Computational Neuroscience

  1. Modeling Single Neurons
    1. The Neuron as an Electrical Circuit
      1. Equivalent circuit model
        1. Membrane resistance
          1. Membrane capacitance
            1. Intracellular resistance
            2. Cable Theory
              1. The Cable Equation
                1. Derivation and assumptions
                  1. Steady-state solutions
                    1. Transient solutions
                    2. Solutions for simple geometries
                      1. Infinite cable
                        1. Semi-infinite cable
                          1. Finite cable
                          2. Length constant
                            1. Time constant
                            2. Passive Dendritic Trees
                              1. Signal attenuation
                                1. Electrotonic length
                                  1. Voltage attenuation
                                  2. Dendritic filtering
                                    1. Frequency response
                                      1. Signal integration
                                      2. Input resistance
                                        1. Dendritic democracy
                                      3. Simplified Neuron Models
                                        1. Leaky Integrate-and-Fire Model
                                          1. Model equations
                                            1. Membrane equation
                                              1. Threshold condition
                                              2. Threshold and reset mechanism
                                                1. Spike generation
                                                  1. Reset potential
                                                  2. Input-output relationships
                                                    1. f-I curves
                                                      1. Gain modulation
                                                    2. Quadratic Integrate-and-Fire Model
                                                      1. Nonlinear dynamics
                                                        1. Type I excitability
                                                          1. Bifurcation analysis
                                                          2. Exponential Integrate-and-Fire Model
                                                            1. Exponential spike initiation
                                                              1. Spike slope factor
                                                                1. Threshold dynamics
                                                                2. Adaptation currents
                                                                  1. Spike-frequency adaptation
                                                                    1. Burst firing
                                                                  2. Izhikevich Model
                                                                    1. Two-variable model
                                                                      1. Rich firing patterns
                                                                        1. Computational efficiency
                                                                      2. Biophysically Detailed Models
                                                                        1. The Hodgkin-Huxley Model
                                                                          1. Model equations
                                                                            1. Membrane current equation
                                                                              1. Ionic current equations
                                                                              2. Modeling Ion Channels
                                                                                1. Sodium channels
                                                                                  1. Activation and inactivation
                                                                                    1. Voltage dependence
                                                                                    2. Potassium channels
                                                                                      1. Delayed rectifier
                                                                                        1. Voltage dependence
                                                                                        2. Leak channels
                                                                                          1. Linear conductance
                                                                                        3. Gating Variables
                                                                                          1. Activation variable m
                                                                                            1. Inactivation variable h
                                                                                              1. Potassium variable n
                                                                                                1. Voltage dependence
                                                                                                  1. Time constants
                                                                                                    1. Steady-state values
                                                                                                    2. Voltage Clamp Experiments
                                                                                                      1. Ionic current isolation
                                                                                                        1. Conductance measurements
                                                                                                        2. Space Clamp Experiments
                                                                                                          1. Uniform voltage control
                                                                                                        3. Multi-Compartment Models
                                                                                                          1. Compartmental Modeling
                                                                                                            1. Dividing neurons into segments
                                                                                                              1. Compartment coupling
                                                                                                              2. Simulating complex morphologies
                                                                                                                1. Dendritic trees
                                                                                                                  1. Axonal arbors
                                                                                                                  2. NEURON simulator
                                                                                                                    1. Model specification
                                                                                                                      1. Simulation protocols
                                                                                                                    2. Ion Channel Kinetics
                                                                                                                      1. Markov models
                                                                                                                        1. State transitions
                                                                                                                          1. Rate constants
                                                                                                                          2. Single-channel recordings
                                                                                                                            1. Channel noise
                                                                                                                          3. Analysis of Firing Patterns
                                                                                                                            1. Firing Rate
                                                                                                                              1. Instantaneous firing rate
                                                                                                                                1. Time-averaged firing rate
                                                                                                                                  1. Population firing rate
                                                                                                                                  2. Interspike Interval Analysis
                                                                                                                                    1. ISI distributions
                                                                                                                                      1. Coefficient of variation
                                                                                                                                        1. Serial correlations
                                                                                                                                        2. Phase-Plane Analysis
                                                                                                                                          1. Nullclines
                                                                                                                                            1. V-nullcline
                                                                                                                                              1. Recovery variable nullcline
                                                                                                                                              2. Fixed points
                                                                                                                                                1. Stability analysis
                                                                                                                                                  1. Eigenvalue analysis
                                                                                                                                                  2. Limit cycles
                                                                                                                                                    1. Oscillatory behavior
                                                                                                                                                      1. Hopf bifurcations
                                                                                                                                                    2. Spike-Frequency Adaptation
                                                                                                                                                      1. Calcium-activated potassium channels
                                                                                                                                                        1. Sodium channel inactivation
                                                                                                                                                          1. Adaptation time scales
                                                                                                                                                          2. Bursting
                                                                                                                                                            1. Intrinsic bursting
                                                                                                                                                              1. Network-induced bursting
                                                                                                                                                                1. Burst mechanisms
                                                                                                                                                                2. Tonic Firing
                                                                                                                                                                  1. Regular spiking
                                                                                                                                                                    1. Fast spiking
                                                                                                                                                                      1. Chattering
                                                                                                                                                                      2. Bifurcation analysis
                                                                                                                                                                        1. Saddle-node bifurcations
                                                                                                                                                                          1. Hopf bifurcations
                                                                                                                                                                            1. Parameter space exploration

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                                                                                                                                                                        3. Mathematical and Physical Foundations

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                                                                                                                                                                        5. Synaptic Plasticity and Learning