For calculating input and transfer impedances at an instant of time Usage involves first defining a location either for the current stimulus or else the voltage measuring electrode, then computing the global transfer and input impedance function at a particular frequency, then retrieving values of the complex transfer and input impedance at particular locations.
The default calculation (the only calculation prior to version 5.3) is defined by di/dv only. i.e it assumes conductances depend only locally on v and does not take into account the impedance contributions of gating state differential equations. Specifically, the cable equation, c*dv/dt = i(v), where the d2v/dx2 compartmental terms are in i, yields the linearized impedance matrix [(jwc - di/dv)v = i0 ] * exp(jwt) The di/dv terms, apart from the axial terms, are defined by the current calculation in the BREAKPOINT blocks of the membrane mechanisms.
In version 5.3 the calculation was extended to take into account effects of other gating states. The calculation is currently limited to systems that can be solved with the CVode method but can be extended to systems solvable by the DASPK method. ie. currently one cannot deal with the extracellular mechanism or :class:`LinearMechanism`. It would be easy to implement the LinearMechanism extension and that would be the only way one could handle non-local interactions such as gap junctions. The extension takes into account not only di/dv but also di/ds, ds'/dv and ds'/ds contributions to the impedance where s are all the other states of the membrane mechanisms. i.e the system can be written
|c 0| * d/dt |v| = |i(v,s)|
|0 1| |s| |f(v,x)|
which is formally similar to the original. E.g. the hh mechanism has a sodium channel defined by
ina = gnabar*m^3*h*(v - ena)
m' = (minf - m)/mtau
h' = (hinf - h)/htau
the only thing contributed (one compartment) by the default method is
(jwc + gnabar*m^3*h) * v = i0
but the full linearized method contributes a matrix of terms like
(jwc + gnabar*m^3*h) gnabar*3*m^2*h*(v-ena) gnabar*m^3*(v-ena)
-d((minf - m)/mtau)/dv (jw - 1/mtau)
-d((hinf - h)/htau)/dv (jw - 1/htau)
associated with the vector of states (v, m, h)
The extended full impedance calculation is invoked with an extra argument to the :meth:`Impedance.compute` function. One should also review the :meth:`Impedance.deltafac` method which defines the accuracy of the calculation.
.. method:: Impedance.loc
Syntax:
``.loc(x, sec=section)``
Description:
A fixed current stimulus or voltage electrode location
at position 0<=x<=1 of the specified section.
This is needed for the transfer impedance calculation. Note that
transfer impedances obey the relation
\ ``v(x)/i(loc) == v(loc)/i(x)`` where *loc* is the fixed location and
x ranges over every position of every section.
.. method:: Impedance.compute
Syntax:
``.compute(freq)``
``.compute(freq, 1)``
``.compute(freg, 1, maxiter=500)``
Description:
Transfer impedance between location specified above and any other
location is computed. Also the input impedance at all locations
is computed -- \ ``v(x)/i(x)``
Frequency specified in Hz.
All membrane conductances are computed and used in the
calculation as if \ :func:`fcurrent()` was called.
The compute call is expensive but as a rule of thumb is not
as expensive as \ :func:`fadvance()`.
Since version 5.3, when the second argument is 1, an extended impedance
calculation is performed which takes into account the effect of
differential gating states. ie. the linearized cy' = f(y) system is used
where y is all the membrane potentials plus all the states in KINETIC and
DERIVATIVE blocks of membrane mechanisms. Currently, the system must
be computable with the Cvode method, i.e.extracellular and
LinearMechanism are not allowed. See :meth:`Impedance.deltafac`
Note that the extended impedance calculation may involve a singular matrix
because of the negative resistance contributions of excitable channels.
If the extended impedance calculation has been chosen (second arg = 1)
then parallel gap junction effects will be taken into account.
But for parallel gap junctions, there are several qualifications:
One and only one rank can have a stimulus location. :meth:`Impedance.loc`
can be used with an arg of -1 to remove the stimulus location on
a rank.
Every rank must participate in the call to compute (because of the use of
MPI collective calls to carry out the impedance calculation). Note that only the
freq arg value on the rank that has a location matters. If not all ranks have the
second arg value of 1, the machine will hang in an MPI collective call.
Not more than 5 types of gap junction POINT_PROCESS mechanisms can be instantiated.
If any POINT_PROCESS instance participates in a gap junction
(via :meth:`ParallelContext.target_var`) then all instances of that type
must participate in gap junctions.
Only :meth:`Impedance.transfer` and :meth:`Impedance.transfer_phase` can be used
to access the impedance values.
