# Global solutions with infinitely many blowups in a mean-field neural network

Lorenzo<EMAIL_ADDRESS>[ Thibaud
<EMAIL_ADDRESS>[ Department of Mathematics,
University of Texas, Austin,  Department of Neuroscience, University of Texas,
Austin,

###### Abstract

We recently introduced idealized mean-field models for networks of integrate-
and-fire neurons with impulse-like interactions—the so-called delayed
Poissonian mean-field models. Such models are prone to blowups: for a strong
enough interaction coupling, the mean-field rate of interaction diverges in
finite time with a finite fraction of neurons spiking simultaneously. Due to
the reset mechanism of integrate-and-fire neurons, these blowups can happen
repeatedly, at least in principle. A benefit of considering Poissonian mean-
field models is that one can resolve blowups analytically by mapping the
original singular dynamics onto uniformly regular dynamics via a time change.
Resolving a blowup then amounts to solving the fixed-point problem that
implicitly defines the time change, which can be done consistently for a
single blowup and for nonzero delays. Here we extend this time-change analysis
in two ways: First, we exhibit the existence and uniqueness of explosive
solutions with a countable infinity of blowups in the large interaction
regime. Second, we show that these delayed solutions specify “physical”
explosive solutions in the limit of vanishing delays, which in turn can be
explicitly constructed. The first result relies on the fact that blowups are
self-sustaining but nonoverlapping in the time-changed picture. The second
result follows from the continuity of blowups in the time-changed picture and
incidentally implies the existence of periodic solutions. These results are
useful to study the emergence of synchrony in neural network models.

60G99,

60K15, 35Q92, 35D30, 35K67, 45H99,

mean-field neural network models; blowups in parabolic partial differential
equation; regularization by time change; singular interactions; delayed
integral equations; inhomogeneous renewal processes,

second keyword,

###### keywords:

[class=MSC2020]

###### keywords:

and

##  1 Introduction

###  1.1 Background

In this work, we consider idealized neural-network models called delayed
Poissonian mean-field (dPMF) models introduced in [22]. These dPMF models are
variations of classical mean-field models [5, 4, 7], whose dynamics are also
prone to blowups. From a modeling perspective, blowups correspond to the
occurrence of synchronous, macroscopic spiking events in a neural network.
Understanding the emergence of these synchronous events is of interest for
studying the maintenance of precise temporal information in neural networks
[19, 15, 3]. However, blowups resist direct analytical treatment in classical
mean-field models [10, 9, 13, 18, 17]. This observation is the primary
motivation justifying the introduction of dPMF dynamics, whose blowups prove
analytically tractable.

In [22], we conjectured dPMF dynamics as the mean-field limit of finite-size
particle systems with singular interactions (see Fig. 1). In a finite-size
system of $N$ particles, each particle $i$, $1\leq i\leq N$, represents a
neuron with time-dependent state variable $X_{N,i,t}$. These state variables
evolve jointly according to a $\mathbbm{R}^{N}$-valued continuous-time process
$t\mapsto\\{X_{N,i,t}\\}_{1\leq i\leq N}$. Specifically, the network dynamics
is parametrized by the drift value $\nu>0$, the refractory period
$\epsilon>0$, and the interaction parameter $\lambda>0$ as follows: $(i)$
Whenever a process $X_{N,i,t}$ hits the spiking boundary at zero, it
instantaneously enters its inactive refractory state. $(ii)$ At the same time,
all the other active processes $X_{N,j,t}$ (which are not in the inactive
refractory state) are respectively updated by amounts $-w_{N,ij,t}$, where
$w_{N,ij,t}$ is independently drawn from a normal law with mean and variance
equal to $\lambda/N$. $(iii)$ After an inactive (refractory) period of
duration $\epsilon>0$, the process $X_{N,i,t}$ restarts its autonomous
stochastic dynamics from the reset state $\Lambda>0$. $(iv)$ In between
spiking/interaction times, the autonomous dynamics of active processes follow
independent drifted Wiener processes with negative drift $-\nu$.
Correspondingly, an initial condition for the network is specified by the
starting values of the active processes, i.e., $X_{N,i,0}>0$ if $i$ is active,
and the last inactivation time of the inactive processes, i.e.,
$-\epsilon\leq\rho_{i,0}\leq 0$, if $i$ is inactive. Following the above
definition, the finite-size versions of dPMF dynamics exhibit two key features
(see Fig. 1): $(1)$ recurrent interactions implement self-excitation whose
strength is quantified by the interaction parameter $\lambda$; $(2)$ owing to
the post-spiking refractory period $\epsilon>0$, individual neuronal processes
follow delayed dynamics. Parenthetically, the inclusion of a negative drift
$\nu$ is necessary to ensure that neurons will spike in finite time,
independent of the initial conditions and of the coupling strength.

By analogy with [7, 8], dPMF dynamics are deduced from finite-size ones by
conjecturing propagation of chaos in the infinite-size limit $N\to\infty$,
which is supported by numerical simulations (see Fig. 2). The propagation of
chaos states that for exchangeable initial conditions, the processes
$X_{N,i,t}$, $1\leq i\leq N$, become i.i.d. in the limit of infinite-size
networks $N\to\infty$, so that each individual process follows a mean-field
dynamics [21]. For exchangeable initial conditions, a representative dPMF
process $X_{t}=\lim_{N\to\infty}X_{N,i,t}$ is governed by the deterministic
cumulative drift $\Phi$ that amalgamates the contribution of the autonomous
drift $-\nu$ and the mean-field contribution of neuronal interactions via
$\lambda$. Specifically, we have $\Phi(t)=\nu
t+\lambda\mathbb{E}\left[M_{t}\right]$, where the process $M_{t}$ counts the
successive first-passage times of the representative process $X_{t}$ to the
zero spiking threshold. Such a process is defined as

$\displaystyle
M_{t}=\sum_{n>0}\mathbbm{1}_{[0,t]}(\rho_{n})\,,\quad\mathrm{with}\quad\rho_{n+1}=\inf\left\\{t>r_{n}=\rho_{n-1}+\epsilon\,\big{|}\,X_{t}\leq
0\right\\}\,,$ (1)

with initial conditions bearing on $X_{0}$ or $\rho_{0}$ depending on the
representative process being initially active or inactive. We will later
specify such initial conditions in detail. Note that in the above definition,
the first-passage times $\rho_{n}$, $n\geq 1$, denote successive inactivation
times, whereas $r_{n}=\rho_{n}+\epsilon$, $n>0$, denotes the corresponding
sequence of reset times. For mediating all interactions, the function
$t\mapsto\mathbb{E}\left[M_{t}\right]$ plays a central role in our analysis.
We denote this function by $F$ and observe that $F$ is defined as an
increasing _càdlàg_ function. Remember that a function is said to be _càdlàg_
if it is right-continuous with left limits. Then, the hallmark of dPMF
dynamics is that the drift $\Phi$ impacts the representative neuron
stochastically via a process $Z_{t}=\Phi(t)+W_{\Phi(t)}$, where $W$ is a
canonical, driving Wiener process. Accordingly, dPMF dynamics are referred to
as Poissonian because the process $Z_{t}$ represents the diffusive limit of a
Poisson counting process with
$\mathbb{E}\left[Z_{t}\right]=\mathbb{V}\left[Z_{t}\right]$ for all $t\geq 0$.
With these conventions, the stochastic dPMF dynamics of a representative
process is given by

$\displaystyle
X_{t}=X_{0}-\int_{0}^{t}\mathbbm{1}_{\\{X_{s^{-}}>0\\}}\mathrm{d}Z_{s}+\Lambda
M_{t-\epsilon}\,.$ (2)

The above equation fully defines dPMF dynamics: The integral term indicates
that when the neuron is active, for $X_{t}>0$, its state evolves according to
the Poissonian diffusion process $Z_{t}$. The delayed term indicates that
after hitting the spiking boundary at zero at time $\tau$, the process remains
inactive until it resets at $\Lambda$ after a duration $\epsilon$: $X_{t}=0$
for $\tau\leq t<\tau+\epsilon$.

Because of their self-exciting nature, dPMF dynamics are prone to
blowups/synchronous events for large enough interaction parameter and/or for
initial conditions that are concentrated near the zero spiking boundary.
Blowups occur at those times when $f$, the instantaneous spiking rate of a
representative neuron, diverges. Formally, the firing rate $f$ is defined in
the distribution sense as the Radon-Nikodym derivative of $F$ with respect to
the Lebesgue measure: $f=\mathrm{d}F/\mathrm{d}t$. Then, assuming $f$ to be
finite in the left vicinity of $T_{1}$, $T_{1}$ is a blowup time if
$\lim_{t\to T_{1}^{-}}f(t)=\infty$. In turn, synchronous events occur at those
times $T_{1}$ for which a finite fraction of processes spikes simultaneously
or equivalently, for which a representative neuron spikes with nonzero
probability: $\pi_{1}=\mathbb{P}\left[X_{T_{1}}=0\right]>0$. Formally, this
corresponds to $F$ admitting a jump discontinuity at $T_{1}$ so that
$F(T_{1})-F(T_{1}^{-})=\lambda\pi_{1}$.

Figure 1:  Delayed Poissonian network dynamics. a. Schematic representation of
the delayed Poissonian network dynamics in a finite-size particel system with
$N=3$ interacting processes. In between spiking events, neuronal dynamics are
that of independent Wiener processes with negative drift $-\nu<0$. Neuron $2$
spikes and inactivates at time $t_{1}$, which leads to updating neuron $1$ and
$3$ via weights $w_{21,t_{1}}$ and $w_{23,t_{1}}$ i.i.d. with normal law
$\mathcal{N}(\lambda/N,\lambda/N)$. This causes neuron $1$ to spike and
inactivate, leading to another instantaneous update of neuron $3$ via a random
weight $w_{13,t_{1}}$. Then, neuron $3$ spikes and inactivates at a later time
$t_{2}$, but this has no impact on neurons $1$ and $2$ as they remain inactive
during a refractory period of duration $\epsilon>t_{2}-t_{1}$. At the end of
their refractory period, neurons reset at $\Lambda>0$. Finally, the
spontaneous spiking of neuron $1$ in $t_{3}$ cause both neurons $2$ and $3$ to
spike at the same time. b. Schematic representation of the conjectured delayed
Poissonian mean-field (dPMF) dynamics in the limit of infinite-size networks
$N\to\infty$. When $N\to\infty$, it is conjectured that the neuronal dynamics
become independent and that interactions are mediated by the mean-field
deterministic cumulative drift $\Phi(t)=\nu t+\lambda F(t)$, where the
increasing function $F$ denotes the cumulative spiking or inactivation rate.
Specifically, it is conjectured that neuronal dynamics follow independent
time-changed Wiener processes with Poissonian attributes parametrized by
$\Phi$. For time $t<T_{1}$, the (Radon-Nikodym) derivative of $\Phi$ remains
finite and the spiking of representative processes are determined via first-
passage time problems with regular boundary. Accordingly, two representative
processes have zero probability to spike synchronously. However, for large
enough interaction parameter $\lambda$, it is possible that the (Radon-
Nikodym) derivative of $\Phi$ locally diverges and that a finite fraction
$\pi_{1}$, $0<\pi_{1}<1$, of the processes synchronously spike as depicted in
$T_{1}$. We refer to such a possibility as a full-blowup event. This
corresponds to a jump discontinuity of size $\lambda\pi_{1}$ in $\Phi$ and to
singular stochastic updates of the representative processes with weight
$w_{1,T_{1}}$ and $w_{2,T_{1}}$ i.i.d. normal law
$\mathcal{N}(\lambda\pi_{1},\lambda\pi_{1})$.

The main interest of considering dPMF dynamics is that blowups and synchronous
events are analytically tractable under some reasonable restrictions about the
initial conditions. Specifically, denoting by $\mathcal{M}(I)$ the set of
positive measure on an interval $I\subset\mathbbm{R}$, we assume as in [22]
that:

######  Assumption 1.

The initial conditions for dPMF dynamics is specified by

$\displaystyle\mathbb{P}\left[X_{0}\in\mathrm{d}x|X_{0}>0\right]=p_{0}(\mathrm{d}x)\quad\mathrm{and}\quad\mathbb{P}\left[\rho_{0}\in\mathrm{d}t|X_{0}=0\right]=f_{0}(\mathrm{d}t)\,,$

where $(p_{0},f_{0})$ in
$\mathcal{M}((0,\infty))\times\mathcal{M}([-\epsilon,0))$ is normalized, i.e.,
$\|p_{0}\|_{1}+\|f_{0}\|_{1}=1$ and such that $p_{0}$ is locally
differentiable in zero with $\lim_{x\to 0^{+}}p_{0}(x)=0$ and $\lim_{x\to
0^{+}}\partial_{x}p_{0}(x)/2<1/\lambda$.

The above initial conditions guarantee that there exists a dPMF dynamics
locally solving (2) and that this dynamics is initially smooth in the sense
that $F$ is an infinitely differentiable function in the right vicinity of
zero. Such a smooth solution can be maximally continued on a possibly infinite
interval $[0,T_{1})$, where $T_{1}$ marks the occurrence of the first blowup.
In [22], we show that for all $0<t<T_{1}$, the density $x\mapsto
p(x,t)=\mathbb{P}\left[X_{t}\in\mathrm{d}x\,|X_{t}>0\right]/\mathrm{d}x$ is
smooth in the right vicinity of zero with $\lim_{x\to 0^{+}}p(x)=0$ and
$\lim_{x\to 0^{+}}\partial_{x}p(t,x)/2<1/\lambda$. As the later limits are
always well defined and finite, we will simply refer to their values as $p(0)$
and $\partial_{x}p(t,0)/2$ for conciseness. With this in mind, we show in [22]
that for all $0<t<T_{1}$, the instantaneous spiking rate is given by

$\displaystyle
f(t)=\frac{\nu\partial_{x}p(0,t)}{2-\lambda\partial_{x}p(0,t)}\,.$

This leads to introducing the following criterion for first-blowup times:

######  Definition 1.1.

Under Assumption 1, the first blowup time is defined as

$\displaystyle T_{1}=\sup\\{t>0\,|\,\partial_{x}p(t,0)<2/\lambda\\}\,.$

The above criterion only bears on the local divergence of the firing rate $f$
and is silent about the possible occurrence of a synchronous event. To further
characterize blowups, we introduce in [22] the so-called full-blowup
criterion:

######  Assumption 2.

At the blowup time $T_{1}$, the density function $x\mapsto p(x,t)$ satisfies

$\displaystyle\lim_{t\to T_{1}^{-}}\partial^{3}_{x}p(t,0)>8/\lambda\,.$

The full-blowup criterion, which generically holds with respect to the choice
of initial conditions, allowed us to characterize blowups in dPMF dynamics as
follows:

######  Proposition 1.1.

Under Assumptions 1 and 2, the instantaneous rate blows up as $f(t)\sim
1/\sqrt{T_{1}-t}$ when $t\to T_{1}^{-}$ and triggers a synchronous event in
$T_{1}$ of size

$\displaystyle\pi_{1}=\inf\left\\{p>0\,|\,p>\mathbb{P}\left[\tau_{X_{T_{1}}}<\lambda
p\right]\right\\}>0\,,$ (3)

where $\tau_{X_{T_{1}}}$ denotes the first-passage time in zero of a Wiener
process with unit negative drift and started with random initial condition
$X_{T_{1}^{-}}$.

Thus, for generic initial conditions, blowups correspond to left Hölder
singularity of exponent $1/2$ in the cumulative function $F$, followed by a
jump discontinuity whose size $\lambda\pi_{1}$ can be determined as the
solution of the self-consistent problem (3). In [22], we show that the latter
problem directly follows from the requirement of conservation of probability
during blowups.

