![](bg5.png)
Figure 3: R vs
ω
2π
for the 3
rd
order model - Plots the-
oretical and numerically computed reliability versus
ω
2π
,
with c
2
= 0.015, c
1
= 0.075, I
0
= 6.5, T
r
= 105ms, T
0
=
80ms,T = 180ms. Dotted lines are lower and upper bounds
on reliability from the (16). Solid line is plotting R
emp
cal-
culated by running simulation of (1), error bars show ±std.
We estimated I
th
= 6 as the minimum hieght of a r(t) pulse,
that make the neuron spike.
In Figure 3, we plot our reliability bounds (16)
along with reliability computed numerically through
simulation of the 3rd order model. We see that our
bounds predict the reliability well except for some er-
rors in the low frequency range. This is due to the fact
that we did not consider higher order dynamics of the
3
rd
order model. However, our bounds qualitatively
predict the trend of reliability well even for the higher
order model.
In general, our analytical bounds are applicable to
higher dimensional models as long as the model 1.
does not generate a spike if there is no pulse in r(t),
and 2. has a threshold behavior. The second condi-
tion is true for most of neurons that satisfy the first
condition. Our analysis may also be extended to in-
clude neurons that spike without any driver input, but
in this manuscript we neglect such dynamics.
4 DISCUSSION
In this manuscript, we studied the reliability of a relay
neuron. A relay neuron receives two inputs: a driving
input, r(t), and a modulating input, u(t). The neuron
generates one output, V (t), which relays r(t) condi-
tioned on u(t). Our goal was to precisely determine
how the modulating input impacts relay reliability. To
calculate relay reliability, we used LTI systems theo-
retic tools to derive the analytical bounds (16) on relay
reliability as a function of different input and model
parameters. Specifically, (16) implies that if the mod-
ulating input is of the form u(t) = c
1
+c
2
sin(ωt), then
increasing c
1
or c
2
decreases reliability. However, in-
creasing ω increases reliability. In addition, our reli-
ability curve (see Figure 3) suggests that on increas-
ing ω, reliability first increases slowly and then in-
creases rapidly and plateaus. (16) is powerful as it
characterizes the multiple dependencies of reliability
on u(t), r(t) and relay neuron model parameters. Fur-
thermore, analytic bounds from (16) contain results
obtained through simulation of the 3
rd
order models
of a relay neuron. Our bounds captured reliability
under both the depolarized and hyperpolarized (not
shown due to limited space) states of the 3rd order
neuron and shows the generality of our analysis.
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