# Interacting Hawkes processes with multiplicative inhibition

Céline Duval***Université de Paris, Laboratoire MAP5, UMR CNRS 8145\. E-mail:
<EMAIL_ADDRESS>Eric Luçon†††Université de Paris, Laboratoire MAP5,
UMR CNRS 8145\. E-mail<EMAIL_ADDRESS>and Christophe Pouzat ‡‡‡IRMA,
Université de Strasbourg, CNRS UMR 7501. E-mail:
<EMAIL_ADDRESS>
###### Abstract

In the present work, we introduce a general class of mean-field interacting
nonlinear Hawkes processes modelling the reciprocal interactions between two
neuronal populations, one excitatory and one inhibitory. The model
incorporates two features: inhibition, which acts as a multiplicative factor
onto the intensity of the excitatory population and additive retroaction from
the excitatory neurons onto the inhibitory ones. We give first a detailed
analysis of the well-posedness of this interacting system as well as its
dynamics in large population. The second aim of the paper is to give a
rigorous analysis of the longtime behavior of the mean-field limit process. We
provide also numerical evidence that inhibition and retroaction may be
responsible for the emergence of limit cycles in such system.

Keywords. Multivariate nonlinear Hawkes processes, Inhibition, Mean-field
systems, Neural networks.
AMS Classification. 60G55, 92B20, 60F99.

##  1 Introduction

###  1.1 Biological description

Neurons are peculiar network forming cells whose (biological) function is to
receive signals from other neurons, ’integrate’ these signals and transmit the
’integration result’ to other neurons or _effector_ cells like muscles [37].
Since most neurons are large cells by biological standards (e.g. the
motoneurons controlling the muscles of our big toes are about a meter long)
they use a specific mechanism to reliably transmit signals from one of their
extremities to another [13]: a propagating electrical wave of _short_ duration
($\sim$ 1-2 ms) called the _action potential_ or _spike_. Action potentials
are _all-or-none_ events (a necessary condition for getting reliable
transmission across long distances) that are triggered when the neuron’s
_membrane potential_ (i.e. the electrical potential difference between the
inside and the outside of the neuron) is ’large enough’. To simplify, between
two successive action potentials the neuron ’sums’ its inputs, changing
thereby its membrane potential and, when this membrane potential crosses a
more or less sharp threshold, the neuron fires a spike that propagates [37].
Neurons come in two main categories: i) _excitatory_ neurons make their
_postsynaptic_ partners more likely to spike; ii) _inhibitory_ neurons make
their _postsynaptic_ partners less likely to spike. Although, we are dealing
with biological material, implying that there are many sub-types within both
of these categories [5, 37], we are interested in studying a simplified model
of the _neocortex_ –the outmost and most recently evolved part of the
vertebrate brain–and we are going to consider just two neuronal types,
excitatory and inhibitory with the actual neocortical proportions of 80% and
20% [5].

###  1.2 Our model

In this paper, a general class of mean-field interacting Hawkes processes is
considered, modeling the reciprocal interactions between a population $A$ of
excitatory neurons and a population $B$ of inhibitory neurons. Hawkes
processes, originally introduced in [32], have met a recent interest in the
mathematical neuroscience literature, for their ability to model the
dependence of the activity of a neuron in the history of the network. We adopt
here a usual simplifying hypothesis on the network, that is to suppose all-to-
all connections between neurons, within a large population (mean-field
framework). We refer to Section 1.3.2 below for a detailed account of the use
of Hawkes processes in the modeling of neuronal networks.

More precisely, consider a population of neurons of size $N\geq 1$, that is
divided into population $A$ (that is considered to be excitatory) with size
$N_{A}:=\alpha N$ with $\alpha\in(0,1)$ (e.g. $\alpha=0.8$ [5]) and a
population $B$ with size $N_{B}=(1-\alpha)N$ which is considered to be
inhibitory. A particular instance of the model we describe in the paper (and
its main motivation) is then given in terms of a family of counting processes
$(Z^{1}_{t},\ldots,Z^{N_{A}}_{t})$ (population $A$) and
$(Z^{N_{A}+1}_{t},\ldots,Z^{N}_{t})$ (population $B$) with coupled conditional
stochastic intensities given respectively by $\lambda^{A}$ and $\lambda^{B}$
as follows

$\begin{cases}\lambda_{t}^{A}:&=\Phi_{A}\left(\frac{1}{N}\underset{1\leqslant
j\leqslant N_{A}}{\sum}\int_{0}^{t^{-}}h_{1}(t-u){{\rm
d}Z_{u}^{j}})\right)\Phi_{B\to A}\left(\frac{1}{N}\underset{N_{A}+1\leqslant
j\leqslant N}{\sum}\int_{0}^{t^{-}}h_{2}(t-u){{\rm d}Z_{u}^{j}}\right),\\\
\lambda_{t}^{B}:&=\Phi_{B}\left(\frac{1}{N}\underset{N_{A}+1\leqslant
j\leqslant N}{\sum}\int_{0}^{t^{-}}h_{3}(t-u){{\rm
d}Z_{u}^{j}})\right)+\Phi_{A\to B}\left(\frac{1}{N}\underset{1\leqslant
j\leqslant N_{A}}{\sum}\int_{0}^{t^{-}}h_{4}(t-u){{\rm
d}Z_{u}^{j}}\right).\end{cases}$ (1)

We refer to Section 2.1 below for precise definitions and main hypotheses
concerning Model (1). The dynamics given by (1) is of Hawkes type: the
intensity of each particle depends on the history of the whole system, through
memory kernels $h_{i}$, $i=1,\ldots,4$ and firing rate functions $\Phi_{A}$
and $\Phi_{B}$ (a classical example, so-called linear, being when
$\Phi_{A}(x)=\mu_{A}+x$ and $\Phi_{B}(x)=\mu_{B}+x$, where
$\mu_{A},\mu_{B}\geq 0$ are constant intensities modeling spontaneous activity
in populations $A$ and $B$ in absence of interaction). The main novelty of the
Model (1) (in comparison with previous Hawkes models with inhibition, see
Section 1.3.2 below) lies in the multiplicative nature of the influence of
inhibitory population $B$ onto population $A$, through the inhibition kernel
$\Phi_{B\to A}$ (think for now as $\Phi_{B\to A}$ being a decreasing
nonnegative function on $[0,+\infty)$, with $\Phi_{B\to A}(0)=1$ and
$\Phi_{B\to A}(x)\xrightarrow[x\to\infty]{}0$): activity of population $A$
should decrease as activity of population $B$ rises. In this respect, a
natural control case is when $\Phi_{B\to A}\equiv 1$ where $\lambda_{t}^{A}$
in (1) reduces to a simple, possibly nonlinear, Hawkes process [23] decoupled
from population $B$. The model secondly incorporates retroaction from
population $A$ onto population $B$, that is supposed to be mostly additive,
although possibly modulated by a nonlinear feedback kernel $\Phi_{A\to B}$.

###  1.3 Stochastic models for inhibition

Two different asymptotic behaviors for (1) will be successively considered in
the paper: its behavior in large population ($N\to\infty$) and in large time
($t\to\infty$). The large population analysis goes back to well-proven
techniques for mean-field systems and, as we will see below, the natural limit
of (1) will be described in terms of an inhomogeneous Poisson process with
nonlinear intensity. Similar mean-field analysis have been successfully
carried out for a large class of models (that do not restrict to interacting
point processes), incorporating various relevant features in neuroscience,
such as refractory periods [14, 42], spatial inhomogeneities [16, 3],
excitability [36]. At this point, due to the nonlinear (and often non
Markovian) character of the mean-field limit, one major difficulty concerns
the characterization of its dynamics as $t\to\infty$, this difficulty being
all the more present when one wants to incorporate inhibition. Whereas there
is a vast literature on mean-field limits as $N\to\infty$, a significantly
fewer number of models are amenable to a rigorous longtime analysis. In this
respect, we will show that the present Model (1) is sufficiently tractable for
a detailed rigorous analysis as $t\to\infty$. This allows us in particular to
highlight some biologically relevant effects of inhibition on the system, see
Section 1.4.

The point of the current section is to motivate these theoretical difficulties
in existing models in neuroscience and to compare them with (1).

####  1.3.1 Voltage-based models

A signal gets transmitted from one neuron to the next by the activation of a
synapse: the action potential of the _presynaptic_ neuron reaches the
presynaptic terminal, this triggers the release of packets of
neurotransmitters (small molecules like glutamate for the excitatory synapses
and GABA for the inhibitory ones) that diffuse in the small space between the
pre- and post-synaptic neurons and bind to _receptor-channels_ located in the
membrane of the postsynaptic neuron [37]. Channels are macromolecules spanning
the cell membrane and forming a pore through the latter; the pore can be
closed or opened. After transmitter binding to these receptor-channels, the
latter open and let specific ions flow through their pore (mainly sodium for
excitatory synapses and chloride for the inhibitory ones). These ion fluxes or
_currents_ will induce a change of the postsynaptic neuron membrane potential.

A simple quantitative neuronal model compatible with this biological
description was introduced in 1907 by Lapicque [34]. It is now known as the
_Lapicque_ or the _integrate and fire_ model and it describes the membrane
potential dynamics of ’point-like’ neurons (their spatial extension is not
explicitly modeled) as follows [47, 9]:

$C_{m}\frac{{\rm d}V_{i}}{{\rm
d}t}=-\frac{V_{i}}{R_{m}}+\sum_{j\in\mathcal{S}_{i,E}}I_{j\rightarrow
i}(t)+\sum_{k\in\mathcal{S}_{i,I}}I_{k\rightarrow i}(t)\quad\text{if}\quad
V_{i}(t)<V_{thr}\,,$ (2)

where $i$ is the neuron index; $C_{m}$ is the neuron capacitance; $R_{m}$ is
the neuron membrane resistance; $\mathcal{S}_{i,E}$, respectively
$\mathcal{S}_{i,I}$, are the indices of excitatory, respectively inhibitory,
neurons presynaptic to $i$; $I_{j\rightarrow i}(t)\geqslant 0$, respectively
$I_{k\rightarrow i}(t)\leqslant 0$, are the synaptic currents due to neuron
$j$, respectively $k$, at time $t$; $V_{thr}$ is the ’threshold’ voltage.
Every time $V_{i}(t)=V_{thr}$ an action potential is emitted and $V_{i}$ is
reset to 0. Very often the $I_{j\rightarrow i}(t)$ are set to:

$I_{j\rightarrow i}(t)=w_{j\rightarrow i}\sum_{l}\delta(t-t_{j,l})\,,$ (3)

where $w_{j\rightarrow i}$ is referred to as the _synaptic weight_ ; $\delta$
stands for the Dirac delta function; the $t_{j,l}$ are the successive spike
times of neuron $j$. In this model, when there are no inputs, the membrane
potential relaxes towards 0 with a time constant $\tau=R_{m}C_{m}$. A
presynaptic spike from an excitatory neuron $j$ generates an instantaneous
upward ’kick’ of amplitude $w_{j\rightarrow i}$, while a presynaptic spike
from an inhibitory neuron $k$ generates an instantaneous downward ’kick’ of
amplitude $w_{k\rightarrow i}$.

The actual action potential is not explicitly modeled–it was not doable at the
time of Lapicque since the biophysics of this phenomenon was not
understood–but is replaced by a point event. Notice that with the synaptic
input description illustrated by (3) the current generated in the postsynaptic
neuron by a given synapse does not depend on the membrane voltage of the
former. This constitutes a crude approximation of the actual biophysics of
synaptic current generation. A much better approximation [33, 48]–but harder
to work with analytically–is provided by using:

$I_{j\rightarrow i}(t)=g_{j\rightarrow
i}(V_{rev}-V_{i})\,\sum_{l}\delta(t-t_{j,l})\,,$ (4)

where $g_{j\rightarrow i}$ is the _synaptic conductance_ , $V_{rev}$ is the
synaptic current _reversal potential_ –as its name says this is the voltage at
which the current changes sign–, it is negative or null for inhibitory
synapses and larger than the threshold voltage for excitatory ones. Taking
$V_{rev}=0$ for the inhibitory inputs we see an important and empirically
correct feature appearing: the downward ’kick’ generated by the activation of
an inhibitory input becomes _proportional_ to the membrane voltage $V_{i}$;
the larger the latter, the larger the kick.

The biophysical and molecular events leading to the action potential emission
and propagation are by now well understood [37] and are known to involve the
_stochastic_ opening and closing of _voltage-gated ion channels_ : the
probability of finding a channel closed or opened depends on the membrane
voltage. This stochastic dynamics of the channels can lead to fluctuations in
the action potential emission or propagation time when a given (deterministic)
stimulation is applied repetitively to a given neuron [49, 51]. Similarly, the
synaptic receptor-channels do fluctuate between open and close states, but
that’s the binding of the transmitter rather than the membrane potential that
influences the probability of finding the channel in the open state. An even
(much) larger source of fluctuations at the synapse results from the variable
number of transmitter packets that get released upon a presynaptic action
potential arrival [37], even if the same presynaptic neuron is repetitively
activated in the same conditions. The result of all these fluctuation sources
is a rather ’noisy’ aspect of the membrane potential of cortical neurons that
legitimates the use of ’stochastic units’ as building blocks of neural network
models [48, 51]. Historically, the first stochastic units were built by adding
a Brownian motion process term to the right hand side of (2) [28, 9, 45].

Among the vast literature concerning rigorous mean-field approaches in the
context of integrate and fire systems and their derivations, we refer in
particular to e.g. [22, 8] and references therein. This approach leads to
Chapman-Kolmogorov / Fokker-Planck equations that are hard to work with
analytically and numerically [11], the main difficulty being the existence of
blowups in finite time for the system, due to the possibility of having a
macroscopic number of neurons simultaneously close to the firing threshold
$V_{thr}$. Several extensions with random firing rates have been proposed in
the literature, [24, 21] and references therein, but the longtime analysis of
such models (especially when it concerns the effect of inhibition) remains not
fully understood (see nonetheless the recent significant advances in this
direction in [21]).

####  1.3.2 Hawkes processes

The model we consider in (1) enters into the framework of Hawkes processes
[32, 23]. The point of view adopted here is to model directly the stochastic
intensity [7, 17] by identifying a neuron’s sequence of strereotyped action
potentials with the point process $(t_{j,l})_{j,l}$ in (4). In this context,
the Hawkes framework generically reads

$\lambda_{t}^{i}=\Phi_{i}\left(\sum_{j\in\mathcal{S}_{i,E}}\int_{0}^{t}h_{j\rightarrow
i}(t-u){\rm d}Z_{u}^{j}\right)\,,$ (5)

where $\lambda_{t}^{i}$ is the intensity of neuron $i$, $\Phi_{i}$ a positive
function, $Z_{j,t}$ is the counting process associated with neuron $j$,
$h_{j\rightarrow i}(t)$ is the _synaptic kernel_ associated with the synapse
between neurons $j$ and $i$. To qualitatively match the case of the integrate
and fire model (2), we would take $\Phi_{i}(x)=\mu_{0}^{i}+x$ where
$\mu_{i}^{0}$ is the basal rate of neuron $j$ and $h_{j\rightarrow
i}(t)=(w_{j\rightarrow i}/\tau)\,\exp-t/\tau$. It is not really possible to
account for the vast recent literature on mean-field analysis of Hawkes
processes in neuroscience, we refer to e.g. [6, 23, 25, 16, 14] and references
therein for further details. Note that this framework allows for a large
versatility in the modeling, since it can accommodate for various features
such as age-dependent behaviors/refractory periods [14, 15, 42] or spatially-
structured dynamics [16].

Intensity-based models such as (5) lead to much better looking equations, that
are also easier to work with numerically [31, 17, 23]: in the mean-field case,
one naturally obtains large population limits in terms of inhomogeneous
Poisson processes with intensity solving convolution equations [23]. A
particularly important instance of (5) that is explicitly solvable concerns
the linear case where $\Phi_{i}(x)=\mu_{i}+x$ with nonnegative synaptic
kernels $h_{i\to j}$: the corresponding convolution equation for the mean-
field intensity becomes linear and one can explicitly compute its limit as
$t\to\infty$, [23], Th. 10 and 11.

As far as inhibition is concerned, following the approach of Section 1.3.1,
one would rightfully criticize the use of a Hawkes process by pointing out the
difficulty, not to say the impossibility, of modifying it to accommodate
inhibition without loosing the nice analytical properties of linear Hawkes
processes [17, 6]. In this context, the main strategy followed in the
literature so far has been to model inhibition by introducting kernels
$h_{j\to i}(\cdot)$ in (5) taking negative values [20, 41, 42, 43, 4] or
positive kernels $h$ (e.g. of Erlang type) multiplied by random and possibly
negative coefficients [25, 26, 40]. This is what we could qualify as an
_additive_ inhibition, in the sense that the intensity of a neuron is the
(temporal) additive superposition of several memory kernels $h_{i\to j}$, in
the same spirit as the scheme suggested by (3) for integrate and fire models.
In order to keep $\lambda_{t}^{i}$ positive, one crucially needs here to
modulate this superposition by a positive (necessarily nonlinear) synaptic
kernel $\Phi_{i}$, (e.g. $\Phi_{i}(x)=\mu^{i}+x_{+}$ where $x_{+}=\max(x,0)$).
The major counterpart is that one looses the nice analytical properties which
makes the longtime analysis of linear Hawkes models tractable.

The strategy of the present Model (1) is different, as it relies on a
_multiplicative inhibition_ , in the spirit of the scheme provided by (4) for
integrate and fire models: we view the inhibitory inputs as ’modulations’ of
the excitatory ones and describe inhibition as a multiplicative factor between
0 and 1 applied to the excitatory inputs. In particular, the multiplicative
structure of (1) no longer requires the synaptic kernels to take negative
values: we may and we will only consider nonnegative kernels $h_{i}$,
$i=1,\ldots,4$. In this way, the non-negativity of the intensity is then
automatically preserved, together with the ’nice’ properties of the canonical
linear Hawkes process: we may still take advantage of the positivity of the
synaptic kernels and the respective monotonicities of $\Phi_{A}$ and
$\Phi_{B\to A}$ (positive and decreasing) to obtain asymptotic results similar
to the linear case.

###  1.4 The effect of inhibition: quenching of supercriticality and
oscillations

We briefly review here the biological features concerning the effects of
inhibition that one can retrieve from Model (1). We refer to the sequel for
more details. The first main observation that one can make from Model (1) is
that inhibition has an ability to quench supercriticality: Model (1) can
accommodate with situations where population $A$ alone (that is isolated from
population $B$, i.e. $\Phi_{B\to A}\equiv 1$) is supercritical, in the sense
that the intensity of $A$ diverges as $t\to\infty$, whereas the addition of
inhibition and retroaction forces the system to go back to subcriticality. We
refer in particular to Theorem 3.6 below, where it can be shown rigorously
that (1) cannot be supercritical in presence of both inhibition and
retroaction. This corresponds to a biological situation commonly observed,
where epileptic seizures may be observed when inhibition is altered (see [37]
and Remark 3.4 below).

Another particularly interesting feature of Model (1) is that inhibition may
lead to oscillations, a ubiquitous phenomenon in the nervous system [10].
Emergence of oscillations in mean-field processes is a longstanding issue in
the literature, that is not specific to neuroscience applications, as it
reflects a common feature of self-organization in statistical physics (see
e.g. [18, 19, 38, 29]), biological models of synchrony [12, 1] or neuroscience
[36, 35, 25]. We mention here specifically [25] where global oscillations have
been proven for interacting Hawkes processes with circular connectivity.

###  1.5 About simulations

The Python code, together with a detailed documentation, that has been used
for the simulations made in the present paper is available at
https://plmlab.math.cnrs.fr/xtof/hawkes-x-hawkes.

###  1.6 Organization of the paper

The paper is organized as follows: the precise definitions and main hypotheses
are given in Section 2, where a generalized version (7) of (1) is introduced.
We first address well-posedness results for the microscopic system (Section
2.2) and for its mean-field limit (Section 2.3) as well as propagation of
chaos estimates. The second purpose of the paper it to give a detailed
analysis, in the particular case of Model (1), of the behavior as $t\to\infty$
of the limiting mean-field process in Section 3. Focus is put on the
presence/absence of inhibition and retroaction: the easier cases of decoupled
dynamics are first treated in Section 3.2. The model with full connectivity
(that is with inhibition and feedback) is considered in Section 3.3.
Fluctuations results and some resulting statistical tests for inhibition is
given in Section 4. Examples of dynamics that fulfil the hypotheses of Section
3 are then given in Section 5. We give finally in Section 6 intuition and
numerical evidence that inhibition and retroaction may lead to oscillations,
for some particular instances of the inhibition kernel $\Phi_{B\to A}$. Proofs
of the main results are gathered in Section 7 and Section 8.

