u_{j}^{(N)}\\}}\mathds{1}_{\cup_{j\in[|r,N_{+}|]}\mathtt{E}_{\frac{1}{2}m}^{(j)}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.\mathds{1}_{x+\sqrt{\frac{1}{2}}W_{\tau_{N_{+}}^{(r)}}\in-\frac{3}{4}\log
N+[z,z+v)}\right)$
$\displaystyle\ll_{\upsilon}e^{2(x-z)}2^{r-N}N^{\frac{3}{2}}{\mathbb{P}}\left(GEv(r,0,0,x,z)\cap\left\\{\cup_{j\in[|r,N_{+}|]}\mathtt{E}_{\frac{1}{2}m}^{(j)}\right\\}\right)$

As $\mathtt{E}_{\frac{m}{2}}^{(l)}$ is measurable with respect to
$\sigma({\mathcal{N}}_{t}^{\mathbb{C}},\,t\in[|2^{l},2^{l+1}-1|])$, by
applying [CMN18, Corollary A.6] and using the Fact 4 we get, for $r$ large
enough and $N$ large enough depending on $r$:

$\displaystyle{\mathbb{P}}\left(GEv(r,0,0,x,z)\cap\left\\{\cup_{j\in[|r,N_{+}|]}\mathtt{E}_{\frac{1}{2}m}^{(j)}\right\\}\right)$
$\displaystyle\ll_{\upsilon}|xz|e^{2(x-z)}N^{-\frac{3}{2}}\sum_{j\geq
r}\sqrt{{\mathbb{P}}(\mathtt{E}_{\frac{1}{2}m}^{(j)})}$
$\displaystyle\ll|xz|e^{2(x-z)}N^{-\frac{3}{2}}\sum_{j\geq
r}e^{-(m^{2}/8)2^{j-\mathtt{k}_{+}}}$
$\displaystyle\ll|xz|e^{2(x-z)}N^{-\frac{3}{2}}e^{-2^{r-\mathtt{k}_{+}}m^{2}/8},$

since $r\geq\mathtt{k}_{+}$.

By combining this inequality with (B.41) and (B.3.2), we get:

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime},x,z,x^{\prime},z^{\prime})\ll_{\upsilon}|xzx^{\prime}z^{\prime}|e^{2(x-z+x^{\prime}-z^{\prime})}2^{2(r-N_{+})}$
$\displaystyle\qquad+\sum_{m\geq
1}|xzx^{\prime}z^{\prime}|e^{2(x-z+x^{\prime}-z^{\prime})}2^{r-N_{+}}2^{2m2^{-e^{\sqrt{\log
r}}}}(1+(m+1)2^{-e^{\sqrt{\log
r}}})^{3}2^{r-N}e^{-(m^{2}/8)2^{r-\mathtt{k}_{+}}}$
$\displaystyle\leq|xzx^{\prime}z^{\prime}|e^{2(x-z+x^{\prime}-z^{\prime})}2^{2(r-N_{+})}\left[1+\left(\sum_{m\geq
1}2^{2m}(1+m)^{3}e^{-m^{2}/8}\right)e^{-(1/8)(2^{r-\mathtt{k}_{+}}-1)}\right]$
$\displaystyle\ll|xzx^{\prime}z^{\prime}|e^{2(x-z+x^{\prime}-z^{\prime})}2^{2(r-N_{+})},$
(B.42)

which concludes the case $\mathtt{k}_{+}\leq r$.

When $\mathtt{k}_{+}\geq r$: Using the decomposition (B.36) and Fact 2 and
noticing that
$\sum_{l=\mathtt{k}_{+}}^{+\infty}m2^{-\frac{\Delta^{(l)}}{4}}\leq
m2^{-e^{\sqrt{\log r}}/8}$ for $r$ large enough, we can affirm that

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime},x,z,x^{\prime},z^{\prime})$
$\displaystyle\leq\sum_{m\geq
0}{\mathbb{P}}\left(Ev(r,0,0,x,z),\,\cup_{j\in[|\mathtt{k}_{+},N_{+}|]}\mathtt{E}_{m}^{(j)}\right)$
(B.43) $\displaystyle\sup_{l_{\mathtt{k}_{+}}^{(N)}\leq x^{\prime}+w\leq
u_{\mathtt{k}_{+}}^{(N)}}{\mathbb{P}}\left(Ev(\mathtt{k}_{+},(m+1)2^{-e^{\sqrt{\log
r}}/8},\mathtt{C},x^{\prime}+w,z^{\prime})\right).$ (B.44)

By a similar computation as what we have done in the case $\mathtt{k}_{+}\leq
r$, we get:

$\displaystyle{\mathbb{P}}\left(Ev(r,0,0,x,z),\,\cup_{j\in[|\mathtt{k}_{+},N_{+}|]}\mathtt{E}_{m}^{(j)}\right)\ll_{\upsilon}|xz|e^{2(x-z)}2^{r-N_{+}-(m^{2}/8)}.$
(B.45)

On the other hand, by using Eq. (B.29), for any
$w^{\prime}\in[l_{\mathtt{k}_{+}}^{(N)},u_{\mathtt{k}_{+}}^{(N)}]$, we obtain:

$\displaystyle{\mathbb{P}}\left(Ev(\mathtt{k}_{+},(m+1)2^{-e^{\sqrt{\log
r}}/8},\mathtt{C},w^{\prime},z^{\prime})\right)$
$\displaystyle\ll_{\upsilon}e^{2(w^{\prime}-z^{\prime})}2^{\mathtt{k}_{+}-N_{+}}e^{2(m+1)2^{-\frac{r}{400}}}N^{\frac{3}{2}}{\mathbb{P}}\left(GEv(\mathtt{k}_{+},(m+1)2^{-e^{\sqrt{\log
r}}/8}+\sup_{\mathtt{k}_{+}\leq j\leq
N}|\mathtt{C}_{j}|,\mathtt{C},w^{\prime},z^{\prime})\right)$
$\displaystyle\leq
e^{2(w^{\prime}-z^{\prime})}2^{\mathtt{k}_{+}-N_{+}}e^{2(m+1)2^{-\frac{r}{400}}}N^{\frac{3}{2}}{\mathbb{P}}\left(GEv(\mathtt{k}_{+},2(m+1)2^{-e^{\sqrt{\log
r}}/8},\mathtt{C},w^{\prime},z^{\prime})\right),$ (B.46)

where we used that $\sup_{j\geq\mathtt{k}_{+}}|\mathtt{C}_{j}|\leq
2^{-\Delta^{(\mathtt{k})}}\leq\frac{1}{2}(m+1)2^{-e^{\sqrt{\log r}}/8}$.

For $\mathtt{k}_{+}\leq N/4$, we can use [CMN18, Corollary A.6] to deduce, for
$w^{\prime}\leq u_{\mathtt{k}^{+}}^{(N)}=-(\mathtt{k}_{+})^{\alpha_{-}}$,

$\displaystyle{\mathbb{P}}\left(Ev(\mathtt{k}_{+},(m+1)2^{-e^{\sqrt{\log
r}}/8},\mathtt{C},w^{\prime},z^{\prime})\right)$ $\displaystyle\ll
e^{2(w^{\prime}-z^{\prime})}2^{\mathtt{k}_{+}-N_{+}}e^{2(m+1)2^{-e^{\sqrt{\log
r}}/8}}(1+2(m+1)2^{-e^{\sqrt{\log r}}/8})^{3}|w^{\prime}z^{\prime}|$
$\displaystyle\ll_{\upsilon}|w^{\prime}z^{\prime}|e^{2(w^{\prime}-z^{\prime})}2^{\mathtt{k}_{+}-N_{+}}e^{3(m+1)-(\mathtt{k}_{+})^{\alpha_{-}}}\
.$ (B.47)

For $\mathtt{k}_{+}\geq N/4$, we bound the probability of the $GEv$ event by
$1$ and use the fact that $w^{\prime}\leq-0.9(\mathtt{k}_{+})^{\alpha_{-}}$
(the factor $0.9$ coming from the case $\mathtt{k}\leq
N/2\leq\mathtt{k}_{+}$). We then get

$\displaystyle P\left(Ev(\mathtt{k}_{+},(m+1)2^{-e^{\sqrt{\log
r}}/8},\mathtt{C},w^{\prime},z^{\prime})\right)$
$\displaystyle\ll_{\upsilon}|w^{\prime}z^{\prime}|e^{2(w^{\prime}-z^{\prime})}2^{\mathtt{k}_{+}-N_{+}}e^{2(m+1)2^{-e^{\sqrt{\log
r}}/8}}N^{\frac{3}{2}}$
$\displaystyle\ll_{\upsilon}|z^{\prime}|e^{-2z^{\prime}}2^{\mathtt{k}_{+}-N_{+}}e^{2m+2}e^{-(\mathtt{k}_{+})^{\alpha_{-}}}(N^{3/2}e^{-0.8(\mathtt{k}_{+})^{\alpha_{-}}})$
$\displaystyle\ll|z^{\prime}|e^{-2z^{\prime}}2^{\mathtt{k}_{+}-N_{+}}e^{2m+2}e^{-(\mathtt{k}_{+})^{\alpha_{-}}}.$

This again implies (B.47). Finally, by combining this equation with (B.45) and
(B.43), we get

