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Problems for 16 use none none maker toadd none for nonepdf-417 printing 1. Show tha none for none t Eq. (16.

8a) can be expressed equivalently as dm/dt (m1 m)/ m. (Of course, the other equations of this group can be put in this form too.) 2.

Why will the threshold by somewhat higher than the second zerocrossing in the inset in Fig. 16.18b .

Microsoft Office Official Website P R O B L E M S F O R C H A P T E R 16 3. Use Eqs. none for none (16.

7a) (16.7c) and (16.9) to write a complete expression that can be solved (numerically) to determine the resting potential.

4. Plot Eq. 16.

11 using an initial voltage of 75 mV instead of 65 mV and use this plot to estimate the threshold. How does the value differ from that indicated in Fig. 16.

18b First determine the holding current to add to Istim to obtain 75 mV. The plotted current must be corrected for this holding current. 5.

Use any computer language or a mathematical modeling program such as MATLAB, MATHCAD, or MATHEMATICA to write code that integrates the Hodgkin Huxley system of equations with a stimulus current to generate an action potential (many simulations here used MATHCAD). Do not use a modeling program such as NEURON or GENESIS because these programs already contain the code you should write. 6.

For the case where Na channels do not inactivate (Fig. 16.2b, trace c), use the Hodgkin Huxley equations to calculate the final value to which the voltage settles after the action potential.

7. Consider a squid axon where the Na channels have been completely blocked. A depolarizing current of 20 mA cm 2 is applied for 50 ms.

Sketch the voltage response and use the Hodgkin Huxley equations to estimate voltage at key time points both during the stimulus and after it is turned off.. Appendix 1 Expansions and series A1.1 Taylor series Any functio n can be approximated in the vicinity of a particular point by a tangent line through that point. This approximation takes the form. F x x % none none F x x dF x dx (A1:1). F ( x). This is ill none none ustrated graphically in Fig. A1.1.

The deterioration of this approximation with distance is clear. As x increases the curvature pulls the function away from the line. The approximation can be improved by incorporating the second derivative.

F x x % F x x x x + x dF x x2 d2 F x dx 2 dx2 (A1:2). Fig: A1:1: An arbitrary function is approximated in the vicinity of x by a line through that point with a slope equal to the derivative at that point (Eq. (A1.1)).

. Now we are none for none using a parabola through the point at x instead of a line. This works better but it too fails as x grows. This idea is generalized with the Taylor expansion, which can approximate any continuous function at arbitrary distances from x to any desired degree of accuracy, provided that all the derivatives are defined at x, as follows.

F x x F x x dF x x2 d2 F x x3 d3 F x 2 dx2 6 dx3 dx 1 X xn dn F x F x n! dxn n 1. (A1:3). Taylor expa nsions of functions to first or second order are encountered very often and are extremely useful in theoretical analysis. The expansions of the following functions are especially common. They were calculated directly from Eq.

(A1.3) to third order. They are generally valid for small values of x.

ex 1 x none for none x2 x3 2 6 x2 x3 2 3 (A1:4). ln 1 x x (A1:5). p x x2 3x3 1 x 1 48 2 8 (A1:6). A1.3 G E O M E T R I C S E R I E S A1.2 The binomial expansion This takes the form N . N X i 0 N! i N i N i !i!. (A1:7). This is eas none none ily checked by trying out small values of N. For arbitrarily large N the combinatoric term N!/(N i)!i! represents the number of ways of selecting i times from N factors..

A1.3 Geometric series It is easy to check by multiplying out the product that n n 1 1 1 2 3 1 (A1:8). The sum of a geometric series is the second factor on the left, so n X i 0 1 n 1 1 . (A1:9). Differentiating this expression with respect to and then multiplying by gives n X i 0 i i 1 n 1 1 2. n 1 n 1 1 (A1:10). For n ! 1 and 0 < < 1, we have 1 X i 0 1 1 . (A1:11). 1 X i 0 i i 1 2. (A1:12). from Eq. (A1.9) and Eq. (A1.10), respectively. Appendix 2 Matrix algebra A2.1 Linear transforms Matrices si none for none mplify the mathematical analysis of problems with multiple linear equations and variables. For example, four linear equations with four unknowns take the form. a11 x1 a1 2 x2 a13 x3 a14 x4 y1 a21 x1 a22 x2 a23 x3 a24 x4 y2 a31 x1 a32 x2 a33 x3 a34 x4 y3 a41 x1 a42 x2 a43 x3 a44 x4 y4 (A2:1a) (A2:1b) (A2:1c) (A2:1d).
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