# Electromagnetic Debye mass within Gribov-Zwanziger action

Aritra Bandyopadhyay  Institut für Theoretische Physik, Universität
Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

###### Abstract

In the present study we have investigated the electromagnetic Debye mass by
computing the static limit of the temporal component of the one-loop photon
polarization tensor involving effective quarks in the loop. These effective
quarks have been considered within the Gribov-Zwanziger action, thereby
incorporating the necessary non-perturbative effects. As an academic exercise,
we have also shown one of the applications of this Gribov modified Debye mass
by using it to evaluate the real and imaginary parts of the heavy quark
potential in a QED system. This computation of the electromagnetic Debye mass
and the corresponding estimations of the heavy quark potential can be seen as
stepping stones in the direction of further practical QCD calculations in the
non perturbative domain.

Relativistic heavy-ion collisions, Debye Mass, Quantum electrodynamics, Heavy
Quarks

##  I Introduction

Due to the continuous influx of the experimental results from the ongoing
ultra-relativistic heavy-ion collision experiments at RHIC and LHC, the
existence of the deconfined quark matter at very high temperatures and/or
densities is now a well known fact [1, 2]. The characteristics of quark matter
can be further explored by studying some important observables that serve as
the signatures of quark matter. One such very important signature of quark
matter is the set of n-point correlation functions. The fact that they are
connected to a plethora of useful observables like dilepton production rate,
transport coefficients and Debye screening mass, makes their investigation all
the more essential in this work. We will however confine our interest mostly
in exploring the electromagnetic (EM) Debye screening mass within a particular
scenario as discussed in the following.

A strongly coupled quark matter can only be explained using pure perturbative
methods like Hard Thermal Loop (HTL) resummation [3, 4] in the limit of very
high temperature ($T>T_{c}$, $T_{c}\sim 160$ MeV being the pseudocritical
temperature of the QCD phase diagram). Although in those relatively higher
temperatures we have grasped enough in-depth understanding of the various
collective modes originating from the quark matter using resummed perturbation
theories (pt), e.g. HTLpt [5, 6, 7], for relatively lower temperatures close
to $T_{c}$, one must include the necessary non-perturbative physics. In recent
years such efforts have been done using the Gribov-Zwanziger (GZ) action [8,
9], which takes into account the non-perturbative physics related to
confinement associated with the magnetic scale $g^{2}T$ ($g$ being the strong
coupling) of the collective modes [10, 11, 12]. GZ action introduces a new
Gribov parameter $\gamma_{G}$ in the gluon propagator $D^{\mu\nu}(P)$, i.e.

$\displaystyle
D^{\mu\nu}(P)=\left[\delta^{\mu\nu}-(1-\xi)\frac{P^{\mu}P^{\nu}}{P^{2}}\right]\frac{P^{2}}{P^{4}+\gamma_{G}^{4}}\,,$
(1)

which moves the poles of the gluon propagator to a nonphysical location,
thereby making the theory a confining one. A perturbative form of the
parameter $\gamma_{G}$ up to one loop order can be self-consistently
determined using a gap equation at asymptotically high temperatures and it is
given as [13]

$\displaystyle\gamma_{G}=\frac{D-1}{D}\frac{N_{c}}{4\sqrt{2}\pi}g^{2}T,$ (2)

where $D$ is the dimension of the theory and $N_{c}$ represents the number of
colors. This modified gluon propagator within GZ action has already been used
in several avenues of exploration to study various properties of quark matter.
These studies have produced various exciting results in several fronts, e.g.
Yang-Mills thermodynamics [14], quark collective modes [15], dilepton
production rate [16], transport properties [17], heavy quark dynamics [18]
etc. More recently, a refined GZ approach has surfaced which has also produced
some interesting results [19, 20, 21, 22, 23]. The in-medium quark propagation
within GZ action generates a novel massless spacelike collective mode aside
from the usual quark and plasmino modes, termed as the Gribov mode [15].
Appearance of this novel mode demands further analysis of the necessary
modifications in the various signatures of the quark matter, some of which
have already been done [16]. In the present work we have studied the EM Debye
mass using the effective quark propagator which includes this novel Gribov
mode and report its consequences.

Static electric fields get screened at large distances because of the
interaction between electrically charged particle and electromagnetic fields
in quark matter. EM Debye mass ($m_{D}$) is basically the inverse screening
length which is related with the corresponding vanishing of the static
potential [24, 25, 26]. As an important spectral property, both EM and QCD
Debye mass have also been studied within various scenarios in the presence of
an external electromagnetic [27, 28, 29, 30, 31] and gravitational [32]
fields. Moreover, the QCD Debye mass is significant in the context of the non-
perturbative physics related to the magnetic scale associated with the GZ
action, as it is the earlier studies of the Debye mass corrections of the
order of the magnetic scale $\mathcal{O}(g^{2}T)$, which indicated that the
background physics is fundamentally non-perturbative [33, 34, 35, 36, 37]. To
the best of our knowledge, the present study incorporates GZ effective quark
propagator in the computation of the Debye mass for the first time. As an
application of the EM Debye mass computed in this work, we have also estimated
the real and imaginary parts of the heavy quark potential in a QED system,
where the in-medium effects are assumed to solely reside in the Debye mass.
Heavy quarks (HQ) are also crucial in the context of characterizing quark
matter as they interact less within the medium because of their relatively
large masses. Discovery of the bound states like $J/\psi~{}(c\bar{c})$ and
$\Upsilon~{}(b\bar{b})$ have prompted several interesting studies of the HQ
potential under various circumstances [38, 39, 40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 50], much of which have essentially been translated into distinctive
refinements of the Debye mass. Hence the effect of a Gribov modified Debye
mass on the HQ potential is certainly compelling to say the least, as recently
showed in Ref [51] where the authors have considered one-loop effective static
gluon propagator. Finally we must emphasise here that the present estimations,
being the first of its kind, is to an extent simplified and rudimentary in
some aspects. But this computations should act as a stepping stone for
further, more involved QCD calculations in near future, which would serve
better practical purposes.

This paper is organized as follows. In section II we have computed the
electromagnetic (EM) Debye mass with GZ modified effective quark propagator
and discussed corresponding results. Next in section III, we have explored the
effect of GZ modified quark propagation on the HQ potential, as one of the
important applications of the EM Debye mass. Finally we summarize our results
in section IV.

