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 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)