# U(1) dynamics in neuronal activities

Chia-Ying Lin  Department of Physics, National Tsing Hua University, Hsinchu
300044, Taiwan  Ping-Han Chen  Department of Physics, National Tsing Hua
University, Hsinchu 300044, Taiwan  Hsiu-Hau Lin<EMAIL_ADDRESS>Department of Physics, National Tsing Hua University, Hsinchu 300044, Taiwan
Wen-Min Huang<EMAIL_ADDRESS>Department of Physics, National Chung
Hsing University, Taichung 402204, Taiwan

(April 30, 2021)

###### Abstract

Neurons convert the external stimuli into action potentials, or spikes, and
encode the contained information into the biological nerve system. Despite the
complexity of neurons and the synaptic interactions in between, the rate
models are often adapted to describe neural encoding with modest success.
However, it is not clear whether the firing rate, the reciprocal of the time
interval between spikes, is sufficient to capture the essential feature for
the neuronal dynamics. Going beyond the usual relaxation dynamics in Ginzburg-
Landau theory for statistical systems, we propose the neural activities can be
captured by the U(1) dynamics, integrating the action potential and the
“phase” of the neuron together. The gain function of the Hodgkin-Huxley neuron
and the corresponding dynamical phase transitions can be described within the
U(1) neuron framework. In addition, the phase dependence of the synaptic
interactions is illustrated and the mapping to the Kinouchi-Copelli neuron is
established. It suggests that the U(1) neuron is the minimal model for single-
neuron activities and serves as the building block of the neuronal network for
information processing.

##  I Introduction

Figure 1: Neurons with different network structures: (A) Recurrent neuronal
network often observed in biological nerve systems. (B) Feed forward network
with input, output and hidden layers, largely put to practice in deep
learning.

Human beings rely on their nervous systems to detect external stimuli and take
proper reactions afterwards. Neurons are the fundamental units in the nervous
system and deserve careful and thorough characterizations for their responses
to external stimuliGalizia2013 ; Gerstner2014 ; Izhikevich2007 ; Dayan2001 .
However, because neurons display considerable diversity in morphological and
physiological properties, it is rather challenging to pin down the essential
degrees of freedom even when studying the single-neuron dynamics. While the
neural encoding and decodingDayan2001  are not fully understood yet, the
action potentials, or the spikes, of the neurons upon external stimuli are the
apparent means to pass the information onto the nervous system. In
consequence, the firing rate of the neuronal spiking is often used for data
analysis and modelingWilson1972 ; Ermentrout1994 ; Gerstner1995 ; Shriki2003 ;
Ostojic2011 . The firing-rate models are appealing due to their simplicity and
accessibility for numerical simulations.

Although the firing-rate models are helpful descriptions for neural circuits,
it is not clear whether the neuronal spiking alone is sufficient to capture
the essential features of neuronal activities. At the microscopic scale, the
electric activities of a single neuron arise from a variety of ionic flows
passing the relevant ion channels embedded on the membrane of the nerve cell
Choe2002 ; Naylor2016 . The conductance-based approach, such as the Hodgkin-
Huxley model Hodgkin1952 , provides an effective description for the emergence
of action potentials by the inclusion of gating variables of the ion channels
involvedGerstner2014 . Above some current threshold, the neuronal spikes
starts to appear. Hodgkin proposed to classify the neurons into type I or type
II depending on whether the firing rate changes continuously or
discontinuously above the current thresholdHodgkin1948 . While the
conductance-based model with ion-channel dynamics explains the emergence of
the neuronal spikes and provides a better description for biological details,
it blurs the priority of various degrees of freedom in a single neuron,
rendering a clear understanding of neuronal dynamics intractable.

The dynamical transitions in a single neuron above the current threshold posts
another challenge. Both the biologically based models, such as the Hodgkin-
Huxley model and its generalizations, or the reduced neuron models, including
the Ermentrout-Kopell model Ermentrout1986 , the FitzHugh-Nagumo model
Fitzhugh1961 , the Izhikevich modelIzhikevich2003 ; Izhikevich2004  and so on,
exhibit rich types of dynamical phase transitions above the firing threshold.
There are three major types of dynamical phase transitions found in single-
neuron dynamics: saddle-node on invariant circle (SNIC), supercritical Hopf
bifurcation and subcritical Hopf bifurcationGerstner2014 ; Izhikevich2007 .
ErmentroutErmentrout1996  showed that the type I neurons undergo a SNIC
dynamical transition at the threshold, while IzhikevichIzhikevich2007  pointed
out that type II neurons may go through all three different bifurcations. It
would be great to build a theoretical framework, capturing the essential
degrees of freedom for spiking neurons and incorporating all types of
dynamical phase transitions systematically.

In this Article, we propose the essential degrees of freedom for a spiking
neuron are the membrane potential and the temporal sequence. It is rather
remarkable that both can be integrated into a unified theoretical framework
described by a single complex dynamical variable and the U(1) dynamics emerges
naturally. The real part of the complex dynamical variable represents the
potential of the neuron and the phase describes the temporal sequence during
the firing process. When describing the neuronal dynamics of the complex
dynamical variable, our U(1) neuron model not only reproduces the action
potentials from the Hodgkin-Huxley neuron but also capture the gain function
of the firing rate in response to the external current.

It is known the classification of spiking neurons is closely related to the
bifurcation of neuronal dynamics. The gain function of the type I neuron is
continuous at the current threshold while the type II neuron exhibits a
discontinuous jump at the threshold. The U(1) neuron described by the complex
variable provides a natural explanation for the classification. The
bifurcations on the complex plane, either in radial or phase directions, leads
to various transitions among resting, excitable, firing states of a single
neuron. In short, the U(1) neuron not only captures the single-neuron activity
upon external stimuli, but also provides a coherent understanding for the
dynamical phase transition between different types of neuronal activities.

The major impact of the U(1) neuron is not to provide a realistic description
of a spiking neuron (although it can be done as shown in the later
paragraphs), just like the Fermi liquid theory is not targeting on providing a
precise quantitative description for metals. The key is to grab the essential
features in neuronal dynamics so that model building for different purposes
can be facilitated with these ingredients. For instance, within the U(1)
neuron description, we find the phase dynamics of spiking neurons is
nonuniform during the firing process and the firing rate is thus dictated by
the bottleneck (phase regime with smaller angular velocity). In addition, it
is known that the neuron reacts differently when stimulated in different
firing processes. Going beyond the usual Kumamoto-like interactions, the U(1)
neuron framework provides a systematic approach to describe the phase
dependence of the synaptic interactions on the presynaptic and postsynaptic
neurons. With the phase dependence in mind, the refractory effect can be
incorporated seamlessly into the U(1) neuron framework. In fact, we show that
the Kinouchi-Copelli neuronal network is equivalent to the discrete version of
the U(1) neuronal network and the spontaneous asynchronous firing (SAF) state,
crucial for information processing, can be realized when the synaptic strength
is strong enough.

The remainder of the paper is organized as follows. In Section II, we first
compare the artificial and biological neuronal networks and point out the
importance of neuronal dynamics. In Section III, we discuss the mode-locking
phenomena in biological neurons. In Section IV, we go beyond the usual
Ginzburg-Landau theory and construct the theoretical foundation for the U(1)
neuron. In Section V, we demonstrate how the Hodgkin-Huxley neuron can be
described with the theoretical framework of the U(1) neuron. In Section VI, we
reveal the importance of phase dependence in the synaptic interactions and
establish the equivalence of the Kinouchi-Copelli neuron and the discrete
version of the U(1) neuron. Finally, we extend the single-neuron approach to
neuronal network and show that the spontaneous asynchronous firing phase,
beneficial for information processing, in the Kinouchi-Copelli neuronal
network.

Figure 2: Mode-locking in the Hodgkin-Huxley neuron. (A) The firing rate of
the Hodgkin-Huxley neuron exhibits roust plateaus at $\nu=n\nu_{\rm ac}$,
where $n$ is an integer, in the presence of an external ac drive with
frequency $\nu_{\rm ac}=25$ Hz. (B) Action potentials at $I=0,5,15$ $\mu$A/cm2
exhibits different mode-locking behaviors with $\nu/\nu_{\rm ac}=1,2,3$
respectively.

##  II Artificial and Biological Neurons

We first briefly explain the difference between artificial and biological
neurons as shown in Figure 1. In biological nerve system, recurrent neuronal
network is often found, while the artificial neural networks used in deep
learningGoodfellow2016 ; Haykin2009  usually belong to the feed-forward type
with well defined input, output and hidden layers. Despite the difference
between the network structure, the neuronal dynamics within the rate-model
framework is captured by the set of coupled non-linear differential equations,

$\displaystyle\tau\frac{dA_{n}}{dt}=-A_{n}+G\left(\sum_{m}W_{nm}A_{m}+I_{n}\right),$
(1)

where $A_{n}$ denotes the firing rate of the single neuron or the activity of
the neuronal population at the $n$-th node, $I_{n}$ is the external current
injected into the $n$-th neuron and $W_{nm}$ is the synaptic weight from the
$m$-th neuron to the $n$-th neuron. The relaxation time $\tau$ is the typical
time scale for a single neuron returning to its resting state when the
external stimulus is turned off. The gain function $G$ relates the firing rate
with the stimuli, including the external current and the synaptic currents
from connected neurons. The variety of individual neurons is ignored here for
simplicity but can be included without technical difficulty.