##  2 Well-posedness and large population behavior

###  2.1 Model and assumptions

Let $\alpha\in(0,1)$ and $N=N_{A}+N_{B}\gg 1$ with $N_{A}=\alpha N$ and
$N_{B}=(1-\alpha)N$ the respective size of population $A$ and $B$. The neurons
within population $A$ (resp. population $B$) are indexed by $i=1,\ldots,N_{A}$
(resp. $i=N_{A}+1,\ldots,N$). Consider on a filtered probability space
$(\Omega,\mathcal{F},\left(\mathcal{F}_{t}\right)_{t\geq 0},\mathbf{P})$ an
independent family of i.i.d. Poisson measures $(\pi_{i}({\rm d}s,{\rm
d}z),i\in\\{1,\ldots,N\\})$ with intensity measure ${\rm d}s\times{\rm d}z$ on
$[0,\infty)\times[0,\infty)$. Let $(x,y)\mapsto F(x,y)$ and $(x,y)\mapsto
G(x,y)$ two nonnegative functions defined on $(0,\infty)^{2}$. Consider the
family of càdlàg $(\mathcal{F}_{t})_{t\geq 0}$ point processes
$(Z_{t}^{i})_{t\geq 0,i=1,…,N}$ given by

$Z_{t}^{i}=\int_{0}^{t}\int_{0}^{\infty}\mathbf{1}_{z\leqslant\lambda_{s}^{i}}\pi_{i}({\rm
d}s,{\rm d}z),i=1,…,N,$ (6)

where the intensity $\lambda^{i}$, $i=1,…,N$, is given as

$\begin{cases}\lambda_{t}^{i}=\lambda_{t}^{A}:=F\left(\frac{1}{N}\underset{1\leqslant
j\leqslant N_{A}}{\sum}\int_{0}^{t^{-}}h_{1}(t-u){{\rm
d}Z_{u}^{j}}),\frac{1}{N}\underset{N_{A}+1\leqslant j\leqslant
N}{\sum}\int_{0}^{t^{-}}h_{2}(t-u){{\rm
d}Z_{u}^{j}})\right),&i=1,\ldots,N_{A},\\\
\lambda_{t}^{i}=\lambda_{t}^{B}:=G\left(\frac{1}{N}\underset{N_{A}+1\leqslant
j\leqslant N}{\sum}\int_{0}^{t^{-}}h_{3}(t-u){{\rm
d}Z_{u}^{j}}),\frac{1}{N}\underset{1\leqslant j\leqslant
N_{A}}{\sum}\int_{0}^{t^{-}}h_{4}(t-u){{\rm
d}Z_{u}^{j}}\right),&i=N_{A}+1,\ldots,N.\end{cases}$ (7)

In the sequel, we sometimes write $Z_{t}^{i,A}$ (resp. $Z_{t}^{i,B}$) in place
of $Z_{t}^{i}$ for $i=1,\ldots,N_{A}$ (resp. $i=N_{A}+1,\ldots,N$) in order to
specify the population the $i$th neuron belongs to. Let $p\geq 1$ and $T>0$.
For any function $h$ on $[0,\infty)$ such that $\left|h\right|^{p}$ is locally
integrable (resp. integrable) denote by $\left\|h\right\|_{p,T}$ (resp.
$\left\|h\right\|_{p}$) as the $L^{p}$ norm of $h$ on $[0,T]$ (resp. on
$[0,\infty)$):

$\displaystyle\left\|h\right\|_{p,T}$
$\displaystyle:=\left(\int_{0}^{T}\left|h(s)\right|^{p}{\rm
d}s\right)^{\frac{1}{p}},\hskip 56.9055pt\mbox{resp. }\ \
\left\|h\right\|_{p}:=\left(\int_{0}^{\infty}\left|h(s)\right|^{p}{\rm
d}s\right)^{\frac{1}{p}}.$

For any locally bounded (resp. bounded) function $h$, denote in a same way
$\left\|h\right\|_{\infty,T}:=\sup_{s\in[0,T]}\left|h(s)\right|$ and
$\left\|h\right\|_{\infty}:=\sup_{s\geq 0}\left|h(s)\right|.$ For any
Lipschitz-continuous function $f$ on $\mathbb{R}$, we denote by
$\left\|f\right\|_{L}$ the Lispchitz constant of $f$. Throughout the paper,
Assumptions 2.1 and 2.2 below will be required.

######  Assumption 2.1 (Assumptions on $h_{i}$, $i=1,\ldots,4$).

The kernels $h_{i}$, $i=1,\ldots,4$ on $[0,+\infty)$ are such that

$H(t):={\alpha}h_{1}(t)+{(1-\alpha)}h_{2}(t)+{(1-\alpha)}h_{3}(t)+{\alpha}h_{4}(t),\quad
t\geq 0,$ (8)

is locally integrable, i.e. $\|H\|_{1,T}<\infty$, $T>0$.

At this point, note that we need not assume that $h_{i}\geq 0$, since we are
only concerned for now with well-posedness results concerning both the
particle system (7) and its mean-field limit as well as propagation of chaos
results. The positivity of the kernels will only be assumed for the study of
the longtime behavior of the mean-field process in Section 3. However, in view
of the discussion of Section 1.3.2, the fact that the $h_{i}$ might take
negative values is not something that we have in mind in this paper.

######  Assumption 2.2 (Assumptions on $F$ and $G$).

Suppose that $F$ and $G$ are nonnegative functions such that the following
holds: there exist constants $c_{i}\geq 0,i=1,\ldots,5$ such that

$\displaystyle\sup_{x,y\geq 0}\frac{F(x,y)}{c_{1}(x+y)+c_{2}}$
$\displaystyle\leq 1,$ (9) $\displaystyle\sup_{x\neq x^{\prime}\geq 0,y\geq
0}\frac{\left|F(x,y)-F(x^{\prime},y)\right|}{c_{3}\left|x-x^{\prime}\right|}$
$\displaystyle\leq 1,$ (10) $\displaystyle\sup_{x\geq 0,y\neq y^{\prime}\geq
0}\frac{\left|F(x,y)-F(x,y^{\prime})\right|}{(c_{4}x+c_{5})\left|y-y^{\prime}\right|}$
$\displaystyle\leq 1$ (11)

Suppose also that $G$ is globally Lipschitz in both variables: for some
$c_{6}>0$

$\left|G(x,y)-G(x^{\prime},y^{\prime})\right|\leq
c_{6}\left(\left|x-x^{\prime}\right|+\left|y-y^{\prime}\right|\right),\
x,x^{\prime},y,y^{\prime}\geq 0.$ (12)

Note that an immediate consequence of (12) is that $G$ is sublinear: for
$c_{7}:=G(0,0)$,

$G(x,y)\leq c_{7}+c_{6}\left(x+y\right),\ x,y\geq 0.$ (13)

######  Example 2.3.

Any globally Lipschitz function $F$ (with Lipschitz constant
$\left\|F\right\|_{L}$) satisfies Assumption 2.2 for
$c_{1}=c_{3}=c_{5}=\left\|F\right\|_{L}$, $c_{2}=F(0,0)$ and $c_{4}=0$.

In place of Assumption 2.2, we will sometimes specify to a more restrictive
model, which corresponds to (1):

######  Assumption 2.4 (Multiplicative model).

Suppose that $F$ and $G$ satisfy

$F(x,y)=\Phi_{A}(x)\Phi_{B\to A}(y),\ G(x,y)=\Phi_{B}(x)+\Phi_{A\to B}(y),$
(14)

where $\Phi_{A}$, $\Phi_{B\to A}$, $\Phi_{B}$ and $\Phi_{A\to B}$ are
nonnegative functions, each of them globally Lipschitz with $\Phi_{B\to A}$
bounded (and with no loss of generality we assume $0\leq\Phi_{B\to A}\leq 1$).

Assumption 2.4 is a particular case of Assumption 2.2, for
$c_{1}=c_{3}:=\left\|\Phi_{A}\right\|_{L}$, $c_{2}:=\Phi_{A}(0)$,
$c_{4}:=\left\|\Phi_{B\to A}\right\|_{L}\left\|\Phi_{A}\right\|_{L}$ and
$c_{5}:=\left\|\Phi_{B\to A}\right\|_{L}\Phi_{A}(0)$. Although it is not
required for the moment, the examples to have in mind to model inhibition is
$\Phi_{B\to A}$ being nonincreasing with $\Phi_{B\to A}(y)\to 0$ as
$y\to\infty$ and $\Phi_{A\to B}$ nondecreasing with $\Phi_{A\to
B}(y)\to+\infty$ as $y\to\infty$. A further particular example is

######  Example 2.5.

A case of particular interest is for $F$ and $G$ satisfying Assumption 2.4
with $\Phi_{A}$ and $\Phi_{B}$ being linear:

$\Phi_{A}(x)=\mu_{A}+x,\ \Phi_{B}(x)=\mu_{B}+x,\ x\geq 0,$

where $\mu_{A},\mu_{B}\geq 0$. In this example, suppose in addition that
$h_{i}\geq 0$ for $i=1,\ldots,4$.

Assumption 2.4 and Example 2.5 (for which $c_{1}=c_{3}=c_{4}=1$ and
$c_{2}=c_{2}=\mu_{A}$) are considered in detail in Section 3. In the remaining
of this Section, we only require Assumption 2.2.

###  2.2 Well-posedness of the particle system

The first result concerns well-posedness of the particle system (6):

######  Proposition 2.6.

Suppose Assumptions 2.1 and 2.2 hold. Then, conditional on the Poisson
measures $(\pi_{i})_{i=1,\ldots,N}$, there exists a pathwise unique Hawkes
process $(Z^{i})_{1\leqslant i\leqslant N}$ as in (6) such that
$\sum_{i=1}^{N}\mathbf{E}[Z_{t}^{i}]<\infty$, for all $t\geqslant 0$.

In the case where $F$ is globally Lipschitz (Example 2.3), existence and
uniqueness of (6) can be proven directly through a fixed-point procedure as in
[23]. Then, Proposition 2.6 is a direct consequence of [23], Theorem 6, up to
minor notational changes. The additional technical difficulty in our case
(consider e.g. Assumption 2.4) is that even though $\Phi_{A}$ and $\Phi_{B\to
A}$ are globally Lipschitz, the product $\Phi_{A}\Phi_{B\to A}$ is not and the
usual result of [23] no longer applies. The key point to note is that $0\leq
F(x,y)\leq c_{1}(x+y)+c_{2}$, one can stochastically dominate our process by a
_linear Hawkes process_ that is known to exist. Therefore, the proof relies on
a thinning procedure (similar to [20]) and is carried out in Section 7.1. For
similar ideas see [42] Prop. 1.4 or [20] Prop. 2.

###  2.3 The mean-field limit

Following a standard procedure for mean-field systems [46], it is easy to
derive the limit nonlinear process of (6) as $N\to\infty$. In the Hawkes
setting, it reduces to an inhomogeneous Poisson process [23]. Informally
speaking, as $N\to\infty$, the empirical mean in (7) concentrates around an
expectation and it is natural to introduce the following: let
$(\pi,\widetilde{\pi})$ be independent Poisson point processes on
$[0,+\infty)^{2}$ with intensity ${\rm d}s{\rm d}z$. Consider

$\begin{cases}\bar{Z}_{t}^{A}&=\int_{0}^{t}\int_{0}^{\infty}\mathbf{1}_{z\leqslant
F\left(\alpha\int_{0}^{s}h_{1}(s-u){\rm
d}\mathbf{E}(\bar{Z}_{u}^{A}),(1-\alpha)\int_{0}^{s}h_{2}(s-u){\rm
d}\mathbf{E}\left(\bar{Z}_{u}^{B}\right)\right)}\pi({\rm d}s,{\rm d}z),\\\
\bar{Z}_{t}^{B}&=\int_{0}^{t}\int_{0}^{\infty}\mathbf{1}_{z\leqslant
G\left((1-\alpha)\int_{0}^{s}h_{3}(s-u){\rm
d}\mathbf{E}\left(\bar{Z}_{u}^{B}\right),\alpha\int_{0}^{s}h_{4}(s-u){\rm
d}\mathbf{E}\left(\bar{Z}_{u}^{{A}}\right)\right)}\widetilde{\pi}({\rm
d}s,{\rm d}z),\end{cases}\ t\geq 0\ .$ (15)

The nonlinearity of the process $\bar{Z}:=(\bar{Z}^{A},\bar{Z}^{B})$ lies in
the fact that it interacts with its own expectation
$(m_{t}^{A},m_{t}^{B}):=\left(\mathbb{E}\left(\bar{Z}_{t}^{A}\right),\mathbb{E}\left(\bar{Z}_{t}^{B}\right)\right)$
which solves

$\begin{cases}m_{t}^{A}&=\int_{0}^{t}{F\left(\alpha\int_{0}^{s}h_{1}(s-u){\rm
d}m_{u}^{A},(1-\alpha)\int_{0}^{s}h_{2}(s-u){\rm d}m_{u}^{B}\right)}{\rm
d}s,\\\ m_{t}^{B}&=\int_{0}^{t}G\left((1-\alpha)\int_{0}^{s}h_{3}(s-u){\rm
d}m_{u}^{B},\alpha\int_{0}^{s}h_{4}(s-u){\rm d}m_{u}^{A}\right){\rm
d}s.\end{cases}$ (16)

The following result is a well-posedness result for both (15) and (16):

######  Proposition 2.7.

Under Assumptions 2.1 and 2.2, for all $T>0$, there exists a unique
nondecreasing in both coordinates locally bounded solution
$(m_{t}^{A},m_{t}^{B})_{t\in[0,T]}$ to (16) that is of class $\mathcal{C}^{1}$
on $[0,T]$. Moreover, for any given independent Poisson point processes $\pi$
and $\widetilde{\pi}$ on $[0,+\infty)$, there exists a pathwise unique process
$(\bar{Z}_{t}^{A},\bar{Z}_{t}^{B})$ to (15) such that
$(\mathbb{E}(\bar{Z}_{t}^{A}),\mathbb{E}(\bar{Z}_{t}^{B}))$ is locally
bounded.

Proposition 2.7 is proven in Section 7.2.

We now turn to the propagation of chaos result: there exists a well-chosen
coupling such that the particle system (7) and the nonlinear process (15) are
close. Suppose that $H$ is locally square integrable:

$\|H\|_{2,T}<\infty,\ T>0.$ (17)

Define
$(\bar{Z}_{t}^{i})_{i=1,\ldots,N}=\left((\bar{Z}_{t}^{i})_{i=1,\ldots,N_{A}},(\bar{Z}_{t}^{i})_{i=N_{A}+1,\ldots,N}\right)$
solution to (15) driven by the same Poisson measures
$(\pi_{i})_{i=1,\ldots,N}$ as for (6). The main convergence result is the
following:

######  Proposition 2.8.

Under Assumption 2.2 and (17), for all $T>0$, there exists some constant $C>0$
(depending on $T$ and the parameters of the model) such that

$\sup_{i=1,\ldots,N}\mathbf{E}\left[\sup_{t\in[0,T]}\left|Z_{t}^{i}-\bar{Z}_{t}^{i}\right|\right]\leq\frac{C}{\sqrt{N}}.$
(18)

######  Remark 2.9.

Under the Assumptions of Proposition 2.8, assuming additionally (see (21) for
the definitions of $\kappa_{i}$, $i=1,\ldots,4$) that
$\max\\{(c_{3}+c_{4}\kappa_{1}\|\lambda^{A}\|_{\infty}+c_{5})(\kappa_{1}+\kappa_{2}),c_{6}(\kappa_{3}+\kappa_{4})\\}<1$,
we have furthermore that the constant $C$ in (18) can be chosen linear in $T$:
$C=\widetilde{C}T$, where $\widetilde{C}>0$ does not depend on $T$. This
constraint requires an explicit estimate on $\sup_{t\geq 0}\lambda_{t}^{A}$.
Although we will obtain in Section 3 that $\sup_{t\geq
0}\lambda_{t}^{A}<\infty$ under fairly general assumptions, it may be
difficult to derive an explicit upper bound for $\sup_{t\geq
0}\lambda_{t}^{A}$ in general. Nonetheless, this can easily be achieved in the
model of Example 2.5, under non optimal conditions: since $\Phi_{B\to A}\leq
1$, population $A$ is stochastically bounded by a linear Hawkes process with
memory kernel $\alpha h_{1}$, that is subcritical if
$\kappa_{1}:=\alpha\left\|h_{1}\right\|_{L^{1}}<1$, we obtain immediately the
uniform a priori bound $\sup_{t\geq
0}\lambda^{A}_{t}\leq\frac{\mu_{A}}{1-\kappa_{1}}$ (see Lemma 23 of [23]). In
that case, the quantities $c_{j},\ 1\leqslant j\leqslant 6,$ are explicit and
the latter condition simplifies into
$\max\\{(1+\kappa_{1}\frac{\mu_{A}}{1-\kappa_{1}})(\kappa_{1}+\kappa_{2}),\
(\kappa_{3}+\kappa_{4})\\}<1$.

Proposition 2.8 is proven in Section 7.3.

##  3 Long time dynamics of the mean-field limit in the inhibition model

We are now interested in the behavior as $t\to\infty$ of the macroscopic
intensities $\lambda_{t}^{A}:=\frac{{\rm d}}{{\rm d}t}m_{t}^{A}$ and
$\lambda_{t}^{B}:=\frac{{\rm d}}{{\rm d}t}m_{t}^{B}$, where
$(m_{A,t},m_{t}^{B})$ solves (16). We specify here the analysis to the model
described in Assumption 2.4: for the choice of $F$ and $G$ in (14), we see
from (16) that $(\lambda_{A},\lambda_{B})$ solves

$\begin{cases}\lambda_{t}^{A}&=\Phi_{A}\left(\alpha\int_{0}^{t}h_{1}(t-u)\lambda_{u}^{A}{\rm
d}u\right)\Phi_{B\to
A}\left((1-\alpha)\int_{0}^{t}h_{2}(t-u)\lambda_{u}^{B}{\rm d}u\right),\\\
\lambda_{t}^{B}&=\Phi_{B}\left((1-\alpha)\int_{0}^{t}h_{3}(t-u)\lambda_{u}^{B}{\rm
d}u\right)+\Phi_{A\to B}\left(\alpha\int_{0}^{t}h_{4}(t-u)\lambda_{u}^{A}{\rm
d}u\right).\end{cases}$ (19)

In the specific case of Example 2.5, (19) becomes

$\begin{cases}\lambda_{t}^{A}&=\left(\mu_{A}+\alpha\int_{0}^{t}h_{1}(t-u)\lambda_{u}^{A}{\rm
d}u\right)\Phi_{B\to
A}\left((1-\alpha)\int_{0}^{t}h_{2}(t-u)\lambda_{u}^{B}{\rm d}u\right),\\\
\lambda_{t}^{B}&=\mu_{B}+(1-\alpha)\int_{0}^{t}h_{3}(t-u)\lambda_{u}^{B}{\rm
d}u+\Phi_{A\to B}\left(\alpha\int_{0}^{t}h_{4}(t-u)\lambda_{u}^{A}{\rm
d}u\right).\end{cases}$ (20)

Here that we assume that $h_{i}\geq 0$ for $i=1,\ldots,4$. By Proposition 2.7,
there is a unique solution $(\lambda^{A},\lambda^{B})$ in
$\mathcal{C}([0,T],\left(\mathbb{R}^{+}\right)^{2})$ to the system (19):
existence is provided by $(\lambda_{t}^{A},\lambda_{t}^{B}):=(m_{t}^{\prime
A},m_{t}^{\prime B})$ where $(m^{A},m^{B})$ solves (16) and uniqueness holds
since for any solution $(\lambda^{A},\lambda^{B})$, the pairwise
$(m_{t}^{A},m_{t}^{B}):=\left(\int_{0}^{t}\lambda_{s}^{A}{\rm
d}s,\int_{0}^{t}\lambda_{s}^{B}{\rm d}s\right)$ is the unique solution to
(16).

To simplify notations below, set

$\displaystyle\kappa_{1}:=\alpha\left\|h_{1}\right\|_{1},\
\kappa_{2}:=(1-\alpha)\left\|h_{2}\right\|_{1},\
\kappa_{3}:=(1-\alpha)\left\|h_{3}\right\|_{1},\
\kappa_{4}=\alpha\left\|h_{4}\right\|_{1}.$ (21)

It appears that the longtime behavior of system (19) (and of its particular
case (20)) depends strongly on the connectivity between populations $A$ and
$B$: crucial criteria are the absence/presence of inhibition from $B$ to $A$
($\kappa_{2}=0$ or $\kappa_{2}>0$) and the absence/presence of retroaction
from $A$ to $B$ ($\kappa_{4}=0$ or $\kappa_{4}>0$). The analysis of the three
simpler cases where the system is not fully-connected (one among $\kappa_{2}$
or $\kappa_{4}$ is zero) is detailed in Section 3.2 below. Some features are
commonly observed in all three cases, but we have chosen a separate exposition
for purpose of clarity. The more complicated case with full connectivity (that
is $\kappa_{2}>0$ and $\kappa_{4}>0$), reveals richer dynamical patterns and
is analyzed separately in Section 3.3.

###  3.1 Hypotheses

In addition to Assumptions 2.1 and 2.4, we require the following:

######  Assumption 3.1.

Suppose that $\Phi_{B\to A}$ is non-increasing with $\Phi_{B\to A}(x)\to 0$ as
$x\to\infty$ and, without loss of generality, that $\Phi_{B\to A}(0)=1$.
Similarly, suppose that $\Phi_{A\to B}$ is nondecreasing with $\Phi_{A\to
B}(0)=0$ and $\Phi_{A\to B}(x)\to+\infty$ as $x\to\infty$. Finally, assume
that

$h_{i}(u)\geq 0,\ \text{ and }h_{i}(u)\xrightarrow[u\to\infty]{}0,\ u\geq 0,\
i=1,\ldots,4.$ (22)

###  3.2 Decoupled cases

We analyze in this section the long-time behavior of (19) and (20) in the
simpler cases where $\kappa_{2}=0$ or $\kappa_{4}=0$.

