COURSE SYLLABUS
Computational Neuroscience
Neurl UA 302
Psych UA 300
Fall 2018

Thursdays: 2-4pm
Meyer Hall (6 Washington Place), room 760

Last updated: Nov 28, 2018

UNDER CONSTRUCTION


Course Objective

The objective of this course is to master select topics in computational neuroscience, focusing in particular on dynamics of neural activity. This is an interdisciplinary field of science, crossing the boundaries between psychology, biology, physics and engineering. This course is intended for neural science majors and psychology majors that are on track for careers in science and medicine. This course is also appropriate for students in the psychology masters degree program. Topics include: the membrane equation, synaptic input, spike generation, convolution and receptive fields, recurrent neural nets, neural integrators and working memory. The best way to learn something is to do it. So we will cover each topic by implementing simulations, using spreadsheets (Excel or Numbers) to simulate the dynamics of neural activity over time.

Prerequisites

Intro to Neuroscience (NEURL-UA 100) or Cognitive Neuroscience (PSYCH-UA 25) or Perception (PSYCH-UA 22).

Assignments and Grading

There will be a series of computational lab assignments (see below), using Excel, each of which will be submitted with a written lab report. Some of the work on these assignments will be done in class (bring your computer to class).

There will also be a final project, again using Excel, that will be presented in class. You will have the option of working in pairs on the final project. Some of the options for final project topics include: V1 direction-selectivity, working memory, Hodgkin-Huxley model of spike generation.

Your grade will be determined by: final project (40%), computational lab assignments (40%), attendance and class participation (20%).

Readings

Readings are available online through the links provided in the schedule below. These include mostly lecture notes that I've written over the years for teaching computational neuroscience. Some of them are a bit "mathy" but it'll be ok - we'll go through them together.

Schedule

Date Topic Reading
9/13 & 9/20 The membrane equation Membrane equation handout
9/27, 10/4, 10/11 Synaptic input and computing with synapses Synaptic input handout
10/18, 10/25, 11/1 Linear systems, convolution, and V1 physiology Linear systems handout
11/8 & 11/15 Neural integrators and neural oscillators Primer on neural integrators and neural oscillators
11/22 No class: Thanksgiving
11/29, 12/6, 12/13 Poisson and integrate-and-fire models of spike generation Poisson handout
Integrate and fire handout


Assignment 1 (due 9/27)

Read the membrane equation handout. Use Eq. 5 to implement the membrane equation in Excel (or Numbers). Simulate the membrane potential over time for current steps, and make graphs that look like those in Fig. 2 (using the parameters in the figure caption). We call this the step response. Make another series of graphs for current inputs that vary sinusoidally over time. We call this the frequency response. You will note that the frequency response is also sinusoidal (i.e., if the injected current varies sinusoidally over time then the membrane potential also varies sinusoidally over time).

Use your simulations to answer the following questions:

(1) What happens to the step response when you double the value of C, when you double the value of g, and when you double both C and g?

(2) What happens to the frequency response when you change the frequency of the injected current? In what way does the frequency of the sinusoidal membrane potential depend on the frequency of injected current? In what way does the amplitude of the sinusoidal membrane potential depend on the frequency of injected current? In what way does the phase of the sinusoidal membrane potential depend on the frequency of injected current?

Write a lab report consisting of a couple pages of text and a few figures showing the results of your simulations. Copy/paste the graphs from Excel/Numbers into either Word or Pages. Please make the figures legible and comprehensible: label the axes, add figure captions, etc. Make a PDF of your lab report and send it to me by email. I will accept only PDF files.

Assignment 2 (due 10/18)

Read the synaptic input handout.

Part 1. Use Eq. 3 on p. 2 to make a graph that looks like that in Fig. 1. You will have only 1 synapse. Simulate the membrane potential over time for synaptic conductance steps. In each simulation the synaptic conductance is set to a fixed value starting at t=0. Choose several different values for the synaptic conductance. Measure the steady state membrane potential (when it gets close to being constant over time after it has had time to ramp up). Then plot the steady-state membrane potential as a function of the different values of synaptic conductance. Repeat for two different values of the leak conductance.

Part 2. Use the equations of p. 7 to make a graph that looks like the solid curve in Fig. 4B. The inputs are the four firing rates denoted re1, re2, ri1, and ri2 on p. 7, two of which are depicted in the two curves in Fig. 4A. They are sinusoids of two different frequencies (2 Hz and 6 Hz), each of which is offset by a spontaneous firing rate of 100 spikes/sec. Use the equations on p. 7 to compute the synaptic conductances (ge1, ge2, ge3, and ge4). Then use the membrane equation (on p. 7) with these 4 synaptic conductances and the parameters listed in the Fig. 4 caption to compute Fig. 4D. Change the values of beta1 and beta2 to get different linear sums of the two input firing rates. [Note that you should not use any of the equations on p. 8. to do the simulations.] It might be helpful to look at this derivation.