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime})$
$\displaystyle\ll_{\upsilon}|xzz^{\prime}|e^{2(x-z-z^{\prime})}2^{r-N_{+}}2^{\mathtt{k}_{+}-N_{+}}e^{-(\mathtt{k}_{+})^{\alpha_{-}}}\sum_{m\geq
0}e^{3m+3-(m^{2}/8)}$
$\displaystyle\ll|xzz^{\prime}|e^{2(x-z-z^{\prime})}2^{r-N_{+}}2^{\mathtt{k}_{+}-N_{+}}e^{-(\mathtt{k}_{+})^{\alpha_{-}}}$

which concludes the proof of Lemma B.4. ∎

###### Proof of Lemma B.5.

We can use (B.40) in order to get (for $r$ large enough and $N$ large enough
depending on $r$):

$\displaystyle{\mathbb{P}}(Ev(r,0,0,x,z)){\mathbb{P}}(Ev(r,2^{-e^{\sqrt{\log
r}}/8},\mathtt{C}^{(r)},x^{\prime},z^{\prime}))$ $\displaystyle\leq
e^{2(x-z+x^{\prime}-z^{\prime})}e^{4\upsilon+2^{1-e^{\sqrt{\log
r}}/8}}2^{2(r-N_{+})}N^{3}{\mathbb{P}}(GEv(r,0,0,x,z)){\mathbb{P}}(GEv(r,2^{1-e^{\sqrt{\log
r}}/8},\mathtt{C}^{(r)},x^{\prime},z^{\prime})),$

and then, by the majorization of the second term of (B.3.2) which is involved
in (B.42):

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime},x,z,x^{\prime},z^{\prime})$
$\displaystyle\leq$ $\displaystyle
e^{2(x-z+x^{\prime}-z^{\prime})}e^{4\upsilon+2^{1-e^{\sqrt{\log
r}}/8}}2^{2(r-N_{+})}N^{3}{\mathbb{P}}(GEv(r,0,0,x,z)){\mathbb{P}}(GEv(r,2^{1-e^{\sqrt{\log
r}}/8},\mathtt{C}^{(r)},x^{\prime},z^{\prime}))$
$\displaystyle+\mathcal{O}_{\upsilon}\left(2^{2(r-N_{+})}e^{-\frac{2^{r-\mathtt{k}_{+}}-1}{8}}\right).$

Hence, we have:

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime})$ $\displaystyle\leq
e^{2(x-z+x^{\prime}-z^{\prime})}e^{4\upsilon+2^{1-e^{\sqrt{\log
r}}/8}}2^{2(r-N_{+})}$ $\displaystyle\qquad\times
N^{3}{\mathbb{P}}(GEv(r,0,0,x,z)){\mathbb{P}}(GEv(r,2^{1-e^{\sqrt{\log
r}}/8},\mathtt{C}^{(r)},x^{\prime},z^{\prime}))$
$\displaystyle+\mathcal{O}_{\upsilon}\left(2^{2(r-N_{+})}e^{-\frac{2^{r-(r/2)-3e^{\sqrt{\log
r}}/8-100\log^{2}(r/2)}-1}{8}}\right).$

By applying [BRZ19, Lemmas 2.1,2.3] with the same argument as in Lemma C.1,
using the fact that $||E||_{1}$ goes to zero when $r$ goes to infinity, we get

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime})$ $\displaystyle\leq
e^{2(x-z+x^{\prime}-z^{\prime})}e^{4\upsilon}2^{2(r-N_{+})}N^{3}{\mathbb{P}}(Event_{r,N}(x,z)){\mathbb{P}}(Event_{r,N}(x^{\prime},z^{\prime}))(1+\eta_{r})$
$\displaystyle\qquad+\mathcal{O}_{\upsilon}\left(2^{2(r-N)}e^{-\frac{2^{r/3}-1}{8}}\right),$

where $\eta_{r}$ goes to zero when $r$ goes to infinity. Using (B.16), we
obtain Lemma B.4. ∎

###### Proof of Lemma B.6 in the general case.

The general case needs to uses exactly the same arguments as in the general
case of the proof of Lemma B.4. This time, for $\mathtt{k}_{+}\leq l\leq N-1$
and $p\leq 2^{\kappa^{(\mathtt{k})}}-1$, the random variables
$J_{l,p}^{(\mathtt{k})}(\theta)$ and $J_{l,p}^{(\mathtt{k})}(\theta^{\prime})$
are not rigorously independent. However, we observe that for
$\mathtt{k}_{+}\leq l\leq N_{+}$, the absolute value of their correlations,
decreases exponentially with $l$. Indeed, if
$C_{l,p}^{(\mathtt{k})}(\theta,\theta^{\prime}):={\mathbb{E}}\left(J_{l,p}^{(\mathtt{k})}(\theta)\overline{J_{l,p}^{(\mathtt{k})}(\theta^{\prime})}\right)$,
then

$\displaystyle\left|C_{l,p}^{(\mathtt{k})}(\theta,\theta^{\prime})\right|$
$\displaystyle=\frac{1}{{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}}}\left|\sum_{j=0}^{2^{l-\kappa^{(\mathtt{k})}}-1}e^{i(\theta-\theta^{\prime})j}\right|\leq\frac{4}{2^{l}||\theta-\theta^{\prime}||}\ll
2^{\mathtt{k}-l}\ .$

Since
${\mathbb{E}}\left(J_{l,p}^{(\mathtt{k})}(\theta)J_{l,p}^{(\mathtt{k})}(\theta^{\prime})\right)=0$,
and the vector
$(J_{l,p}^{(\mathtt{k})}(\theta),J_{l,p}^{(\mathtt{k})}(\theta^{\prime}))$ is
centered complex Gaussian, one checks, by computing covariances, that it is
possible to write

$J_{l,p}^{(\mathtt{k})}(\theta)=\frac{C_{l,p}^{(\mathtt{k})}(\theta,\theta^{\prime})}{C_{l,p}^{(\mathtt{k})}(\theta^{\prime},\theta^{\prime})}J_{l,p}^{(\mathtt{k})}(\theta^{\prime})+J^{(\mathtt{k}),ind}_{l,p}(\theta,\theta^{\prime}),$

where the two terms of the sums are independent, with an expectation of the
square equal to zero. Note that

$C^{(\mathtt{k})}_{l,p}:=C^{(\mathtt{k})}_{l,p}(\theta^{\prime},\theta^{\prime})=\frac{2^{l-\kappa^{(\mathtt{k})}}}{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}}=(2^{\kappa^{(\mathtt{k})}}+p)^{-1},$

does not depend on $\theta^{\prime}$. Moreover, we have by Pythagoras’
theorem:

${\mathbb{E}}[|J_{l,p}^{(\mathtt{k}),ind}(\theta,\theta^{\prime})|^{2}]={\mathbb{E}}[|J_{l,p}^{(\mathtt{k})}(\theta)|^{2}]-\left|\frac{C_{l,p}^{(\mathtt{k})}(\theta,\theta^{\prime})}{C_{l,p}^{(\mathtt{k})}}\right|^{2}{\mathbb{E}}[|J_{l,p}^{(\mathtt{k})}(\theta^{\prime})|^{2}]=C_{l,p}^{(\mathtt{k})}-\frac{|C_{l,p}^{(\mathtt{k})}(\theta,\theta^{\prime})|^{2}}{C_{l,p}^{(\mathtt{k})}}.$

Using this decomposition of $J_{l,p}^{(\mathtt{k})}(\theta)$ and the
measurability of the different quantities with respect to the
$\sigma$-algebras of the form $\mathcal{G}_{j}$, we deduce that one can write:

$\displaystyle(R_{2^{l}}^{(2^{\mathtt{k}_{+}},\kappa^{(\mathtt{k})})}(\theta))_{l\geq\mathtt{k}_{+}}=(R_{2^{l}}^{(2^{\mathtt{k}_{+}},ind)}+E_{2^{l}}^{(2^{\mathtt{k}_{+}})})_{l\geq\mathtt{k}_{+}},$
(B.48)

with $(R_{2^{l}}^{(2^{\mathtt{k}_{+}},ind)})_{l\geq\mathtt{k}_{+}}$ is a
Gaussian process, independent of
$(R_{2^{l}}^{(2^{\mathtt{k}_{+}},\kappa^{(\mathtt{k})})}(\theta^{\prime}))_{l\geq\mathtt{k}_{+}}$,
and distributed as
$(\sqrt{\frac{1}{2}}W_{\tau^{(\mathtt{k}_{+},\mathtt{k}_{+})}_{l}-\mathtt{C}^{(\mathtt{k})}_{l}})_{l\geq\mathtt{k}_{+}}$
with
$\tau^{(\mathtt{k}_{+},\mathtt{k}_{+})}_{j}=\sum_{l=\mathtt{k}_{+}}^{j-1}\sum_{p=0}^{2^{\kappa^{(\mathtt{k})}}-1}(2^{\kappa^{(\mathtt{k})}}+p)^{-1}$
and