Even with the simple relaxation dynamics, the coupled non-linear differential
equations are already extremely complicated. Because the synaptic dynamics
(how $W_{nm}$ evolves with time) is typically much slower than the neuronal
dynamics, the stationary solution plays a significant role in some cases,

$\displaystyle A_{n}=G\left(\sum_{m}W_{nm}A_{m}+I_{n}\right)$ (2)

Note that the true dynamics drops out completely in the above relations. In
the feed-forward network, these are exactly the relations between input and
output neurons commonly used in the artificial neural network. The synaptic
weight $W_{nm}$ can be adjusted by employing appropriate algorithm to minimize
the cost function but the evolution of $W_{nm}$ at different epochs does not
represent the true dynamics of the neuronal network.

The deep neural networkGoodfellow2016  enjoys great success in recent years
and makes strong impacts in many areas in science and technology. However, as
explained in the previous paragraph, it explains the slower process such as
learning but does not include the reactive information processing at the
shorter time scale. If the neuronal dynamics is properly included, shall the
neuronal network with both types of dynamics exhibits different class of
intelligence? The first step to answer this important question is to capture
the essential features in neuronal dynamics before constructing the network
with complicated structures. It will become evident later that the activity
(firing rate) $A_{n}$ is insufficient to describe the dynamics of a single
neuron and more degrees of freedom must be included to account for proper
synaptic interactions.

##  III Mode-Locking in a Single Neuron

Mode lockingJensen1984 ; Flaherty1978 ; Fletcher1978  is a common phenomena in
physical and biological systemsGuevara1981 ; Aihara1986 ; Takahashi1990 ;
Gray1996 ; Szucs2001  with non-linear dynamics and may play an important role
in neural information processing. In a wide variety of neuron models including
Hodgkin-Huxley model, FitzHugh-Nagumo model, Izhikevich model and some
integrate-and-fire models, the firing rate $\nu$ is locked to the integer
multiples of the oscillatory frequency $\nu_{\rm ac}$ of the external ac
current. As shown in Figure 2, the Hodgkin-Huxley neuron shows robust mode-
locking behavior in the presence of the time-dependent current stimulus,

$I(t)=I+I_{\rm ac}(t)=I+I_{\rm ac}\sin(2\pi\nu_{\rm ac}t),$ (3)

where $I$ and $I_{\rm ac}$ represent the strengths of dc and ac currents
respectively. It is rather remarkable that, even with a moderate $I_{\rm ac}$,
the gain function of the Hodgkin-Huxley neuron changes drastically and robust
firing-rate plateaus appear with $\nu=n\nu_{\rm ac}$, where $n=0,1,2,\cdots$.
The action potentials on the firing-rate plateaus indicate clear mode-locking
behaviors as shown in Figure 2(b).

The mode-locking phenomena have been known in the neuroscience community for
quite a long time but its deeper implication seems neglected. First of all,
the mode-locking phenomena provide a stable method to transfer information
between neurons in the presence of stochastic noises. Furthermore, the mode-
locking phenomena manifest the underlying non-linear phase dynamics, or the
so-called U(1) dynamics.

Figure 3: Definition of the U(1) phase. (A) The action potential of a spiking
neuron is marked by its potential minimum $V_{\rm min}$ (point $a,e$), average
$\overline{V}$ (point $b,d$) and maximum $V_{\rm max}$ (point $c$). (B) The
points from $a$ to $e$ corresponds to the U(1) phase
$\varphi=\pi,3\pi/2,0,\pi/2,\pi$ respectively. A complete firing process can
thus be viewed as a winding process of $\Delta\varphi=2\pi$ in the phase
dynamics.

To unveil the underlying phase dynamics, we introduce a complex dynamical
variable to describe the neuronal dynamics,

$z(t)=r(t)\exp[i\varphi(t)],$ (4)

where $r(t)$ denotes the firing amplitude and $\varphi(t)$ represents the
phase during the firing process. We propose the essential degrees of freedom
for a spiking neuron are the membrane potential $V(t)=r(t)\cos\varphi(t)$ and
the phase $\varphi(t)$ as shown in Figure 3. The phases $\varphi=0,\pi$
correspond to the potential maximum and minimum respectively. When a spiking
neuron completes a full firing cycle, it can be viewed as a winding process of
$\Delta\varphi=2\pi$ in the phase dynamics.

In general, the angular velocity $\Omega(\varphi)$ is not constant and depends
on the phase,

$\frac{d\varphi}{dt}=\Omega(\varphi).$ (5)

As a demonstrating example, we extract the nonuniform phase dynamics of the
Hodgkin-Huxley neuron with our U(1) neuron framework. The phase dynamics
reveals lots of interesting features as shown in Figure 4. First of all, the
angular velocity $\Omega(\varphi)$ is nonuniform, showing a bottleneck (small
angular velocity) starting around $\varphi=\pi$ and a whirlwind (large angular
velocity) slight below $\varphi=0$. A non-trivial correlation between membrane
potential $V(t)$ and the phase $\varphi(t)$ is thus established: the phase
dynamics is fast near potential maximum while it slows down near potential
minimum.

Upon changing the external current stimulus, the overall shape of
$\Omega(\varphi)$ remains more or less the same, indicating the nonuniform
phase dynamics we found here is an intrinsic property of the Hodgkin-Huxley
neuron. Zooming into the finer differences caused by different current
stimuli, a larger current increases the angular velocity slightly near the
bottleneck regime while slows down the phase dynamics around the whirlwind
regime. It means that an increase of the injected current suppresses the non-
uniformity of the angular velocity.

Figure 4: Nonuniform phase dynamics in the Hodgkin-Huxley neuron. The angular
velocity $\Omega(\varphi)$ exhibits a highly non-trivial dependence on the
U(1) phase. In the regime of current stimuli from 6.3 $\mu$A/cm2 (red) to 46.3
$\mu$A/cm2 (blue), one finds the overall shape remains more or less the same.
However, because the firing rate is dictated by the bottleneck regime, a
slight increase of the angular velocity here will boost up the firing rate
significantly.  Figure 5: Dynamical phase transitions in the U(1) neuron:
SNIC, supercritical Hopf, subcritical Hopf. The upper panels present the
topological structures of the limit cycles and the fixed points in the
vicinity of the dynamical phase transitions. The firing amplitude and rate of
the neuron versus the external current stimulus are illustrated in the bottom
panels.

The firing rate $\nu$ can be determined from the phase dynamics as well.
Because the complete firing cycle corresponds to a $2\pi$ phase winding, the
firing rate of a spiking neuron is

$\nu=\frac{1}{\Delta
t_{f}}=\left[\oint\frac{d\varphi}{\Omega(\varphi)}\right]^{-1},$ (6)

where $\Delta t_{f}$ is the time duration for adjacent firing events. From the
above rate-phase relation, it is clear that the firing rate is dominated by
the angular velocity in the bottleneck regime. Thus, a slight increase of
$\Omega(\varphi)$ in the bottleneck gives rise to a significant upsurge in the
firing rate $\nu$. On the other hand, the relatively large changes in the
whirlwind regime for different current stimuli are irrelevant to the firing
rate.

The nonuniform phase dynamics revealed by the U(1) neuron framework presents a
natural explanation for the observed mode-locking phenomena. It is worth
emphasizing that a uniform angular velocity $\Omega(\varphi)=\omega$ can be
gauged away by redefining the complex dynamical variable $z(t)\to
z(t)\exp(-i\omega t)$ but the phase-dependent angular velocity
$\Omega(\varphi)$ is an intrinsic property of the neuron and cannot be gauged
away. The simple yet stimulating findings presented in Figure 4 encourage us
to go beyond the rate models and to integrate both membrane potential $V(t)$
and the phase $\varphi(t)$ together into a coherent theoretical framework.

