UNIVERSITY OF CALIFORNIA, SAN DIEGO Fractional Diffusion: Numerical Methods and Applications in Neuroscience A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Bioengineering with a specialization in Multi-Scale Biology by Nirupama Bhattacharya Committee in charge: Professor Gabriel A. Silva, Chair Professor Henry Abarbanel Professor Gert Cauwenberghs Professor Todd Coleman Professor Marcos Intaglietta 2014 Copyright Nirupama Bhattacharya, 2014 All rights reserved. The dissertation of Nirupama Bhattacharya is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2014 iii DEDICATION Thank you to my dearest family and friends for all your love and support over the years. iv TABLE OF CONTENTS Signature Page . iii Dedication . iv Table of Contents . .v List of Figures . viii List of Tables . .x Acknowledgements . xi Vita ......................................... xii Abstract of the Dissertation . xiii Chapter 1 Introduction . .1 1.1 Fractional Calculus, Fractional Differential Equations, and Applications in Science and Engineering . .1 1.2 Fractional Diffusion . .2 Chapter 2 From Random Walk to the Fractional Diffusion Equation . .5 2.1 Introduction to the Random Walk Framework . .5 2.2 Einstein’s Formalism . .7 2.3 Continuous Time Random Walk and the Fractional Diffusion Equation . .9 2.4 Fundamental Solution . 15 Chapter 3 The Fractional Diffusion Equation and Adaptive Time Step Memory 18 3.1 Introduction . 18 3.2 The Grunwald-Letnikov¨ Derivative . 22 3.2.1 Derivation of the Fractional Diffusion Equation . 22 3.2.2 Definition of the Grunwald-Letnikov¨ Derivative . 24 3.2.3 Discretization of the Fractional Diffusion Equation . 28 3.3 Adaptive Time Step Memory as an Arithmetic Sequence . 30 3.3.1 Derivation . 30 3.3.2 Comparison to Short Memory Methods . 34 3.4 Adaptive Time Step as a Power Law . 34 3.4.1 Set Theoretic Implementation . 38 3.4.2 Numerical Implementation . 41 3.5 A Smart Adaptive Time Step Memory Algorithm . 42 v 3.5.1 Approximating the Discrete Grunwald-Letnikov¨ Se- ries as a Continuous Time Integral . 44 3.5.2 Extension of y(g;m) to the Positive Real Domain . 46 Chapter 4 Stability and Complexity Analyses of Finite Difference Algorithms for the Time-fractional Diffusion Equation . 51 4.1 Introduction . 51 4.2 Stability Analysis of the Two-dimensional Fractional FTCS Discretization . 54 4.2.1 Full Implementation . 54 4.2.2 Adaptive Memory Algorithm . 66 4.2.3 Stability Results . 74 4.3 Complexity Analysis . 76 4.3.1 Full Implementation . 76 4.3.2 Adaptive Memory Algorithm . 80 4.3.3 Linked List Adaptive Timestep Algorithm . 85 4.3.4 Empirical Results . 90 4.4 Error . 92 4.5 Conclusion . 92 Chapter 5 An Efficient Finite Difference Approach to Solving the Time- fractional Diffusion Equation . 95 5.1 Introduction . 95 5.2 Methods . 96 5.2.1 Crank-Nicholson in Time . 98 5.2.2 Time-splitting into Two One-dimensional Steps . 103 5.2.3 Adaptive Timestep . 108 5.3 Analysis . 109 5.3.1 Consistency . 109 5.3.2 Stability . 111 5.3.3 Order of Accuracy . 115 5.4 Results . 117 5.4.1 Accuracy . 117 5.4.2 Computational Time . 119 5.5 Conclusion . 120 Chapter 6 Fractional Diffusion of IP3 in the Induction of LTD . 122 6.1 Introduction . 122 6.2 Modified Model . 125 6.2.1 Calcium Dynamics . 126 6.2.2 IP3 Dynamics . 131 6.2.3 Calcium Buffers . 134 6.2.4 Parameters . 135 vi 6.2.5 A Note on Numerical Methodology . 137 6.3 Preliminary Results and Discussion . 138 6.3.1 Modified Model With Fractional Diffusion . 138 6.3.2 Modified Model With Regular Diffusion . 139 6.4 Conclusion . 143 Chapter 7 Conclusion and Future Directions . 144 7.1 Fractional Diffusion and Effects on LTD . 145 7.2 Calcium Signaling . 147 7.3 ATP Signaling in Astrocyte Networks . 148 Bibliography . 150 vii LIST OF FIGURES Figure 2.3.1: The shape of various Levy´ distributions in Fourier space. 11 Figure 2.3.2: The shape of various Levy´ distributions in Laplace space. 13 Figure 2.4.1: The fundamental solution for a set g = 0:75, at various times. 16 Figure 2.4.2: The fundamental solution for a set time = 1 second, for various g values. 17 Figure 3.2.1: Simulation results for g = 0:5; 0:75; 0:9; 1:0 (for traces from top to bottom) in one dimensional space (panel A) and time (panel B). As the subdiffusion regime is approached the profile becomes more and more hypergaussian. 30 Figure 3.3.1: Short memory and adaptive memory methods for estimating the the Grunwald-Letnikov¨ discretization. 