$\mathtt{C}^{(\mathtt{k})}_{l}:=\sum_{t=\mathtt{k}_{+}}^{l-1}\sum_{p=0}^{2^{\kappa^{(\mathtt{k})}}-1}\frac{|C_{t,p}^{(\mathtt{k})}(\theta,\theta^{\prime})|^{2}}{C_{t,p}^{(\mathtt{k})}}$

and $(E_{2^{l}}^{(2^{\mathtt{k}_{+}})})_{l\geq\mathtt{k}_{+}}$ defined by

$\displaystyle
E_{2^{l+1}}^{(2^{l})}=\Re\Bigg{(}\sigma\sum_{p=0}^{2^{\kappa^{(\mathtt{k})}}-1}e^{i\psi_{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}}(\theta)}\frac{C_{l,p}^{(\mathtt{k})}(\theta,\theta^{\prime})}{C_{l,p}^{(\mathtt{k})}}J_{l,p}^{(\mathtt{k})}(\theta^{\prime})\Bigg{)}.$
(B.49)

Note that $\mathtt{C}^{(\mathtt{k})}_{l}$ and $E_{2^{l+1}}^{(2^{l})}$ here
represent quantities which are different from those denoted in the same way in
the proof of Lemma B.4. Furthermore notice that

##### Fact 1:

For any $l\geq\mathtt{k}_{+}$,

$\displaystyle|E_{2^{l+1}}^{(2^{l})}|\leq|E|_{2^{l+1}}^{(2^{l})}:=\sum_{p=0}^{2^{\kappa^{(\mathtt{k})}}-1}|C_{l,p}^{(\mathtt{k})}(\theta,\theta^{\prime})|(2^{\kappa^{(\mathtt{k})}}+p)|J_{l,p}^{(\mathtt{k})}(\theta^{\prime})|$

is measurable with respect to the sigma field
$\sigma\left({\mathcal{N}}_{t}^{\mathbb{C}},\,t\in[|2^{l},2^{l+1}-1|]\right)$.

##### Fact 2:

The process $(R_{2^{l}}^{(2^{\mathtt{k}_{+}},ind)})_{l\geq\mathtt{k}_{+}}$ is
independent of the couple
$(R_{2^{l}}^{(2^{\mathtt{k}_{+}},\kappa^{(\mathtt{k})})}(\theta^{\prime}),|E|_{2^{l}}^{(2^{\mathtt{k}_{+}})})_{l\geq\mathtt{k}_{+}}$.

##### Fact 3:

$\sup_{l\geq\mathtt{k}_{+}}|\mathtt{C}_{l}^{(\mathtt{k})}|\ll
2^{-\kappa^{(\mathtt{k})}}$ if $r$ is large enough. Indeed we have

$\displaystyle\sup_{l\geq\mathtt{k}_{+}}|\mathtt{C}_{l}^{(\mathtt{k})}|$
$\displaystyle\ll\sum_{l=\mathtt{k}_{+}}^{\infty}\sum_{p=0}^{2^{\kappa^{(\mathtt{k})}}-1}(2^{\mathtt{k}-l})^{2}(2^{\kappa^{(\mathtt{k})}}+p)\ll\sum_{l=\mathtt{k}_{+}}^{\infty}2^{2\mathtt{k}-2l+2\kappa^{(\mathtt{k})}}\ll\sum_{l=\mathtt{k}+3\kappa^{(\mathtt{k})}}^{\infty}2^{2(\mathtt{k}+3\kappa^{(\mathtt{k})}-l)-4\kappa^{(\mathtt{k})}}\ll
2^{-2\kappa^{(\mathtt{k})}}.$

It means that the process
$(R_{2^{l}}^{(2^{\mathtt{k}_{+}},ind)})_{l\geq\mathtt{k}_{+}}$ is very
”similar” to
$\sqrt{\frac{1}{2}}(W_{\tau_{l}^{(\mathtt{k}_{+},\mathtt{k}_{+})}})_{l\geq\mathtt{k}_{+}}$.

##### Fact 4:

$|E|$ is small. For any $m\geq 0$, $l\geq\mathtt{k}_{+}$, we introduce the
event

$\mathtt{E}_{m}^{(l)}:=\\{|E|_{2^{l+1}}^{(2^{l})}\geq
2^{-\frac{\kappa^{(\mathtt{k})}}{4}}m\\}\ .$

For some universal constants $c,c^{\prime}>0$, and $m\geq 1/2$, the
probability of $\mathtt{E}_{m}^{(l)}$ is smaller than

$\displaystyle{\mathbb{P}}\left(|E|_{2^{l+1}}^{2^{l}}\geq
2^{-\frac{\kappa^{(\mathtt{k})}}{4}}m\right)\leq
2^{\kappa^{(\mathtt{k})}}{\mathbb{P}}\left(c2^{\mathtt{k}-l}|J_{l,0}^{(\mathtt{k})}(\theta^{\prime})|\geq
m2^{-\frac{9}{4}\kappa^{(\mathtt{k})}}\right)$ $\displaystyle\ll
2^{\kappa^{(\mathtt{k})}}e^{-c^{\prime}m^{2}2^{2(l-\mathtt{k}-\frac{7}{4}\kappa^{(\mathtt{k})})}}$
$\displaystyle\ll
e^{-c^{\prime}m^{2}2^{2(l-\mathtt{k}-2\kappa^{(\mathtt{k})})}}.$

Then for $r$ large and $l\geq\mathtt{k}_{+}$,

${\mathbb{P}}\left(|E|_{2^{l+1}}^{2^{l}}\geq
2^{-\frac{\kappa^{(\mathtt{k})}}{4}}m\right)\ll
e^{-m^{2}2^{l-\mathtt{k}_{+}}}\ .$

Using the decomposition (B.48) and Fact 2 and noticing that
$\sum_{l=\mathtt{k}_{+}}^{N}m2^{-\frac{\kappa^{(\mathtt{k})}}{4}}\leq
m2^{-\frac{r}{400}}$ for $r$ large enough, we can affirm that

$\displaystyle{\mathbb{P}}_{N}^{(\Delta,\kappa)}(\theta,\theta^{\prime})$
(B.50) $\displaystyle\leq$ $\displaystyle\sum_{m\geq
0}{\mathbb{P}}\left(Ev(r,1,E^{(\mathtt{k}_{+})},x,z),\,\cup_{j\in[|\mathtt{k}_{+},N-1|]}\mathtt{E}_{m}^{(j)}\right)$
$\displaystyle\quad\quad\sup_{l_{\mathtt{k}_{+}}^{(N)}-1\leq
x^{\prime}+w^{\prime}\leq
u_{\mathtt{k}_{+}}^{(N)}+1}{\mathbb{P}}\left(Ev(\mathtt{k}_{+},1+(m+1)2^{-\frac{r}{400}},\mathtt{C}^{(\mathtt{k})}+E^{(\mathtt{k}_{+})},x^{\prime}+w^{\prime},z^{\prime})\right).$

Here, by abuse of notation, we refer to the same event $Ev(k,a,E,z)$ as in Eq.
(B.3.1) but for the new time clock
$\tau_{.}^{(\mathtt{k}_{+},\mathtt{k}_{+})}$. By the same arguments used to
prove (B.42) we have:

$\displaystyle{\mathbb{P}}\left(Ev(r,1,E^{(\mathtt{k}_{+})},x,z),\,\cup_{j\in[|\mathtt{k}_{+},N-1|]}\mathtt{E}_{m}^{(j)}\right)\ll_{\upsilon}e^{2(x-z)}|xz|2^{r-N_{+}-(m^{2}/8)}.$
(B.51)

On the other hand, by using the inequality (B.29) deduced from the Girsanov
transform (which still holds for the time clock
$\tau^{(\mathtt{k}_{+},\mathtt{k}_{+})}_{j}=\sum_{l=\mathtt{k}_{+}}^{j-1}\sum_{p=0}^{2^{\kappa^{(\mathtt{k})}}-1}(2^{\kappa^{(\mathtt{k})}}+p)^{-1}$),
we obtain for any
$x^{\prime}+w^{\prime}\in[l_{\mathtt{k}_{+}}^{(N)}-1,u_{\mathtt{k}_{+}}^{(N)}+1]$:

$\displaystyle{\mathbb{P}}\left(Ev(\mathtt{k}_{+},1+(m+1)2^{-\frac{r}{400}},\mathtt{C}^{(\mathtt{k})}+E^{(\mathtt{k}_{+})},x^{\prime}+w^{\prime},z^{\prime})\right)$
$\displaystyle\ll_{\upsilon}2^{\mathtt{k}_{+}-N_{+}}e^{1+2(m+1)2^{-\frac{r}{400}}}N^{\frac{3}{2}}e^{2(x^{\prime}+w^{\prime}-z^{\prime})}$
$\displaystyle\qquad\times{\mathbb{P}}\left(GEv(\mathtt{k}_{+},1+2(m+1)2^{-\frac{r}{400}},\mathtt{C}^{(\mathtt{k})}+E^{(\mathtt{k}_{+})},x^{\prime}+w^{\prime},z^{\prime}))\right),$
(B.52)

where we used that
$\sup_{j\geq\mathtt{k}_{+}}|\mathtt{C}^{(\mathtt{k})}_{j}|\leq
2^{-\kappa^{(\mathtt{k})}}\leq(m+1)2^{-\frac{r}{400}}$. We bound the
probability of the $GEv$ event by $1$ and use the fact that
$x^{\prime}+w^{\prime}\leq 1+u_{k_{0}}^{(N)}\leq
1-(N_{+}-\mathtt{k}_{+})^{\alpha_{-}}-\frac{3}{4}\log N$. We then get

