Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science,Department of Mechanical Engineering, Department of Biological Engineering,Harvard-MIT Division of Health Sciences and Technology2.791J/2.794J/6.021J/6.521J/20.370J/20.470J/HST.541JQuantitative Physiology: Cells and TissuesHomework Assignment #8 Issued: November 16, 2006Due: November 30, 2006Exercise 1. Does the time constant of a cylindrical cell depend on its dimensions? Does the spaceconstant of a cylindrical cell depend on its dimensions?Exercise 2. The space constant decreases as the specific membrane conductance is increased. Givea physical explanation for this result.Exercise 3. What are the distinctions among the conductance variables Gm, Gm, and gm?Exercise 4. A squid axon is placed in a large volume of sea water so that you may assume thatro ri. The following data are given: resistivity (specific resistance) of squid axoplasm, ρi=30 Ω·cm; diameter of axon is 500 µm; space constant, λC= 6 mm; thickness of membrane d = 50˚A; capacitance per unit area of membrane Cm= 1 µF/cm2.a) Find the conductance of the axon per unit length, gm.b) Find the conductance of the axon per unit area, Gm.c) Find λCfor unmyelinated axons whose membranes have specific properties (i.e., Gmand ρi)that are identical to those of the squid axon but whose diameter are: 1 mm; 0.1 mm; 0.01mm; 0.001 mm.d) Find τMfor the squid axon and for the same axons considered in part (c).1Problem 1. Constant currents I1and I2are applied to the exteriors of Axons 1 and 2, respec-tively, and the resulting time-independent changes in membrane potential are vm1(z) and vm2(z),respectively. I1and I2are adjusted so that vm1(0) = vm2(0) = 10 mV. This change in potential issufficiently small so that the membrane voltage-current characteristic may be assumed to be linear.You may also assume that ro rifor both axons and that rois the same for both axons. Thegeometries and parameters for Axons 1 and 2 are given below, where a is the axon radius, ρiis thecytoplasmic resistivity, and Gmis the specific membrane conductance.0 z (cm)Axon 1−0.1Axon 2I1I2Axon a ρiGm# (µm) (Ω·cm) (S/cm2)1 100 100 5×10−32 10 100 (1/8)×10−3a) Let vm1(−0.1) and vm2(−0.1) be the membrane potential changes at z = −0.1 cm for thetwo axons. Determine the value of the ratio A = vm1(−0.1)/vm2(−0.1).b) Determine the value of the ratio B = I1/I2.Problem 2. A large unmyelinated axon is immersed in oil, and five different arrangements ofelectrodes for delivering current stimuli and for measuring potential responses are attached to theaxon as shown below.2+−+−+−+−+−abcdeie(t)v(t)5λCie(t)v(t)5λCλC5λC5λCλC5λCλCie(t)v(t)ie(t)v(t)ie(t)v(t)The stimulus current, a brief positive pulse at t = 0, is the same for each arrangement of electrodes.Assume that the duration of the pulse is much shorter than the membrane time constant of the cell,and that the cell’s membrane can be modeled as a linear cable. The space constant of the cell isλC. In arrangement d, the potential is recorded and the current is delivered at the same longitudinalposition. Assume that the electrodes have infinitesimal spatial extent in parts a-d and have lengthmuch larger than λCin part e.For each of the different arrangements (parts a-e in the figure) determine which of the followingwaveforms for v(t) represents the deviation of the measured potential from its resting value. Foreach waveform, the horizontal axis corresponds to v(t) = 0, and the vertical axis to t = 0. If nowaveform applies, answer None. Explain the basis of your choice in each case.v1(t)v2(t)v3(t)v4(t)v5(t)v6(t)v7(t)v8(t)v9(t)v10(t)v11(t)v12(t)v13(t)v14(t)3Problem 3. A large invertebrate axon is immersed in oil and stimulated with a constant currentIe> 0. The current is delivered using a pair of electrodes that are either extracellular or intracel-lular, as illustrated in the following figure.IezL0ExtracellularexcitationIntracellularexcitationIezL0The current Ieis sufficiently small that the cell behaves as a linear cable. The specific resistancesof the internal and external conductors are 100 kΩ/cm and 10 kΩ/cm, respectively. The specificconductance and capacitance of the membrane are 10 mS/cm and 0.1 µF/cm, respectively. Themembrane potential is allowed to come to steady state. [Remember: Vmis positive when theinside of the cell is positive with respect to the outside of the cell. Also, the reference direction forthe longitudinal currents is in the +z direction.]Determine if the following statements are true or false.a) The intracellular potential at z = L is greater than the intracellular potential at z = 0.a1) True or False for extracellular excitation?a2) True or False for intracellular excitation?b) The extracellular potential at z = L is greater than the extracellular potential at z = 0.b1) True or False for extracellular excitation?b2) True or False for intracellular excitation?c) The external longitudinal current Io(z) is equal to 0 for z < 0.c1) True or False for extracellular excitation?c2) True or False for intracellular excitation?d) The membrane potential at z = L, Vm(L), is greater than the membrane potential at z = 0,Vm(0).d1) True or False for extracellular excitation?d2) True or False for intracellular excitation?e) The internal longitudinal current at a point just to the right of the rightmost electrode, i.e.,Ii(L+), is less than zero.e1) True or False for extracellular excitation?e2) True or False for intracellular excitation?4Problem 4. Electrical responses for an unmyelinated axon are measured in two configurations —unclamped and space clamped — as shown in the following figure.The stimulus current and voltage responses are shown in the following figure.2 4 6 8 102 4 6 8 102 4 6 8 100.20.40.60.810.110.010The current stimulus has an amplitude Iefor t < 0 and an amplitude of 0 for t ≥ 0. Each ofthe voltage responses are shown twice: with a linear ordinate (middle panel) and on a logarithmicscale (lower panel). For each configuration, Ieis adjusted so that the membrane potential exceedsthe resting potential by 1 mV for t < 0. This potential change is so small that the voltage-currentcharacteristic of an incremental portion of the axon membrane can be considered to be representedas a constant conductance and a capacitance in parallel.Part a. Which trace, solid or dashed, corresponds to the response of the space clamped axon?Briefly explain.Part