Page 115 - 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

Myelin
Initial Paranode Internode Terminal Paranode NoR
Axon

k-1 rm k-1 cm k k cm k+1 rm k+1
cm
... rm ...

k-1 k k+1
rf rf rf
Patch k

Fig. 2 – Two‑dimensional diagram of myelinated axon and the circuit corresponding to a myelinated segment. The regions where axon is surrounded
by myelin sheath are called internode and remaining parts without myelin are called Node of Ranvier (NoR). A myelinated segment consists of intern‑
odal region with constant myelin thickness, initial and terminal paranodes, where number of myelin wraps depends on the distance from and to NoR,
respectively.

In the simplistic model of the myelinated axon, leakage Since solving this equation is hard, we employ the i‑ nite
resistance and capacitance are constant. However, due difference method, by partitioning our model into
to the shape of the myelin sheath, paranodal regions do patches.
have different leakage resistance and capacitance values.
Therefore, rather than taking and constant, we take 2.2 The discrete axonal cable

them as a function of the distance from the start of the
myelinated segment. Hence, We consider a multi‑compartment model of the myeli‑
nated axon to solve the inhomogeneous heat low equa‑
∼ ( ) (3)
tion described in Section 2.1. Since this problem is hard
∼ ( ) (4) to solve analytically, we attempt to solve this problem us‑
ing the inite element method. By compartmentalizing
The lumped circuit approximation, valid due to the small the myelinated segment, where we assume the resistance
dimensions of the myelinated segments compared to the and capacitance values are constant for the given com‑
wavelengths of the dominant components forming the ac‑ partment, we obtain a numerical solution to the heat low
tion potential, helps us to assume any ield effect is in‑ problem.
stantaneous throughout the entire segment. Therefore,
to calculate attenuation values in the steady state, partial Here, we divide a length‑ myelinated segment into
derivative of voltage with respect to the time variable be‑ compartments, where is a large integer [20]. Each of
comes zero, i.e., = 0. Hence, the resulting equation these pieces consists of three circuit elements, i.e., a for‑

becomes ward resistance, , leakage resistance, and leakage
capacitance, .

2
d
2
= ( ) . (5) The equivalent circuit is shown in Fig. 2. Looking at Fig.
d 2
ℎ
2, we see the ( − 1) , ℎ and ( + 1) ℎ compartments
of the myelinated segment, which we divided into com‑
This is the inhomogeneous heat low equation for the
partments. The length of each compartment is Δ = .
steady state [19]. Apart from rare simple cases, the
cable equation cannot be solved analytically but Note that the smaller Δ is, the more accurate the inite
numerically [13].
element method will be.
© International Telecommunication Union, 2021 103