Cellular biophysics and modeling : a primer on the computational biology of excitable cells / Greg Conradi Smith. -- Cambridge ; New York, NY : Cambridge University Press, c2019. – (59.5922/S648)

Contents

Preface    xi

1  Introduction    1

1.1  Why Study Biophysics?   1

1.2  Neurons are Brain Cells    2

1.3  Cellular Biophysics    3

1.4  Dynamical Systems Modeling    5

1.5  Benefits and Limitations of Mathematical Models

1.6  Minimal Models and Graphical Methods    7

1.7  Biophysics and Dynamics Together    8

1.8  Discussion    9

Solutions   11

Notes    11

Part I  Models and Ordinary Differential Equations    13

2  Compartmental Modeling    15

2.1  Physical Dimensions and Material Balance    15

2.2  A Model of Intracellular Calcium Concentration    16

2.3  The Initial Value Problem and its Solution    17

2.4  Checking the Solution    19

2.5  Interpreting the Solution   19

2.6  Calcium Dynamics and Disease    22

2.7  Appendix    24

2.8  Discussion    25

Supplemental Problems    27

Solutions    33

Notes    39

3  Phase Diagrams    42

3.1  Phase Diagram for a Single Compartment Model   42

3.2  Stable and Unstable Steady States    44

3.3  Phase Diagram of a Nonlinear ODE    45

3.4  Classifying Steady States    47

3.5  Stability Analysis Requiring Higher Derivatives    49

3.6  Scalar ODEs with Multiple Stable Steady States   50

3.7  Discussion    51

Supplemental Problems    55

Solutions    57

Notes    58

4  Ligands, Receptors and Rate Laws    59

4.1  Mass Action Kinetics    59

4.2  Reaction Order and Physical Dimensions of Rate Constants    60

4.3  Isomerization - ODEs and a Conserved Quantity    61

4.4  Isomerization - Phase Diagram and Solutions    63

4.5  Bimolecular Association of Ligand and Receptor    65

4.6  Sequential Binding    69

4.7  Sigmoidal Binding Curves    70

4.8  Binding Curves and Hill Functions    72

4.9  Discussion    74

Supplemental Problems    75

Solutions   77

Notes   79

5  Function Families and Characteristic Times    81

5.1  Functions and Relations    81

5.2  Scaling and Shifting of Functions    82

5.3  Qualitative Analysis of Functions    84

5.4  Characteristic Times    88

5.5  Discussion    90

Supplemental Problems    93

Solutions    94

Notes   96

6  Bifurcation Diagrams of Scalar ODEs    98

6.1  A Single-Parameter Family of ODEs    98

6.2  Fold Bifurcation    99

6.3  Transcritical Bifurcation    101

6.4  Pitchfork Bifurcations    102

6.5  Bifurcation Types and Symmetry    105

6.6  Structural Stability   106

6.7  Further Reading    108

Supplemental Problems    109

Solutions    110

Notes   111

Part II  Passive Membranes    113

7  The Nernst Equilibrium Potential    115

7.1  Cellular Compartments and Electrical Potentials    115

7.2  Nernst Equilibrium Potential    116

7.3  Derivation of the Nernst Equation    119

7.4  Calculating Nernst Equilibrium Potentials    121

7.5  Chemical Potential    122

7.6  Discussion    124

Supplemental Problems    129

Solutions    130

Notes    130

8  The Current Balance Equation    132

8.1  Membrane Voltage    132

8.2  Ionic Fluxes and Currents    132

8.3  Ionic Currents and Voltage    133

8.4  Applied Currents and Voltage    134

8.5  The Current Balance Equation    135

8.6  Constitutive Relation for Ionic Membrane Current    137

8.7  The Phase Diagram for Voltage of Passive Membranes    139

8.8  Exponential Time Constant for Membrane Voltage    140

8.9  Discussion    143

Supplemental Problems    147

Solutions    149

Notes   153

9  GHK Theory of Membrane Permeation    154

9.1  Goldman-Hodgkin-Katz Theory - Assumptions    154

9.2  Physical Dimensions of the GHK Current Equation    155

9.3  The Goldman-Hodgkin-Katz Current Equation    156

9.4  Limiting Conductances Implied by GHK Theory    157

9.5  Derivation of the GHK Current Equation    159

9.6  Further Reading and Discussion    161

Supplemental Problems    164

Solutions    165

Notes    168

Part III  Voltage-Gated Currents    169

10 Voltage-Gated Ionic Currents    171

10.1  Voltage-Dependent Gating and Permeation Block    171

10.2  The L-Type Calcium Current Icav    173

10.3  The Inward Rectifying Potassium Current Ikir    176

10.4  The Hyperpolarization-Activated Cation Current/sag    177

10.5  The Depolarization-Activated Potassium Current Ikv    177

