Page 117 - 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

= Δ (2 ). (14) Information transmission through axons is affected by

noise sources including channel noise due to random luc‑
tuations of voltage‑gated ion channels at the node of Ran‑
Using Eq. (25), we can calculate electric permittivity of vier, thermal noise (Johnson noise) produced by mem‑
axolemma as
brane resistance, membrane capacitance, and cytoplas‑
mic resistance, and crosstalk noise caused by axon‑axon
+ interactions such as ephaptic coupling [22, 23]. Concern‑
= ln ( ) (15)
2 Δ ing the internodal channel, we do not consider channel
+ and crosstalk noise, since the former is effective at the
= ln ( ) . (16) nodes of Ranvier and the latter is speci ic to neuron types

[24]. Here, thermal noise is the only noise source that af‑
fects the internodaltransmission in general; therefore, we
use it to offer an upper bound for the per‑use rate of the
Similar to the leakage resistance calculations, we can
reach the leakage capacitance by modeling the myelin internodal channel. Thus, the power spectral density of
turns as concentric cylindrical capacitors with inner radii thermal noise voltage is given by
+ × and outer radii as + + × . The resultant

capacitance would be ( ) = 2 Re{ ( )}, (20)

2 Δ where is the Boltzmann constant, T is the temperature
= . (17) in Kelvin and is the total impedance of the system [23].
∑ ln ( + + ) Since, depends on frequency, thermal noise is also fre‑
+
0
quency dependent.
We can calculate the leakage capacitance of a segment Action potential, the input to the channel, is represented
with length Δ in case of partial myelin turns as by voltage variable ( ). The power spectral density of
signal ( ) over the inite time interval [0, ] is given by
2 Δ
= ⌊ ⌋ . (18) 1 2
(∑ ln ( + +⌊ ⌋ )) + ln + +⌈ ⌉ ( ) = lim | ( )| , (21)
→∞
+⌊ ⌋ +⌈ ⌉
0
where ( ) is the Fourier transform of ( ). The input
signal has power constraint that
Similar to the leakage resistance, we can ind the leakage
capacitance at the ℎ segment by using Eq. (18) with ∫ ( ) ≤ , (22)
instead of as in Eq. (13).

Outer resistance where is the average transmitted signal power over
given bandwidth .
Outer resistance, or extracellular resistance, is the resis‑ Hence, the rate per channel use of a myelinated segment
tance of the luid between axons [21]. Here, we assume under thermal noise is bounded by
that outer resistance is due to the extracellular luid en‑
closed by the endoneurium. /2
2
| ( )| ( )
< ∫ log (1 + )d , (23)
We calculate the outer resistance similar to the forward 2 ( )

resistance, i.e., − /2
Δ where ( ) is the channel gain function and is chan‑

= ( − ( − ) ) , (19) nel bandwidth. The channel capacity of a myelinated seg‑

2
2
ℎ

ment can be obtained by water‑ illing. However, due to
where is the speci ic outer resistance, is the radius the frequency‑dependent gain function, irst, we need to

of the endoneurium and is the thickness of the en‑ calculate the effective noise spectral density, i.e.,
ℎ
doneurium. Since endoneurium thickness and radius are

accepted to be constant through the axon, , similar to ( )

, is independent of . ( ) = | ( )| 2 . (24)

3. INTERNODAL CHANNEL CAPACITY Thus, the capacity of the channel is found as
( )
In this section, we ind the rate per channel use in intern‑ = Δ ∑ log (1 + ), (25)

2
odal regions of myelinated axons. =1 ( )

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