Spijker, S. [Promotor]Smit, A.B. [Promotor]Vries, T.J. de [Copromotor]Oever, M.C. van den [Copromotor

Szidarovszky
					, 
						Ferenc

Lubbers, B.R.

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VU Research PortalMolecular mechanisms controlling relapse to drug seeking in rodent models ofaddictionLubbers, B.R.2014document versionPublisher's PDF, also known as Version of recordLink to publication in VU Research Portalcitation for published version (APA)Lubbers, B. R. (2014). Molecular mechanisms controlling relapse to drug seeking in rodent models of addiction.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.E-mail address:vuresearchportal.ub@vu.nlDownload date: 21. May. 2021Molecular mechanisms controlling relapseto drug seeking in rodent models of addictionBart LubbersMolecular mechanisms controlling relapse to drug seeking    in rodent models of addiction         Bart LubbersOOOHNOOHOOCH3

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Molecular mechanisms controlling relapse to drug seeking in rodent models of addiction

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ESE - Salento University Publishing

- 31 -linear. Letf(s) = As + B•~k\: (!.k) = L ~i X ki + bk 'i=lthen the solution of the optimization pr~blem (27) is apiece-wise linear' function Qk • In this case the reduced gamecan be solved easily as i t is shown in [lOJ, pp. 43-44 ••5. Multiproduct oligopoly gameIn this paragraph we will consider the game having thesets of strategies~ = [o, ~CLlJend pa.y-off functionsx ••• x (30 )nL xP) ,••• ,{,=l(31)where (ll) lM))~ = Xk '···'Xk 'n n9 Cfm) = 0, L ri-1) x ••• x 0, L: LiMl , R(fm)CRl fort=l e=1k=1,2 •••• ,n and 1Il=1 , 2, •• , M. This game can come up if thefactories manufacture differ~nt p~ducts and sell them on thesame market. Let M be the number of products, and 1et x~m) ,~) be the p~duction level and capacity limit of factory kf~m p~duct m. If f donotes the unit price of p~duct m, thanmit is assumed that f is a function of the total productionmlevels of the different products. The function y~ is tbe- J2 -an~ ~3lng tn~ aOOV6 _erminology tne incorneV~ fa." .ory k ..... oJ.ven oy function (Jl).3imilar lnterprètation c~~ be given to the other applicationssh~/~ ~ the section àealing with the classical oligopoly gamebut ìiffe=ent aualities of water ~~d waste-water have to be•introduceci.':"..e :'ollc.'I(ing ~sul t is basic in the theory of mul tiproàuctecono~es, acd it is a generalization of par~ cl in the proofof Theorem 4.are concave and,Lemma 11., Let g, be a vector'-vector' function such thatrv:of Rb(~) is a convex set in the nonnegative orthantM7I,l~IC R • Assume that the components of g,continuously differentiable. Let J--be the Jacobian matrixof g. Ii !I-~ ~ .9(~I,'!'(~) + II l~)- is nonnegative semidefinite for arbitrarythen the functionh(~) = ~T gl~)lS concave.(J2)FroofL Let \J denote the gradient operation. Then simplecalculations show thatT TV h(~) = g ~) + ~ II (~) •Since the components of g, are concave, we have(JJ)and the condition given for the Jacobian J--implies- )) -'"- (ii. - ~l· ~(~)(;y - ~).(34 )The ~nequalities (3)) ,(34) •ana imply'"l- {g(iL) - gl~\} ~consequently- ~ l,wh::.cÌ! and equation ()2) c·~veo the inequalityThus function h is concave.As a corollary to this general r~sult we can prove themain resul t of this see H '1n.•'lheorem 7..!. :'et and let J--oe t!leJacobian of f. Assume that fune tions f and ~ (l ::.. k ~ !l) arecontinuous)the components of f are continuously differentiableand concave, ~ is convex and for arbitrary ~ ~ ~(f) tr-eT+ ~t~) is nonnegative semidefinite. Then thegame has at least one equilibrium point.Preof. Since ~ ~s a closed, convex, boundeQ subset ofconcave in ~k' the game satisfies all conditionsNikaiQo-Isoda theorem. Thus the game has at leastequilibrium point.