9.40 | Spring 2018 | Undergraduate

# Introduction to Neural Computation

## Lecture Notes

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

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)

Spring 2018
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