Figure 2: Simulations of finite-size interacting particle systems. We simulate
a finite network of neurons which follows the interacting dynamics depicted in
Fig. 1 and for varying size $N=10,100,10000$. Parameters: $\Lambda=1$,
$\nu=1$, $\lambda=20$ and $\epsilon=0$. Initial conditions: all neurons start
at reset value. In each panel: $(1)$ the top graphics represents a raster plot
of the neuronal activity, where the spiking of neuron $i$ at time $t$ is
marked by a point at coordinate $(i,t)$. Synchronous event corresponds to
vertically aligned points which are apparent for all conditions. $(2)$ the
bottom left graphics represents the histogram of time interval between
synchronous events. $(3)$ the bottom right graphics represents the spectral
density of the overall network activity. Making the approximation that the
full blowup has size $\pi_{1}=1$ and neglecting reset besides at $S_{1}$
yields the mean-field predictions: $1/T_{1}\simeq 16.18$ with
$T_{1}=(S_{1}-\lambda H(S_{1},\Lambda))/\nu\simeq 0.07143$, where $S_{1}$ is
the smallest solution of $h(S_{1},\Lambda)=1/\lambda$. The level of synchrony
increases with the size of the system $N$, supporting that the particle system
admits a (periodic) mean-field dynamics in the limit $N\to\infty$.

###  1.2 Motivation

Unfortunately, the blowup analysis of [22] is only local in the sense that it
allows one to resolve a single blowup for generic initial conditions. However,
this local analysis provides one with natural blowup exit conditions. In
principle, these blowup exit conditions can also serve as initial conditions
to analyze the next blowup, if any. Numerical simulations suggest that for
large enough interaction parameters $\lambda>\Lambda$, explosive solutions
exhibit repeated blowup episodes at regular time intervals (see Fig. 2).
Following on this observation, the first goal of this work is to extend the
analysis of [22] to show the existence of global dPMF solutions that are
defined over the entire half-line $\mathbbm{R}^{+}$ and that exhibit a
countable infinity of blowups. This program involves proving that for large
enough interaction parameters, iteratively applying the blowup analysis of
[22] produces synchronous events with sizes that remain bounded away from
zero, at time intervals that also remain bounded away from zero. Moreover, the
blowup analysis of [22] is only concerned with dPMF dynamics for positive
refractory period $\epsilon>0$. With positive refractory period $\epsilon>0$,
explosive dPMF dynamics are always well posed but at the analytical cost of
being determined as delayed dynamics. For small enough $\epsilon>0$, we do not
expect the refractory period to impact dPMF dynamics, except for ensuring
their well-posedness in the presence of blowups. This suggests defining
explosive Poissonian mean-field (PMF) dynamics in the absence of a refractory
period as the limit dPMF dynamics obtained when $\epsilon\to 0^{+}$.
Accordingly, the second goal of this work is to prove the existence and
uniqueness of such limit dPMF dynamics. This will only be possible for large
enough interaction parameters, when global dPMF dynamics sustain isolated
blowups for small enough $\epsilon>0$. PMF dynamics with zero refractory
period $\epsilon=0$ are of interest for analysis as their blowups can be
resolved just as for $\epsilon>0$, whereas their inter-blowup dynamics are
nondelayed.

###  1.3 Time-change approach

Figure 3:  Time-changed picture. a. In the absence of blowups, the cumulative
drift $\Phi$ specifies an increasing function which can be interpreted as an
invertible time change. Denoting the inverse time change by $\Psi=\Phi^{-1}$
and given a representative process $X_{t}$ of the dPMF dynamics, the process
$Y_{\sigma}=X_{\Psi(\sigma)}$ follows a noninteracting drifted Wiener dynamics
with negative unit drift and nonhomogeneous backward-delay function
$\eta:\sigma\mapsto\sigma-\Phi(\Psi(\sigma)-\epsilon)$. Thus, in the time-
change picture, all the interactions at play in dPMF dynamics are encoded by
the time-dependence of delays. b. In the absence of blowups, full blowups are
marked by a flat section $[U_{1},S_{1})$ of the inverse time change $\Psi$,
where $U_{1}$ is the blowup trigger time and $U_{1}$ is the blowup exit time.
This corresponds to the backward-delay function $\eta$ having a jump
discontinuity of size $S_{1}-U_{1}$ in $U_{1}$. This also means that the
forward-delay function $\tau$ specifying reset times is constant over the
interval $[U_{1},S_{1})$, so that all processes involved in the blowup episode
can reset at the same time $R_{1}$.

In [22], we analytically characterized blowup generation in dPMF dynamics by
mapping the original time-homogenous, nonlinear dynamics onto a time-
inhomogeneous, linear dynamics (see Fig. 3). Such a mapping is operated by
considering the cumulative drift $\Phi$ as an implicitly defined time-change
function

$\displaystyle\sigma=\Phi(t)=\nu t+\lambda F(t)\,,\quad\mathrm{with}\quad
F(t)=\int_{0}^{t}f(s)\,\mathrm{d}s\,,$ (4)

which is only assumed to be a _càdlàg_ increasing function. Due to the
Poisson-like attributes of the neuronal drives, the time change $\Phi$
parametrizes the dPMF dynamics of a representative process as
$X_{t}=Y_{\Phi(t)}$, where the time-changed dynamics $Y_{\sigma}$ obeys a
linear, noninteracting dynamics. The latter dynamics is that of a Wiener
process absorbed at zero, with constant negative unit drift and with reset at
$\Lambda$, but with time-inhomogeneous refractory period specified via a
$\Phi$-dependent, backward-delay function $\sigma\mapsto\eta[\Phi](\sigma)$
(see Fig. 3). Independent of the presence of blowups, the functional
dependence of $\eta$ on the time change $\Phi$ is given by

$\displaystyle\eta(\sigma)=\sigma-\Phi(\Psi(\sigma)-\epsilon)\,,$ (5)

where $\Psi=\Phi^{-1}$ refers to the inverse of the strictly increasing time
change $\Phi$. Given a backward-delay function $\eta$, the transition kernel
of the process $Y_{\sigma}$ denoted by $(\sigma,x)\mapsto
q(\sigma,x)=\mathrm{d}\mathbb{P}\left[Y_{\sigma}\in\mathrm{d}x\,|\,Y_{\sigma}>0\right]/\mathrm{d}x$
satisfies the time-changed PDE problem

$\displaystyle\partial_{\sigma}q$ $\displaystyle=$
$\displaystyle\partial_{x}q+\frac{1}{2}\partial^{2}_{x}q+\frac{\mathrm{d}}{\mathrm{d}\sigma}[G(\sigma-\eta(\sigma))]\delta_{\Lambda}\,,$
(6)

with absorbing and conservation conditions respectively given by

$\displaystyle
q(\sigma,0)=0\quad\mathrm{and}\quad\partial_{\sigma}G(\sigma)=\partial_{x}q(\sigma,0)/2\,.$
(7)

In equations (6) and (7), $G$ denotes the $\eta$-dependent cumulative flux of
$Y_{\sigma}$ through the zero threshold. By definition of the time change
$\Phi$, which is such that $X_{t}=Y_{\Phi(t)}$, $G$ is related to the
cumulative flux $F$ via $F=G\circ\Phi$. A key result of [22] is that as long
as the delay function $\eta$ remains bounded, the PDE problem defined by (6)
and (7) admits a unique solution parametrized by an unconditionally smooth
cumulative function $G=G[\eta]$. This result holds even in the presence of
blowups for the time-changed versions of the initial conditions given in
Assumption 1. These time-changed initial conditions are specified as follows:

######  Definition 1.2.

Given normalized initial conditions $(p_{0},f_{0})$ in
$\mathcal{M}(\mathbbm{R}^{+})\times\mathcal{M}([-\epsilon,0))$, the initial
conditions for the time-changed problem are defined by $(q_{0},g_{0})$ in
$\mathcal{M}(\mathbbm{R}^{+})\times\mathcal{M}([\xi_{0},0))$ such that

$\displaystyle q_{0}=p_{0}\quad\mathrm{and}\quad
g_{0}=\frac{\mathrm{d}G_{0}}{\mathrm{d}\sigma}\quad\mathrm{with}\quad
G_{0}=(\mathrm{id}-\nu\Psi_{0})/\lambda\,,$

where the function $\Psi_{0}$ and the number $\xi_{0}$ are given by:

$\displaystyle\Psi_{0}(\sigma)$ $\displaystyle=$
$\displaystyle\inf\left\\{t\geq 0\,\bigg{|}\,\nu
t+\lambda\int_{0}^{t}f_{0}(s)\,\mathrm{d}s>\sigma\right\\}\,,$
$\displaystyle\xi_{0}$ $\displaystyle=$
$\displaystyle-\nu\epsilon-\lambda\int_{-\epsilon}^{0}f_{0}(t)\,\mathrm{d}t<0\,.$

In light of (12), the cumulative function $G=G[\Phi]$ actually depends on
$\Phi$ via $\eta=\eta[\Phi]$. This realization allows one to interpret the
implicit definition of $\Phi$ given in (4) as a self-consistent equation for
admissible time changes:

$\displaystyle\Phi(t)=\nu t+\lambda G[\Phi](\Phi(t))\,.$ (8)

In [22], we show that such an interpretation specifies $\Phi$ as the solution
of a certain fixed-point problem. This fixed-point problem turns out to be
most conveniently formulated in term of the inverse time change
$\Psi=\Phi^{-1}$, assumed to be a continuous, nondecreasing function. Given an
inverse time change $\Psi$, the time change $\Phi$ can be recovered as the
right-continuous inverse of $\Psi$. In the time-changed picture, blowups
happen if the inverse time change $\Psi$ becomes locally flat and a
synchronous event happens if $\Psi$ remains flat for a finite amount of time.
Informally, flat sections of $\Psi$ unfold blowups by freezing time in the
original coordinate $t$, while allowing time to pass in the time-changed
coordinate $\sigma$. This unfolding of blowups in the time-changed picture is
the key to analytically resolve blowups in dPMF dynamics. Concretely, this
amounts to showing the existence and uniqueness of solutions to the following
fixed-point problem:

######  Definition 1.3.

Given time-changed initial conditions $(q_{0},g_{0})$ in
$\mathcal{M}(\mathbbm{R}^{+})\times\mathcal{M}([\xi_{0},0))$, an admissible
inverse time change $\Psi$ satisfies the fixed-point problem

$\displaystyle\forall\;\sigma\geq
0\,,\quad\Psi(\sigma)=\left\\{\begin{array}[]{ccc}\left(\sigma-\lambda\int_{0}^{\sigma}g_{0}(\xi)\mathrm{d}\xi\right)/\nu&\quad\mathrm{if}&-\xi_{0}\leq\sigma<0\,,\vspace{5pt}\\\
\sup_{0\leq\xi\leq\sigma}\big{(}\xi-\lambda
G[\eta](\xi)\big{)}/\nu&\quad\mathrm{if}&\sigma\geq 0\,.\end{array}\right.$
(11)

where $G[\eta]$ is the smooth cumulative flux uniquely specified by the time-
inhomogeneous backward-delay function $\eta:\mathbbm{R}^{+}\to\mathbbm{R}^{+}$
(see Definition 1.4). The fixed-point nature of the problem follows from the
definition of the backward-delay function $\eta$ as the $\Psi$-dependent time-
wrapped version of the constant delay $\epsilon$:

$\displaystyle\eta(\sigma)=\sigma-\Phi(\Psi(\sigma)-\epsilon)\quad\mathrm{with}\quad\Phi(t)=\inf\left\\{\sigma\geq\xi_{0}\,\big{|}\,\Psi(\sigma)>t\right\\}\,,$
(12)

for which we consistently have $\eta(0)=-\Phi(-\epsilon)=\xi_{0}$.

Figure 4: Blowup resolution in the time-change picture. Schematic explosive
dPMF dynamics considering a purely spatial initial condition with unit mass
concentrated at $\Lambda$: $q_{0}=\delta_{\Lambda}$. Initially, the dPMF
dynamics solving the time-changed PDE is that of a noninteracting Wiener
process with negative unit drift and delayed reset at $\Lambda$. For
$\sigma>0$, such dynamics admits a time-dependent density $x\mapsto
q(\sigma,x)$ that is smooth on $(0,\infty)$, except for a slope discontinuity
at the reset site $\Lambda$. Moreover, the density $x\mapsto q(\sigma,x)$ is
locally differentiable in $0^{+}$ and the time-changed rate of inactivation
$g(\sigma)$ is determined as the instantaneous flux:
$g(\sigma)=\partial_{x}q(\sigma,0)/2$. For large enough $\lambda$, the flux
$g$ crosses the level $1/\lambda$ as a locally strictly convex increasing
function at time $S_{1}$. In principle, this would correspond to the inverse
time change $\Phi:\sigma\mapsto(\sigma-\lambda G(\sigma))/\nu$ admitting a
strict local maximum at $S_{1}$. However, such a behavior is not allowed as it
would indicate that $T_{1}=\Phi(S_{1})$ is a time-reversal point for the
original dPMF dynamics. This reveals $T_{1}$ as a blowup time for the original
dynamics and $S_{1}$ as a blowup trigger time for the time-changed dynamics.
During a blowup episode, the original time freezes, while the time changed
dynamics is allowed to proceed unimpeded. This corresponds to imposing that
the inverse time change be flat via the definition
$\Phi(\sigma)=\sup_{0\leq\xi\leq\sigma}\big{(}\xi-\lambda
G[\eta](\xi)\big{)}/\nu$. Accordingly, during a blowup episode, the backward-
delay function $\eta$ increases with slope one and no reset can occur. In
other words, during a blowup episode, the time-changed dynamics loses its
reset character and probability mass is gradually lost by inactivation in
zero. Eventually, this entails a decrease in the inactivation rate $g(\sigma)$
and an increase in $\sigma-\lambda G[\eta](\sigma)\big{)}/\nu$, up to a time
$U_{1}$ when it reaches the value $T_{1}=\Phi(S_{1})$ anew. The time $U_{1}$
marks the blowup exit time and necessarily satisfies
$U_{1}-S_{1}=\lambda\pi_{1}$ where $\pi_{1}$ is the fraction of processes that
inactivate during the blowup episode. Moreover, it is also such that the mean
value of $g$ over the blowup episode is exactly $1/\lambda$. Finally, the
inactivated fraction $\pi_{1}$ is reset all at once at a later time $R_{1}$,
referred to as the blowup reset time for which $\Phi(R_{1})=T_{1}+\epsilon$.
This instantaneous reset corresponds to a jump discontinuity in the backward-
delay function of size $-\lambda\pi_{1}$.

In [22], we further showed that the cumulative flux $G[\eta]$, introduced in
the PDE problem defined by equations (6) and (7), is most conveniently
characterized in terms of an integral equation obtained via renewal analysis:

######  Definition 1.4.

Given a time-inhomogeneous backward-delay function
$\eta:\mathbbm{R}^{+}\to\mathbbm{R}^{+}$, the cumulative flux $G$ is given as
the unique solution to the quasi-renewal equation

$\displaystyle
G(\sigma)=\int_{0}^{\infty}H(\sigma,x)q_{0}(x)\,\mathrm{d}x+\int_{0}^{\sigma}H(\sigma-\tau,\Lambda)\,\mathrm{d}G(\tau-\eta(\tau))\,,$
(13)

where by convention we set
$G(\sigma)=G_{0}(\sigma)=\int_{0}^{\sigma}g_{0}(\xi)\mathrm{d}\xi$ if
$-\xi_{0}\leq\sigma<0$. The integration kernel featured in (13) is specified
as $\sigma\mapsto H(\sigma,x)=\mathbb{P}\left[\tau_{x}\leq\sigma\right]$,
where $\tau_{x}$ is the first-passage time to zero of a Wiener process started
in $x>0$ and with negative unit drift.

