Lec # | Learning Objectives | Lecture Notes |
---|---|---|

1 | - To understand how the timescale of diffusion relates to length scales
- To understand how concentration gradients lead to currents (Fick’s First Law)
- To understand how charge drift in an electric field leads to currents (Ohm’s Law and resistivity)
| Overview and Ionic Currents (PDF - 1.7MB) |

2 | - To understand how neurons respond to injected currents
- To understand how membrane capacitance and resistance allows neurons to integrate or smooth their inputs over time (RC model)
- To understand how to derive the differential equations for the RC model
- To be able to sketch the response of an RC neuron to different current inputs
- To understand where the ‘batteries’ of a neuron come from
| RC Circuit and Nernst Potential (PDF - 2.7MB) |

3 | - To be able to construct a simplified model neuron by replacing the complex spike generating mechanisms of the real neuron (HH model) with a simplified spike generating mechanism
- To understand the processes that neurons spend most of their time doing which is integrating inputs in the interval between spikes
- To be able to create a quantitative description of the firing rate of neurons in response to current inputs
- To provide an easy-to implement model that captures the basic properties of spiking neurons
| Nernst Potential and Integrate and Fire Models (PDF - 4.1MB) |

4 | - To be able to draw the circuit diagram of the HH model
- Understand what a voltage clamp is and how it works
- Be able to plot the voltage and time dependence of the potassium current and conductance
- Be able to explain the time and voltage dependence of the potassium conductance in terms of Hodgkin-Huxley gating variables
| Hodgkin Huxley Model Part 1 (PDF - 6.3MB) |

5 | Hodgkin Huxley Model Part 2 (PDF - 3.3MB) | |

6 | - To be able to draw the ‘circuit diagram’ of a dendrite
- Be able to plot the voltage in a dendrite as a function of distance for leaky and non-leaky dendrite, and understand the concept of a length constant
- Know how length constant depends on dendritic radius
- Understand the concept of electrotonic length
- Be able to draw the circuit diagram a two-compartment model
| Dendrites (PDF - 3.2MB) |

7 | - Be able to add a synapse in an equivalent circuit model
- To describe a simple model of synaptic transmission
- To be able to describe synaptic transmission as a convolution of a linear kernel with a spike train
- To understand synaptic saturation
- To understand the different functions of somatic and dendritic inhibition
| Synapses (PDF - 3.1MB) |

8 | - To understand the origin of extracellular spike waveforms and local field potentials
- To understand how to extract local field potentials and spike signals by low-pass and high-pass filtering, respectively
- To be able to extract spike times as a threshold crossing
- To understand what a peri-stimulus time histogram (PSTH) and a tuning curve is
- To know how to compute the firing rate of a neuron by smoothing a spike train
| Spike Trains (PDF - 2.6MB) |

9 | - To be able to mathematically describe a neural response as a linear filter followed by a nonlinear function.
- A correlation of a spatial receptive field with the stimulus
- A convolution of a temporal receptive field with the stimulus
- To understand the concept of a Spatio-temporal Receptive Field (STRF) and the concept of ‘separability’
- To understand the idea of a Spike Triggered Average and how to use it to compute a Spatio-temporal Receptive Field and a Spectro-temporal Receptive Field (STRF).
| Receptive Fields (PDF - 2.1MB) |

10 | - Spike trains are probabilistic (Poisson Process)
- Be able to use measures of spike train variability
- Fano Factor
- Interspike Interval (ISI)
- Understand convolution, cross-correlation, and autocorrelation functions
- Understand the concept of a Fourier series
| Time Series (PDF - 4.5MB) |

11 | - Fourier series for symmetric and asymmetric functions
- Complex Fourier series
- Fourier transform
- Discrete Fourier transform (Fast Fourier Transform - FFT)
- Power spectrum
| Spectral Analysis Part 1 (PDF - 4.3MB) |

12 | - Fourier Transform Pairs
- Convolution Theorem
- Gaussian Noise (Fourier Transform and Power Spectrum)
- Spectral Estimation
- Filtering in the frequency domain
- Wiener-Kinchine Theorem
- Shannon-Nyquist Theorem (and zero padding)
- Line noise removal
| Spectral Analysis Part 2 (PDF - 3.1MB) |

13 | - Brief review of Fourier transform pairs and convolution theorem
- Spectral estimation
- Windows and Tapers
- Spectrograms
- Multi-taper spectral analysis
- How to design the best tapers (DPSS)
- Controlling the time-bandwith product
- Advanced filtering methods
| Spectral Analysis Part 3 (PDF - 2.2MB) |

14 | - Derive a mathematically tractable model of neural networks (the rate model)
- Building receptive fields with neural networks
- Vector notation and vector algebra
- Neural networks for classification
- Perceptrons
| Rate Models and Perceptrons (PDF - 3.9MB) |

15 | - Perceptrons and perceptron learning rule
- Neuronal logic, linear separability, and invariance
- Two-layer feedforward networks
- Matrix algebra review
- Matrix transformations
| Matrix Operations (PDF - 4.0MB) |

16 | - More on two-layer feed-forward networks
- Matrix transformations (rotated transformations)
- Basis sets
- Linear independence
- Change of basis
| Basis Sets (PDF - 2.8MB) |

17 | - Eigenvectors and eigenvalues
- Variance and multivariate Gaussian distributions
- Computing a covariance matrix from data
- Principal Components Analysis (PCA)
| Principal Components Analysis (PDF - 4.8MB) |

18 | - Mathematical description of recurrent networks
- Dynamics in simple autapse networks
- Dynamics in fully recurrent networks
- Recurrent networks for storing memories
- Recurrent networks for decision making (winner-take-all)
| Recurrent Networks (PDF - 2.2MB) |

19 | - Recurrent neural networks and memory
- The oculomotor system as a model of short term memory and neural integration
- Stability in neural integrators
- Learning in neural integrators
| Neural Integrators (PDF - 2.0MB) |

20 | - Recurrent networks with lambda greater than one
- Attractors
- Winner-take-all networks
- Attractor networks for long-term memory (Hopfield model)
- Energy landscape
- Hopfield network capacity
| Hopfield Networks (PDF - 2.7MB) |