Quantitative Physiology: Cells and Tissues2.791J/2.794J/6.021J/6.521J/BE.370J/BE.470J/HST.541JRecitation 14: November 2, 2006Austin CheProblem 4.12. A propagated action potential is recorded at one position along an axon. Five pointsin time during this action potential are identified in Figure 1.tt0t1t2t3t4Vm(t)Figure 1. Propagated action potential.• At t0the membrane is in its resting state, i.e. Vm(t0) = Vom.• At t1the membrane potential has a point of inflection, i.e. [∂2Vm/∂t2]t1= 0.• At t2the membrane potential is zero, i.e. Vm(t2) = 0.• At t3the membrane potential has its maximum value.• At t4the membrane potential has its minimum value.1t0t1t2t3t4abcdefghijkNoneTable 1. Organization for answers to Problem 4.12.For each of the times t0, t1, t2, t3, and t4determine which, if any, of the following statements applyin a precise manner. If none applies precisely then indicate none. Indicate your answers with acheck mark in the appropriate places in a table such as Table 2.a) The sodium conductance equals the potassium conductance.b) The membrane current is zero and changes from inward to outward.c) The membrane current is zero and changes from outward to inward.d) The capacitance current is zero.e) The total ionic current is zero.f) The sodium conductance has a maximum value.g) The potassium conductance has a maximum value.h) The sodium conductance has a minimum value.i) The potassium conductance has a minimum value.j) The magnitude of the potential difference between the membrane potential and the sodiumequilibrium potential is a minimum.k) The magnitude of the potential difference between the membrane potential and the potassiumequilibrium potential is a minimum.2Problem 4.28. Each of the panels in Figure 2 shows action potentials computed from the Hodgkin-Huxley model of a space-clamped axon in response to a current pulse of duration 0.5 ms andof amplitude 20 µA/cm2. In each panel the dashed curve shows the response for the standard−80−60−40−2002040600 2 4 6 8 10 12 14 16 18 20Membrane potential (mV)Time (ms)−80−60−40−2002040600 2 4 6 8 10 12 14 16 18 20Membrane potential (mV)Time (ms)−80−60−40−2002040600 2 4 6 8 10 12 14 16 18 20Membrane potential (mV)Time (ms)−80−60−40−2002040600 2 4 6 8 10 12 14 16 18 20Membrane potential (mV)Time (ms)(a) (b)(c) (d)Figure 2. Waveforms of action potentials computed from the Hodgkin-Huxley model.parameters of the Hodgkin-Huxley model. Each of the solid curves was obtained by computingthe response of the Hodgkin-Huxley model with identical parameters as the dashed curve exceptthat one parameter of the model was changed. For each of the waveforms a)-d), determine whichone of the following statements is consistent with the computation.(1) The leakage conductance was reduced from GL= 0.3 to 0.01 mS/cm2.(2) The temperature was increased from 6.3 to 10◦C.(3) The membrane capacitance was increased from Cm= 1 to 1.1 µF/cm2.(4) The intracellular sodium concentration was reduced from 50 to 25 mmol/L.3t0t1t2t3t4abc√d√ √ √e√fghij√k√None√Table 2. Properties at different points in a propagated actionpotential (Problem 4.12).SolutionsProblem 4.12. From the core conductor modelJm(z, t) =12πa(ri+ ro)∂2Vm(z, t)∂z2.Since the action potential is propagating at constant velocity ν,Jm(z, t) =12πa(ri+ ro)ν2∂2Vm(z, t)∂t2.Therefore, Jm(z, t) = 0 when ∂2Vm(z, t)/∂t2= 0. Also the capacitance current isJC(z, t) = Cm∂Vm(z, t)∂t.Therefore, JC(z, t) = 0 when ∂Vm(z, t)/∂t = 0. Taking these results into account leads to theanswers shown in Table 2.Problem 4.28.a. Ans. is (3). The action potential shows a slower depolarization at the onset and a slowerrepolarization. This is consistent with an increase in the membrane capacitance.b. Ans. is (4). The peak action potential goes to a higher potential which is consistent with anincrease in the Nernst equilibrium potential for sodium.c. Ans. is (1). The resting potential is decreased and the undershoot of the action potentialhas been reduced implying that the resting potential is closer to the potassium equilibriumpotential. This is consistent with a reduction in the leakage conductance.d. Ans. is (2). The action potential shows a more rapid depolarization at the onset and a fasterrepolarization. This is consistent with an increase in the