####  3.2.1  The fully decoupled case

Suppose that $\kappa_{2}=\kappa_{4}=0$, there is no inhibition and no
retroaction. The dynamics of (19) is somehow trivial since populations $A$ and
$B$ behave independently from one another. Nonetheless consider this case as a
control model for the other cases.

######  Proposition 3.2.

Suppose that Assumptions 2.1, 2.4 and 3.1 are satisfied with
$\kappa_{2}=\kappa_{4}=0$. If $\left\|\Phi_{A}\right\|_{L}\kappa_{1}<1$ (resp.
$\left\|\Phi_{B}\right\|_{L}\kappa_{3}<1$), then population $A$ (resp. $B$) is
subcritical: $\sup_{t\geq 0}\lambda_{t}^{A}<\infty$ (resp. $\sup_{t\geq
0}\lambda_{t}^{B}<\infty$). Moreover, in the linear case (20),

  1. 1.

if $\kappa_{1}<1$ (resp. $\kappa_{3}<1$), then population $A$ (resp. $B$) is
subcritical and
$\lambda_{t}^{A}\xrightarrow[t\to\infty]{}\frac{\mu_{A}}{1-\kappa_{1}}$ (resp.
$\lambda_{t}^{B}\xrightarrow[t\to\infty]{}\frac{\mu_{B}}{1-\kappa_{3}}$),

  2. 2.

if $\kappa_{1}>1$ and $\mu_{A}>0$ (resp. $\kappa_{3}>1$ and $\mu_{B}>0$), then
population $A$ (resp. $B$) is supercritical and
$\lambda_{t}^{A}\xrightarrow[t\to\infty]{}+\infty$ (resp.
$\lambda_{t}^{B}\xrightarrow[t\to\infty]{}+\infty$).

###### Proof of Proposition 3.2.

This case is an immediate consequence of [23], Th. 10 and 11. Note also that
this result can be retrieved easily from the estimates given in Section 8
below. ∎

####  3.2.2 The case with inhibition and no retroaction

Suppose that $\kappa_{2}>0$ and $\kappa_{4}=0$: there is inhibition from $B$
to $A$ but no retroaction from $A$ to $B$. In this case, population $B$
behaves independently from $A$.

######  Proposition 3.3.

Suppose that Assumptions 2.1, 2.4 and 3.1 are satisfied with $\kappa_{2}>0$
and $\kappa_{4}=0$. If $\left\|\Phi_{B}\right\|_{L}\kappa_{3}<1$, then
population $B$ is subcritical: $\sup_{t\geq 0}\lambda_{t}^{B}<\infty$.
Moreover, in the particular case of (20),

  1. 1.

If $\kappa_{3}<1$, then population $B$ is subcritical:
$\lambda_{t}^{B}\xrightarrow[t\to\infty]{}\frac{\mu_{B}}{1-\kappa_{3}}$.
Moreover,

    1. (a)

if $\kappa_{1}\Phi_{B\to
A}\left(\frac{\kappa_{2}\mu_{B}}{1-\kappa_{3}}\right)<1$, then population $A$
is subcritical and
$\lambda_{t}^{A}\xrightarrow[t\to\infty]{}\frac{\mu_{A}\Phi_{B\to
A}\left(\frac{\kappa_{2}\mu_{B}}{1-\kappa_{3}}\right)}{1-\kappa_{1}\Phi_{B\to
A}\left(\frac{\kappa_{2}\mu_{B}}{1-\kappa_{3}}\right)},$

    2. (b)

if $\kappa_{1}\Phi_{B\to
A}\left(\frac{\kappa_{2}\mu_{B}}{1-\kappa_{3}}\right)>1$ and $\mu_{A}>0$, then
population $A$ is supercritical:
$\lambda_{t}^{A}\xrightarrow[t\to\infty]{}+\infty$.

  2. 2.

If $\kappa_{3}>1$ and $\mu_{B}>0$: population $B$ is supercritical and
$\lambda_{t}^{B}\xrightarrow[t\to\infty]{}+\infty$. Regardless of the values
of $\mu_{A},\kappa_{1},\kappa_{2}>0$, we have that
$\lambda_{t}^{A}\xrightarrow[t\to\infty]{}0$.

Proposition 3.3 reveals a first effect of inhibition: when $\kappa_{2}=0$, (no
inhibition, Proposition 3.2) the critical parameter for $\kappa_{1}$ is
$\kappa_{1,c}^{\text{ isolated}}:=1$. With a subcritical inhibition
($\kappa_{3}<1$), the critical parameter for $\kappa_{1}$ becomes (since
$\Phi_{B\to A}\leq 1$) $\kappa_{1,c}^{\text{ inhib}}:={1}/{\Phi_{B\to
A}\left(\frac{\kappa_{2}\mu_{B}}{1-\kappa_{3}}\right)}\geq\kappa_{1,c}^{\text{
isolated}}$, the inequality being strict when $\Phi_{B\to A}$ strictly
decreases. Note that $\kappa_{1,c}^{\text{ inhib}}$ increases as either the
intrinsic activity of population $B$ increases ($\mu_{B}$ increases), or
population $B$ gets closer to criticality ($\kappa_{3}\nearrow 1$), or in
presence of a longer memory effect of population $B$ onto population $A$
($\kappa_{2}$ increases).

######  Remark 3.4.

This allows in particular for regimes where $\kappa_{1,c}^{\text{
inhib}}>\kappa_{1}>\kappa_{1,c}^{\text{ isolated}}=1$: in absence of
inhibition, population $A$ may be supercritical, but inhibition brings back
population $A$ into subcriticality. This situation is all the more spectacular
when population $B$ is supercritical ($\kappa_{3}>1$): supercritical
inhibition simply kills the dynamics of $A$. This result reflects biological
observations: epilepsy is a chronic condition characterized by recurrent
seizures. Episodes of seizures, involving abnormal synchronous firing of large
groups of neurons [37], constitute an actual example of population $A$
becoming supercritical when recurrent inhibition from population $B$ is
altered or suppressed.

Proof of Proposition 3.3 is given in Section 8.1.2.

####  3.2.3 The case with retroaction and no inhibition

Suppose $\kappa_{2}=0$ and $\kappa_{4}>0$: there is no inhibition from $B$ to
$A$ but presence of a retroaction from $A$ to $B$. In this case, population
$A$ behaves independently from $B$ and has a sole excitatory influence on $B$.

######  Proposition 3.5.

Suppose that Assumptions 2.1, 2.4 and 3.1 are satisfied with $\kappa_{2}=0$
and $\kappa_{4}>0$. If $\left\|\Phi_{A}\right\|_{L}\kappa_{1}<1$, then
population $A$ is subcritical: $\sup_{t\geq 0}\lambda_{t}^{A}<\infty$.
Moreover, in the particular case of (20),

  1. 1.

If $\kappa_{1}<1$, then population $A$ is subcritical:
$\lambda_{t}^{A}\xrightarrow[t\to\infty]{}\frac{\mu_{A}}{1-\kappa_{1}}$.
Moreover,

    1. (a)

if $\kappa_{3}<1$, then population $B$ is subcritical and
$\lambda_{t}^{B}\xrightarrow[t\to\infty]{}\frac{\mu_{B}}{1-\kappa_{3}}+\frac{1}{1-\kappa_{3}}\Phi_{A\to
B}\left(\frac{\kappa_{4}\mu_{A}}{1-\kappa_{1}}\right)$,

    2. (b)

if $\kappa_{3}>1$ and $\mu_{B}>0$, then population $B$ is supercritical:
$\lambda_{t}^{B}\xrightarrow[t\to\infty]{}+\infty$.

  2. 2.

If $\kappa_{1}>1$ and $\mu_{A}>0$: both populations $A$ and $B$ are
supercritical, $\lambda_{t}^{A}\xrightarrow[t\to\infty]{}+\infty$ and
$\lambda_{t}^{B}\xrightarrow[t\to\infty]{}+\infty$.

As anticipated, whenever $A$ is supercritical, so is population $B$ whatever
its own intrinsic behavior. Proposition 3.5 is proven in Section 8.1.3.

###  3.3 The fully-coupled case

Consider now the more interesting case with full connectivity, i.e.
$\kappa_{2}>0$ and $\kappa_{4}>0$: both inhibition from $B$ to $A$ and
retroaction from $A$ to $B$ are present. We first propose a general result for
Model (19) in Theorem 3.6 and then specialize to the linear Model (20).

####  3.3.1 A general result of subcriticality

######  Theorem 3.6.

Consider Model (19) satisfying Assumptions 2.1, 2.4 and 3.1 with
$\kappa_{2}\kappa_{4}>0$. Then, the following holds true:

  1. 1.

Whatever the values of $\kappa_{i}$, $i=1,\ldots,4$, $\lambda_{t}^{A}$ does
not tend to $+\infty$ as $t\to\infty$.

  2. 2.

If $\lambda_{t}^{B}\xrightarrow[t\to\infty]{}+\infty$, it holds that
$\lambda_{t}^{A}\xrightarrow[t\to\infty]{}0.$

We retrieve the same phenomenon described in Remark 3.4 in an even more
universal way: in the regime of full connectivity $\kappa_{2}>0$ and
$\kappa_{4}>0$, population $A$ cannot be supercritical, even for arbitrary
large values of $\kappa_{1}$ (possibly larger than $1$, when population $A$,
isolated from $B$, is supercritical). Note that Theorem 3.6 only states a weak
form of subcriticality: we only prove that
$\lim\inf_{t\to\infty}\lambda_{t}^{A}<\infty$. In the whole generality of the
hypotheses of Theorem 3.6, one cannot rule out the possibility of having
$\lim\sup_{t\to\infty}\lambda_{t}^{A}=+\infty$, although this has not been
observed numerically.

Proof of Theorem 3.6 is given in Section 8.2.1.

####  3.3.2  Specifying to the linear model

We now provide more precise estimates in the case of the linear Model (20) of
Example 2.5. The large time asymptotic of (20) depends crucially on the
behavior of the population $B$. The critical parameter here is $\kappa_{3}$:
$\kappa_{3}<1$ (resp. $\kappa_{3}>1$) means that population $B$, isolated from
population $A$, is subcritical (resp. supercritical). Let us mention briefly
the easiest second case $\kappa_{3}>1$:

######  Proposition 3.7.

Consider Model (20) satisfying Assumptions 2.1, 2.4 and 3.1 with
$\kappa_{2}>0$ and $\kappa_{4}>0$. Suppose also that $\mu_{B}>0$ and that
population $B$, in isolation, is supercritical, i.e. $\kappa_{3}>1.$ Then, the
following is true: $\lambda_{t}^{B}\xrightarrow[t\to\infty]{}+\infty$ and
$\lambda_{t}^{A}\xrightarrow[t\to\infty]{}0.$

Proposition 3.7 is a straightforward consequence of Theorem 3.6: one has
simply to check that $\lambda_{t}^{B}\xrightarrow[t\to\infty]{}+\infty$ in
this case. This follows directly from
$\lambda_{t}^{B}\geq\mu_{B}+(1-\alpha)\int_{0}^{t}h_{3}(t-u)\lambda_{u}^{B}{\rm
d}u$ (recall that $\Phi_{A\to B}\geq 0$). Details are given in Section 8.2.1
for the sake of completeness.

From now on, we restrict to the case $\kappa_{3}<1$. Note that the arguments
leading to Theorem 3.10 in the fully coupled case $\kappa_{2}\kappa_{4}>0$ can
easily be adapted to the decoupled cases $\kappa_{2}\kappa_{4}=0$ mentioned
earlier (with a considerably simpler analysis in these cases). Nonetheless, we
have chosen a separate exposition for purpose of clarity. Before stating the
main result of this paragraph, we introduce some notations. We are interested
here in a priori bounds concerning

$\begin{cases}\underline{\ell}_{A}&:=\liminf_{t\to\infty}\lambda_{t}^{A}\in[0,\infty],\hskip
28.45274pt\bar{\ell}_{A}:=\limsup_{t\to\infty}\lambda_{t}^{A}\in[0,\infty],\\\
\underline{\ell}_{B}&:=\liminf_{t\to\infty}\lambda_{t}^{B}\in[0,\infty],\hskip
28.45274pt\bar{\ell}_{B}:=\limsup_{t\to\infty}\lambda_{t}^{B}\in[0,\infty].\end{cases}$
(23)

For fixed $\kappa_{1},\kappa_{2}$, define the interval

$\mathcal{I}_{\kappa_{1},\kappa_{2}}:=\left\\{x\geq 0,\ \kappa_{1}\Phi_{B\to
A}\left(\kappa_{2}x\right)<1\right\\}.$ (24)

Under Assumption 3.1, $\mathcal{I}_{\kappa_{1},\kappa_{2}}=[0,+\infty)$ if
$\kappa_{1}<1$ and if $\kappa_{1}\geq 1$
$\mathcal{I}_{\kappa_{1},\kappa_{2}}=(x^{\ast}_{\kappa_{1},\kappa_{2}},+\infty)$
for some $x^{\ast}_{\kappa_{1},\kappa_{2}}>0$. Define the following functions

$\displaystyle\Psi_{1}$
$\displaystyle:=x\in\mathcal{I}_{\kappa_{1},\kappa_{2}}\mapsto\frac{\mu_{A}\Phi_{B\to
A}\left(\kappa_{2}x\right)}{1-\kappa_{1}\Phi_{B\to
A}\left(\kappa_{2}x\right)},$ (25) $\displaystyle\Psi_{2}$
$\displaystyle:=x\geq 0\mapsto\frac{\mu_{B}}{1-\kappa_{3}}+\frac{\Phi_{A\to
B}\left(\kappa_{4}x\right)}{1-\kappa_{3}},$ (26)

as well as

$\displaystyle\Phi:=\Psi_{2}\circ\Psi_{1}:$ $\displaystyle
x\in\mathcal{I}_{\kappa_{1},\kappa_{2}}\mapsto\frac{1}{1-\kappa_{3}}\left(\mu_{B}+\Phi_{A\to
B}\left(\frac{\kappa_{4}\mu_{A}\Phi_{B\to
A}\left(\kappa_{2}x\right)}{1-\kappa_{1}\Phi_{B\to
A}\left(\kappa_{2}x\right)}\right)\right).$ (27)

######  Lemma 3.8.

The function $\Phi$ given by (27) is continuous, nonincreasing and has a
unique fixed-point $\ell\in\mathcal{I}_{\kappa_{1},\kappa_{2}}$. This fixed-
point satisfies

$\frac{\mu_{B}}{1-\kappa_{3}}\leq\ell.$ (28)

###### Proof of Lemma 3.8.

It is a direct consequence of Assumption 3.1, since $\Psi_{1}$ is
nonincreasing and $\Psi_{2}$ is nondecreasing. If $\kappa_{1}\geq 1$,
$\Phi(x)\xrightarrow[x\to x^{\ast}_{\kappa_{1},\kappa_{2}}]{}+\infty$, as
$\lim_{+\infty}\Phi_{A\to B}=+\infty$. Since $\Phi_{A\to B}\geq 0$, the a
priori bound (28) is straightforward. ∎

Finally, define the following domain

$\displaystyle\mathcal{U}_{\Phi}$
$\displaystyle:=\left\\{(u,v)\in\left(\mathcal{I}_{\kappa_{1},\kappa_{2}}\right)^{2},\,\Phi(v)\leq
u\leq\ell\leq v\leq\Phi(u),\ \frac{\mu_{B}}{1-\kappa_{3}}\leq u\right\\}.$
(29)

The set $\mathcal{U}_{\Phi}$ is nonempty, as
$(\ell,\ell)\in\mathcal{U}_{\Phi}$, by (28).

######  Assumption 3.9.

The set $\mathcal{U}_{\Phi}$ is a singleton:
$\mathcal{U}_{\Phi}=\left\\{(\ell,\ell)\right\\}.$

Hereafter in Proposition 5.1 we provide sufficient conditions for Assumption
3.9 as well as examples in Section 5 where these are satisfied. The main
convergence result of this paragraph is the following:

######  Theorem 3.10.

Consider Model (20) satisfying Assumptions 2.1, 2.4 and 3.1 with
$\kappa_{2}>0$, $\kappa_{4}>0$, $\kappa_{3}<1$ and $\mu_{A}>0$. Then the
following is true:

  1. 1.

Population $B$ is subcritical: $\sup_{t\geq 0}\lambda_{t}^{B}<\infty$ and one
has the a priori bounds (recall the definition of
$\mathcal{I}_{\kappa_{1},\kappa_{2}}$ in (24)):

$\underline{\ell}_{B}\geq\frac{\mu_{B}}{1-\kappa_{3}}\text{ and
}\bar{\ell}_{B}\in\mathcal{I}_{\kappa_{1},\kappa_{2}}.$ (30)

  2. 2.

If $\lambda_{t}^{A}$ converges as $t\to\infty$ to $\ell_{A}$, then
$\lambda_{t}^{B}$ converges as well, to $\ell_{B}=\Psi_{2}(\ell_{A})$ where
$\Psi_{2}$ is given in (26).

  3. 3.

Suppose furthermore that Assumption 3.9 is true. Then there is an equivalence
between

    1. (a)

$\underline{\ell}_{B}$ verifies

$\underline{\ell}_{B}\in\mathcal{I}_{\kappa_{1},\kappa_{2}}$ (31)

    2. (b)

$\lambda_{t}^{B}$ as a finite limit as $t\to\infty$.

If this is true, the limit $\lim_{t\to\infty}\lambda_{t}^{B}:=\ell_{B}=\ell$
where $\ell$ is the fixed-point of $\Phi$ given by (27). Moreover,
$\sup_{t\geq 0}\lambda_{t}^{A}<\infty$ (population $A$ is subcritical) and
$\lambda_{t}^{A}\xrightarrow[t\to\infty]{}\ell_{A}:=\Psi_{1}(\ell_{B})$.

Proof of Theorem 3.10 is given in Section 8.2.2.

####  3.3.3 Remarks on the main convergence result

Theorem 3.10 comes with two hypotheses: firstly, Assumption 3.9 for which we
give sufficient conditions in Proposition 5.1 and is verified for several
examples in Section 5, and secondly, a technical a priori estimate (31) on
$\underline{\ell}_{B}$. Several remarks concerning these two hypotheses are
necessary: firstly, from
$\underline{\ell}_{B}\geq\frac{\mu_{B}}{1-\kappa_{3}}$ and the fact
$\Phi_{B\to A}$ is nonincreasing, we obtain immediately that (31) is satisfied
under the following sufficient condition:

$\kappa_{1}\Phi_{B\to
A}\left(\frac{\mu_{B}\kappa_{2}}{1-\kappa_{3}}\right)<1.$ (32)

This condition is the same subcriticality condition met in Proposition 3.3. We
retrieve the phenomenon mentioned in Remark 3.4: inhibition has a tendency to
quench supercriticality of population $A$. Numerical results however suggest
that the convergence of Theorem 3.10 may remain true (and hence so is (31))
even though (32) is violated (see Figure 2 below). Remark also that the
situation when $\underline{\ell}_{B}\notin\mathcal{I}_{\kappa_{1},\kappa_{2}}$
would imply, by (30), $\underline{\ell_{B}}<\bar{\ell}_{B}$ and thus
correspond to a case where neither $\lambda^{B}$ nor $\lambda^{A}$ converge,
but oscillate. Giving some necessary and sufficient condition for (31) (and in
particular giving some sufficient condition for
$\underline{\ell}_{B}\notin\mathcal{I}_{\kappa_{1},\kappa_{2}}$, hence proving
the existence of oscillations for (20)) would require to have some precise a
priori upper bounds on $\underline{\ell}_{B}$, that we do not have for the
moment (what we easily get is a lower bound on $\underline{\ell}_{B}$, not a
upper bound). Our conjecture is that (31) remains true as long as $\Phi_{B\to
A}$ is convex (independently of (32)), although we do not have a proof for
this statement. We give in Figure 4 a numerical example of
$\underline{\ell}_{B}\notin\mathcal{I}_{\kappa_{1},\kappa_{2}}$, in a case
when $\Phi_{B\to A}$ is not convex.

A crucial observation of the proof of Theorem 3.10 is that, under (31), we
always have

$(\underline{\ell}_{B},\bar{\ell}_{B})\in\mathcal{U}_{\Phi}.$ (33)

This fact, which does not rely on Assumption 3.9, is of independent interest,
especially in cases where we do not have convergence. In presence of
oscillations for (20) (see Section 6 for further details), (33) provides lower
and upper bounds on the size of the limit cycle. A first consequence of (33)
is that whenever $\lambda_{B}(t)$ converges as $t\to\infty$,
$\underline{\ell}_{B}=\bar{\ell}_{B}$ so that (recall the definition (29) of
$\mathcal{U}_{\Phi}$) necessarily $\underline{\ell}_{B}=\bar{\ell}_{B}=\ell$.
This, together with the previous remark that
$\underline{\ell}_{B}\notin\mathcal{I}_{\kappa_{1},\kappa_{2}}$ would lead to
oscillations, tells us that Theorem 3.10 gives a complete picture of the
possible limit for $(\lambda_{t}^{A},\lambda_{t}^{B})$: if one among
$(\lambda_{t}^{A},\lambda_{t}^{B})$ converges as $t\to\infty$, both do, and
their limit are necessarily given by $\ell_{A}=\Psi_{1}(\ell_{B})$ and
$\ell_{B}=\ell$, unique fixed-point of $\Phi$ in (27).