Definitions 1.3 and 1.4 fully specify the dPMF fixed-point problem in the
time-change picture. In this time-changed picture, the occurrence of a (first)
blowup involves two distinct times: the blowup trigger time $S_{1}$ and the
blowup exit time $U_{1}$, so that a synchronous event of size $\pi_{1}$
corresponds to $\Psi$ remaining flat on $[S_{1},U_{1})$ with
$S_{1}=U_{1}+\lambda\pi_{1}$ (see Fig. 4). The definition of the blowup
trigger time $S_{1}$ directly follows from Proposition 1.1, which can be
recast in the time-changed picture as:

######  Definition 1.5.

Under Assumption 1, the first blowup trigger time is defined as

$\displaystyle S_{1}=\inf\\{\sigma>0\,|\,g(\sigma)>1/\lambda\\}\,.$ (14)

where $g(\sigma)=\partial_{\sigma}G(\sigma)=\partial_{x}q(\sigma,0)/2$ is the
smooth instantaneous flux in zero.

In turn, the full-blowup assumption 2 admits a more natural but equivalent
formulation in the time-changed picture:

######  Assumption 3.

At the blowup trigger time $S_{1}$, the smooth instantaneous flux $g$ is such
that $\partial_{\sigma}g(S_{1})>0$.

The above formulation of the full-blowup assumption is natural in view of the
fixed-point problem 1.3. Indeed, such an assumption implies that the function
$\sigma\mapsto\sigma-\lambda G(\sigma)$ becomes decreasing on a finite
interval to the left of $S_{1}$. Thus, as a solution to the fixed-point
problem (11), $\Psi$ must remain flat on a nonempty maximum interval
$[S_{1},U_{1})$, which corresponds to a synchronous event of size
$\pi_{1}=(S_{1}-U_{1})/\lambda$ (see Fig. 4). Incidentally, this observation
sheds light on Definition 1.1 characterizing full blowups. During a full
blowup episode, $\Psi$ remains flat so that the backward-delay function is
constant on $[S_{1},U_{1})$. As a result, the renewal-type integral term in
(13) vanishes and $G$ is revealed as the cumulative flux of a linear time-
changed dynamics $Y_{\sigma}$, but without birth term due to reset. This
amounts to stopping the clock for the original time coordinate $t$, while
letting the clock run in the changed coordinate $\sigma$. Such a stoppage can
only be maintained until $\Psi$ exceeds its blowup trigger value $\Psi(S_{1})$
(see Fig. 4), justifying the definition of blowup exit times $U_{1}$ as:

######  Definition 1.6.

The first blowup exit time satisfies

$\displaystyle
U_{1}=\inf\\{\sigma>0\,|\,\Psi(\sigma)>\Psi(S_{1})\\}=S_{1}+\lambda\pi_{1}\,,$
(15)

where $\pi_{1}$ is determined by the self-consistent condition (3).

The above definition of the blowup exit time is consistent with the
characterization of the blowup size $\pi_{1}$ given in (3). Indeed, one can
check that

$\displaystyle
U_{1}=\inf\\{\sigma>0\,|\,\Psi(\sigma)>\Psi(S_{1})\\}=\inf\\{\sigma>0\,|\,\sigma-
S_{1}>\lambda(G(\sigma)-G(S_{1}))\\}\,.$

Then, introducing the reduced variable $p=(\sigma-S_{1})/\lambda$, we
consistently have

$\displaystyle U_{1}$ $\displaystyle=$ $\displaystyle
S_{1}+\lambda\inf\\{p>0\,|\,p>G(S_{1}+\lambda p)-G(S_{1})\\}\,,$
$\displaystyle=$ $\displaystyle
S_{1}+\lambda\inf\left\\{p>0\,|\,p>\mathbb{P}\left[\tau_{X_{T_{1}}}<\lambda
p\right]\right\\}\,,$

where the last equality follows from the fact that the process $Y_{\sigma}$
does not reset on $[S_{1},U_{1})$. For nonzero vanishing period $\epsilon>0$,
the processes that inactivate during a blowup episode reset after a period of
duration $\epsilon$ elapses in original time coordinate. Thus, the reset time
of a representative process $R_{1}$ shall satisfy
$\Psi(R_{1})=\Psi(S_{1})+\epsilon$. The latter relation uniquely specifies
$R_{1}$ if $\Phi=\Psi^{-1}$ is continuous in
$T_{1}+\epsilon=\Psi(S_{1})+\epsilon$, which holds true if no blowup happens
in $T_{1}+\epsilon$ (see Fig. 4). Following on [22], without any continuity
assumption, the blowup reset time $R_{1}$ is generally defined as the leftmost
solution of $\Psi(R_{1})=\Psi(S_{1})+\epsilon$:

######  Definition 1.7.

The first blowup reset time satisfies

$\displaystyle
R_{1}=\inf\\{\sigma>0\,|\,\Psi(\sigma)\geq\Psi(S_{1})+\epsilon\\}\,.$ (16)

The definitions of the blowup trigger time $S_{1}$, the exit time $U_{1}$, and
the reset time $R_{1}$ illuminate how the time-changed process $Y_{\sigma}$
resolves a blowup episode by alternating two types of dynamics (see Fig. 4).
Before a full blowup, the process $Y_{\sigma}$ follows a linear diffusion with
absorption in zero and resets at $\Lambda$. These resets occur with time-
inhomogeneous delays, which depends on the inverse time change $\Psi$. At the
blowup onset $S_{1}$, $\Psi$ becomes locally flat, indicating that the
original time $t=\Psi(\sigma)$ freezes, thereby stalling resets. As a result,
after the blowup onset in $S_{1}$, the dynamics of $Y_{\sigma}$ remains that
of an absorbed linear diffusion but without resets. Such a dynamics persists
until a self-consistent blowup exit condition is met in
$S_{1}+\lambda\pi_{1}$. This condition states that it must take
$\lambda\pi_{1}$ time-changed units for a fraction $\pi_{1}$ of processes to
inactivate during a blowup episode. Finally, the original time
$t=\Psi(\sigma)$ resume flowing past $U_{1}$ and the inactivated fraction
$\pi_{1}$ is reset at $\Lambda$ at time $R_{1}$. Importantly, observe that the
time-change process $Y_{\sigma}$ always behaves regularly in the sense that
$G$ remains a smooth function throughout the blowup episode.

###  1.4 Results

In principle, dPMF dynamics could be continued past a blowup episode, and
possibly even extended to the whole half-line $\mathbbm{R}^{+}$. Simulating
the particle systems conjectured to approximate dPMF dynamics suggests that
these dynamics can sustain repeated synchronization events over the whole
half-line $\mathbbm{R}^{+}$ (see Fig. 2). This leads to conjecture the
existence of solutions $\Psi$ globally defined on $\mathbbm{R}^{+}$ with a
countable infinity of blowup episodes with associated sequence of blowup times
$\\{S_{k},U_{k},R_{k}\\}_{k\in\mathbbm{N}}$. Extending results from [22] to
show the existence of such global solutions requires showing:

  * •

_Unconditional explosiveness_ : $(i)$ the so-called non-false-start exit
condition $\partial_{\sigma}q(U_{k},0)/2<1/\lambda$ and $(ii)$ the full-blowup
condition $\partial^{2}_{\sigma}\Psi(S_{k}^{-})<0$ are satisfied for all
blowup episodes, not just for the first one.

  * •

_Full-time domain_ : the blowup times $T_{k}=\Psi(S_{k})=\Psi(U_{k})$ do not
have an accumulation point, i.e., $\lim_{k\to\infty}T_{k}=\infty$, which means
that the explosive solution is defined over the whole half-line
$\mathbbm{R}^{+}$.

The main difficulties in establishing the two above properties is that blowup
times may not be well ordered, i.e., $R_{k}>S_{k+1}$ for some $k>0$ and that
blowup sizes $\pi_{k}$ may exhibit vanishing sizes:
$\lim_{k\to\infty}\pi_{k}=0$. However, one can exclude these possibilities in
the strong interaction regime $\lambda\gg\Lambda$, for which the stable
repetition of synchronization events can be understood intuitively:

$(i)$ Consider an initial condition of the form
$q_{0}=\pi_{0}\delta_{\Lambda}$ with $\pi_{0}=1$, which corresponds to a full
reset at time $R_{0}=0$ and which satisfies Assumption 1. For a full blowup to
occur next, the post-reset flux $g$ needs to satisfy the full-blowup
Assumption 3 at some trigger time $S_{1}$, where we must have
$g(S_{1})=1/\lambda$ and $\partial_{\sigma}g(S_{1})>0$. For large enough
$\lambda$, we expect this flux $g$ to be primarily due to processes that do
not reset more than once on $[0,S_{1}]$. In other words, we have
$g(\sigma)\simeq h(\sigma,\Lambda)$, where $h(\cdot,\Lambda)$ denotes the
first-passage density of a Wiener process started at $\Lambda$ with unit
negative drift and with a zero boundary. The function $h(\cdot,\Lambda)$ is
known to be a strictly increasing convex function on some nonempty interval
$[0,\sigma^{\dagger})$ with $h(0,\Lambda)=0$. Thus, choosing a large enough
$\lambda>\Lambda$ guarantees that the full-blowup Assumption 3 will be met at
some finite time $S_{1}>0$ with $S_{1}<\sigma^{\dagger}$. One can check the
scaling $S_{1}\propto 1/\ln\lambda$, so that the fraction of inactive
processes at trigger time $S_{1}$, denoted $P(S_{1})$, can be made arbitrarily
small for large enough $\lambda$. Actually, provided the refractory period is
such that $\epsilon=O(1/\lambda)$, one can find an upper bound for $P(S_{1})$
of the form $O(1/\lambda)$.

$(ii)$ Past the trigger time $S_{1}$, a fraction $1-P(S_{1})$ of processes are
susceptible to synchronize during the blowup episode. It turns out that the
actual size of the blowup $\pi_{1}<1-P(S_{1})$ can be made arbitrary close to
$1-P(S_{1})$ for large enough $\lambda$. Again, this is because we still have
$g(\sigma)\simeq h(\sigma,\Lambda)$ past the trigger time $S_{1}$, which
satisfies $S_{1}<\sigma^{\dagger}$. This observation, together with the
monotonicity properties of $h(\cdot,\Lambda)$, implies that
$g(\sigma)>1/\lambda$ on $[\sigma^{\dagger},\sigma^{\star}]$, where
$\sigma^{\star}$ is the unique maximizer of $h(\cdot,\Lambda)$. This means
that the blowup episode includes at least the nonempty time interval
$[\sigma^{\dagger},\sigma^{\star}]$, during which at least a fraction
$\Delta_{H}=\int_{\sigma^{\dagger}}^{\sigma^{\star}}h(\sigma,\Lambda)\mathrm{d}\sigma>0$
synchronizes. Thus, we must have $\pi_{1}\geq\Delta_{H}>0$, so that the
duration of the blowup $\lambda\pi_{1}\geq\lambda\Delta_{H}$ can be made
arbitrary long for large enough $\lambda$. The key is then to observe that the
tail behavior of $h(\cdot,\Lambda)$ indicates that inactivation during blowup
asymptotically happens with hazard rate $1/2$ for large times. This
observation allows one to deduce an exponential lower bound for the blowup
size of the form, e.g.,
$\pi_{1}\geq\big{(}1-P(S_{1})\big{)}\big{(}1-O(e^{-\lambda\Delta_{H}/4})\big{)}>1/2$.

$(iii)$ Altogether, less than a fraction $1-\pi_{1}\leq O(1/\lambda)$ of
processes survive the blowup at exit time $U_{1}=S_{1}+\lambda\pi_{1}$ with
$\pi_{1}>1/2$. The distribution of these surviving processes is such that
$\partial_{x}q(U_{1},0)/2\simeq h(U_{1},\Lambda)\leq
O(e^{-U_{1}/2})=O(e^{-\lambda/4})<1/\lambda$ for large enough $\lambda$ so
that the no-false-start Assumption 1 can be met for large enough
$\lambda>\Lambda$. Moreover, we expect the fraction of surviving processes at
$U_{1}$, i.e., $\|q(U_{1},\cdot)\|_{1}$, to be so small that they cannot
trigger a blowup alone. In other words, the assumption of well-ordered blowups
holds: the fraction of inactivated processes $\pi_{1}$ must reset at some time
$R_{1}>U_{1}$ before any subsequent blowup can occur. Then, choosing
$\pi_{1}=1-O(1/\lambda)$ close enough to one ensures that $g$ remains
primarily shaped by those processes that reset at time $R_{1}$, so that
$g(\sigma)\simeq\pi_{1}h(\sigma-R_{1},\Lambda)$. In turn, if the well-ordered
property of blowups holds, we can apply $(i)$ anew but starting from $R_{1}$
with a controlled probability mass $\pi_{1}$, rather than the full mass
$\pi_{0}=1$.

$(iv)$ To prove the well ordered property of blowups, one has to establish
some _a priori_ bounds about the boundary flux $g$ with respect to the $L_{1}$
norm $\|q(U_{1},\cdot)\|_{1}$. Establishing such _a priori_ bounds requires to
restrict the set of considered initial conditions to the so-called set of
natural initial conditions. In a nutshell, these are those—not necessarily
normalized—initial distributions such that all the active and inactive
processes at the starting time have been reset at $\Lambda$ at some point in
the past. For bearing on processes started away from zero at $\Lambda$,
natural initial conditions cannot lead to large transient variations in
boundary fluxes. As a result, one can show that in the absence of reset, $|g|$
and $|\partial_{\sigma}g|$ are uniformly upper bounded by
$O(\|q(U_{1},\cdot)\|_{1})$. Then, the well-ordered property of blowups
follows from showing that $|g|=O(\|q(U_{1},\cdot)\|_{1})<1/\lambda$ in the
absence of the blowup reset. This requires that one has at least that
$\|q(U_{1},\cdot)\|_{1}=O(1/\lambda)$, consistent with the fact that
$\|q(U_{1},\cdot)\|_{1}\leq 1-\pi_{1}=O(1/\lambda)$. but not precise enough to
conclude without explicit knowledge of the bounding coefficients..

$(v)$ Establishing that the no-false-start condition (Assumption 1), the full-
blow up condition (Assumption 2), and the well ordered condition can be all
met requires to estimate coefficients in the various $\lambda$-dependent
bounds at play. Then, the strategy is to use these explicit bounds to show
that given a large enough $\lambda>\Lambda$, there exists a threshold mass
$\Delta_{\pi}$, $1/2<\Delta_{\pi}<1$, such that for all blowups of size
$\pi_{1}\geq\Delta_{\pi}$, the next full blowup also satisfies
$\pi_{2}>\Delta_{\pi}$. From there, a direct induction argument on the number
of blowups shows that there is a unique inverse time change
$\Psi:\mathbbm{R}^{+}\mapsto\mathbbm{R}^{+}$ dynamics as long as the initial
mass $\pi_{0}\geq\Delta_{\pi}$. The fact that the domain of $\Psi$ is the full
half-line $\mathbbm{R}^{+}$ follows from $\Psi$ having a countable infinity of
blowup with size at least $\Delta_{\pi}>1/2$. The fact that the domain of
$\Phi=\Psi^{-1}$ is also the full half-line $\mathbbm{R}^{+}$ follows from the
fact that well ordered blowups must be separated by the duration of the
nonzero refractory period $\epsilon>0$.

Making the arguments above rigorous leads to our first main result:

######  Theorem 1.8.

Consider some normalized natural initial conditions
$(p_{\epsilon,0},f_{\epsilon,0})$ in
$\mathcal{M}(\mathbbm{R}^{+})\times\mathcal{M}([\xi_{0},0))$ such that
$p_{\epsilon,0}$ contains a Dirac-delta mass $\pi_{\epsilon,0}$, with
$0<\pi_{\epsilon,0}\leq 1$, at reset value $\Lambda$.

For small enough $\epsilon>0$, large enough $\lambda>\Lambda$, and large
enough $\pi_{\epsilon,0}<1$, there exists a unique explosive dPMF dynamics
parametrized by the time change
$\Phi_{\epsilon}:\mathbbm{R}^{+}\rightarrow\mathbbm{R}^{+}$, with a countable
infinity of blowups. These blowups occurs at consecutive times
$T_{\epsilon,k}$, $k\in\mathbbm{N}$, with size
$\pi_{\epsilon,k}=(\Phi_{\epsilon}(T_{\epsilon,k})-\Phi_{\epsilon}(T_{\epsilon,k}^{-}))/\lambda$,
and are such that $\pi_{\epsilon,k}$ and $T_{\epsilon,k+1}-T_{\epsilon,k}$ are
both uniformly bounded away from zero.

