:
Y V)
• for any , 0 <n <£<7 , it holds wg C wn (where wg is a restriction of w
• for any w° € W it holds lim d(wg,w0) = o .
For any 7 € (0,1), we denote by S7 and S the sets of v(7) -local solutions and global solutions to (2.2) , respectively.
Let hw/v(Y) be the Hausdorff metric in the space of all non-empty closed subsets of the metric space (W/v(Y),dw/v{l)).
Definition 2.3. We define a Volterra on the system v set-valued map ^ : W ^ Q,(W) to be locally contracting at a point 7 € [0,1) on the system v , if for any wY € W/v(j), one can find: an element w° € wY and q< 1 such that for any r> 0 there exists 5 > 0 such that for all wl1+&,w21+& € Bw/v(j+6) (w^+s, r) (w^+s € ^Y+sw0) , satisfying in the case 7> 0 for any £ € (0,7) the inclusion wY+s ,w"Y+6 C w0 , where (w0 C ^g (w°, A)), it holds true that
hW/v(Y+6)(^Y+6Twli1+6, ^j+6wY+6) < qdW/v(Y+6)(w\+6,W2Y+6).
Definition 2.4. We define a Volterra set-valued map ^ : W ^ Q,(W) to be locally contracting on the system v , if it is locally contracting for any 7 € [0,1) with the constants q h 5(r) independent of 7 € [0,1).
Theorem 2.1. Let the set-valued map ^ is a locally contracting Volterra map on the system v .
Then the inclusion (2.2) has a local solution and each local solution is extendable to a global or maximally extended solution.
Proof. We construct the solution in the following way. We choose r1 = (1 — q)-1pw(w0, ^w°) + + 1 and find all 5 > 0 that satisfy the theorem condition with r = r1. For 5^2 sup{5} , we have
hw/v(61 )(^w\^w2) < qdw/v(61)(w1 ,w2)
at any w1,w1 CBw/V(6i)(w°1 , which implies (Bw/u(6i)(w01 ,n)) CBw/V(6i)(w°1, r{). By the Nadler theorem (see e.g. [21]), the mapping ^ has a fixed point w61 in the ball Bw/V(61)(w01 ,r\). This fixed point is a v(51) -local solution to (2.2).
Choose some solution w61 to the equation (2.2) and the corresponding radius r2 = (1 — q)-1dw/v(61)(^(w61 ,w0). We find all possible 5> 0 that satisfy the theorem condition with r = r2 .For 52 = 2 sup{5} at any wl,w2 C Bw/U(61 +62)(w01 ,r2) n w61 we have
hw/v(61+62)(^wl, ^w2) < qdw/v(61+62)(w1,w2).
1953
