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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:
- 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.
- 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.
- Solve the cable equation with an implicit method, solve the
Hodgkin-Huxley channel equations with an explicit method at the same
time points.
- 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
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2002-11-15