Furthermore, we understand from (33) the point of Assumption 3.9: it gives
immediately the convergence of $\lambda^{B}$. In this respect, one may find
that Assumption 3.9 is quite strong: but it is trivially verified in the
uncoupled cases $\kappa_{2}\kappa_{4}=0$, as the function $\Phi$ is then
constant, and it is satisfied for several nontrivial examples in the fully
coupled case (see Section 5 below). However, we have numerical evidence (see
Figure 3) that Assumption 3.9 is in fact not necessary for the convergence of
$(\lambda^{A},\lambda^{B})$. In this context, finding a general necessary and
sufficient condition for convergence of $(\lambda^{A},\lambda^{B})$ seems to
be a tricky question.

######  Remark 3.11.

From Theorem 3.6: under hypotheses of full connectivity, population $A$ cannot
be supercritical (in the sense of
$\liminf_{t\to\infty}\lambda_{t}^{A}<\infty$), regardless the parameters of
the model. Theorem 3.10 provides a stronger version of this statement:
$\sup_{t\geq 0}\lambda_{t}^{A}<\infty$, although we do not provide an explicit
control on $\sup_{t\geq 0}\lambda_{t}^{A}$ (explicit estimates are only on
$\limsup_{t\to\infty}\lambda^{A}_{t}$). Recall however what we have said in
Remark 2.9: using $\Phi_{B\to A}\leq 1$, population $A$ is bounded by a linear
Hawkes process with memory kernel $\alpha h_{1}$. Therefore, in the particular
case of $\kappa_{1}<1$ (which is in particular stricter than (32)), we have
directly that $\sup_{t\geq 0}\lambda^{A}_{t}\leq\frac{\mu_{A}}{1-\kappa_{1}}$
(see Lemma 23 of [23]). Deriving similarly an explicit bound in the general
case (for example under (32) only) is unclear.

##  4 Fluctuations results and a test of inhibition

In this section, we are interested in fluctuation results corresponding to the
convergence given in Proposition 2.8 and the derivation of corresponding
statistical tests for inhibition.

###  4.1 Hierarchy of intensities under inhibition and/or retroaction

We restrict to the framework of Model (20) satisfying Assumptions 2.1, 2.4 and
3.1. Suppose also that Assumption 3.9 as well as (32) are satisfied.

The previous analysis shows that under these hypotheses convergence of
$\lambda_{t}^{A}$ as $t\to\infty$ holds in all cases (i.e. both when
$\kappa_{2}\kappa_{4}=0$ and when $\kappa_{2}\kappa_{4}>0$). Recall that
Assumption 3.9 holds trivially in the decoupled cases $\kappa_{2}\kappa_{4}=0$
since $\Phi$ defined in (27) is then constant (and in such cases, the
conditions $\kappa_{3}<1$ and (32) for subcriticality are sharp).

The first main observation of this paragraph is that there is a natural
ordering between the different limits $\lim_{t\to\infty}\lambda_{t}^{A}$,
depending on the presence or absence of inhibition and retroaction: when $A$
is isolated ($\kappa_{2}=0$), we obtain from Proposition 3.2 that

$\lambda_{t}^{A}\xrightarrow[t\to\infty]{}\frac{\mu_{A}}{1-\kappa_{1}}:=\ell_{A}^{\text{(I)}}.$

When inhibition from $B\to A$ is added, without feedback from $A\to B$
($\kappa_{2}>0,\kappa_{4}=0$), the conclusion of Proposition 3.3 is

$\lambda_{t}^{A}\xrightarrow[t\to\infty]{}\frac{\mu_{A}\Phi_{B\to
A}\left(\frac{\kappa_{2}\mu_{B}}{1-\kappa_{3}}\right)}{1-\kappa_{1}\Phi_{B\to
A}\left(\frac{\kappa_{2}\mu_{B}}{1-\kappa_{3}}\right)}:=\ell_{A}^{\text{(II)}}.$

Finally, in the fully coupled case $\kappa_{2}\kappa_{4}>0$, one obtains from
Theorem 3.10 that

$\lambda_{t}^{A}\xrightarrow[t\to\infty]{}\frac{\mu_{A}\Phi_{B\to
A}\left(\kappa_{2}\ell\right)}{1-\kappa_{1}\Phi_{B\to
A}\left(\kappa_{2}\ell\right)}:=\ell_{A}^{\text{(III)}},$

for $\ell$ the unique fixed-point to the function $\Phi$ given in (27). Using
that $\Phi_{B\to A}$ is non increasing and bounded by $1$ together with (28),
for $\kappa_{4}>0$, the following natural order between the previous limits
holds true

$\ell_{A}^{\text{(I)}}>\ell_{A}^{\text{(II)}}>\ell_{A}^{\text{(III)}}.$ (34)

Note that one has
$\ell_{A}^{\text{(III)}}=\ell_{A}^{\text{(III)}}(\kappa_{1},\kappa_{2},\kappa_{3},\kappa_{4})\xrightarrow[\kappa_{4}\to
0]{}\ell_{A}^{\text{(II)}}$.

###  4.2 Central limit results

We take advantage of the strict ordering observed in (34) in order to
experimentally test the presence/absence of inhibition in a neuronal network
modeled by (20). For (34) to be useful, the convergence results seen before
must be accompanied with fluctuation results. Fluctuation theorems for Hawkes
processes (both in large population $N\to\infty$ and in large time
$t\to\infty$) have already been considered in the literature [23, 25] and some
of the existing results directly apply to our case: for instance, in the
setting of Example 2.5 with $\kappa_{1}<1$, $\kappa_{3}<1$ and $\kappa_{2}=0$
(resp. $\kappa_{4}=0$), Theorem 10 of [23] provides Gaussian limits for the
quantities
$\left(\sqrt{m_{t}^{A}}\left(\frac{Z^{i,A}}{m_{t}^{A}}-1\right)\right)_{i\in
I^{A}}$ (resp.
$\big{(}\sqrt{m_{t}^{B}}(\frac{Z^{i,B}}{m_{t}^{B}}-1)\big{)}_{i\in I^{B}}$),
when $(t,N)\to(\infty,\infty)$ and where $I^{A}$ (resp. $I^{B}$) is a subset
of $\\{1,\ldots,N_{A}\\}$ (resp. $\\{N_{A}+1\ldots,N\\}$) and $(m^{A},m^{B})$
is the solution of (16). Note nonetheless that the proof in [23] appears to be
specific to the case of linear Hawkes processes so that its direct application
to our model with full connectivity is doubtful.

An existing result more useful to our concern may be found in [25], Th. 2,
where a central limit theorem for $K$ macroscopic populations of interacting
nonlinear Hawkes processes is proven. In particular, a closer look to the
proof of Theorem 2 of [25] permits to realize that it does not rely on the
specific shape of the intensities of the underlying Hawkes processes but only
on the existence of a propagation of chaos estimates similar to Proposition
2.8. This result can easily transpose to our setting:

######  Proposition 4.1 (Ditlevsen, Löcherbach, [25], Th. 2).

Consider Model (20) satisfying Assumptions 2.1, 2.4, 3.1 and 3.9, either in
the uncoupled cases $\kappa_{2}\kappa_{4}=0$ or with full coupling
$\kappa_{2}\kappa_{4}>0$. Require also that

$\max\\{(1+\kappa_{1}\frac{\mu_{A}}{1-\kappa_{1}})(\kappa_{1}+\kappa_{2}),\
(\kappa_{3}+\kappa_{4})\\}<1.$ (35)

Then, for any subsets $I^{A}_{k_{A}}\subset\\{1,\ldots,N_{A}\\}$ and
$I^{B}_{k_{B}}\subset\\{N_{A}+1,\ldots,N\\}$ of fixed cardinality $k_{A}$ and
$k_{B}$ (independent of $N$)

$\left(\sqrt{m_{t}^{A}}\left(\frac{Z_{t}^{i,A}}{m_{t}^{A}}-1\right)\ ,\
\sqrt{m_{t}^{B}}\left(\frac{Z_{t}^{j,B}}{m_{t}^{B}}-1\right)\right)_{i\in
I^{A}_{k_{A}},\ j\in
I^{B}_{k_{B}}}\xrightarrow[\overset{(t,N)\to(\infty,\infty),}{\frac{t}{N}\to
0}]{d}\mathcal{N}(0,I_{k_{A}+k_{B}}).$ (36)

###### Sketch of proof.

This is a straightforward consequence of [25], Th 2: it suffices to note that
in all cases, $m^{A}_{t}/t$ and $m^{B}_{t}/t$ are bounded away from 0 and
$\infty$ (recall Propositions 3.2 and 3.3 and Theorem 3.10). Condition (35)
ensures that the constant appearing in Proposition 2.8 does not increase
faster than $Ct$, for a positive constant $C$ (see Remark 2.9). Putting
everything together one has simply to follow the lines of [25], Th 2 to
conclude. ∎

####  4.2.1 Discussion on the statistical significance of the CLT

Here we discuss the practical significance of (36). Denote by $\ell$ the limit
intensity of either population $A$ or $B$ appearing in the convergence results
of Propositions 3.2, 3.3 and Theorem 3.10. An estimator $\widehat{\ell}_{N,T}$
for this quantity may be given through the following approximations:

$\ell\underset{T\to\infty}{\approx}{\lambda_{T}}\underset{{N\to\infty}}{\approx}\frac{Z_{t}^{i}}{T}:=\widehat{\ell}_{N,T},\quad
1\leqslant i\leqslant N.$

Therefore, $\widehat{\ell}_{N,T}$ can be computed from the observation of one
neuron (recall that in the mean field limit, neurons within the same
population are interchangeable) in a $N$-particle system of Hawkes processes
observed in a large time $T$. From a statistical viewpoint, a statement as in
(36) cannot be exploited directly as the unknown parameter $\ell$ does not
explicitly appear and secondly the normalization of $Z^{i}_{t}$ by $m_{t}$ is
unknown. Replacing $\frac{Z^{i}_{t}}{m_{t}}$ by
$\frac{Z^{i}_{t}}{\widehat{m_{t}}}$ where $\widehat{m}_{t}=Z^{i^{\prime}}_{t}$
for $i\neq i^{\prime}$ for instance or
$\widehat{m}_{t}=2(Z^{i}_{t}-Z^{i}_{\frac{t}{2}})$ is not satisfactory as
information on $\ell$ will not emerge. Instead, what is required here (for
instance to derive testing procedures and confidence intervals) is a result of
the form
$\sqrt{T}\left(\widehat{\ell}_{N,T}-\ell\right)\xrightarrow[(T,N)\to(\infty,\infty)]{d}K$
for some distribution $K$.

The simple case of an isolated subcritical population $A$ with linear
intensity (Model (20) and Proposition 3.2, with $\kappa_{2}=0$) is of
particular interest in order to build the test considered in the next
paragraph. In this case, $\ell=\frac{\mu_{A}}{1-\kappa_{1}}$, for
$\kappa_{1}=\alpha\|h_{1}\|_{1}<1$ and the following decomposition holds

$\displaystyle\sqrt{T}\left(\widehat{\ell}_{N,T}-\ell\right)$
$\displaystyle=\sqrt{\frac{m_{T}}{T}}\sqrt{m_{T}}\left(\frac{Z^{i}_{T}}{m_{T}}-1\right)+\sqrt{T}\left(\frac{m_{T}}{T}-\frac{\mu_{A}}{1-\kappa_{1}}\right):=\sqrt{\frac{m_{T}}{T}}I_{N,T}(1)+I_{T}(2).$

From (36) (see also Theorem 10 of [23]), it immediately follows that
$I_{N,T}(1)\xrightarrow[]{d}\mathcal{N}(0,1)$ as $(T,N)\to(\infty,\infty)$,
$\frac{T}{N}\to 0$. Controlling the deterministic term $I_{2}(T)$ requires to
evaluate the rate at which $m_{T}/T$ converges to its limit
$\mu_{A}/(1-\kappa_{1}).$ This has been studied, in the linear case, in Lemma
5 of [2] which implies that $I_{T}(2)=o(1)$ if
$\int_{0}^{\infty}\sqrt{t}h(t){\rm d}t<\infty$. Then, assuming
$\int_{0}^{\infty}\sqrt{t}h(t){\rm d}t<\infty$ and using that Proposition 3.2
implies that $\frac{m_{T}}{T}\xrightarrow[T\to\infty]{}\ell$, we derive

$\displaystyle\sqrt{T}\left(\widehat{\ell}_{N,T}-\ell\right)\xrightarrow[{(T,N)\to(\infty,\infty)}\atop{\frac{T}{N}\to
0}]{d}\mathcal{N}\left(0,\ell\right)\quad\mbox{and}\quad\sqrt{\frac{T}{\widehat{\ell}_{N,T}}}\left(\widehat{\ell}_{N,T}-\ell\right)\xrightarrow[(T,N)\to(\infty,\infty)\atop{\frac{T}{N}\to
0}]{d}\mathcal{N}\left(0,1\right).$ (37)

####  4.2.2 A test of inhibition

Neurophysiologists studying synaptic function routinely isolate one type of
synaptic coupling with specific compounds (very often toxins synthesized by
plants, bacteria, spiders, etc. [37]). Using picrotoxin or bicuculline (two
plant toxins) they can for instance suppress inhibition, that is, force our
kernel $h_{2}$ to become uniformly zero. Based on (37), we can then propose
the following procedure in order to test if inhibition on a population $A$ is
present. In a experimental preparation, neurons from the $A$ population can be
continuously recorded in three successive conditions: control (no toxin
applied); inhibition blocked (with picrotoxin or bicuculline); ’wash’ (the
physiologists’ term to designate a return to the control condition after
removal of a toxin). Collecting for a duration $T$ the activities of the
neurons of population $A$ and estimate their intensities with
$\widehat{\ell}^{Control}_{N,T}$, $\widehat{\ell}^{Toxin}_{N,T}$ and
$\widehat{\ell}^{Wash}_{N,T}$ respectively.

Denote by $(\ell^{{Control}},\ell^{Toxin})$ the _true_ intensities of
population $A$ with and without inhibition blocked. If population $B$ has no
effect onto population $A$, one should have $\ell^{Control}=\ell^{Toxin}$,
otherwise following (34) one should observe $\ell^{Control}>\ell^{Toxin}$.
This suggests to consider the following testing procedure:

$\displaystyle\begin{cases}H_{0}\ :\ \mbox{Population $B$ has no inhibitive
effect on population $A$},\\\ H_{1}\ :\ \mbox{Population $B$ has an inhibitive
effect on population $A$},\end{cases}$

or equivalently

$\displaystyle\begin{cases}H_{0}\ :\ \ell^{Control}=\ell^{Toxin},\\\ H_{1}\ :\
\ell^{Control}>\ell^{Toxin}.\end{cases}$ (38)

Consider the test statistics
$R_{N,T}=\widehat{\ell}^{Control}_{N,T}-\widehat{\ell}^{Toxin}_{N,T},$ where
$\widehat{\ell}^{Control}_{N,T}$ and $\widehat{\ell}^{Toxin}_{N,T}$ are
independent. Using (37) and that under $H_{0}$ it holds
$\ell^{Control}=\ell^{Toxin}$, we write:

$\displaystyle\sqrt{T}(\widehat{\ell}^{Control}_{N,T}-\widehat{\ell}^{Toxin}_{N,T})$
$\displaystyle=\sqrt{T}(\widehat{\ell}^{Control}_{N,T}-\ell^{Control})-\sqrt{T}(\widehat{\ell}^{Toxin}_{N,T}-\ell^{Toxin})\xrightarrow[(T,N)\to(\infty,\infty)\atop{\frac{T}{N}\to
0}]{d,\ H_{0}}\mathcal{N}(0,2\ell),$

from which we derive

$\displaystyle\sqrt{\frac{T}{\widehat{\ell}^{Control}_{N,T}+\widehat{\ell}^{Toxin}_{N,T}}}(\widehat{\ell}^{Control}_{N,T}-\widehat{\ell}^{Toxin}_{N,T})$
$\displaystyle\xrightarrow[(T,N)\to(\infty,\infty)\atop{\frac{T}{N}\to 0}]{d,\
H_{0}}\mathcal{N}(0,1).$

Therefore, we get a test for the set of assumptions (38) with asymptotic level
$a\in(0,1)$

$\Psi^{(a)}_{N,T}=\mathbf{1}_{\left\\{\widehat{\ell}^{Control}_{N,T}-\widehat{\ell}^{Toxin}_{N,T}\geqslant
s_{a}\right\\}}\quad\mbox{where}\quad
s_{a}=\sqrt{\frac{\widehat{\ell}^{Control}_{N,T}+\widehat{\ell}^{Toxin}_{N,T}}{T}}q^{1-a}_{\mathcal{N}(0,1)},$

where $q^{r}_{\mathcal{N}(0,1)}$ the quantile of order $r$ of the distribution
$\mathcal{N}(0,1)$. Note that under $H_{1}$ the same decomposition holds if
one adds the additional diverging term $\sqrt{T}(\ell^{Control}-\ell^{Toxin})$
that ensures the consistency of the test $\Psi^{(a)}_{N,T}$.

##  5 Examples in the fully-coupled subcritical linear case

We place ourselves here in the context of Theorem 3.10 concerning Model (20).
Recall in particular that we assume full connectivity $\kappa_{2}\kappa_{4}>0$
with population $B$ subcritical: $\kappa_{3}<1$. The aim of this paragraph is
twofold: first, to give general sufficient conditions for Assumption 3.9 used
in Theorem 3.10 and second, to give instances of kernels $\Phi_{A\to B}$ and
$\Phi_{B\to A}$ that fulfil the hypotheses of Theorem 3.10 (and this for which
the convergence result described in Theorem 3.10 is valid). Focus will be made
on particular choices of inhibition kernels $\Phi_{B\to A}$: in all the
examples below, we have simply chosen $\Phi_{A\to B}(x)=x$. To simplify
notations in the sequel, define the following parameters

$a:=\frac{\mu_{B}\kappa_{2}}{1-\kappa_{3}}\quad\mbox{and}\quad
b:=\frac{\kappa_{2}\kappa_{4}}{1-\kappa_{3}}.$ (39)

Informally, the parameter $a$ measures the influence of population $B$ onto
population $A$ whereas $b$ assesses the impact of the coupling between both
populations.

###  5.1 Verifying Assumption 3.9

First, we provide general sufficient conditions for Assumption 3.9. Recall the
definitions of $\Phi$ and $\mathcal{U}_{\Phi}$ in (27) and (29).

######  Proposition 5.1.

Suppose that Assumption 3.1 is satisfied together with $\kappa_{3}<1$. Then,

  1. 1.

General sufficient conditions: Assumption 3.9 is true under one of the
following conditions:

    1. (a)

$\Phi$ is a contraction on
$\mathcal{I}_{\kappa_{1},\kappa_{2}}\cap\left[\frac{\mu_{B}}{1-\kappa_{3}},+\infty\right)$:

$\left|\Phi(x)-\Phi(y)\right|<\left|x-y\right|,\quad\quad x\neq y.$ (40)

    2. (b)

$\Phi^{2}:=\Phi\circ\Phi$ has a unique fixed-point in
$\mathcal{I}_{\kappa_{1},\kappa_{2}}\cap\left[\frac{\mu_{B}}{1-\kappa_{3}},+\infty\right)$.

  2. 2.

Suppose now that $\Phi$ is strictly decreasing (that is the case when e.g.
$\Phi_{A\to B}$ and $\Phi_{B\to A}$ are strictly increasing and decreasing
respectively). Then Assumption 3.9 is satisfied if and only if

$\displaystyle u<\Phi\circ\Phi(u),\text{ on
}\left(\max\left(x_{\kappa_{1},\kappa_{2}}^{\ast},\frac{\mu_{B}}{1-\kappa_{3}}\right),\ell\right).$
(41)

######  Remark 5.2.

Condition (41) is true in particular if $\Phi\circ\Phi$ is strictly concave on
$\left(\max\left(x_{\kappa_{1},\kappa_{2}}^{\ast},\frac{\mu_{B}}{1-\kappa_{3}}\right),\ell\right)$
or if $\Phi$ is differentiable in $\ell$,
$\left|\Phi^{\prime}(\ell)\right|\leq 1$ and $\Phi^{2}$ is strictly convex on
$\left(\max\left(x_{\kappa_{1},\kappa_{2}}^{\ast},\frac{\mu_{B}}{1-\kappa_{3}}\right),\ell\right)$.