##  II EM Debye Mass with GZ action

Being related to the vanishing static potential, the electromagnetic Debye
mass can be determined from the poles of the effective photon propagator at
zero frequency (also known as the static limit). In the static limit, the
electromagnetic Debye screening mass is defined as

$\displaystyle m_{D}^{2}=\Pi_{00}(\omega=0,p\rightarrow 0),$ (3)

where $\Pi_{00}$ is the temporal part of the photon self energy
$\Pi_{\mu\nu}$. At one-loop order, the photon self energy diagram is shown in
Fig. 1, which can be written as

$\displaystyle\Pi_{\mu\nu}(P)=-i\sum_{f}q_{f}^{2}\int\frac{d^{4}K}{(2\pi)^{4}}\textsf{Tr}_{c}\left[\gamma_{\mu}S(K)\gamma_{\nu}S(Q)\right],$
(4)

where $Q=K-P$. The corresponding leading order result for the electromagnetic
Debye mass yields -

$\displaystyle m_{D}^{b}=\frac{eT}{\sqrt{3}}+\mathcal{O}(g^{2}T),$ (5)

where superscript $b$ denotes the bare nature of the result.

Figure 1: Photon self-energy diagram with GZ modified quarks in the loop
(denoted by the double lines).

For the present work, we will be using the GZ modified effective propagator
$S^{*}(K/Q)$ in Eq. (4) (see Fig. 1). Using the helicity representation, the
GZ modified effective fermion propagator can be written as [15, 16]

$\displaystyle iS^{*}(K)$
$\displaystyle=\frac{1}{2}\frac{(\gamma_{0}-\gamma\cdot\hat{\bf{k}})}{D_{+}}+\frac{1}{2}\frac{(\gamma_{0}+\gamma\cdot\hat{\bf{k}})}{D_{-}},$
(6)

where $D_{\pm}$ are explicitly given as

$\displaystyle D_{+}$
$\displaystyle=k_{0}-k-\frac{2g^{2}C_{F}}{(2\pi)^{2}}\sum_{\pm}\int
dr~{}r~{}\tilde{n}_{\pm}(r,\gamma_{G})\times$
$\displaystyle\left[Q_{0}(\tilde{\omega}_{1}^{\pm},k)+Q_{1}(\tilde{\omega}_{1}^{\pm},k)+Q_{0}(\tilde{\omega}_{2}^{\pm},k)+Q_{1}(\tilde{\omega}_{2}^{\pm},k)\right],$
(7) $\displaystyle D_{-}$
$\displaystyle=k_{0}+k-\frac{2g^{2}C_{F}}{(2\pi)^{2}}\sum_{\pm}\int
dr~{}r~{}\tilde{n}_{\pm}(r,\gamma_{G})\times$
$\displaystyle\left[Q_{0}(\tilde{\omega}_{1}^{\pm},k)-Q_{1}(\tilde{\omega}_{1}^{\pm},k)+Q_{0}(\tilde{\omega}_{2}^{\pm},k)-Q_{1}(\tilde{\omega}_{2}^{\pm},k)\right].$
(8)

Here $r$ is the loop momentum in the quark self energy and the modified
frequencies are defined as $\tilde{\omega}_{1}^{\pm}\equiv
E_{\pm}^{0}(k_{0}+r-E_{\pm}^{0})/r$ and $\tilde{\omega}_{2}^{\pm}\equiv
E_{\pm}^{0}(k_{0}-r+E_{\pm}^{0})/r$ with $E_{\pm}^{0}=\sqrt{r^{2}\pm
i\gamma_{G}^{2}}$. Modified distribution functions are given by
$\tilde{n}_{\pm}(r,\gamma_{G})\equiv n_{B}\\!\left(\sqrt{r^{2}\pm
i\gamma_{G}^{2}}\right)+n_{F}(r)$, $n_{B}$ and $n_{F}$ being the Bose-Einstein
and Fermi-Dirac distribution functions respectively. $Q_{0}$ and $Q_{1}$ are
the Legendre functions of the second kind, defined as

$\displaystyle Q_{0}(k_{0},k)$
$\displaystyle\equiv\frac{1}{2k}\ln\frac{k_{0}+k}{k_{0}-k},$ (9)
$\displaystyle Q_{1}(k_{0},k)$
$\displaystyle\equiv\frac{1}{k}(1-k_{0}Q_{0}(k_{0},k)).$ (10)

Figure 2: Three different poles of the GZ modified quark propagator are shown
for two different temperatures, two extremes within the temperature range we
have worked in the present study. These plots show the significant temperature
dependence of the different poles.

As it would be of relevance later in our evaluation, we would also like to
write down the corresponding spectral functions for $D_{\pm}$, which can be
expressed as [16]

$\displaystyle\rho_{\pm}^{G}(k_{0},k)$
$\displaystyle=\frac{k_{0}^{2}-k^{2}}{2m_{q}^{2}(\gamma_{G})}\left[\delta(k_{0}\mp\omega_{+})+\delta(k_{0}\pm\omega_{-})\right.$
$\displaystyle\left.+\delta(k_{0}\pm\omega_{G})\right],$ (11)

where $D_{+}$ has poles at $\omega_{+}$, $-\omega_{-}$, and $-\omega_{G}$ and
$D_{-}$ has poles at $\omega_{-}$, $-\omega_{+}$, and $\omega_{G}$ with a
prefactor, $(\omega^{2}-p^{2})/2m_{q}^{2}(\gamma_{G})$ as the residue
respectively. $m_{q}(\gamma_{G})$ is the Gribov modified thermal quark mass,
given as [15, 16]

$\displaystyle
m_{q}^{2}(\gamma_{G})=\frac{g^{2}C_{F}}{4\pi^{2}}\sum_{\pm}\int\limits_{0}^{\infty}dr\,\frac{r^{2}}{E_{\pm}^{0}}\,\tilde{n}_{\pm}(r,\gamma_{G}).$
(12)

One can immediately notice that the spectral function that appears in Eq. (11)
has no Landau cut contribution, unlike usual HTL spectral functions. This
signifies the absence of the high temperature quasigluons, which are usually
incorporated through the perturbative methods.

With these necessary information in our hand, we can now proceed to evaluate
the temporal component of the photon self energy as

$\displaystyle\Pi_{00}=-i\sum_{f}q_{f}^{2}\int\frac{d^{4}K}{(2\pi)^{4}}\textsf{Tr}_{c}\left[\gamma_{0}S^{*}(K)\gamma_{0}S^{*}(Q)\right].$
(13)

After performing the color and Dirac traces, Eq. (13) yields

$\displaystyle\Pi_{00}$
$\displaystyle=iN_{c}\sum_{f}q_{f}^{2}\int\frac{d^{4}K}{(2\pi)^{4}}\left[\frac{1+\hat{\bf
k}\cdot\hat{\bf q}}{D_{+}(K)D_{+}(Q)}+\right.$
$\displaystyle\left.\frac{1+\hat{\bf k}\cdot\hat{\bf
q}}{D_{-}(K)D_{-}(Q)}+\frac{1-\hat{\bf k}\cdot\hat{\bf
q}}{D_{+}(K)D_{-}(Q)}+\frac{1-\hat{\bf k}\cdot\hat{\bf
q}}{D_{-}(K)D_{+}(Q)}\right].$ (14)

Implementing the static limit further simplifies the expression as, i.e.

