Computational Neuroscience

  1. Mathematical and Physical Foundations
    1. Linear Algebra
      1. Vectors
        1. Vector operations
          1. Vector spaces
            1. Basis vectors
            2. Matrices
              1. Matrix representation
                1. Matrix algebra
                2. Matrix operations
                  1. Addition and multiplication
                    1. Transpose and inverse
                    2. Eigenvalues
                      1. Characteristic equation
                        1. Eigenvalue problems
                        2. Eigenvectors
                          1. Eigenvector computation
                            1. Principal components
                            2. Diagonalization
                              1. Similarity transformations
                                1. Jordan normal form
                                2. Singular value decomposition
                                  1. SVD applications
                                    1. Dimensionality reduction
                                  2. Calculus
                                    1. Derivatives
                                      1. Partial derivatives
                                        1. Chain rule
                                          1. Gradient vectors
                                          2. Integrals
                                            1. Definite integrals
                                              1. Multiple integrals
                                              2. Vector calculus
                                                1. Divergence and curl
                                                  1. Line and surface integrals
                                                2. Differential Equations
                                                  1. Ordinary Differential Equations
                                                    1. First-order ODEs
                                                      1. Separable equations
                                                        1. Linear equations
                                                        2. Higher-order ODEs
                                                          1. Second-order linear equations
                                                            1. Systems of ODEs
                                                            2. Phase plane analysis
                                                              1. Nullclines
                                                                1. Fixed points
                                                                  1. Stability analysis
                                                                2. Partial Differential Equations
                                                                  1. Diffusion equation
                                                                    1. Heat equation
                                                                      1. Cable equation
                                                                      2. Wave equation
                                                                        1. Laplace equation
                                                                        2. Numerical Integration Methods
                                                                          1. Euler Method
                                                                            1. Forward Euler
                                                                              1. Backward Euler
                                                                              2. Runge-Kutta Methods
                                                                                1. Fourth-order Runge-Kutta
                                                                                  1. Adaptive step size
                                                                                  2. Stability and accuracy
                                                                                    1. Numerical stability
                                                                                      1. Convergence analysis
                                                                                  3. Probability and Statistics
                                                                                    1. Probability Distributions
                                                                                      1. Discrete distributions
                                                                                        1. Binomial distribution
                                                                                          1. Poisson distribution
                                                                                          2. Continuous distributions
                                                                                            1. Uniform distribution
                                                                                              1. Gaussian distribution
                                                                                                1. Exponential distribution
                                                                                              2. Bayesian Inference
                                                                                                1. Bayes' theorem
                                                                                                  1. Prior distributions
                                                                                                    1. Likelihood functions
                                                                                                      1. Posterior distributions
                                                                                                      2. Bayesian estimation
                                                                                                        1. Model comparison
                                                                                                        2. Stochastic Processes
                                                                                                          1. Markov processes
                                                                                                            1. Markov chains
                                                                                                              1. Transition matrices
                                                                                                              2. Random walks
                                                                                                                1. Brownian motion
                                                                                                                  1. Diffusion processes
                                                                                                                  2. Poisson processes
                                                                                                                    1. Point processes
                                                                                                                      1. Renewal processes
                                                                                                                    2. Statistical Inference
                                                                                                                      1. Hypothesis testing
                                                                                                                        1. Confidence intervals
                                                                                                                          1. Maximum likelihood estimation
                                                                                                                        2. Information Theory
                                                                                                                          1. Entropy
                                                                                                                            1. Shannon entropy
                                                                                                                              1. Differential entropy
                                                                                                                                1. Conditional entropy
                                                                                                                                2. Mutual Information
                                                                                                                                  1. Information gain
                                                                                                                                    1. Redundancy
                                                                                                                                    2. Information transmission in neurons
                                                                                                                                      1. Channel capacity
                                                                                                                                        1. Noise and information
                                                                                                                                        2. Coding theory
                                                                                                                                          1. Optimal codes
                                                                                                                                            1. Error correction
                                                                                                                                          2. Physics of Electrical Circuits
                                                                                                                                            1. Basic Circuit Elements
                                                                                                                                              1. Resistors
                                                                                                                                                1. Ohm's law
                                                                                                                                                  1. Power dissipation
                                                                                                                                                  2. Capacitors
                                                                                                                                                    1. Capacitance
                                                                                                                                                      1. Energy storage
                                                                                                                                                      2. Inductors
                                                                                                                                                        1. Inductance
                                                                                                                                                          1. Magnetic energy
                                                                                                                                                          2. Batteries
                                                                                                                                                            1. Voltage sources
                                                                                                                                                              1. Current sources
                                                                                                                                                            2. Circuit Laws
                                                                                                                                                              1. Ohm's Law
                                                                                                                                                                1. Kirchhoff's Laws
                                                                                                                                                                  1. Kirchhoff's current law
                                                                                                                                                                    1. Kirchhoff's voltage law
                                                                                                                                                                  2. Circuit Analysis
                                                                                                                                                                    1. RC circuits
                                                                                                                                                                      1. Charging and discharging
                                                                                                                                                                        1. Time constants
                                                                                                                                                                        2. RLC circuits
                                                                                                                                                                          1. Resonance
                                                                                                                                                                            1. Damping
                                                                                                                                                                            2. AC circuit analysis
                                                                                                                                                                              1. Impedance
                                                                                                                                                                                1. Phase relationships

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                                                                                                                                                                          4. Modeling Single Neurons