Biophysics of Computation: Information Processing in Single Neurons

Cover......Page 1
Contents......Page 10
Preface......Page 20
List of Symbols......Page 22
Introduction......Page 26
1.1 Structure of the Passive Neuronal Membrane......Page 30
1.1.2 Membrane Capacity......Page 31
1.2 A Simple RC Circuit......Page 33
1.3.1 Filtering by RC Circuits......Page 37
1.4 Synaptic Input......Page 39
1.5 Synaptic Input Is Nonlinear......Page 44
1.5.2 Synaptic Interactions among Excitation and Shunting Inhibition......Page 45
1.6 Recapitulation......Page 48
2 Linear Cable Theory......Page 50
2.1 Basic Assumptions Underlying One-Dimensional Cable Theory......Page 51
2.1.1 Linear Cable Equation......Page 55
2.2.1 Infinite Cable......Page 57
2.2.2 Finite Cable......Page 58
2.3.1 Infinite Cable......Page 62
2.3.2 Finite Cable......Page 68
2.4 Neuronal Delays and Propagation Velocity......Page 69
2.5 Recapitulation......Page 72
3 Passive Dendritic Trees......Page 74
3.1.1 What Happens at Branch Points?......Page 78
3.2 Equivalent Cylinder......Page 80
3.3 Solving the Linear Cable Equation for Branched Structures......Page 83
3.3.2 Compartmental Modeling......Page 84
3.4 Transfer Resistances......Page 85
3.4.1 General Definition......Page 86
3.4.3 Properties of K[sub(ij)]......Page 87
3.4.4 Transfer Resistances in a Pyramidal Cell......Page 89
3.5.1 Electrotonic Distance......Page 91
3.5.2 Voltage Attenuation......Page 92
3.5.3 Charge Attenuation......Page 95
3.5.4 Graphical Morphoelectrotonic Transforms......Page 96
3.6.1 Experimental Determination of T[sub(m)]......Page 100
3.6.2 Local and Propagation Delays in Dendritic Trees......Page 102
3.6.3 Dependence of Fast Synaptic Inputs on Cable Parameters......Page 105
3.7 Recapitulation......Page 107
4 Synaptic Input......Page 110
4.2 Synaptic Transmission Is Stochastic......Page 112
4.2.1 Probability of Synaptic Release p......Page 114
4.2.2 What Is the Synaptic Weight?......Page 116
4.3 Neurotransmitters......Page 117
4.4 Synaptic Receptors......Page 119
4.5.1 Synaptic Reversal Potential in Series with an Increase in Conductance......Page 121
4.5.2 Conductance Decreasing Synapses......Page 123
4.6 Excitatory NMDA and Non-NMDA Synaptic Input......Page 124
4.7 Inhibitory GABAergic Synaptic Input......Page 130
4.8 Postsynaptic Potential......Page 131
4.8.1 Stationary Synaptic Input......Page 132
4.8.2 Transient Synaptic Input......Page 134
4.8.3 Infinitely Fast Synaptic Input......Page 135
4.9 Visibility of Synaptic Inputs......Page 136
4.10 Electrical Gap Junctions......Page 137
4.11 Recapitulation......Page 140
5 Synaptic Interactions in a Passive Dendritic Tree......Page 142
5.1.1 Absolute versus Relative Suppression......Page 143
5.1.2 General Analysis of Synaptic Interaction in a Passive Tree......Page 146
5.1.3 Location of the Inhibitory Synapse......Page 148
5.1.4 Shunting Inhibition Implements a \"Dirty\" Multiplication......Page 149
5.1.6 Functional Interpretation of the Synaptic Architecture and Dendritic Morphology: AND-NOT Gates......Page 153
5.1.7 Retinal Directional Selectivity and Synaptic Logic......Page 155
5.2.1 Sensitivity of Synaptic Input to Spatial Clustering......Page 157
5.2.2 Cluster Sensitivity for Pattern Discrimination......Page 161
5.3 Synaptic Microcircuits......Page 163
5.4 Recapitulation......Page 165
6 The Hodgkin-Huxley Model of Action Potential Generation......Page 167