$\displaystyle m_{D}^{2}$ $\displaystyle=\Pi_{00}(\omega=0,p\rightarrow 0)$
$\displaystyle=2iN_{c}\sum_{f}q_{f}^{2}\int\frac{d^{4}K}{(2\pi)^{4}}\left[\frac{1}{D_{+}(K)^{2}}+\frac{1}{D_{-}(K)^{2}}\right],$
$\displaystyle=-2N_{c}\sum_{f}q_{f}^{2}T\sum_{k_{0}}\int\frac{d^{3}k}{(2\pi)^{3}}\left[\frac{1}{D_{+}(K)^{2}}+\frac{1}{D_{-}(K)^{2}}\right].$
(15)

To perform the Matsubara frequency sum, we will use the spectral
representation of the corresponding propagators, i.e. $D_{\pm}$. This process
gives the following expressions for the frequency sums :

$\displaystyle T\sum_{k_{0}}\frac{1}{D_{\pm}(K)^{2}}$
$\displaystyle=\int\limits_{-\infty}^{\infty}\left(dk_{0}\right)^{2}\frac{1-2n_{F}(k_{0})}{2k_{0}}\rho^{G}_{\pm}(K)^{2},$
(16)

where $n_{F}$ is the Fermi-Dirac distribution function. Further, using this
template for the frequency sum with the spectral function given in Eq. (11),
we get the final expression for the EM Debye mass as :

$\displaystyle m_{D}^{2}$
$\displaystyle=2N_{c}\sum_{f}q_{f}^{2}\int\frac{d^{3}k}{(2\pi)^{3}}$
$\displaystyle\sum_{\omega=\omega_{\pm},\omega_{G}}\left[\left(\frac{\omega^{2}-k^{2}}{2m_{q}^{2}(\gamma_{G})}\right)^{2}\frac{n_{F}(\omega)}{\omega}\right].$
(17)

One can notice that the modified EM Debye mass receives contributions from all
the three individual collective modes of excitation originating from the quark
propagation within a Gribov modified medium. Eq. (17) is the primary analytic
result of the present work which we will explore further in the following.
With Eq. (17) in our hand, we move on to numerically evaluate the
electromagnetic Debye mass, for which we need the values of different poles as
functions of temperature. For each value of the temperature we solve for the
zeroes of $D_{\pm}$ to get the corresponding $T$ dependent values of the
poles, i.e. $\omega_{+}(k),\omega_{-}(k),\omega_{G}(k)$. As a demonstration,
in Fig. 2 we present these poles for two extreme values of temperatures we
have worked with, i.e. $T=50$ MeV and $T=400$ MeV. One can notice the drastic
variation of the poles for different temperatures. For lower temperatures,
both the Gribov mode $\omega_{G}$ (dotted curve) and the plasmino mode
$\omega_{-}$ (dashed curve) rapidly approach the free mass-less propagation
(not separately shown in Fig. 2) and eventually merges with it simultaneously.
So the medium effects for the lower regime of temperatures are confined within
the very low momentum limit. Whereas, for relatively larger temperatures like
$T=400$ MeV, the Gribov and plasmino modes merge with the free mass-less
propagation at relatively much higher values of the momentum, i.e. $k\sim 0.7$
GeV. This implies that at higher values of temperatures the effect of
incorporating the GZ action in any observable will be much more significant
than that of the lower values of temperatures. Since the perturbative Landau
cut contribution also contributes at relatively higher temperatures due to the
quasigluons captured in the process, it is physically plausible that the
Gribov mode contribution is compensating for the excluded Landau cut
contribution.

Figure 3: Comparison between the bare and the GZ modified Debye mass while
varying with temperature

Figure 4: Comparison between the bare and the GZ modified real part of the HQ
potential while varying with distance.

Once we get the values of the $\omega_{\pm}(k)$ and $\omega_{G}(k)$ for
different values of temperature, it is relatively straightforward to
numerically evaluate the electromagnetic Debye mass as a function of the
temperature using Eq. (17). In Fig. 3 we present the main result of this
study, where we compare the GZ modified electromagnetic Debye mass ($m_{D}$)
with the bare result ($m_{D}^{b}$), given in Eq. (5). We have limited our
comparison to $T$ as low as $0.05$ GeV, keeping in mind that the perturbative
value of $\gamma_{G}$ used for our purpose has been obtained within the
assumption of asymptotically high temperatures. It can be inferred from Fig.
3, that although at lower temperatures there is not much difference between
the bare and the Gribov modified results, at higher temperatures the GZ
modified electromagnetic Debye mass clearly dominates the corresponding bare
result. This result is also in accordance with the behavior noticed in Fig. 2,
which showed the enhanced significance of the Gribov and plasmino modes with
increasing temperature. For high temperatures, the considerable enhancement of
the GZ modified result over the bare result is the most important takeaway in
the context of our present study.

##  III Application to Heavy Quark potential

Figure 5: Comparison between the bare and the GZ modified imaginary part of
the HQ potential while varying with distance.

As an application of the Debye mass evaluated in section II, in this section
we will estimate the corresponding modifications on the real and imaginary
parts of the HQ potential in a QED medium. Considering the full Cornell
potential $V(r)=-\frac{\alpha}{r}+\sigma r$, containing both the Coulomb and
the string part, the real and imaginary parts of the HQ potential is dependent
on the temporal component of the retarded propagator and hence can be obtained
in terms of the Debye mass. The simplified versions of the same are given in a
modified coordinate space $\hat{r}=rm_{D}$ by taking the short-distance limit
$\hat{r}\ll 1$ as [48, 46, 49] :

$\displaystyle{\rm Re}\,V(\hat{r};T,B)$ $\displaystyle=$
$\displaystyle\left(\frac{2\sigma}{m_{D}}-\alpha\,m_{D}\right)\frac{e^{-\hat{r}}}{\hat{r}}$
(18) $\displaystyle-$
$\displaystyle\frac{2\sigma}{m_{D}\,\hat{r}}+\frac{2\sigma}{m_{D}}-\alpha\,m_{D}~{},$
$\displaystyle{\rm Im}\,V(\hat{r};T,B)$ $\displaystyle=$ $\displaystyle
T\left(\frac{\alpha\,{\hat{r}^{2}}}{3}-\frac{\sigma\,{\hat{r}}^{4}}{30\,m_{D}^{2}}\right)\log\left(\frac{1}{\hat{r}}\right).$
(19)

As one can see from Eqs. (18) and (19), in this simplified approximation the
non-perturbative contribution through the GZ action in the heavy quark
potential is being incorporated through the Debye mass $m_{D}$.

