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.