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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.

Subsections

- Basic Use: A short overview
- The `chanmode` Field: Modes of Operation
- Tabulated Calculations
  - The `calc_mode` field.
  - The Table Properties.
- Interpreting the Mode of Operation
- Exercise: Speed Comparison
- Communication with Other Elements
  - Messages from and to external objects
  - The '`findsolvefield`' command
- A Practical Example 1: `Hsolve` and the `efield` Object
- A Practical Example 2: Experimental Setups with `Hsolve`
  - Chronology of Commands
  - Setup of Experiments
    - Current Injection
    - Input to Synaptic Channels

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