In Figs. 4 and 5 we show the variations of respectively the real and imaginary
parts of the HQ potential with respect to distance. For both the cases we have
chosen two different temperatures, one relatively lower at $T=200$ MeV and the
other in the higher temperature regime at $T=400$ MeV. It can be observed that
for higher temperature the modification in the HQ potential due to the GZ
modified quark propagation is more prominent, mimicing the results from
section II. Furthermore, one can explicitly notice from both Figs. 4 and 5,
that for larger values of temperature, the Gribov modified result diverges
from the bare result rapidly, i.e. at a lower value of the distance $r$. It is
worth noting that in this work, we have just focused on the modifications of
the HQ potential, and have not explored other quantities related to HQ
potential, e.g. binding energy or dissociation temperature. However, through
these preliminary observations we can rightfully conclude that once a more
complete study of QCD Debye mass incorporating the GZ modified quarks sees the
light of the day, it would be worth exploring the physical quantities such as
HQ dissociation, which depends on the HQ potential.

##  IV Summary and Outlook

The present study explores the electromagnetic Debye mass as an important
signature of quark matter, by incorporating the GZ modified quark propagator
in the medium. This spontaneously includes a novel Gribov mode into the
collective excitations of the quark matter which gets reflected in the
corresponding observables computed, i.e. EM Debye mass and HQ potential.
Absence of the Landau cut in the GZ modified quark spectral function means
there are no explicit high temperature contributions from the quasigluon
sector in the present scenario. However the spacelike Gribov mode compensates
for this and encompasses medium effects up to considerably high momenta for
higher temperatures. As an evidence of this fact, in the case of the EM Debye
mass we find that with increasing temperature, the GZ modified Debye mass
starts to dominate its bare counterpart due to the role played by the
increasing contributions from the Gribov and plasmino mode. This enhanced EM
Debye mass is expected to influence further prominent effects in corresponding
dynamic signatures of quark matter, e.g. quarkonia suppression. In view of
this we have also provided an estimation about the modifications on the HQ
potential, albeit in a QED medium as an application of the GZ modified EM
Debye mass.

To the best of our knowledge, this is the first effort towards evaluating the
Debye mass within the GZ action. To get a first qualitative estimate about the
modification of the Debye mass, we have chosen to evaluate the EM Debye mass.
Naturally the next step, which would serve better practical purposes, will be
to explore the QCD Debye mass. For that case we need to compute more number of
diagrams due to the self coupling of gluon which requires some nontrivial
effort. It will be quite interesting to see how the QCD Debye mass with GZ
modified quarks compares with and excels the previously obtained non-
perturbative results of the same that include the $\mathcal{O}(g^{2}T)$
corrections [52, 37]. This would also help us to accurately estimate the HQ
potential in a QCD medium and corresponding other relevant physical
observable. Also for our present preliminary study, we have only chosen the
asymptotic value of the Gribov parameter $\gamma_{G}$, solved from the gap
equation. However, some recent studies [14, 17, 18] have also used modified
versions of the Gribov parameter, which can be more appropriate to capture the
physics for relatively lower values of temperature. These modifications would
also provide some interesting takes in calculating the Debye Mass more
precisely.

## Acknowledgements

The author would like to thank Sanjukta Paul and Arghya Mukherjee for helpful
discussions. This work was supported by the postdoctoral research fellowship
of Alexander von Humboldt Foundation.

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# The interplay of common noise and finite pulses on biological neurons

Afifurrahman1∗111*Corresponding author, Mohd Hafiz Mohd2, Farah Aini Abdullah3
1Department of Mathematics Universitas Islam Negeri Mataram  Mataram 83116
Indonesia
2,3School of Mathematical Sciences  Universiti Sains Malaysia  11800 USM
Penang  Malaysia
*Email<EMAIL_ADDRESS>Received (date), Revised (date), Accepted for publication (date). Copyright ©2020 Published by Indonesian Biomathematical Society, e-ISSN: 2549-2896, DOI:10.5614/cbms.2020.1.1.(Paper’s number)

###### Abstract

The response of neurons is highly sensitive to the stimulus. The stimulus can
be associated with a direct injection in vitro experimentation (e.g., time
dependent and independent inputs); or post-synaptic potentials resulting from
the interaction of many neurons. A typical incoming stimulus resembles a noise
which in principle can be described as a random variable. In computational
neuroscience, the noise has been extensively studied for different setups. In
this study, we investigate the effect of noisy inputs in a minimal network of
two identical leaky integrate-and-fire (LIF) neurons interacting with finite
pulses. In particular, we consider a Gaussian white noise as a standard
function for stochastic modelling of neurons, while taking into account the
pulse width as an elementary component for the signal transmission. By
exploring the role of noise and finite pulses, the two neurons show a
synchronous spiking behaviour characterized by fluctuations in the inter-spike
intervals. Above some critical values the synchronous regime collapses onto
asynchronous dynamics. The abrupt change in such dynamics is accompanied by a
hysteresis, i.e., the coexistence of synchronous and asynchronous firing
behaviour.

Keywords: LIF neuron, stochastic process, finite pulses, inter-spike interval,
synchronization.

2010 MSC classification number: 37N25, 92B25, 92C42

##  1 Introduction

A single neuron reacts to the incoming stimuli in unexpected ways. As for
instance, it might fires a pulse (spike) at constant rates, if the neuron
operates above a threshold. The periodic firing behaviour is probably the
simplest dynamical phenomena observed in the neuronal models [4, 11, 23, 33].

The variability of neuron’s response to the incoming stimulus is encoded in
the so-called inter-spike interval (ISI), i.e., the time-length between any
two consecutive spikes [34, 35]. In a situation where the periodic regime
appears, the coefficient of variations of ISIs is zero. Here, one can easily
predict a sequence of spiking times once the neuron fires its first spike.

