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 easyto 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 HodgkinHuxley 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 nonleaky 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 twocompartment 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 lowpass and highpass filtering, respectively
 To be able to extract spike times as a threshold crossing
 To understand what a peristimulus 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 Spatiotemporal 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 Spatiotemporal Receptive Field and a Spectrotemporal 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, crosscorrelation, 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
 WienerKinchine Theorem
 ShannonNyquist 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
 Spectrograms
 Multitaper spectral analysis
 How to design the best tapers (DPSS)
 Controlling the timebandwith 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
 Twolayer feedforward networks
 Matrix algebra review
 Matrix transformations

Matrix Operations (PDF  4.0MB)

16

 More on twolayer feedforward 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 (winnertakeall)

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
 Winnertakeall networks
 Attractor networks for longterm memory (Hopfield model)
 Energy landscape
 Hopfield network capacity

Hopfield Networks (PDF  2.7MB)