###### Proof of Proposition 5.1.

The case 1a. is a straightforward consequence of the definition of
$\mathcal{U}_{\Phi}$ in (29) together with (40). Turn now to case 1b.: suppose
that $\Phi$ is such that $\Phi^{2}$ has a unique fixed-point, which is
necessarily $\ell$. From the definition (29) of $\mathcal{U}_{\Phi}$ and the
fact that $\Phi$ is nonincreasing it follows that, for any
$(u,v)\in\mathcal{U}_{\Phi}$,

$\Phi^{2}(u)\leq\Phi\left(v\right)\leq u\leq\ell\leq
v\leq\Phi\left(u\right)\leq\Phi^{2}(v).$

An immediate recursion from the latter inequality shows that
$\left(\Phi^{2n}(u)\right)_{n\geq 0}$ is (positive) non-increasing and hence
converges to a fixed-point of $\Phi^{2}$, which by uniqueness is $\ell$. Thus,
$\ell\leq\Phi^{2n}(u)\leq u\leq\ell$, which gives in particular $u=\ell$ and
since $\ell\leq v\leq\Phi(u)=\Phi\left(\ell\right)=\ell$, this gives the
result. Turn now to the proof when $\Phi$ strictly decreases: the domain
$\mathcal{U}_{\Phi}$ boils down to

$\displaystyle\mathcal{U}_{\Phi}$
$\displaystyle:=\left\\{\max\left(x_{\kappa_{1},\kappa_{2}}^{\ast},\frac{\mu_{B}}{1-\kappa_{3}}\right)\leq
u\leq\ell,\ \Phi^{-1}(u)\leq v\leq\Phi(u)\right\\}.$ (42)

So that Assumption 3.9 is satisfied if and only if $\Phi^{-1}(u)>\Phi(u)$ on
$\left(\max\left(x_{\kappa_{1},\kappa_{2}}^{\ast},\frac{\mu_{B}}{1-\kappa_{3}}\right),\ell\right)$,
which gives the result. ∎

###  5.2 Polynomial inhibition

Let $\beta,\tau>0$ and consider in this paragraph

$\Phi_{B\to A}(x)=\frac{1}{1+\tau x^{\beta}}\quad\mbox{and}\quad\Phi_{A\to
B}(x)=x,\quad x\geq 0.$ (43)

Thenn the function $\Phi$ writes

$\Phi(x)=\frac{a}{\kappa_{2}}+\frac{\mu_{A}b}{\kappa_{2}\left(1-\kappa_{1}+\tau\left(\kappa_{2}x\right)^{\beta}\right)},\quad
x\in\mathcal{I}_{\kappa_{1},\kappa_{2}},$ (44)

where the domain $\mathcal{I}_{\kappa_{1},\kappa_{2}}$ is given by

$\mathcal{I}_{\kappa_{1},\kappa_{2}}=\begin{cases}[0,+\infty)&\text{ if
}\kappa_{1}<1,\\\
\left(x_{\kappa_{1},\kappa_{2}}^{\ast},+\infty\right)=\left(\frac{1}{\kappa_{2}}\left(\frac{\kappa_{1}-1}{\tau}\right)^{\frac{1}{\beta}},+\infty\right)&\text{
if }\kappa_{1}\geq 1.\end{cases}$

######  Proposition 5.3.

Consider Model (20) with $\Phi_{B\to A}$ and $\Phi_{A\to B}$ given by (43),
satisfying $\kappa_{3}<1$, $\kappa_{2}\kappa_{4}>0$, $\mu_{A}>0$,
$\kappa_{1}<1+\tau a^{\beta}$ and Assumptions 2.1 and (22). Then, in the
following cases:

  1. 1.

when $\beta\in(0,1]$,

  2. 2.

when $\beta>1$:

    1. (a)

if $\kappa_{1}<1$, at least when $\tau$ is sufficiently small (weak
inhibition) or when $\tau$ is sufficiently large (strong inhibition).

    2. (b)

if $\kappa_{1}\geq 1$, at least when $\tau$ is sufficiently large (strong
inhibition),

the conclusions of Theorem 3.10 are valid, i.e.
$(\lambda^{A}_{t},\lambda^{B}_{t})\xrightarrow[t\to\infty]{}\left(\ell_{A},\ell_{B}\right):=\left(\frac{\mu_{A}}{1-\kappa_{1}+\tau\left(\kappa_{2}\ell\right)^{\beta}},\ell\right)$,
where $\ell$ is the unique fixed-point of $\Phi$ given by (44) on
$\left[\frac{\mu_{B}}{1-\kappa_{3}},+\infty\right)$. In the case $\beta=1$,
$\ell_{B}=\ell$ can be explicitly computed as

$\ell=\frac{-(1-\kappa_{1}-\tau a)+\left(\left(1-\kappa_{1}-\tau
a\right)^{2}+4\tau\left((1-\kappa_{1})a+\mu_{A}b\right)\right)^{\frac{1}{2}}}{2\tau\kappa_{2}}.$
(45)

######  Remark 5.4.

Note that $\beta\in[0,1]$ corresponds to the case where $\Phi_{B\to A}$ (and
hence $\Phi$ itself) is convex. We stress the fact that, from a numerical
point of view, convergence is actually valid beyond the hypotheses of
Proposition 5.3, especially in cases when the hypothesis (32)
$\kappa_{1}<1+\tau a^{\beta}$ is no longer valid, see Figure 2.

The case $\beta>1$ seems to reveal richer dynamical patterns, especially in
the regime where $\beta\gg 1$, where (43) is of sigmoid type and approaches
(for fixed $\tau$) the indicator function $\mathbf{1}_{[0,1]}$ as
$\beta\to\infty$, see Section 6 for further details.

###### Proof of Proposition 5.3.

Note that the subcriticality condition (32) writes here $\kappa_{1}<1+\tau
a^{\beta}$, which is equivalent to $\frac{\mu_{B}}{1-\kappa_{3}}\geq
x^{\ast}_{\kappa_{1},\kappa_{2}}$ and hence (31) is true. Suppose first
$\beta\in(0,1]$ and define $\rho(x):=1-\kappa_{1}+\tau x^{\beta}$. We show
that $\Phi\circ\Phi$ is concave, from which we derive by Proposition 5.1 point
2 that Assumption 3.9 is valid (see Remark 5.2). Showing that $\Phi\circ\Phi$
is concave is equivalent to showing that $(\Phi\circ\Phi)^{\prime}$ is
decreasing. Simple computations show that

$(\Phi\circ\Phi)^{\prime}(x)=\mu^{2}\frac{\rho^{\prime}(\kappa_{2}x)}{\rho^{2}(\kappa_{2}x)}\frac{\rho^{\prime}\left(a+\frac{\mu_{A}b}{\rho(\kappa_{2}x)}\right)}{\rho^{2}\left(a+\frac{\mu_{A}b}{\rho(\kappa_{2}x)}\right)},$

therefore $(\Phi\circ\Phi)^{\prime}$ has the same monotonicity as $F(y)G(y),\
y\geqslant a,$ where $F(y):=\frac{\rho^{\prime}(y)}{\rho^{2}(y)}$ is positive
and
$G(y):=\frac{\rho^{\prime}\left(a+\frac{\mu}{\rho(y)}\right)}{\rho^{2}\left(a+\frac{\mu}{\rho(y)}\right)}$
is positive, where we set $\mu=\mu_{A}b$. A sufficient condition for $H$ to be
decreasing is that $\frac{F^{\prime}}{F}+\frac{G^{\prime}}{G}<0$. First,
tedious computations enable to write for $y\geqslant a$,

$\displaystyle H(y):$
$\displaystyle=y\rho(y)\left(\frac{F^{\prime}(y)}{F(y)}+\frac{G^{\prime}(y)}{G(y)}\right),$
$\displaystyle=(\beta-1)\rho(y)-\tau\beta
y^{\beta}\left(2+\frac{\mu}{a\rho(y)+\mu}(\beta-1)-2\beta\frac{\mu}{a\rho(y)+\mu}\frac{\tau\left(a+\frac{\mu}{\rho(y)}\right)^{\beta}}{\rho\left(a+\frac{\mu}{\rho(y)}\right)}\right),$
$\displaystyle=T_{1}(y)+T_{2}(y)$

where

$\displaystyle T_{1}(y):$
$\displaystyle=-(1-\beta)\left(\rho(y)+\frac{\beta\mu\tau
y^{\beta}}{a\rho(y)+1}\right),$ $\displaystyle T_{2}(y):$
$\displaystyle=-2\tau\beta
y^{\beta}\left(1+\frac{\mu}{a\rho(y)+\mu}(\beta-1)-\beta\frac{\mu}{a\rho(y)+\mu}\frac{\tau\left(a+\frac{\mu}{\rho(y)}\right)^{\beta}}{\rho\left(a+\frac{\mu}{\rho(y)}\right)}\right).$

As $\beta\leqslant 1$, we immediately have that $T_{1}\leqslant 0.$ To control
the second term $T_{2}$, we write it as follows, using the definition of
$\rho$,

$\displaystyle T_{2}(y)$ $\displaystyle=-2\tau\beta
y^{\beta}\frac{\mu}{a\rho(y)+\mu}\left(1+\frac{a\rho(y)}{\mu}+(\beta-1)-\beta\frac{\rho\left(a+\frac{\mu}{\rho(y)}\right)+\kappa_{1}-1}{\rho\left(a+\frac{\mu}{\rho(y)}\right)}\right),$
$\displaystyle=-2\tau\beta
y^{\beta}\frac{\mu}{a\rho(y)+\mu}\left(\frac{a\rho(y)}{\mu}+\beta\frac{1-\kappa_{1}}{\rho\left(a+\frac{\mu}{\rho(y)}\right)}\right),$
$\displaystyle=-2\tau\beta
y^{\beta}\frac{\mu}{a\rho(y)+\mu}\frac{1}{\rho\left(a+\frac{\mu}{\rho(y)}\right)}\left(\frac{a\rho(y)}{\mu}{\rho\left(a+\frac{\mu}{\rho(y)}\right)}+\beta(1-{\kappa_{1}})\right).$

Note that $T_{2}\leqslant 0$ holds true whenever $\kappa_{1}\leqslant 1$ but
requires an additional justification if $\kappa_{1}\in]1,1+\tau a^{\beta}[$.
For that, we study the function
$T_{3}(y):=\frac{a\rho(y)}{\mu}\rho\left(a+\frac{\mu}{\rho(y)}\right)+\beta(1-\kappa_{1}),\
y\geqslant a$, which, as $\rho$ is increasing and $\mu\geqslant 0$, has the
same monotonicity as the function $g:z\mapsto
z\rho\left(a+\frac{1}{z}\right)$, $z>0$. Using that $\kappa_{1}<1+\tau
a^{\beta}$ and $\beta\leqslant 1$, it holds that

$\displaystyle g^{\prime}(z)$
$\displaystyle=\frac{1}{z^{\beta}}\left(z^{\beta}(1-\kappa_{1})+\tau(1+az)^{\beta-1}(1-\beta+az)\right)\geqslant\frac{\tau}{z^{\beta}}\left(-(az)^{\beta}+(1+az)^{\beta-1}(1-\beta+az)\right)\geqslant
0.$

It follows that $g$ and $T_{3}$ are increasing and that $T_{3}$ attains its
minimum at $y=a$. We conclude using that for all $y\geqslant a$ and
$\kappa_{1}<1+\tau a^{\beta}$

$\displaystyle T_{3}(y)\geqslant T_{3}(a)$
$\displaystyle=\frac{a\rho(a)}{\mu}\rho\left(a+\frac{\mu}{\rho(a)}\right)+\beta(1-\kappa_{1})\geqslant\frac{a\rho(a)}{\mu}\left(-\tau
a^{\beta}+\tau\left(a+\frac{\mu}{\rho(a)}\right)^{\beta}\right)-\beta\tau
a^{\beta}$
$\displaystyle=\tau\frac{a\rho(a)^{1-\beta}}{\mu}\left((a\rho(a)+\mu)^{\beta}-({a\rho(a)})^{\beta}-\beta({a\rho(a)})^{\beta-1}\right)\geqslant
0$

as $z\mapsto(z+\mu)^{\beta}-z^{\beta}-\beta z^{\beta-1}$ is increasing (as
$\beta\leqslant 1$ and $\mu\geqslant 0$) and is positive if $z=0$. Gathering
everything together, we conclude that $T_{2}$ is negative, therefore $H$ is
negative and $\Phi\circ\Phi$ is concave. Assumption 3.9 is fulfilled and the
conclusions of Theorem 3.10 are valid.

Turn now to the case $\beta>1$: we have
$\Phi^{\prime}(x)=-\mu_{A}b\frac{\tau\beta(\kappa_{2}x)^{\beta-1}}{\left(1-\kappa_{1}+\tau(\kappa_{2}x)^{\beta}\right)^{2}}$,
so that for $x\in\left[\frac{\mu_{B}}{1-\kappa_{3}},\ell\right)$,
$\left|\Phi^{\prime}(x)\right|\leq\frac{\mu_{A}b\beta\left(\kappa_{2}\ell\right)^{\beta-1}\tau}{\left(1-\kappa_{1}+\tau
a^{\beta}\right)^{2}}$. Suppose first that $\kappa_{1}<1$. Note that since
$\ell\geq\frac{\mu_{B}}{1-\kappa_{3}}$, one obtains from (44) the following a
priori bound on $\ell$:
$\ell\leq\frac{a}{\kappa_{2}}+\frac{\mu_{A}b}{\kappa_{2}\left(1-\kappa_{1}+\tau
a^{\beta}\right)}$, quantity that remains bounded as $\tau\to 0$ and
$\tau\to\infty$. Putting everything together, this yields that
$\sup_{x\in\left[\frac{\mu_{B}}{1-\kappa_{3}},\ell\right)}\left|\Phi^{\prime}(x)\right|$
can be made strictly smaller than $1$ provided $\tau\to 0$ or $\tau\to\infty$.
In these regimes, $\Phi$ is a contraction and Assumption 3.9 holds, by
Proposition 5.1, which gives the result. The same argument works in the case
$\kappa_{1}\geq 1$, considering now only the limit of strong inhibition
$\tau\to\infty$. Computing the fixed-point $\ell$ in the case $\beta=1$ is
basic analysis from (44). ∎

Figure 1: Trajectories of the mean-field $(\lambda^{A},\lambda^{B})$ and their
microscopic counterparts $(\lambda_{N}^{A},\lambda_{N}^{B})$, when
$h_{i}=\mathbf{1}_{[0,\theta_{i}]}$ for the Model (43) for $\alpha=0.8$,
$\beta=1$, $\kappa_{1}=1.5$, $\kappa_{2}=0.5$, $\kappa_{3}=0.5$,
$\kappa_{4}=1$, $\mu_{A}=10$, $\mu_{B}=1$, $\tau=1$, on $[0,T]$ with $T=50$:
the conditions of Proposition 5.3 are satisfied (note in particular that
$1<\kappa_{1}<1+\tau a=2$). The limit of $(\lambda^{A},\lambda^{B})$ match
their theoretical counterparts (recall (45)) $\ell_{B}\approx 7.844$ and
$\ell_{A}=\Psi_{1}(\ell_{B})\approx 2.922$. Figure 2: Trajectories of the
mean-field $(\lambda^{A},\lambda^{B})$ and their microscopic counterparts
$(\lambda_{N}^{A},\lambda_{N}^{B})$, when $h_{i}=\mathbf{1}_{[0,\theta_{i}]}$
for the Model (43) for the same parameters as Figure 1, except for
$\kappa_{1}$ where now $\kappa_{1}=4$, so that condition (32) no longer holds.
We still observe convergence to the correct theoretical value (and (31) is
nonetheless true).

###  5.3 Exponential inhibition

Consider in this paragraph the case ($\tau>0$)

$\Phi_{B\to A}(x)=\exp\left(-\tau x\right)\quad\mbox{and}\quad\Phi_{A\to
B}(x)=x,\quad x\geq 0.$ (46)

The function $\Phi$ rewrites in this case as

$\Phi(x)=\frac{a}{\kappa_{2}}+\frac{b\mu_{A}\exp\left(-\tau\kappa_{2}x\right)}{\kappa_{2}\left(1-\kappa_{1}\exp\left(-\tau\kappa_{2}x\right)\right)}$
(47)

defined on

$\mathcal{I}_{\kappa_{1},\kappa_{2}}=\begin{cases}[0,+\infty)&\text{ if
}\kappa_{1}<1,\\\
\left(\frac{\ln\left(\kappa_{1}\right)}{\tau\kappa_{2}},+\infty\right)&\text{
if }\kappa_{1}>1.\end{cases}$

The sufficient subcriticality condition (32) becomes here

$\kappa_{1}e^{-\tau a}<1.$ (48)

######  Proposition 5.5.

Consider Model (20) with $\Phi_{B\to A}$ and $\Phi_{A\to B}$ given by (46),
satisfying $\kappa_{3}<1$, $\kappa_{2}\kappa_{4}>0$, $\mu_{A}>0$,
$\kappa_{1}e^{-\tau a}<1$ and Assumptions 2.1 and (22). Then, the conclusions
of Theorem 3.10 are valid, i.e.
$(\lambda^{A}_{t},\lambda^{B}_{t})\xrightarrow[t\to\infty]{}\left(\ell_{A},\ell_{B}\right):=\left(\frac{\mu_{A}e^{-\tau\kappa_{2}\ell}}{1-\kappa_{1}e^{-\tau\kappa_{2}\ell}},\ell\right)$,
where $\ell$ is the unique fixed-point of $\Phi$ given by (47) on
$\left[\frac{\mu_{B}}{1-\kappa_{3}},+\infty\right)$, at least in the following
cases:

  1. 1.

when $\kappa_{1}<1$, at least when $\tau>0$ is sufficiently small (weak
inhibition) and when $\tau$ is sufficiently large (strong inhibition),

  2. 2.

when $\kappa_{1}\geq 1$, at least when $\tau>\tau_{\ast}$, for some
$\tau_{\ast}>\frac{\ln(\kappa_{1})}{a}>0$ depending on the parameters of the
system (strong inhibition).

###### Proof of Proposition 5.5.

The derivative of $\Phi$ on
$\mathcal{I}_{\kappa_{1},\kappa_{2}}\cap\left[\frac{\mu_{B}}{1-\kappa_{3}},+\infty\right)$
writes
$\Phi^{\prime}(x)=\frac{-b\mu_{A}\tau\exp\left(-\tau\kappa_{2}x\right)}{\left(1-\kappa_{1}\exp\left(-\tau\kappa_{2}x\right)\right)^{2}}.$
If $\kappa_{1}<1$, $\mathcal{I}_{\kappa_{1},\kappa_{2}}=[0,+\infty)$, the
maximum of $x\mapsto\left|\Phi^{\prime}(x)\right|$ on
$\left[\frac{\mu_{B}}{1-\kappa_{3}},+\infty\right)$ is attained for
$x=\frac{\mu_{B}}{1-\kappa_{3}}$ and is given by

$r(\tau):=\frac{b\mu_{A}\tau\exp(-\tau a)}{\left(1-\kappa_{1}\exp(-\tau
a)\right)^{2}}$ (49)

In particular, $r(\tau)<1$ as $\tau\to 0$ and $\tau\to+\infty$, imply that
$\Phi$ is a contraction, and hence Item 2 of Proposition 5.1 is true. More
precisely, since
$r(\tau)\leq\frac{b\mu_{A}\tau}{\left(1-\kappa_{1}\right)^{2}}$, $r(\tau)<1$
as least when $\tau<\frac{\left(1-\kappa_{1}\right)^{2}}{b\mu_{A}}$. Secondly,
since $r(\tau)\leq\frac{2b\mu_{A}}{ea(1-\kappa_{1})^{2}}\exp\left(-\frac{\tau
a}{2}\right)$, $r(\tau)<1$ at least when
$\frac{2}{a}\ln\left(\frac{2b\mu_{A}}{ea(1-\kappa_{1})^{2}}\right)<\tau$.

In the case $\kappa_{1}>1$, suppose that (48) holds, so that
$\frac{\mu_{B}}{1-\kappa_{3}}>\frac{\ln(\kappa_{1})}{\tau\kappa_{2}}=x_{\kappa_{1},\kappa_{2}}^{\ast}$.
Once again, the maximum of $\left|\Phi^{\prime}(x)\right|$ on
$\mathcal{I}_{\kappa_{1},\kappa_{2}}\cap\left[\frac{\mu_{B}}{1-\kappa_{3}},+\infty\right)$
is attained for $x=\frac{\mu_{B}}{1-\kappa_{3}}$ and given by (49), which goes
to $0$ as $\tau\to\infty$ and hence $r(\tau)<1$ for some
$\tau_{\ast}>\frac{\ln(\kappa_{1})}{a}$. ∎

######  Remark 5.6.

As already underlined, Assumption 3.9 is only sufficient for the convergence
of Theorem 3.10, not necessary: in the exponential case of Section 5.3, Figure
3 shows a situation where $\Phi\circ\Phi(u)<u$ for $u<\ell$ (and hence
$\mathcal{U}_{\Phi}$ is nontrivial). Nonetheless, convergence of
$(\lambda^{A},\lambda^{B})$ still occurs. The conjecture one may draw here is
that convergence mentioned in Theorem 3.10 holds as long as $\Phi$ is convex,
although we do not have a proof of this statement.

(a) Graph of $\Phi\circ\Phi$, where $\Phi$ is given by (47). Here,
$\ell\approx 7.567$ and $\Phi\circ\Phi(u)<u$ for $u\leq\ell$: Assumption 3.9
does not hold.

(b) Trajectories of the mean-field $(\lambda^{A},\lambda^{B})$ and their
microscopic counterparts $(\lambda_{N}^{A},\lambda_{N}^{B})$.

