Biophysics of Computation Information Processing in Single Neurons

by Koch, Christof

• ISBN13: 9780195181999
• ISBN10: 0195181999
• eBook ISBN(s): 9780190292850
• Format: Paperback
• Copyright: 2004-10-28
• Publisher: Oxford University Press
• More Book Details

Summary

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 Neuronschallenges 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 howinformation 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 dendritictrees; 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 andcomputer engineering, and physics.

Table of Contents

Preface	xix
List of Symbols	xxi
Introduction	1	(4)
1 The Membrane Equation	5	(20)
1.1 Structure of the Passive Neuronal Membrane	5	(3)
1.1.1 Resting Potential	6	(1)
1.1.2 Membrane Capacity	6	(2)
1.1.3 Membrane Resistance	8	(1)
1.2 A Simple RC Circuit	8	(4)
1.3 RC Circuits as Linear Systems	12	(2)
1.3.1 Filtering by RC Circuits	12	(2)
1.4 Synaptic Input	14	(5)
1.5 Synaptic Input Is Nonlinear	19	(4)
1.5.1 Synaptic Input, Saturation, and the Membrane Time Constant	20	(1)
1.5.2 Synaptic Interactions among Excitation and Shunting Inhibition	20	(3)
1.5.3 Gain Normalization in Visual Cortex and Synaptic Input	23	(1)
1.6 Recapitulation	23	(2)
2 Linear Cable Theory	25	(24)
2.1 Basic Assumptions Underlying One-Dimensional Cable Theory	26	(6)
2.1.1 Linear Cable Equation	30	(2)
2.2 Steady-State Solutions	32	(5)
2.2.1 Infinite Cable	32	(1)
2.2.2 Finite Cable	33	(4)
2.3 Time-Dependent Solutions	37	(7)
2.3.1 Infinite Cable	37	(6)
2.3.2 Finite Cable	43	(1)
2.4 Neuronal Delays and Propagation Velocity	44	(3)
2.5 Recapitulation	47	(2)
3 Passive Dendritic Trees	49	(36)
3.1 Branched Cables	53	(2)
3.1.1 What Happens at Branch Points?	53	(2)
3.2 Equivalent Cylinder	55	(3)
3.3 Solving the Linear Cable Equation for Branched Structures	58	(2)
3.3.1 Exact Methods	59	(1)
3.3.2 Compartmental Modeling	59	(1)
3.4 Transfer Resistances	60	(6)
3.4.1 General Definition	61	(1)
3.4.2 An Example	62	(1)
3.4.3 Properties of Kij	62	(2)
3.4.4 Transfer Resistances in a Pyramidal Cell	64	(2)
3.5 Measures of Synaptic Efficiency	66	(9)
3.5.1 Electrotonic Distance	66	(1)
3.5.2 Voltage Attenuation	67	(3)
3.5.3 Charge Attenuation	70	(1)
3.5.4 Graphical Morphoelectrotonic Transforms	71	(4)
3.6 Signal Delays in Dendritic Trees	75	(7)
3.6.1 Experimental Determination of τm	75	(2)
3.6.2 Local and Propagation Delays in Dendritic Trees	77	(3)
3.6.3 Dependence of Fast Synaptic Inputs on Cable Parameters	80	(2)
3.7 Recapitulation	82	(3)
4 Synaptic Input	85	(32)
4.1 Neuronal and Synaptic Packing Densities	87	(1)
4.2 Synaptic Transmission Is Stochastic	87	(5)
4.2.1 Probability of Synaptic Release ρ	89	(2)
4.2.2 What Is the Synaptic Weight?	91	(1)
4.3 Neurotransmitters	92	(2)
4.4 Synaptic Receptors	94	(2)
4.5 Synaptic Input as Conductance Change	96	(3)
4.5.1 Synaptic Reversal Potential in Series with an Increase in Conductance	96	(2)
4.5.2 Conductance Decreasing Synapses	98	(1)
4.6 Excitatory NMDA and Non-NMDA Synaptic Input	99	(6)
4.7 Inhibitory GABAergic Synaptic Input	105	(1)
4.8 Postsynaptic Potential	106	(5)