6.1 Basic Assumptions......Page 169
6.2.1 Potassium Current I[sub(K)]......Page 171
6.2.2 Sodium Current I[sub(Na)]......Page 173
6.3 Generation of Action Potentials......Page 176
6.3.1 Voltage Threshold for Spike Initiation......Page 177
6.3.2 Refractory Period......Page 179
6.4 Relating Firing Frequency to Sustained Current Input......Page 181
6.5 Action Potential Propagation along the Axon......Page 184
6.5.1 Empirical Determination of the Propagation Velocity......Page 185
6.6 Action Potential Propagation in Myelinated Fibers......Page 189
6.7 Branching Axons......Page 193
6.8 Recapitulation......Page 196
7 Phase Space Analysis of Neuronal Excitability......Page 197
7.1 The FitzHugh-Nagumo Model......Page 198
7.1.1 Nullclines......Page 200
7.1.2 Stability of the Equilibrium Points......Page 201
7.1.3 Instantaneous Current Pulses: Action Potentials......Page 204
7.1.4 Sustained Current Injection: A Limit Cycle Appears......Page 206
7.1.5 Onset of Nonzero Frequency Oscillations: The Hopf Bifurcation......Page 208
7.2.1 Abrupt Onset of Oscillations......Page 210
7.2.2 Oscillations with Arbitrarily Small Frequencies......Page 211
7.4 Recapitulation......Page 216
8 Ionic Channels......Page 218
8.1 Properties of Ionic Channels......Page 219
8.1.1 Biophysics of Channels......Page 220
8.1.2 Molecular Structure of Channels......Page 223
8.2 Kinetic Model of the Sodium Channel......Page 225
8.3 From Stochastic Channels to Deterministic Currents......Page 227
8.3.1 Probabilistic Interpretation......Page 228
8.3.2 Spontaneous Action Potentials......Page 233
8.4 Recapitulation......Page 235
9 Beyond Hodgkin and Huxley: Calcium and Calcium-Dependent Potassium Currents......Page 237
9.1 Calcium Currents......Page 238
9.1.1 Goldman-Hodgkin-Katz Current Equation......Page 239
9.1.2 High-Threshold Calcium Current......Page 240
9.1.3 Low-Threshold Transient Calcium Current......Page 241
9.1.4 Low-Threshold Spike in Thalamic Neurons......Page 242
9.1.6 Calcium as a Measure of the Spiking Activity of the Neuron......Page 244
9.2.1 Transient Potassium Currents and Delays......Page 246
9.2.2 Calcium-Dependent Potassium Currents......Page 248
9.3 Firing Frequency Adaptation......Page 249
9.5 An Integrated View......Page 251
9.6 Recapitulation......Page 255
10 Linearizing Voltage-Dependent Currents......Page 257
10.1 Linearization of the Potassium Current......Page 258
10.2 Linearization of the Sodium Current......Page 261
10.3 Linearized Membrane Impedance of a Patch of Squid Axon......Page 264
10.4.2 Temporal Differentiation......Page 268
10.4.3 Electrical Tuning in Hair Cells......Page 269
10.5 Recapitulation......Page 270
11 Diffusion, Buffering, and Binding......Page 273
11.1.1 Random Walk Model of Diffusion......Page 274
11.1.2 Diffusion in Two or Three Dimensions......Page 276
11.1.3 Diffusion Coefficient......Page 277
11.2 Solutions to the Diffusion Equation......Page 278
11.2.2 Time-Dependent Solution for an Infinite Cable......Page 280
11.2.3 Square-Root Relationship of Diffusion......Page 281
11.3 Electrodiffusion and the Nernst-Planck Equation......Page 284
11.3.2 An Approximation to the Electrodiffusion Equation......Page 286
11.4 Buffering of Calcium......Page 288
11.4.1 Second-Order Buffering......Page 289
11.4.2 Higher Order Buffering......Page 290
11.5 Reaction-Diffusion Equations......Page 291
11.5.1 Experimental Visualization of Calcium Transients in Diffusion-Buffered Systems......Page 292