Ranks do not have to participate in the calls to the those two
methods since no MPI collective calls are involved. After
:meth:`Impedance.compute` is called, the transfer impedance is available at any
cell location and multiple calls from a rank are allowed. Note that if the stimulus
location is at location x and the transfer impedance is obtained at location x and
y, the input impedance is known only at location x (equal to the transfer impedance)
and the voltage ratio is known only at x and y. Note that the voltage ratio at
x is trivially 1.0, and the voltage at y, given that x is voltage clamped to a 1mV
sine wave with freq, is transfer(y)/transfer(x) . Unfortunately this is the opposite
of the definition given for :meth:`Impedance.ratio` which voltage clamped y
and recorded at x. I regret
the original convention which was an artifact of
:meth:`Impedance.compute` with args (freq, 0) calculating at one time, not only all the transfer
impedances, but also all the input impedances at every location. The problem with
the original convention for :meth:`Impedance.ratio`, and also with
:meth:`Impedance.input`, when the second :meth:`Impedance.compute` arg is 1,
is that their use necessitates a solve with a moved input stimulus location
specified by their argument. This is very inconvenient in a parallel context, as
that solve would require the participation of all the ranks where all the args except
one would have to be -1. An error message will be generated if one attempts to use the
ratio or input methods in the context of parallel gap junctions when nhost > 1.
Impedance calculations with parallel gap junctions use the
Jacobi iterative method to solve the linear matrix equation.
This method converges linearly and the number of iterations
required is proportional to the gap junction strength. Up to 500 iterations
are allowed before an error message is generated. Iteration stops when no state
changes more than 1e-9 after an iteration. It is expected that the number of
iterations will be quite modest with realistic gap junction conductances (a dozen
or so). A third argument to .compute specifies the maximum number of iterations
(default 500).
.. warning::
There are many limitations to the extended linearization of the
complete system. It basically handles only voltage sensitive
density channels where the gating states are defined by
DERIVATIVE or KINETIC blocks. Prominent limitations are:
extracellular mechanism not allowed.
LinearMechanism not allowed.
Because we are not doing the complete full df/dy calculation, there
may be interactions between states that are not computed.
An example is where ion concentration
equations are voltage sensitive in one mechanism and then the ionic
current is concentration sensitive in another mechanism. ie. the
typical way NEURON deals with ionic concentration coupling to current
is not handled.
.. method:: Impedance.transfer
Syntax:
``.transfer(x, sec=section)``
Description:
absolute amplitude of the transfer impedance between the position
specified in the \ ``loc(x)`` call above and 0<=x<=1 of the
specified section at the freq specified by a previous
compute(freq). The value returned can be thought of as either
\ ``|v(loc)/i(x)| or |v(x)/i(loc)|``
Probably the more useful way of thinking about it is to assume
a current stimulus of 1nA injected at x and the voltage in mV
recorded at loc.
Return value has the units of
Megohms and can be thought of as the amplitude of the voltage (mV)
at one location that would result from the injection of 1nA at the
other.
.. method:: Impedance.input
Syntax:
``.input(x, sec=section)``
Description:
absolute amplitude of \ ``v(x)/i(x)`` of the specified section
.. method:: Impedance.ratio
Syntax:
``.ratio(x)``
Description:
\ ``|v(loc)/v(x)|`` Think of it as voltage clamping to 1mV at x at some
frequency and recording the voltage at loc.
.. method:: Impedance.transfer_phase
Syntax:
``.transfer_phase(x)``
Description:
phase of transfer impedance. The phase is modulo 2Pi in the range
-Pi to +Pi so as one moves away from the loc remember that the
actual phase can become less than -Pi. If the amplitude is very
small the phase may be inaccurate and cannot be computed at all
if the amplitude is 0.
.. method:: Impedance.input_phase
Syntax:
``.input_phase(x)``
Description:
phase of input impedance.
Note: Impedance makes heavy use of memory since four complex
vectors are allocated with size equal to the total number of
segments. After compute is called two of these vectors holds
the input and transfer impedance for a given loc, freq, and
neuron state. Because
of the way results of calculations are stored it is very efficient
to access amp and phase; reasonably efficient to change freq or loc,
and inefficient to vary neuron state, eg, diameters. The last case
implies at least the overhead of a call like \ :func:`fcurrent()`.(actually
the present implementation calls \ :func:`fcurrent()` on every \ ``compute()`` call
but that could be fixed if increased performance was needed).
.. method:: Impedance.deltafac
Syntax:
``fac = imp.deltafac()``
``fac = imp.deltafac(fac)``
Description:
Gets or sets and gets the factor used in computing the numerical derivatives
during calculation of the extended full impedance. Jacobian elements are
calculated via the formula ``f(s+delta) - f(s))/delta`` where
delta is defined by fac * the state tolerance scale factor for cvode.
Note that default state tolerance scale factors are 1.0 except when
specifically declared in mod files or changed by calling
:meth:`CVode.atolscale`. The default delta factor is 0.001 which is consistent
with the factor used by the default impedance calculation. Note that the
factor for the default impedance calculation cannot be changed.