$\displaystyle
P\left(Ev(\mathtt{k}_{+},1+(m+1)2^{-\frac{r}{400}},\mathtt{C}^{(\mathtt{k})}+E^{(\mathtt{k}_{+})},x^{\prime}+w^{\prime},z^{\prime})\right)$
$\displaystyle\ll
2^{\mathtt{k}_{+}-N_{+}}e^{2(m+1)2^{-\frac{r}{400}}}N^{\frac{3}{2}}e^{2(x^{\prime}+w^{\prime}-z^{\prime})}$
$\displaystyle\ll_{\upsilon}2^{-2z^{\prime}}2^{\mathtt{k}_{+}-N_{+}}e^{2m+2}N^{3/2}e^{-2(N_{+}-\mathtt{k}_{+})^{\alpha_{-}}-\frac{3}{2}\log
N}$ $\displaystyle\ll
2^{-2z^{\prime}}2^{\mathtt{k}_{+}-N_{+}}e^{2m+2}e^{-(N_{+}-\mathtt{k}_{+})^{\alpha_{-}}},$

Finally, by combining this equation with (B.51) and (B.50), we get

$\displaystyle{\mathbb{P}}_{N}^{(\Delta,\kappa)}(\theta,\theta^{\prime})$
$\displaystyle\ll_{\upsilon}|xz|2^{2(x-z-z^{\prime})}2^{r-N_{+}}2^{\mathtt{k}_{+}-N}e^{-(N_{+}-\mathtt{k}_{+})^{\alpha_{-}}}\sum_{m\geq
0}e^{2m+2-(m^{2}/8)}$
$\displaystyle\ll|xz|2^{2(x-z-z^{\prime})}2^{r-N_{+}}2^{\mathtt{k}_{+}-N}e^{-(N_{+}-\mathtt{k}_{+})^{\alpha_{-}}}\
,$

which concludes the proof of Lemma B.6. ∎

###  B.4 Short time barriers

We will also need the analogue of Lemmas B.4 and B.5 with $\mathfrak{I}_{N}$
replaced by $\mathfrak{I}_{t,f}$. Thus, let

$\displaystyle{\mathbb{P}}_{t,f}(\theta,\theta^{\prime})={\mathbb{P}}_{t,f}(\theta,\theta^{\prime},x,z,x^{\prime},z^{\prime}):={\mathbb{P}}(\mathfrak{I}_{t,f}(\theta,x,z)\cap\mathfrak{I}_{t,f}(\theta^{\prime},x^{\prime},z^{\prime})).$
(B.53)

We have the following.

######  Lemma B.8 (Time of branching $\mathtt{k}\leq N/2$).

For any $\upsilon\in(0,1)$, $r$ large enough, $C>1$, $t$ large enough
depending on $r$, $z,z^{\prime}\in[-\sqrt{t}/C,C\sqrt{t}]$, $\mathtt{k}\leq t$
and
$2^{-\mathtt{k}}\leq\frac{||\theta-\theta^{\prime}||}{2\pi}<2^{-(\mathtt{k}-1)}$,
we have

$\displaystyle{\mathbb{P}}_{t,f}(\theta,\theta^{\prime})$
$\displaystyle\ll\left\\{\begin{array}[]{ll}\frac{|zz^{\prime}xx^{\prime}|}{t^{3}}2^{2r-2t+2(x-z+x^{\prime}-z^{\prime})},&\text{when
}\mathtt{k}_{+}\leq r,\\\
\frac{|zz^{\prime}x|}{t^{3/2}}e^{2(x-z-z^{\prime})}2^{r-t}2^{\mathtt{k}_{+}-t}e^{-\mathtt{k}_{+}^{\alpha_{-}}},&\text{when
}\mathtt{k}_{+}\geq r.\end{array}\right.$ (B.56)

######  Lemma B.9 (Time of branching $\mathtt{k}\leq r/2$).

For any $\upsilon\in(0,1)$, $r\in{\mathbb{N}}$ large enough, $r$ large enough,
$C>1$, $t$ large enough depending on $r$,
$z,z^{\prime}\in[-\sqrt{t}/C,C\sqrt{t}]$, $\mathtt{k}\leq\frac{r}{2}$,
$2^{-\mathtt{k}}\leq\frac{||\theta-\theta^{\prime}||}{2\pi}\leq
2^{-(\mathtt{k}-1)}$ and $t$ large enough depending on $r$, we have

$\displaystyle{\mathbb{P}}_{t,f}(\theta,\theta^{\prime})$
$\displaystyle\leq(1+\eta_{r,\upsilon})2^{2r-2t}e^{2(x-z+x^{\prime}-z^{\prime})}{\mathbb{P}}(Event_{r,N}(x,z)){\mathbb{P}}(Event_{r,N}(x^{\prime},z^{\prime})),$

where

$\underset{\upsilon\rightarrow
0}{\limsup}\,\underset{r\rightarrow\infty}{\limsup}\,\eta_{r,\upsilon}=0.$

##  Appendix C Appendix: Classical estimates on Gaussian walks

The following estimates are classical and extend those in [CMN18] by keeping
track of the dependence in starting and ending points. In the following, the
process $\left(W_{s};s\in{\mathbb{R}}_{+}\right)$ is a standard Brownian
motion. We use the notation from B.2.

The following is classical. Since the proof is short, we bring it.

######  Lemma C.1.

Notation as in Section B.2, with $x,z$ in the appropriate range. Then,

$(1-\eta_{N,r,k_{1},v})\sqrt{\frac{2}{\pi}}|xz|v\leq
N^{\frac{3}{2}}{\mathbb{P}}(Event_{r,N})\leq(1+\eta_{N,r,k_{1},v})\sqrt{\frac{2}{\pi}}|xz|v,$
(C.1)

where

$\lim_{v\to
0}\lim_{r\to\infty}\lim_{k_{1}\to\infty}\lim_{N\to\infty}\eta_{N,r,k_{1}}=0.$

###### Proof.

We show the lower bound, the upper bound is similar. Recall that

$\displaystyle Event_{r,N}=Event_{r,N}(x,z):=$ $\displaystyle\left\\{\forall
j\in[|r,N_{+}|],\,l_{j}^{(N)}\leq x+\sqrt{\frac{1}{2}}W_{\tau^{(r)}_{j}}\leq
u_{j}^{(N)},x+\sqrt{\frac{1}{2}}W_{\tau^{(r)}_{N_{+}}}\in\tau^{(r)}_{N_{+}}-\frac{3}{4}\log
N+[z,z+\upsilon)\right\\}.$

For $r,k_{1}$ large, we have that $|\tau^{(r)}_{j}-(j-r)\log 2|\leq 1$. Set,
comparing with (B.12) and (B.15), for $k\in[r,N_{+}+1]$,

$\displaystyle
u_{k}^{+,(N)}:=-2+\left\\{\begin{array}[]{ll}-k^{2\alpha_{-}},&\text{if
}k\leq\lfloor N/2\rfloor,\\\ -(N-k)^{2\alpha_{-}}-\frac{3}{4}\log N,&\text{if
}\lfloor N/2\rfloor<k\leq N,\end{array}\right.$ (C.4)

and

$\displaystyle
l_{k}^{+,(N)}:=2+\left\\{\begin{array}[]{ll}-k^{\alpha_{+}-\alpha_{-}},&\text{if
}k\leq\lfloor N/2\rfloor,\\\ -(N-k)^{\alpha_{+}-\alpha_{-}}-\frac{3}{4}\log
N,&\text{if }\lfloor N/2\rfloor<k\leq N.\end{array}\right.$ (C.7)

The crucial fact is that for $N$ large we have that for all $k\in[r,N_{+}]$,

$\inf_{\theta\in[(k-1)\vee r,k+1]}u_{k}^{(N)}\geq
u_{k}^{+,(N)},\quad\sup_{\theta\in[(k-1)\vee r,k+1]}l_{k}^{(N)}\leq
l_{k}^{+,(N)}.$ (C.8)

In particular, we have that

$\displaystyle Event_{r,N}^{\prime}:=$ $\displaystyle\left\\{\forall
t\in[r,\tau^{(r)}_{N_{+}}],\,l_{t}^{+,(N)}\leq x+\sqrt{\frac{1}{2}}W_{t}\leq
u_{t}^{+,(N)},x+\sqrt{\frac{1}{2}}W_{\tau^{(r)}_{N_{+}}}\in\tau^{(r)}_{N_{+}}-\frac{3}{4}\log
N+[z,z+\upsilon)\right\\}$ $\displaystyle\subset Event_{r,N}.$

The conclusion of the lemma follows from a variant of [BRZ19, Lemma 2.1], and
our assumptions on $x,z$. ∎

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# Influence of sodium inward current on dynamical behaviour of modified
Morris-Lecar model

Hammed Olawale Fatoyinbo<EMAIL_ADDRESS>Sishu Shankar Muni
<EMAIL_ADDRESS>Afeez Abidemi<EMAIL_ADDRESS>School of Fundamental
Sciences, Massey University, Palmerston North, New Zealand  Department of
Mathematical Sciences, Federal University of Technology, Akure, PMB 704, Ondo
State, Nigeria

###### Abstract

This paper presents a modified Morris-Lecar model by incorporating the sodium
inward current. The dynamical behaviour of the model in response to key
parameters is investigated. The model exhibits various excitability properties
as the values of parameters are varied. We have examined the effects of
changes in maximum ion conductances and external current on the dynamics of
the membrane potential. A detailed numerical bifurcation analysis is
conducted. The bifurcation structures obtained in this study are not present
in existing bifurcation studies of original Morris-Lecar model. The results in
this study provides the interpretation of electrical activity in excitable
cells and a platform for further study.

