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" ) The frequency locking (synchronization) fZ\Z$  ( "*"*"*"""bl""*"" b  " <3   $$$$$$        ! ;  !   5. Conclusion  We have developed a mathematical models of BP, HR and BT circadian oscillations using the coupled LC oscillators approach. Coupled LC oscillator-model can have variety of stationary and nonstationary solutions which depend on the coupling K, the limit cycle radius a and the frequency differencies "kj = |k  j|. Weakly coupled oscillators behave as independent units but with coupled phases. They are subjected each to the influence of the external disturbancies (Fext (t)) which can change circadian organization of the organism and become an important cause of morbidity. {ZZZZ Z  ( "*"*"*"""*"e""         ,    ~           " *1     d 70  `W  aX  Self-organization  Self-organization in biological systems relies on functional interactions between populations of structural units (molecules, cells, tissues, organs, or organisms). . Te$  Synchronization"  There are several types of synchronization : Phase synchronization (PS), Lag synchronization (LS), Complete synchronization (CS), and Generalized synchronization (GS) (usually observed in coupled chaotic systems)H043o.L                    IA2Relationship between entropy and self-organization33( 2zThe relationship between entropy and self-organization tries to relate organization to the 2nd Law of Thermodynamics order is a necessary result of far-from-equilibrium (dissipative) systems trying to maximise stress reduction. This suggests that the more complex the organism then the more efficient it is at dissipating potentials, a field of study sometimes called 'autocatakinetics' and related to what has been called 'The Law of Maximum Entropy Production'. Thus organization does not 'violate' the 2nd Law (as often claimed) but seems to be a direct result of it. D>Z[ u JBWhat are dissipative systems ?(   Systems that use energy flow to maintain their form are said to be dissipative (e.g. living systems ). Such systems are generally open to their environment. LO   6Biological signals    QI-Transport phenomena (an elementary approach) <.((( .  b jX = Xv X = X/V density V = SL volume L = vt jXS = X/t X = (mass, energy, momentum, charge, ...)    " * " "*"G"*M"n    [RJ,Transport phenomena (an elementary approach)>-((( -F Continuity equation "tX + div jX = 0 Transport equation jX = - X grad X X(from kinetic theory) ~ v! ! - mean free pathZZ#ZZTZ  ",*,",*,",",*,", " , " , * , " , * , " , * , " , * , " , * ,",  ", 4 SK&Transport phenomena (kinetic approach)4'(( ( '   The net flux through the middle plane in one direction is j = (j2  j1)/6 = -  grad  = v!/6;?C          ZbTL8Transport phenomena Mass, momentum, and energy transport&9$( 9Diffusion(mass transport) ,  UM8Transport phenomena Mass, momentum, and energy transport&9$( 9"Heat transver (energy transport) 6# 3L     VN8Transport phenomena Mass, momentum, and energy transport&9$( 9 Viscosity (momentum transport) ,! ɱ2    81 ENTROPY PRODUCTION At the very core of the second law of thermodynamics we find the basic distinction between  reversible and  irreversible processes (1). This leads ultimately to the introduction of entropy S and the formulation of the second law of thermodynamics. The classical formulation due to Clausius refers to isolated systems exchanging neither energy nor matter with the outside world. The second law then merely ascertains the existence of a function, the entropy S, which increases monotonically until it reaches its maximum at the state of thermodynamic equilibrium, (2.1) It is easy to extend this formulation to systems which