Page 116 - ITU Journal Future and evolving technologies Volume 2 (2021), Issue 3 – Internet of Bio-Nano Things for health applications

ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 3

2.3 Multilayer cell membrane

In the previous section, we described how the inite el‑
ement method transforms the cable theory into a cir‑
cuit with different compartments. In this section, we
present how circuit parameters for each compartment a
are calculated. There are three circuit elements we need d a d m d l
to investigate, namely, forward resistance, , leakage axoplasm

resis‑ tance, , and leakage capacitance, . Schwann

Cell myelin
Forward resistance layer
ℎ
The forward resistance for the compartment is ob‑ intracranial axolemma
tained as fluid
Δ

= (6)
2
Fig. 3 – Cross‑sectional view of myelinated axon.
where is the speci ic forward resistance of the axon.

Note that = ∀ , as forward resistance is only due myelin layers are connected in series, we can calculate the

to the cytoplasmic resistance of axon, not on the myelin total axonal leakage resistance of a segment with length
covers spanning the axon. Δ and myelin cover from axoplasm to the intracranial
luid as
Leakage resistance + +

= ∑ 2 Δ ln ( + ). (10)
We calculate the leakage resistance of an unmyelinated 0
segment of thickness Δ as
In the paranodal region, myelin turns wrapping a segment
are not constant. If the last myelin layer is a partial turn,

= 2 Δ , (7) i.e., not covering the whole axolemma, we can still use Eq.
(10) with a slight modi ication such that
where is the axon radius and is the speci ic leakage ⌊ ⌋

resistance. + +

= ∑ ( 2 Δ ln ( + )) + (11)
In order to calculate the leakage resistance of a myeli‑ 0
nated segment, we irst need to use the result in Eq. (7) to ( − ⌊ ⌋) ln ( + + ⌈ ⌉ ) (12)

obtain the resistivity of axolemma, i.e., 2 Δ + ⌈ ⌉

1 where the last part is due to the partial wrap. Note that
= (8) here is not an integer. Rather, − ⌊ ⌋ gives us the ratio
2 Δ ln ( + )
of the partial cover.

= (9) Finally, we can calculate the leakage resistance of the ℎ
ln ( + )
piece of a length , radius and the paranodal region
where is the axolemma thickness and we assumed ax‑ length myelinated segment with Eq. (11) by switching

olemma to be cylindrical. to as
In case axolemma is covered with multiple myelin turns, ⎧ Δ , Δ ≤
we can assume each turn to be cylindrical resistors with {
inner thickness + × and outer thickness + + = ⎨ ( − Δ ) , Δ ≥ − (13)

× where is the myelin layer thickness. Note { .
⎩ ,
that myelin tissue shows remarkable similarities with the
axolemma. Hence, we model myelin as axolemma en‑
capsulating some intracranial luid, where is the to‑ Leakage capacitance
tal layer thickness including the encapsulated intracranial
luid and the myelin itself, while is only the thickness Here we can pursue an approach similar to leakage resis‑

of myelin, equal to the axolemma thickness. The cross‑ tance. Leakage capacitance of an unmyelinated segment
sectional view of our model is depicted in Fig. 3. Since all of thickness Δ calculated as

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