###### keywords:

Excitable cells , Ion conductance , Morris-Lecar model , Period-doubling
bifurcation

###### MSC:

[2010] 37N25 , 74H60 , 37M20 , 37C75 , 92C30

##  1 Introduction

The variation in concentration of ions across the cell membrane results in
fluxes of ions through the voltage-gated ion channels. This
electrophysiological process in the cell membrane plays a fundamental role in
understanding the electrical activities in excitable cells such as neurons
[1], muscle cells [2] and hormones [3]. The temporal variation of the cell
membrane potential due to external stimulation is known as an action
potential. Different ion channels play different roles in the generation of an
action potential. Depending on the cell, the opening of ${\rm Na}^{+}$ (${\rm
Ca}^{2+}$) ion channels causes influx of ${\rm Na}^{+}$ (${\rm Ca}^{2+}$) and
the membrane potential becomes more positive, hence the membrane is
depolarised. When the K+ channels are open, there is efflux of K+ which
results in the repolarisation of the cell. Later, the membrane potential
becomes more negative than the resting potential and the membrane is
hyperpolarised. At this stage, the membrane will not respond to stimulus until
it returns to the resting potential [4, 5, 6]. From the viewpoint of
mathematics, numerous mathematical models have been developed to study the
nonlinear dynamics involved in the generation of an action potential in the
cell membrane. They are often modelled by a nonlinear system of ordinary
differential equations (ODEs). Among the famous works is the one by Hodgkin
and Huxley [7] on the conduction of electrical impulses along a squid giant
axon. In their experiments, it was reported that action potentials depends on
the influx of ${\rm Na}^{+}$ and ${\rm Ca}^{2+}$. This work laid foundation
for other electrophysiological models. Other well-known models are the
FitzHugh-Nagumo model [8, 9], the Morris-Lecar (ML) model [10], the Chay model
[11], and the Smolen and Keizer model [12].

ML model describes the electrical activities of a giant barnacle muscle fibre
membrane. Despite being a model for muscle cell, it has been widely used in
modelling electrical activities in other excitable cells mostly in neurons
[13, 14, 15, 16]. Based on experimental observations, ML model is formulated
on the assumption that the electrical activities in barnacle muscle depend
largely on fluxes of ${\rm Ca}^{2+}$ and ${\rm K}^{+}$ rather than ${\rm
Na}^{+}$. On this basis, their model consists of three ODEs. It is observed
that the ${\rm Ca}^{2+}$ current activates faster than the ${\rm K}^{+}$
current and the charging capacitor [17]. Thus, the model is further reduced to
two ODEs by setting the ${\rm Ca}^{2+}$ activation to quasi-steady state.

The two-dimensional ML model has been extensively used in many single-cell
[18, 19, 20, 21] and network of cells [22, 23, 24, 25] studies despite it is
an approximation of the three-dimensional ML model. In spite of little
attention to the three-dimensional model, it has been used in modelling
electrophysiological studies. For example, Gottschalk and Haney [26]
investigated how the activity of the ion channels are regulated by
anaesthetics. The three-dimensional ML model was used by Marreiros et al [27]
for modelling dynamics in neuronal populations using a statistical approach.
Also, González-Miranda [28] investigated pacemaker dynamics in ML model using
the three-dimensional model. Gall and Zhou [29] considered four-dimensional ML
model by including the second inward sodium ${\rm Na}^{+}$ current.

In recent years, experimental and computational analyses have suggested that
sodium ${\rm Na}^{+}$ currents are relevant in the depolarisation of action
potential in some muscle cells [30, 31, 32]. Motivated by these results, in
this present paper we propose to investigate the influence of including sodium
inward currents on variation of membrane voltage of a single excitable cell.
Bifurcation analysis is often used to investigate the mode of transition of
electrical activities of excitable cell. It helps us to identify the key
parameters that cause changes in the dynamical behaviour qualitatively [33]. A
lot of studies on bifurcation analyses have been carried out on the two-
dimensional [18, 34, 35, 36] and three-dimensional ML models [28], however, to
our knowledge apart from the work of [29] there appears no work in the
literature that has considered the bifurcation analysis of the four-
dimensional ML model. The external current is considered as the bifurcation
parameter by [29] whereas in this present paper we focus on the maximum
conductances of ion currents as the bifurcation parameters. As a consequence,
we show some additional results that are not present in the existing results
of ML model.

The paper is organised as follows. In Section 2, we present the model
equations and the dynamics of the model upon variation of model parameters. A
detailed bifurcation analysis is carried out in Section 3. Finally, the
conclusion is presented in Section 4.

##  2 Model Equation

The modified ML model consists of ODEs [29]

$\displaystyle\textrm{C}\frac{dV}{dt}$ $\displaystyle=I_{\rm ext}-I_{\rm
L}-I_{\rm Ca}-I_{\rm K}-I_{\rm Na},$ (1) $\displaystyle\frac{dm}{dt}$
$\displaystyle=\lambda_{m}(V)(m_{\infty}(V)-m),$ (2)
$\displaystyle\frac{dn}{dt}$ $\displaystyle=\lambda_{n}(V)(n_{\infty}(V)-n),$
(3) $\displaystyle\frac{dw}{dt}$
$\displaystyle=\lambda_{w}(V)(w_{\infty}(V)-w),$ (4)

where $V$ is the membrane potential, $m$, $n$ and $w$ are the fraction of open
calcium, potassium and sodium channels, respectively. The ionic currents in
(1) are defined as

$\displaystyle I_{\rm L}$ $\displaystyle=g_{\rm L}(V-v_{\rm L}),\hskip
5.69054ptI_{\rm Ca}=g_{\rm Ca}m(V-v_{\rm Ca}),$ (5) $\displaystyle I_{\rm K}$
$\displaystyle=g_{\rm K}n(V-v_{\rm K}),\hskip 5.69054ptI_{\rm Na}=g_{\rm
Na}w(V-v_{\rm Na}),$

where $g_{\rm L}$, $g_{\rm Ca}$, $g_{\rm K}$, and $g_{\rm Na}$ are the maximum
conductances of the leak, calcium, potassium, and sodium channels,
respectively. Also $v_{\rm L}$, $v_{\rm Ca}$, $v_{\rm K}$, and $v_{\rm Na}$
are the Nerst reversal potentials of the leak, calcium, potassium, and sodium
channels, respectively, while $I_{\rm ext}$ is the external current and ${\rm
C}$ is the membrane capacitance. The equivalent circuit representation of the
cell membrane with four ionic channels, $I_{\rm L}$, $I_{\rm Ca}$, $I_{\rm
K}$, and $I_{\rm Na}$, is shown in Fig. 1.

Fig. 1: Equivalent circuit representation of the cell membrane with four ionic
channels.

The fraction of open calcium, potassium and sodium channels at steady state,
denoted by $m_{\infty}$, $n_{\infty}$ and $w_{\infty}$, are defined as

$\displaystyle m_{\infty}(V)$
$\displaystyle=0.5\left(1+\tanh\left(\frac{V-\bar{v}_{1}}{\bar{v}_{2}}\right)\right),$
$\displaystyle n_{\infty}(V)$
$\displaystyle=0.5\left(1+\tanh\left(\frac{V-\bar{v}_{3}}{\bar{v}_{4}}\right)\right),$
$\displaystyle w_{\infty}(V)$
$\displaystyle=0.5\left(1+\tanh\left(\frac{V-\bar{v}_{5}}{\bar{v}_{6}}\right)\right).$

The voltage-dependent rate constants associated with calcium, potassium and
sodium channels are

$\displaystyle\lambda_{m}(V)$
$\displaystyle=\psi_{m}\cosh\left(\frac{V-\bar{v}_{1}}{2\bar{v}_{2}}\right),$
$\displaystyle\lambda_{n}(V)$
$\displaystyle=\psi_{n}\cosh\left(\frac{V-\bar{v}_{3}}{2\bar{v}_{4}}\right),$
$\displaystyle\lambda_{w}(V)$
$\displaystyle=\psi_{w}\cosh\left(\frac{V-\bar{v}_{5}}{2\bar{v}_{6}}\right).$

Unless otherwise stated, parameter values are as listed in [29]:
$\textrm{C}=1$, $I_{\text{ext}}=50$, $\textrm{g}_{\text{L}}=2$,
$v_{\textrm{L}}=-50$, $\textrm{g}_{\textrm{Ca}}=4$, $v_{\text{Ca}}=100$,
$\mathrm{g_{K}}=8$, $v_{K}=-70$, $\textrm{g}_{\textrm{Na}}=2$,
$v_{\textrm{Na}}=55$, $v_{1}=-1$, $v_{2}=15$, $v_{3}=10$, $v_{4}=14.5$,
$v_{5}=5$, $v_{6}=15$, $\psi_{m}=1$, $\psi_{n}=0.0667$, $\psi_{w}=0.033$.

