HOMEWORK #1: N.B.: There will only be two homeworks this year!

Checklist:
  (a) due by Friday, 2/23/07, under my door (CSB 171)
  (b) use this web page as the first page(s)
  (c) staple all pages together
  (d) one page max per answer (less is better!)
  (e) each answer on a separate page
  (f) figures on same page as answer
  (g) word-processor text with hand-drawn figures preferred
  (h) list names in your working group (if any besides you)

1. A cell has channels that only conduct, Z, a singly-charged positive ion. Assume that the concentration of the ion inside is 20 mM and the concentration outside is 60 mM, that the resting membrane potential is -70 mv, and that the cell is at body temperature T = 37 deg C. Using the Nernst equation, show calculations for (a) equilibrium potential for this ion, (b) how equilibrium potential would change if cell was room temperature T = 20 deg C, (c) (from now on, back at body temp) which way the ions will flow if the channel is briefly opened, (d) how this will affect the membrane potential, (e) What equilibrium potential would be if cell bath was changed to 100 mM ZCl ('Z chloride': net charge added is zero). Finally, (f) give the equation for calculating what the membrane potential will be given two ions with differential in/out concentrations (each with their own ion-selective channel).

2. There are a number of different types of neurons in the cerebral cortex as determined by anatomical and physiological features. First, summarize the different types as classified by anatomical features (e.g., dendritic and axonal arbor shapes, spines or not, connections, neurochemical features). Second, summarize the main cell types on the basis of physiological properties (e.g., firing patterns, receptive field properties). List known correlations between anatomical and physiological types (note that this is difficult because it is hard to study both detailed anatomy and detailed physiology in the same cells, particularly in vivo).

3. Ionic current flows at synapses on dendrites propagate surprisingly slowly given that changes in intracellular potentials are conducted through cellular fluids virtually instantaneously. (a) Draw a simple circuit diagram of an unbranched portion of a dendrite. (b) Explain intuitively why there appears to be a propagation delay from one end to the other. (c) Consider quantitatively what happens when a current impulse -- a sudden increase and then immediate decrease to zero -- of current is injected into the distal end of a dendrite: draw one graph containing two curves showing how the transmembrane potential changes with time near the current injection site, and at the cell body (further away). (d) Draw a circuit diagram for a spine attached to a dendrite with one compartment for the spine head and one for the spine neck. (e) Write down (nothing more!) the cable equation that decribes current flow in a cable in the limit of infinitesimal compartments.

4. We described the behavior of a binary, asynchronously updated attractor ("Hopfield") network. (a) Construct a fully-connected network with 3 units and a symmetric weight matrix, and then list all the possible states, their respective energies, and say which ones are stable. (b) Briefly, explain the proof of why a symmetric weight matrix is required to guarantee stable states (explicitly, to guarantee that no single unit update can ever increase the energy). (c) The binary attractor network has its possible states at the corners of a cube (hypercube); illustrate where the stable states are in a continuous formulation of an attractor network found in the second attractor paper (reading number 02.02).

5. (a) Why does the Linsker update equation contain only pre-synaptic terms given that a Hebb rule is typically described as changing the weight according to the correlation of pre- and post-synaptic activity? (b) Explain how Linsker ran his simulations -- that is, what was his starting state, how did he update his weights, and specifically, what was his input? (c) Explain how the Linsker learning rule can be thought of as a matrix operating on a vector to yield another vector. Start with the update equation for one weight, explain where the matrix comes from and what its dimensions are, and say explicitly what the input and output vectors are. (d) Say what the eigenvectors and eigenvalues tell us about how learning proceeds. (e) There is no tendency to form orientation columns in a feedfoward Linsker model like the one from class. How did Linsker get columns to form? (reading number 02.05).

6. We went over the update equations for a single-compartment integrate-and-fire model described in Wilson and Bower (1989) from the readings. (a) Draw the equivalent electrical circuit diagram for a one-compartment cell in this model. (b) Starting at rest, an inhibitory GABA input to a GABA-A (K+) channel at a synapse on a neuron is activated soon after a nearby GABA input to a GABA-A (Cl-) channel synaptic input to the same neuron is activated. At the first instant of opening, will the GABA-A channel conduct more, the same, or less current than in the situation where it is activated by itself? Explain your answer by giving a qualitative discussion of the relevant equation and include terms for the GABA-A and GABA-B channel. (c) A convolution is used to sum up the effect of synaptic inputs at one connection across time (which means that that linear superposition of conductances is assumed). Give a reason (there are many possibilities!) why linear superposition at a connection would not hold for a real neuron. (d) Briefly state the main differences between an integrate-and-fire model and a Hodgkin-Huxley model (both single-compartment).

7. (a) Briefly describe the general mechanism by which NMDA channels at a synapse detect correlations between the activity in the pre- and post-synaptic cells and give plausible pre- and post-synaptic mechanisms for how the increased synaptic strength seen during LTP could be generated. (b) Summarize an in vitro experiment that demonstrates the newly-described phenomenon of spike-timing-dependent plasticity (STDP) and say how it differs from the original concept of LTP. (c) There is some evidence that spikes are actively propagated back into the dendrites. Find the paper that describes spike backpropagation and describe how it is thought to work.

8. (a) The main streams of information at the level of the primate dLGN appear to be left versus right eye, sustained X-like versus transient Y-like versus koniocellular, and ON versus OFF. Briefly summarize the current view of the main streams of visual information at the level of primate V1 as recently updated by Sincich and Horton, and Yabuta et al. (in readings). (b) Illustrate the retinotopic maps in V1, V2, V3d/VP, V4d/V4v, V4t, and MT in the left hemisphere of a macaque monkey. For each map, include the horizontal and vertical meridians, mark the upper and lower field with plus and minus, and indicate the center of gaze with a star. You don't have to illustrate shared borders twice. (c) What is "visual field sign" (Sereno et al., 1994) and how is it determined?

9. (a) What is the major difference between simple, complex, hypercomplex, and simple-hypercomplex cells? (b) Describe how the spike-triggered averaging method (reverse correlation) works for simple cells (e.g., ref) for determining how neurons respond to visual stimuli and then say why it doesn't generally work with complex and hypercomplex cells. (c) Ohzawa examined the effect of binocular disparity and receptive field position on the responses of cells in cat striate cortex. How do simple and complex cells differ from each other in their response to binocular bars at different disparities and at different receptive field positions?