classes compute input loc transfer deltafac input_phase ratio transfer_phaseFor 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
which is formally similar to the original. E.g. the hh mechanism has a sodium channel defined by|c 0| * d/dt |v| = |i(v,s)| |0 1| |s| |f(v,x)|
the only thing contributed (one compartment) by the default method isina = gnabar*m^3*h*(v - ena) m' = (minf - m)/mtau h' = (hinf - h)/htau
but the full linearized method contributes a matrix of terms like(jwc + gnabar*m^3*h) * v = i0
associated with the vector of states (v, m, h)(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)
The extended full impedance calculation is invoked with an extra argument to the compute function. One should also review the deltafac method which defines the accuracy of the calculation.
v(x)/i(loc) == v(loc)/i(x)where
locis the fixed location and x ranges over every position of every section.
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 deltafac
Note that the extended impedance calculation may involve a singular matrix because of the negative resistance contributions of excitable channels.
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.
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.
v(x)/i(x)of the currently accessed section
|v(loc)/v(x)|Think of it as voltage clamping to 1mV at x at some frequency and recording the voltage at loc.
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
the present implementation calls
fcurrent() on every
but that could be fixed if increased performance was needed).
fac = imp.deltafac()
fac = imp.deltafac(fac)