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
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
Impedance.compute()function. One should also review the
Impedance.deltafac()method which defines the accuracy of the calculation.
- 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.
.compute(freg, 1, maxiter=500)
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
fcurrent()was called. The compute call is expensive but as a rule of thumb is not as expensive as
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
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.
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
ParallelContext.target_var()) then all instances of that type must participate in gap junctions.
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
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
Impedance.ratio()which voltage clamped y and recorded at x. I regret the original convention which was an artifact of
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
Impedance.ratio(), and also with
Impedance.input(), when the second
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).
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.
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.
- absolute amplitude of
v(x)/i(x)of the specified section
|v(loc)/v(x)|Think of it as voltage clamping to 1mV at x at some frequency and recording the voltage at loc.
- 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.
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
fcurrent().(actually the present implementation calls
compute()call but that could be fixed if increased performance was needed).
fac = imp.deltafac()
fac = imp.deltafac(fac)
- 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))/deltawhere 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
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.