4.8.1 Stationary Synaptic Input	107	(2)
4.8.2 Transient Synaptic Input	109	(1)
4.8.3 Infinitely Fast Synaptic Input	110	(1)
4.9 Visibility of Synaptic Inputs	111	(1)
4.9.1 Input Impedance in the Presence of Synaptic Input	112	(1)
4.10 Electrical Gap Junctions	112	(3)
4.11 Recapitulation	115	(2)
5 Synaptic Interactions in a Passive Dendritic Tree	117	(25)
5.1 Nonlinear Interaction among Excitation and Inhibition	118	(14)
5.1.1 Absolute versus Relative Suppression	118	(3)
5.1.2 General Analysis of Synaptic Interaction in a Passive Tree	121	(2)
5.1.3 Location of the Inhibitory Synapse	123	(1)
5.1.4 Shunting Inhibition Implements a "Dirty" Multiplication	124	(4)
5.1.5 Hyperpolarizing Inhibition Acts Like a Linear Subtraction	128	(1)
5.1.6 Functional Interpretation of the Synaptic Architecture and Dendritic Morphology: AND-NOT Gates	128	(2)
5.1.7 Retinal Directional Selectivity and Synaptic Logic	130	(2)
5.2 Nonlinear Interaction among Excitatory Synapses	132	(6)
5.2.1 Sensitivity of Synaptic Input to Spatial Clustering	132	(4)
5.2.2 Cluster Sensitivity for Pattern Discrimination	136	(2)
5.2.3 Detecting Coincident Input from the Two Ears	138	(1)
5.3 Synaptic Microcircuits	138	(2)
5.4 Recapitulation	140	(2)
6 The Hodgkin-Huxley Model of Action Potential Generation	142	(30)
6.1 Basic Assumptions	144	(2)
6.2 Activation and Inactivation States	146	(5)
6.2.1 Potassium Current IK	146	(2)
6.2.2 Sodium Current INa	148	(3)
6.2.3 Complete Model	151	(1)
6.3 Generation of Action Potentials	151	(5)
6.3.1 Voltage Threshold for Spike Initiation	152	(2)
6.3.2 Refractory Period	154	(2)
6.4 Relating Firing Frequency to Sustained Current Input	156	(3)
6.5 Action Potential Propagation along the Axon	159	(5)
6.5.1 Empirical Determination of the Propagation Velocity	160	(4)
6.5.2 Nonlinear Wave Propagation	164	(1)
6.6 Action Potential Propagation in Myelinated Fibers	164	(4)
6.7 Branching Axons	168	(3)
6.8 Recapitulation	171	(1)
7 Phase Space Analysis of Neuronal Excitability	172	(21)
7.1 The FitzHugh-Nagumo Model	173	(12)
7.1.1 Nullclines	175	(1)
7.1.2 Stability of the Equilibrium Points	176	(3)
7.1.3 Instantaneous Current Pulses: Action Potentials	179	(2)
7.1.4 Sustained Current Injection: A Limit Cycle Appears	181	(2)
7.1.5 Onset of Nonzero Frequency Oscillations: The Hopf Bifurcation	183	(2)
7.2 The Morris-Lecar Model	185	(6)
7.2.1 Abrupt Onset of Oscillations	185	(1)
7.2.2 Oscillations with Arbitrarily Small Frequencies	186	(5)
7.3 More Elaborate Phase Space Models	191	(1)
7.4 Recapitulation	191	(2)
8 Ionic Channels	193	(19)
8.1 Properties of Ionic Channels	194	(6)
8.1.1 Biophysics of Channels	195	(3)
8.1.2 Molecular Structure of Channels	198	(2)
8.2 Kinetic Model of the Sodium Channel	200	(2)
8.3 From Stochastic Channels to Deterministic Currents	202	(8)
8.3.1 Probabilistic Interpretation	203	(5)
8.3.2 Spontaneous Action Potentials	208	(2)
8.4 Recapitulation	210	(2)
9 Beyond Hodgkin and Huxley: Calcium and Calcium-Dependent Potassium Currents	212	(20)
9.1 Calcium Currents	213	(8)
9.1.1 Goldman-Hodgkin-Katz Current Equation	214	(1)
