Biophysics of Computation - Information Processing in Single Neurons

Description

Neural network research often builds on the fiction that neurons are simple linear threshold units, completely neglecting the highly dynamic and complex nature of synapses, dendrites, and voltage-dependent ionic currents. Biophysics of Computation: Information Processing in Single Neurons challenges this notion, using richly detailed experimental and theoretical findings from cellular biophysics to explain the repertoire of computational functions available to single neurons. The author shows how individual nerve cells can multiply, integrate, or delay synaptic inputs and how information can be encoded in the voltage across the membrane, in the intracellular calcium concentration, or in the timing of individual spikes. Key topics covered include the linear cable equation; cable theory as applied to passive dendritic trees and dendritic spines; chemical and electrical synapses and how to treat them from a computational point of view; nonlinear interactions of synaptic input in passive and active dendritic trees; the Hodgkin-Huxley model of action potential generation and propagation; phase space analysis; linking stochastic ionic channels to membrane-dependent currents; calcium and potassium currents and their role in information processing; the role of diffusion, buffering and binding of calcium, and other messenger systems in information processing and storage; short- and long-term models of synaptic plasticity; simplified models of single cells; stochastic aspects of neuronal firing; the nature of the neuronal code; and unconventional models of sub-cellular computation. Biophysics of Computation: Information Processing in Single Neurons serves as an ideal text for advanced undergraduate and graduate courses in cellular biophysics, computational neuroscience, and neural networks, and will appeal to students and professionals in neuroscience, electrical and computer engineering, and physics. Table of contents : 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 B......Page 578 D......Page 579 E......Page 580 H......Page 581 L......Page 582 N......Page 583 P......Page 584 S......Page 585 Z......Page 587 Cover Contents Preface List of Symbols Introduction 1 The Membrane Equation 1.1 Structure of the Passive Neuronal Membrane 1.1.1 Resting Potential 1.1.2 Membrane Capacity 1.1.3 Membrane Resistance 1.2 A Simple RC Circuit 1.3 RC Circuits as Linear Systems 1.3.1 Filtering by RC Circuits 1.4 Synaptic Input 1.5 Synaptic Input Is Nonlinear 1.5.1 Synaptic Input, Saturation, and the Membrane Time Constant 1.5.2 Synaptic Interactions among Excitation and Shunting Inhibition 1.5.3 Gain Normalization in Visual Cortex and Synaptic Input 1.6 Recapitulation 2 Linear Cable Theory 2.1 Basic Assumptions Underlying One-Dimensional Cable Theory 2.1.1 Linear Cable Equation 2.2 Steady-State Solutions 2.2.1 Infinite Cable 2.2.2 Finite Cable 2.3 Time-Dependent Solutions 2.3.1 Infinite Cable 2.3.2 Finite Cable 2.4 Neuronal Delays and Propagation Velocity 2.5 Recapitulation 3 Passive Dendritic Trees 3.1 Branched Cables 3.1.1 What Happens at Branch Points? 3.2 Equivalent Cylinder 3.3 Solving the Linear Cable Equation for Branched Structures 3.3.1 Exact Methods 3.3.2 Compartmental Modeling 3.4 Transfer Resistances 3.4.1 General Definition 3.4.2 An Example 3.4.3 Properties of K[sub(ij)] 3.4.4 Transfer Resistances in a Pyramidal Cell 3.5 Measures of Synaptic Efficiency 3.5.1 Electrotonic Distance 3.5.2 Voltage Attenuation 3.5.3 Charge Attenuation 3.5.4 Graphical Morphoelectrotonic Transforms 3.6 Signal Delays in Dendritic Trees 3.6.1 Experimental Determination of T[sub(m)] 3.6.2 Local and Propagation Delays in Dendritic Trees 3.6.3 Dependence of Fast Synaptic Inputs on Cable Parameters 3.7 Recapitulation 4 Synaptic Input 4.1 Neuronal and Synaptic Packing Densities 4.2 Synaptic Transmission Is Stochastic 4.2.1 Probability of Synaptic Release p 4.2.2 What Is the Synaptic Weight? 