•'f k ~s continuous and Lemma 11. implies that , o ~s'kof theone•- 34 -Remark. The theorem does not g~ve numerical methods forthe determination of the equilibrium point. But in the linearcase a very efficient algorithm can be constructed which is a"generalization of the method of M. Manas given for the one--product case.Let us assume that'. ( (l) (M))I,.u. S ,,, •• ,8ML ~m) x~m) + Bkm=lN;= La~m)m=l(k=1,2, ••• ,n),where (m)sn, (m)- L xk •k=l1et us intr'Oduce thefollowing notations: ~(m)Finally let us assume thatn= L~m)k=lA + AT- -- -(l.m)~M, fl. = al.. -l .-,U. "" , m-is nonnegative semidefinite.Under the above condi tions the game has at least one equilibriumpoint, and sinceH (H ii)!. = !l'···'!nonly if•'f k is concave in ~k' a vec torois an equilibrium point of the game if and/' O for (m) ft - O-xk--Cl fk(~ii)> O for lffi)ii ~m) (~ k,m) (35)él x Cm) xk ----k- O for O < x~m)ft < ~m)-,•here ii (Uht (Mllt)~ = \xk , ••• ,xk ( k= l , 2 , ••• , n). Le t- 35 -Wl ::> O- X,K =/ , . , x'Y-' (\ (36 )I v " . '>- ... ~V') k •)z,K \I ~ - otherwisev--\..• ..; ..;' xV-) ..- • (t<)-v-. ~V·l , k ---Iv,eIl;:> o otherwise,-then oy calculating the partial derivatives of -t k 'N e caneasily verify that the conditions (35) are equivalent to theset of equations (see [lO] pp. 46-47)b,u. + + zrl - O (37)for J-l-=1,2, ••• ,M; k=1,2, ••• ,n , wherenslw) = L x~m).k=lThe above system can be written ~n a simpler form if weintroduce the following notations:"'~ = ( xpl , ••• ,x~M) , ••• ,x~l), •••• ,x~M)ra = ( III CM) (l) (Ml)T_ Al' ••• , Al t ••• t An , •••• , An"'( \.Il (M) (l) CM)) •! = vI'···' vI'···' vn , ••• , vn(38 )_ ( ll)~ - zl' ... , (MI2 1 t···'(l)zn , ••• ,- 36 -Y:. = ( w?\ t ',l) ll) W~M) ) T... , Wl • ... , wn , ••• t-. ll) '1'[M) III L(M) ) ~.{,- l Lì ,••• , Ll • ... , Ln ', ••• ,- n( bI' ••• t t'M' ••• ' 11 •••• , bM )To ---;x.Ful"thermore let P denote the Mn x Mn matl"ix--I • ATA A • \I ••• I-- - - ••p • + • •- • •- •r• '1'A•••A A~- -- - -then the relations (36). (37), (38) have the forrr:r x + b - a - v T Z = O-- - - - - -Z + W - ~-- -T T T Ox Z v W - V Z- - -- - - - - -(39)x, v, Z, w- - - -:::, O •- --Thus we have proven the following r'esul1;.Lemma 12•. A vector ::.s an equilibrium point , .'01 tl1elinear oligopoly game with nonnegative definite* Ji *if and only if there exist vector's :!., Y!. , ~matrix A. +--such 1;hatT,r.-conditions (39) are satisfied with ii~=!.,y.= Jt iiV • W = W- - -and z = z* •- -In a furthel" special case the uruqueness of the equilibl"iu:npoint is assured, as it is shown in the following theorem.Theorem 8. Assume that matrix A is symmetric, negative-definite. Then the game has a unique e qui li brium poir.t.Proof., Let us consider the quadratic :programnung problem- 37 -o , x ~ I:- (4 0 )- ---- --,xT rn- p (~ ,. max~ x - - al y. >- •--- - --P::-rs t we pr-ove tha <: problem (4 O) i s astri:; t::' -,- :; onvex-progI'a.mJJUng problen:. h is sufficient tO prove tilat n:atrix !'-,-'f" - ( )L!R,·m~s ne(;at~ve ue =~te • .uet ~ = l:!1' ••••1!.n ~ , 'Nherefor k=1.2 ••••• n. Then~ r: ~_.--~ L T L In• "- ,1,;, p 1,;, - ti " l:!k + 1;.~ A u.- n I ---- =k - -~ - -"- - - vk=l i-l j-l-- --n I n,n, rnL T i ~ - L,- u, A u + I u, A u_ < \)-K - -k -~ -,- - vk-' i-l j-l-,!,-- --for u i- O. 1:-- -."-~s symmetric, then obviously f ~s ..a..sosymmetric.llext we ODserve that conùitions (39) without equat~on':f..T ~ = O are the Kuhn-Tucker condi tions of che quadraticprogr~ng problem (see G. Hadley (31). and since it isconvex. the Kuhn-Tucker condi tions ar'e necessary and sufficientconàitions for tile optimality. The fact that the matrixnegative ùefini te implies that problem (40) ilas a uniquep--~s•solution. and since the game has an equilibrium point whicnmust satisfy system D9) we conclude that the unique solutionof (40) gives the ~que solution of (39). which is the ~queequilibrium point of the game.Remark. The numerical solution of problem (40)obtained by standard methods (see G. Hadley (31).can De- 38 -Fina.l1y we remark that the statements of Lemma 12. andTheorem 8. can be extended for the multiproduct group equi1ibriumprob1em, but the detai1s ar'e not discussed here.