exchange energy and matter with the outside world. (see fig. 2.1). Fig. 2.1. The exchange of entropy between the outside and the inside. P 92 To extend thermodynamics to non-equilibrium processes we need an explicit expression for the entropy production. Progress has been achieved along this line by supposing that even outside equilibrium entropy depends only on the same variables as at equilibrium. This is the assumption of  local equilibrium (2). Once this assumption is accepted we obtain for P, the entropy production per unit time, (2.3) : dtSi =  J F where the Jp are the rates of the various irreversible processes involved (chemical reactions, heat flow, diffusion. . .) and the F the corresponding generalized 266 Chemistry 1977 forces (affinities, gradients of temperature, of chemical potentials . . .). This is the basic formula of macroscopic thermodynamics of irreversible processes. P  "*"*RV  b` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>>  (    6  `}  T Click to edit Master title style! !  0  `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0@ ^ `  P*   0 ^   R*   0$Ɔ ^ `  R* H  0޽h ? 3380___PPT10.  Default Design 0 zrp (    0d P    P*    0     R*  d  c $ ?    0  0  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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MMM̙___PPT10i.rA+D=' b= @B +} b p$(  pr p S 2 `}   r p S 3 `  H p 0޽h ? 33___PPT10i.y08+D=' b= @B +} b p$(  r  S T9 `}   r  S (: `  H  0޽h ? 33___PPT10i.ྡ+D=' b= @B + b 40(  4x 4 c $hZ'g{   x 4 c $  H 4 0޽h ? 33___PPT10u..`+Z+D=' b= @B +* b *)044L )(  L~ L s *D `}   ~ L  BtCDELF$vd @5Nj )@@BL\gtto h+C@      "B   @cD L  |BCDEF@vd @((6HWg`F% *o Oh&2?KWpp(5{?oMid_QQXZ @             "B   MAR L  <BCDEpF0vd @@u1KeSA  g77{+&'@         "B    L  r;kBCDEF<vd @%%OuSeS\\Gq++) !4D N]htT!^ kjj{'R- \K_5@            "B   2F   L  BC@DEFDvd @++z:]%Vl S1|Gzz(;C5c@|<:4.,$@NqpqpyX?'xS}} @              "B   t   L <h@*' C1   L <(= C2   L <dR2W C3   L <`6lS C4   L <:\  C5   L <8S W:  C6  L BCDE4F J}RqWbbXxD'' @    E L BCDE4F PTnz& @    R[RG L BC2DE@F E2M&b^ S(^OY~v${  @     ]< L BC@DE(F 8+|(@'n., @   ` L  BC3DEF^vd @==kVlnY?#R1|!V1 %  p# 11\e+lnU""|"G #|  +31(8!,0@                    `"B    L BCpDE@F  z%t1t?nJ`dFw=;;/Mp @     0 L <kx< C1  L  ~B=CDEFBvd @((3V4<7&i : j{$2U|@Xr=74a  3 @             `"B    2 L 0c s  ES XB L 0DpS XB L 0D"XB L 0DXB L 0D}-LB L c $DCCXB L 0DjC^R L 6GVHIV -c dR L <ZGHӔI  db L@ <ZG&HIp~s -   L <s MZ C7 2 !L 6 , E7 2 "L 6| SF  E3 2 #L 6D &C E2 2 $L 6 @ E1 2 %L 6  )F  E6 2 &L 6VC E5 2 'L 6 )@ E4 LB (L c $D@CCLB )L c $DVLB *L c $D@ LB +L@ c $D@ LB ,L c $D S) RB -L s *D I ^b .L 6GuHKIh,I /L <Ԕm=  P Graph theory   0L <c Vg 34$   1L <   FGraph with weighted edges ! network<$"  $ 2L <Ԕ9 K RNetwork theory  3L <"  IGraph  4L <&  G = {gij} connectivity matrix$0   H L 0޽h ?OLLL LLLLLL &L!L.L 33___PPT10i.jc@"J+D=' b= @B + b D0(  Dx D c $P.'g   x D c $$// `  H D 0޽h ? 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ZXdbmo___PPT10i.}`,~+D=' b= @B + b ^(  ~  s *<ג `}    0 TA  ?z   H  0޽h ? 3333̙3___PPT10i.}P)+D=' b= @B +U b l d |(  | | c $ܒ@}   ,Entropy production in a driven LC oscillator.-((  - | c $/%   Biological systems are generically out of equilibrium. In an environment with constant temperature the source of non-equilibrium are usually mechanical (external forces) or chemical (imbalanced reactions) stimuli with stochastic character of the non-equilibrium processes.H7b833 B | 0d% s  Stochastic (Langevin) description of a driven LC oscillator representing stochastic trajectory (dts(t), s(t))) in (r, )-phase space@d n;(        | | s *\ 7 " dtr(t) = (a2  r2)r + (t), dt(t) =  (t) Gaussian white noise < (t) (t )> = 2d(t  t ) u    "  "  P              H | 0޽h ? 