Figure 3: The case of Model (46) with $h_{i}=\mathbf{1}_{[0,\theta_{i}]}$ and
$\alpha=0.8$, $\kappa_{1}=0.95$, $\kappa_{2}=1$, $\kappa_{3}=0.5$,
$\kappa_{4}=1$, $\mu_{A}=10$, $\mu_{B}=1$, $\tau=0.2$, on $[0,T]$ with $T=50$.
Note that $\kappa_{1}<1$ so that (31) is satisfied. The intensities converge
(Figure 3b) whereas Assumption 3.9 does not hold (Figure 3a): Assumption 3.9
is sufficient for convergence, not necessary.

##  6 Towards oscillations: inhibition through a sigmoidal kernel

Consider again the Model (20) with full connectivity $\kappa_{2}\kappa_{4}>0$.
In most of the examples considered in Section 5, $\Phi_{B\to A}$ (and hence
$\Phi$ itself) is convex. As already said, although we do not have a rigorous
proof, we do not expect anything else but convergence of the intensities
$(\lambda^{A},\lambda^{B})$ in the convex case (even for large values of
$\kappa_{1}$ where (32) no longer holds or in cases when Assumption 3.9 is no
longer valid, recall Figure 3).

The case where $\Phi_{B\to A}$ is of sigmoid type (hence no longer convex) is
more intriguing. In the numerical examples below, the following two examples
will be considered: for fixed $R>0$ and $\beta>0$ a possibly large parameter,

$\displaystyle\Phi_{B\to A}(x)$
$\displaystyle=\frac{1}{1+\left(\frac{x}{R}\right)^{\beta}},$ (50)
$\displaystyle\Phi_{B\to A}(x)$
$\displaystyle=\frac{1}{2}-\frac{1}{\pi}\arctan\left(\beta\left(x-R\right)\right).$
(51)

The kernel (50) is the same example as (43) (take $\tau=1/{R^{\beta}}$). Both
are smooth approximations as $\beta\to\infty$ of the indicator function

$\Phi_{B\to A}(x)=\mathbf{1}_{[0,R]}(x).$ (52)

However, There is every reasons to believe that the main features developed in
this section should not depend specifically on the choice of the sigmoid
kernel, as long as it is sufficiently close to (52). In this paragraph we
illustrate, both with non-rigorous intuition and numerical examples that the
lack of convexity of $\Phi_{B\to A}$ is likely to induce oscillations for the
solution $(\lambda^{A},\lambda^{B})$ to (20) (recall Remark 3.4). The
existence of limit cycles for (20) is only made explicit based on numerical
simulations. A rigorous proof is lacking and will be the object of a future
work.

###  6.1 The case of an indicator

In order to give some intuition about the possible emergence of limit cycles
for (20), replace for the moment the kernels $\Phi_{B\to A}$ in (50) or (51)
by their common limit (52) as $\beta\to\infty$. Obviously, none of the
rigorous results developed in the previous sections should apply (at least
easily) to the case (52), as this kernel is not even continuous. Even the
well-posedness of the particle system (6) or its mean-field limit (15) is
unclear in this case, not to mention the analysis made in Section 3 to derive
the limit of the mean-field system as $t\to\infty$. This being said, suppose
anyway that one would be able to consider (52) as a proper candidate for the
inhibition kernel $\Phi_{B\to A}$. We place ourselves in the framework where
$\mu_{A}$ is large and $\mu_{B}$ small (possibly even $0$). In ’real life’,
our $\mu_{A}$ and $\mu_{B}$ account for the following two features: i) the
weaker one, the intrinsic properties of the neurons (that make them more or
less prone to spike spontaneously) [37]; ii) the stronger one, the excitatory
inputs from the other brain regions (other neocortical area and _mostly_
thalamic inputs [50, 5]). There is strong anatomical evidence that the latter
are stronger onto excitatory neurons than onto inhibitory ones (reviewed in
[50]), justifying the relevance of considering $\mu_{A}\gg\mu_{B}$.

The intuition for the emergence of oscillations is the following: considering
(20) at $t\approx 0$, as $\mu_{A}\gg\mu_{B}\approx 0$, the intensity of
population $A$ (resp. $B$) should be high (resp. small). Then, in presence of
feedback from $A$ onto $B$ ($\kappa_{4}>0$), the spiking activity of
population $A$ propagates to population $B$ so that the intensity of $B$
increases. But the higher the activity of $B$ is, the larger
$X_{B}(t):=(1-\alpha)\int_{0}^{t}h_{2}(t-u)\lambda_{u}^{B}{\rm d}u$ (that is
the term within the inhibition kernel $\Phi_{B\to A}(\cdot)$ in the first
equation of (20)) becomes. Once $X_{B}(t)$ has reached the threshold $R>0$ of
(52), the activity of $A$ gets killed (recall that $\kappa_{2}>0$). Hence,
supposing for instance that the memory kernel $h_{4}$ has compact support with
length $\theta_{4}$, after a time $\theta_{4}$, population $B$ will no longer
feel the influence of population $A$: intensity of $B$ will go back to
$\mu_{B}\approx 0$. Thus, $X_{B}(t)$ crosses back the threshold $R$:
population $A$ goes back to its normal high activity, and so on: oscillations
should appear.

For this to happen, the threshold $R$ should not be too large, otherwise the
term $X_{B}(t)$ will not be able to reach $R$. We can actually pursue the
intuition given by (52) in deriving informally a phase transition. Suppose
again that the formalism developed in Section 3.3 is valid for (52) (which is
not). Note first that the subcriticality condition (32) simply boils down here
to

$\kappa_{1}<1$ (53)

that is population $A$ alone is subcritical. The fixed-point function $\Phi$
becomes

$\Phi(x)=\frac{a}{\kappa_{2}}+\frac{\mu_{A}b}{\kappa_{2}(1-\kappa_{1})}\mathbf{1}_{[0,R]}(\kappa_{2}x).$
(54)

Recall that the whole point of Section 3.3 is to compute the fixed-point of
$\Phi$. Straightforward computations enable to distinguish two cases:

  * •

Either $\frac{R-a}{b}\geq\frac{\mu_{A}}{1-\kappa_{1}}$: Then $\Phi$ has a
unique fixed-point
$\ell=\frac{a}{\kappa_{2}}+\frac{\mu_{A}b}{\kappa_{2}(1-\kappa_{1})}$ (which
should be the limit of $\lambda^{B}$ if the results of Section 3.3 would apply
to (52)). In this case, the limit of $\lambda^{A}$ should be
$\ell_{A}=\Psi_{1}(\ell)=\frac{\mu_{A}\mathbf{1}_{[0,R]}(\kappa_{2}\ell)}{1-\kappa_{1}\mathbf{1}_{[0,R]}(\kappa_{2}\ell)}=\frac{\mu_{A}}{1-\kappa_{1}}$.

  * •

Either $\frac{R-a}{b}<\frac{\mu_{A}}{1-\kappa_{1}}$: in this case, there is no
fixed-point for $\Phi$. But the main conclusion of Theorem 3.10 that any
possible limit for $\lambda^{B}$ is necessarily given by $\ell$ fixed-point of
$\Phi$. Hence, if one believes in the conclusion of Theorem 3.10 for (52), we
would necessarily have $\underline{\ell}_{B}<\bar{\ell}_{B}$: this is the
regime where one expects oscillations.

Of course, once again, this phase transition is only formal, as (52) does
enter into the hypotheses used in the paper. What is more, this formal
argument is not stable by perturbation: if one replaces (52) by any continuous
sigmoid function with the same threshold $R$, Theorem 3.10 does apply in this
case and $\Phi$ has always a fixed-point and the formal argument above becomes
meaningless. It appears nonetheless that this formal phase transition is
actually meaningful and visible in simulations in the sigmoid case, see
Section 6.2.1.

###  6.2  The case of a sigmoid kernel

Having in mind the intuition provided by the previous paragraph, let us go
back to the case where $\Phi_{B\to A}$ is defined as a smooth Lipschitz
sigmoid function, such as in (50) of (51). Here, every results of the present
paper apply. In view of Theorem 3.10, there are only two ways to capture
oscillations:

  1. 1.

either to ensure that (31) no longer holds, i.e. situations where
$\kappa_{1}\gg 1$ so that (with the notations of the previous sections, recall
(24))

$\underline{\ell}_{B}<x_{\kappa_{1},\kappa_{2}}^{\ast}<\bar{\ell}_{B}.$ (55)

This corresponds to a situation where population $A$, in isolation from $B$
would be highly supercritical (but still subcritical with inhibition and
retroaction, see Theorem 3.6). The technical difficulty is that we lack some a
priori upper bounds on $\underline{\ell}_{B}$ in order to give some rigorous
sufficient conditions for (55). We give nonetheless a numerical illustration
of this case in Figure 4.

Figure 4: Trajectories of the mean-field $(\lambda^{A},\lambda^{B})$ and their
microscopic counterparts $(\lambda_{N}^{A},\lambda_{N}^{B})$, when $\Phi_{B\to
A}$ is given by (51), with $R=1$, $\beta=20$,
$h_{i}=\mathbf{1}_{[0,\theta_{i}]}$, $\alpha=0.8$, $\kappa_{1}=5$,
$\kappa_{2}=\kappa_{3}=\kappa_{4}=0.5$, $\mu_{A}=7$ and $\mu_{B}=0$, on
$[0,T]$ with $T=50$. Note that $\kappa_{1}\Phi_{B\to
A}\left(\kappa_{2}\underline{\ell}_{B}\right)\approx 4.81>1$ so that (31) does
not hold: we have oscillations.

  2. 2.

or either in situations where (31) is valid (e.g. by supposing (32); note that
in simulations below, we take $\kappa_{1}<1$ so that (31) is verified
immediately), but where Assumption 3.9 no longer holds (recall that Assumption
3.9 is sufficient for convergence, not necessary, but oscillations necessarily
occur when Assumption 3.9 is not true). The difficulty (already met in the
proof of Proposition 5.3) is that checking Assumption 3.9 requires nontrivial
algebraic estimates that may be, in general, out of reach from any rigorous
treatment. Therefore, we only check Assumption 3.9 numerically here.

####  6.2.1 Phase transition and emergence of oscillations

The first observation of this paragraph is that the phase transition derived
formally in the previous section is (numerically) accurate: take
$h_{i}=\mathbf{1}_{[0,\theta_{i}]}$, $\theta_{i}>0$, $i=1,\ldots,4$,
$\alpha=0.8$ and for simplicity $\kappa_{i}=0.5$ for $i=1,\ldots,4$ and
$\mu_{B}=0$. Consider $\beta$ in (50) (or in (51)) sufficiently large so that
$\Phi_{B\to A}$ is close enough to the indicator function (52) (in pratice,
interesting dynamical features already appears for $\beta\approx 10$). With
these parameters, the phase transition described in Section 6.1 simply boils
down to $R\geq\mu_{A}$ or $R<\mu_{A}$. The corresponding simulations are
displayed in Figures 5 and 6 below, made for the choice of (50) with
$\beta=1000$ and $R=1$: for $\mu_{A}=0.99$, observe numerically in Figure 5
that $\mathcal{U}_{\Phi}$ is trivial (Assumption 3.9 is true) and therefore
Theorem 3.10 applies: we have convergence of $(\lambda^{A},\lambda^{B})$. On
the contrary, for $\mu_{A}=1.01$, $\mathcal{U}_{\Phi}$ is no longer reduced to
$(\ell,\ell)$ (Assumption 3.9 is not valid) and oscillations appear (Figure
6). These oscillations are all the more effective when $\mu_{A}$ gets larger,
see Figure 7.

(a) Verifying the hypotheses of Theorem 3.10: representation of $\Phi$ (in
blue) and $\Phi^{-1}$ (in red) on the interval $\left(0,\ell\right)$. We see
here that $\Phi<\Phi^{-1}$ so that, by (41), $\mathcal{U}_{\Phi}$ is trivial
so that Assumption 3.9 is true. Here $\kappa_{1}<1$ and (31) is trivial.

(b) Trajectories of the mean-field $(\lambda^{A},\lambda^{B})$ and their
microscopic counterparts $(\lambda_{N}^{A},\lambda_{N}^{B})$.

Figure 5: The case $\Phi_{B\to A}$ given by (50), with $R=1$, $\beta=1000$,
$h_{i}=\mathbf{1}_{[0,\theta_{i}]}$, $\alpha=0.8$,
$\kappa_{1}=\kappa_{2}=\kappa_{3}=\kappa_{4}=0.5$ and $\mu_{B}=0$. Here we
take $\mu_{A}=0.99$. The conclusions of Theorem 3.10 hold in this case (Figure
5a) and we have convergence of the intensities as $t\to\infty$ (Figure 5b).

(a) Representation of $\Phi$ (in blue) and $\Phi^{-1}$ (in red) on the
interval $\left(0,\ell\right)$. We see here that $\mathcal{U}_{\Phi}$ is
nontrivial (recall (42)): Assumption 3.9 no longer holds.

(b) Trajectories of the mean-field $(\lambda^{A},\lambda^{B})$ and their
microscopic counterparts $(\lambda_{N}^{A},\lambda_{N}^{B})$.

Figure 6: The case $\Phi_{B\to A}$ given by (50), with $R=1$, $\beta=1000$,
$h_{i}=\mathbf{1}_{[0,\theta_{i}]}$, $\alpha=0.8$,
$\kappa_{1}=\kappa_{2}=\kappa_{3}=\kappa_{4}=0.5$ and $\mu_{B}=0$. Here we
take $\mu_{A}=1.01$. Assumption 3.9 is no longer valid (Figure 6a) and we
observe oscillations (Figure 6b). Figure 7: Trajectories of the mean-field
$(\lambda^{A},\lambda^{B})$ and their microscopic counterparts
$(\lambda_{N}^{A},\lambda_{N}^{B})$, when $\Phi_{B\to A}$ is given by (50),
with $R=1$, $\beta=1000$, $h_{i}=\mathbf{1}_{[0,\theta_{i}]}$, $\alpha=0.8$,
$\kappa_{1}=\kappa_{2}=\kappa_{3}=\kappa_{4}=0.5$, $\mu_{A}=3$ and
$\mu_{B}=0$, on $[0,T]$ with $T=30$.

Note that this phase transition does not intrinsically depend on the specific
choice of the sigmoid kernel $\Phi_{B\to A}$. Replacing (50) by (51) leads to
similar patterns, even for smaller values of $\beta$, see Figure 8. However
the precision of the phase transition around $\mu_{A}^{c}:=R$ degrades as
$\beta$ is not too large.

####  6.2.2 The role of memory kernels in oscillations

Figure 8: Trajectories of the mean-field $(\lambda^{A},\lambda^{B})$ and their
microscopic counterparts $(\lambda_{N}^{A},\lambda_{N}^{B})$, when $\Phi_{B\to
A}$ is given by (51), with $R=1$, $\beta=10$,
$h_{i}=\mathbf{1}_{[0,\theta_{i}]}$, $\alpha=0.8$, $\kappa_{1}=0.5$,
$\kappa_{2}=1$, $\kappa_{3}=0.1$, $\kappa_{4}=1.0$, $\mu_{A}=25$ and
$\mu_{B}=0$, on $[0,T]$ with $T=50$.

The point we want to raise here is the importance of the choice of the memory
kernels $h_{i}$ in the emergence of oscillations. Previous simulations were
made with compactly supported $h_{i}=\mathbf{1}_{[0,\theta_{i}]}$. Several
other choices for the $h_{i}$ can be made, such as e.g. exponential
$h_{i}(u)=e^{-\frac{u}{\theta_{i}}}$ or Erlang kernels
$h_{i}(u)=e^{-\frac{u}{\theta_{i}}}\frac{u^{n}}{n!}$ [26]. This last choice of
exponential kernels has the nice ability to transform the convolution equation
governing the mean-field intensity $\lambda(t)$ of a single population of
Hawkes processes into an ODE governing the mean activity
$X(t)=\int_{0}^{t}e^{-\frac{t-s}{\theta}}\lambda(s){\rm d}s$ (see e.g. [26]
for further details). The same procedure can be applied here: set
$h_{i}(u)=e^{-\frac{u}{\theta_{i}}}$, $i=1,\ldots,4$ and write
$X_{1}(t):=\alpha\int_{0}^{t}h_{1}(t-s)\lambda^{A}(s){\rm d}s$,
$X_{2}(t):=(1-\alpha)\int_{0}^{t}h_{2}(t-s)\lambda^{B}(s){\rm d}s$,
$X_{3}(t):=(1-\alpha)\int_{0}^{t}h_{3}(t-s)\lambda^{B}(s){\rm d}s$,
$X_{4}(t):=\alpha\int_{0}^{t}h_{4}(t-s)\lambda^{A}(s){\rm d}s$, one easily
sees that the vector $(X_{1},X_{2},X_{3},X_{4})$ solves the system of coupled
ODEs

$\begin{cases}\frac{{\rm d}}{{\rm
d}t}X_{1}(t)&=-\frac{1}{\theta_{1}}X_{1}(t)+\alpha\left(\mu_{A}+X_{1}(t)\right)\Phi_{B\to
A}\left(X_{2}(t)\right),\\\ \frac{{\rm d}}{{\rm
d}t}X_{2}(t)&=-\frac{1}{\theta_{2}}X_{2}(t)+(1-\alpha)\left(\mu_{B}+X_{3}(t)+X_{4}(t)\right),\\\
\frac{{\rm d}}{{\rm
d}t}X_{3}(t)&=-\frac{1}{\theta_{3}}X_{3}(t)+(1-\alpha)\left(\mu_{B}+X_{3}(t)+X_{4}(t)\right),\\\
\frac{{\rm d}}{{\rm
d}t}X_{4}(t)&=-\frac{1}{\theta_{4}}X_{4}(t)+\alpha\left(\mu_{A}+X_{1}(t)\right)\Phi_{B\to
A}\left(X_{2}(t)\right).\end{cases}$ (56)

Nonetheless, no oscillations have been numerically observed for (56): for the
same set of parameters, limit cycles that are present in the compact case
$h_{i}=\mathbf{1}_{[0,\theta_{i}]}$ disappear with exponential kernels,
replaced by damped oscillations. To be more precise, oscillations disappear
whenever both $h_{2}$ and $h_{4}$ are exponential. We interpret this situation
as follows: for the intuition described in Section 6.1 to work, it is
necessary for population $B$ that it forgets rapidly the influence of
population $A$ or that inhibition of $B$ onto $A$ has finite memory, that is
one among $h_{2}$ or $h_{4}$ has compact support. Concerning $h_{4}$, the
influence of population $A$ onto $B$ is made through
$X_{4}=\alpha\int_{0}^{t}h_{4}(t-s)\lambda^{A}(s){\rm d}s$ which consists in
integrating on the whole history of population $A$. Therefore, if $h_{4}$ has
infinite support, $X_{4}$ conserves the influence of nontrivial activity of
$A$ for arbitrary long times and even if the activity of $A$ may sometimes
vanish, it may not be sufficient for a significant decrease in the feedback
from $A$ to $B$. The same argument applies for $h_{2}$: if $h_{2}$ has long
memory, $X_{2}$ keeps the trace of periods when activity of $B$ was
significant, therefore $X_{2}$ cannot go back to its initial state where
$\mu_{B}\approx 0$.

##  7 Proofs of well-posedness results and propagation of chaos

We prove here the results of Section 2.1.

###  7.1 Well-posedness of the particle system

We prove Proposition 2.6. Similarly to the proof of Proposition 2.1 [20], we
proceed by an inductive thinning procedure on the Poisson point processes
$\left(\left(\pi_{i}\right)_{i=1,\ldots,N_{A}},\left(\widetilde{\pi}_{i}\right)_{i=N_{A}+1,\ldots,N}\right)$.

##### Initialization

Define $\Lambda_{0}^{A}:=F(0,0)$ and $\Lambda_{0}^{B}:=G(0,0)$ and

$\displaystyle U_{1}^{(i)}$ $\displaystyle:=\inf\left\\{u>0,\
\int_{(0,u]}\int_{(0,\Lambda_{0}^{A}]}\pi_{i}({\rm d}s,{\rm d}z)>0\right\\},\
i=1,\ldots,N_{A},$ $\displaystyle U_{1}^{(i)}$
$\displaystyle:=\inf\left\\{u>0,\
\int_{(0,u]}\int_{(0,\Lambda_{0}^{B}]}\widetilde{\pi}_{i}({\rm d}s,{\rm
d}z)>0\right\\},\ i=N_{A}+1,\ldots,N$

as well as

$U_{1}:=\min(U_{1}^{(i)},i=1,\ldots,N).$ (57)

Each $U_{1}^{(i)}$ for $i=1,\ldots,N_{A}$ (resp. for $i=N_{A}+1,\ldots,N$),
$U_{1}^{(i)}$ is the first atom of an homogeneous Poisson process with
parameter $\Lambda_{0}^{A}$ (resp. $\Lambda_{0}^{B}$). Hence, almost surely,
$U_{1}^{(i)}$ and thus $U_{1}$ is strictly positive and finite. Moreover, by
independence of the Poisson point processes
$\left(\left(\pi_{i}\right)_{i=1,\ldots,N_{A}},\left(\widetilde{\pi}_{i}\right)_{i=N_{A}+1,\ldots,N}\right)$,
the variables $U_{1}^{(i)},i=1,\ldots,N$ are almost surely distinct so that

$I_{1}:=\left\\{i\in\left\\{1,\ldots,N\right\\},U_{1}^{(i)}=U_{1}\right\\}$
(58)

is defined without ambiguity. Since any solution to (6) has constant
intensities $F(0,0)$ and $G(0,0)$ until the first atom, the above construction
yields that $U_{1}$ given by (57) is necessarily the first atom of
$(Z_{t}^{i})_{i=1,\ldots,N}$ and that it concerns the process $Z^{i}$ for
$i=I_{1}$.