###  2.1 Changes to Excitable Dynamics as a Parameter is Varied

As seen in previous studies [28, 18], variation of parameters can result in
changes to dynamical behaviour of the model, for example, transitions from
rest state to periodic oscillations and vice versa. Here, we investigate the
effects of maximum conductance on the dynamical behaviour of model (1)–(4).
The model is integrated numerically using the standard fourth-order
Runge–Kutta method using a step size of 0.05 in the numerical software XPPAUT
[37]. The dynamics of the membrane potential $V$ upon varying ${\rm Na}^{+}$
current conductance $g_{\rm Na}$ is shown in Fig. 2. For the range of values
of $g_{\rm Na}$ considered, the system either converge to a rest state or
oscillatory state. For extremely low values of $g_{\rm Na}$, a single action
potential is observed. In particular, the time evolution and its corresponding
phase space for $g_{\rm Na}=-20$ are shown in Figs. 2(a) and 2(b),
respectively. Upon increasing $g_{\rm Na}$, periodic oscillations of action
potentials are observed in the system, see Fig. 2(c). The periodic
oscillations correspond to a closed loop in the phase space, see Fig. 2(d).
The closed loop is also known as a limit cycle or periodic orbit. Further
increasing $g_{\rm Na}$, the system stabilises to a steady state, see Figs.
2(e) and 2(f). Similar behaviours are observed when $g_{\rm K}$ and $g_{\rm
Ca}$ are varied (results not shown). A detailed bifurcation analysis is given
in Sec. 3 to further understand how the dynamical properties of model (1)–(4)
change as parameter values is varied.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 2: Numerical simulations of the membrane potential $V$ for (a) $g_{\rm
Na}=-20$; (c) $g_{\rm Na}=-10$; (e) $g_{\rm Na}=1.8$. Their corresponding
phase space are (b), (d) and (f), respectively.

##  3 Numerical Bifurcation Analysis

With the aid of bifurcation analysis, we examine the dynamical behaviour of
model (1)–(4) as different model parameters are varied in turn. The
bifurcation diagrams are produced in XPPAUT and edited in MATLAB. The
continuation parameters used in XPPAUT are $\verb|NTST=100|$,
$\verb|NMAX=2000|$, $\verb| Method=stiff|$, $\verb|EPSL=1e-7|$ ,
$\verb|EPSU=1e-7|$, $\verb|EPSS=1e-7|$, $\verb|ITMX=20|$, $\verb|ITNW=20|$,
$\verb|DSMIN=1e-05|$, $\verb|DSMAX=0.05|$. The abbreviations and labels for
the bifurcation points are given in Table 1.

Table 1: Abbreviations and notations of bifurcation points. Bifurcation | Abbreviation
---|---
Hopf bifurcation | HB
Saddle Node bifurcation | SN
Saddle Node bifurcation of Cycles | SNC
Homoclinic bifurcation | HC
Period Doubling bifurcation | PD

###  3.1 Influence of $g_{\rm Na}$

Here, we vary $g_{\rm Na}$ to explore the effects of ${\rm Na}^{+}$ current on
the dynamical behaviour of model (1)–(4). Fig. 3 is a bifurcation diagram of
the membrane potential $V$ upon varying $g_{\rm Na}$ with other parameters
fixed. For the range of values of $g_{\rm Na}$ considered, there exists a
unique equilibrium. The system has a stable equilibrium except between two
Hopf bifurcations where the equilibrium is unstable. As seen in Fig. 3(a), the
system loses stability through a subcritical Hopf bifurcation ${\rm HB}_{1}$
at $g_{\rm Na}\approx-13.305$ and regains stability at another subcritical
Hopf bifurcation ${\rm HB}_{2}$ at $g_{\rm Na}\approx 0.69436$. The unstable
limit cycle generated at ${\rm HB}_{1}$ gain stability through a saddle-node
bifurcation of cycle ${\rm SNC}_{1}$ at $g_{\rm Na}\approx-13.4394$, and loses
stability at a period-doubling bifurcation ${\rm PD}_{1}$. The unstable limit
cycle branch regains stability through another ${\rm SNC}_{3}$ at $g_{\rm
Na}\approx-13.1223$. The stable double-period limit cycle branch emanated from
the ${\rm PD}_{1}$ loses stability at another period doubling bifurcation
${\rm PD}_{2}$ at $g_{\rm Na}\approx-13.4323$, and it regains stability
through a ${\rm SNC}_{2}$ at $g_{\rm Na}\approx-13.2516$ before converging to
the first unstable limit cycle branch at $g_{\rm Na}\approx-13.1223$, see
Fig.3(b). Upon further increasing the value of $g_{\rm Na}$, the limit cycle
loses stability in a ${\rm SNC}_{4}$ at $g_{\rm Na}\approx 1.10527$ before it
ends in a ${\rm HB}$ point at $g_{\rm Na}\approx 0.69436$.

Continuation of ${\rm PD}_{2}$ bifurcation results in another stable limit
cycle that loses stability at a period doubling bifurcation ${\rm PD}_{4}$,
the period of this limit cycle is double the period of the limit cycle of
${\rm PD}_{2}$. Continuing this process results in a cascade of PD
bifurcations of limit cycles, and this may lead to chaotic dynamics in the
system [38, 39]. Table 2 shows the values and period of the period doubling
bifurcations that arise as $g_{\rm Na}$ is varied. The projection of the
periodic trajectories for Period-1, 2, 4, 8, 16 and 32 onto $(V,n,m)$ phase
space is illustrated in Fig. 4. All the double-period unstable limit cycles
generated at each PD points undergo SNC bifurcations before they converge to
the limit cycle emanated from the first ${\rm HB}$ bifurcation.

(a)

(b)

(c)

Fig. 3: (a) Bifurcation diagram of the membrane potential $\mathrm{V}$ with $g_{\rm Na}$ as bifurcation parameter. The remaining parameter values are fixed as in Sec. 2. (b)-(c) are enlargements of (a). Continuous [dashed] curves correspond to stable [unstable] solutions. Black [mangenta] curves correspond to equilibria [periodic oscillations]. HB: Hopf bifurcation; SN: saddle-node bifurcation (of an equilibrium); SNC: saddle-node bifurcation of a periodic orbit; PD: period-doubling bifurcation. Table 2: Summary of the parameter values and period of Period doubling bifurcations that arise as $g_{\rm Na}$ is varied. Bifurcation |  $g_{\rm Na}$ | Period
---|---|---
${\rm PD}_{1}$ | -13.4334 | 36.0272
${\rm PD}_{2}$ | -13.4323 | 72.1846
${\rm PD}_{4}$ | -13.4321 | 144.489
${\rm PD}_{8}$ | -13.4320 | 289.001
${\rm PD}_{16}$ | -13.4320 | 578.025
${\rm PD}_{32}$ | -13.4320 | 1156.05

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4: Phase-space of (1)–(4) showing the cascade of period-doubling
bifurcations. (a) Period-1 (b) Period-2 (c) Period-4 (d) Period-8 (e)
Period-16 (f) Period-32, respectively.

###  3.2 Influence of $g_{\rm K}$ and $g_{\rm Ca}$

Fig. 5(a) shows the bifurcation diagram of the membrane potential $V$ as
$g_{\rm K}$ is varied. For the values of $g_{\rm K}$ considered, there exists
a unique equilibrium. For extremely low values and high values of $g_{\rm K}$,
the equilibrium is stable. Increasing $g_{\rm K}$, the system loses stability
through a subcritical Hopf bifurcation ${\rm HB}_{1}$ at $g_{\rm K}\approx
10.029$ and this leads to emergence of an unstable limit cycle which becomes
stable through a saddle node bifurcation of cycles ${\rm SNC}_{1}$ at $g_{\rm
K}\approx 9.345$. As $g_{\rm K}$ increases further, the stable limit cycle
changes stability in another saddle node bifurcation of cycles ${\rm SNC}_{2}$
at $g_{\rm K}\approx 46.598$. The unstable limit cycle ends in another
subcritical Hopf bifurcation ${\rm HB}_{2}$ at $g_{\rm K}\approx 42.583$.
Bistability is observed, that is, a stable limit cycle coexists with a stable
equilibrium when $9.345\leq g_{\rm K}\leq 10.029$ and $42.583\leq g_{\rm
K}\leq 46.598$.

(a)

(b)

Fig. 5: Bifurcation diagrams of the membrane potential $V$ with (a)
$\mathrm{g_{K}}$ (b) $g_{\text{Ca}}$ as the bifurcation parameters and other
parameters are fixed as in Sec. 2. The labels and other conventions are as in
Fig. 3.

Next, we vary the value of the parameter $g_{\rm Ca}$. Fig. 5(b) shows the
bifurcation diagram of the membrane potential $V$ as $g_{\rm Ca}$ is varied.
As $g_{\rm Ca}$ is varied, the system loses stability through a subcritical
Hopf bifurcation ${\rm HB}_{1}$ at $g_{\rm Ca}\approx 1.6191$ and this results
in emergence of unstable limit cycle which becomes stable through a saddle
node bifurcation of cycles ${\rm SNC}_{1}$ at $g_{\rm Ca}\approx 1.5974$. As
$g_{\rm Ca}$ increases further, the stable limit cycle loses stability in
another saddle-node bifurcation ${\rm SNC}_{2}$ at $g_{\rm Ca}\approx 3.2579$
and the unstable limit cycle ends in a subcritical Hopf bifurcation ${\rm
HB}_{2}$ at $g_{\rm Ca}\approx 2.8938$. Between the two subcritical Hopf
bifurcations, there exists a unique unstable equilibrium point. For
$1.5974\leq g_{\rm Ca}\leq 1.6191$ and $2.8938\leq g_{\rm Ca}\leq 3.2579$, a
stable limit cycle coexists with a stable equilibrium and the system is
bistable. For these values of $g_{\rm Ca}$, a stable limit cycle coexists with
a stable equilibrium.

