Table 1.
Commonly used abbreviations.
Table 2.
Commonly used symbols.
Figure 1.
Excitability model template: The equivalent circuit represents the simplified electro-dynamics of an excitable membrane.
is the intra-cellular stimulation current.
is the capacitive current. The direction of the latter is for a case of depolarizing the membrane’s voltage (i.e. the inside of the cell wall becoming more positive). The algebraic sum of all the ionic and all axial currents is represented by
where
stands for the algebraic difference (divergence) of in- and out-going axial currents.
Table 3.
Definition and notation for the key HHM variables.
Table 4.
Gate-state dynamics parameters.
Figure 2.
The MRG’02 myelinated axon model (See also Table 4) Box: Equivalent circuit for current injection into the center RN (#1).
Table 5.
MRG’02 double-cable model-axon electrical parameters.
Figure 3.
Propagating AP’s and spatial profile of the membrane voltage & intracellular potential
(at the end of stimulation, please also see Fig. 2);
is the 1D axonal spatial coordinate.
The peaks of at the Ranvier nodes are due to the direct exposure to the extracellular medium, which is unlike that of the myelinated sections in the double-cable MRG’02 model.
Table 6.
MRG’02 double-cable model-axon geometric parameters, in .
Figure 4.
LAP energy-optimal and
for the LM: for
respectively 10
and 5
; the time constant
was varied as indicated in the legend; membrane capacity was constant -
= 1
, while membrane (leak) conductance
was respectively 0.2, 1 and 5
; The 3 solutions shown correspond to the nominal
= 1
(cyan trace) or 5-fold shorter (thin red dash-dot), or 5-fold longer (thick dashed black)
respectively; (thin dashed black) rectangular pulse with amplitude
.
Figure 5.
LAP optimal waveforms and
for the 0D IM: The 3 solutions shown correspond to the nominal IM opposing current (cyan trace), twice higher (thin red dash-dot), or twice lower (thick dashed black)
respectively.
The approximation of the ionic current is used for a case of very short duration (
= 10
) and the
approximation is used for a case of long duration (
= 5
). It is important to notice that - as with the
model above,
, where
(see the Box) Box: Resting-state
and asymptotic-state
ionic currents for the 0D IM; Markers are inserted at the resting and threshold membrane-voltage points, respectively
= −70,
= −55 and
= −50
.
Figure 6.
LAP optimal waveforms and
for the 0D HHM: The
approximation of the ionic current is used for a case of very short duration (
= 10
) and the
approximation is used for a case of long duration (
= 5
) (see the Box).
As with the IM, bvp4c was used to numerically solve the BVP of eqn. (34). The figure follows a quite similar format to Fig. 5. can also be assumed higher or lower. All the maximal ionic conductances in the HHM (see also Table 3) are temperature-dependent and are linearly proportional to the coefficient
. The 3 solutions shown correspond to the ionic current at
(cyan trace), twice higher (thin red dash-dot), or twice lower (thick dashed black)
respectively. From eqn. (42) we can see that
= 1.6047 (half the nominal) at
, and
= 6.4188 (twice the nominal) for at
. Box: Resting-state
and asymptotic-state
ionic currents for the 0D HHM; Markers are inserted at the resting and threshold membrane-voltage points, respectively
= −77
,
= −64.55
and
= −52.35
.
Figure 7.
The LAP vs or with numerical optimisation for the 0D IM, with = 2
: see also Fig. 5 which shows that an initial guess
, based on the linear-growth rate
is still valid with
= 2
dand
= −50
.
panel A: discrete-time IM and FHOC panel B: continuous-time IM and FHOC, using CVODES adjoint sensitivity analysis capabilities upper plots: (dashed black) a rectangular pulse with amplitude ; (thick cyan) the LAP
; (thick black) the best FHOC
lower plots: (dashed black) linear-growth evolution of the membrane potential from
at
to
at
; (dotted gray) the desired threshold value
= −50 mV; (thick cyan) the resulting LAP
; (thick black) the resulting FHOC
.
Figure 8.
The MRG’02 model: Toward upper bounds on : the figure presents a family of ionic current
approximations at the target site (
), for a set of durations
.
For each of the durations it is assumed that the membrane voltage trajectory evolves according to a linear ramp from rest
to threshold
(the unknown). For each
value on the horizontal (independent-variable) axis of the figure, a
ramp was assumed and the corresponding ionic current
was computed, based on approximate gate states (see the Box). Note: for the sake of better visibility, a
gain is applied to the approx.
for the case of
= 5
. Box: For a chosen
= 5
and as
is linearly ramped up, for each gate state the plots show the ratio
, where
is given by eqn. (46) to its asymptotic value - both functions of
. Legend for gate states: opening
and closing
gates for the fast
ion-channel subtype;
persistent
channel gates;
slow
gates.
Figure 9.
The actually computed as a function of
: Notice how the computed
value is rather similar (almost matched) between the linear and exponential cases, for
respectively 2 and 5 ms; and between the
-order and linear cases, for
respectively 0.2 and 0.5 ms. see also Fig. 10.
Table 7.
Minimal values for the MRG’02 model, obtained for each
trajectory class.
Figure 10.
The energy and charge-transfer
values as a function of
: The linear-ramp voltage profile yields the best
performance for most of the durations.
As in Fig. 8 notice that the and
values are quite similar for the linear and exponential cases, for
respectively 2 and 5 ms; and also for the
-order and linear cases, for
respectively 0.2 and 0.5 ms. Toward the
values electrode impedance of 1
is assumed. Contrasted:
stands for the square (or rectangular) stimulation waveform.
Table 8.
Minimal values for the MRG’02 model, obtained for each
trajectory class.
Figure 11.
Optimal waveforms ,
= 20, 200
: The figure also provides the corresponding optimal
-like linear-growth-related current
(dashed black), as well as the components of
- respectively the
(blue traces) and
(red traces) current trajectories.
Figure 12.
Optimal waveforms : see also Fig. 11. Notes:
Since here , where
is given by eqn. (41), from eqn. (6)
. The figure is optimized to present clearly both
and
(*1) The dashed trace at the bottom plots
as a function of
(*2) Toward equally good plot visibility, for all durations
, the waveforms
are rubber-banded to take the same graph width as the 1 ms-waveform. This is illustrated by the scale bars for the shortest duration
= 20
. (*3) The vertical scale is the same for all plots, except for the logarithmic offset, as defined by pt. (*1) above.
Table 9.
Minimal values for the MRG’02 model, obtained for each
trajectory class.
Figure 13.
Propagating AP due to an optimal (rectangular) waveform,
= 100
: For the shortest durations, the plain rectangular waveform outperforms by
the ones associated to the linear-ramp voltage profile.
One can see clearly that the steep rise of the waveform yields an early superlinear ramping of the membrane voltage. However, the rectangular waveform requires a lot more charge
to be transferred (see Fig. 10).