Unfortunately, the response of real neurons is far from being regular and
periodic. Instead, the irregular firing activity is the most prominent feature
and it might be related to brain disorders according to some experimental
studies [3, 36, 37]. The irregularity is characterized by the variability on
ISIs, which means that the fluctuation is not zero at all [38]. Investigating
the necessary conditions in which irregular (and regular) behaviours exist is
upmost important as it allows us to understand how the brain perceives the
world.

Multiple sources are mainly responsible for the emergence of irregular
behaviour. In the cortical neurons, for example, the intrinsic noise resulting
from the fast activation of ion channels may contribute to the variability on
ISIs [24]. The noise has been extensively studied for different setups and
conditions. It is believed that the noise is not only slows down but also
improves the reliability of neuronal firing, as well as facilitates the
temporal patterning of synchronization and chaos [8, 16, 26, 27, 39, 42].

Apart from the noise effect, the irregular firing behaviour is also induced by
some chemical-type currents which highlight a significant impact of synaptic
transmission [7]. The real neurons communicate by sending and receiving pulse
that lasts about 1-2 ms time course [11]. A recent study reports the pulse
width is responsible for the irregular response of neurons in auditory systems
[45]. It also has been suggested that the persistence of irregular dynamics in
coupled oscillators is strongly depends on the duration of the pulse [1, 20]
making it plausible for neuronal modelling.

In this paper, we investigate numerically the stochastic neural networks
taking into account the pulse width as an additional parameterisation for the
synaptic coupling, which has never been considered in any other similar works
(see [17, 26, 27]). Moreover, compared to the previous studies, our object of
study is a one-dimensional leaky integrate-and-fire (LIF) neuron [43, 44]. The
model is quite simple yet the dynamics are very rich.

On top of that, the earlier works on stochastic LIF model were mostly dealing
with a single neuron and concern only to the theoretical perspectives of it
while less attention has been given to the emerging dynamical states in
macroscopic scales [18, 30, 40, 41]. Here, We are working with a network of
two noisy LIF neurons interacting via excitatory synapses to test the
robustness of irregular firing activities. The synaptic transmission amongst
them is modelled as a smooth function of exponential shape. Due to the
randomness of stochastic input, we limit ourselves to a numerical analysis.

More precisely, in Sec. 2 we define the model including the Gaussian white
noise and finite width pulse. In the same section, we introduce the schemes
used to perform numerical simulation and the corresponding statistical
quantities, namely the synchrony measure, to assess the emerging dynamical
regimes. In the Sec. 3 we present some findings regarding the effect of noise
on single neuron, which later on generalized for the two neurons interacting
through exponential pulses. Finally, in Section 4 we discuss the limitations
of this study and highlight some problems for the future.

##  2 Model formulation

The starting point of this paper is a general stochastic system (Equation 2)
introduced in [26]. Instead of electrical coupling, we incorporate the
synaptic type into the model by employing the finite width pulses. The unit
cell is modelled as a LIF neuron defined in [19]. We use a simple network
architecture of two cells interacting via excitatory synapse with the same
coupling strength as studied in [27]. More specifically, our systems (see
Figure 1) consist of two identical and mutually coupled LIF neurons given by
the following differential equations:

$\dot{u}=a-u+\xi_{u}(t)+\frac{\mu}{2}e_{u},\qquad\dot{v}=a-v+\xi_{v}(t)+\frac{\mu}{2}e_{v},$
(1)

combined with the resetting rules: if $u\geq u_{th}\Rightarrow u=u_{r}$ and
$v\geq v_{th}\Rightarrow v=v_{r}$, respectively. The variables $u$ and $v$
represent the membrane potentials for each neuron, $\dot{u}$ and $\dot{v}$ are
the corresponding derivatives with respect to time $t$, while $a$ is constant
current. We assume that all variables and parameters are dimensionless. The
parameter $\mu>0$ is the coupling strength describing the intensity of
excitatory postsynaptic potential. We introduce a scaling factor of $1/2$ to
account for the full connectivity in the present model of two interconnected
neurons. In principle, the model can be generalized to $N$ number of fully
coupled oscillators, so the appropriate scaling would be $1/N$ [11].

Figure 1: A schematic representation for model (1)

Following ref [1, 31], the fields $e_{u}$ and $e_{v}$ are the linear
superpositions of exponential spikes emitted by the sending neurons.
Mathematically,

$\dot{e}_{u}=-\alpha\left(e_{u}-\sum_{k}\delta\left(t-t^{v}_{k}\right)\right),\qquad\dot{e}_{v}=-\alpha\left(e_{v}-\sum_{k}\delta\left(t-t^{u}_{k}\right)\right),$
(2)

where $\alpha$ denotes the inverse width of the incoming excitatory pulses,
and both $t^{u}_{k},t^{v}_{k}$ are the emission times of the $k$th pulse sent
by the neuron $u$ and $v$, respectively. Whenever $u,v$ reaches the threshold
$u_{th},v_{th}$, then the two neurons are reset to $u_{r},v_{r}$. In the limit
of $\alpha\rightarrow\infty$, the Equations (2) are simply the sum of $\delta$
pulses, i.e.,

$e_{u}=\sum_{k}\delta\left(t-t^{v}_{k}\right),\qquad
e_{v}=\sum_{k}\delta\left(t-t^{u}_{k}\right).$ (3)

The functions $\xi_{u}(t)$ and $\xi_{v}(t)$ are Gaussian white noise with the
statistical property: $\mathbf{E}\left[\xi(t)\right]=0$ for all time $t$, and
the autocorrelations
$\mathbf{E}\left[\xi(t^{\prime})\xi(t^{\prime\prime})\right]=\sigma^{2}\delta\left(t^{\prime}-t^{\prime\prime}\right)$
for $t^{\prime}\neq t^{\prime\prime}$, where $\mathbf{E}$ denotes the expected
value. Here, $\sigma$ is the noise amplitude and $\delta(t)$ is the delta
function. For simplicity, the functions $\xi_{u}(t)=\xi_{v}(t)=\xi(t)$, which
assumes the same noisy inputs for the two neurons.

###  2.1 Numerical analysis

The Euler-Maruyama scheme [28] is implemented to solve simultaneously the main
Equations (1)-(2). Notice that, as the integration time step $dt\rightarrow 0$
the trajectory is smoothened since the fluctuations term vanishes with $dt$.
For convenience, we choose $dt=10^{-3}$.