##  IV The U(1) Neuron

Extending the conventional Ginzburg-Landau theory for the complex dynamical
variable $z(t)$, its dynamical equation contains two parts,

$\frac{dz}{dt}=-\partialderivative{L(z,\overline{z})}{\overline{z}}+iR(z,\overline{z})z,$
(7)

where the Lyapunov function $L(z,\overline{z})$ is a real-valued potential
with U(1) symmetry and $R(z,\overline{z})$ is another real-valued function
describing the nonuniform rotation of the phase. It can be shown that the
Lyapunov function $L(z,\overline{z})$ decreases throughout the temporal
evolution, seeking the potential minimum representing the free energy in
thermal equilibrium. Thus, the radial dynamics of the firing amplitude $r(t)$
is relatively simple and, in most cases, can be understood with the usual
Ginzburg-Landau theory with slight modifications. However, the presence of the
nonuniform phase dynamics $R(z,\overline{z})$ go beyond the relaxation
dynamics (seeking for specific potential minima) and gives rise to interesting
non-equilibrium phenomena as anticipated in the excitable neuronal systems.

To make the radial and phase dynamics explicit, one can choose the biased
double-well potential in the Ginzburg-Landau theory (for both the first-order
and the second-order phase transtions) supplemented with the Fourier expansion
for the phase dynamics,

$\displaystyle L(z,\overline{z})$ $\displaystyle=$
$\displaystyle\frac{v_{2}}{2}|z|^{2}+\frac{v_{3}}{3}|z|^{3}+\frac{v_{4}}{4}|z|^{4},$
(8) $\displaystyle R(z,\overline{z})$ $\displaystyle=$
$\displaystyle\omega+\frac{1}{2}\sum_{n=1}^{\infty}(c_{n}z^{n}+\overline{c}_{n}\overline{z}^{n}).$
(9)

Here $v_{2}$, $v_{4}$ are real while $c_{n}$ are in general complex. Note
that, unlike the Lyapunov function, the U(1) symmetry does not hold for the
phase-rotation function $R(z,\overline{z})$. Separating the complex Eq. (7)
into amplitude and phase parts, the dynamical equations read

$\displaystyle\frac{dr}{dt}$ $\displaystyle=$ $\displaystyle
F(r)=v_{2}r+v_{3}r^{2}+v_{4}r^{3},$ (10) $\displaystyle\frac{d\varphi}{dt}$
$\displaystyle=$
$\displaystyle\Omega(\varphi)=\omega+\sum_{n=1}^{\infty}|c_{n}|r^{n}\cos(n\varphi+\phi_{n}).$
(11)

where $|c_{n}|$ and $\phi_{n}$ are the amplitudes and phases of the complex
numbers $c_{n}$. Note that the radial dynamics contains no phase dependence
and $F(r)$ serves as the conservative force driving the firing amplitude to
the potential minima at $r=r^{*}$ satisfying the equilibrium condition
$F(r^{*})=0$. The phase dynamics can be rather unconventional because the
angular velocity $\Omega(\varphi)$ is nonuniformMatthews1990 , a direct
consequence from the spike-like action potential.

The above dynamical equations for firing amplitude and phase provides a
coherent understanding of different types of dynamical phase transitions in
spiking neurons. Three major types of dynamical phase transitions are
illustrated in Figure 5: the saddle-node-onto-invariant-cycle (SNIC)
transition is associated with bifurcations in the phase dynamics, while the
supercritical and subcritical Hopf transitions are driven by bifurcations in
the radial dynamics for the firing amplitude. While these dynamical phase
transitions are found in various neuron models in the literature, it is rather
satisfying that all three types of transitions emerge naturally within the
U(1) neuron framework.

Let us focus on the SNIC transition first. As shown in Figure 5, in the
presence of a finite amplitude $r^{*}$, the equilibrium condition
$\Omega(\varphi^{*})=0$ gives rise to a pair of saddle-node fixed points in
phase dynamics. Below the current threshold, the saddle-node structure makes
the neuron excitable. Upon increasing current injection, the saddle-node pair
gets close, merges into a critical point and eventually disappears on the
limit cycle. Because the limit cycle already exists below the current
threshold, the firing amplitude exhibits a discontinuous jump. It is known in
statistical physics that the dynamics slows down indefinitely in the vicinity
of a critical point. In consequence, in near the SNIC transition, the time
duration between adjacent firing events diverges, indicating a vanishing
firing rate. Thus, the firing rate changes continuously across the current
threshold, equating to the type I neuron.

On the other hand, in the regime where the angular velocity remains positive
$\Omega>0$, the dynamical transitions are driven by the competitions among the
equilibrium points in the radial dynamics determined by $F(r^{*})=0$. In the
supercritical Hopf transition, a stable limit cycle (non-zero $r^{*}$
solution) appears at the current threshold, rendering the resting state
($r^{*}=0$ solution) unstable. Because the limit cycle grows out from the
resting state, the amplitude changes continuously across the current threshold
as shown in Figure 5. Because there is no critical point involved here, the
firing rate associated with the emergent limit cycle is finite in general
cases. Thus, the gain function exhibits a discontinuous jump at the current
threshold and corresponds to the type II neuron.

The subcritical Hopf transition arises when a pair of limit cycles appears at
the current threshold. Due to the simultaneous presence of the stable limit
cycle (firing state) and the stable fixed point (resting state), the neuron is
bistable. As shown in Figure 5, both the firing rate and amplitude are
discontinuous at the current threshold. Although neurons undergo the
subcritical Hopf transition can be classified as the type II, their dynamics
are more complicated in comparison with the supercritical Hopf transition
where neurons are either in the resting state or the firing state.

Figure 6: Dynamical phase transitions driven by the $F(r)$ evolution. As the
fitting function $s(I)$ changes with the external current, the cubic function
slides through different configurations from (A) to (G). The black and red
points indicate the radius of the stable and unstable limit cycle
respectively. Configurations from (A) to (D) show the birth of a pair of limit
cycles and the unstable limit cycle swallows the stable fixed point at the
origin subsequently. Compared with the previous classification scheme, it
belongs to the subcritical Hopf transition. Configurations from (E) to (G)
illustrate the process of a stable limit cycle shrinks to an unstable fixed
point, leading to a stable fixed point subsequently. This dynamical transition
belongs to the supercritical Hopf.

##  V Fitting Hodgkin-Huxley Neuron

In this section, we demonstrate how the Hodgkin-Huxley neuron can be described
by the U(1) neuron in a wide range of parameter regime. The bifurcation
diagramXie2008  of the Hodgkin-Huxley neuron undergoes a subcritical Hopf
transition first at the current threshold at $I=9.780$ $\mu$A/cm2, followed by
another supercritical Hopf transition at $I=154.527$ $\mu$A/cm2. These
dynamical phase transitions can be captured within the U(1) neuron framework.

Let us focus on the radial dynamics of the firing amplitude first. Because the
resting state ($r^{*}=0$) is always present, the potential flow in the
Ginzburg-Landau theory can be constructed in the following way. Introducing a
cubic function $f(r)$ with three zeroes at $r=0,p_{1},p_{2}$,

$f(r)=A(r-0)(r-p_{1})(r-p_{2}),$ (12)

where $A$ is some positive constant. The conservative force $F(r)$ in the
radial dynamics can be expressed as

The parameters $p_{1}$ and $p_{2}$ can be chosen as the amplitude of the
stable and unstable limit cycle, and $a$ should be negative because of the
limit of the ionic sources. Then the force in the amplitude equation is the
coordinate transformation concerning the external input current $I$ that
switch neurons between firing states and resting states:

$\begin{split}F(r)=&\>f[s(I)]-f[r+s(I)]\\\
=&A\Big{[}p_{1}p_{2}-2p_{1}s(I)-2p_{2}s(I)+3s^{2}(I)\\\
&+3s(I)r-(p_{1}+p_{2})r+r^{2}\Big{]}r.\end{split}$ (13)

Here $s(I)$ depends on the external current and serves as a fitting function
to reproduce the correct dynamical phase transitions in the targeted parameter
regime.

The solutions for $F(r^{*})=0$ describes the fixed point or the amplitudes of
the limit cycles,

$\displaystyle r^{*}_{0}=$ $\displaystyle\>0,$ (14) $\displaystyle r^{*}_{\rm
stable}=$ $\displaystyle\frac{1}{2}\Bigg{[}p_{1}+p_{2}-3s(I)$
$\displaystyle+\sqrt{p_{1}^{2}-2p_{1}p_{2}+p_{2}^{2}+2p_{1}s(I)+2p_{2}s(I)-3s(I)}\Bigg{]},$
$\displaystyle r^{*}_{\rm unstable}=$
$\displaystyle\frac{1}{2}\Bigg{[}p_{1}+p_{2}-3s(I)$
$\displaystyle-\sqrt{p_{1}^{2}-2p_{1}p_{2}+p_{2}^{2}+2p_{1}s(I)+2p_{2}s(I)-3s(I)}\Bigg{]}$

Note that only non-negative solutions are physical because the firing
amplitude cannot be negative. The $r^{*}=0$ solution is always present as
anticipated. The other two solutions appear in pair and represent a pair of
stable and unstable limit cycles.