In the following, we prove an equivalent version of the above theorem for the
time-changed dynamics, i.e., Theorem 3.2. The proof of Theorem 3.2 only relies
on a few properties of the smooth first-passage density function
$h(\cdot,\Lambda)$, which plays the key role throughout this analysis.
Although we do not refer to these properties explicitly in the proof, we list
them as follows:

######  Property 1.

The function $h(\cdot,\Lambda)$ is both strictly convex and log-concave on
some finite interval $[0,\sigma^{\dagger}]$ with $h(0)=0$.

######  Property 2.

The hazard rate function of $h(\cdot,\Lambda)$ is asymptotically bounded away
from zero and has at most exponential growth, i.e.:

$\displaystyle\liminf_{\sigma\to\infty}\;\frac{h(\sigma,\Lambda)}{\int_{\sigma}^{\infty}h(\xi,\Lambda)}\geq
r>0\,\quad\mathrm{and}\quad\limsup_{\sigma\to\infty}\;\ln\left(\frac{h(\sigma,\Lambda)}{\int_{\sigma}^{\infty}h(\xi,\Lambda)}\right)<\infty\,.$

Given some large enough Dirac-delta mass $\pi_{k}$ at reset time $R_{k}$,
Property 1 ensures that a full blowup occurs for large enough $\lambda$ at
trigger time $S_{k+1}>R_{k}$ and with blowup size $\pi_{k+1}$ lower bounded by
some constant $\Delta_{H}$. This corresponds to the “ignition” phase of the
blowup. Given that $\pi_{k+1}\geq\Delta_{H}$, Property 2 ensures that an
exponentially small fraction of processes survives the blowup and that this
fraction cannot lead to a subsequent blowup without reset, i.e., reignition.
This corresponds to the “combustion” phase of the blowup. A potentially useful
consequence of the above observations is that we expect our blowup analysis to
extend to more generic diffusion processes. Indeed, the existence and
uniqueness of explosive dPMF dynamics shall hold for all dynamics derived from
autonomous diffusion processes whose first-passage time to a constant level
has smooth densities satisfying Property 1 and 2. This class of diffusion
processes includes the Ornstein-Uhlenbeck process, which forms the basis of a
widely used class of models in computational neuroscience, called leaky
integrate-and-fire neurons [16, 6].

In stating Theorem 1.8, we do note give a precise criterion for how to jointly
choose the interaction parameter $\lambda>\Lambda$, the refractory period
$\epsilon>0$, and the initial mass $\pi_{0}$ to yield explosive dPMF dynamics.
This is because the precise criterion given in the proof of Theorem 1.8 is not
tight, our main goal being to establish existence and uniqueness of global
solution in the large interaction regime. Actually, we expect explosive dPMF
dynamics to arise as soon as $\lambda>\Lambda$ for small $\epsilon\geq 0$ and
we expect this emergence to be essentially independent of the size of the
initial mass $\pi_{0}$. One can see the latter point by considering the limit
case of PMF dynamics with zero refractory period in the time change picture.
In the absence of blowups, plugging $\epsilon=0$ in Definition 1.3 yields
valid, nondelayed dynamics with explicitly known instantaneous boundary flux
$g$ via renewal analysis [1]. Moreover, classical arguments from renewal
analysis show that $g$ is analytic away from zero and that
$\lim_{\sigma\to\infty}g(\sigma)=1/\Lambda$, independent of the initial
conditions. Thus, it is natural to expect that PMF dynamics exhibit a
countable infinity of blowups (at least) as soon as $\lambda>\Lambda$.

One general caveat is that checking the mean-field conjecture becomes
exceedingly costly when $\lambda\simeq\Lambda$. Indeed we find that the
convergence to an asymptotically independent regimes for large network size
$N\to\infty$ drastically slows down when $\lambda\simeq\Lambda$ —a numerical
evidence of critical slowing down [20]. However, the conjectured PMF dynamics
can always be analyzed in the time-changed picture by setting $\epsilon=0$ in
Definition 1.3. One specific caveat remains that it is not clear why blowups
are resolved in PMF dynamics via the mechanism implied by Definition 1.3 with
$\epsilon=0$. This is by contrast with dPMF dynamics for which the nonzero
refractory period $\epsilon>0$ makes it clear that blowup episodes correspond
to halting reset at the time-changed picture. In view of this, our second main
result is to justify that PMF dynamics “physically” resolve blowups by showing
that these dynamics are recovered from dPMF ones in the limit $\epsilon\to
0^{+}$.

######  Theorem 1.9.

Consider some normalized natural initial condition $p_{0}$ in
$\mathcal{M}(\mathbbm{R}^{+})$ such that $p_{0}$ contains a Dirac-delta mass
$\pi_{0}$, with $0<\pi_{0}\leq 1$, at reset value $\Lambda$.

For large enough $\lambda>\Lambda$ and large enough $\pi_{0}<1$, the unique
explosive PMF dynamics $\Phi:\mathbbm{R}^{+}\rightarrow\mathbbm{R}^{+}$
satisfies $\Phi^{-1}=\lim_{\epsilon\to 0^{+}}\Phi^{-1}_{\epsilon}$ with
compact convergence over $\mathbbm{R}^{+}$, where $\Phi_{\epsilon}$ are dPMF
dynamics with same initial condition $p_{0}$. Moreover, we also have
convergence of the blowup times and blowup sizes: $\lim_{\epsilon\to
0^{+}}T_{\epsilon,k}=T_{k}$ and $\lim_{\epsilon\to
0^{+}}\pi_{\epsilon,k}=\pi_{k}$.

In the following, we prove an equivalent version of the above theorem for the
time-changed dynamics, i.e., Theorem 4.4. Given an explosive dPMF dynamics
parametrized by $\Phi_{\epsilon}=\Psi_{\epsilon}^{-1}$, proving Theorem 4.4
amounts to showing that the $\epsilon$-dependent backward-delay function

$\displaystyle\eta_{\epsilon}(\sigma)=\sigma-\Phi_{\epsilon}(\Psi_{\epsilon}(\sigma)-\epsilon)$
(17)

is well behaved in the limit $\epsilon\to 0^{+}$. By well behaved, we mean
that $(1)$ $\eta_{\epsilon}$ converges to zero in between blowups on intervals
$[R_{k},S_{k}]$, consistent with the nondelayed nature of PMF dynamics, or
$(2)$ $\eta_{\epsilon}$ converges toward the unit slope function
$\sigma\mapsto\sigma-S_{k}$ on blowup intervals $[S_{k},U_{k}]$, consistent
with halting reset at $S_{k}$. Such an alternative covers all cases as we
expect instantaneous reset after blowup exit: $U_{k}=R_{k}$ for $\epsilon=0$.
This corresponds to assuming a limit of the form

$\displaystyle\eta(\sigma)=\sigma-\Phi(\Psi(\sigma))\,,$

where crucially, $\Phi$ is the right-continuous inverse of $\Psi$, just as if
one consider the fixed-point problem of Definition 1.3 with $\epsilon=0$. The
main difficulty is that (17) does not behave continuously for standard
topologies when it bears on functions $\Phi_{\epsilon}$ with jump
discontinuities. One route to address this point is to consider topologies
specially designed to deal with _càdlàg_ functions such as the Skorokhod
topologies [2, 9]. However, the continuity of the fixed-point problem
specified in Definition 1.3 with respect to such topologies does not appear
straightforward.

A more pedestrian route is to leverage the fact that for small enough
$\epsilon>0$, the time change $\Phi_{\epsilon}$ has only a countable infinity
of blowups for large enough interaction parameter $\lambda>\Lambda$ and large
enough initial reset mass $\pi_{0}<1$. Under these standard conditions, one
can show that the associated blowup times
$(S_{\epsilon,k},U_{\epsilon,k},R_{\epsilon,k})$, $k$ in $\mathbbm{N}$, can be
uniformly controlled with respect to $\epsilon\to 0^{+}$. Such control can be
established by considering separately the three stages of a time-change blowup
dynamics: $(1)$ the blowup trigger stage on
$[R_{\epsilon,k},S_{\epsilon,k+1})$, $(2)$ the blowup resolution stage on
$[S_{\epsilon,k+1},U_{\epsilon,k+1})$, and $(3)$ the blowup reset stage
$[U_{\epsilon,k+1},R_{\epsilon,k+1})$. The key observation is that each of
these stages has a duration that continuously depends on $\epsilon$ and on the
$L_{1}$ norm of the current state of the dynamics at the start of the stage.
Note that these current states represent natural initial conditions in
$\mathcal{M}((0,\infty))\times\mathcal{M}([-\eta_{\epsilon}(\sigma),0))$
specified by
$\big{(}q_{\epsilon}(\sigma,\cdot),\\{g_{\epsilon}(\xi-\sigma)\\}_{-\eta_{\epsilon}(\sigma)\leq\xi<0}\big{)}$
for $\sigma=R_{\epsilon,k},S_{\epsilon,k+1},U_{\epsilon,k+1}$. Both the
continuity of the trigger time $S_{\epsilon,k+1}$ and of the exit time
$U_{\epsilon,k+1}$ follow from nondegeneracy conditions, namely that the full-
blowup condition $\partial_{\sigma}g(S_{k+1})>0$ and the non-false-start
condition $\partial_{\sigma}g(U_{k+1})<1/\lambda$ holds uniformly with respect
to $\epsilon$. The continuity of the reset time $R_{k+1}$ is straightforward
for well-ordered blowups. Thus, to propagate these continuity results by
induction on the number of blowups, one just has to show that for each stage,
considering the terminal dynamical state as a function of the initial
dynamical state specifies a continuous mapping in the $L_{1}$ norm.
Technically, establishing such $L_{1}$ continuity results, jointly with
$\epsilon$-continuity, is primarily made possible by the availability of
certain uniform bounds for the heat kernel at times bounded away from zero and
infinity.

The above discussion lays out our strategy to prove that PMF dynamics with
zero refractory period represents “physical” solutions, which is one of our
main objectives. PMF dynamics are of special interest because they admit an
explicit iterative construction in terms of well-known analytic functions.
Moreover, PMF can be easily simulated within the PDE setting. Such simulations
confirm the prediction from particle-system simulations that explosive PMF
solutions become asymptotically periodic (see Fig. 2). In that respect, we
conclude by mentioning that the existence of periodic PMF dynamics directly
follows from our $L_{1}$ continuity analysis with $\epsilon=0$. This is a
consequence of Schauder’s fixed-point theorem [12] applied to the inter-exit-
time mapping

$\displaystyle\mathcal{U}:\big{(}q(U_{k},\cdot),\pi_{k}\big{)}\mapsto\big{(}q(U_{k+1},\cdot),\pi_{k+1}\big{)}$

defined on $\mathcal{C}$, the convex set of normalized natural initial
conditions with large enough blowup mass $\pi_{k}$ (given some large enough
$\lambda>\Lambda$). Note that we automatically have
$\|q(U_{k},\cdot)\|_{1}+\pi_{k}=\|q(U_{k+1},\cdot)\|_{1}+\pi_{k+1}=1$ when
$\epsilon=0$ so that the set $\mathcal{C}$ is stable by $\mathcal{U}$. Our
analysis shows that the mapping $\mathcal{U}$ is continuous with respect to
the $L_{1}$ norm. Then, all there is left to show in order to apply Schauder’s
fixed-point theorem is that the mapping $\mathcal{U}$ is compact in
$\mathcal{C}$ with respect to the $L_{1}$ norm. But this follows directly from
the fact that blowups are lower bounded by some $\Delta_{\pi}>1/2$. The latter
point implies that the image of $\mathcal{U}$ is included in the image of a
compact operator defined in terms of the heat kernel for times at least $1/2$.
Thus there must be a fixed point to $\mathcal{U}$ in $\mathcal{C}$, which
corresponds to a periodic PMF dynamics. Studying the general contractive
property of PMF and dPMF dynamics is beyond the scope of this work.

###  1.5 Structure

In Section 2, we recall results from [22] that allows one to resolve a single
dPMF blowup episode in the time-changed picture. In Section 3, we leverage
these results to show that dPMF solution can be continued past blowup episodes
to specify explosive solutions over the whole half-line $\mathbbm{R}^{+}$ in
the large interaction regime. In Section 4, we show that explosive PMF
dynamics, which can be defined explicitly for zero refractory period
$\epsilon=0$, are recovered from dPMF dynamics in the limit $\epsilon\to
0^{+}$. Appendices A and B contain two technical lemmas that are needed to
demonstrate the result of Section 4.

##  2 Regularization of explosive dynamics via change of time

In this section, we define the time-changed dynamics obtained by change of
variable and introduce the delay functions that parametrize such a dynamics.
We then recall the results from renewal theory that characterize the
cumulative flux function associated with a time-changed dynamics Finally, we
utilize these results to justify the definition of the fixed-point formulation
of dPMF dynamics given in Definition 1.3.

###  2.1 Definition of the time-inhomogeneous linear dynamics

Following [22], our approach is based on representing possibly explosive dPMF
dynamics $X_{t}=Y_{\Phi(t)}$ as time-changed versions of nonexplosive dynamics
$Y_{\sigma}$. A good choice for the time change function $\Phi$ is one for
which the dynamics $Y_{\sigma}$ is simple enough to be analytically tractable.
The Poisson-like attributes of dPMF dynamics suggest defining $\Phi$
implicitly as the integral function of the drift:

$\displaystyle t\mapsto\Phi(t)=\nu t+\lambda F(t)\,.$ (18)

Such a definition equates blowups/synchronous events with
singularities/discontinuities in the time change $\Phi$, while allowing for
the corresponding time-changed dynamics $Y_{\sigma}$ to always be devoid of
blowups. As a result, showing the existence and uniqueness of a dPMF dynamics
$X_{t}$ reduces to showing the existence and uniqueness of a time change
$\Phi$ satisfying relation (18). We shall seek such time changes $\Phi$ in a
set of candidate time changes denoted by $\mathcal{T}$. To define
$\mathcal{T}$, we utilize the fact that the cumulative function
$F:\mathbbm{R}^{+}\to\mathbbm{R}^{+}$ featured in (18) can be safely assumed
to be a nondecreasing function. This leads to the following definition:

######  Definition 2.1.

The set of candidate time change $\mathcal{T}$ is the set of _càdlàg_
functions $\Phi:[-\epsilon,\infty)\to[\xi_{0},\infty)$, such that their
difference quotients are lower bounded by $\nu$: for all $y,x\leq-\epsilon$,
$x\neq y$, we have

$\displaystyle w_{\Phi}(y,x)=\frac{\Phi(y)-\Phi(x)}{y-x}\geq\nu\,.$

As alluded to in the introduction, it is actually most convenient to consider
dPMF dynamics as parametrized by the inverse time change:

######  Definition 2.2.

Given a time change $\Phi$ in $\mathcal{T}$, the inverse time change
$\Phi^{-1}:[\xi_{0},\infty)\to[-\epsilon,\infty)$ is defined as the continuous
function

$\displaystyle\sigma\mapsto\Psi(\sigma)=\Phi^{-1}(\sigma)=\inf\left\\{t\geq
0\,|\,\Phi(t)>\sigma\right\\}\,.$

In order to specify the time changed dynamics $Y_{\sigma}$, we further need to
introduce the so-called backward-delay function $\eta$, which represents the
time-wrapped version of the refractory period $\epsilon$:

######  Definition 2.3.

Given a time change $\Phi$ in $\mathcal{T}$, we define the corresponding
backward-delay function $\eta:[0,\infty)\to\mathbbm{R}^{+}$ by

$\displaystyle\eta(\sigma)=\sigma-\Phi\left(\Psi(\sigma)-\epsilon\right)\,,\quad\sigma\geq
0\,.$

We denote the set of backward functions
$\\{\eta[\Phi]\\}_{\Phi\in\mathcal{T}}$ by $\mathcal{W}$.

