402_502_W11_hw3 - the long-term behavior oF the neuron when...

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AMATH 402/502 Homework 3 DUE at time and place posted on website 1 In this problem we will study the integrate-and-fre model oF a neuron. The membrane potential V ( t ) oF the neuron obeys the Following di±erential equation: τ dV dt = V rest - V + RI ( t ) where V rest is the resting membrane potential, R > 0 is membrane resistance, τ > 0 is a time constant, and I ( t ) is the current injected into the cell. When the voltage reaches its threshold, V th > V rest , the neuron is said to “spike” (fre an action potential) and the voltage is instantaneously reset to V reset < V rest . ²or this problem, we will consider the e±ect oF injecting a constant positive current, I ( t ) = μ , on a neuron that is initially at rest, V (0) = V rest . a. ²irst suppose we do not inject any current into the cell μ = 0. Sketch the trajectory oF the cell’s voltage For initial condition V (0) = V rest , as well as perturbed initial conditions V th > V (0) > V rest and V rest > V (0) > V reset . b. Show that a biFurcation occurs at a critical μ * , and determine this value. Describe
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Unformatted text preview: the long-term behavior oF the neuron when 0 < < * and when > * . In each case sketch a sample trajectory assuming that the neuron is initially at rest, V (0) = V rest . Sketch the biFurcation diagram oF fxed points V * vs. For 0. c. Compute and sketch a plot oF the the Frequency oF spiking that is, the number oF spikes produced per unit time as a Function oF . 2 Problem 3.6.3 3 Problem 3.6.4. or this problem, only consider imperFections oF order up to O (1) in x : we are looking For perturbations that are small in a neighborhood oF the biFurcation point (0,0). 4 Problem 4.5.1 5 Problem 5.1.10 (a,b,c) only 6 Problem 5.1.11 For systems (a,b,c) oF 5.1.10 only 7 Problem 5.2.1 The homework will be graded statistically. Late homework is not accepted. Your homework should be neat and readable (the TA is allowed to subtract points For presentation)....
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This note was uploaded on 02/28/2011 for the course AMATH 402 taught by Professor Staff during the Spring '08 term at University of Washington.

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