Figure 7: Fitting the firing amplitude of the Hodgkin-Huxley neuron. The
firing amplitude of the effective U(1) neuron (orange) fits that of the
Hodgkin-Huxley neuron (black) rather well. The firing amplitude of the
unstable limit cycle (light blue) is also shown for reference. Figure 8: The
action potentials of (A) the effective U(1) neuron and (B) the Hodgkin-Huxley
neuron are almost identical. The external current is $I=26.28$ $\mu$A/cm2 for
both neurons. The voltage of the resting state $V_{\rm rest}=-65$ mV is
shifted to zero for visual clarity. Figure 9: Gain function of the U(1)
neuron. The gain function of the Hodgkin-Huxley neuron (blue) is well captured
by that of the U(1) neuron with Fourier expansion to the seventh order in
phase dynamics. However, the U(1) neuron to the second order delivers an
approximate gain function with the same trend, indicating lower-order terms
already secure the qualitative behavior of the neuronal dynamics. Figure 10:
Kinouchi-Copelli neuron as the U(1) neuron in discrete time steps. (A) Without
external stimuli, the neuron is excitable with a stable fixed point (state 0)
denoting the resting state. (B) In the presence of external stimuli, by either
external or synaptic currents, the neuron fires and the phase swings to
$\varphi=0$ with maximum potential. (C) After firing, the neuron enters the
refractory states denoted by states $2,3,\cdots,q-1$ and eventually returns to
the rest state.

The remaining task to map the Hodgkin-Huxley neuron into the U(1) neuron is to
equate firing amplitudes in both descriptions, $r_{\rm stable}[s(I)]=r_{\rm
HH}(I)$ by numerical fitting. Keeping the lower-order terms in the fitting
function $s(I)$,

$s(I)=-15.4037+0.146443I+6.70533\ln{I},$ (15)

it is sufficient to match the firing amplitude rather well as shown in Figure
7. The coefficients $c_{n}$ in the phase dynamics can be found in numerical
fitting as well. Keeping the Fourier expansion to the seventh order in phase
dynamics, the action potential of the Hodgkin-Huxley neuron is almost
identical to the effective U(1) neuron presented in Figure 8.

To capture the spike-like profile of the action potential, it is necessary to
include higher order terms in the phase dynamics. However, in the cases where
the precise shape of the action potential is irrelevant, the precision of the
Fourier expansion can be relaxed. As shown in Figure 9, the gain function of
the U(1) neuron up to the seventh order in phase dynamics matches that of the
Hodgkin-Huxley neuron rather well. However, the gain function up to the second
order in phase dynamics still keeps the qualitative trend with reasonable
compromise in quantitative precision.

Figure 11: Spontaneous asynchronous firing in Kinouchi-Copelli neuronal
netwrok. (A) Given a weak neuronal activity as the initial configuration, a
larger branching ratio ($\sigma=1.97$) drives the Kinouchi-Copelli neuronal
network to the spontaneous asynchronous firing (SAF) phase with stable
neuronal activities, while a smaller branching ratio ($\sigma=1.84$) leads to
the quiescent state with diminishing neuronal activities. (B) Snapshots of the
Kinouchi-Copelli neuronal network in SAF and quiescent phases.

##  VI Phase-dependent synaptic interactions

With the established theoretical framework for a single neuron, we move on to
investigate the synaptic interactions between connected neurons. As the
nonuniform angular velocity in phase dynamics plays a significant role for the
single neuron, we anticipate that the synaptic interactions also carry
nontrivial phase dependence.

Suppose the phases of presynaptic and postsynaptic neurons are labels as
$\varphi_{1}(t)$ and $\varphi_{2}(t)$ respectively. The Kuramoto
interactionAcebron2005  carries the phase dependence
$\cos(\varphi_{1}-\varphi_{2})$ and tends to synchronize both neurons. In
terms of the complex dynamical variables, the synaptic interaction
$V_{K}(z_{1},\overline{z}_{1},z_{2},\overline{z}_{2})$ can be written as

$V_{K}=K_{12}(\overline{z}_{1}z_{2}+\overline{z}_{2}z_{1}),$ (16)

where $K_{12}$ denotes the synaptic strength of the Kuramoto interaction
between the neurons. Note that the above interaction is U(1) symmetric, i.e.
it is invariant under a constant phase shift for all neurons.

The above synaptic interaction is simple and widely used in the generalized
Kuramoto modelsAcebron2005  for studying synchronization phenomena. But, the
U(1) symmetry imposes a rather unrealistic constraint on the neuronal
dynamics. A constant phase shift means a temporal shift in the firing process,
which is certainly not invariant for any realistic neurons. Furthermore, even
though synchronization phenomena is widely observed in many biological
systems, it is faulty for information processing. In fact, Parkinson’s disease
is correlated with excessively strong oscillatory synchronization in the brain
areas such as thalamus and basal gangliaPare1990 ; Nini1995 ; Tass2010 , while
a healthy brain is asynchronous in these areas.

Let us try to model the phase dependence of the synaptic interactions from
realistic neuronal properties. When the presynaptic neuron fires, it provides
a synaptic current and enhances the chance for the postsynaptic neuron to
fire. This process can be approximated by the phase factor $1+\cos\varphi_{1}$
qualitatively because it reaches the maximum at $\varphi_{1}=0$ (spike) and
vanishes at $\varphi_{1}=\pi$ (hyperpolarized). It is also known that the
postsynaptic neuron is sensitive to external stimuli in the hyperpolarized
period while almost insensitive around the spike. Thus, there is another phase
factor $1-\cos\varphi_{2}$ arisen from the postsynaptic neuron. Combing the
phase factors for presynaptic and postsynaptic neurons together, we anticipate
the synaptic interaction carries the overall phase dependence
$(1+\cos\varphi_{1})(1-\cos\varphi_{2})$. In terms of the complex dynamical
variables, the synaptic interaction
$V(z_{1},\overline{z}_{1},z_{2},\overline{z}_{2})$ can be written as

$V=W_{12}(2|z_{1}|+z_{1}+\overline{z}_{1})(2|z_{2}|-z_{2}-\overline{z}_{2}),$
(17)

where $W_{12}$ denotes the strength of the synaptic interaction between the
neurons. After taking the realistic neuronal properties into account, the
synaptic interaction is no longer U(1) symmetric and the non-physical
constraints are lifted. It is rather interesting that the above synaptic
interaction is similar to the Bardeen-Cooper-Schrieffer interaction, as the
U(1) symmetry is also broken in superconductors. The dynamical behaviors of
neuronal networks with this type of phase-dependent synaptic interactions
avoid the ultimate fate of synchronization, effective for information
processing, and remain open for further investigations.

The refractory effects in spiking neurons also lead to phase-dependent
synaptic interactions. Kinouchi and Copelli proposed a $q$-state neuron model
to account for the observed refractory effectsKinouchi2006 . As shown in
Figure 10, the resting state is labeled as state 0 and the firing state is
labeled as state 1. The other states $2,3,\cdots,q-1$ are refractory and thus
insensitive to external stimuli. Because the U(1) neuron provides the general
theoretical framework for nonuniform phase dynamics, it is not surprising that
the Kinouchi-Copelli neuron can be mapped to the U(1) neuron as well.

In the absence of external stimuli, the Kinouchi-Copelli neuron is excitable,
described by a pair of saddle and node on the limit cycle as shown in Figure
10. The resting state corresponds to the stable node, labeled as state 0. When
the neuron is activated by external stimuli, the angular velocity $F(\varphi)$
lifts up and the neuron fires. The phase swings to $\varphi=0$ (spike) with
maximum potential, labeled as state 1. The change of the angular velocity upon
external stimuli resembles the Hodgkin-Huxley neuron studied in the previous
section. After firing, the neuron enters the refractory period and gradually
relaxes back to the resting state. Cutting the refractory period into $q-1$
time steps of equal intervals, the refractory states $2,3,\cdots,q-1$ are
defined accordingly. In short, the Kinouchi-Copelli neuron can be viewed as a
discrete version of the U(1) neuron going through the SNIC transition upon
external stimuli.

Now we turn to the phase dependence of the synaptic interaction caused by the
refractory effect. For simplicity, we place these neurons on a two dimensional
grid to form the Kinouchi-Copelli neuronal network (KCNN). The phase
dependence of the presynaptic neuron take the usual form of pulse coupling:
when the presynaptic neuron fires (state 1), it gives rise to a finite
probability $p$ to activate the postsynaptic neuron. In a two dimensional
grid, a postsynaptic neurons are connected to 4 presynaptic neurons. Thus, it
is convenient to introduce the brach ratio $\sigma$ as the sum of all
activation probabilities, $\sigma=4p$, to parameterize the dynamical behaviors
of the KCNN. The postsynaptic neuron also brings about another phase
dependence due to refractory effects: only when the postsynaptic neuron is in
the resting state (state 0), it can be activated to fire upon external
stimuli.