4.3 Neurotransmitters 4.4 Synaptic Receptors 4.5 Synaptic Input as Conductance Change 4.5.1 Synaptic Reversal Potential in Series with an Increase in Conductance 4.5.2 Conductance Decreasing Synapses 4.6 Excitatory NMDA and Non-NMDA Synaptic Input 4.7 Inhibitory GABAergic Synaptic Input 4.8 Postsynaptic Potential 4.8.1 Stationary Synaptic Input 4.8.2 Transient Synaptic Input 4.8.3 Infinitely Fast Synaptic Input 4.9 Visibility of Synaptic Inputs 4.9.1 Input Impedance in the Presence of Synaptic Input 4.10 Electrical Gap Junctions 4.11 Recapitulation 5 Synaptic Interactions in a Passive Dendritic Tree 5.1 Nonlinear Interaction among Excitation and Inhibition 5.1.1 Absolute versus Relative Suppression 5.1.2 General Analysis of Synaptic Interaction in a Passive Tree 5.1.3 Location of the Inhibitory Synapse 5.1.4 Shunting Inhibition Implements a "Dirty" Multiplication 5.1.5 Hyperpolarizing Inhibition Acts Like a Linear Subtraction 5.1.6 Functional Interpretation of the Synaptic Architecture and Dendritic Morphology: AND-NOT Gates 5.1.7 Retinal Directional Selectivity and Synaptic Logic 5.2 Nonlinear Interaction among Excitatory Synapses 5.2.1 Sensitivity of Synaptic Input to Spatial Clustering 5.2.2 Cluster Sensitivity for Pattern Discrimination 5.2.3 Detecting Coincident Input from the Two Ears 5.3 Synaptic Microcircuits 5.4 Recapitulation 6 The Hodgkin-Huxley Model of Action Potential Generation 6.1 Basic Assumptions 6.2 Activation and Inactivation States 6.2.1 Potassium Current I[sub(K)] 6.2.2 Sodium Current I[sub(Na)] 6.2.3 Complete Model 6.3 Generation of Action Potentials 6.3.1 Voltage Threshold for Spike Initiation 6.3.2 Refractory Period 6.4 Relating Firing Frequency to Sustained Current Input 6.5 Action Potential Propagation along the Axon 6.5.1 Empirical Determination of the Propagation Velocity 6.5.2 Nonlinear Wave Propagation 6.6 Action Potential Propagation in Myelinated Fibers 6.7 Branching Axons 6.8 Recapitulation 7 Phase Space Analysis of Neuronal Excitability 7.1 The FitzHugh-Nagumo Model 7.1.1 Nullclines 7.1.2 Stability of the Equilibrium Points 7.1.3 Instantaneous Current Pulses: Action Potentials 7.1.4 Sustained Current Injection: A Limit Cycle Appears 7.1.5 Onset of Nonzero Frequency Oscillations: The Hopf Bifurcation 7.2 The Morris-Lecar Model 7.2.1 Abrupt Onset of Oscillations 7.2.2 Oscillations with Arbitrarily Small Frequencies 7.3 More Elaborate Phase Space Models 7.4 Recapitulation 8 Ionic Channels 8.1 Properties of Ionic Channels 8.1.1 Biophysics of Channels 8.1.2 Molecular Structure of Channels 8.2 Kinetic Model of the Sodium Channel 8.3 From Stochastic Channels to Deterministic Currents 8.3.1 Probabilistic Interpretation 8.3.2 Spontaneous Action Potentials 8.4 Recapitulation 9 Beyond Hodgkin and Huxley: Calcium and Calcium-Dependent Potassium Currents 9.1 Calcium Currents 9.1.1 Goldman-Hodgkin-Katz Current Equation 9.1.2 High-Threshold Calcium Current 9.1.3 Low-Threshold Transient Calcium Current 9.1.4 Low-Threshold Spike in Thalamic Neurons 9.1.5 N-Type Calcium Current 9.1.6 Calcium as a Measure of the Spiking Activity of the Neuron 9.2 Potassium Currents 9.2.1 Transient Potassium Currents and Delays 9.2.2 Calcium-Dependent Potassium Currents 9.3 Firing Frequency Adaptation 9.4 Other Currents 9.5 An Integrated View 9.6 Recapitulation 10 