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Contents

Abstract                                                               ii

Acknowledgments                                           iv

List of Figures                                                       viii

List of Tables                                                       ix

1 Introduction                                                      1

1.1  Motivation................................. 1

1.2  Neuronal Modeling Methods....................... 3

1.3  Model Reduction in Computational Neuroscience ........... 5

1.4  Implications for Network Simulations.................. 9

2 Linear Model Reduction                                       12

2.1  The Isopotential Cell........................... 12

2.2 The Quasi-Active Isopotential Cell ................... 15

2.3  Active and Quasi-Active Branched Neurons .............. 20

2.3.1  Nonlinear Cable Equation.................... 21

2.3.2  Linearizing the Cable Equation................. 24

2.4 Linear Model Reduction Techniques................... 31

2.4.1  Balanced Truncation....................... 31

2.4.2 Iterative Rational Krylov Algorithm .............. 34

2.5  Balanced Truncation Model Reduction Results............. 38

2.5.1 Dimension Reduction Ratio................... 38

2.5.2  Application to Synaptic Scaling................. 41

2.5.3 Application to Dendritic Resonance............... 45

2.6  IRKA Model Reduction Results..................... 46

2.7 A Quasi-Integrate-and-Fire Model.................... 49

2.7.1 Thresholding at the Soma.................... 50

2.7.2 Thresholding at Multiple Sites.................. 54

2.8  Discussion................................. 57

3 Nonlinear Model Reduction                                   59

3.1  Nonlinear Cable Equation........................ 60

3.2  The Reduced Cable Equation...................... 63

3.2.1 Proper Orthogonal Decomposition................ 63



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