9.1.2 High-Threshold Calcium Current	215	(1)
9.1.3 Low-Threshold Transient Calcium Current	216	(1)
9.1.4 Low-Threshold Spike in Thalamic Neurons	217	(2)
9.1.5 N-Type Calcium Current	219	(1)
9.1.6 Calcium as a Measure of the Spiking Activity of the Neuron	219	(2)
9.2 Potassium Currents	221	(3)
9.2.1 Transient Potassium Currents and Delays	221	(2)
9.2.2 Calcium-Dependent Potassium Currents	223	(1)
9.3 Firing Frequency Adaptation	224	(2)
9.4 Other Currents	226	(1)
9.5 An Integrated View	226	(4)
9.6 Recapitulation	230	(2)
10 Linearizing Voltage-Dependent Currents	232	(16)
10.1 Linearization of the Potassium Current	233	(3)
10.2 Linearization of the Sodium Current	236	(3)
10.3 Linearized Membrane Impedance of a Patch of Squid Axon	239	(4)
10.4 Functional Implications of Quasi-Active Membranes	243	(2)
10.4.1 Spatio-Temporal Filtering	243	(1)
10.4.2 Temporal Differentiation	243	(1)
10.4.3 Electrical Tuning in Hair Cells	243	(2)
10.5 Recapitulation	245	(3)
11 Diffusion, Buffering, and Binding	248	(32)
11.1 Diffusion Equation	249	(4)
11.1.1 Random Walk Model of Diffusion	249	(2)
11.1.2 Diffusion in Two or Three Dimensions	251	(1)
11.1.3 Diffusion Coefficient	252	(1)
11.2 Solutions to the Diffusion Equation	253	(6)
11.2.1 Steady-State Solution for an Infinite Cable	255	(1)
11.2.2 Time-Dependent Solution for an Infinite Cable	255	(1)
11.2.3 Square-Root Relationship of Diffusion	256	(3)
11.3 Electrodiffusion and the Nernst-Planck Equation	259	(4)
11.3.1 Relationship between the Electrodiffusion Equation and the Cable Equation	261	(1)
11.3.2 An Approximation to the Electrodiffusion Equation	261	(2)
11.4 Buffering of Calcium	263	(3)
11.4.1 Second-Order Buffering	264	(1)
11.4.2 Higher Order Buffering	265	(1)
11.5 Reaction-Diffusion Equations	266	(3)
11.5.1 Experimental Visualization of Calcium Transients in Diffusion-Buffered Systems	267	(2)
11.6 Ionic Pumps	269	(1)
11.7 Analogy between the Cable Equation and the Reaction-Diffusion Equation	270	(6)
11.7.1 Linearization	271	(1)
11.7.2 Chemical Dynamics and Space and Time Constants of the Diffusion Equation	272	(4)
11.8 Calcium Nonlinearities	276	(1)
11.9 Recapitulation	277	(3)
12 Dendritic Spines	280	(28)
12.1 Natural History of Spines	281	(4)
12.1.1 Distribution of Spines	281	(1)
12.1.2 Microanatomy of Spines	282	(3)
12.1.3 Induced Changes in Spine Morphology	285	(1)
12.2 Spines only Connect	285	(1)
12.3 Passive Electrical Properties of Single Spines	286	(9)
12.3.1 Current Injection into a Spine	286	(2)
12.3.2 Excitatory Synaptic Input to a Spine	288	(3)
12.3.3 Joint Excitatory and Inhibitory Input to a Spine	291	(2)
12.3.4 Geniculate Spine Triad	293	(2)
12.4 Active Electrical Properties of Single Spines	295	(3)
12.5 Effect of Spines on Cables	298	(2)
12.6 Diffusion in Dendritic Spines	300	(6)
12.6.1 Solutions of the Reaction-Diffusion Equation for Spines	301	(3)
12.6.2 Imaging Calcium Dynamics in Single Dendritic Spines	304	(2)
12.7 Recapitulation	306	(2)
13 Synaptic Plasticity	308	(22)
13.1 Quantal Release	310	(2)
13.2 Short-Term Synaptic Enhancement	312	(5)
