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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