As $w_{\Phi}\geq\nu$ for all $\Phi$ in $\mathcal{T}$, it is clear that for all
$\eta$ in $\mathcal{W}$, we actually have $\eta\geq\nu\epsilon$, so that all
delays are bounded away from zero. Time-wrapped-delay function $\eta$ in
$\mathcal{W}$ will serve to parametrize the time-changed dynamics obtained via
$\Phi$ in $\mathcal{T}$. In [22], we showed that these time-changed dynamics
are that of a modified Wiener process $Y_{\sigma}$ with negative unit drift,
inactivation on the zero boundary, and reset at $\Lambda$ after a refractory
period specified by $\eta$. Consequently, we defined time-changed dynamics as
the processes $Y_{\sigma}$ solutions to the following stochastic evolution:

######  Definition 2.4.

Denoting the canonical Wiener process by $W_{\sigma}$, we define the time-
changed processes $Y_{\sigma}$ as solutions to the stochastic evolution

$\displaystyle
Y_{\sigma}=-\sigma+\int_{0}^{\sigma}\mathbbm{1}_{\\{Y_{\xi^{-}}>0\\}}\,\mathrm{d}W_{\xi}+\Lambda
N_{\sigma-\eta(\sigma)}\,,\quad\mathrm{with}\quad
N_{\sigma}=\sum_{n>0}\mathbbm{1}_{[\xi_{0},t]}(\xi_{n})\,,$ (19)

where the process $N_{\sigma}$ counts the successive first-passage times
$\xi_{n}$ of the process $Y_{\sigma}$ to the absorbing boundary:

$\displaystyle\xi_{n+1}=\inf\left\\{\sigma>0\,\big{|}\,\sigma-\eta(\sigma)>\xi_{n},Y_{\sigma}\leq
0\right\\}\,.$

A time-changed process $Y_{\sigma}$ is uniquely specified by imposing
elementary initial conditions, which take an alternative formulation: either
the process is active $Y_{0}=x>0$ and $N_{0}=0$, either the process has
entered refractory period at some earlier time $\xi$ so that $Y_{0}=0$ and
$N_{\sigma}=\mathbbm{1}_{\sigma\geq\xi}$ for $\xi_{0}\leq\sigma<0$. Generic
initial conditions are given by considering that $(x,\xi)$ is sampled from
some probability distribution on
$\\{(0,\infty)\times\\{0\\}\\}\cup\\{\\{0\\}\times[\xi_{0},0)\\}$. In all
generality, this amounts to choosing a normalized pair of distributions
$(q_{0},g_{0})$ in $\mathcal{M}((0,\infty))\times\mathcal{M}([\xi_{0},0))$.
Given generic initial conditions $(q_{0},g_{0})$ in
$\mathcal{M}((0,\infty))\times\mathcal{M}([\xi_{0},0))$, the dynamics defined
by 2.4 is well posed as long as the backward-delay function
$\eta\geq\nu\epsilon$ is locally bounded, which is always the case for valid
time change $\Phi$. Moreover, the corresponding density function
$(\sigma,x)\mapsto q(\sigma,x)=\mathrm{d}\mathbb{P}\left[0<Y_{\sigma}\leq
x\right]/\mathrm{d}x$ must solve the PDE problem defined by (6) and (7) in the
introduction. To establish the existence and uniqueness of global explosive
solutions, we will actually consider a restricted class of initial conditions,
referred to as natural conditions. We postpone the introduction of this so-
called notion of natural initial conditions to Section 3.1.

Interestingly, interactions are entirely mediated by the time-delayed counting
process $N_{\sigma-\eta(\sigma)}$ in the linear time-changed dynamics (19).
This motivates introducing the so-called backward-time function
$\sigma\mapsto\xi(\sigma)$, which marks the inactivation time of the processes
being reset at time $\sigma$, and the associated forward-time function
$\sigma\mapsto\tau(\sigma)$, which marks the reset time of the processes
inactivated at time $\sigma$.

######  Definition 2.5.

Given a time change $\Phi$ in $\mathcal{T}$, we define the backward-time
function by

$\displaystyle\xi:\mathbbm{R}^{+}\to[\xi_{0},\infty)\,,\quad\xi(\sigma)=\sigma-\eta(\sigma)=\Phi\left(\Psi(\sigma)-\epsilon\right)\,,$

and the forward-time function by

$\displaystyle\tau:[\xi_{0},\infty)\to\mathbbm{R}^{+}\,,\quad\tau(\sigma)=\inf\\{\tau>0\,|\,\xi(\tau)=\tau-\eta(\tau)\geq\sigma\\}\,.$
(20)

When unambiguous, we will denote these functions by $\xi$ and $\tau$ refer to
them as the “functions $\xi$ and $\tau$” to differentiate from when $\xi$ and
$\tau$ play the role of real variables. By construction, the function $\xi$
and $\tau$ are nondecreasing _càdlàg_ and nondecreasing _càglàd_ functions,
respectively. In particular, both functions can admit discontinuities and flat
regions. In [22], we justify the definition of the function $\tau$ as the
left-continuous inverse of the function $\xi$ with a detailed analysis of the
delayed reset mechanism in the dynamics (19). We also show there that if we
denote the sequence of inactivation times by $\\{\xi_{k}\\}_{k\geq 0}$, then
the corresponding sequences of reset times $\\{\tau_{k}\\}_{k\geq 0}$
satisfies $\tau_{k}=\tau(\xi_{k})$, $k\geq 0$.

###  2.2 Results from time-inhomogeneous renewal analysis

Given a backward-delay function $\eta$ in $\mathcal{W}$, one can consider the
PDE problem defined by (6) and (7) independently of the requirement that
relation (18) be satisfied. Then, the main hurdle to solving this PDE problem
is due to the presence of an inhomogeneous, delayed reset term featuring the
cumulative flux function $G$. Luckily, assuming the cumulative function $G$
known allows one to express the full solution of the inhomogeneous PDE in
terms of its homogeneous solutions. These homogeneous solutions are known in
closed form [14]:

$\displaystyle\kappa(\sigma,y,x)=\frac{e^{-\frac{(y-x+\sigma)^{2}}{2\sigma}}}{\sqrt{2\pi\sigma}}\left(1-e^{-\frac{2xy}{\sigma}}\right)\,.$
(21)

Applying Duhamel’s principle yields the following integral representation for
the density function $(\sigma,x)\mapsto
q(\sigma,x)=\mathrm{d}\mathbb{P}\left[0<Y_{\sigma}\leq x\right]$

$\displaystyle
q(\sigma,x)=\int_{0}^{\infty}\kappa(\sigma,y,x)q_{0}(x)\,\mathrm{d}x+\int_{0}^{\sigma}\kappa(\sigma-\tau,x,\Lambda)\,\mathrm{d}G(\xi(\tau))\,,$
(22)

where $\xi=\mathrm{id}-\eta$ and where $G$ remains to be determined. The above
integral representation shows that $G$ is the key determinant parametrizing
the time-changed process $Y_{\sigma}$.

Just as $Y_{\sigma}$, $G$ only depends on the backward-delay functions $\eta$,
or equivalently on the functions $\xi$ or $\tau$. These dependences can be
made explicit by adapting results from renewal analysis [22]. Informally, this
follows from writing the cumulative flux function as the uniformly convergent
series

$\displaystyle
G(\sigma)=\mathbb{E}\left[\sum_{k=1}^{\infty}\mathbbm{1}_{\left\\{\xi_{k}<\sigma\right\\}}\right]=\sum_{k=1}^{\infty}\mathbb{P}\left[\xi_{k}\leq\sigma\right]\,,$
(23)

and realizing that for all $k\geq 1$,
$\mathbb{P}\left[\xi_{k}\leq\sigma\right]$ can be expressed in terms of the
first-passage time cumulative distribution [14]:

$\displaystyle H(\sigma,x)=\frac{\mathbbm{1}_{\\{\sigma\geq
0\\}}}{2}\left(\mathrm{Erfc}\left(\frac{x-\sigma}{\sqrt{2\sigma}}\right)+e^{2x}\mathrm{Erfc}\left(\frac{x+\sigma}{\sqrt{2\sigma}}\right)\right)\,.$
(24)

Specifically, considering the initial condition $q_{0}=\delta_{x}$ for
simplicity, we have

$\displaystyle\mathbb{P}\left[\xi_{k}\leq\sigma\,|\,Y_{0}=x\right]=H^{(k)}(\sigma,x)\,,$

where the functions $H^{(k)}(\sigma,x)$, $k\geq 1$, are defined as
inhomogeneous iterated convolution:

$\displaystyle
H^{(k)}(\sigma,x)=\int_{0}^{\infty}H(\sigma-\tau(\xi))\,\mathrm{d}H^{(k-1)}(\xi,x)\,,\quad
H^{(1)}(\sigma,x)=H(\sigma,x)\,.$

These iterated functions $H^{(k)}(\sigma,x)$ take a particularly simple form
for constant backward-delay function $\eta=d=-\xi_{0}$. Indeed, by
divisibility of the first-passage distributions

$\displaystyle\mathbb{P}\left[\xi_{k}\leq\xi\,|\,Y_{0}=x\right]=H(\xi-(k-1)d,x+(k-1)\Lambda)\,,$

so that for the elementary initial condition $q_{0}=\delta_{x}$, the
cumulative flux function is

$\displaystyle
G_{d}(\sigma,x)=\sum_{k=1}^{\infty}H(\xi-(k-1)d,x+(k-1)\Lambda)\,.$

Due to the quasi-renewal character of the dynamics $Y_{\sigma}$, the
cumulative flux $G$ admits an alternative characterization in terms of an
integral equation. Because of the presence of inhomogeneous delays, this
integral equation differs from standard convolution-type renewal equations.
Specifically, we show in [22] that for generic initial conditions:

######  Proposition 2.1.

Given a backward-delay function $\eta$ in $\mathcal{W}$, the cumulative flux
function $G$ is the unique solution of the renewal-type equation:

$\displaystyle
G(\sigma)=\int_{0}^{\infty}H(\sigma,x)q_{0}(x)\,\mathrm{d}x+\int_{0}^{\sigma}H(\sigma-\tau,\Lambda)\,\mathrm{d}G(\xi(\tau))\,.$
(25)

As a solution to (25), it is clear that independent of the functions $\eta$,
$\xi$, $\tau$, the cumulative function $G$ inherits all the regularity
properties of $\sigma\mapsto H(\sigma,\Lambda)$, i.e., $G$ must be a smooth
function for $\sigma>0$. In particular, one can differentiate (25) with
respect to $\sigma$ and obtain a new renewal-type equation for the
instantaneous flux $g$:

$\displaystyle
g(\sigma)=\int_{0}^{\infty}h(\sigma,x)q_{0}(x)\,\mathrm{d}x+\int_{0}^{\sigma}h(\sigma-\tau,\Lambda)\,\mathrm{d}G(\xi(\tau))\,,$
(26)

and where $h(\cdot,x)$ denotes the density function of the cumulative function
$H(\cdot,x)$:

$\displaystyle
h(\sigma,x)=\partial_{\sigma}H(\sigma,x)=\mathbbm{1}_{\\{\sigma\geq
0\\}}\frac{xe^{-(x-\sigma)^{2}/2\sigma}}{\sqrt{2\pi\sigma^{3}}}\,.$ (27)

Alternatively, (26) can be directly obtained from differentiating (22) with
respect to $x$ in zero and using that fact that
$h(\sigma,x)=\partial_{y}\kappa(\sigma,0,x)/2$. In [22], renewal-type
equations such as (25) and (26) play a crucial role in showing the local
existence of explosive dPMF solutions. Here, these will provide us with _a
priori_ estimates that are useful to globally define explosive dPMF solutions
over the whole half-line $\mathbbm{R}^{+}$.

Finally, for constant delay, the renewal-type equation (25) turned into an
time-homogeneous equation by a change of variable:

$\displaystyle
G(\sigma)=\int_{0}^{\infty}H(\sigma,x)q_{0}(x)\,\mathrm{d}x+\int_{0}^{\sigma}H(\sigma-\tau-d,\Lambda)\,\mathrm{d}G(\tau)\,.$
(28)

Deriving the above equation with respect to $\sigma$ leads to a convolution
equation for the the flux $g_{d}=\partial_{\sigma}G_{d}$. This equation can be
solved algebraically in the Laplace domain yielding:

$\displaystyle\hat{g}_{d}(u,x)=\int_{0}^{\infty}g_{d}(\sigma,x)e^{-u\sigma}\,d\sigma=\frac{e^{x\left(1-\sqrt{1+2u}\right)-du}}{1-e^{\Lambda\left(1-\sqrt{1+2u}\right)-du}}\,.$

Finally, the above explicit expression allows one to evaluate the long-term
behavior of the flux as $\lim_{\sigma\to\infty}g_{d}(\sigma,x)=\lim_{u\to
0}u\hat{g}_{d}(u,x)$ [11], which gives the following useful result:

######  Proposition 2.2.

For constant delay $d$, the flux function $g_{d}$ admit a limit that is
independent of the initial conditions:

$\displaystyle\lim_{\sigma\to\infty}g_{d}(\sigma)=\frac{1}{\Lambda+d}\,.$

Observe that the above result is valid for $d=0$, in which case
$\lim_{\sigma\to\infty}g(\sigma)=1/\Lambda>1/\lambda$ as soon as
$\lambda>\Lambda$. This shows that for zero refractory period $\epsilon$ and
for $\lambda>\Lambda$, a blowup must occur in finite time. However, it is
unclear how to resolve the ensuing blowup episode, since defining spiking
avalanches can be ambiguous when $\epsilon=0$. We resolve this ambiguity by
viewing PMF dynamics as the limit of dPMF dynamics. That is, we define PMF by
formally setting $\epsilon=0$ in the dPMF fixed-point problem. We then show
that, for large enough interaction parameter $\lambda>\Lambda$, the resulting
PMF dynamics are the limit of dPMF dynamics as $\epsilon\to 0^{+}$.

###  2.3 Fixed-point problem and local solutions

Renewal analysis allows one to prove the existence, uniqueness, and regularity
of the time-changed dynamics $Y_{\sigma}$ assuming the backward-delay function
$\eta$ known. However, $\eta$ is actually an unknown of the problem for being
ultimately defined in terms of the time change $\Phi$ via Definition 2.3.
Moreover, the time-changed dynamics $Y_{\sigma}$ exists for any $\eta$ in
$\mathcal{W}$, whereas given some initial conditions, we expect that a unique
time change $\Phi$ in $\mathcal{T}$ parametrizes a dPMF dynamics. In [22], we
characterize this unique time change $\Phi$ by imposing that the defining
relation (18) be satisfied. Observing that $F$ is the cumulative flux of
$X_{t}=Y_{\Phi}(t)$, relation (18) can be naturally reformulated as:

$\displaystyle\Phi(t)=\nu t+\lambda F(t)=\nu t+\lambda
G(\Phi(t))\quad\Leftrightarrow\quad\nu t=\Phi(t)-\lambda G(\Phi(t))\,.$ (29)

Assuming $G$ known, one can specify the sought-after time change
$\Phi(t)=\sigma$ by solving $\nu t=\sigma-\lambda G(\sigma)$ for $\sigma$
given arbitrary values of $t\geq 0$. However, one has to be mindful about the
existence and multiplicity of solutions. By construction, we have $G(0)=0$, so
that we can always set $\Phi(0)=0$. Then, we can uniquely specified
$\Phi(t)=\sigma$ for increasing value of $t$ as long as
$\sigma\mapsto\sigma-\lambda G(\sigma)$ is an increasing, one-to-one function.
Let us consider the smallest time $S_{1}$ after which the smooth function
$\sigma\mapsto\sigma-\lambda G(\sigma)$ fails to be an increasing, one-to-one
function. Then, $\Phi(t)$ is uniquely defined within the range $[0,S_{1})$, or
rather the inverse time change $\Psi=\Phi^{-1}$ is uniquely defined as

$\displaystyle\Psi(\sigma)=(\sigma-\lambda
G(\sigma))/\nu\,,\quad\forall\sigma\;\in\;[0,S_{1})\,.$ (30)

If $S_{1}$ is finite, we must have $g(S_{1})=1/\lambda$ so that $S_{1}$ is a
blowup trigger time as defined in (14), and setting $T_{1}=\Psi(S_{1})$, the
equation $\nu T_{1}=\sigma-\lambda G(\sigma)$ must admit multiple solutions
for $\sigma$. This indicates that $\Phi$ has a jump discontinuity in $T_{1}$,
with $\Phi(T_{1})=U_{1}>S_{1}=\Phi(T_{1}^{-})$. To be consistent, $U_{1}$
shall be the smallest solution such that $U_{1}>S_{1}$ and
$\sigma\mapsto\sigma-\lambda G(\sigma)$ is increasing, one-to-one in the right
vicinity of $U_{1}$. This corresponds to defining $U_{1}$ as an blowup exit
time as in (16) and imposing that

$\displaystyle\Psi(\sigma)=\Psi(S_{1})\,,\quad\forall\sigma\;\in\;[S_{1},U_{1})\,.$
(31)

In principle, one can hope to continue defining the time change value
$\Phi(t)$ by extending the same reasoning for times $t\geq T_{1}$, leading to
defining an intertwined sequence of blowup trigger times $\\{S_{k}\\}_{k\geq
1}$ and blowup exit time $\\{U_{k}\\}_{k\geq 1}$, with $U_{1}\leq S_{1}\leq
U_{2}\leq S_{2}\ldots$. Such a construction produces a global solution over
$\mathrm{R}^{+}$ if the resulting sequences are devoid of accumulation points:
$\lim_{k\to\infty}S_{k}=\lim_{k\to\infty}U_{k}=\infty$. In this work, we are
primarily concerned with showing that the latter limits hold for large enough
interactions and natural initial conditions so that there are no caveats due
to accumulation of blowup times.