The initial conditions for the membrane potential are drawn randomly from a
uniform distribution in a closed interval $[0,\epsilon]$ i.e., $(u(0),v(0))\in
U_{[0,\epsilon]}$ while the fields are initially set to 0. Table 1 summarizes
all parameters being used for the numerical simulation. Both reset potential
$(u_{r},v_{r})$ and threshold $(u_{th},v_{th})$ have been chosen following the
hypothetical values in the deterministic version of LIF neurons [19]. The
spike emission of LIF neuron is achieved when the constant current is larger
than the threshold, which in this case: $a>1$. Here we choose $a=1.5$ as
initially set in [11]. We also assume a relatively weak coupling strength in
the order of magnitude $10^{-3}$ as introduced in [27]. Finally, the noise
amplitudes and the pulse widths have been selected according to the
computational experimentations [1, 26, 30].

Parameters |  | References
---|---|---
Threshold: $u_{th},v_{th}$ | 1 | [19]
Reset potential: $u_{r},v_{r}$ | 0 | [19]
Constant current: $a$ | 1.5 | [11]
Coupling strength: $\mu$ | $2\times 10^{-3}$ | [27]
Noise amplitude: $\sigma$ | $[0.1,1.4]$ | [26]
Inverse pulse width: $\alpha$ | $[20,95]$ | [1]
Table 1: The default parameters used during the simulation for Eqs.(1)-(2).

In order to quantify the degree of synchronization in coupled systems (1)-(2),
we make use of the average technique [26]

$R=\langle\sqrt{\left(v-u\right)^{2}+\left(e_{v}-e_{u}\right)^{2}}\rangle.$
(4)

The number $R$ is a synchronization error and the angular bracket
$\langle\star\rangle$ denotes time averaging. If the two neurons are
completely synchronized then, $R=0$. Complete synchronization refers to a
dynamical regime where all neurons within the network are firing exactly at
the same time.

##  3 Main Results

###  3.1 Single neuron

In the absence of any synaptic interaction ($\mu=0$), the two neurons are
indeed identical and evolve independently in time. In this case, the fields
(2) become irrelevant and hence the problem of coupled systems (1) can be
reduced to a single neuron. The time evolution for a single neuron is governed
by the following stochastic system:

$\dot{y}=a-y+\xi(t),\hskip 5.69046pt{\text{if}}\hskip 2.84544pty\geq
1\Rightarrow y=0,$ (5)

where $y$ is the membrane potential. The Equation (5) is a special case of the
well-known Ornstein-Uhlenbeck process. To simulate the trajectory of
stochastic LIF neuron (5), it is reformulated as follows

$dy=\left(a-y\right)dt+\sigma\sqrt{dt}x$ (6)

where $x$ is a random number, drawn from a zero-mean Gaussian distribution
with the unit variance. The same procedure of numerical integration is also
adjusted for Equation (6) as in the Section 2.1.

The firing rate and coefficient of variations are statistical quantities
frequently used to characterize the spiking behaviour for single neuron [11,
29]. The firing rate refers to the number of spikes divided by the length of
integration time. The explicit formula is given by

$\nu=\lim_{t\rightarrow\infty}\frac{n_{t}}{t}$ (7)

where $n_{t}$ is the number of spikes emitted by the neuron over a time
interval $t$.

The coefficient of variations ($CV$) is a microscopic measure of the dynamics
based on the variability of inter-spike intervals. The inter-spike interval is
time difference of two consecutive spikes, i.e., ISI$=t_{k+1}-t_{k}$. The $CV$
of ISIs is estimated by a ratio

$CV=\frac{\Delta}{\eta}$ (8)

where $\Delta$ is standard deviation of ISIs and $\eta$ is the corresponding
mean ISIs.

When $\sigma=0$, meaning that no external noise is injected to the neuron, the
Equation 6 is just a linear deterministic system:

${d{y}=(a-y)dt.}$ (9)

The only relevant parameter here is $a$. Note that, the neuron emits a spike
if $a$ is larger than threshold 1, otherwise there is no spike emitted. The
neuron’s free-noise trajectory is obtained by integrating Eq. (9) with an
initial condition $y(0)=0$ as shown in Figure 2a. Once the membrane potential
passes a threshold of 1, it is reset to 0 while at the same time, the spike is
produced.

The periodic firing behaviour of the neuron implies that $t_{1}$ defines a
period of oscillation or ISI (for a detailed discussion on how to derive
$t_{1}$ explicitly see [2]). The next spikes consecutively occur at
$t_{2}=2t_{1},t_{3}=3t_{1},…t_{i}=it_{1}$ where $i$ is a positive integer.

Figure 2: (a)-(b) Time series of noise-free ($\sigma=0$) and noisy trajectory
($\sigma=0.5$) for LIF neuron (see Eqs. (5)-(6)) with a fixed $a=1.5$. When
the membrane potential $y(t)$ hits the threshold 1 (red dashed line), it is
reset to 0. (c) The probability density for white noise when $\sigma=0.5$
(green) and $\sigma=1$ (black) generated from the simulation. The solid red
curves correspond to $f(x)$ in the Equation 10.

Now, let us consider the dynamics of noisy neuron utilizing $\sigma\neq 0$.
The noise is characterized by a probability density function

$f(x)=\frac{1}{\sigma\sqrt{2\pi
dt}}\exp{\left(-\frac{1}{2dt}\left(\frac{x}{\sigma}\right)^{2}\right)}.$ (10)

Since the neuron is stimulated by the noise current with an amplitude
$\sigma=0.5$ (green colour in Figure 2c), the inter-spike interval fluctuates
irregularly in time (Figure 2b). We can no longer predict the exact timing of
the spikes because the spike arrivals are independent of one another. However,
we may characterize the distribution of the spikes by splitting the length of
integration time $t$ into $K$ bins such that

$\tau=\frac{t}{K}.$ (11)

If $K$ is large (meaning that $\tau$ becomes sufficiently small)222This
condition implies that any two or more spikes cannot occur together, instead,
either exactly there is a single spike or no spike at all., the spike count in
each bin is apt to be either 0 or 1. As a result, the distribution of the
spikes is characterized by

$\mathbf{E}\left(\\{0,1\\}\right)=\frac{n_{t}}{K}=\nu\tau,\qquad\mathbf{V}\left(\\{0,1\\}\right)=\nu\tau-(\nu\tau)^{2},$
(12)

where $\mathbf{E}$ and $\mathbf{V}$ stand for the expectation and variance of
the spike events, respectively. The second equality in the expected value
results from the Equation 7 and 11.

Figure 3: Firing rate $\nu$ versus $\sigma$, coefficient of variations $CV$
versus $\sigma$.