##### Recursion

Suppose we have constructed $0<U_{1}<\ldots<U_{k}$ and
$(I_{1},\ldots,I_{k})\in\left\\{1,\ldots,N\right\\}^{k}$ such that on the
event $\left\\{U_{k}<\infty\right\\}$, $0<U_{1}<\ldots<U_{k}$ are the $k$
first atoms of the aggregated process

$Z=\sum_{i=1}^{N}Z^{i}$ (59)

and for all $j=1,\ldots,k$, $I_{j}$ is the index in
$\left\\{1,\ldots,N\right\\}$ responsible for the $j$th atom. Let

$Z_{A}^{(k)}:=\sum_{\begin{subarray}{c}j=1,\ldots,k,\\\
I_{j}\in\left\\{1,\ldots,N_{A}\right\\}\end{subarray}}\delta_{U_{j}}\ \text{
and }Z_{B}^{(k)}:=\sum_{\begin{subarray}{c}j=1,\ldots,k,\\\
I_{j}\in\left\\{N_{A}+1,\ldots,N\right\\}\end{subarray}}\delta_{U_{j}}$

be the point measures corresponding to atoms which concern population $A$ and
$B$ respectively. Define

$\displaystyle\Lambda_{k}^{A}(t)$
$\displaystyle:=F\left(\frac{1}{N}\int_{(0,U_{k}]}h_{1}(t-s){\rm
d}Z_{A,s}^{(k)},\frac{1}{N}\int_{(0,U_{k}]}h_{2}(t-s){\rm
d}Z_{B,s}^{(k)}\right),$ $\displaystyle\Lambda_{k}^{B}(t)$
$\displaystyle:=G\left(\frac{1}{N}\int_{(0,U_{k}]}h_{3}(t-s){\rm
d}Z_{B,s}^{(k)},\frac{1}{N}\int_{(0,U_{k}]}h_{4}(t-s){\rm
d}Z_{A,s}^{(k)}\right)$

as well as

$\displaystyle U_{k+1}^{(i)}$ $\displaystyle:=\inf\left\\{u>0,\
\int_{(U_{k},u]}\int_{(0,\Lambda_{k}^{A}(s)]}\pi_{i}({\rm d}s,{\rm
d}z)>0\right\\},\ i=1,\ldots,N_{A},$ $\displaystyle U_{k+1}^{(i)}$
$\displaystyle:=\inf\left\\{u>0,\
\int_{(U_{k},u]}\int_{(0,\Lambda_{k}^{B}(s)]}\widetilde{\pi}_{i}({\rm d}s,{\rm
d}z)>0\right\\},\ i=N_{A}+1,\ldots,N$

Define finally $U_{k+1}:=\min\left(U_{k+1}^{(i)},i=1,\ldots,N\right).$ For
$\varepsilon>0$, denote by

$\displaystyle\mathcal{R}_{\varepsilon}^{A}$
$\displaystyle:=\left\\{(s,z),s\in(U_{k},U_{k}+\varepsilon],\
z\in(0,\Lambda_{k}^{A}(s)]\right\\},$
$\displaystyle\mathcal{R}_{\varepsilon}^{B}$
$\displaystyle:=\left\\{(s,z),s\in(U_{k},U_{k}+\varepsilon],\
z\in(0,\Lambda_{k}^{B}(s)]\right\\}$

For all $i=1,\ldots,N_{A}$, conditionally on $\mathcal{F}_{U_{k}}$,
$\pi_{i}(\mathcal{R}_{\varepsilon}^{A})$ follows a Poisson law with parameter
$\int_{U_{k}}^{U_{k}+\varepsilon}\Lambda_{k}^{A}(t){\rm d}t$, so that

$\mathbf{P}\left(\pi_{i}(\mathcal{R}_{\varepsilon}^{A})<\infty\right)=\mathbf{E}\left(\mathbf{P}\left(\int_{U_{k}}^{U_{k}+\varepsilon}\Lambda_{k}^{A}(t){\rm
d}t<\infty|\mathcal{F}_{U_{k}}\right)\right).$

Then, using (9) and Fubini Theorem

$\displaystyle\int_{U_{k}}^{U_{k}+\varepsilon}\Lambda_{k}^{A}(t){\rm d}t$
$\displaystyle\leq
c_{2}\varepsilon+\frac{c_{1}}{N}\left\\{\left\|h_{1}\right\|_{1}+\left\|h_{2}\right\|_{1}\right\\}k<\infty.$

Hence, almost surely for all $i=1,\ldots,N_{A}$,
$\pi_{i}(\mathcal{R}_{\varepsilon}^{A})<\infty$ and the same reasoning applies
to $\pi_{i}(\mathcal{R}_{\varepsilon}^{B})$ for all $i=N_{A}+1,\ldots,N$. This
gives that almost surely for all $i=1,\ldots,N$, $U_{k+1}^{(i)}>U_{k}$ and
thus $U_{k+1}>U_{k}$. Again by independence of the $(\pi_{i})_{i=1,\ldots,N}$,
the $U_{k+1}^{(i)},i=1,\ldots,N$ are all distinct and
$I_{k+1}:=\left\\{i\in\left\\{1,\ldots,N\right\\},U_{k+1}^{(i)}=U_{k+1}\right\\}$
is uniquely defined. By definition (6), the above construction yields that
$U_{k+1}$ is necessarily the $k$th atom of $(Z_{t}^{i})_{i=1,\ldots,N}$ and
that it concerns the process $Z^{i}$ for $i=I_{k+1}$. This completes the proof
of the existence of a pathwise unique Hawkes process as defined in (6). We now
establish absence of explosion of this process.

##### Stochastic domination.

Introduce the following linear processes:

$\begin{cases}Z_{t}^{i,\text{lin}}&=\int_{0}^{t}\int_{0}^{\infty}\mathbf{1}_{z\leqslant\lambda_{s}^{i,\text{lin}}}\pi_{i}({\rm
d}s,{\rm d}z),\ i=1,\ldots,N_{A},\\\
Z_{t}^{i,\text{lin}}&=\int_{0}^{t}\int_{0}^{\infty}\mathbf{1}_{z\leqslant\lambda_{s}^{i,\text{lin}}}\widetilde{\pi}_{i}({\rm
d}s,{\rm d}z),\ i=N_{A}+1,\ldots,N,\end{cases}$ (60)

where (recall (9) and (13))

$\begin{cases}\lambda_{s}^{i,\text{lin}}=\lambda_{s}^{A,\text{lin}}&:=c_{1}\left(\frac{1}{N}\displaystyle{\sum_{j=1}^{N_{A}}}\int_{0}^{s^{-}}h_{1}(s-u){{\rm
d}Z_{u}^{j,\text{lin}}}+\frac{1}{N}\displaystyle{\sum_{j=N_{A}+1}^{N}}\int_{0}^{s^{-}}h_{2}(s-u){{\rm
d}Z_{u}^{j,\text{lin}}}\right)+c_{2},\ 1\leqslant i\leqslant N_{A},\\\
\lambda_{s}^{i,\text{lin}}=\lambda_{s}^{B,\text{lin}}&:=c_{6}\left(\frac{1}{N}\underset{N_{A}+1\leqslant
j\leqslant N}{\sum}\int_{0}^{s^{-}}h_{3}(s-u){{\rm
d}Z_{u}^{j,\text{lin}}}+\frac{1}{N}\underset{1\leqslant j\leqslant
N_{A}}{\sum}\int_{0}^{s^{-}}h_{4}(s-u){{\rm
d}Z_{u}^{j,\text{lin}}}\right)+c_{7},\ N_{A}+1\leqslant i\leqslant
N.\end{cases}$ (61)

Existence and pathwise uniqueness of a process
$\left(Z^{i,\text{lin}}\right)_{i=1,\ldots,N}$ such that
$\sum_{i=1}^{N}\mathbf{E}\left[\sup_{t\in[0,T]}Z_{t}^{i,\text{lin}}\right]<\infty$
for all $T>0$ is a direct consequence of well-known existing results (see e.g.
[23]).

We prove here that the point processes $Z^{i},i=1,\ldots,N$ are stochastically
dominated by $Z^{i,\text{lin}},i=1,\ldots,N$ given in (60). More precisely, we
show that for all $i=1,\ldots,N$, each atom of $Z^{i}$ is also an atom of
$Z^{i,\text{lin}}$. We proceed by recursion: we claim that (recall (7))

$\begin{cases}\lambda_{t}^{A}&\leq\lambda_{t}^{A,\text{lin}},\\\
\lambda_{t}^{B}&\leq\lambda_{t}^{B,\text{lin}},\end{cases}\ t\in(0,U_{1}].$
(62)

Indeed, by construction of $Z^{i},i=1,\ldots,N$, there are no atoms before
$U_{1}$ so that, for $t\in\left(0,U_{1}\right)$, $\lambda_{t}^{A}=F(0,0)\leq
c_{2}\leq\lambda_{t}^{A,\text{lin}}$ and $\lambda_{t}^{B}=G(0,0)\leq
c_{7}\leq\lambda_{t}^{B,\text{lin}}$. Suppose that
$I_{1}\in\left\\{1,\ldots,N_{A}\right\\}$ (recall the definition of $I_{1}$ in
(58)), the other case $I_{1}\in\left\\{N_{A}+1,\ldots,N\right\\}$ is treated
similarly. By definition of $U_{1}$, there exists an atom $(U_{1},z)$ of
$\pi_{I_{1}}$ with $0\leq z\leq\lambda_{U_{1}}^{A}$ so that by (62), $0\leq
z\leq\lambda_{U_{1}}^{A,\text{lin}}$ so that by (60), $U_{1}$ is also an atom
of $Z_{I_{1}}^{\text{lin}}$.

Suppose now that $U_{1},\ldots,U_{k}$ (which are by definition atoms of
$Z_{I_{1}},\ldots,Z_{I_{k}}$ respectively) are also atoms of
$Z_{I_{1}}^{\text{lin}},\ldots,Z_{I_{k}}^{\text{lin}}$. In particular, for
$t\in[U_{k},U_{k+1})$, recalling

$\displaystyle\lambda_{t}^{A}$ $\displaystyle=\Lambda_{k}^{A}(t)\leq
c_{1}\left\\{\frac{1}{N}\int_{(0,U_{k}]}h_{1}(t-s){\rm
d}Z_{A,s}^{(k)}+\frac{1}{N}\int_{(0,U_{k}]}h_{2}(t-s){\rm
d}Z_{B,s}^{(k)}\right\\}+c_{2}\leq\lambda_{t}^{A,\text{lin}},$

where we used (9) and the fact that the number of atoms among
$U_{1},\ldots,U_{k}$ that concern population $A$ (resp. $B$) is the same for
both processes $(Z^{i})_{i=1,\ldots,N}$ and
$(Z^{i,\text{lin}})_{i=1,\ldots,N}$. The same reasoning proves that
$\lambda_{t}^{B}\leq\lambda_{t}^{B,\text{lin}}$ for $t\in[U_{k},U_{k+1})$.
Hence, this proves again that $U_{k+1}$ (which is an atom of $Z_{I_{k+1}}$) is
also an atom of $Z_{I_{k+1}}^{\text{lin}}$, which proves the claim.

##### Absence of explosion

A direct consequence of the previous claim is
$\mathbf{E}\left[\sup_{t\in[0,T]}Z_{t}^{i}\right]\leq\mathbf{E}\left[\sup_{t\in[0,T]}Z_{t}^{i,\text{lin}}\right]$
so that (recall (59))

$\mathbf{E}\left[\sup_{t\in[0,T]}Z_{t}\right]\leq C_{T},T>0$ (63)

for some constant $C_{T}>0$. This yields in particular that almost surely
$U_{\infty}:=\lim_{k\to\infty}U_{k}=+\infty$. Indeed, set
$\Omega_{M}:=\left\\{U_{\infty}\leq M\right\\}$ for all $M>0$. Then,
$\mathbf{E}\left[Z_{U_{k}}\mathbf{1}_{\Omega_{M}}\right]=k\mathbf{P}\left(\Omega_{M}\right)$
on the one hand and
$\mathbf{E}\left[Z_{U_{k}}\mathbf{1}_{\Omega_{M}}\right]\leq\mathbf{E}\left[Z_{M}\right]\leq
C_{M}$ on the other hand, by (63). Hence,
$\mathbf{P}\left(\Omega_{M}\right)\leq\frac{C_{M}}{k}$, for all $k\geq 0$.
Letting $k\to\infty$, $\mathbf{P}\left(\Omega_{M}\right)=0$ for all $M>0$
which gives that $\mathbf{P}\left(U_{\infty}=\infty\right)=1$.

Uniqueness of a trajectory follows from the inductive procedure. This
concludes the proof of Proposition 2.6.

###  7.2 Proof of well-posedness of the mean-field limit

We prove here Proposition 2.7.

####  7.2.1 Well-posedness of (16)

Consider first the dynamics of the mean-value $(m_{t}^{A},m_{t}^{B})$ given by
(16). Let us begin with an a priori estimate: for any solution $(m_{A},m_{B})$
that is $\mathcal{C}^{1}$ on $[0,T]$ (if it exists), such a solution
satisfies, for $\lambda_{t}^{A}:=(m_{t}^{A})^{\prime}$,
$\lambda_{t}^{B}:=(m_{t}^{B})^{\prime}$,

$\begin{cases}\lambda_{t}^{A}&=F\left(\alpha\int_{0}^{t}h_{1}(s-u)\lambda_{u}^{A}{\rm
d}u,(1-\alpha)\int_{0}^{t}h_{2}(t-u)\lambda_{u}^{B}{\rm d}u\right),\\\
\lambda_{t}^{B}&=G\left((1-\alpha)\int_{0}^{t}h_{3}(s-u)\lambda_{u}^{B}{\rm
d}u,\alpha\int_{0}^{t}h_{4}(t-u)\lambda_{u}^{A}{\rm d}u\right).\end{cases}$

In particular $\lambda_{u}^{A}\geq 0$ and $\lambda_{u}^{B}\geq 0$ for all
$u\in[0,T]$. Using (9) and (13), we obtain that, for $t\in[0,T]$

$\displaystyle\lambda_{t}^{A}$ $\displaystyle\leq
c_{2}+c_{1}\left\\{\alpha\int_{0}^{t}h_{1}(t-v)\lambda_{v}^{A}{\rm
d}v+(1-\alpha)\int_{0}^{t}h_{2}(t-v)\lambda_{v}^{B}{\rm d}v\right\\},$
$\displaystyle\lambda_{t}^{B}$ $\displaystyle\leq
c_{7}+c_{6}\left\\{(1-\alpha)\int_{0}^{t}h_{3}(t-v)\lambda_{v}^{B}{\rm
d}v+\alpha\int_{0}^{t}h_{4}(t-v)\lambda_{v}^{A}{\rm d}v\right\\}.$

By summation, we obtain, for $\Lambda(t):=\lambda_{t}^{A}+\lambda_{t}^{B}$,
that for some constant $C>0$

$\Lambda(t)\leq C\left(1+\int_{0}^{t}H(t-v)\Lambda(v){\rm d}v\right).$ (64)

An application of [23], Lemma 23 (i) leads to the following a priori bound:
there exists of a constant $C_{0}>0$, depending on $T,F,G$ and $H$ such that

$\sup_{t\in[0,T]}\left\\{\lambda_{t}^{A}+\lambda_{t}^{B}\right\\}=\sup_{t\in[0,T]}\left\\{(m_{t}^{A})^{\prime}+(m_{t}^{B})^{\prime}\right\\}\leq
C_{0}.$ (65)

We now turn to the proof of uniqueness of a solution to (16): consider two
solutions $(m_{A},m_{B})$ and $(\widetilde{m}^{A},\widetilde{m}^{B})$. Set
$(v_{A}(t),v_{B}(t)):=\left(\int_{0}^{t}\left|{\rm
d}(m_{u}^{A}-\widetilde{m}_{u}^{A})\right|,\int_{0}^{t}\left|{\rm
d}(m_{u}^{B}-\widetilde{m}_{u}^{B})\right|\right)$ and
$v(t):=v_{A}(t)+v_{B}(t)$. Then,

$\displaystyle v_{A}(t)$ $\displaystyle=\int_{0}^{t}\left|{\rm
d}(m_{u}^{A}-\widetilde{m}_{u}^{A})\right|\leq\int_{0}^{t}\bigg{|}F\left(\alpha\int_{0}^{u}h_{1}(u-v){\rm
d}m_{v}^{A},(1-\alpha)\int_{0}^{u}h_{2}(u-v){\rm d}m_{v}^{B}\right)$
$\displaystyle\hskip 56.9055pt-F\left(\alpha\int_{0}^{u}h_{1}(u-v){\rm
d}m_{v}^{A},(1-\alpha)\int_{0}^{u}h_{2}(u-v){\rm
d}\widetilde{m}_{v}^{B}\right)\bigg{|}{\rm d}u$
$\displaystyle+\int_{0}^{t}\bigg{|}F\left(\alpha\int_{0}^{u}h_{1}(u-v){\rm
d}m_{v}^{A},(1-\alpha)\int_{0}^{u}h_{2}(u-v){\rm
d}\widetilde{m}_{v}^{B}\right)$ $\displaystyle\hskip
56.9055pt-F\left(\alpha\int_{0}^{u}h_{1}(u-v){\rm
d}\widetilde{m}_{v}^{A},(1-\alpha)\int_{0}^{u}h_{2}(u-v){\rm
d}\widetilde{m}_{v}^{B}\right)\bigg{|}{\rm d}u$

Using (10) and (11), we obtain

$\displaystyle v_{A}(t)$
$\displaystyle\leq(1-\alpha)\int_{0}^{t}\left(c_{4}\alpha\int_{0}^{u}h_{1}(u-v){\rm
d}m_{v}^{A}+c_{5}\right)\left|\int_{0}^{u}h_{2}(u-v)({\rm d}m_{v}^{B}-{\rm
d}\widetilde{m}_{v}^{B})\right|{\rm d}u$ $\displaystyle\hskip
56.9055pt+c_{3}\alpha\int_{0}^{t}\left|\int_{0}^{u}h_{1}(u-v)({\rm
d}m_{v}^{A}-{\rm d}\widetilde{m}_{v}^{A})\right|{\rm d}u.$

Using here the uniform bound (65) to control the first term above, we obtain

$\displaystyle v_{A}(t)$ $\displaystyle\leq(1-\alpha)\left(c_{4}\alpha
C_{0}\left\|h_{1}\right\|_{1,T}+c_{5}\right)\int_{0}^{t}h_{2}(t-u)v_{B}(u){\rm
d}u+c_{3}\alpha\int_{0}^{t}h_{1}(t-u)v_{A}(u){\rm d}u.$ (66)

Similarly, using the Lipschitz continuity of $G$ (12), we obtain secondly

$\displaystyle v_{B}(t)$ $\displaystyle=\int_{0}^{t}\left|{\rm
d}(m_{u}^{B}-\widetilde{m}_{u}^{B})\right|\leq
c_{6}(1-\alpha)\int_{0}^{t}h_{3}(t-u)v_{B}(u){\rm
d}u+c_{6}\alpha\int_{0}^{t}h_{4}(t-u)v_{A}(u){\rm d}u.$ (67)

Gathering (66) and (67) and applying Gronwall’s inequality (recall [23], Lemma
23, (i)) gives uniqueness of a solution to (16).