13.2.1 Facilitation Is an Increase in Release Probability	312	(3)
13.2.2 Augmentation and Posttetanic Potentiation	315	(1)
13.2.3 Synaptic Release and Presynaptic Calcium	316	(1)
13.3 Long-Term Synaptic Enhancement	317	(3)
13.3.1 Long-Term Potentiation	317	(3)
13.3.2 Short-Term Potentiation	320	(1)
13.4 Synaptic Depression	320	(1)
13.5 Synaptic Algorithms	321	(6)
13.5.1 Hebbian Learning	322	(1)
13.5.2 Temporally Asymmetric Hebbian Learning Rules	323	(1)
13.5.3 Sliding Threshold Rule	324	(1)
13.5.4 Short-Term Plasticity	324	(3)
13.5.5 Unreliable Synapses: Bug or Feature?	327	(1)
13.6 Nonsynaptic Plasticity	327	(2)
13.7 Recapitulation	329	(1)
14 Simplified Models of Individual Neurons	330	(20)
14.1 Rate Codes, Temporal Coding, and All of That	331	(4)
14.2 Integrate-and-Fire Models	335	(6)
14.2.1 Perfect or Nonleaky Integrate-and-Fire Unit	335	(3)
14.2.2 Forgetful or Leaky Integrate-and-Fire Unit	338	(1)
14.2.3 Other Variants	339	(1)
14.2.4 Response Time of Integrate-and-Fire Units	340	(1)
14.3 Firing Rate Models	341	(4)
14.3.1 Comparing the Dynamics of a Spiking Cell with a Firing Rate Cell	343	(2)
14.4 Neural Networks	345	(3)
14.4.1 Linear Synaptic Interactions Are Common to Almost All Neural Networks	345	(1)
14.4.2 Multiplicative Interactions and Neural Networks	346	(2)
14.5 Recapitulation	348	(2)
15 Stochastic Models of Single Cells	350	(24)
15.1 Random Processes and Neural Activity	352	(3)
15.1.1 Poisson Process	352	(2)
15.1.2 Power Spectrum Analysis of Point Processes	354	(1)
15.2 Stochastic Activity in Integrate-and-Fire Models	355	(11)
15.2.1 Interspike Interval Histogram	356	(2)
15.2.2 Coefficient of Variation	358	(2)
15.2.3 Spike Count and Fano Factor	360	(2)
15.2.4 Random Walk Model of Stochastic Activity	362	(3)
15.2.5 Random Walk in the Presence of a Leak	365	(1)
15.3 What Do Cortical Cells Do?	366	(6)
15.3.1 Cortical Cells Fire Randomly	366	(2)
15.3.2 Pyramidal Cells: Integrator or Coincidence Detector	368	(3)
15.3.3 Temporal Precision of Cortical Cells	371	(1)
15.4 Recapitulation	372	(2)
16 Bursting Cells	374	(7)
16.1 Intrinsically Bursting Cells	374	(2)
16.2 Mechanisms for Bursting	376	(1)
16.3 What Is the Significance of Bursting?	377	(2)
16.4 Recapitulation	379	(2)
17 Input Resistance, Time Constants, and Spike Initiation	381	(21)
17.1 Measuring Input Resistances	381	(4)
17.1.1 Membrane Chord Conductance	383	(1)
17.1.2 Membrane Slope Conductance	383	(2)
17.2 Time Constants for Active Systems	385	(2)
17.3 Action Potential Generation and the Question of Threshold	387	(7)
17.3.1 Current-Voltage Relationship	387	(1)
17.3.2 Stability of the Membrane Voltage	387	(2)
17.3.3 Voltage Threshold	389	(2)
17.3.4 Current Threshold	391	(1)
17.3.5 Charge Threshold	392	(2)
17.3.6 Voltage versus Current Threshold	394	(1)
17.4 Action Potential	394	(1)
17.5 Repetitive Spiking	395	(5)
17.5.1 Discharge Curve	395	(1)
17.5.2 Membrane Potential during Spiking Activity	396	(4)
17.6 Recapitulation	400	(2)
18 Synaptic Input to a Passive Tree	402	(26)
18.1 Action of a Single Synaptic Input	403	(4)
18.1.1 Unitary Excitatory Postsynaptic Potentials and Currents	403	(2)