However, one can specify the inverse time change $\Phi$ independent of the
possible accumulation of blowup times. Following on the informal discussion
above, this implicit specification of $\Phi(t)=\sigma$ as a solution $\nu
t=\sigma-\lambda G(\sigma)$ is best formulated in terms of the inverse time
change $\Psi$ featuring in (30) and (31). Then to circumvent possible caveats
of accumulating blowup times, we leverage the fact that $\Psi$ must be a
nondecreasing function to write relation (18) as a fixed-point problem,
whereby $\Psi$ is defined as a running maximum function. The fixed-point
nature of $\eqref{eq:Phi}$ naturally follows from the fact that the cumulative
function $G=G[\Psi]$ also depends on $\Psi$ via $\eta=\eta[\Phi]=\eta[\Psi]$.
Specifically, we show in [22] that:

######  Proposition 2.3.

The inverse time-change $\Psi$ parametrizes a dPMF dynamics if and only if it
solves the fixed-point problem

$\displaystyle\forall\;\sigma\geq
0\,,\quad\Psi(\sigma)=\sup_{0\leq\xi\leq\sigma}\big{(}\xi-\lambda
G[\Psi](\xi)\big{)}/\nu\,.$ (32)

The full formulation of the fixed-point problem given in Definition 1.3
follows directly from the above proposition together with the renewal-type
equation (25) for the cumulative flux $G$ and Definitions 2.3 and 2.5 for the
backward functions $\eta$ and $\xi$. The fixed point problem 1.3 allows one to
characterize the occurrence of a single blowup in the time changed-picture for
generic initial conditions satisfying Assumption 1 and under full-blowup
Assumption 3. This local result is stated as follows in [22]:

######  Theorem 2.6.

Under Assumption 1, there is a unique smooth solution $\Psi$ to the fixed-
point problem 1.3 up to the possibly infinite time

$\displaystyle S_{1}=\inf\left\\{\sigma>0\,\big{|}\,\Psi^{\prime}(\sigma)\leq
0\right\\}=\inf\left\\{\sigma>0\,\big{|}\,g[\Psi](\sigma)\geq
1/\lambda\right\\}>0\,.$ (33)

######  Theorem 2.7.

Under Assumption 1 and 3, the solution $\Psi$ to the fixed-point problem 1.3
can be uniquely extended as a constant function on $[S_{1},U_{1}]$ with
$U_{1}=S_{1}+\lambda\pi_{1}$ where $\pi_{1}$ satisfies
$0<\pi_{1}<\|q(S_{1},\cdot)\|_{1}\leq 1$ and is defined as

$\displaystyle\pi_{1}=\inf\left\\{p\geq
0\,\bigg{|}\,p>\int_{0}^{\infty}H(\lambda
p,x)q(S_{1},x)\,\mathrm{d}x\right\\}\,.$ (34)

We show our two main results in Section 3 and in Section 4. In preparation of
these, let us state a few properties that solutions to the fixed-point problem
1.3 must satisfy. The first such property shows that for any solutions $\Psi$,
the associated backward-delay functions $\eta[\Psi]$ must be uniformly
bounded.

######  Proposition 2.4.

For all solutions $\Psi$ to the fixed-point problem 1.3, the associated
backward-delay functions $\eta$ are bounded by $\lambda+\nu\epsilon$.

###### Proof.

Solutions $\Psi$ to the fixed-point problem 1.3 are necessarily continuous by
smoothness of the associated cumulative function $G[\Psi]$. Moreover by
definition of the backward function $\xi$, such solutions must satisfy
$\Psi(\xi(\sigma))=\Psi(\sigma)-\epsilon$. This implies that

$\displaystyle\nu\epsilon=\nu\big{(}\Psi(\sigma)-\Psi(\xi(\sigma))\big{)}=\sigma-\xi(\sigma)-\lambda\big{(}G(\sigma)-G(\xi(\sigma))\big{)}\,.$

We conclude by remembering that $\eta=\mathrm{Id}-\xi$ and that by
conservation of probability, we must have $G(\sigma)-G(\xi(\sigma))\leq 1$. ∎

The second property states that global inverse time change $\Psi$ provide us
with global time changes $\Phi=\Psi^{-1}:\mathbbm{R}^{+}\to\mathbbm{R}^{+}$
for the dPMF dynamics. Suppose that the fixed-point problem 1.3 admits an
inverse time change $\Psi$ as a solution. Such a solution cannot explode in
finite time as it is nondecreasing with bounded difference quotient:
$w_{\Psi}\leq 1/\nu$. Let us further assume that it is a global solution,
i.e., that $\Psi$ is defined on the whole set $\mathbbm{R}^{+}$. Then, by
boundedness of the backward-delay function $\eta=\mathrm{Id}-\xi$, it must be
that $S_{\infty}=\lim_{\sigma\to\infty}\xi(\sigma)=\infty$. For finite
refractory period $\epsilon>0$, it turns out that this observation is enough
to show the function $\Phi=\Psi^{-1}$ is also a global time change, as stated
in the following proposition.

######  Proposition 2.5.

For $\epsilon>0$, solutions to the fixed-point problem 1.3 that are defined
over $\mathbbm{R}^{+}$ necessarily satisfy
$T_{\infty}=\lim_{\sigma\to\infty}\Psi(\sigma)=\infty$.

###### Proof.

Given a solution $\Psi:\mathbbm{R}^{+}\to\mathbbm{R}^{+}$ to the fixed-point
problem 1.3, suppose that
$T_{\infty}=\lim_{\sigma\to\infty}\Psi(\sigma)<\infty$. Then, as $\Psi$ and
$\Phi=\Psi^{-1}$ are nondecreasing, for all $\epsilon>0$

$\displaystyle
S_{\infty}=\lim_{\sigma\to\infty}\xi(\sigma)=\lim_{\sigma\to\infty}\Phi(\Psi(\sigma)-\epsilon)=\lim_{t\to
T_{\infty}^{-}}\Phi(t-\epsilon)\leq\Phi(T_{\infty}^{-}-\epsilon)<\infty\,.$

This contradicts the fact that $\eta=\mathrm{Id}-\xi$ is uniformly bounded on
$\mathbbm{R}^{+}$. Thus we must have $T_{\infty}=\infty$. ∎

Note that the above result only holds for nonzero refractory period
$\epsilon>0$. Establishing a similar result for vanishing refractory period
$\epsilon=0$ will require a detailed analysis, which we will conduct only for
large enough interaction parameters $\lambda>\Lambda$.

##  3 Global blowup solutions

In this section, we establish the existence and uniqueness of explosive dPMF
dynamics over the whole half-line $\mathbbm{R}^{+}$ in the large interaction
regime. First, we introduce useful _a prori_ estimates about the fluxes
associated to the fixed-point solutions $\Psi$ for a natural class of initial
conditions. Second, we utilize these estimates to show that in the large
interaction regime, a large enough blowup is followed by a subsequent blowup
in a well ordered fashion, in the sense that this next blowup can only happen
after all the processes that previously blew up have reset. Third, we show
that the size of the next blowup is lower bounded away from zero, which allows
one to establish the existence and uniqueness of explosive dPMF dynamics over
the the whole half-line $\mathbbm{R}^{+}$ by induction on the number of blowup
episodes.

###  3.1 _A_ priori bounds for natural initial conditions

Given normalized initial conditions $(q_{0},g_{0})$ in
$\mathcal{M}((0,\infty))\times\mathcal{M}([\xi_{0},0))$, assume that the
fixed-point problem 1.3 admits a solution $\Psi$ up to some possibly infinite
time $S_{\infty}$. Then, the associated instantaneous flux function $g$
satisfies the renewal-type equation (26) on $[0,S_{\infty})$. We aim at
utilizing this characterization to show that for a restricted class of initial
conditions, $g$ is uniformly bounded on $[0,S_{\infty})$, with an upper bound
that is independent of the coupling parameter $\lambda$ and that scales with
the $L_{1}$ norms of the initial conditions. Informally, the restricted
initial conditions at stake are these distributions of time-changed processes
that have been reset at some time in the past. We will refer to this
restricted set of initial conditions as natural initial conditions, since they
turn out to have the same regularities as generic dPMF solutions. We formally
define natural initial conditions as follows:

######  Definition 3.1.

Natural initial conditions $(q_{0},g_{0})$ in
$\mathcal{M}((0,\infty))\times\mathcal{M}([\xi_{0},0))$ are those for which
there exists a measure $\mu_{0}$ in $\mathcal{M}((0,\infty))$ such that:

$(i)$ The measure $q_{0}$ corresponds to the density

$\displaystyle
q_{0}(x)=\int_{0}^{\infty}\frac{\kappa(\tau,x,\Lambda)}{1-H(\tau,\Lambda)}\,\mu_{0}(\mathrm{d}\tau)\,,$

$(ii)$ The measure $g_{0}$ corresponds to the density

$\displaystyle
g_{0}(\sigma)=\int_{0}^{\infty}\frac{h(\sigma+\tau,\Lambda)}{1-H(\tau,\Lambda)}\,\mu_{0}(\mathrm{d}\tau)\,.$

$(iii)$ We have the degenerate normalization condition

$\displaystyle\int_{0}^{\infty}\mu_{0}(\mathrm{d}\tau)+\int_{\xi_{0}}^{0}g_{0}(\tau)\,\mathrm{d}\tau\leq
1\,.$

We derive the above formal definition for natural initial conditions by
considering absorbed, drifted Wiener dynamics that have started at $\Lambda$
at some time in the past with cumulative rate function $R$. Such dynamics
admit density functions under the integral form

$\displaystyle
q(\sigma,x)=\int_{-\infty}^{0}\kappa(\sigma-\xi,x,\Lambda)\,\mathrm{d}R(\xi)\,,$

where integration by part guarantees convergence for all cumulative rate
functions with, say, polynomial growth. Note that we only requires $R$ to be a
nondecreasing _càdlàg_ function for being a cumulative rate function. The
instantaneous flux function associated to this density is

$\displaystyle
g(\sigma)=\partial_{x}q(\sigma,0)/2=\int_{-\infty}^{0}h(\sigma-\xi,\Lambda)\,\mathrm{d}R(\xi)\,.$

From there, the parametrization of natural initial conditions is recovered by
setting

$\displaystyle\mu_{0}(\mathrm{d}\tau)=\mathrm{d}R(-\tau)(1-H(\tau,\Lambda))\,,$

and recognizing that $q_{0}(x)=q(0,x)$ for $x>0$ and
$g_{0}(\sigma)=g(\sigma)$, $-\xi_{0}\leq\sigma<0$. Bearing the above remarks
in mind, we have the following interpretation for the parametrizing measure
$\mu_{0}$: given a fraction of active processes distributed according to
$q_{0}$ at time $\sigma=0$, $\mu_{0}$ represents the distribution of times
elapsed since their last resets. In particular, definition $(i)$ implies that
$\int_{0}^{\infty}\mu_{0}(\tau)\,\mathrm{d}\tau=\int_{0}^{\infty}q_{0}(x)\,\mathrm{d}x$.
Furthermore, by definition $(i)$ and $(ii)$ and the remarks made above, we
have

$\displaystyle\lim_{\sigma\to
0^{-}}g_{0}(\sigma)=\partial_{\sigma}q_{0}(0)/2\,,$

so that conservation of probability holds as in (7). The above observation
directly shows that given any natural initial conditions $(q_{0},g_{0})$ in
$\mathcal{M}((0,\infty))\times\mathcal{M}([\xi_{0},0))$, the current state of
a solution dPMF dynamics specifies natural initial conditions

$\displaystyle\big{(}q(\sigma,\cdot),\\{g(\xi-\sigma)\\}_{-\eta(\sigma)\leq\xi<0}\big{)}\quad\in\quad\mathcal{M}((0,\infty))\times\mathcal{M}([-\eta(\sigma),0))\,.$

In other word, natural initial conditions are stabilized by dPMF dynamics.
Finally, note that we do not require normalization to one in $(iii)$ so that
natural initial conditions can represent a smaller-than-one fraction of the
processes. This latter point will be useful to discuss the following _a
priori_ estimates for the instantaneous flux associated to solutions of the
fixed-point equation 1.3:

######  Proposition 3.1.

For natural initial conditions $(q_{0},g_{0})$ in
$\mathcal{M}((0,\infty))\times\mathcal{M}([\xi_{0},0))$, we have

$\displaystyle\|g\|_{[0,\infty)}\leq
A_{\Lambda}\,\left(\int_{0}^{\infty}q_{0}(x)\,\mathrm{d}x+\int_{\xi_{0}}^{0}g_{0}(\sigma)\,\mathrm{d}\sigma\right)=A_{\Lambda}\,\|(q_{0},g_{0})\|_{1}\,.$

for a constant $A_{\Lambda}$ that only depends on $\Lambda$.