10.6  Qualitative Features of Current-Voltage Relations    179

10.7  Further Reading and Discussion    180

Supplemental Problems    181

Solutions    182

Notes    183

11  Regenerative Ionic Currents and Bistability    185

11.1  Regenerative Currents and Membrane Bistability    185

11.2  Response of a Bistable Membrane to Applied Current Pulses    188

11.3  Membrane Currents and Fold Bifurcations    188

11.4  Bifurcation Diagram for the Bistable Icav + IL Membrane    190

11.5  Overlaying Trajectories on the Bifurcation Diagram    191

11.6  Bistable Membrane Voltage Mediated by Ikir    191

11.7  Further Reading and Discussion    193

Supplemental Problems    197

Solutions    197

Notes   198

12  Voltage-Clamp Recording    199

12.1  Current-Clamp and Voltage-Clamp Recording    199

12.2  Modeling Delayed Activation of Ionic Currents    203

12.3  Voltage Clamp and Transient Ionic Currents    206

12.4  Modeling Transient Ionic Currents    209

12.5  Further Reading and Discussion    211

Supplemental Problems    213

Solutions    213

Notes   215

13  Hodgkin-Huxley Model of the Action Potential    216

13.1  The Squid Giant Axon    216

13.2  The Hodgkin-Huxley Model    219

13.3  Excitability in the Hodgkin-Huxley Model   221

13.4  Repetitive Spiking (Oscillations)   224

13.5  Further Reading and Discussion    225

Supplemental Problems    229

Solutions    230

Notes   230

Part IV  Excitability and Phase Planes    233

14  The Morris-Lecar Model    235

14.1  The Morris-Lecar Model    235

14.2  The Reduced Morris-Lecar Model    237

14.3  The Morris-Lecar Phase Plane    239

14.4  Phase Plane Analysis of Membrane Excitability    241

14.5  Phase Plane Analysis of Membrane Oscillations    244

14.6  Further Reading and Discussion    248

Supplemental Problems    249

Solutions    251

Notes    251

15  Phase Plane Analysis    252

15.1  The Phase Plane for Two-Dimensional Autonomous ODEs    252

15.2  Direction Fields of Two-Dimensional Autonomous ODEs    255

15.3  Nullclines for Two-Dimensional Autonomous ODEs    256

15.4  How to Sketch a Phase Plane    258

15.5  Phase Planes and Steady States    263

15.6  Discussion    265

Supplemental Problems    268

Solutions    269

Notes    273

16  Linear Stability Analysis    275

16.1  Solutions for Two-Dimensional Linear Systems    275

16.2  Real and Distinct Eigenvalues - Saddles and Nodes    278

16.3  Complex Conjugate Eigenvalues - Spirals    281

16.4  Criterion for Stability    284

16.5  Further Reading and Discussion    285

Supplemental Problems    290

Solutions    291

Notes    293

Part V  Oscillations and Bursting    295

17  Type II Excitability and Oscillations (Hopf Bifurcation)    297

17.1  Fitzhugh-Nagumo Model    297

17.2  Phase Plane Analysis of Resting Steady State   300

17.3  Loss of Stability with Increasing y (Depolarization)    303

17.4  Analysis of Hopf Bifurcations    304

17.5  Limit Cycle Fold Bifurcation    310

17.6  Further Reading and Discussion    313

Supplemental Problems    315

Solutions    316

Notes    317

18  Type I Excitability and Oscillations (SNIC and SHO Bifurcations)    319

18.1  Saddle-Node on an Invariant Circle    319

18.2  Saddle Homoclinic Bifurcation    323

18.3  Square-Wave Bursting    324

18.4  Calcium-Activated Potassium Currents as Slow Variable    328

18.5  Further Reading and Discussion    331

Supplemental Problems    335

Solutions    336

Note    337

19  The Low-Threshold Calcium Spike    338

19.1  Post-Inhibitory Rebound Bursting    338

19.2  Fast/Slow Analysis of Post-Inhibitory Rebound Bursting    342

19.3  Rhythmic Bursting in Response to Hyperpolarization    343

19.4  Fast/Slow Analysis of Rhythmic Bursting    344

19.5  Minimal Model of the Low-Threshold Calcium Spike    346

19.6  Further Reading and Discussion    349

Solutions    351

Notes    351

20  Synaptic Currents    353

20.1  Electrical Synapses    353

20.2  Electrical Synapses and Synchrony    355

20.3  Chemical Synapses    356

20.4  Phase Plane Analysis of Instantaneously Coupled Cells    357

20.5  Reciprocally Coupled Excitatory Neurons    362

20.6  Further Reading and Discussion    363

Supplemental Problems    365

Solutions    367

Note   367

Afterword    368

References    371

Index    380