NumericalDiffEqs - Revision history

Carver at 21:31, 17 February 2009

2009-02-17T21:31:38Z

2009-02-17T21:31:13Z

2009-02-17T18:29:49Z

‎Equilibrium Potentials

2009-02-17T16:54:52Z

‎Equilibrium Potentials

2009-02-17T16:52:23Z

‎Equilibrium Potentials

2009-02-13T00:23:09Z

‎Numerical Solution of Differential Equations

2009-02-13T00:08:21Z

‎Numerical Solution of Differential Equations

2009-02-13T00:07:20Z

‎Equilibrium Potentials

2009-02-12T23:40:09Z

‎Numerical Solution of Differential Equations

2009-02-12T23:39:39Z

‎Numerical Solution of Differential Equations

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 Today's lab is the next tutorial, the Action Potential Tutorial.
 * '''Homework:''' Do as much of the tutorial as you can and make a PowerPoint out of three observations.
 Recall the equation for the membrane potential in the Hodgkin-Huxley model:

← Older revision		Revision as of 21:31, 17 February 2009
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	== Equilibrium Potentials ==		== Equilibrium Potentials ==
		+
		+	Today's lab is the next tutorial, the Action Potential Tutorial.
		+
		+	* '''Homework:''' Do as much of the tutorial as you can and make a PowerPoint out of three observations.

	Recall the equation for the membrane potential in the Hodgkin-Huxley model:		Recall the equation for the membrane potential in the Hodgkin-Huxley model:

← Older revision		Revision as of 18:29, 17 February 2009
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	The equilibrium potential is the weighted average of the reversal potentials -- weighted by the corresponding conductances. Note that the weights add to one.		The equilibrium potential is the weighted average of the reversal potentials -- weighted by the corresponding conductances. Note that the weights add to one.
		+
		+	Now we can talk about the [[Media:Lab_F.pdf\|homework due last night]].

	== Numerical Solution of Differential Equations ==		== Numerical Solution of Differential Equations ==

← Older revision		Revision as of 16:54, 17 February 2009
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	== Equilibrium Potentials ==		== Equilibrium Potentials ==
		+
		+	Recall the equation for the membrane potential in the Hodgkin-Huxley model:
		+
		+	<math> C \frac{dV}{dt} = I(t) - g_L (V(t) - E_L) - g_{Na} m^3(t) h(t) (V(t) - E_{Na}) - g_K n^4(t) (V(t) - E_K) </math>

	You can find the equilibrium potential for given channel conductances. Set the injected current to zero I = 0. Then fix and absorb the gating variables and maximal conductances <math> (g_L, g_{Na}, g_{K}) </math> into single conductance variables <math> (G_L, G_{Na}, G_{K}) </math>.		You can find the equilibrium potential for given channel conductances. Set the injected current to zero I = 0. Then fix and absorb the gating variables and maximal conductances <math> (g_L, g_{Na}, g_{K}) </math> into single conductance variables <math> (G_L, G_{Na}, G_{K}) </math>.

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 For example
-<big> <big> <math> G_{K} = n^4 g_K </math> </big> </big>
+<big> <big> <math> G_{K} = g_K n^4 </math> </big> </big>
 Then the differential equation for the membrane is

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 <big> <big> <big> <math> y_k = y_{k-1} + dt \ f(y_{k-1}) </math> </big> </big> </big>
-−
 Click here for [[Media:intuit.hoc|code]] for visualizing the numerical solution of differential equations.

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 Here k is the rate constant, 1/k is the time constant, 1/k is <math> \frac{C}{g_L} = RC </math> in the notation above.  A leaky cell is what is called an RC circuit -- a resistor and capacitor together in a circuit.  The time constant of an RC circuit is RC.  The bigger k, the higher the rate of convergence, and the smaller the time constant 1/k.  The time constant is the time it takes the solution to decay to 1/e of its value.
-Solution of differential equations happens at discrete times: <math> y_k </math>, separated by small time intervals dt.
+Solution of differential equations happens at discrete times: <big> <math> y_k </math> </big>, separated by small time intervals dt.
 The simplest way of solving this equation is with Euler's method:

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 The simplest way of solving this equation is with Euler's method:
-<big> <big> <big> <math> y_k = y_{k-1} + dt (-k y_{k-1}) </math> </big> </big> </big>
+<big> <big> <big> <math> y_k = y_{k-1} + dt \ (-k y_{k-1}) </math> </big> </big> </big>
 This is a special case of the general formula for Euler's method applied to the (vector) differential equation
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 <math> \frac{dy}{dt} = f(y) </math>
-<big> <big> <big> <math> y_k = y_{k-1} + dt / f(y_{k-1}) </math> </big> </big> </big>
+<big> <big> <big> <math> y_k = y_{k-1} + dt \ f(y_{k-1}) </math> </big> </big> </big>
 Click here for [[Media:intuit.hoc|code]] for visualizing the numerical solution of differential equations.