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# Impedance

class `Impedance`

  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.

`Impedance.``loc`()
  Syntax:
    `.loc(x)`
  Description:
    A fixed current stimulus or voltage electrode location at position 0<=x<=1 of the currently accessed 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.

`Impedance.``compute`()
  Syntax:

    `.compute(freq)`

    `.compute(freq, 1)`

  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 `fcurrent()` was called. The compute call is expensive but as a rule of thumb is not as expensive as `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 `Impedance.deltafac()`

    Note that the extended impedance calculation may involve a singular matrix because of the negative resistance contributions of excitable channels.

  Warning

  The calculation is incomplete in the presence of nonlocal mechanisms such as gap junctions.

  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. Gap junctions mentioned above are a case in point. Others are 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.

`Impedance.``transfer`()
  Syntax:
    `.transfer(x)`
  Description:

    absolute amplitude of the transfer impedance between the position specified in the `loc(x)` call above and 0<=x<=1 of currently accessed 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.

`Impedance.``input`()
  Syntax:
    `.input(x)`
  Description:
    absolute amplitude of `v(x)/i(x)` of the currently accessed section

`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.

`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.

`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 `fcurrent()`.(actually the present implementation calls `fcurrent()` on every `compute()` call but that could be fixed if increased performance was needed).

`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 `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.

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