Numerical solution of passive models

Once we get beyond a single compartment model, it is much easier to simulate the behavior rather than attempting to explicitly solve the differential equations that you derive. In particular, once that are more than 2 compartments and once channels and synapses are added, simulation is just about the only generally applicable tool for studying behavior. There are many simulation programs available and we will talk about some of them later in the term; notably GENESIS and NEURON, both of which are specifically designed for neural simulations. Both of these simulators solve differential equations, but their interface essentially removes the actual equations from you. We will use a general purpose differential equation solver in which we must explicitly provide the relevant equations. The simulator is called XPPAUT and runs on any UNIX system with X windows. A version that is similar but with some interface differences is avaliable for Windows 95/NT.

The first model we will simulate is the single compartment injected by a current (3)

$$C_M A \frac{dV}{dt} = A\frac{V_m-V}{R_M} + A\bar{I}(t)$$

where A is the area of the membrane compartment. We can divide this whole thing by A to get rid of it. We are interested in a scale of millivolts, milliseconds and picoamps. Dividing by A C_M we obtain:

$$\frac{dV}{dt} = (\frac{V_m-V}{R_M}+\bar{I}(t))/C_M.$$

We use the following values for the parameters: $R_M=10000\Omega-cm^2$,$C_M=1\mu F/cm^2$. The with units of milliseconds and millivolts, we can set C_M=1 and obtain:

$$\frac{dV}{dt} = (1000\frac{V_m-V}{R_M}+\bar{I}(t))/C_M$$

where $\bar{I}(t)$ is measured in $\mu A/cm^2.$ We can view the reciprocal of the resistance as a conductance measured in Siemens/cm². The factor of 1000 comes from the conversion to millivolts and from the factor of 10^-6 from the capacitance. We typically measure the conductance, g_M=1/R_M in $\mu S/cm^2$ so that we finally obtain:

$$\frac{dV}{dt} = (g_M(V_m-V)+\bar{I}(t))/C_M$$

where g_M=10³/R_M is the membrane conductance in $\mu S/cm^2.$ All units are now in terms of millivolts, milliseconds, and microamps, microsiemens, microfarads per square centimeter.