Although the phase dependence of the synaptic interactions in the KCNN is not
the same as that in Eq. (17), the broken U(1) symmetry is manifest. Therefore,
spontaneous asynchronous firing (SAF) phase is anticipated when the synaptic
weight is strong enough. This is indeed true. We perform numerical simulations
for the KCNN with different branching ratios as shown in Figure 11. The
initial configurations are randomly set with sparsely distributed activated
neurons. When the branching ratio is larger than some critical value, the
firing activity is enhanced and the neuronal activity is described by the SAF
phase. On the other hand, when the branching ratio is smaller than the
critical, the firing activities are suppressed, entering the quiescent phase
with diminishing neuronal activities. The dynamical phase transition and
associated statistical analysis for the KCNN will be presented in detail
elsewhere.

Unlike the synchronous phase, the SAF phase can encode and decode information
effectively and is of vital importance for studying neuronal networks. To
achieve this goal, the phase dependence of the synaptic interactions must be
properly taken into account and the U(1) neuron framework provides a natural
and convenient approach to tackle this daunting challenge.

In conclusion, we propose that the rate models of spiking neurons miss the
essential ingredient of neuronal dynamics and construct the U(1) neuron
framework to integrate the dynamics of firing amplitude and phase coherently.
We show that the dynamics of the Hodgkin-Huxley neuron, including action
potentials, the gain function and associated dynamical phase transitions, is
well captured within the U(1) neuron framework. Meanwhile, the nonuniform
phase dynamics is unveiled with the bottleneck-whirlwind feature. We emphasize
the importance of the phase-dependent synaptic interactions and map the
Kinouchi-Copelli neuron as a discrete version of the U(1) neuron with
refractory effects. Placing these neurons on a two dimensional grid to form a
neuronal network, our numerical simulations show that the SAF phase, crucial
for information processing, can be realized. Our findings suggest that the
U(1) neuron is the minimal model for single-neuron activities and serves as
the building block of neuronal networks for information processing.

## Acknowledgements

The authors would like to express their gratitude to Ching-Lung Hsu, Chin-Yuan
Lee, Georg Northoff and Wei-Ping Pan for helpful discussions and comments. We
acknowledge supports from the Ministry of Science and Technology through
grants MOST 109-2112-M-007-026-MY3, MOST 109-2124-M-007-005 and MOST
110-2112-M-005-011. Financial supports and friendly environment provided by
the National Center for Theoretical Sciences in Taiwan are also greatly
appreciated.

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# A Simple Model for Quantum Gravity:
the one-dimensional case

Ricardo<EMAIL_ADDRESS>
Universidade Federal do ABC

###### Abstract

Group theory is used to construct a model for quantum gravity and exact
results are obtained in one dimension. Comparison with Regge calculus shows
that the results agree in the strong cosmological constant limit for an open
curve. In the closed case, the average edge length has a minimum value and
this give us a clue for higher dimensions.

Keywords: Quantum Gravity, Group Theory, Regge Calculus

##  1 Introduction

Consider euclidean gravity in 1D, that is, a curve representing time, $x$. As
the intrinsic curvature is zero, we will work with the action

$S=\lambda\int dx\,\sqrt{|\det g|}=\lambda L,$

where $\lambda$ is the cosmological constant222Classically, $\lambda=0$ for
pure gravity in 1D and 2D, but in the quantum theory $\lambda$ is necessary to
obtain convergence., $L=\int dx\,\sqrt{|\det g|}$ is the total length and we
set $\hbar=c=1$.

Since the only observable here is $L$, we are interested in your average
value, $\langle L\rangle$. So we have to calculate the partition function

$Z=\int Dg\,\exp(-S)=\int Dg\,\exp(-\lambda L)$ (1)

and then take the derivative

$\langle L\rangle=-\frac{1}{Z}\frac{dZ}{d\lambda}=-\frac{d(\ln Z)}{d\lambda},$
(2)

where $g$ is the metric, $Dg$ is a measure, and $Z$ is a path integral, that
is, a sum over all possible metrics. Even in 1D, to calculate $Z$ exactly is
tricky (see Sec. 2.2) and we often rely on numerical methods.

One of the most important examples of numerical calculation is the quantum
Regge calculus [1], where they approximate a curve by $N$ edges and use their
lengths, $l_{n}$ for $n=1,2,\ldots N$, as the integration variables.

Usually, they choose the measure $\prod_{n=1}^{N}dl_{n}l_{n}^{\sigma+1}$ with
some appropriate power $\sigma>-2$ to make the integrals convergent. They also
work with an infinity interval $\epsilon\leq l_{n}<\infty$, where $\epsilon>0$
is an ultraviolet cutoff.

In order to obtain the continuum limit, they make $N$ as big as they can and
$\epsilon$ small as they can too. And in higher dimensions, they have to
guarantee that the triangle inequalities between the edges are satisfied. This
is put in the measure by hand.

There are other choices for the measure, including non-local measures, but we
beg to disagree with them333As $l_{n}=2\ell\tan(\phi_{n}/2)$, if we choose to
work with $l_{n}$, then the Haar measure becomes
$\prod_{n}d\phi_{n}/\pi=\prod_{n}4\ell\,dl_{n}/\pi(4\ell^{2}+l_{n}^{2})$ which
is a Cauchy distribution, not a power.. We believe that the measure must be
unique (up to normalization) as the Haar measure of group theory.

##  2 One-Dimensional Model

Instead of working with lengths and their infinity intervals, inequalities,
etc. We would like to use angles and their finite intervals, which in 1D is
related to the rotation group, SO(2), or equivalently the U(1) group.

To do this, we will associate an angle, $\phi_{n}$, for each pair of equal
lengths, $l_{n}/2$, in a circle of radius $\ell$, as in the open polygonal
line represented in Fig. (1).

Figure 1: Open polygon tangent to the U(1) circle.

Notice that we can align this polygon into a straight line or approximate it
to any curve we want, but this is just the extrinsic curvature.

###  2.1 Open Case

For the moment, the sum of the angles are not constrained, the polygonal line
can turn around the circle freely. But each angle is restricted to
$0\leq\phi_{n}<\pi$ because the length $l_{n}=2\ell\tan(\phi_{n}/2)$ lies in
the interval $[0,\infty)$.

Thus we have the (unique) normalized Haar measure

$\prod_{n=1}^{N}\left(\int_{0}^{\pi}\frac{d\phi_{n}}{\pi}\right)$

and the total length or perimeter is given by

$L=\sum_{n=1}^{N}l_{n}=2\ell\sum_{n=1}^{N}\tan\frac{\phi_{n}}{2}.$ (3)

Obviously, we can choose the interval to be $[-\pi,\pi)$, or $[0,2\pi)$, etc.
But we don’t want to complicate things at this stage by using the absolute
value $|\tan(\phi_{n}/2)|$ in $L$ (see Sec. 2.2).

Since the angles are independent, this quantum curve is very easy to
calculate. The partition function, Eq. (1), separates into $N$ copies of the
same integral [2]

$\displaystyle
Z=\prod_{n=1}^{N}\left(\int_{0}^{\pi}\frac{d\phi_{n}}{\pi}\right)\exp(-\lambda
L)=\left\\{\int_{0}^{\pi}\frac{d\phi}{\pi}\exp\left[-2\lambda\ell\tan\left(\frac{\phi}{2}\right)\right]\right\\}^{N}$
$\displaystyle=\left\\{\frac{2}{\pi}[\operatorname{ci}(2\lambda\ell)\sin(2\lambda\ell)-\operatorname{si}(2\lambda\ell)\cos(2\lambda\ell)]\right\\}^{N},$
(4)

where

$\operatorname{si}(x)=\int_{0}^{x}\frac{\sin
t}{t}dt-\frac{\pi}{2},\quad\text{and}\quad\operatorname{ci}(x)=\gamma+\ln
x+\int_{0}^{x}\frac{\cos t-1}{t}dt,$

are the sine and cosine integrals, respectively. Here $\gamma=0.577\ldots$ is
the Euler constant.

Thus the average edge length $\langle l\rangle=\langle L\rangle/N$ is

$\langle
l\rangle=2\ell\left[\frac{1+\text{ti}(2\lambda\ell)\tan(2\lambda\ell)}{\text{ti}(2\lambda\ell)-\tan(2\lambda\ell)}\right],$
(5)

where we have defined
$\text{ti}(x)=\operatorname{si}(x)/\operatorname{ci}(x)$.

Eq. (5) is exact, see Fig. (2), but it can be approximated, using series [2]
for $\operatorname{si}(x)$ and $\operatorname{ci}(x)$, as

$\langle
l\rangle\approx\begin{cases}-4\ell[\gamma+\ln(2\lambda\ell)]/\pi,&\lambda\ell\ll
1,\\\ 1/\lambda,&\lambda\ell\gg 1.\end{cases}$ (6)

In the strong approximation, $\lambda\ell\gg 1$, Eq. (6) is equal to $\langle
l\rangle=(2+\sigma)/\lambda$, with the measure
$\prod_{n}dl_{n}l_{n}^{\sigma+1}$ used in Regge calculus [1], for $\sigma=-1$.