###  3.3 Influence of $I_{\rm ext}$

Apart from maximum conductance of ionic channels, the influence of external
current is highly important while investigating the dynamics of action
potentials in electrophysiological studies. Here, we consider the effects of
$I_{\rm ext}$ using two parameter sets. For set I, the parameter values are as
listed in Section 2. Fig. 6(a) is a bifurcation diagram of the membrane
potential $V$ with the applied current $I_{\rm ext}$ as a bifurcation
parameter, other parameters fixed. For very low value of $I_{\rm ext}$, a
unique stable equilibrium point exists. Upon increasing $I_{\rm ext}$, the
system changes stability through a saddle node bifurcation ${\rm SN}_{1}$ at
$I_{\rm ext}\approx 30.52$ and the unstable branch fold back via another
saddle node bifurcation ${\rm SN}_{2}$ at $I_{\rm ext}\approx-39.57$. Between
the two SN bifurcations, the system has three equilibria: one stable (lower
branch) and two unstable (upper and middle branch), see Fig. 6(a). The upper
unstable branch changes stability at a subcritical Hopf bifurcation ${\rm HB}$
at $I_{\rm ext}\approx 6.656$ before the system returns to a rest state as
$I_{\rm ext}$ increases. The unstable limit cycle emanated from ${\rm HB}$
fold back and changes to a stable limit cycle through a saddle node
bifurcation of cycles ${\rm SNC}_{1}$ at $\text{I}_{\text{ext}}\approx 26.84$.
The limit cycle loses stability at another ${\rm SNC}_{2}$ at
$\text{I}_{\text{ext}}\approx 22.99$ before it terminates at $I_{ext}\approx
23.79$.

(a)

(b)

Fig. 6: (a) Bifurcation diagram of the membrane potential $V$ with
$\text{I}_{\text{ext}}$ as the bifurcation parameter. and other parameters are
fixed as in Sec. 2. The labels and other conventions are as in Fig. 3.

For set II, $v_{6}=3$ while other parameters are fixed as in Section 2. A
bifurcation diagram of the membrane potential $V$ with $I_{\rm ext}$ as
bifurcation parameter is shown in Fig. 7(a). For $I_{\rm ext}<-8.7715$, there
exists a unique stable equilibrium point. Upon increasing $I_{\rm ext}$, the
system loses stability through a subcritical Hopf bifurcation ${\rm HB}_{1}$
at $I_{\rm ext}\approx 33.29650$. The unstable limit cycle emanated from ${\rm
HB}_{1}$ ends in an homoclinic bifurcation ${\rm HC}_{1}$ at $I_{\rm
ext}\approx 33.2911$, see Fig. 7(b). The curve of the homoclinic orbit is
shown in Fig. 8(a). Increasing $I_{\rm ext}$ slightly there appears a saddle-
node bifurcation ${\rm SN}_{1}$ at $I_{\rm ext}\approx 33.2026$, the unstable
branch fold back at another saddle-node bifurcation ${\rm SN}_{2}$ at $I_{\rm
ext}\approx-8.7715$.

As $I_{\rm ext}$ increases further, the system passes through another saddle
node bifurcation ${\rm SN}_{3}$ at $I_{\rm ext}\approx 0.8353$. For $I_{\rm
ext}\in[{\rm SN}_{2},{\rm SN_{3}}]$, there exist three equilibria; one stable
and two unstable. The branch of ${\rm SN}_{3}$ bifurcation folds at another
saddle-node bifurcation $\text{SN}_{4}$ at $I_{\rm ext}\approx-1.7961$, and
the unstable upper branch becomes stable in another subcritical Hopf
bifurcation $\text{HB}_{2}$. For $I_{\rm ext}\in[{\rm SN}_{4},{\rm HB_{2}}]$,
there exist five equilibria; one stable and four unstable equilibria. Also,
for $I_{\rm ext}\in[{\rm HB}_{2},{\rm SN_{3}}]$, there exist five equilibria;
two stable and three unstable. For this parameter values, the system is
bistable, that is, coexistence of two stable equilibria. To the right of ${\rm
SN}_{1}$, the system has a unique stable equilibrium.

(a)

(b)

(c)

Fig. 7: (a) Bifurcation diagram of membrane potential $V$ with
$I_{\text{ext}}$ as a bifurcation parameter. (b)–(c) are enlargements of (a).
and other parameters are fixed as in Sec. 2. The labels and other conventions
are as in Fig. 3.

(a)

(b)

Fig. 8: The curves of homoclinic orbits of the periodic oscillation emanated
at (a) ${\rm HB}_{1}$; (b) ${\rm HB}_{2}$.

The unstable limit cycle generated at the Hopf bifurcation ${\rm HB}_{2}$ fold
back at $I_{\rm ext}\approx 10.80$ and slightly after the fold point appears a
period-doubling bifurcation ${\rm PD}_{1}$ at $I_{\rm ext}\approx 10.77$. At
$\text{PD}_{1}$, the limit cycle bifurcates into unstable double-period and
unstable limit cycles, and they both end in an homoclinic bifurcation, see
Fig. 7(c). The curve of the homoclinic orbit is shown in Fig. 8(b).
Continuation from the period-doubling ${\rm PD}_{1}$ results in period-
doubling bifurcation ${\rm PD}_{2}$, subsequently, the ${\rm PD}_{2}$ results
in period-doubling bifurcation ${\rm PD}_{4}$. Table 3 shows the parameter
values for the period-doubling and homoclinic bifurcations and their
corresponding periods as $I_{\rm ext}$ is varied. The projections of periodic
trajectories for period-1, 2, 4 onto $(V,n,m)$ phase space are shown in Fig.
9.

Table 3: Summary of the parameter values and period of period doubling and homoclinic bifurcations that arise as $I_{\rm ext}$ is varied. Bifurcation point |  $I_{\text{ext}}$ | Period
---|---|---
${\rm PD}_{1}$ | 10.7705 | 33.5585
${\rm PD}_{2}$ | 10.7584 | 67.1396
${\rm PD}_{4}$ | 10.7555 | 134.353
${\rm HC}_{1}$ | 33.2911 | 2.61499E+08
${\rm HC}_{2}$ | -4.05553 | 3.95045E+09

(a)

(b)

(c)

Fig. 9:  Phase-space of (1)–(4) showing the period-doubling bifurcations in
response to variation of $I_{ext}$. (a) Period-1 (b) Period-2 (c) Period-4,
respectively.

###  3.4 Two Parameter Bifurcation Analysis

In this section we perform two parameter bifurcation analysis of (1)–(4) in
$(I_{\rm ext},g_{\rm K})$ plane. The bifurcation diagram shown in Fig. 10 is
produced via numerical continuation software MATCONT [40]. The description for
the codimension-2 bifurcations are explained in Table. 4. Fig. 10 is divided
into regions with respect to different types of dynamical behaviour and we
have assigned each region a number, see Table 5. In the remainder of this
section we describe the dynamics of model (1)–(4) as $I_{\rm ext}$ and $g_{\rm
K}$ are varied.

Fig. 10: Two parameter bifurcation diagram of (1)–(4) in the $(I_{\rm ext},g_{\rm K})$-plane for parameter set II in Section 3.3 and other parameter values as in Section 2. The values of $g_{\rm K}$ in ${\rm l}_{1}$, ${\rm l}_{2}$, ${\rm l}_{3}$, ${\rm l}_{4}$ are $60$, $26$, $8$ and $3.5$, respectively. The blue, red and magenta curves are the loci of Hopf bifurcation, saddle-node bifurcation, and Period doubling bifurcation. The labels for the codimension-2 bifurcations are explained in Table 4. The invariant sets that exist in each region are listed in Table 5. Table 4: Abbreviations of codimension-two bifurcations. Bifurcation | Abbreviation
---|---
Cusp bifurcation |  $\text{CP}_{i}$ $i=1,2,3$
Bogdanov-Takens bifurcation |  $\text{BT}_{i}$ $i=1,2$
Generalized Hopf bifurcation |  $\text{GH}_{i}$ $i=1,2,3$
Zero-Hopf bifurcation | ZH
Generalised Period Doubling bifurcation |  $\text{GPD}_{i}$ $i=1,2$
1:2 Resonance | R2
Flip-flop bifurcation | LPPD
Table 5: Summary of the six different combinations of equilibria and limit cycles that arise in Fig.10 and its magnifications, Figs. 12(a), 12(b), and 13(a). Region | Existence of equilibria and limit cycles
---|---
I | One stable equilibrium, no limit cycles (rest state).
II | One unstable equilibrium, one stable limit cycle.
| One stable equilibrium, two unstable equilibria,
III | no limit cycles.
| Two stable equilibria, one unstable equilibrium,
IV | no limit cycles.
| One stable equilibrium, four unstable equilibria,
V | one unstable limit cycle.
| Two stable equilibria, three unstable equilibria,
VI | one unstable limit cycle.

