Contents

• • 1. Introduction
    • 1.1 Elements of Neuronal Systems
      • 1.1.1 The Ideal Spiking Neuron
      • 1.1.2 Spike Trains
      • 1.1.3 Synapses
    • 1.2 Elements of Neuronal Dynamics
      • 1.2.1 Postsynaptic Potentials
      • 1.2.2 Firing Threshold and Action Potential
    • 1.3 A Phenomenological Neuron Model
      • 1.3.1 Definition of the Model SRM₀
      • 1.3.2 Limitations of the Model
    • 1.4 The Problem of Neuronal Coding
    • 1.5 Rate Codes
      • 1.5.1 Rate as a Spike Count (Average over Time)
      • 1.5.2 Rate as a Spike Density (Average over Several Runs)
      • 1.5.3 Rate as a Population Activity (Average over Several Neurons)
    • 1.6 Spike Codes
      • 1.6.1 Time-to-First-Spike
      • 1.6.2 Phase
      • 1.6.3 Correlations and Synchrony
      • 1.6.4 Stimulus Reconstruction and Reverse Correlation
    • 1.7 Discussion: Spikes or Rates?
    • 1.8 Summary
  
  
• I. Single Neuron Models
  • 2. Detailed Neuron Models
    • 2.1 Equilibrium potential
      • 2.1.1 Nernst potential
      • 2.1.2 Reversal Potential
    • 2.2 Hodgkin-Huxley Model
      • 2.2.1 Definition of the model
      • 2.2.2 Dynamics
    • 2.3 The Zoo of Ion Channels
      • 2.3.1 Sodium Channels
      • 2.3.2 Potassium Channels
      • 2.3.3 Low-Threshold Calcium Current
      • 2.3.4 High-threshold calcium current and Ca²⁺-Activated Potassium Channels
      • 2.3.5 Calcium Dynamics
    • 2.4 Synapses
      • 2.4.1 Inhibitory Synapses
      • 2.4.2 Excitatory Synapses
    • 2.5 Spatial Structure: The Dendritic Tree
      • 2.5.1 Derivation of the Cable Equation
      • 2.5.2 Green's Function (*)
      • 2.5.3 Non-linear Extensions to the Cable Equation
    • 2.6 Compartmental Models
    • 2.7 Summary
  • 3. Two-Dimensional Neuron Models
    • 3.1 Reduction to two dimensions
      • 3.1.1 General approach
      • 3.1.2 Mathematical steps (*)
    • 3.2 Phase plane analysis
      • 3.2.1 Nullclines
      • 3.2.2 Stability of Fixed Points
      • 3.2.3 Limit cycles
      • 3.2.4 Type I and type II models
    • 3.3 Threshold and excitability
      • 3.3.1 Type I models
      • 3.3.2 Type II models
      • 3.3.3 Separation of time scales
    • 3.4 Summary
  • 4. Formal Spiking Neuron Models
    • 4.1 Integrate-and-fire model
      • 4.1.1 Leaky Integrate-and-Fire Model
      • 4.1.2 Nonlinear integrate-and-fire model
      • 4.1.3 Stimulation by Synaptic Currents
    • 4.2 Spike response model (SRM)
      • 4.2.1 Definition of the SRM
      • 4.2.2 Mapping the Integrate-and-Fire Model to the SRM
      • 4.2.3 Simplified Model SRM₀
    • 4.3 From Detailed Models to Formal Spiking Neurons
      • 4.3.1 Reduction of the Hodgkin-Huxley Model
      • 4.3.2 Reduction of a Cortical Neuron Model
      • 4.3.3 Limitations
    • 4.4 Multi-compartment integrate-and-fire model
      • 4.4.1 Definition of the Model
      • 4.4.2 Relation to the Model SRM₀
      • 4.4.3 Relation to the Full Spike Response Model (*)
    • 4.5 Application: Coding by Spikes
    • 4.6 Summary
  • 5. Noise in Spiking Neuron Models
    • 5.1 Spike train variability
      • 5.1.1 Are neurons noisy?
      • 5.1.2 Noise sources
    • 5.2 Statistics of spike trains
      • 5.2.1 Input-dependent renewal systems
      • 5.2.2 Interval distribution
      • 5.2.3 Survivor function and hazard
      • 5.2.4 Stationary renewal theory and experiments
      • 5.2.5 Autocorrelation of a stationary renewal process
    • 5.3 Escape noise
      • 5.3.1 Escape rate and hazard function
      • 5.3.2 Interval distribution and mean firing rate
    • 5.4 Slow noise in the parameters
    • 5.5 Diffusive noise
      • 5.5.1 Stochastic spike arrival
      • 5.5.2 Diffusion limit (*)
      • 5.5.3 Interval distribution
    • 5.6 The subthreshold regime
      • 5.6.1 Sub- and superthreshold stimulation
      • 5.6.2 Coefficient of variation C_V
    • 5.7 From diffusive noise to escape noise
    • 5.8 Stochastic resonance
    • 5.9 Stochastic firing and rate models
      • 5.9.1 Analog neurons
      • 5.9.2 Stochastic rate model
      • 5.9.3 Population rate model
    • 5.10 Summary
  