Linearizing Voltage-Dependent Currents 10.1 Linearization of the Potassium Current 10.2 Linearization of the Sodium Current 10.3 Linearized Membrane Impedance of a Patch of Squid Axon 10.4 Functional Implications of Quasi-Active Membranes 10.4.1 Spatio-Temporal Filtering 10.4.2 Temporal Differentiation 10.4.3 Electrical Tuning in Hair Cells 10.5 Recapitulation 11 Diffusion, Buffering, and Binding 11.1 Diffusion Equation 11.1.1 Random Walk Model of Diffusion 11.1.2 Diffusion in Two or Three Dimensions 11.1.3 Diffusion Coefficient 11.2 Solutions to the Diffusion Equation 11.2.1 Steady-State Solution for an Infinite Cable 11.2.2 Time-Dependent Solution for an Infinite Cable 11.2.3 Square-Root Relationship of Diffusion 11.3 Electrodiffusion and the Nernst-Planck Equation 11.3.1 Relationship between the Electrodiffusion Equation and the Cable Equation 11.3.2 An Approximation to the Electrodiffusion Equation 11.4 Buffering of Calcium 11.4.1 Second-Order Buffering 11.4.2 Higher Order Buffering 11.5 Reaction-Diffusion Equations 11.5.1 Experimental Visualization of Calcium Transients in Diffusion-Buffered Systems 11.6 Ionic Pumps 11.7 Analogy between the Cable Equation and the Reaction-Diffusion Equation 11.7.1 Linearization 11.7.2 Chemical Dynamics and Space and Time Constants of the Diffusion Equation 11.8 Calcium Nonlinearities 11.9 Recapitulation 12 Dendritic Spines 12.1 Natural History of Spines 12.1.1 Distribution of Spines 12.1.2 Microanatomy of Spines 12.1.3 Induced Changes in Spine Morphology 12.2 Spines only Connect 12.3 Passive Electrical Properties of Single Spines 12.3.1 Current Injection into a Spine 12.3.2 Excitatory Synaptic Input to a Spine 12.3.3 Joint Excitatory and Inhibitory Input to a Spine 12.3.4 Geniculate Spine Triad 12.4 Active Electrical Properties of Single Spines 12.5 Effect of Spines on Cables 12.6 Diffusion in Dendritic Spines 12.6.1 Solutions of the Reaction-Diffusion Equation for Spines 12.6.2 Imaging Calcium Dynamics in Single Dendritic Spines 12.7 Recapitulation 13 Synaptic Plasticity 13.1 Quantal Release 13.2 Short-Term Synaptic Enhancement 13.2.1 Facilitation Is an Increase in Release Probability 13.2.2 Augmentation and Posttetanic Potentiation 13.2.3 Synaptic Release and Presynaptic Calcium 13.3 Long-Term Synaptic Enhancement 13.3.1 Long-Term Potentiation 13.3.2 Short-Term Potentiation 13.4 Synaptic Depression 13.5 Synaptic Algorithms 13.5.1 Hebbian Learning 13.5.2 Temporally Asymmetric Hebbian Learning Rules 13.5.3 Sliding Threshold Rule 13.5.4 Short-Term Plasticity 13.5.5 Unreliable Synapses: Bug or Feature? 13.6 Nonsynaptic Plasticity 13.7 Recapitulation 14 Simplified Models of Individual Neurons 14.1 Rate Codes, Temporal Coding, and All of That 14.2 Integrate-and-Fire Models 14.2.1 Perfect or Nonleaky Integrate-and-Fire Unit 14.2.2 Forgetful or Leaky Integrate-and-Fire Unit 14.2.3 Other Variants 14.2.4 Response Time of Integrate-and-Fire Units 14.3 Firing Rate Models 14.3.1 Comparing the Dynamics of a Spiking Cell with a Firing Rate Cell 14.4 Neural Networks 14.4.1 Linear Synaptic Interactions Are Common to Almost All Neural Networks 14.4.2 Multiplicative Interactions and Neural Networks 14.5 Recapitulation 15 Stochastic Models of Single Cells 15.1 Random Processes and Neural Activity 15.1.1 Poisson Process 15.1.2 Power Spectrum Analysis of Point Processes 15.2 Stochastic Activity in Integrate-and-Fire Models 15.2.1 Interspike Interval Histogram 15.2.2 Coefficient of Variation 15.2.3 Spike Count and Fano Factor 15.2.4 Random Walk Model of Stochastic Activity 15.2.5 Random Walk in the Presence of a Leak 15.3 What Do Cortical Cells Do? 15.3.1 Cortical Cells Fire Randomly 15.3.2 Pyramidal Cells: Integrator or Coincidence Detector 15.3.3 Temporal Precision of Cortical Cells 15.4 Recapitulation 16 Bursting Cells 16.1 Intrinsically Bursting Cells 16.2 Mechanisms for Bursting 16.3 What Is the Significance of Bursting? 