###### Proof.

Considering natural initial conditions allows one to utilize the Markov
property of the first-passage kernels $\kappa$ to write:

$\displaystyle\int_{0}^{\infty}h(\sigma,x)q_{0}(x)\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{1}{2}\partial_{y}\kappa(\sigma,0,x)q_{0}(x)\,\mathrm{d}x\,,$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\left(\int_{0}^{\infty}\frac{1}{2}\partial_{y}\kappa(\sigma,0,x)\kappa(\tau,x,\Lambda)\,\mathrm{d}x\right)\frac{\mu_{0}(\tau)\,\,\mathrm{d}\tau}{1-H(\tau,\Lambda)}\,,$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\left(\int_{0}^{\infty}\frac{1}{2}\partial_{y}\kappa(\tau+\sigma,0,\Lambda)\,\mathrm{d}x\right)\frac{\mu_{0}(\tau)\,\,\mathrm{d}\tau}{1-H(\tau,\Lambda)}\,,$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{h(\tau+\sigma,\Lambda)}{1-H(\tau,\Lambda)}\mu_{0}(\tau)\,\mathrm{d}\tau\,.$

Injecting the above in equation (26), we deduce the inequality

$\displaystyle|g(\sigma)|$ $\displaystyle\>\leq\>$
$\displaystyle\bigg{\|}\frac{h(\cdot+\sigma,\Lambda)}{1-H(\cdot,\Lambda)}\bigg{\|}_{\infty}\int_{0}^{\infty}\mu_{0}(\tau)\,\mathrm{d}\tau+$
$\displaystyle\quad\quad\bigg{\|}\int_{0}^{\sigma}h(\sigma-\tau,\Lambda)\,\mathrm{d}G(\xi(\tau))\bigg{\|}_{\infty}\,,$
$\displaystyle\>\leq\>$
$\displaystyle\bigg{\|}\frac{h(\cdot+\sigma,\Lambda)}{1-H(\cdot,\Lambda)}\bigg{\|}_{\infty}\int_{0}^{\infty}\mu_{0}(\tau)\,\mathrm{d}\tau+$
$\displaystyle\quad\quad\bigg{\|}\frac{h(\cdot,\Lambda)}{1-H(\cdot,\Lambda)}\bigg{\|}_{\infty}\int_{0}^{\sigma}(1-H(\sigma-\tau,\Lambda))\,\mathrm{d}G(\xi(\tau))\,,$

where both uniform norms are finite. Indeed, for all $\sigma\geq 0$,
L’Hospital rule yields that

$\displaystyle\lim_{\tau\to\infty}\frac{h(\tau+\sigma,\Lambda)}{1-H(\tau,\Lambda)}=\frac{e^{-\sigma/2}}{2}\,,$

which shows that the infinity norms are finite at fixed $\sigma$. Moreover,
exploiting the fact that $h$ is unimodal with maximum value
$h^{\star}_{\Lambda}$ at $\sigma^{\star}=(-3+\sqrt{9+4\Lambda})/2$ and that
$H(\cdot,\Lambda)$ is increasing, we have for $\sigma\geq\sigma^{\star}$

$\displaystyle\bigg{\|}\frac{h(\cdot+\sigma,\Lambda)}{1-H(\cdot,\Lambda)}\bigg{\|}_{\infty}\leq\bigg{\|}\frac{h(\cdot,\Lambda)}{1-H(\cdot,\Lambda)}\bigg{\|}_{\infty}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}M<\infty\,,$

whereas for $\sigma<\sigma^{\star}$, we have

$\displaystyle\bigg{\|}\frac{h(\cdot+\sigma,\Lambda)}{1-H(\cdot,\Lambda)}\bigg{\|}_{\infty}\leq\frac{\|h(\cdot,\Lambda)\|_{\infty}}{1-H(\sigma^{\star},\Lambda)}=\frac{h^{\star}_{\Lambda}}{1-H(\sigma^{\star},\Lambda)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}N<\infty\,.$

Thus, we have

$\displaystyle|g(\sigma)|$ $\displaystyle\leq$
$\displaystyle\max(M,N)\int_{0}^{\infty}q_{0}(x)\,\mathrm{d}x+M\int_{0}^{\sigma}(1-H(\sigma-\tau,\Lambda))\,\mathrm{d}G(\xi(\tau))\,,$
(35)

where the constant $M$ and $N$ only depends on $\Lambda$. To bound the
integral term in the inequality above, we make two observations: First, we
write the renewal-type equation (25) for the cumulative flux $G$ under the
form

$\displaystyle\int_{0}^{\sigma}H(\sigma-\tau,\Lambda))\,\mathrm{d}G(\xi(\tau))=G(\sigma)-G(0)-\int_{0}^{\infty}H(\sigma,x)q_{0}(x)\,\mathrm{d}x\,.$

Second, we write the conservation of probability for processes originating
from $\mu_{0}$ as

$\displaystyle
G(\sigma)-G(\xi(\sigma))-\big{[}G(0)-G(\xi_{0})\big{]}=P(\sigma)-P(0)\,,$

where $P$ denotes the fraction of processes originating from $\mu_{0}$ that is
inactive. These two observations allows one to express the integral term in
inequality (35) as

$\displaystyle\int_{0}^{\sigma}(1-H(\sigma-\tau,\Lambda))\,\mathrm{d}G(\xi(\tau))=P(0)-P(\sigma)+\int_{0}^{\infty}H(\sigma,x)q_{0}(x)\,\mathrm{d}x\,.$

Therefore, as $\|H\|_{\infty}\leq 1$, we have

$\displaystyle\|g\|_{[0,\infty)}$ $\displaystyle\leq$
$\displaystyle\max(M,N)\int_{0}^{\infty}q_{0}(x)\,\mathrm{d}x+M\left(P(0)+\int_{0}^{\infty}q_{0}(x)\,\mathrm{d}x\right)\,.$

We conclude by observing that by conservation of probability mass
$P(0)\leq\|(q_{0},g_{0})\|_{1}$, so that

$\displaystyle\|g\|_{[0,\infty)}$ $\displaystyle\leq$ $\displaystyle
3\max(M,N)\|(q_{0},g_{0})\|_{1}\,.$

∎

The arguments of the above proof can be adapted to obtain a uniform bound to
$\partial_{\sigma}g$, which consitutes another useful _a priori_ estimate :

######  Proposition 3.2.

For natural initial conditions $(q_{0},g_{0})$ in
$\mathcal{M}((0,\infty))\times\mathcal{M}([\xi_{0},0))$, we have

$\displaystyle\|\partial_{\sigma}g\|_{[0,\infty)}\leq
B_{\Lambda}\,\left(\int_{0}^{\infty}q_{0}(x)\,\mathrm{d}x+\int_{\xi_{0}}^{0}g_{0}(\sigma)\right)=B_{\Lambda}\,\|(q_{0},g_{0})\|_{1}\,.$

for a constant $B_{\Lambda}$ that only depends on $\Lambda$.

###### Proof.

The proof proceeds in exactly the same fashion as the proof of Proposition
(3.1) but starting from the renewal-type equation

$\displaystyle\partial_{\sigma}g(\sigma)=\int_{0}^{\infty}\partial_{\sigma}h(\sigma,x)q_{0}(x)\,\mathrm{d}x+\int_{0}^{\sigma}\partial_{\sigma}h(\sigma-\tau,\Lambda)\,\mathrm{d}G(\xi(\tau))\,,$

where we observe that

$\displaystyle\lim_{\tau\to\infty}\frac{|\partial_{\sigma}h(\tau+\sigma,\Lambda)|}{1-H(\tau,\Lambda)}=\frac{e^{-\sigma/2}}{4}\,.$

∎

The above _a priori_ bounds will be crucial to exhibit global solutions to the
fixed-point problem 2.3 that exhibit an infinite number of blowups. Their main
use will be in showing that for blowup of large enough size $0<\pi_{1}<1$, the
small fraction $1-\pi_{1}$ of surviving active processes cannot trigger
another blowup before the original fraction $\pi_{1}$ has reset. In other
words, this will guarantee that blowups are well ordered.

###  3.2 Well-ordered blowups for large interactions

Given natural initial conditions $(q_{0},g_{0})$ in
$\mathcal{M}((0,\infty))\times\mathcal{M}([\xi_{0},0))$, consider a solution
to the fixed-point problem 1.3 with associated instantaneous flux $g$. By
Propositions 3.1 and 3.2, we know that the flux $g$ and its derivative
$\partial_{\sigma}g$, are both uniformly bounded on the domain of the
solution. Here, we aim at refining the bounding analysis of $g$ to ensure that
for large enough synchronization events, blowups are well ordered in the sense
that no blowup can occur before the reset of processes that synchronized in
the past. Large synchronization events inactivate so many processes that the
remaining processes are too few to trigger a blowup on their own.

To prove this, we distinguish between two contributions to the current state
of the dynamics, according to whether processes have inactivated in the most
recent blowup or not. Specifically, assume that a blowup of size $\pi_{1}$
triggers at $S_{1}\geq 0$ with exit time $U_{1}$ and reset time $R_{1}$. At
time $\sigma\geq R_{1}$, we can always split the current state of the dynamics
into a contribution from the synchronized processes that reset at $R_{1}$
(marked by a subscript $\pi_{1}$) and a contribution from the other processes
(marked by a subscript $\cancel{\pi_{1}}$). This means that there are
$(q_{\pi_{1}},g_{\pi_{1}})$ and $(q_{\cancel{\pi_{1}}},g_{\cancel{\pi_{1}}})$
in $\mathcal{M}((0,\infty))\times\mathcal{M}([\xi(\sigma),\sigma))$ such that
for all $\sigma\geq R_{1}$

$\displaystyle
q(\sigma,\cdot)=q_{\pi_{1}}(\sigma,\cdot)+q_{\cancel{\pi_{1}}}(\sigma,\cdot)\quad\mathrm{and}\quad
g(\sigma)=g_{\pi_{1}}(\sigma)+g_{\cancel{\pi_{1}}}(\sigma)\,,$ (36)

with $q_{\pi_{1}}(R_{1},\cdot)=\pi_{1}\delta_{\Lambda}$ and
$g_{\pi_{1}}(\sigma)=0$ for $\sigma\leq R_{1}$ and with
$q_{\cancel{\pi_{1}}}(R_{1},\cdot)=q(R_{1}^{-},\cdot)$ and
$g_{\cancel{\pi_{1}}}(\sigma)=g(\sigma)$ for $U_{1}\leq\sigma<R_{1}$. Given a
fixed backward function $\xi$, it is clear that the governing renewal-type
equation (26) applies to both contribution $(q_{\pi_{1}},g_{\pi_{1}})$ and
$(q_{\cancel{\pi_{1}}},g_{\cancel{\pi_{1}}})$ separately. As a result,
Propositions 3.1 and 3.2 also apply to both contributions separately, allowing
to control the role of each contribution in triggering the next possible
blowup.

The next proposition states this point concisely by making use of the real
numbers $\sigma^{\dagger}$ and $h^{\dagger}$ defined by

$\displaystyle\sigma^{\dagger}=\inf\\{\sigma>0\,|\,\partial^{2}_{\sigma}h(\sigma,\Lambda)=0\\}\quad\mathrm{and}\quad
h^{\dagger}_{\Lambda}=h(\sigma^{\dagger},\Lambda)\,.$

Observe that $\sigma^{\dagger}$ and $h^{\dagger}_{\Lambda}$ only depends on
$\Lambda$, and that $\sigma^{\dagger}$ satisfies
$0<\sigma^{\dagger}<\sigma^{\star}$, where $\sigma^{\star}$ is the unique
maximizer of $h(\cdot,\Lambda)$:
$h(\sigma^{\star},\Lambda)=h^{\star}_{\Lambda}>h^{\dagger}_{\Lambda}$.
Moreover, by definition of $\sigma^{\dagger}$, observe that
$\partial_{\sigma}h(.,\Lambda)$ is increasing on $[0,\sigma^{\dagger}]$. Then,
equipped with $h^{\star}_{\Lambda}$ and $\sigma^{\dagger}$, the following
proposition states that for large enough interaction parameters $\lambda$, a
sufficiently large blowup $\pi_{1}$ will trigger a next blowup in a well
ordered fashion.

######  Proposition 3.3.

Given an interaction parameter $\lambda$ with
$\lambda>l_{\Lambda}=1/h^{\dagger}_{\Lambda}$, there exists a constant
$C_{\Lambda}$ that only depends on $\Lambda$ such that a full blowup of size
$\pi_{1}$ with

$\displaystyle\pi_{1}>\max\left(\frac{l_{\Lambda}}{\lambda},1-\frac{C_{\Lambda}}{\lambda}\right)\,,$
(37)

is followed by another full blowup in finite time. Moreover, if the first
blowup happens at time $S_{1}$, the next blowup time $S_{2}$ is such that
$S_{1}\leq R_{1}<S_{2}\leq R_{1}+\sigma^{\dagger}$, where we have defined the
reset time $R_{1}=\tau(U_{1})$.

###### Proof.

The proof proceeds in four steps: $(i)$ we show that a blowup cannot happen
before the fraction $\pi_{1}$ of processes reset, $(ii)$ we show that the
fraction $\pi_{1}$ of reset processes is enough to trigger a blowup, $(iii)$
we show the full blowup condition, and $(iv)$ we conclude by specifying the
choice of $l_{\Lambda}$ and $C_{\Lambda}$.

$(i)$ Suppose that the last blowup triggers at time $S_{1}$ with blowup size
$\pi_{1}$, then at blowup exit time $U_{1}=S_{1}+\lambda\pi_{1}$, the exit
states

$\displaystyle\big{(}q(U_{1},\cdot),\\{g(\sigma)\\}_{\xi(U_{1})\leq\sigma<U_{1}}\big{)}$

determine natural initial conditions. These are such that
$G(U_{1})-G(S_{1})=\int_{S_{1}}^{U_{1}}g(\sigma)\,\mathrm{d}\sigma=\pi_{1}$,
representing the fraction of processes to be instantaneously reset at time
$R_{1}=\tau(S_{1})$. Let us consider the part of the exit initial conditions
excluding this fraction, i.e., the active processes that either survive the
blowup or that are already inactive before the blowup:

$\displaystyle(q_{\cancel{\pi_{1}},0},g_{\cancel{\pi_{1}},0})=\big{(}q(U_{1},\cdot),\\{g(\sigma)\\}_{\xi(U_{1})\leq\sigma<S_{1}}\big{)}\,,\quad\mathrm{with}\quad\|(q_{\cancel{\pi_{1}},0},g_{\cancel{\pi_{1}},0})\|_{1}\leq
1-\pi_{1}\,.$

Choosing $C_{\Lambda}<1/(4A_{\Lambda})$ in (37) implies that
$1-\pi_{1}<1/(4A_{\Lambda}\lambda)$ so that by Proposition 3.1, the
instantaneous flux $g_{\cancel{\pi_{1}}}$ due to the linear dynamics of the
processes arising from the partial initial conditions
$(q_{\cancel{\pi_{1}},0},g_{\cancel{\pi_{1}},0})$ is bounded above with
$\|g_{\cancel{\pi_{1}}}\|_{[U_{1},\infty)}\leq 1/(4\lambda)$. This implies
that no blowup can occur before resetting the fraction of inactivated
processes $\pi_{1}$ as the blowup condition ($g(\sigma)=1/\lambda$) cannot be
met. Moreover, before the reset time $R_{1}$, the inverse time change $\Psi$
only depends on the partial partial initial conditions
$(q_{\cancel{\pi_{1}},0},g_{\cancel{\pi_{1}},0})$ so that we have

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\Psi(R_{1})-\Psi(U_{1})\,,$ $\displaystyle=$
$\displaystyle\big{(}R_{1}-U_{1}-\lambda(G(R_{1})-G(U_{1})\big{)}/\nu\,,$
$\displaystyle\geq$
$\displaystyle(1-\lambda\|g_{\cancel{\pi_{1}}}\|_{[U_{1},\infty)})(R_{1}-U_{1})/\nu\,,$
$\displaystyle\geq$ $\displaystyle(R_{1}-U_{1})/(2\nu)\,.$

Thus the reset of the processes inactivated during the last blowup must happen
in finite time at $R_{1}$ with

$\displaystyle\nu\epsilon<R_{1}-U_{1}<2\nu\epsilon\,.$ (38)

$(ii)$ We denote by $g_{\pi_{1}}$ the instantaneous flux due the fraction
$\pi_{1}$ of the processes that reset at forward time $R_{1}$. For all
$\sigma\geq R_{1}$, this flux satisfies

$\displaystyle\pi_{1}h(\sigma-R_{1},\Lambda)\leq
g_{\pi_{1}}(\sigma)\leq\pi_{1}\sum_{n=0}^{\infty}h(\sigma-
R_{1},n\Lambda)=\pi_{1}\tilde{g}(\sigma-R_{1},\Lambda)\,,$ (39)

where left term is the flux due to the fraction $\pi_{1}$ excluding any reset
after $R_{1}$, whereas the right term is the flux due to the fraction
$\pi_{1}$ with instantaneous reset after $R_{1}$. Remembering that by
definition $h^{\star}_{\Lambda}=\sup_{\sigma\geq
0}h(\sigma,\Lambda)>h^{\dagger}_{\Lambda}$, choosing
$\lambda>1/(h^{\dagger}_{\Lambda}\pi_{1})$ implies that the blowup condition
$g(\sigma)\geq 1/\lambda$ will be met at some time $S_{2}>R_{1}$. The upper
bound flux term $\tilde{g}$ in inequality (39) only depends on $\Lambda$ and
is uniformly bounded with respect $\sigma$ in $\mathbbm{R}^{+}$. Moreover,
$\tilde{g}$ satisfies the renewal equation

