Difference between revisions of "NumericalDiffEqs"

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(→‎Equilibrium Potentials)
(→‎Equilibrium Potentials)
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== Equilibrium Potentials ==
 
== Equilibrium Potentials ==
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Recall the equation for the membrane potential in the Hodgkin-Huxley model:
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<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>.

Revision as of 16:54, 17 February 2009

Equilibrium Potentials

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

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)

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 (g_L, g_{Na}, g_{K}) into single conductance variables (G_L, G_{Na}, G_{K}) .

For example

G_{K} = g_K n^4

Then the differential equation for the membrane is

C \frac{dV}{dt} = - G_L (V - E_L) - G_{Na} (V - E_{Na}) - G_K (V - E_K)

The equilibrium potential is the value of V such that \frac{dV}{dt} is zero. You can plug in \frac{dV}{dt} = 0 and solve for V_{equilibrium}.

V_{equilibrium} = \frac{G_L E_L + G_{Na} E_{Na} + G_K E_K}{G_L + G_{Na} + G_K}

The equilibrium potential is the weighted average of the reversal potentials -- weighted by the corresponding conductances. Note that the weights add to one.

Numerical Solution of Differential Equations

Remember the equation for the cell with only leak channels.

C \frac{dV}{dt} = I(t) - g_L(V - E_L)

Let's simplify: suppose there is no injected current and that the reversal potential for the leak channels is E_L = 0 . Then our equation is

\frac{dV}{dt} = - \frac{g_L}{C} V

Using different letters for the variables (because this is done in the software linked below):

\frac{dy}{dt} = - k y

Here k is the rate constant, 1/k is the time constant, 1/k is \frac{C}{g_L} = RC 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: y_k , separated by small time intervals dt.

The simplest way of solving this equation is with Euler's method:

y_k = y_{k-1} + dt \ (-k y_{k-1})

This is a special case of the general formula for Euler's method applied to the (vector) differential equation

\frac{dy}{dt} = f(y)

y_k = y_{k-1} + dt \ f(y_{k-1})

Euler's equation is the simplest way to solve a differential equation numerically. However it is often not the preferred method: often you need to take much smaller time steps with Euler than with some other methods, so it takes longer to get as good a solution. Still if you are doing something complicated, like solving an equation with noise, or Bayesian filtering (to compute likelihood), an argument can be made that a simpler method is desirable -- at least as a first step.

Click here for code for visualizing the numerical solution of differential equations.

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