  
• II. Population Models
  • 6. Population Equations
    • 6.1 Fully Connected Homogeneous Network
    • 6.2 Density Equations
      • 6.2.1 Integrate-and-Fire Neurons with Stochastic Spike Arrival
      • 6.2.2 Spike Response Model Neurons with Escape Noise
      • 6.2.3 Relation between the Approaches
    • 6.3 Integral Equations for the Population Activity
      • 6.3.1 Assumptions
      • 6.3.2 Integral equation for the dynamics
    • 6.4 Asynchronous firing
      • 6.4.1 Stationary Activity and Mean Firing Rate
      • 6.4.2 Gain Function and Fixed Points of the Activity
      • 6.4.3 Low-Connectivity Networks
    • 6.5 Interacting Populations and Continuum Models
      • 6.5.1 Several Populations
      • 6.5.2 Spatial Continuum Limit
    • 6.6 Limitations
    • 6.7 Summary
  • 7. Signal Transmission and Neuronal Coding
    • 7.1 Linearized Population Equation
      • 7.1.1 Noise-free Population Dynamics (*)
      • 7.1.2 Escape noise (*)
      • 7.1.3 Noisy reset (*)
    • 7.2 Transients
      • 7.2.1 Transients in a Noise-Free Network
      • 7.2.2 Transients with Noise
    • 7.3 Transfer Function
      • 7.3.1 Signal Term
      • 7.3.2 Signal-to-Noise Ratio
    • 7.4 The Significance of a Single Spike
      • 7.4.1 The Effect of an Input Spike
      • 7.4.2 Reverse Correlation - the Significance of an Output Spike
    • 7.5 Summary
  • 8. Oscillations and Synchrony
    • 8.1 Instability of the Asynchronous State
    • 8.2 Synchronized Oscillations and Locking
      • 8.2.1 Locking in Noise-Free Populations
      • 8.2.2 Locking in SRM₀ Neurons with Noisy Reset (*)
      • 8.2.3 Cluster States
    • 8.3 Oscillations in reverberating loops
      • 8.3.1 From oscillations with spiking neurons to binary neurons
      • 8.3.2 Mean field dynamics
      • 8.3.3 Microscopic dynamics
    • 8.4 Summary
  • 9. Spatially Structured Networks
    • 9.1 Stationary patterns of neuronal activity
      • 9.1.1 Homogeneous solutions
      • 9.1.2 Stability of homogeneous states
      • 9.1.3 `Blobs' of activity: inhomogeneous states
    • 9.2 Dynamic patterns of neuronal activity
      • 9.2.1 Oscillations
      • 9.2.2 Traveling waves
    • 9.3 Patterns of spike activity
      • 9.3.1 Traveling fronts and waves (*)
      • 9.3.2 Stability (*)
    • 9.4 Robust transmission of temporal information
    • 9.5 Summary
  
  
• III. Models of Synaptic Plasticity
  • 10. Hebbian Models
    • 10.1 Synaptic Plasticity
      • 10.1.1 Long-Term Potentiation
      • 10.1.2 Temporal Aspects
    • 10.2 Rate-Based Hebbian Learning
      • 10.2.1 A Mathematical Formulation of Hebb's Rule
    • 10.3 Spike-Time Dependent Plasticity
      • 10.3.1 Phenomenological Model
      • 10.3.2 Consolidation of Synaptic Efficacies
      • 10.3.3 General Framework (*)
    • 10.4 Detailed Models of Synaptic Plasticity
      • 10.4.1 A Simple Mechanistic Model
      • 10.4.2 A Kinetic Model based on NMDA Receptors
      • 10.4.3 A Calcium-Based Model
    • 10.5 Summary
  • 11. Learning Equations
    • 11.1 Learning in Rate Models
      • 11.1.1 Correlation Matrix and Principal Components
      • 11.1.2 Evolution of synaptic weights
      • 11.1.3 Weight Normalization
      • 11.1.4 Receptive Field Development
    • 11.2 Learning in Spiking Models
      • 11.2.1 Learning Equation
      • 11.2.2 Spike-Spike Correlations
      • 11.2.3 Relation of spike-based to rate-based learning
      • 11.2.4 Static-Pattern Scenario
      • 11.2.5 Distribution of Synaptic Weights
    • 11.3 Summary
  • 12. Plasticity and Coding
    • 12.1 Learning to be Fast
    • 12.2 Learning to be Precise
      • 12.2.1 The Model
      • 12.2.2 Firing time distribution
      • 12.2.3 Stationary Synaptic Weights
      • 12.2.4 The Role of the Firing Threshold
    • 12.3 Sequence Learning
    • 12.4 Subtraction of Expectations
      • 12.4.1 Electro-Sensory System of Mormoryd Electric Fish
      • 12.4.2 Sensory Image Cancellation
    • 12.5 Transmission of Temporal Codes
      • 12.5.1 Auditory Pathway and Sound Source Localization
      • 12.5.2 Phase Locking and Coincidence Detection
      • 12.5.3 Tuning of Delay Lines
  
  
• Bibliography
• Index

© Cambridge University Press
This book is in copyright. No reproduction of any part of it may take place without the written permission of Cambridge University Press.