At the microscopic level, channels are stochastic event with probabities of turning on and off. With the advent of patch clamping and other experimental procedures, it is possible to record from a single channel while stepping through many different voltages. Here I will look at some simple and then more complex models. First, let's consider for example the potassium channel for the Morris-Lecar model. The current for this is:

I_{K} = g n (V - E_{K})

The gate ** n ** just satisfies the first order kinetic scheme:

The rates are voltage dependent. However, for a fixed value of the
voltage, they are fixed. We simulate the stochastic process as
follows. We choose a time step, ** dt ** and choose a random
number, ** r ** between 0 and 1. If we are in the closed state ** C ** and
** alpha dt > r ** then we jump to the open state. Otherwise, we
stay at the closed state. Similarly, if we are in the open state ** O
** and ** beta dt > r ** then we jump to the closed state.

For the Morris -Lecar equation, we are given only ** tau ** and
** ninf ** so that we obtain:

alpha = ninf/tau

beta = (1-ninf)/tau

XPP has a built-in routine to handle random switching called the **
Markov ** object. For an ** n-state ** system, you define an **
n ** dimensional square matrix. The entries of row ** j **
contain the rates of transition from state ** j ** to any other
state ** k **. The diagonals are set to zero since they are really
just the probabilities of staying where you are so given the other
entries, these diagonals are determined.

Simulate the Morris-Lecar channel given the kinetics:

tau(v) = 1/(phi cosh((v-12)/34.8))

Range Over: | z |

Steps | 100 |

Start: | 0 |

End: | 0 |

and press ** OK ** to repeat the simulation 100 times. XPP keeps
track of the mean and variance over the trials at each time
point. When complete, click on ** stocHastic Mean ** and then on
** Esc ** to get to the main menu. Click on ** Restore ** to see
a rough approximation of the deterministic limit.

Even as few as 100 channels is pretty damn close to the deterministic limit. The law of mass action is pretty awesome. Note the little negative blip occuring after the stimulus. This is because the channels take some time to turn off and the potential has been brought back to -100 mV which is below the equilibrium potential for potassium.

One can consider the channel as having 5 states: ** 0000, 0001, 0011,
0111, 1111 ** where only yhe last state is conducting. Here is a
kinetic model for the channel:

with ** C1=0000 ** and ** O = 1111 ** being the completely
closed and open states respectively. We can use the standard HH
kinetics for potassium to simulate a voltage clamp with a holding
potential of -100 mV and then a step to +20 mV for 20 milliseconds.
The following XPP file implements a stochastic simulation of this
channel using a 5 state markov process with the rates defined by the
usual HH equations. Try integrating this a few times. Look at the
mean after 10 trials (equivalent to 10 channels) and 100 trials. It
looks pretty much like the ML. Note that the channel is much faster
and thus looks much more like the mean field equations over the time
course of 20 msec.

The sodium channel is more complicated. The following kinetic scheme becaomes the standard HH sodium model in the mean field limit:

which is an EIGHT!! state model. In the last 15 years, more careful experiments shed doubt on this model and Patlak has devised a more realistic model for this which has the following kinetic scheme:

The transitions to the inactive state ** I1 ** are all voltage
independent and can occur from three different points in the
system. The alpha's and beta's are all as in the standard HH model and
the remaining parameters are ** k1=0.24, k2=0.4, k3=1.5 ** all in
msec^{-1}. You should simulate this for a 20 msec step from -100 mV
to 20 mV and look at the sodium current which is

Here is an XPP file if you don't want to try it yourself. The result of a simulation of 1, 10, and 100 channels (trials) is shown here:

Note that the current is scaled and has no particular units.

- Write down and simulate a single channel model for the Morris-Lecar calcium channel
- Write down a kinetic scheme for the T-type calcium channel
assuming the the channel has the following current:
I _{T}= g m^{2}h (V-E_{ca})(Hint: it will look alot like the 8 state sodium channel but with fewer states.)

- Simulate a three-state model which has the following transitions:
- closed -> open 10/sec
- open -> closed 100/sec
- open -> inactive 50/sec
- inactive -> open 5/sec