Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

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3 Theoryalready presented in section 3.7, will enable us to estimate the parameter set, forming the basis ofthe considered HMM-SDE model.In this case we can stick to [16] and [17].In order to start, we can assume a Hidden Markov Model in the way that the system is arrangedin an initial state, and probabilistically jumps to other states (Markov jump process) which arehidden within the scope of any observation, while it is characterized by a diffusion process withinthese hidden states (metastable states). Thereby for every such state we have a system of SDEs ofover-damped Langevin type, in the following manner:with:• X: Random variable,• q(t): Markov jump process with states 1...M,Ẋ(t) = −∂ x U (q(t)) (x(t)) + σ (q(t)) Ẇ(t). (3.68)• W(t): Standard Brownian motion or Wiener process,• Σ = (σ (1) ,...,σ (M) ) contains noise intensities.• U = (U (1) ,...,U (M) ) contains interaction potentials.In the context of this work (in the framework of the rhodopsin-transducin-system) we can expectour system to have at least three hidden states:1. Diffusion of the transducin proteins in the solution.2. Diffusion of the transducin proteins on the membrane.3. Diffusion of the transducin proteins on the membrane, bound to rhodopsin proteins.Further assumptions, we should make are:1. Assumption:The Potentials U (q) are of harmonic form:while F (q)P2. Assumption:U (q) = 1 2 F(q) (x − µ(q)PP)2 +U (q)0 , (3.69)is the potential slope, U (q)0the initial value and µ (q)P the mean of the potential.Let us apply that the jump process is fixed to one state, q(t) = q.3.10.1 Propagation of the Probability DensityNow consider a statistical probability density function ρ(x,t) of an ensemble of solutions for theequation (3.68), and for different realizations of the stochastic process W(t). As we have seen38
3.10 Hidden Markov Models with Stochastic Differential Equation Output (HMM-SDE)earlier (see subsection 3.9.4), we can present for this purpose an equivalent representation of thedynamics in terms of the Fokker-Planck equation (see equation (3.60)):∂∂t ρ(x,t) = ∂ ∂x [ρ(x,t) ∗ ∂ ∂x U (q) (x)] + 1 ∂ 22 B(q) ρ(x,t), (3.70)∂x2 where B (q) = (σ (q) ) 2 ∈ R 1 denotes the variance of the white noise (for R d with d ∈ Z it is a positiveself-adjoint matrix).In terms of harmonic potentials like (3.69) this partial differential equation (PDE) can be solvedanalytically, whenever the initial density function can be represented as a superposition of Gaussiandistributions likeρ(x,t) = A(t)exp(−(x − µ(t)) · Σ · (x − µ(t)) T ), (3.71)where µ(t) denotes the time-dependent mean of the random variable X(t).This leads to a system of ordinary differential equations (ODE) for the time-dependent parameters{µ(t),Σ(t),A(t)}:˙µ = −F (q)P (x − µ(q) P ),˙Σ = 2F (q)P Σ − 2B(q) Σ 2 ,Ȧ= (F (q)P − B(q) Σ)A. (3.72)The explicit solution of this system of equations on the time-interval (t, t + τ) is given byµ(t + τ)Σ(t + τ)= µ (q)P= [F (q)P+ exp(−F(q) P τ)(µ(t) − µ(q) P ),−1B (q) − exp(−2F (q) −1B (q) − Σ −1 )] −1 ,pτ)(F (q)PA(t + τ) = 1 √ π√Σ(t + τ). (3.73)What we can see in (3.73) is that the time-dependent mean µ(t) converges to the potential meanfor big times τ.µ (q)PIn the case here, we are interested in the probability of output O(t j+1 ) in metastable state q t j+1,under the condition that the system has been in the state {O(t j ), q t j} at time t j . For this reason wecan use (3.73) with x(t j ) = O t jand Σ(t j ) −1 = 0 which leads to the output probability distributionρ(O t j+1|q t j,O t j) = A(t j+1 )exp[−(O t j+1− µ(t j+1 ))Σ(t j+1 )(O t j+1− µ(t j+1 )) T ], (3.74)with the solutions, given by the following relations:39