In view of the Wigner-Inönü contraction, it is easy to understand what is
happening here: as the radius, $\ell$, of the U(1) circle becomes greater, the
length of the arc, $\ell\phi_{n}$, becomes more and more close to the edge
length, $l_{n}$, that is associated with the translation group, $T^{1}$. Thus
$d\phi_{n}\approx dl_{n}/\ell$ and we get $\sigma\approx-1$.

Note that $\ell$ has nothing to do with the continuum limit, it is a constant
related to the U(1) manifold and we _do not_ take this to zero. The continuum
limit is obtained solely as the number of edges, $N$, goes to infinity. Since
we have defined $\langle l\rangle=\langle L\rangle/N$, then $N$ disappears, as
in Statistical Mechanics.

In this aspect, quantum gravity is completely different from quantum field
theory on a lattice, where we make $N\to\infty$ simultaneously with the
lattice size $a\to 0$. Remember that in QFT all lengths $l_{n}$ are equal to
the lattice size, $a$.

However in quantum gravity we don’t have control of the lengths, they
fluctuate from a minimum value to infinity, see Fig. (2).

Figure 2: Average edge length for the exact value, Eq. (5), is plotted as a
continuous curve (in blue) together with the approximations, Eq. (6), for weak
(in red) and strong cosmological constant (in green).

###  2.2 Closed Case

A more complicated problem, but a little bit more realistic than the open
case, is the closed polygon, with the topology of the circle $S^{1}$, because
of the constraint

$\sum_{n=1}^{N}\phi_{n}=2\pi,$ (7)

where $N=3,4,5\ldots$ to close the triangle, quadrilateral, etc. For
simplicity, we will restrict the problem to just one wind around the circle.

In this case, the partition function, Eq. (4), is replaced by

$Z={\cal
N}\prod_{n=1}^{N}\left(\int_{0}^{\pi}\frac{d\phi_{n}}{\pi}\right)\delta\left(\sum_{n=1}^{N}\phi_{n}-2\pi\right)\exp\left(-2\lambda\ell\sum_{n=1}^{N}\tan\frac{\phi_{n}}{2}\right),$
(8)

where ${\cal N}$ is a normalization constant. For $N=3,4,5,\ldots$ edges, one
can choose ${\cal N}=2\pi,\frac{3\pi}{2},\frac{24\pi}{11},\ldots$ to make
$Z=1$ when $\lambda=0$. Obviously, the observable $\langle l\rangle$ do not
depends on ${\cal N}$.

The first problem is that we can not use perturbation theory here. Usually, we
expand the exponential in Taylor series, using the dimensionless coupling,
$\lambda\ell$. But the integral of the tangent function diverges
$\int_{0}^{\pi}d\phi\,\tan(\phi/2)=\infty$. (Remember that the lengths are not
restricted, only the sum of the angles is).

One can try to regularize this integral choosing a cutoff. For example,
reducing the interval from $[0,\pi]$ to $[0,\pi-\epsilon]$ with some small
$\epsilon>0$. But at each order in $\lambda\ell$ the divergence becomes more
severe. This situation is similar to the QFT versions of gravity [3], where
the degree of divergence increase order by order.

Since we are using the Haar measure, a natural alternative to perturbation
theory is to use the character expansion for these functions. Nevertheless,
the well-known character expansion for the delta function

$\delta(\phi)=\frac{1}{2\pi}\sum_{p=-\infty}^{\infty}\chi_{p}(\phi),$ (9)

where $\chi_{p}(\phi)=\exp(-ip\phi)$ and $p\in\mathbb{Z}$, _doesn’t work_ when
there are several variables and a constant.444For example, try to calculate
$\int_{-\pi}^{\pi}d\phi_{1}\int_{-\pi}^{\pi}d\phi_{2}(\phi_{1}+\phi_{2})\delta(\phi_{1}+\phi_{2}-\pi)/\pi$
which is, obviously, equal to $\pi$, using the _wrong_ expansion
$\delta(\phi_{1}+\phi_{2}-\pi)\neq\sum_{p}\chi_{p}(\phi_{1}+\phi_{2}-\pi)/2\pi$,
you will get zero.

One way out is to expand the delta function using a product of characters

$\delta\left(\sum_{n}\phi_{n}-2\pi\right)=\sum_{p,q,\ldots,r}c_{pq\ldots
r}\chi_{p}(\phi_{1})\chi_{q}(\phi_{2})\ldots\chi_{r}(\phi_{N}),$ (10)

where the coefficients $c_{pq\ldots r}$ can be found, using the orthogonality
relation

$\int_{-\pi}^{\pi}\frac{d\phi}{2\pi}\chi_{p}(\phi)\chi_{q}^{*}(\phi)=\delta_{pq},$
(11)

to be

$c_{pq\ldots
r}=\int_{-\pi}^{\pi}\frac{d\phi_{1}}{2\pi}\int_{-\pi}^{\pi}\frac{d\phi_{2}}{2\pi}\ldots\int_{-\pi}^{\pi}\frac{d\phi_{N}}{2\pi}\delta\left(\sum_{n}\phi_{n}-2\pi\right)\chi_{p}^{*}(\phi_{1})\chi_{q}^{*}(\phi_{2})\ldots\chi_{r}^{*}(\phi_{N}).$

These integrals are not hard to calculate, because the first integral turns
the delta into a Heaviside function $H$ and then the interval is
reduced555Alternatively, one can calculate $c_{pq\ldots r}$ using the integral
representation for the Dirac delta function
$\delta(\phi)=\int_{-\infty}^{\infty}\frac{dk}{2\pi}\exp(ik\phi)$, which works
fine with several variables and a constant inside the argument, $\phi$..

For example, with $N=3$ we have

$\displaystyle c_{pqr}$
$\displaystyle=\int_{-\pi}^{\pi}\frac{d\phi_{1}}{2\pi}\int_{-\pi}^{\pi}\frac{d\phi_{2}}{2\pi}\int_{-\pi}^{\pi}\frac{d\phi_{3}}{2\pi}\delta(\phi_{1}+\phi_{2}+\phi_{3}-2\pi)\chi_{p}^{*}(\phi_{1})\chi_{q}^{*}(\phi_{2})\chi_{r}^{*}(\phi_{3})$
$\displaystyle=\frac{1}{8\pi^{3}}\int_{-\pi}^{\pi}d\phi_{1}\chi_{p}^{*}(\phi_{1})\int_{-\pi}^{\pi}d\phi_{2}\chi_{q}^{*}(\phi_{2})\exp[-ir(\phi_{1}+\phi_{2})]H(\phi_{1}+\phi_{2}-\pi)$
$\displaystyle=\frac{1}{8\pi^{3}}\int_{0}^{\pi}d\phi_{1}\exp[i(p-r)\phi_{1}]\int_{\pi-\phi_{1}}^{\pi}d\phi_{2}\exp[i(q-r)\phi_{2}]$
$\displaystyle=\frac{i(-1)^{q-r}}{8\pi^{3}(q-r)}\int_{0}^{\pi}d\phi_{1}\\{\exp[i(p-q)\phi_{1}]-\exp[i(p-r)\phi_{1}]\\}$
$\displaystyle=\frac{(p-q)(-1)^{p-q}+(q-r)(-1)^{q-r}+(r-p)(-1)^{r-p}}{8\pi^{3}(p-q)(q-r)(r-p)}.$
(12)

Note the cyclic symmetry between the indices $p$, $q$, and $r$. Care must be
taken with Eq. (12) when a pair or triple of indices are equal, this case must
be seen as a limit.

Similarly, for each exponential we have the character expansion

$\exp[-2\lambda\ell|\tan(\phi/2)|]=\sum_{p=-\infty}^{\infty}d_{p}\chi_{p}(\phi),$
(13)

where the coefficients $d_{p}$ are

$d_{p}=\int_{-\pi}^{\pi}\frac{d\phi}{2\pi}\exp[-2\lambda\ell|\tan(\phi/2)|]\chi_{p}^{*}(\phi),$

which is valid for $\lambda\geq 0$. Notice that we are working now with an
absolute value, because the orthogonality relation, Eq. (11), only works with
the complete interval $[-\pi,\pi)$, or $[0,2\pi)$, etc. However, by the
symmetry, this equation can be reduced from a complex Fourier integral to a
cosine Fourier integral without the modulus

$d_{p}=\int_{0}^{\pi}\frac{d\phi}{\pi}\exp[-2\lambda\ell\tan(\phi/2)]\cos(p\phi),$
(14)

which shows that $d_{-p}=d_{p}$ and $d_{p}$ is real.