11.6 Ionic Pumps......Page 294
11.7 Analogy between the Cable Equation and the Reaction-Diffusion Equation......Page 295
11.7.1 Linearization......Page 296
11.7.2 Chemical Dynamics and Space and Time Constants of the Diffusion Equation......Page 297
11.8 Calcium Nonlinearities......Page 301
11.9 Recapitulation......Page 302
12 Dendritic Spines......Page 305
12.1.1 Distribution of Spines......Page 306
12.1.2 Microanatomy of Spines......Page 307
12.2 Spines only Connect......Page 310
12.3.1 Current Injection into a Spine......Page 311
12.3.2 Excitatory Synaptic Input to a Spine......Page 313
12.3.3 Joint Excitatory and Inhibitory Input to a Spine......Page 316
12.3.4 Geniculate Spine Triad......Page 318
12.4 Active Electrical Properties of Single Spines......Page 320
12.5 Effect of Spines on Cables......Page 323
12.6 Diffusion in Dendritic Spines......Page 325
12.6.1 Solutions of the Reaction-Diffusion Equation for Spines......Page 326
12.6.2 Imaging Calcium Dynamics in Single Dendritic Spines......Page 329
12.7 Recapitulation......Page 331
13 Synaptic Plasticity......Page 333
13.1 Quantal Release......Page 335
13.2.1 Facilitation Is an Increase in Release Probability......Page 337
13.2.2 Augmentation and Posttetanic Potentiation......Page 340
13.2.3 Synaptic Release and Presynaptic Calcium......Page 341
13.3.1 Long-Term Potentiation......Page 342
13.4 Synaptic Depression......Page 345
13.5 Synaptic Algorithms......Page 346
13.5.1 Hebbian Learning......Page 347
13.5.2 Temporally Asymmetric Hebbian Learning Rules......Page 348
13.5.4 Short-Term Plasticity......Page 349
13.6 Nonsynaptic Plasticity......Page 352
13.7 Recapitulation......Page 354
14 Simplified Models of Individual Neurons......Page 355
14.1 Rate Codes, Temporal Coding, and All of That......Page 356
14.2.1 Perfect or Nonleaky Integrate-and-Fire Unit......Page 360
14.2.2 Forgetful or Leaky Integrate-and-Fire Unit......Page 363
14.2.3 Other Variants......Page 364
14.2.4 Response Time of Integrate-and-Fire Units......Page 365
14.3 Firing Rate Models......Page 366
14.3.1 Comparing the Dynamics of a Spiking Cell with a Firing Rate Cell......Page 368
14.4.1 Linear Synaptic Interactions Are Common to Almost All Neural Networks......Page 370
14.4.2 Multiplicative Interactions and Neural Networks......Page 371
14.5 Recapitulation......Page 373
15 Stochastic Models of Single Cells......Page 375
15.1.1 Poisson Process......Page 377
15.1.2 Power Spectrum Analysis of Point Processes......Page 379
15.2 Stochastic Activity in Integrate-and-Fire Models......Page 380
15.2.1 Interspike Interval Histogram......Page 381
15.2.2 Coefficient of Variation......Page 383
15.2.3 Spike Count and Fano Factor......Page 385
15.2.4 Random Walk Model of Stochastic Activity......Page 387
15.2.5 Random Walk in the Presence of a Leak......Page 390
15.3.1 Cortical Cells Fire Randomly......Page 391
15.3.2 Pyramidal Cells: Integrator or Coincidence Detector......Page 393
15.3.3 Temporal Precision of Cortical Cells......Page 396
15.4 Recapitulation......Page 397
16.1 Intrinsically Bursting Cells......Page 399
16.2 Mechanisms for Bursting......Page 401
16.3 What Is the Significance of Bursting?......Page 402
16.4 Recapitulation......Page 404
17.1 Measuring Input Resistances......Page 406
17.1.2 Membrane Slope Conductance......Page 408
17.2 Time Constants for Active Systems......Page 410