33___PPT10i.'+D=' b= @B + b $ (    c $9   jNon- equilibrium entropy J  S BC.DEFbAA@@%5 [rq6Ieh@wI @d2w%n([.[.7+-% wwqqdedeT3f\@.,$@%  09v[eLM%.0@                     `Granitec"$  wR XB  0D3Ԕ}XB @ 0D3ԔGS   < vdtSe6    <$% p  dtSiF33 3333 33 v   6h03 '  *dtS = dtSi + dtSe e" 0 4     &   67  $Si(t) = - +"rdr p(r,t) lnp(r,t) a" <si(t)> si(t) = - lnp(r,t), dtse(t) = dtq(t)/ T = (a2  r2)r dtr, D = T p(r,t) is the probability to find the LCO in the state r p(r,t) is the solution of the the Fokker-Planck equation with a given initial condition p(r,0) = p0(r)t "*"*"  "*"*"* "*"*"* " (              '     5  d   6DR  h"tp(r,t) = - "rj(r,t) = - "r[(a2  r2)r - D"r]p(r,t)5"* "* "*"*"*"*"Z        H  0޽h ? 33___PPT10i.m>+D=' b= @B +?b VN (  4  C xcv @33 "`    M  r  S dd `    H  0޽h ? 33___PPT10i.hu+D=' b= @B +b `0(  `x ` c $$j,Tt  x ` c $jm  H ` 0޽h ? 33___PPT10i.d01+D=' b= @B +b d0(  dx d c $+B#style.visibility<*%(D' =-s6Bcircle(in)*<3<*D' b=%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bcircle(in)*<3<*Dd' b=%(D ' =%(D' =4@BBBB%(D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D' =+4 8?`CB ppt_yBCB1+ppt_h/2B*Y3>B ppt_y<*D' =1:Bhidden*o3>+B#style.visibility<*%(Dd' b=%(D ' =%(D' =4@BBBB%(D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D' =+4 8?`CB ppt_yBCB1+ppt_h/2B*Y3>B ppt_y<*D' =1:Bhidden*o3>+B#style.visibility<*%(Dd' b=%(D ' =%(D' =4@BBBB%(D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D' =+4 8?`CB ppt_yBCB1+ppt_h/2B*Y3>B ppt_y<*D' =1:Bhidden*o3>+B#style.visibility<*%(Dd' b=%(D ' =%(D' =4@BBBB%(D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D' =+4 8?`CB ppt_yBCB1+ppt_h/2B*Y3>B ppt_y<*D' =1:Bhidden*o3>+B#style.visibility<*%(D{' b=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D'  =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D$' b=%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*+p+0+ ++0+ +$ b @$(  r  S 4W `}   r  S `< `  H  0޽h ? 3380___PPT10.iO$ b P$(  r  S ı `}   r  S hű `  H  0޽h ? 3380___PPT10.iӕP b p\0(  \x \ c $tʱ `}   x \ c $H˱ `  H \ 0޽h ? ZXdbmo___PPT10i.ZxP+D=' b= @B + b @6(  @~ @ s *б `}   x @ c $ѱ  H @ 0޽h ? 3_j___PPT10i.u@ +D=' b= @B +} b $$(  $r $ S ڱ `}   r $ S ۱ `   H $ 0޽h ? 33___PPT10i.͌:+D=' b= @B +} b ($(  (r ( S  `}   r ( S  `   H ( 0޽h ? 33___PPT10i.Ό@"+D=' b= @B +$ b l(  l~ l s *H `}    l 0@d ,$D  0 0D Every living cell, organ, or organism generates signals for internal and external communication. In-out relationship is generated by a biological process (electrochemical, mechanical, biochemical or hormonal). The received signal is usually very distorted by the transmission channel in the body. 2 6 3 J 8 ( 6 1 EH l 0޽h ? 3F433  ___PPT10l .o +x wD@ ' b= @B D' = @BA?%,( < +O%,( < +Dn' b=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l2%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*l2D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*l2Dn' b=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l2h%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*l2hD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*l2hDn' b=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*lh%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*lhD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*lhDn' b=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*lD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*lDn' b=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*lD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*lDn' b=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*lD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*lDn' b=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*lE%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*lED' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*lE+ b Hc(  H~ H s *B   ~ H s *h `   H BC{DEF{zvq@ c"$ `w  H ZBChDExF^^;8r]66j>V\+%1&8BVevtbL6!