Figure 3 shows a more quantitative description of neural activity. There one
can see, the firing rate $\nu$ increases with the noise amplitude $\sigma$
meaning that the neuron is highly sensitive to the random stimulus. Such
observation is in line with the experimental study, particularly for the
neurons in the cerebral cortex [6].

The same scenario is also shown when looking at the plot for the coefficient
of variations $CV$ as a function of $\sigma$. The fluctuation of the inter-
spike intervals is an effect of noisy inputs that raises by increasing the
parameter $\sigma$. Such dynamics is robust, not only in the case of LIF
neuron but also found in the biophysically Hodgkin-Huxley neuronal model [21].
The noise-induced spike times variability serves as a basic mechanism for the
brain functioning such as information processing and perception [9].

###  3.2 Noise-induced synchronization in two connected neurons

We start our analysis of the noise effect on a network of two neurons
($\mu>0$) interacting through finite width pulses. All parameters are kept the
same as in Section 2.1 with the initial conditions are drawn randomly in a
such way that $(u(0),v(0))\in U_{[0,1]}$ . The model (1) assumes a weak
coupling and so we fix the coupling strength as $\mu=2\times 10^{-3}$. The
only relevant parameters are $\sigma$ and $\alpha$. The simulation is
performed over 10000-time units after discarding 2000-time units of transients
to let enough of the systems settle onto the attractor.

Figure 4: Synchronization error $R$ as a function of noise amplitude $\sigma$
for different pulse widths $\alpha^{-1}$ : black triangles ($\alpha=20$),
green circles ($\alpha=60$), and blue diamonds ($\alpha=95$). The two panels
inset are the time series of the membrane potentials for $\sigma=0.4$ and
$\sigma=1$, respectively, when $\alpha=20$.

We turn our attention on Figure 4 where we have plotted the synchrony measure
upon varying the parameter $\sigma$ for a fixed $\alpha$. If $\alpha=20$, the
synchronization error is approximately $R\approx 0.5$ while the noise
amplitude is within the interval range $\sigma\in[0.1,0.59)$. In this regime,
the two neurons are asynchronously firing the spike as shown by the time
series of membrane potential $u$ (solid black line) and $v$ (red dashed line):
see the upper panel inset generated for $\sigma=0.4$.

Around $\sigma\approx 0.6$ the synchronization error drops abruptly to 0
meaning that the two neurons are now completely synchronized. The time series
analysis reveals that the ISIs fluctuate due to the presence of noise (lower
panel inset). For future reference, we define $\sigma_{\tau}$ as a transition
point at which the systems collapse onto complete synchronization (CS).
Formally,

$\sigma_{\tau}=\min\\{\sigma|R\left(\sigma\right)=0\\}.$ (13)

It is also helpful to define a subset $S_{\tau}\subset S$

$S_{\tau}:=\\{\left(\alpha,\sigma_{\tau}\right)|R\left(\alpha,\sigma_{\tau}\right)=0\\},$
(14)

where $S:=\\{\left(\alpha,\sigma\right)|R\left(\alpha,\sigma\right)=0\\}$. As
instance, we observe the CS regime is self-sustained for $\alpha=20$ within
the range of $\sigma\in[0.6,1.4]$, where $\sigma_{\tau}\approx 0.6$ and
$S_{\tau}=\\{(20,0.6)\\}$. Specifically, the time series of membrane potential
show that $u(t)=v(t)$ for each $t$ and $t\rightarrow\infty$: see the lower
panel inset generated for $\sigma=1$.

Upon increasing $\alpha$ to 60 and 95, the asynchronous firing (AF) behaviour
keeps persist in a finite but smaller interval range $\sigma\in[0.1,0.58)$ and
$\sigma\in[0.1,0.49)$, respectively. Meanwhile, the CS emerges in a wider
interval range $\sigma\in[0.58,1.4]$ and $\sigma\in[0.49,1.4]$. The Figure 5
sketches $S_{\tau}$ for $\alpha\in[22,95]$ which are estimated for very long
integration time. The curve $S_{\tau}$ splits the whole plane into two
regions: complete synchronization (above the curve) and asynchronous dynamics
(below one).

Figure 5: Parameter space $\left(\alpha,\sigma\right)$. The red stars
correspond to $S_{\tau}$.

To get a better insight into the role of noise and finite pulses to neuron
dynamics, we produce a distribution of ISIs over 900000-time units for the
neuron $u$ as displayed in Figure 6. As seen, both noise and finite pulses
give rise to the variations of ISIs. The width of the distributions is almost
ten times wider as $\sigma$ is increased from 0.4 to 1 for each case of
$\alpha$. Meanwhile, it is almost doubled upon shortening the pulse widths.

Figure 6: Distribution of inter-spike intervals for $\sigma=0.4$ (left column)
and $\sigma=1$ (right column). The vertical axis for all panels is in the
logarithmic scale. Each row corresponds to different $\alpha$: upper panel
($\alpha=20$), middle panel ($\alpha=60$) and lower panel ($\alpha=95$).

###  3.3 Coexistence regime

So far, we have explored the behaviours of the coupled neurons (1) by
selecting random initial conditions $(u(0),v(0))\in U_{[0,1]}$ and found a
discontinuous transition upon tuning the parameter $\sigma$. Let us now
investigate the possibility of coexistence regimes that are often accompanying
the transition phenomena.

Figure 7: The coexistence of complete synchronization and asynchronous firing
dynamics for $\alpha=20$. The black triangles, magenta and green crosses
correspond to different initial conditions: fully random (black triangles),
finite but small interval width (magenta and green crosses). The narrow
initial conditions are chosen to be in the order of $\epsilon=10^{-3}$
(magenta crosses) and $\epsilon=10^{-2}$ (green crosses). The time series
corresponds to $\sigma=0.4$ with random initial conditions (upper panel) and
narrow initial conditions (lower panel).

We start by fixing a small interval of width $\epsilon$ at which the initial
conditions for neuron $u$ and $v$ are selected whereas the fields are set to
zero. Figure 7 compares the emerging scenario both for random and narrow
initial conditions. In the interval range of $\sigma\in[0.1,0.6)$, the
$R\approx 0.5$ results from imposing random initial conditions (see the black
triangles). It is a clear indication of asynchronous firing dynamics. However,
one can see that the complete synchronization emerges in the whole range of
parameter $\sigma$ upon choosing a small $\epsilon=10^{-3}$ as signalled by
$R=0$ (see the magenta crosses).

Figure 8: Left to right: Distribution of inter-spike intervals for neuron $u$
when $\alpha=20$ and $\alpha=60$. The vertical axis is in the logarithmic
scale. The colours are corresponding to different noise amplitudes:
$\sigma=0.4$ (black), $\sigma=0.7$ (red), and $\sigma=1$ (green). The data are
all generated for the narrow initial conditions $\epsilon=10^{-3}$.