We now turn to the existence part. We follow here closely the argument of
[23], Lemmas 23 and 24: consider the sequence
$(m_{A}^{0},m_{B}^{0})\equiv(0,0)$ and for $n\geq 0$,

$\begin{cases}m_{t}^{A,n+1}&=\int_{0}^{t}{F\left(\alpha\int_{0}^{s}h_{1}(s-u){\rm
d}m_{u}^{A,n},(1-\alpha)\int_{0}^{s}h_{2}(s-u){\rm d}m_{u}^{B,n}\right)}{\rm
d}s,\\\ m_{t}^{B,n+1}&=\int_{0}^{t}G\left((1-\alpha)\int_{0}^{s}h_{3}(s-u){\rm
d}m_{u}^{B,n},\alpha\int_{0}^{s}h_{4}(s-u){\rm d}m_{u}^{A,n}\right){\rm
d}s,\end{cases}$

Using (9) and (13), we obtain that

$\displaystyle m_{t}^{A,n+1}$ $\displaystyle\leq
c_{2}t+c_{1}\left\\{\alpha\int_{0}^{t}h_{1}(t-u)m_{u}^{A,n}{\rm
d}u+(1-\alpha)\int_{0}^{t}h_{2}(t-u)m_{u}^{B,n}{\rm d}u\right\\},$
$\displaystyle m_{t}^{B,n+1}$ $\displaystyle\leq
c_{7}t+c_{6}\left\\{(1-\alpha)\int_{0}^{t}h_{3}(t-u)m_{u}^{B,n}{\rm
d}u+\alpha\int_{0}^{t}h_{4}(t-u)m_{u}^{A,n}{\rm d}u\right\\},$

so that $M^{n}(t):=m_{t}^{A,n}+m_{t}^{B,n}$ satisfies, for some constant $C>0$

$\displaystyle M^{n+1}(t)\leq Ct+C\int_{0}^{t}H(t-u)M^{n}(u){\rm d}u.$

Applying [23], Lemma 23, (iii), we obtain that $\sup_{t\in[0,T]}\sup_{n\geq
0}M^{n}(t)$ is bounded by some constant $C>0$. Using this uniform estimate and
the hypotheses on $F$ and $G$ gives, in a similar way to the uniqueness part,
that for $\delta_{t}^{n}:=\int_{0}^{t}\left|{\rm
d}(m_{u}^{A,n+1}-m_{u}^{A,n})\right|+\int_{0}^{t}\left|{\rm
d}(m_{u}^{B,n+1}-m_{u}^{B,n})\right|$,

$\delta_{t}^{n+1}\leq C\int_{0}^{t}H(t-u)\delta_{u}^{n}{\rm d}u.$ (68)

By [23], Lemma 23, (ii), this gives that $\sum_{n\geq
0}\delta_{t}^{n}<\infty$, which implies the existence of a locally bounded
trajectory $(m_{t}^{A},m_{t}^{B})$ such that
$\lim_{n\to\infty}\int_{0}^{t}\left|{\rm
d}(m_{u}^{A}-m_{u}^{A,n})\right|+\int_{0}^{t}\left|{\rm
d}(m_{u}^{B}-m_{u}^{B,n})\right|=0$. This provides existence of a solution to
(16). It remains to check that it is $\mathcal{C}^{1}$. We easily see that
$(m_{A}^{n},m_{B}^{n})$ is $\mathcal{C}^{1}$, by induction, with a bound on
$\sup_{u\in[0,T]}\left|(m_{u}^{A,n})^{\prime}\right|+\left|(m_{u}^{B,n})^{\prime}\right|$
that is uniform in $n$. Using this uniform bound and the properties of $F$ and
$G$, we directly see that $((m^{A,n,})^{\prime},(m^{B,n})^{\prime})$ is Cauchy
for the uniform convergence on $[0,T]$, which gives the desired conclusion.

####  7.2.2 Well-posedness of the mean-field process

Now turn to the mean-field limit (15). Suppose that such a process
$(\bar{Z}_{t}^{A},\bar{Z}_{t}^{B})$ exists. Then, taking expectation in (15),
we see that $(\mathbb{E}(\bar{Z}_{t}^{A}),\mathbb{E}(\bar{Z}_{t}^{B}))$ solves
(16) and is thus uniquely defined, by Proposition 2.7. Hence, any solution to
(15) solves necessarily

$\begin{cases}\bar{Z}_{t}^{A}&=\int_{0}^{t}\int_{0}^{\infty}\mathbf{1}_{z\leqslant
F\left(\alpha\int_{0}^{s}h_{1}(s-u){\rm
d}m_{u}^{A},(1-\alpha)\int_{0}^{s}h_{2}(s-u){\rm d}m_{u}^{B}\right)}\pi({\rm
d}s,{\rm d}z),\\\
\bar{Z}_{t}^{B}&=\int_{0}^{t}\int_{0}^{\infty}\mathbf{1}_{z\leqslant
G\left((1-\alpha)\int_{0}^{s}h_{3}(s-u){\rm
d}m_{B,u},\alpha\int_{0}^{s}h_{4}(s-u){\rm
d}m_{u}^{A}\right)}\widetilde{\pi}({\rm d}s,{\rm d}z),\end{cases}$ (69)

which gives uniqueness. Concerning existence, consider the solution
$(\bar{Z}_{A,t},\bar{Z}_{t}^{B})$ to (69). Then
$(\mathbb{E}(\bar{Z}_{t}^{A}),\mathbb{E}(\bar{Z}_{t}^{B}))$ verifies

$\begin{cases}\mathbb{E}\left(\bar{Z}_{t}^{A}\right)&=\int_{0}^{t}{F\left(\alpha\int_{0}^{s}h_{1}(s-u){\rm
d}m_{u}^{A},(1-\alpha)\int_{0}^{s}h_{2}(s-u){\rm d}m_{u}^{B}\right)}{\rm
d}s,\\\
\mathbb{E}(\bar{Z}_{t}^{B})&=\int_{0}^{t}\left\\{G\left((1-\alpha)\int_{0}^{s}h_{3}(s-u){\rm
d}m_{u}^{B},\alpha\int_{0}^{s}h_{4}(s-u){\rm d}m_{u}^{A}\right)\right\\}{\rm
d}s,\end{cases}$

so that
$(\mathbb{E}(\bar{Z}_{t}^{A}),\mathbb{E}(\bar{Z}_{t}^{B}))=(m_{t}^{A},m_{t}^{B})$
since $(m^{A},m^{B})$ solves (16). This gives existence and Proposition 2.7 is
proven.

###  7.3 Proof of propagation of chaos

We prove here Proposition 2.8. Denote by

$\displaystyle\Lambda_{N,t}^{(A,1)}$
$\displaystyle:=\frac{1}{N}\underset{1\leqslant j\leqslant
N_{A}}{\sum}\int_{0}^{t^{-}}h_{1}(t-u){{\rm d}Z_{u}^{j}},\ $
$\displaystyle\Lambda_{N,t}^{(B,2)}$
$\displaystyle:=\frac{1}{N}\underset{N_{A}+1\leqslant j\leqslant
N}{\sum}\int_{0}^{t^{-}}h_{2}(t-u){{\rm d}Z_{u}^{j}},$
$\displaystyle\Lambda_{N,t}^{(B,3)}$
$\displaystyle:=\frac{1}{N}\underset{N_{A}+1\leqslant j\leqslant
N}{\sum}\int_{0}^{t^{-}}h_{3}(t-u){{\rm d}Z_{u}^{j}},\ $
$\displaystyle\Lambda_{N,t}^{(A,4)}$
$\displaystyle:=\frac{1}{N}\underset{1\leqslant j\leqslant
N_{A}}{\sum}\int_{0}^{t^{-}}h_{4}(t-u){{\rm d}Z_{u}^{j}},$

so that (6) becomes

$\begin{cases}Z_{t}^{i}&=\int_{0}^{t}\int_{0}^{\infty}\mathbf{1}_{z\leqslant
F\left(\Lambda_{n,s}^{(A,1)},\Lambda_{N,s}^{(B,2)}\right)}\pi_{i}({\rm
d}s,{\rm d}z),\ 1\leqslant i\leqslant N_{A},\\\
Z_{t}^{i}&=\int_{0}^{t}\int_{0}^{\infty}\mathbf{1}_{z\leqslant
G\left(\Lambda_{N,s}^{(B,3)},\Lambda_{N,s}^{(A,4)}\right)}\widetilde{\pi}_{i}({\rm
d}s,{\rm d}z),\ N_{A}+1\leqslant i\leqslant N.\end{cases}$

Denote also by

$\displaystyle\Lambda_{t}^{(A,1)}$
$\displaystyle:=\alpha\int_{0}^{t}h_{1}(t-u){\rm
d}\mathbb{E}(\bar{Z}_{u}^{A}),\ $ $\displaystyle\Lambda_{t}^{(B,2)}$
$\displaystyle:=(1-\alpha)\int_{0}^{t}h_{2}(t-u){\rm
d}\mathbb{E}\left(\bar{Z}_{u}^{B}\right),$ $\displaystyle\Lambda_{t}^{(B,3)}$
$\displaystyle:=(1-\alpha)\int_{0}^{t}h_{3}(t-u){\rm
d}\mathbb{E}\left(\bar{Z}_{u}^{B}\right),\ $
$\displaystyle\Lambda_{t}^{(A,4)}$
$\displaystyle:=\alpha\int_{0}^{t}h_{4}(t-u){\rm
d}\mathbb{E}\left(\bar{Z}_{u}^{A}\right),$

so that the limiting system (15) becomes

$\begin{cases}\bar{Z}_{t}^{A}&=\int_{0}^{t}\int_{0}^{\infty}\mathbf{1}_{z\leqslant
F\left(\Lambda_{s}^{(A,1)},\Lambda_{s}^{(B,2)}\right)}\pi({\rm d}s,{\rm
d}z),\\\ \bar{Z}_{t}^{B}&=\int_{0}^{t}\int_{0}^{\infty}\mathbf{1}_{z\leqslant
G\left(\Lambda_{s}^{(B,3)},\Lambda_{s}^{(A,4)}\right)}\widetilde{\pi}({\rm
d}s,{\rm d}z).\end{cases}$

Let
$((\bar{Z}^{i,A})_{i=1,\ldots,N_{A}},(\bar{Z}^{i,B})_{i=N_{A}+1,\ldots,N})$ be
the coupling driven by the same Poisson point processes $\pi_{i}$,
$i=1,\ldots,N_{A}$ and $\widetilde{\pi}_{i}$, $i=N_{A}+1,\ldots,N$ as for (6).
With a slight abuse of notations, we use the same $\bar{Z}^{i}$ for
$\bar{Z}^{i,A}$ or $\bar{Z}^{i,B}$ depending on wether $i=1,\ldots,N_{A}$ or
$i=N_{A}+1,\ldots,N$. For all $i=1,\ldots,N$, denote by
$\Delta_{N}^{i}(t):=\int_{0}^{t}\left|d\left(Z^{i}_{u}-\bar{Z}^{i}_{u}\right)\right|$
the total variation distance between processes $Z^{i}$ and $\bar{Z}^{i}$ and
$\delta_{N}(t):=\mathbf{E}\left[\Delta_{N}^{i}(t)\right]$. Firstly, for
$i=1,\ldots,N_{A}$:

$\displaystyle\Delta_{N}^{i}(t)$
$\displaystyle=\int_{0}^{t}\left|\int_{0}^{\infty}\mathbf{1}_{z\leq
F\left(\Lambda_{n,s}^{(A,1)},\Lambda_{N,s}^{(B,2)}\right)}-\mathbf{1}_{z\leq
F\left(\Lambda_{s}^{(A,1)},\Lambda_{s}^{(B,2)}\right)}\right|\pi_{i}({\rm
d}s,{\rm d}z)$

so that

$\displaystyle\delta_{N}(t)$
$\displaystyle\leq\int_{0}^{t}\mathbf{E}\left[\left|F\left(\Lambda_{N,s}^{(A,1)},\Lambda_{N,s}^{(B,2)}\right)-F\left(\Lambda_{s}^{(A,1)},\Lambda_{N,s}^{(B,2)}\right)\right|\right]{\rm
d}s$ (70)
$\displaystyle+\int_{0}^{t}\mathbf{E}\left[\left|F\left(\Lambda_{s}^{(A,1)},\Lambda_{N,s}^{(B,2)}\right)-F\left(\Lambda_{s}^{(A,1)},\Lambda_{s}^{(B,2)}\right)\right|\right]{\rm
d}s.$ (71)

Using (10) in (70) and (11) in (71), we obtain

$\displaystyle\delta_{N}(t)$ $\displaystyle\leq
c_{3}\int_{0}^{t}\mathbf{E}\left[\left|\Lambda_{N,s}^{(A,1)}-\Lambda_{s}^{(A,1)}\right|\right]{\rm
d}s+\int_{0}^{t}\left(c_{4}\Lambda_{s}^{(A,1)}+c_{5}\right)\mathbf{E}\left[\left|\Lambda_{N,s}^{(B,2)}-\Lambda_{s}^{(B,2)}\right|\right]{\rm
d}s.$ (72)

Concentrate on the first term in the sum above: one has

$\displaystyle\int_{0}^{t}\mathbf{E}\left[\left|\Lambda_{N,s}^{(A,1)}-\Lambda_{s}^{(A,1)}\right|\right]{\rm
d}s$
$\displaystyle\leq\int_{0}^{t}\mathbf{E}\left[\left|\frac{1}{N}\underset{1\leqslant
j\leqslant N_{A}}{\sum}\int_{0}^{s^{-}}h_{1}(s-u)({{\rm d}Z_{u}^{j}}-{\rm
d}\bar{Z}_{u}^{j})\right|\right]{\rm d}s$
$\displaystyle+\int_{0}^{t}\mathbf{E}\left[\left|\frac{1}{N}\underset{1\leqslant
j\leqslant N_{A}}{\sum}\int_{0}^{s^{-}}h_{1}(s-u){{\rm
d}\bar{Z}_{u}^{j}}-\alpha\int_{0}^{s}h_{1}(s-u){\rm
d}\mathbb{E}(\bar{Z}_{u}^{A})\right|\right]{\rm d}s,$
$\displaystyle\leq\alpha\int_{0}^{t}h_{1}(t-u)\delta_{N}(u){\rm
d}u+\int_{0}^{t}\mathbf{E}\left[\left|\frac{\alpha}{N_{A}}\underset{1\leqslant
j\leqslant
N_{A}}{\sum}\left(X_{j}(s)-\mathbf{E}\left[X_{j}(s)\right]\right)\right|\right]{\rm
d}s,$

where $X_{j}(s):=\int_{0}^{s^{-}}h_{1}(s-u){{\rm d}\bar{Z}_{u}^{j}}$,
$j=1,\ldots,N_{A}$ is a collection of i.i.d. random variables, with
$Var\left[X_{j}(s)\right]=\int_{0}^{s}h_{1}(s-u)^{2}\lambda_{u}^{A}{\rm d}u$,
so that the Cauchy-Schwarz inequality leads to

$\displaystyle\int_{0}^{t}\mathbf{E}\left[\left|\frac{\alpha}{N_{A}}\underset{1\leqslant
j\leqslant
N_{A}}{\sum}\left(X_{j}(s)-\mathbf{E}\left[X_{j}(s)\right]\right)\right|\right]{\rm
d}s$
$\displaystyle\leq\frac{\alpha}{N_{A}^{\frac{1}{2}}}\int_{0}^{t}\left(\int_{0}^{s}h_{1}(s-u)^{2}\lambda_{u}^{A}{\rm
d}u\right)^{\frac{1}{2}}{\rm d}s,$ $\displaystyle\leq\frac{\alpha
t\left\|\lambda^{A}\right\|_{\infty,t}^{\frac{1}{2}}\left\|h_{1}\right\|_{2,t}}{N_{A}^{\frac{1}{2}}}:=\frac{C_{1}(t)}{\sqrt{N}},$

where
$C_{1}(t)\leqslant{\sqrt{\alpha}\left\|\lambda^{A}\right\|_{\infty}^{\frac{1}{2}}\left\|h_{1}\right\|_{2}}t$.
We obtain finally that

$\displaystyle\int_{0}^{t}\mathbf{E}\left[\left|\Lambda_{N,s}^{(A,1)}-\Lambda_{s}^{(A,1)}\right|\right]{\rm
d}s$ $\displaystyle\leq\alpha\int_{0}^{t}h_{1}(t-u)\delta_{N}(u){\rm
d}u+\frac{C_{1}(t)}{\sqrt{N}}$

Note secondly that

$\displaystyle\Lambda_{s}^{(A,1)}=\alpha\int_{0}^{t}h_{1}(t-u)\lambda_{u}^{A}{\rm
d}u$
$\displaystyle\leq\alpha\left\|\lambda^{A}\right\|_{\infty,T}\left\|h_{1}\right\|_{1,t}:=C_{2}(t),$

where $C_{2}(t)\leqslant\kappa_{1}\left\|\lambda^{A}\right\|_{\infty}$. Hence,
proceeding similarly, there exists $C_{3}(t)>0$ such that

$\displaystyle\int_{0}^{t}\left(c_{4}\Lambda_{s}^{(A,1)}+c_{5}\right)\mathbf{E}\left[\left|\Lambda_{N,s}^{(B,2)}-\Lambda_{s}^{(B,2)}\right|\right]{\rm
d}s$
$\displaystyle\leq(c_{4}C_{2}(t)+c_{5})\left\\{\int_{0}^{t}(1-\alpha)h_{2}(t-u)\delta_{N}(u){\rm
d}u+\frac{C_{3}(t)}{\sqrt{N}}\right\\},$

where
$C_{3}(t)\leqslant(1-\alpha)\|\lambda^{B}\|_{\infty}^{\frac{1}{2}}\|h_{2}\|_{2}$.
Collecting everything into (72), gives the existence of a constant
$C_{4}(t)>0$ such that, for all $i=1,\ldots,N_{A}$,

$\displaystyle\delta_{N}(t)$ $\displaystyle\leq
C_{4}(t)\int_{0}^{t}\left\\{\alpha
h_{1}(t-u)+(1-\alpha)h_{2}(t-u)\right\\}\delta_{N}(u){\rm
d}u+\frac{C_{5}(t)}{\sqrt{N}},$

where $C_{4}(t)=c_{3}+c_{4}C_{2}(t)+c_{5}$ and
$C_{5}(t)=c_{3}C_{1}(t)+(c_{4}C_{2}(t))+c_{5})C_{3}(t)\leqslant Ct.$ We now
turn to the similar treatment of $i=N_{A}+1,\ldots,N$: for such $i$, using the
global Lipschitz continuity of $G$ (12)

$\displaystyle\delta_{N}(t)$ $\displaystyle\leq
c_{6}\int_{0}^{t}\mathbf{E}\left[\left|\Lambda_{N,s}^{(B,3)}-\Lambda_{s}^{(B,3)}\right|+\left|\Lambda_{N,s}^{(A,4)}-\Lambda_{s}^{(A,4)}\right|\right]{\rm
d}s.$

Proceeding in the same way as before, we obtain, for some constant
$C_{6}(t)>0$

$\displaystyle\delta_{N}(t)$ $\displaystyle\leq
c_{6}\int_{0}^{t}\left\\{(1-\alpha)h_{3}(t-u)+\alpha
h_{4}(t-u)\right\\}\delta_{N}(u){\rm d}u+\frac{C_{6}(t)}{\sqrt{N}},$

where
$C_{6}(t)\leqslant({\sqrt{1-\alpha}\left\|\lambda^{B}\right\|_{\infty}^{\frac{1}{2}}\left\|h_{3}\right\|_{2}}+{\sqrt{\alpha}\left\|\lambda^{A}\right\|_{\infty}^{\frac{1}{2}}\left\|h_{4}\right\|_{2}})t$.
This gives finally that, for some constants $C>0$ and $\overline{C}$

$\displaystyle\delta_{N}(t)$
$\displaystyle\leq\overline{C}\int_{0}^{t}H(t-u)\delta_{N}(u){\rm
d}u+C\frac{t}{\sqrt{N}},$

for $H$ given in (8) and
$\overline{C}\leqslant\max\\{(c_{3}+c_{4}\kappa_{1}\|\lambda^{A}\|_{\infty}+c_{5})(\kappa_{1}+\kappa_{2}),c_{6}(\kappa_{3}+\kappa_{4})\\}$.
Observing that
$\Delta^{i}_{N}(t)\geqslant\sup_{[0,t]}|Z^{i}_{u}-\bar{Z}^{i}_{u}|$ and
applying Lemma 23 of [23] gives that

$\displaystyle\sup_{i=1,\ldots,N}\mathbf{E}\left[\sup_{u\in[0,t]}\left|Z_{t}^{i}-\bar{Z}_{t}^{i}\right|\right]\leq\frac{C(t)}{\sqrt{N}}.$
(73)

Additionally, if $\overline{C}<1$, using that $\delta_{N}$ is non decreasing
it holds that $C(t)\leqslant Ct$ for some constant $C$ independent of $t$.
This concludes the proof of Proposition 2.8.

##  8 Proofs for the longtime dynamics

We prove here the results of Section 3. Recall that we consider the system
(19) under Assumption 3.1. Recall notations (23). For $t\geq 0$, introduce
$u_{A}(t):=\sup_{s\geq t}\lambda_{s}^{A}$, $u_{B}(t)=\sup_{s\geq
t}\lambda_{s}^{B}$, $v_{A}(t):=\inf_{s\geq t}\lambda_{s}^{A}$ and
$v_{B}(t):=\inf_{s\geq t}\lambda_{s}^{B}$. Both $t\mapsto u_{A}(t)$ and
$t\mapsto u_{B}(t)$ (resp. $t\mapsto v_{A}(t)$ and $t\mapsto v_{B}(t)$) are
non-increasing (resp. non-decreasing), and we have that
$u_{A}(t)\xrightarrow[t\to\infty]{}\bar{\ell}_{A}$,
$u_{B}(t)\xrightarrow[t\to\infty]{}\bar{\ell}_{B}$,
$v_{A}(t)\xrightarrow[t\to\infty]{}\underline{\ell}_{A}$ and
$v_{B}(t)\xrightarrow[t\to\infty]{}\underline{\ell}_{B}$.

###  8.1 Some a priori estimates and first proofs

####  8.1.1 General estimates

######  Remark 8.1.

We state a bound that is constantly used in the following: for any kernel
$h\geq 0$ and function $\lambda\geq 0$, for $v(u):=\inf_{u\geq s}\lambda(u)$
and any $s\geq 0$,

$\displaystyle\int_{0}^{s}h(s-u)\lambda(u){\rm d}u\geq
v(s/2)\int_{s/2}^{s}h(s-u){\rm d}u=v(s/2)\int_{0}^{s/2}h(u){\rm d}u.$

A first a priori estimate is given by Proposition 8.2:

######  Proposition 8.2.