To evaluate Eq. (14) we will use a representation for the exponential function
as an inverse Mellin transform [4] (this will be useful for 2D and higher)

$\exp(-x)=\int_{\sigma-i\infty}^{\sigma+i\infty}\frac{dz}{2\pi
i}\frac{\Gamma(z)}{x^{z}},$

where $\sigma=\Re(z)>0$.

Now we change the order of integration

$d_{p}=\int_{\sigma-i\infty}^{\sigma+i\infty}\frac{dz}{2\pi
i}\frac{\Gamma(z)}{(2\lambda\ell)^{z}}\int_{0}^{\pi}\frac{d\phi}{\pi}\frac{\cos(p\phi)}{\tan^{z}(\phi/2)}$
(15)

and the $\phi$ integral is

$\int_{0}^{\pi}\frac{d\phi}{\pi}\frac{\cos(p\phi)}{\tan^{z}(\phi/2)}=\sec\left(\frac{\pi
z}{2}\right)f_{p}(z),$ (16)

valid for $0<\sigma<1$, where $f_{p}(z)=f_{-p}(z)$ is a polynomial

$f_{0}(z)=1,\;f_{1}(z)=z,\;f_{2}(z)=z^{2},\;f_{3}(z)=\frac{2z^{3}+z}{3},\;f_{4}(z)=\frac{z^{4}+2z^{2}}{3},\;\ldots$

obtained by the Chebyshev polynomial of the first kind $T_{2p}(x)$ with
$x^{2n}$ replaced by $(z+1)(z+3)...(z+2n-1)/(2^{n}n!)$ or more explicitly

$f_{p}(z)=\begin{cases}1,&p=0\\\
\sum_{k=0}^{p}\frac{(-1)^{k}p(z+1)2^{-2k+2p-1}(-k+2p-1)!\left(\frac{z+3}{2}\right)_{-k+p-1}}{k!(p-k)!(2p-2k)!},&p\geq
1\end{cases}$

where $(x)_{n}=\Gamma(x+n)/\Gamma(x)$ is the Pochhammer symbol and the last
expression can be written as a hypergeometric function
$\frac{2^{2p-1}}{p!}\left(\frac{z+1}{2}\right)_{p}{}_{3}F_{2}\left(\frac{1}{2}-p,-p,-p;1-2p,\frac{1}{2}-p-\frac{z}{2};1\right)$,
but we will not use them.

Substituting Eq. (16) in Eq. (15), we obtain inverse Mellin transforms

$\int_{\sigma-i\infty}^{\sigma+i\infty}\frac{dz}{2\pi
i}\frac{\Gamma(z)\sec(\pi
z/2)z^{n}}{x^{z}}=\begin{cases}\frac{2}{\pi}(\operatorname{ci}x\sin
x-\operatorname{si}x\cos x),&n=0\\\ -\frac{2x}{\pi}(\operatorname{ci}x\cos
x+\operatorname{si}x\sin x),&n=1\\\
\frac{(-2)^{n}}{\pi^{3/2}}G_{n,n+2}^{4,n-1}\biggl{(}\frac{x}{2},\frac{1}{2}|\begin{array}[]{c}0,\ldots,0,\frac{1}{2},0\\\
\frac{1}{2},\frac{1}{2},1,1,1,\ldots,1\end{array}\biggr{)},&n\geq
2\end{cases}$

where $G$ is the generalized Meijer function. So we can read off $d_{p}$
directly from $f_{p}(z)$, for example

$\displaystyle
d_{0}=\frac{2}{\pi}[\operatorname{ci}(2\lambda\ell)\sin(2\lambda\ell)-\operatorname{si}(2\lambda\ell)\cos(2\lambda\ell)],\;d_{1}=-\frac{4\lambda\ell}{\pi}[\operatorname{ci}(2\lambda\ell)\cos(2\lambda\ell)+\operatorname{si}(2\lambda\ell)\sin(2\lambda\ell)],$
$\displaystyle
d_{2}=-\frac{4}{\pi^{3/2}}G_{2,4}^{3,2}\left(\lambda\ell,\frac{1}{2}|\begin{array}[]{c}0,\frac{1}{2}\\\
\frac{1}{2},\frac{1}{2},1,1\end{array}\right),\;d_{3}=\frac{16}{3\pi^{3/2}}G_{3,5}^{3,3}\left(\lambda\ell,\frac{1}{2}|\begin{array}[]{c}0,0,\frac{1}{2}\\\
\frac{1}{2},\frac{1}{2},1,1,1\end{array}\right)$ (21)
$\displaystyle-\frac{4\lambda\ell}{3\pi}[\operatorname{ci}(2\lambda\ell)\cos(2\lambda\ell)+\operatorname{si}(2\lambda\ell)\sin(2\lambda\ell)],\;\ldots$
(22)

Observe that $d_{0}$ has appeared before, in the open case, Eq. (4).

Substituting the expansions from Eqs. (10) and (13) into the partition
function, Eq. (8), and using the orthogonality relation, Eq. (11), gives us
the _exact_ answer to the problem of the closed quantum line

$Z={\cal N}\sum_{p,q,\ldots,r}c_{pq\ldots r}d_{p}d_{q}\ldots d_{r}.$ (23)

In Fig. (3) we plotted the average edge length, $\langle l\rangle=-(1/N)d(\ln
Z)/d\lambda$, using the exact answer, Eq. (23), for the $N=3$ case. Just a few
terms are necessary, because of the rapid convergence of Eq. (23).

Figure 3: Average edge length for the exact value, Eq. (23), is plotted as a
continuous curve (in blue) together with the approximations for weak (left
figure, Eq. (24), in red) and strong cosmological constant (right figure, Eq.
(25), in green). We split the figure in two figures because they are far away
in the $\lambda\ell$ range.

The approximations for weak and strong cosmological constant are calculated in
the next sections.

####  2.2.1 Weak Cosmological Constant

For $\lambda\ell\ll 1$ we can approximate the coefficients $d_{p}$ as (see
appendix A)

$d_{p}\approx\delta_{p0}+\frac{4\lambda\ell(-1)^{p}}{\pi}\left[\gamma-1+\ln(2\lambda\ell)+\sum_{s=0}^{|p|-1}\frac{1}{2[s/2]+1}\right].$

Substituting this approximation in Eq. (23) and keeping terms up to order
$\lambda\ell$, again for $N=3$, we get

$Z=16\pi\sum_{p,q,r}c_{pqr}d_{p}d_{q}d_{r}\approx 1+3\sum_{p}16\pi
c_{p00}\frac{4\lambda\ell(-1)^{p}}{\pi}\left[\gamma-1+\ln(2\lambda\ell)+\sum_{s=0}^{|p|-1}\frac{1}{2[s/2]+1}\right],$

where

$c_{p00}=\frac{1}{16\pi}\begin{cases}1,&p=0\\\
2[(-1)^{p}-1]/p^{2}\pi^{2}-i2(-1)^{p}/p\pi,&p\neq 0\end{cases}$

from Eq. (12).

Since $Z$ is real, the imaginary part cancel out and the real part contributes
only when $p$ is odd, thus

$\displaystyle Z\approx$ $\displaystyle
1+\frac{12\lambda\ell}{\pi}[\gamma-1+\ln(2\lambda\ell)]\left[1+\frac{8}{\pi^{2}}\sum_{t=0}^{\infty}\frac{1}{(2t+1)^{2}}\right]$
$\displaystyle+\frac{96\lambda\ell}{\pi^{3}}\sum_{t=0}^{\infty}\frac{1}{(2t+1)^{2}}\sum_{s=0}^{2t}\frac{1}{2[s/2]+1}$
$\displaystyle\approx$ $\displaystyle
1+\frac{24\lambda\ell}{\pi}[\gamma-1+\ln(4\lambda\ell)],$

where we have used (see appendix B)

$\sum_{t=0}^{\infty}\frac{1}{(2t+1)^{2}}=\frac{\pi^{2}}{8},\qquad\sum_{t=0}^{\infty}\frac{1}{(2t+1)^{2}}\sum_{s=0}^{2t}\frac{1}{2[s/2]+1}=\frac{\pi^{2}}{4}\ln
2.$

Therefore the average edge length is

$\frac{\langle
l\rangle}{\ell}\approx-\frac{8[\gamma+\ln(4\lambda\ell)]}{\pi+24\lambda\ell[\gamma-1+\ln(4\lambda\ell)]}$
(24)

which is the curve plotted in red in Fig. (3).

####  2.2.2 Strong Cosmological Constant

For $\lambda\ell\gg 1$ the partition function is dominated by configurations
that minimize the total length, $L$. So we have to find the minimum of Eq. (3)
with the constraint, Eq. (7), that is,

$\vec{\nabla}L(\phi_{1},\ldots,\phi_{N})=\alpha\vec{\nabla}C(\phi_{1},\ldots,\phi_{N}),$

where

$C(\phi_{1},\ldots,\phi_{N})=\sum_{n}\phi_{n}-2\pi$

and $\alpha$ is a Lagrange multiplier. And the answer is trivial: all angles
are equal, thus $\phi_{n}=2\pi/N$ and $L_{\text{min}}=2\ell N\tan(\pi/N)$ is
the minimum perimeter. Clearly,

$\frac{\langle l_{\text{min}}\rangle}{\ell}=2\tan\left(\frac{\pi}{N}\right)$
(25)

is the minimum average edge length for the polygon with $N$ sides.