For sufficiently large values of $g_{\rm K}$, there are two supercritical Hopf
bifurcations ${\rm HB}_{1}$ and ${\rm HB}_{2}$. Thus for slice ${\rm l}_{1}$
in Fig. 10 there are period solutions in region II. A codimension-1
bifurcation diagram along slice ${\rm l}_{1}$ for which $g_{\rm K}=60$ is
shown in Fig. 11(a). The stable equilibrium solution loses stability through a
Hopf bifurcation ${\rm HB}_{2}$ as ${\rm I}_{\rm ext}$ is varied. A stable
limit cycle emanated from ${\rm HB}_{2}$ ends in another Hopf bifurcation
${\rm HB}_{1}$ before the equilibrium regains stability via ${\rm HB}_{1}$.
Here the system passes through regions I$\to$II$\to$I. As the value of $g_{\rm
K}$ decreases, there appears a generalised Hopf bifurcation, denoted ${\rm
GH}_{1}$, on the Hopf bifurcation locus at $g_{\rm K}\approx 43.9007$. This is
a codimension-2 point where the ${\rm HB}$ locus changes from supercritical
SupHB to subcritical SubHB [33]. Below ${\rm GH}_{1}$, there are two Hopf
bifurcations, a subcritical and a supercritical. Fig. 11(b) is a bifurcation
diagram along slice ${\rm l}_{2}$ in Fig. 10 for which $g_{\rm K}=26$. The
system passes through regions I$\to$II$\to$I as in the previous case (slice
${\rm l}_{1}$) except that the stable equilibrium solution in region I loses
stability through a subcritical Hopf bifurcation ${\rm HB}_{2}$. An unstable
limit cycle emanated from ${\rm HB}_{2}$ changes stability via a saddle-node
bifurcation of limit cycles (SNC), the stable limit cycle ends in a
supercritical Hopf bifurcation ${\rm HB}_{1}$ then to the left of ${\rm
HB}_{1}$ the equilibrium solution regains stability.

(a)

(b)

Fig. 11: (a) A codimension-1 bifurcation diagram along line ${\rm l}_{1}$ with
$g_{\rm K}=60$. (b) A codimension-1 bifurcation diagram along line ${\rm
l}_{2}$ with $g_{\rm K}=26$. The labels and other conventions are as in Fig.
3.

Upon further decrease in the value of $g_{\rm K}$, the loci of saddle-node
bifurcations ${\rm SN}_{1}$ and ${\rm SN}_{2}$ collide and annihilate in a
cusp bifurcation ${\rm CP}_{1}$ at $g_{\rm K}\approx 18.1715$. As $g_{\rm K}$
decreases, a 1:2 resonance bifurcation ${\rm R2}$ and two generalised period-
doubling bifurcations ${\rm GPD}_{1}$ and ${\rm GPD}_{2}$ appear on the locus
of period doubling bifurcation at $g_{\rm K}\approx 12.624$, $15.982$, and
$13.535$, respectively. Also, the loci of saddle-node bifurcations ${\rm
SN}_{3}$ and ${\rm SN}_{4}$ collide and annihilate in a cusp bifurcation ${\rm
CP}_{3}$ at $g_{\rm K}\approx 8.6962$ and the supercritical Hopf bifurcation
${\rm SupHB}$ changes to subcritical Hopf bifurcation in another generalised
Hopf bifurcation ${\rm GH}_{2}$ at $g_{\rm K}\approx 11.3037$, see Fig. 12(a).

(a)

(b)

Fig. 12: (a) An enlargement of Fig. 10. (b) An enlargement of Fig. 10 showing
line ${\rm l}_{3}$ with $g_{\rm K}=8$. The labels and other conventions are as
in Fig. 10 and Table. 4.

As the value of $g_{\rm K}$ is decreased below ${\rm CP}_{3}$, there exist
four saddle-node bifurcations ${\rm SN}_{1}$, ${\rm SN}_{2}$, ${\rm SN}_{3}$
and ${\rm SN}_{4}$, an example is shown in Fig. 12(b) along slice ${\rm
l}_{3}$. The corresponding codimension-1 bifurcation diagram for which $g_{\rm
K}=8$ is shown in Fig. 7(b) and described in Section 3.3. The system passes
through regions ${\rm I}\to{\rm III}\to{\rm V}\to{\rm VI}\to{\rm IV}\to{\rm
I}$ in Fig. 10. The loci of saddle-node bifurcations ${\rm SN}_{2}$ and ${\rm
SN}_{3}$ collide and annihilate in a cusp bifurcation ${\rm CP}_{2}$ at
$g_{\rm K}\approx 18.1715$. As we decrease the value of $g_{\rm K}$ further,
Bogdanov-Takens ${\rm BT}_{1}$ and ${\rm BT}_{2}$ occcur on the loci of
saddle-nodes ${\rm SN}_{2}$ and ${\rm SN}_{1}$ at $g_{\rm K}\approx 7.1062$
and $g_{\rm K}\approx 6.9935$, respectively. The loci of subcritical Hopf
bifurcations emanate from these codimension-2 points. These loci are
tangential to ${\rm SN}_{1}$ and ${\rm SN}_{2}$ at these codimension-2 points.
Observed also are the presence of zero-Hopf bifurcation ZH at $g_{\rm
K}\approx 6.4099$, a codimension-2 where the locus of ${\rm HB}_{2}$
intersects the locus of ${\rm SN}_{4}$, and flip-flop bifurcation at $g_{\rm
K}\approx 6.8379$ on the locus of period doubling bifurcation as $g_{\rm K}$
decreases.

Finally, as $g_{\rm K}$ is decreased further a generalised Hopf bifurcation,
denoted ${\rm GH}_{3}$, occurs on the Hopf bifurcation locus ${\rm HB}_{2}$ at
$g_{\rm K}\approx 4.1025$. Below this codimension-2 point, the only
bifurcations that remain are the two saddle-node bifurcations ${\rm SN}_{1}$
and ${\rm SN}_{2}$. An example is shown in Fig. 13(a) which is an enlargement
of Fig. 13(a). A bifurcation diagram along slice $l_{4}$ for which $g_{\rm
K}=3.5$ is shown in Fig. 13(b). Here the system passes through regions ${\rm
I}\to{\rm IV}\to{\rm I}$.

(a)

(b)

Fig. 13: (a) An enlargement of Fig. 10 showing line ${\rm l}_{4}$. (b) A
codimension-1 bifurcation diagram along line ${\rm l}_{4}$ with $g_{\rm
K}=3.5$. The labels and other conventions are as in Fig. 10 and Table. 4.

##  4 Conclusion

In this present paper, we have studied a 4D-ML model to explore the influence
of second inward ${\rm Na}^{+}$ currents on electrical activities of excitable
tissues. This work is motivated by the results in [32], where it was reported
that voltage-gated ${\rm Na}^{+}$ currents appear to contribute to the
depolarising stage of action potentials in some excitable cells. We focused on
addressing the influence of maximum conductances of ion channels on the
dynamics of the membrane potential. Upon varying the conductance associated
with the ${\rm Na}^{+}$ currents, $g_{\rm Na}$, the model exhibits different
electrical activities.

With the aid of numerical bifurcation analysis, we examined the effects of
parameters on the dynamical behaviour of the model. Our results showed that
increasing the maximum conductance of sodium current $g_{\rm Na}$, the model
transitions from rest state to periodic oscillations. For some values of
$g_{\rm Na}$, the model shows complex behaviour, specifically, it undergoes
cascades of period-doubling bifurcations. It was found that the bifurcation
structure of varying the maximum conductance of potassium current $g_{\rm K}$
is qualitatively similar to that of varying the maximum conductance of calcium
current $g_{\rm Ca}$ except in reverse. That is, increasing the value of
$g_{\rm K}$ results in the same qualitative changes to the dynamics of the
model as decreasing the value of $g_{\rm Ca}$.

We also showed qualitatively the effect of varying the external current
$I_{\rm ext}$ on the dynamical behaviour of the model. Similar bifurcation
diagram has been observed by [29], they discussed the bifurcation diagram in
some detail, although without an explicit determination of the period
oscillations thus their bifurcation diagram seems incomplete. However, in this
work, we give a detailed bifurcation structure. We showed that the unstable
periodic oscillations emanated from the two Hopf bifurcations terminate in
homoclinic bifurcations. We also observed cascades of period-doubling PD
bifurcations for some values of $I_{\rm ext}$. The existence of PD
bifurcations is an indicator that the model can exhibit chaotic behaviour in
some parameter regime.

The codimension-2 bifurcation analysis in $(I_{\rm ext},g_{\rm K})$-plane
gives further details on transitions between different electrical activities
in the model. The electrical activities in the original ML model can be of
Type I or II excitability depending on how the cell transitions from rest
state to periodic oscillations is through a Hopf bifurcation. [18, 36]. In
Type I excitability, the cell transitions from rest to an oscillatory state
via a saddle-node on an invariant circle bifurcation and in Type II
excitability the transition is via a Hopf bifurcation. In this work, the model
exhibits only Type II excitability.

The results in this paper showed that the ${\rm Na}^{+}$ channels may
influence the depolarisation stage of an action potential. It is hope that
this model provides a framework that can aid the understanding of various
electrical activities in excitable cells where the effect of ${\rm Na}^{+}$
ion can be studied. Based on the results of the present paper, more complex
behaviour is expected when two or more cells are coupled together, thus the
dynamics of a network of cells would be addressed in future.

## Conflict of interest

The authors declare that they have no conflict of interest.

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