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Physiology 472/572 - Quantitative Modeling of Biological Systems Homework 5 Questions 1 and 2 require the use of Matlab and the simulation package SoftCell5.3. Leave all parameters at the default values except for the ones mentioned. Results should be given to two significant figures accuracy. 1. Load the simulation 'Hodgkin-Huxley space-clamped model' and use current-clamped mode. a) Find the minimum current density of a pulse of current with duration 1 ms needed to excite an action potential. b) Reset the pulse to 0.5 ms duration and 20 μA/cm2 intensity. Introduce a second pulse of the same duration and intensity. Set the stimulus duration to 30 ms. Find the earliest start time of the second pulse that will excite a second action potential. c) Repeat part b) but with the second pulse set to 50 μA/cm2 intensity. d) Estimate the lengths of the absolute and relative refractory periods of this neuron. [The absolute refractory period is the interval during which a second action potential cannot be initiated, no matter how large a stimulus is applied. For the purpose of this question, assume that the absolute refractory period begins when the membrane potential drops below the resting potential. The relative refractory period is the interval immediately following the absolute refractory period during which initiation of a second action potential is inhibited but not impossible.] 2. Load the simulation 'Hodgkin-Huxley propagated action potential'. a) Estimate the propagation velocity of the action potential. b) Move the S+ electrode to position 1.5 cm. Sketch or print out the membrane potential as a function of position at three different time points during the first 1 ms and briefly explain what is happening. c) Restore the S+ electrode to position 0.2 cm. Click on the axon schematic in the workspace to access the parameters. Gradually increase the extracellular potassium concentration and observe the effects. What is the maximum concentration at which you can observe a propagated action potential? Can you give a precise answer? d) Describe the changes in the properties of the action potential as the potassium concentration is increased up to the level found in c). (continued on next page)3. A simplified model for an excitable membrane is represented by the equation ⎩⎨⎧>+<=+∂∂=∂∂α V if 1 V-α V if V- h(V) where h(V)xV tV22 Here 0 < α < 1 and 0 ≤ V(x,t) ≤ 1. (a) Suppose that V(x,t) has the form of a traveling wave with velocity θ > 0. Use this information to convert the above equation to an ordinary differential equation for V(x,0). (b) Find the general solution to the ordinary differential equation assuming V < α. [Hint: Look for solutions Aeλx. Obtain a quadratic in λ with roots λ1 and λ2. Express the general solution as a linear combination of two exponential functions with two unknown constants.] (c) Find the general solution to the ordinary differential equation assuming V > α. [Hint: Don't use the same names as in (b) for the two unknown constants.] The goal now is to find a traveling wave solution, such that V(0,0) = α, V(x,0) > α when x > 0, V(x,0) → 1 as x → ∞, V(x,0) < α when x < 0, and V(x,0) → 0 as x → -∞. (d) Sketch the likely shape of the solution according to the given information. (e) The solution in (b) applies for x < 0 and the solution in (c) applies for x > 0. Use the above information to eliminate one unknown constant in each solution. (f) Use the fact that both solutions satisfy V(0) = α to solve for the remaining unknown constants in terms of α. (g) Use the fact that the derivatives of the solutions match at x = 0 to find the propagation velocity θ in terms of α. [Parts (d) - (g) are extra credit for


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