-2n;_@UOGU.`fosx|s&?Ob9ojIdT`;5q+}!VG\.A.JHRb[o_hh\_&AD@                               S"  P H (BCDEF<## qrSH8,  &:zHrgZo ||ZZZZ\<$ @             LB H@ c $DS  LB H@ c $D  LB  H@ c $D LB  H@ c $DC X  H 0BF 7 RB  H s *D%    H <u  ES  H <b   TjX$  XB H 0D 9  H < M6  EL ^B H 6D1:JS  H <lkJ Ev j  I H# #"  I H <(? I X   @``B H 01 ? I`B H 01 ? I`B H 01 ?  `B H 01 ?II H <*% E  EX  H <4/@ ` eCurrent density (flux):   H H 0޽h ? 33___PPT10i.  ,1+D=' b= @B + b  L<(  L~ L s *B `}   ~ L s *DC  H L 0޽h ? 33___PPT10i.B+D=' b= @B +  b .&0P(  P~ P s *Q `}   ~ P s *Rp `  Xr P 0 Xr P 0gd Xr P 0  XB P 0D`pXB P 0Dp S X  P 0`(X  P 0x7A J E   P <dV H!"   P <[@   H!" (  P <PZ   j2 = v(r - !)Z """"2 P <g;[ j1 = v(r + !) b " H P 0޽h ? 33___PPT10i.@Q+D=' b= @B +m b |@T(  T~ T s *hv `}   ~ T s *Ly `  RB T s *Dodd RB T s *DjJ`jj R T s *3 d d T <G=?H~3 j 8 T <{C"?#   jD = v[C(x - !)  C(x + !)] / 6 = v( - 2 ! "x C(x)) / 6 jD = - D "xC(x) D = v ! / 3\ "  (((( (((                  T <V wv  H!" RB  T s *D2gg9   T <V tv  F! 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"2 Driven Oscillators   .--. @"Arial-2 Mladen .-.@"Arial-2 Martinis, .-.@"Arial-2 Vesna.-.@"Arial-2 Mikuta .-.@"Arial- 2 9-.-.@"Arial-2 =Martinis .-.@"Arial- 2 zRu.-..@"Arial- 2 .-..@"Arial- 2 er.@"Arial-..@"Arial-2 Bokovi.-..@"Arial-2 InstituteP.@"Arial-.@"Arial- 2 , .-.@"Arial-12 Theoretical Physics Division.-.@"Arial-2 Zagreb, Croatias.--.@"Arial--2 ].MATH/CHEM/COMP, Dubrovnik  .-.@"Arial- 2 ]-.-.@"Arial- 2 ]2006.-Z՜.+,D՜.+,    On-screen ShowgHome ComputergE0 8Arial Wingdings굴림SymbolDefault DesignMicrosoft Equation 3.0Microsoft Office Excel ChartGEntropy Production in a System of Coupled Nonlinear Driven Oscillators=Nonequilibrium thermodynamics of complex biological networksSlide 3Equation of balanceSlide 56 1st law of thermodynamics (Energy balance equation)Slide 72nd law of thermodynamicsCoupled oscillators Biological oscillators Biological network3 Biological homeostasis (Dynamic self-regulation)  What is feedback ? LNegative feedback control stabilizes the system (It is a nonlinear process) Coupled nonlinear oscillators Coupled nonlinear oscillators Coupled nonlinear oscillatorsComplex phase space5Limit Cycle Oscillator(LCO) negative feedback effectLimit cycle propertyOscillating signalLimit cycle in the z - plane-Entropy production in a driven LC oscillatorNon- equilibrium entropy ApplicationsBiological Rhythms (BRs)Biological Clocks (BCs)4Modelling circadian rhythmus as coupled oscillatorsThree coupled oscillators Coupled Limit Cycle Oscillators Consequences Conclusion Slide 33 Slide 34 Slide 35Self-organizationSynchronization3Relationship between entropy and self-organizationWhat are dissipative systems ?Biological signals .Transport phenomena (an elementary approach) -Transport phenomena (an elementary approach)'Transport phenomena (kinetic approach)9Transport phenomena Mass, momentum, and energy transport9Transport phenomena Mass, momentum, and energy transport9Transport phenomena Mass, momentum, and energy transport Slide 47 Slide 48  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles0 8@ _PID_HLINKSA=http://en.wikipedia.org/wiki/Open_system_%28system_theory%29&http://en.wikipedia.org/wiki/Organism1http://en.wikipedia.org/wiki/Dynamic_equilibrium%_ Dean MartinisDean Martinis  !"#$%&'()+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)PicturesRCurrent UserSummaryInformation(PowerPoint Document(*DocumentSummaryInformation8