Introducing Hsolve for Single Cells

If you create a set of coupled compartments with Genesis, you already have discretized your neuron so we will not consider spatial discretization any further. In this section we introduce the Genesis 'hsolve' object. To implement the time discretization, this object implements some of the previously introduced numerical methods and combinations thereof.

So far we have identified the backward-Euler and the Crank-Nicolson rules as implicit finite-difference schemes. The explicit schemes we encountered are the exponential Euler and the forward-Euler. Since we have to solve two types of equations (cable equation and Hodgkin-Huxley equations), these different time-discretization techniques can be applied to the different equations and combined at will:

1. Solve all equations with an explicit method (forward-Euler or exponential-Euler). This isolates all compartments to single equations that can be solved independently. Being the default numerical methodology in Genesis, you are supposed to be familiar with it. See [2] for more details.
2. Solve the cable equation with an explicit method, solve the Hodgkin-Huxley channel equations with an implicit method. This is rather uncommon and will not be treated either.
3. Solve the cable equation with an implicit method, solve the Hodgkin-Huxley channel equations with an explicit method at the same time points.
4. Solve the cable equation with an implicit method and solve the Hodgkin-Huxley channel equations with an implicit method at halve time points.

The implicit solver of Genesis, 'hsolve', implements the last two points in the above enumeration. We cover the use of hsolve in the following sections.