18.1.2 Utility of Measures of Synaptic Efficacy	405	(1)
18.1.3 What Do Unitary EPSPs and EPSCs Tell Us about the Threshold?	406	(1)
18.2 Massive Synaptic Input	407	(4)
18.2.1 Relationship between Synaptic Input and Spike Output Jitter	407	(3)
18.2.2 Cable Theory for Massive Synaptic Input	410	(1)
18.3 Effect of Synaptic Background Activity	411	(7)
18.3.1 Input Resistance	412	(2)
18.3.2 Time Constant	414	(1)
18.3.3 Electroanatomy	415	(1)
18.3.4 Resting Potential	416	(1)
18.3.5 Functional Implications	417	(1)
18.4 Relating Synaptic Input to Output Spiking	418	(5)
18.4.1 Somatic Current from Distal Synaptic Input	418	(2)
18.4.2 Relating ƒout to ƒin	420	(1)
18.4.3 Functional Considerations	421	(2)
18.5 Shunting Inhibition Acts Linearly	423	(3)
18.6 Recapitulation	426	(2)
19 Voltage-Dependent Events in the Dendritic Tree	428	(24)
19.1 Experimental Evidence for Voltage-Dependent Dendritic Membrane Conductances	428	(2)
19.1.1 Fast Dendritic Spikes	429	(1)
19.2 Action Potential Initiation in Cable Structures	430	(7)
19.2.1 Effect of Dendritic Geometry on Spike Initiation	431	(3)
19.2.2 Biophysical Modeling of Antidromic Spike Invasion	434	(3)
19.3 Synaptic Input into Active Dendrites: Functional Considerations	437	(13)
19.3.1 Back-Propagating Spike as Acknowledgment Signal	437	(1)
19.3.2 Implementing Logic Computations with Spikes in Spines	437	(2)
19.3.3 Coincidence Detection with Dendritic Spikes	439	(3)
19.3.4 Nonlinear Spatial Synaptic Interactions Using Active Currents	442	(4)
19.3.5 Graded Amplification of Distal Synaptic Input	446	(4)
19.4 Recapitulation	450	(2)
20 Unconventional Computing	452	(17)
20.1 A Molecular Flip-Flop	452	(6)
20.1.1 Autophosphorylating Kinases	452	(4)
20.1.2 CaM Kinase II and Synaptic Information Storage	456	(2)
20.2 Extracellular Resources and Presynaptic Inhibition	458	(1)
20.3 Computing with Puffs of Gas	459	(3)
20.4 Programming with Peptides	462	(2)
20.5 Routing Information Using Neuromodulators	464	(2)
20.6 Recapitulation	466	(3)
21 Computing with Neurons: A Summary	469	(12)
21.1 Is a Biophysics of Computation Possible?	471	(6)
21.1.1 The Many Ways to Multiply	471	(2)
21.1.2 A Large Number of Biophysical Mechanisms for Computation	473	(1)
21.1.3 Can Different Biophysical Mechanisms Be Selected For?	474	(3)
21.2 Strategic Questions or How to Find a Topic for a Ph.D. Thesis	477	(4)
Appendix A: Passive Membrane Parameters	481	(3)
A.1 Intracellular Resistivity Ri	481	(1)
A.2 Membrane Resistance Rm	482	(1)
A.3 Membrane Capacitance Cm	483	(1)
Appendix B: A Miniprimer on Linear Systems Analysis	484	(3)
Appendix C: Sparse Matrix Methods for Modeling Single Neurons	487	(16)
C.1 Linear Cable Equation	488	(11)
C.1.1 Unbranched Cables and Tridiagonal Matrices	488	(6)
C.1.2 Branched Cables and Hines Matrices	494	(1)
C.1.3 Boundary Conditions	495	(1)
C.1.4 Eigensystems and Model Fitting	496	(1)
C.1.5 Green's Functions and Matrix Inverses	497	(2)
C.2 Nonlinear Cable Equations	499	(4)
C.2.1 Generalized Hodgkin-Huxley Equations	499	(1)
C.2.2 Calcium Buffering	500	(1)
C.2.3 Conclusion	501	(2)
References	503	(50)
Index	553