35 Figure 3.3.2: Comparison of the error between adaptive memory and the short memory as a function of the calculation time. 36 Figure 3.4.1: Comparison of the full discretization, short memory approach, and adaptive memory approach, to the minimal memory implementation. 38 Figure 3.4.2: Overview of the use of linked list data structures for the algorithmic numerical implementation of adaptive memory time step as a power law scheme. See the text for details. 43 Figure 3.5.1: Computing a continuous version of y(g;m) for various values of g.. 47 Figure 4.2.1: Effect of a on Accuracy . 69 Figure 4.2.2: A)For a range of values for a, we plot X as a function of time step n. B,C) X for a = 5; a = 6. There are several obvious discontinuities in the pattern of the oscillations, which occur at the breaks between consecutive s intervals. 72 Figure 4.2.3: Stable Simulations in the Subdiffusion Region . 77 Figure 4.2.4: Unstable Simulations in the Subdiffusion Region . 78 Figure 4.2.5: Stable and Unstable Superdiffusion Simulations . 79 Figure 4.3.1: Empirical data verifying theoretical complexity results. 91 Figure 4.4.1: Error comparison between one-dimensional versions of all the ex- plicit algorithms discussed in Chapter 4. 93 Figure 5.2.1: The five-point stencil of the Crank-Nicholson algorithm applied to the fractional diffusion equation. 101 Figure 5.2.2: The schematic shows how the original Crank-Nicholson setup based on the five-point stencil, is split into two one-dimensional problems where at each step, we are solving a system of equations in one spatial direction only. 105 viii Figure 5.2.3: The values at the current and past real time steps are used to calculate the values at the next virtual timestep. This is in turn used to calculate the values at the next real time step. 106 Figure 5.3.1: The value of the history summation values in Eq. 5.2.15 and Eq. 5.2.21 is compared at A) uNx=2;Ny=2, B) uNx=4;Ny=4 , and C) uNx=8;Ny=8 112 Figure 5.3.2: Stability comparison for the full implementation and the CN-ADI Adaptive Memory method. 116 Figure 5.4.1: The figure shows the percentage error of the solution using the CN-ADI Adaptive Memory algorithm, compared to the full finite difference algorithm, at the center of the 2D grid, which is the most quickly changing part of the simulation. 118 Figure 5.5.1: Comparison of computational times as a function of simulation time, for various numerical algorithms for the fractional diffusion equation. 121 Figure 6.1.1: Schematic of the model developed by Hernjak et. al. 125 Figure 6.1.2: In the modified model presented in this chapter, we collapse every- thing into a single compartment that represents the dendrite along a one-dimensional domain. PF input, which occurs at a single spine, is applied to a single location along the one-dimensional axis. 126 Figure 6.2.1: The behavior of Rchannel as a function of IP3 and calcium concentra- tions. 130 Figure 6.2.2: Schematic of the components contributing to calcium dynamics, including diffusion, calcium buffers, transport across the ER mem- brane, and transport across the plasma membrane. 132 Figure 6.2.3: Schematic of the components contributing to IP3 dynamics, includ- ing a source term that represents the production of IP3 resulting from PF activation, degradation, and diffusion. 134 Figure 6.3.1: Preliminary results for the modified model with fractional diffusion, g = 0:7................................. 140 Figure 6.3.2: Preliminary results for the modified model with regular diffusion. 142 Figure 7.1.1: The IP3 diffusion profiles from the modified model when using fractional diffusion, and using regular diffusion, at t = 0:1 s..... 146 ix LIST OF TABLES Table 4.1: Error values between X and Xapprox for various values of a...... 71 Table 4.2: B(g;n) f ull and B(g;n;a)adap values as parameters a;n are varied. 73 Table 4.3: B f ull (g;n) and Badap (g;n;a)values as parameters a;n are varied. 73 Table 4.4: Linked list data for timestep n = 25 . 86 Table 6.1: Parameter values and initial conditions used in the modified model. 136 x ACKNOWLEDGEMENTS Chapter 3, in part, has been submitted for publication of the material as it may appear in the Journal of Computational Physics, 2015, Christopher L. MacDonald, Nirupama Bhattacharya, Brian P. Sprouse, Gabriel A. Silva. The dissertation author was a co-author of this paper. Chapter 4, in part, is currently being prepared for submission for publication of the material. Nirupama Bhattacharya and Gabriel A. Silva. The dissertation author was the primary investigator and author of this paper. Chapter 5, in part, is currently being prepared for submission for publication of the material.