17.3.2 Stability of the Membrane Voltage......Page 412
17.3.3 Voltage Threshold......Page 414
17.3.4 Current Threshold......Page 416
17.3.5 Charge Threshold......Page 417
17.4 Action Potential......Page 419
17.5.1 Discharge Curve......Page 420
17.5.2 Membrane Potential during Spiking Activity......Page 421
17.6 Recapitulation......Page 425
18 Synaptic Input to a Passive Tree......Page 427
18.1.1 Unitary Excitatory Postsynaptic Potentials and Currents......Page 428
18.1.2 Utility of Measures of Synaptic Efficacy......Page 430
18.1.3 What Do Unitary EPSPs and EPSCs Tell Us about the Threshold?......Page 431
18.2.1 Relationship between Synaptic Input and Spike Output Jitter......Page 432
18.2.2 Cable Theory for Massive Synaptic Input......Page 435
18.3 Effect of Synaptic Background Activity......Page 436
18.3.1 Input Resistance......Page 437
18.3.2 Time Constant......Page 439
18.3.3 Electroanatomy......Page 440
18.3.4 Resting Potential......Page 441
18.3.5 Functional Implications......Page 442
18.4.1 Somatic Current from Distal Synaptic Input......Page 443
18.4.2 Relating f[sub(out)] to f[sub(in)]......Page 445
18.4.3 Functional Considerations......Page 446
18.5 Shunting Inhibition Acts Linearly......Page 448
18.6 Recapitulation......Page 451
19.1 Experimental Evidence for Voltage-Dependent Dendritic Membrane Conductances......Page 453
19.1.1 Fast Dendritic Spikes......Page 454
19.2 Action Potential Initiation in Cable Structures......Page 455
19.2.1 Effect of Dendritic Geometry on Spike Initiation......Page 456
19.2.2 Biophysical Modeling of Antidromic Spike Invasion......Page 459
19.3.2 Implementing Logic Computations with Spikes in Spines......Page 462
19.3.3 Coincidence Detection with Dendritic Spikes......Page 464
19.3.4 Nonlinear Spatial Synaptic Interactions Using Active Currents......Page 467
19.3.5 Graded Amplification of Distal Synaptic Input......Page 471
19.4 Recapitulation......Page 475
20.1.1 Autophosphorylating Kinases......Page 477
20.1.2 CaM Kinase II and Synaptic Information Storage......Page 481
20.2 Extracellular Resources and Presynaptic Inhibition......Page 483
20.3 Computing with Puffs of Gas......Page 484
20.4 Programming with Peptides......Page 487
20.5 Routing Information Using Neuromodulators......Page 489
20.6 Recapitulation......Page 491
21 Computing with Neurons: A Summary......Page 494
21.1.1 The Many Ways to Multiply......Page 496
21.1.2 A Large Number of Biophysical Mechanisms for Computation......Page 498
21.1.3 Can Different Biophysical Mechanisms Be Selected For?......Page 499
21.2 Strategic Questions or How to Find a Topic for a Ph.D. Thesis......Page 502
A.1 Intracellular Resistivity R[sub(i)]......Page 506
A.2 Membrane Resistance R[sub(m)]......Page 507
A.3 Membrane Capacitance C[sub(m)]......Page 508
Appendix B: A Miniprimer on Linear Systems Analysis......Page 509
Appendix C: Sparse Matrix Methods for Modeling Single Neurons......Page 512
C.1.1 Unbranched Cables and Tridiagonal Matrices......Page 513
C.1.2 Branched Cables and Hines Matrices......Page 519
C.1.3 Boundary Conditions......Page 520
C.1.4 Eigensystems and Model Fitting......Page 521
C.1.5 Green\'s Functions and Matrix Inverses......Page 522
C.2.1 Generalized Hodgkin-Huxley Equations......Page 524
C.2.2 Calcium Buffering......Page 525
C.2.3 Conclusion......Page 526
References......Page 528
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