The case of the triangle, $N=3$, is plotted in Fig. (3), as a constant curve
in green, where $\langle l_{\text{min}}\rangle/\ell=2\sqrt{3}\approx
3.464\ldots$ which is the length of an equilateral triangle with an inscribed
circle of radius $\ell$.

Note that the closed polygon is different from the open case, where the open
polygon could shrink to zero length. Here the constraint fixes a minimal
length.

##  3 Conclusions

A path integral, as in Eq. (1), is a “sum over paths”, that in the case of
quantum gravity means an integration in the space of all possible metrics,
$g$. In a 1D lattice, with edge lengths $l_{n}$ for $n=1,\ldots,N$, since $g$
has only one component, $g=(g_{11})$, we can choose $g_{11}(l_{n})$ to be the
same as the length squared, that is, $g_{11}(l_{n})=l_{n}^{2}$. Therefore, it
is natural to consider integrals of $l_{n}$.

As far as we know, there is no general consensus about the measure,
$Dg=dl_{n}f(l_{n})$, where $f(l_{n})$ is a function of only $l_{n}$ for a
local measure666For a non-local measure, $f$ is usually choose as a function
of the first neighbors. In 1D, this is $f(l_{n-1},l_{n},l_{n+1})$.. For
example, in Regge calculus [1] they choose $f(l_{n})=l_{n}^{\sigma+1}$ for
some number $\sigma$. Besides the ambiguity in the value of $\sigma$, there
are some difficulties such as the triangle inequalities, ultraviolet cutoff,
etc.

The main idea of our simple model is to work with the angles, $\phi_{n}$, and
set the unique Haar measure for U(1), which is a compact group. The open
polygon was very easy to calculate, Eq. (4). But in the little more realistic
example of the closed line, Eq. (8), there was some problems.

The first problem is that we can not use perturbation theory with a Boltzmann
weight, $\exp(-\lambda L)=1-\lambda L+\lambda^{2}L^{2}/2+\ldots$, that
diverges $\int Dg\,L=\int Dg\,L^{2}=\ldots=\infty$ at each order in $\lambda$.
This is similar to the QFT expansions for quantum gravity [3], where we have
to absorb a new infinity at each order in Newton’s constant.

Since we are using the Haar measure, an alternative is to expand the Dirac
delta function and the Boltzmann weight in characters, $\chi_{p}(\phi_{n})$,
but then we have another problem: the _global_ constraint, Eq. (7), involves
all the angles $\phi_{n}$ and this can not be treated with the usual delta
function expansion, Eq. (9).

To solve this second problem, we use a new expansion, Eq. (10), that
multiplies all characters, one for each angle $\phi_{n}$. The result in the
closed case is very interesting: the average edge length, $\langle
l\rangle=\langle L\rangle/N$, varies from a minimum value, Eq. (25), to
infinity, see Fig. (3). This is different from Regge calculus, where $\langle
l\rangle\propto 1/\lambda$.

Notice that the radius, $\ell$, of the U(1) circle is related to this minimum
average edge length, $\langle l_{\text{min}}\rangle$, which in turn is
connected to the configuration of maximum symmetry of the polygon. This give
us a hint of what happens in higher dimensions.

For example, in 2D the minimum surface is a regular tetrahedron with a
inscribed sphere of radius $\ell$, which can be related to a shell in the
SU(2) manifold. In 3D, the SU(2) ball must be used and the radius (not a
constant anymore) is related to the height of the solid tetrahedron. And so
on…

The 2D case is under study right now and will be published soon. Inclusion of
matter and fields can be done in the same way as in Regge calculus [1] and
these calculations will be published somewhere.

Finally, one can say that we have to integrate in $\ell$, which bring back the
infinities, because the number of independent variables in our model is one
less than in Regge calculus [1]. However, in the continuum limit $N\to\infty$
this makes no difference at all.

## Acknowledgement

We thank J. Lopez and N. Faustino for helpful discussions.

##  Appendix A Approximation

For small $x=2\lambda\ell$ the coefficients in Eq. (22) can be approximated to

$d_{p}\approx\delta_{p0}+\frac{2x(-1)^{p}}{\pi}(\gamma-1+\ln x+S_{p}),$

where

$S_{0}=0,\;S_{1}=1,\;S_{2}=2,\;S_{3}=\frac{7}{3},\;S_{4}=\frac{8}{3},\;S_{5}=\frac{43}{15},\;S_{6}=\frac{46}{15},\;S_{7}=\frac{337}{105},\;S_{8}=\frac{352}{105},\;\ldots$

This sequence can be written as

$\displaystyle 1$ $\displaystyle=1$ $\displaystyle 2$ $\displaystyle=1+1$
$\displaystyle\frac{7}{3}$ $\displaystyle=1+1+\frac{1}{3}$
$\displaystyle\frac{8}{3}$ $\displaystyle=1+1+\frac{1}{3}+\frac{1}{3}$
$\displaystyle\frac{43}{15}$
$\displaystyle=1+1+\frac{1}{3}+\frac{1}{3}+\frac{1}{5}$
$\displaystyle\frac{46}{15}$
$\displaystyle=1+1+\frac{1}{3}+\frac{1}{3}+\frac{1}{5}+\frac{1}{5}$
$\displaystyle\frac{337}{105}$
$\displaystyle=1+1+\frac{1}{3}+\frac{1}{3}+\frac{1}{5}+\frac{1}{5}+\frac{1}{7}$
$\displaystyle\frac{352}{105}$
$\displaystyle=1+1+\frac{1}{3}+\frac{1}{3}+\frac{1}{5}+\frac{1}{5}+\frac{1}{7}+\frac{1}{7}$
$\displaystyle\vdots$

which is equal to

$S_{p}=\sum_{s=0}^{p-1}\frac{1}{2[s/2]+1},$ (26)

where $[s/2]$ is the integer part of $s/2$. This is also equivalent to

$S_{p}=H_{p}+\frac{1}{p}\sum_{s=0}^{p}[(-1)^{s}\Phi(-1,1,s)+\ln 2],$

where $H_{p}$ is the harmonic number and $\Phi$ is the Lerch transcendent.

##  Appendix B Double series

To find the sum of $S_{p}/p^{2}$, see Eq. (26), for $p=2t+1$ an odd integer,
we note that

$\displaystyle\sum_{t=0}^{\infty}\frac{S_{2t+1}}{(2t+1)^{2}}=\sum_{t=0}^{\infty}\frac{1}{(2t+1)^{2}}\sum_{s=0}^{2t}\frac{1}{2[s/2]+1}$
$\displaystyle=2+\frac{2}{3^{2}}\left(1+\frac{1}{3}\right)+\frac{2}{5^{2}}\left(1+\frac{1}{3}+\frac{1}{5}\right)+\frac{2}{7^{2}}\left(1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}\right)+\ldots-\left(1+\frac{1}{3^{3}}+\frac{1}{5^{3}}+\frac{1}{7^{3}}+\ldots\right)$
$\displaystyle=2\sum_{m=0}^{\infty}\frac{1}{(2m+1)^{2}}\sum_{n=0}^{m}\frac{1}{2n+1}-\sum_{m=0}^{\infty}\frac{1}{(2m+1)^{3}}$
$\displaystyle=2\sum_{m=0}^{\infty}\frac{1}{(2m+1)^{2}}\left[\ln
2+\frac{\gamma}{2}+\frac{\psi(m+3/2)}{2}\right]-\frac{7}{8}\zeta(3),$

where $\psi$ is the digamma function and $\zeta$ is the Riemann zeta function.
Using that $\psi(m+3/2)=\psi(m+1/2)+2/(2m+1)$ and [5]

$\sum_{m=0}^{\infty}\frac{\psi(m+1/2)}{(2m+1)^{2}}=-\frac{1}{8}[\pi^{2}\gamma+7\zeta(3)]$

we obtain the double series

$\sum_{t=0}^{\infty}\frac{1}{(2t+1)^{2}}\sum_{s=0}^{2t}\frac{1}{2[s/2]+1}=\frac{\pi^{2}}{4}\ln
2.$

## References

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  * [2] I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2015).
  * [3] R. Paszko and A. Accioly, Class. Quant. Grav. 27 (2010) 145012.
  * [4] G. Iwata, Prog. Theor. Phys. 24 (1960) 1118.
  * [5] Y. Brychkov, Handbook of Special Functions (CRC Press, 2008).