Part 3. Combine your simulation of part 2 with a shunting synapse so that you can compute a linear summation of firing rates that is divided by another input firing rate. Feel free to choose any value you want/need for the Esh (the reversal potential of the shunting synapse) so it acts like division. Note that Esh will have to be different from Vrest because of the spontaneous firing rates (s) mean that the membrane potential is different from Vrest even when there is no modulation in the input firing rates re1, re2, ri1, and ri2. Make graphs of the membrane potential for each of 2 different values for the shunting synaptic conductance. It might be helpful to look at this example and this corresponding derivation.

Use your simulations to answer the following questions:

(1) For part 1, why doesn't the membrane potential increase linearly with synaptic conductance?

(2) For part 2, and for a fixed pair of values of beta1 and beta2, show that the output membrane potential is a linear sum of the input firing rates. (a) Set the 1st pair of input firing rates (re1 and ri1) to be equal to the spontaneous firing rate (s) with no modulation over time, and measure the membrane potential to the time-varying firing rates of the 2nd pair (re2 and ri2). (b) Measure the membrane potential for the time-varying firing rates of the 1st pair with the 2nd pair equal to the spontaneous firing rate (s). (c) Add those two results from (a) and (b). (d) Measure the membrane potential for the time-varying firing rates of both pairs and compare with the result from (c).

(3) For part 3, how did you pick Esh to get perfect division and in what way did it depend on s? What happens if the Esh is different from that value?

Write a lab report consisting of a couple pages of text and a few figures showing the results of your simulations. Copy/paste the graphs from Excel/Numbers into either Word or Pages. Please make the figures legible and comprehensible: label the axes, add figure captions, etc. Make a PDF of your lab report and send it to me by email. I will accept only PDF files.

Assignment 3 (due 11/8)

Download the Linear systems tutorial. Work through this spreadsheet to understand what it does and how it works. Write up a lab report that answers the questions in red in the tutorial, including graphs.

Assignment 4 (due 11/29)

Read the Primer on Neural Integrators and Neural Oscillators. Use Eq. 1 to recreate Fig. 1. Use the same equation but with the value of lambda changing over time to recreate Fig. 2. Use Eqs. 7, 8, and 9 to recreate Fig. 3.

Write a lab report consisting of a couple pages of text and a few figures showing the results of your simulations. Copy/paste the graphs from Excel/Numbers into either Word or Pages. Please make the figures legible and comprehensible: label the axes, add figure captions, etc. Make a PDF of your lab report and send it to me by email. I will accept only PDF files.

Assignment 5 (due 12/18)

Read the Poisson Handout and the Integrate and Fire Handout. Pick either option 1 or option 2 (not both). Write a lab report consisting of a couple pages of text and a few figures showing the results of your simulations. Copy/paste the graphs from Excel/Numbers into either Word or Pages. Please make the figures legible and comprehensible: label the axes, add figure captions, etc. Make a PDF of your lab report and send it to me by email. I will accept only PDF files.

Option 1

Part 1. Use the algorithm outlined on p. 5-6 of the Poisson Handout (the first approach that relies on subdividing time into a bunch of short intervals) to generate a homogeneous Poisson spike process with a mean firing rate of 100 spikes/sec and duration of 1 sec and a time bin interval of 1 msec (1000 rows), with a 1 in each cell (corresponding to a 1 msec time bin) in which a spike occurs. Compute the spike count as the sum of the spikes. Repeat this a bunch of times (at least 100), to make a histogram of the spike counts to recreate Fig. 1B. Superimpose the theoretical curve (using Eq. 3) with the histogram (as in Fig. 1B). Note that you will need to learn how to use the 'rand' and 'if' functions in Excel and you will need to figure out how to make a histogram in Excel.

Part 2. Use the same algorithm, as explained on p. 7 of the Poisson Handout, to generate an inhomogeneous Poisson process. Make the rate vary sinusoidally over time as in Fig. 4A of the Synaptic Input Handout to general spike trains that look like Fig. 4B of the Synaptic Input Handout. Make a large number of these spike trains (at least 100). Compute the average number of spikes in each 1 msec time bin. Plot the result superimposed with the original sinusoidal rate function that you started with.

Option 2

Implement the leaky integrate and fire model using the 2nd equation on p. 1 of the Integrate and Fire Handout. Use the following parameters: C = 0.2 nF, gleak = 0.02 uS, Eleak = Vrest = -70 mV, Vth = -55 mV, Tref = 4 msec. Use an input current ampitude of 1 nA and count (sum) the number of spikes that you get for a 1 sec current injection. Repeat for input current ampitudes of 0.5, 1, 2, and 3 nA. Make an F-I (firing rate vs current amplitude) graph that looks like Fig. 1 of the handout.

Extra credit

Part 1. Use the Hodgkin-Huxley Handout to simulate action potential generation. Note that your time step will now need to be much shorter than 1 msec because the entire action potential unfolds over a time period of a couple msec.

Part 2. Run the Hodgkin-Huxley simulation for several different different input current amplitudes and plot an F-I curve.

david.heeger@nyu.edu