16.4 Recapitulation 17 Input Resistance, Time Constants, and Spike Initiation 17.1 Measuring Input Resistances 17.1.1 Membrane Chord Conductance 17.1.2 Membrane Slope Conductance 17.2 Time Constants for Active Systems 17.3 Action Potential Generation and the Question of Threshold 17.3.1 Current-Voltage Relationship 17.3.2 Stability of the Membrane Voltage 17.3.3 Voltage Threshold 17.3.4 Current Threshold 17.3.5 Charge Threshold 17.3.6 Voltage versus Current Threshold 17.4 Action Potential 17.5 Repetitive Spiking 17.5.1 Discharge Curve 17.5.2 Membrane Potential during Spiking Activity 17.6 Recapitulation 18 Synaptic Input to a Passive Tree 18.1 Action of a Single Synaptic Input 18.1.1 Unitary Excitatory Postsynaptic Potentials and Currents 18.1.2 Utility of Measures of Synaptic Efficacy 18.1.3 What Do Unitary EPSPs and EPSCs Tell Us about the Threshold? 18.2 Massive Synaptic Input 18.2.1 Relationship between Synaptic Input and Spike Output Jitter 18.2.2 Cable Theory for Massive Synaptic Input 18.3 Effect of Synaptic Background Activity 18.3.1 Input Resistance 18.3.2 Time Constant 18.3.3 Electroanatomy 18.3.4 Resting Potential 18.3.5 Functional Implications 18.4 Relating Synaptic Input to Output Spiking 18.4.1 Somatic Current from Distal Synaptic Input 18.4.2 Relating f[sub(out)] to f[sub(in)] 18.4.3 Functional Considerations 18.5 Shunting Inhibition Acts Linearly 18.6 Recapitulation 19 Voltage-Dependent Events in the Dendritic Tree 19.1 Experimental Evidence for Voltage-Dependent Dendritic Membrane Conductances 19.1.1 Fast Dendritic Spikes 19.2 Action Potential Initiation in Cable Structures 19.2.1 Effect of Dendritic Geometry on Spike Initiation 19.2.2 Biophysical Modeling of Antidromic Spike Invasion 19.3 Synaptic Input into Active Dendrites: Functional Considerations 19.3.1 Back-Propagating Spike as Acknowledgment Signal 19.3.2 Implementing Logic Computations with Spikes in Spines 19.3.3 Coincidence Detection with Dendritic Spikes 19.3.4 Nonlinear Spatial Synaptic Interactions Using Active Currents 19.3.5 Graded Amplification of Distal Synaptic Input 19.4 Recapitulation 20 Unconventional Computing 20.1 A Molecular Flip-Flop 20.1.1 Autophosphorylating Kinases 20.1.2 CaM Kinase II and Synaptic Information Storage 20.2 Extracellular Resources and Presynaptic Inhibition 20.3 Computing with Puffs of Gas 20.4 Programming with Peptides 20.5 Routing Information Using Neuromodulators 20.6 Recapitulation 21 Computing with Neurons: A Summary 21.1 Is a Biophysics of Computation Possible? 21.1.1 The Many Ways to Multiply 21.1.2 A Large Number of Biophysical Mechanisms for Computation 21.1.3 Can Different Biophysical Mechanisms Be Selected For? 21.2 Strategic Questions or How to Find a Topic for a Ph.D. Thesis Appendix A: Passive Membrane Parameters A.1 Intracellular Resistivity R[sub(i)] A.2 Membrane Resistance R[sub(m)] A.3 Membrane Capacitance C[sub(m)] Appendix B: A Miniprimer on Linear Systems Analysis Appendix C: Sparse Matrix Methods for Modeling Single Neurons C. 1 Linear Cable Equation C.1.1 Unbranched Cables and Tridiagonal Matrices C.1.2 Branched Cables and Hines Matrices C.1.3 Boundary Conditions C.1.4 Eigensystems and Model Fitting C.1.5 Green's Functions and Matrix Inverses C.2 Nonlinear Cable Equations C.2.1 Generalized Hodgkin-Huxley Equations C.2.2 Calcium Buffering C.2.3 Conclusion References Index A B C D E F G H I J K L M N O P Q R S T U V W Z