Page 1: Diploma ThesisDepartment for Theore
Page 5 and 6: AbstractAnomalous diffusion is a ub
Page 7 and 8: Zusammenfassung 1Anomale Diffusion
Page 9 and 10: Contents1 Motivation 12 Visual Tran
Page 11 and 12: List of Figures2.1 Anatomy of the e
Page 13 and 14: 1 Motivation„NEC FASCES, NEC OPES
Page 15 and 16: Theoretically we profit from enormo
Page 17 and 18: 2 Visual Transduction”The eye owe
Page 19 and 20: (a)(b)(c)Figure 2.3: (a) The anatom
Page 21 and 22: effect of generating the photorecep
Page 23 and 24: 3 Theory”The theory is the net, t
Page 25 and 26: 3.2 Markov Chains, Markov Processes
Page 31 and 32: 3.3 Hidden Markov ModelsP[X(t + τ)
Page 33 and 34: 3.4 Maximum Likelihood Principlewit
Page 35 and 36: 3.5 Optimization3.5 OptimizationAcc
Page 37 and 38: 3.7 Baum-Welch-AlgorithmAlgorithm 3
Page 39 and 40: 3.8 Two Approaches on Stochastic Sy
Page 41 and 42: 3.9 DiffusionConsider a basin fille
Page 43 and 44: 3.9 Diffusion169].Rearranging (3.34
Page 45 and 46: 3.9 Diffusionwith τ = t −t ′ .
Page 47 and 48: 3.9 DiffusionD = 2k2 B T 2σ . (3.4
Page 49: 3.10 Hidden Markov Models with Stoc
Page 53 and 54: 3.10 Hidden Markov Models with Stoc
Page 55 and 56: 3.11 Hidden Markov Model - Vector A
Page 57 and 58: 3.11 Hidden Markov Model - Vector A
Page 59 and 60: 3.12 Artificial Test Examples for H
Page 69 and 70: 3.13 Global Optimization Methods3.1
Page 71 and 72: 3.13 Global Optimization MethodsEve
Page 73 and 74: 4 Fluorescence Tracking Experiments
Page 75 and 76: 4.2 Fluorescence SpectroscopyAfter
Page 77 and 78: 4.3 Single Molecule Tracking via Wi
Page 79 and 80: 4.4 Total Internal Reflection Fluor
Page 81 and 82: 4.4 Total Internal Reflection Fluor
Page 83 and 84: 4.6 The expected range for the Tran
Page 85 and 86: 5 Modeling of the Experiment”The
Page 87 and 88: 5.1 Experimental Data(a)(b)(c)(d)Fi
Page 89 and 90: 5.2 Model Ansatz and Estimation of
Page 91 and 92: 5.2 Model Ansatz and Estimation of
Page 93 and 94: 5.3 Testing the Modelalgorithm was
Page 95 and 96: 5.5 Estimation of the Noise Intensi
Page 97 and 98: 5.6 Estimation on the Basis of Diff
Page 99 and 100: 5.6 Estimation on the Basis of Diff
Page 101 and 102:
5.6 Estimation on the Basis of Diff
Page 103 and 104:
6 Conclusion and OutlookThe main as
Page 105 and 106:
7 Bibliography[1] R. C. Aster, B. B
Page 107 and 108:
[36] C. U. M. Smith: Elements of Mo
Page 109 and 110:
Index11-cis retinal isomer, 87TM se
Page 111 and 112:
A AppendixA.1 From Copernicus to Ne
Page 113 and 114:
A.2 Important Papers on the Rhodops
Page 115 and 116:
A.3 Probability TheoryDefinition A.
Page 117 and 118:
A.4 Definitions for OptimizationDef
Page 119 and 120:
A.4 Definitions for OptimizationDef
Page 121 and 122:
A.7 The Fluctuation-Dissipation-The
Page 123 and 124:
A.7 The Fluctuation-Dissipation-The
Page 125 and 126:
A.8 Kramers-Moyal Forward Expansion
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A.9 Deriving the Fokker-Planck Equa
Page 129 and 130:
A.9 Deriving the Fokker-Planck Equa
Page 131 and 132:
DanksagungenIch möchte folgenden M
Page 133:
AffirmationHereby I, Arash Azhand,
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Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

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