The scenario is further supported by the time evolution of neuron $u$ (solid
black) and $v$ (red dashed) for $\sigma=0.4$ in the inset panels. The upper
panel correspond to the green crosses where $\epsilon\in(10^{-3},1]$ or
equivalently random initial conditions. Clearly, the two neurons fire the
spike asynchronously. In comparison, the two neurons are identical for
$\epsilon=10^{-3}$ as displayed in the lower panel.

Once again, the complete synchronous state is characterized by fluctuating
ISIs. We sketch the distribution of ISIs for neuron $u$ in Figure 8. For a
relatively low noise intensity (black curve), the ISIs looks homogeneous
compared to the intermediate and strong (red and green curves) noises.
Interestingly, decreasing the pulse width significantly adds the variability
of ISIs.

##  4 Discussion and Conclusion

In this study, we have investigated the effect of noisy inputs on the firing
patterns in a minimal network of two LIF neurons interacting through finite
width pulses. Consistent to the earlier reports [26, 27, 14], the noise evokes
the neuromodulation of synchronous and asynchronous firing dynamics. Some
experimental studies show that both phenomena are robust in certain areas of
the brain such as the visual cortex and hippocampus [5, 12], which might be
associated with brain disease such as seizures and epilepsy [22].

We observe the existence of synchronous behaviour, as indicated by $R=0$. The
main important feature of the regime is the variability of ISIs, while at the
same time the neurons are firing simultaneously. This study thereby suggests
the dependence of the average ISI, not only on the coupling strength as
highlighted in [13], but also on the noise amplitude. Given this situation, it
is impossible to predict the exact timing of the spike compared to the
classical case of “periodic” synchronization333“Periodic” synchronization
refers to the regimes where the fluctuation of ISIs of the whole networks is
zero on average [1, 32].. Consequently, the computation becomes more
complicated as the noise interferes with the system: one must go beyond the
traditional approaches at least in the context of stability analysis [15].

As we have shown, the fluctuation of ISIs changes when the noise is varied. A
sufficient strong noise causes a large excursion on the trajectory and thus
greatly affects the next spiking time once the last spike has fired. This is
possibly the mechanism underlying the appearance of wide ISI distributions
displayed in Figure 6 (e.g., for $\sigma=1$). Upon reducing the noise
amplitudes, the two neurons desynchronize and fire the spike independently as
signalled by $R>0$. In the microscopic level, the regime is characterized by
more uniform ISIs and hence a shorter width of ISI distributions (Figure 6,
$\sigma=0.4$), similar to that have been observed in other neuronal models
[17, 25].

The novel outcome of this study is the role played by the width of exponential
pulses in shaping the ISI distribution. The width of ISI distribution is
lengthened as the pulse duration is shortened (see again Figure 6 and 8),
which also suggesting the dependence of ISI distribution on the finite width
pulses. It is unclear, however, to what extent the pulse width affects the
distribution of ISIs produced by different levels of noisy inputs. This
problem, in principle, can be approached mathematically by deriving an
equation describing the probability distribution of ISIs by means of the
Fokker-Planck equation [18], considering the pulse widths as the additional
parameter.

Another observed phenomenon is the transition from complete synchronization to
asynchronous dynamics. As indicated by the sharp jump of global observable
variable $R$, the transition is discontinuous, a typical first-order
transition commonly found in many living cells including neurons, as usual
mechanism to respond to the external stimulus [10]. In this context, the noise
should be regarded as a crucial component in the mathematical modelling for
the nervous systems. The real neurons are noisy, and the noise might come from
many sources such as thermal noise, ion channel propagation, synaptic
transmissions, network connectivity, etc [9, 46]. Furthermore, the transition
region is accompanied by a hysteresis. Upon choosing different initial
conditions within a small interval range $\epsilon=10^{-3}$, we explore the
emergence of bistable regimes: the new value of $R=0$ (see magenta crosses in
Figure 7) indicates a complete synchronous behaviour.

Our findings highlight the key points for future reference. First, even in
simple stochastic LIF neural networks, the dynamical regimes are intriguing
and are in line with the studies in a higher dimensional system, such as
Hindmarsh-Rose and Morris-Lecar neurons (see again [26, 27]). We hypothesize
the overall scenario emerging in this model is likely to persist in the larger
system sizes. Secondly, the finite pulses is matter: shortening the pulse
width adds the variability of ISIs in stochastic neurons. If confirmed
analytically, these statistics could lead to well-established computational
models and thus better understanding of irregular activity in the brain.

Finally, it is important to note some limitations of this study. We are
working with a minimal network of two globally coupled identical neurons,
while the brain consists of billions of neurons having more complex structures
and intrinsic properties. The works herein can be extended to large-scale
networks in the assumption of two interacting populations of excitatory and
inhibitory neurons, which will be our next object of study. This study is also
limited to test numerically the impacts of noise and finite pulses on neuron
dynamics (see again Figure 5). We argue that the asynchronous firing dynamics
vanishes for a large $\alpha$, therefore further studies are required.

## Acknowledgement

This work was partially supported by the Indonesian Mathematical Society
(IndoMS) through a visiting research program, grant No.
028/Pres/IndoMS/SP/VIII/2022.

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$\displaystyle\dot{f}(\tau,\sigma)\rightarrow\partial_{\tau}f(\tau,\sigma+\alpha(\tau))=\dot{f}(\tau,\sigma+\alpha(\tau))+f^{\prime}(\tau,\sigma+\alpha(\tau))\dot{\alpha}(\tau).$
(111c)

Note that in the last line, we use the dot in $\dot{f}$ to denote the
derivative with respect the the first argument of the function $f$.
Consequently

$\displaystyle\int
d\sigma\frac{\dot{f}(\tau,\sigma)}{f^{\prime}(\tau,\sigma)}\rightarrow\int
d\sigma\left(\frac{\dot{f}(\tau,\sigma+\alpha(\tau))}{f^{\prime}(\tau,\sigma+\alpha(\tau))}+\dot{\alpha}(\tau)\right)=\beta(\tau)+2\pi\dot{\alpha}(\tau)\overset{!}{=}0$
(112)

Clearly, this can be set to zero solving the last equation as a differential
equation for $\alpha(\tau)$. However, since this is a first order equation,
the condition of the form $\alpha(\tau_{0})=0$ can only be accommodated in
general at either the beginning or the end of the circuit, but not both.

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