$\displaystyle\tilde{g}(\sigma,\Lambda)=h(\sigma,\Lambda)+\int^{\sigma}_{0}h(\sigma-\tau,\Lambda)\tilde{g}(\tau,\Lambda)\,\mathrm{d}\tau\,,$

so that for all $0\leq\sigma\leq\sigma^{\star}$, remembering that
$\|\tilde{g}\|_{\infty}\leq A_{\Lambda}$, we have the upper bound

$\displaystyle\tilde{g}(\sigma,\Lambda)\leq
h(\sigma,\Lambda)+\|\tilde{g}\|_{\infty}\int^{\sigma}_{0}h(\sigma-\tau,\Lambda)\,\mathrm{d}\tau\leq
h(\sigma,\Lambda)(1+A_{\Lambda}\sigma^{\star})\,,$

where the last inequality follows from the fact that $h(\cdot,\Lambda)$ is
increasing on $[0,\sigma^{\star}]$. Therefore, defining
$a_{\Lambda}=(1+A_{\Lambda}\sigma^{\star})^{-1}/2<1/2$, we have

$\displaystyle s_{1}<S_{2}-R_{1}<s_{2}\,,$

where the delay times $s_{1}$ and $s_{2}$ only depend on $\Lambda$ and
$\pi_{1}\lambda$ via:

$\displaystyle\quad
s_{1}=\inf\left\\{s>0\,\bigg{|}\,h(s,\Lambda)=\frac{a_{\Lambda}}{\pi_{1}\lambda}\right\\}\;\;\mathrm{and}\;\;s_{2}=\inf\left\\{s>0\,\bigg{|}\,h(s,\Lambda)=\frac{1}{\pi_{1}\lambda}\right\\}\,.$
(40)

$(iii)$ It remains to check the full-blowup condition, i.e.,
$\partial_{\sigma}g(S_{2})>0$. In this perspective, let us first establish a
lower bound for $\partial_{\sigma}g_{\pi_{1}}(S_{2})$. Such a bound is
obtained by differentiating (26) with respect to $\sigma$ for
$R_{1}\leq\sigma\leq R_{1}+\sigma^{\dagger}$, which yields

$\displaystyle\partial_{\sigma}g_{\pi_{1}}(\sigma)=\pi_{1}\partial_{\sigma}h(\sigma-
R_{1},\Lambda)+\int^{\sigma}_{0}\partial_{\sigma}h(\sigma-\tau,\Lambda)\,g_{\pi_{1}}(\xi(\tau))\geq\pi_{1}\partial_{\sigma}h(\sigma-
R_{1},\Lambda)\,,$

where we utilize the fact that $\partial_{\sigma}h(\cdot,\Lambda)$ is
increasing on $[0,\sigma^{\dagger}]$. Thus, under the condition that
$s_{2}\leq\sigma^{\dagger}$, we have

$\displaystyle\partial_{\sigma}g_{\pi_{1}}(S_{2})\geq\pi_{1}\partial_{\sigma}h(S_{2}-R_{1},\Lambda)\geq\pi_{1}\partial_{\sigma}h(s_{1},\Lambda)\,,$

where $s_{1}$ is defined by (40). Moreover, from the explicit expression of
$h(\cdot,\Lambda)$, we evaluate

$\displaystyle\frac{\partial_{\sigma}h(s_{1},\Lambda)}{h(s_{1},\Lambda)}=\partial_{\sigma}\ln
h(s_{1},\Lambda)=\frac{\Lambda^{2}-(3s_{1}+s_{1}^{2})}{2s_{1}^{2}}\,,$

so that by the definition of $s_{1}$ given in (40) we have

$\displaystyle\pi_{1}\partial_{\sigma}h(s_{1},\Lambda)=\frac{\Lambda^{2}-(3s_{1}+s_{1}^{2})}{2s_{1}^{2}}\frac{1}{a_{\Lambda}\lambda}\,.$

To show that the above quantity is nonnegative, it is enough to remember that
under the condition that $s_{2}\leq\sigma^{\dagger}$, we have
$s_{1}<s_{2}\leq\sigma^{\dagger}<\sigma^{\star}$. Moreover, the function
$s_{1}\mapsto(\Lambda^{2}-(3s_{1}+s_{1}^{2}))/(2s_{1}^{2})$ is decreasing and
strictly positive on $(0,\sigma^{\star})$ since it has the same sign as
$\partial_{\sigma}h(\cdot,\Lambda)$. Thus, we have

$\displaystyle\partial_{\sigma}g_{\pi_{1}}(S_{2})\geq\pi_{1}\partial_{\sigma}h(s_{1},\Lambda)\geq\frac{b_{\Lambda}}{\lambda}\,,\quad\mathrm{with}\quad
b_{\lambda}=\frac{\Lambda^{2}-\left(3{\sigma^{\dagger}}+\sigma^{\dagger
2}\right)}{2\sigma^{\dagger 2}a_{\Lambda}}>0\,,$

where $b_{\Lambda}$ only depends on $\Lambda$. Choosing
$C_{\Lambda}<b_{\Lambda}/(2B_{\Lambda})$ in (37) implies that
$1-\pi_{1}<b_{\Lambda}/(2B_{\Lambda}\lambda)$ so that by Proposition 3.2, the
absolute value of the term $|\partial_{\sigma}g_{\cancel{\pi_{1}}}|$ due to
the linear dynamics of the processes arising from the partial initial
conditions $(q_{\cancel{\pi_{1},0}},g_{\cancel{\pi_{1},0}})$ is bounded above
with $\|\partial_{\sigma}g_{\cancel{\pi_{1}}}\|_{[U_{1},\infty)}\leq
b_{\Lambda}/(2\lambda)$. This shows that a full blowup happens in $S_{2}$
since

$\displaystyle\partial_{\sigma}g(S_{2})=\partial_{\sigma}g_{\pi_{1}}(S_{2})+\partial_{\sigma}g_{\cancel{\pi_{1}}}(S_{2})\geq\partial_{\sigma}g_{\pi_{1}}(S_{2})-|\partial_{\sigma}g_{\cancel{\pi_{1}}}(S_{2})|\geq\frac{b_{\Lambda}}{2\lambda}>0\,.$
(41)

$(iv)$ The blowup condition due to reset processes alone requires in $(i)$
that $C_{\Lambda}<1/(4A_{\Lambda})$ and in $(ii)$ that
$\pi_{1}h^{*}>1/\lambda$. The full blowup condition requires in $(iii)$ that
$s_{2}\leq\sigma^{\dagger}$, which is equivalent to
$\pi_{1}h^{\dagger}>1/\lambda$, and that
$C_{\Lambda}<b_{\Lambda}/(2B_{\Lambda})$. This shows that choosing

$\displaystyle
l_{\Lambda}>\frac{1}{h^{\dagger}_{\Lambda}}\quad\mathrm{and}\quad
0<C_{\Lambda}<\frac{1}{2}\min\left(\frac{1}{2A_{\Lambda}},\frac{b_{\Lambda}}{B_{\Lambda}}\right)\,,$
(42)

suffices to ensure that after a full blowup of size $\pi_{1}$ at time $S_{1}$
satisfying (37), the next blowup must happen in a well ordered fashion at time
$S_{2}$ such that $S_{1}\leq R_{1}<S_{2}\leq R_{1}+\sigma^{\dagger}$. ∎

###  3.3 Persistence of blowups for large interactions

Proposition 3.3 exhibits a criterion for a large enough full blowup to be
followed by another full blowup in a well ordered fashion. By well ordered, we
mean that the next blowup must occur after the processes that last blew up
have all reset. In this context, for blowups to occur indefinitely in a
sustained fashion, we further need to check that the size of the next blowup
remains larger than that of the criterion of Proposition 3.3. In this
perspective, it is instructive to find an upper bound to the fraction of
inactive processes at the trigger time of the next blowup $S_{2}$. Indeed,
processes that are inactive at $S_{2}$ will remain so throughout the blowup,
and therefore cannot contribute to the next blowup of size $\pi_{2}$. The next
proposition shows that the fraction of inactive process $P(S_{2})$ can be made
arbitrarily small for large enough $\lambda$. This proposition only considers
interaction parameter values $\lambda>\max(2l_{\Lambda},2/C_{\Lambda})$, so
that the constraint on the full blowup size in Proposition 3.3 reduces to

$\displaystyle\pi_{1}>1-\frac{C_{\Lambda}}{\lambda}>1/2\,.$ (43)

######  Proposition 3.4.

Consider an interaction parameter $\lambda$ such that
$\lambda>\max(2l_{\Lambda},2/C_{\Lambda})$ and a dynamics for which a full
blowup happens at time $S_{1}$ with size $\pi_{1}>1-C_{\Lambda}/\lambda>1/2$.
If $\lambda$ further satisfies that
$\lambda>2\sqrt{2\pi\Lambda}e^{\Lambda^{2}/2C_{\Lambda}}$ and if the
refractory period is such that $\epsilon<C_{\Lambda}/(2\nu)$, then the next
full blowup triggers at time $S_{2}<\infty$ with a fraction of inactive
processes:

$\displaystyle P(S_{2})\leq\frac{C_{\Lambda}}{2\lambda}\,.$

###### Proof.

Let us first consider $s_{2}$, the upper bound to $S_{2}-R_{1}$ defined by
(40). Noticing that $h(\Lambda,\Lambda)=1/(\sqrt{2\pi\Lambda})$, by definition
of $s_{2}$, if $\lambda>\sqrt{2\pi\Lambda}/\pi_{1}$, then we necessarily have
$s_{2}<\Lambda$. Moreover, if $s_{2}<\Lambda$, we have

$\displaystyle\quad\frac{1}{\pi_{1}\lambda}=h(s_{2},\Lambda)=\frac{\Lambda}{\sqrt{2\pi
s_{2}^{3}}}\,e^{-\frac{(\Lambda-
s_{2})^{2}}{2s_{2}}}>\frac{e^{-\frac{\Lambda^{2}}{2s_{2}}}}{\sqrt{2\pi\Lambda}}\,,$

which together with $\pi_{1}>1/2$, implies the following upper bound on
$s_{2}$:

$\displaystyle
s_{2}<\frac{\Lambda^{2}}{2\lambda\ln\left(\pi_{1}\lambda/\sqrt{2\pi\Lambda}\right)}<\frac{\Lambda^{2}}{2\lambda\ln\left(\lambda/\big{(}2\sqrt{2\pi\Lambda}\big{)}\right)}\,.$

Therefore the cumulative flux of inactivated processes between the last blowup
exit time $U_{1}$ and the next blowup trigger time $S_{2}$ satisfies

$\displaystyle G(S_{2})-G(U_{1})$ $\displaystyle\leq$
$\displaystyle\|g\|_{U_{1},R_{1}}(R_{1}-U_{1})+\|g\|_{R_{1},S_{2}}(S_{2}-R_{1})\,,$
$\displaystyle\leq$
$\displaystyle\frac{\nu\epsilon}{2\lambda}+\frac{\Lambda^{2}}{2\lambda\ln\left(\lambda/\big{(}2\sqrt{2\pi\Lambda}\big{)}\right)}\,,$

where we utilize that $\|g\|_{U_{1},R_{1}}<1/(4\lambda)$ and
$R_{1}-U_{1}<2\nu\epsilon$ by (38). Then for all
$\epsilon<C_{\Lambda}/(2\nu)$, assuming that

$\displaystyle\lambda>2\sqrt{2\pi\Lambda}e^{2\Lambda^{2}/C_{\Lambda}}\,,$

yields an upper bound on the fraction of inactive processes at the next blowup
time $S_{2}$:

$\displaystyle P(S_{2})=1-\int_{0}^{\infty}q(S_{2},x)\,dx\leq
G(S_{2})-G(U_{1})\leq\frac{C_{\Lambda}}{2\lambda}\,.$

∎

We are now in a position to show that for large enough interaction parameter
$\lambda$, the next blowup will have a size $\pi_{2}$ that satisfies the
criterion of Proposition 3.3. This is a consequence of the following
observations: For well ordered blowup, the reset of a fraction $\pi_{1}$ of
processes at time $R_{1}$ is enough to trigger a blowup of finite size, which
will be bounded below by a quantity $\Delta H_{\Lambda}/2>0$ that only depends
on $\Lambda$. Then, the duration of the blowup during which reset is halted is
lower bounded by $\lambda\Delta H_{\Lambda}/2>0$. In turn, for large enough
$\lambda$, the probability of inactivation during blowup can be made
exponentially small with respect to $\lambda$ by realizing that inactivation
has an asymptotically constant, nonzero hazard rate function. Altogether,
these observations allow us to form a self-consistent condition guaranteeing
that $\pi_{2}$ satisfies the criterion of Proposition 3.3, leading to the
following result:

######  Proposition 3.5.

Assume that the refractory period satisfies $\epsilon<C_{\Lambda}/\nu$. There
exists a constant $L_{\Lambda}>0$, which only depends on $\Lambda$, such that
for all interaction parameter $\lambda>L_{\Lambda}$, every full blowup at time
$S_{1}$ with size $\pi_{1}\geq 1-C_{\Lambda}/\lambda$ is directly followed by
a full blowup at time $S_{2}$ with size $\pi_{2}\geq 1-C_{\Lambda}/\lambda$.

###### Proof.

The proof proceeds in three steps: $(i)$ we exhibit the asymptotically
exponential regime of processes survival during blowup, $(ii)$ we show that
this exponential regime is met for large enough interaction parameter
$\lambda$, and $(iii)$ we exhibit a sufficient constant $L_{\Lambda}$.

$(i)$ We can always choose $\lambda$ large enough so that

$\displaystyle\lambda\geq\max\big{(}2l_{\Lambda},2/C_{\Lambda},2\sqrt{2\pi\Lambda}e^{\Lambda^{2}/2C_{\Lambda}}\big{)}\,.$
(44)

Then by Proposition 3.3, the occurrence of a full blowup in $S_{1}$ with size
$\pi_{1}\geq 1-C_{\Lambda}/\lambda$ triggers the next full blowup at a time
$S_{2}$ with $R_{1}<S_{2}\leq R_{1}+\sigma^{\dagger}$. By Proposition 3.4, the
fraction of active process $1-P(S_{2})$ at time $S_{2}$ is such that
$1-P(S_{2})>1-C_{\Lambda}/2$. Let us consider a process $Y_{\sigma}$ that is
active at time $S_{2}$ and let us denote by $\xi$ its next inactivation time.
Then for all $\sigma\geq\tau\geq S_{1}$, we have

$\quad\mathbb{P}\left[\xi\geq\sigma\,\Big{|}\,\xi\geq\tau\right]=\frac{\mathbb{P}\left[\xi\geq\sigma\right]}{\mathbb{P}\left[\xi\geq\tau\right]}=\frac{1-H(\sigma,\Lambda)}{1-H(\tau,\Lambda)}\,,$

where the survival probabilities $1-H(\sigma,\Lambda)$ can be written in term
of a hazard rate as

$\displaystyle
1-H(\sigma,\Lambda)=\exp{\left(\int_{0}^{\sigma}\partial_{\tau}\ln\left(1-H(\tau,\Lambda)\right)\,\mathrm{d}\tau\right)}\,.$

The key observation is to notice that the hazard rate appearing above has the
finite limit

$\displaystyle\lim_{\sigma\to\infty}-\partial_{\sigma}\ln\left(1-H(\sigma,\Lambda)\right)=\lim_{\sigma\to\infty}\frac{h(\sigma,\Lambda)}{1-H(\sigma,\Lambda)}=\frac{1}{2}\,,$

showing that for large time, inactivation asymptotically follows a memoryless
exponential law for all active processes from the fraction $1-P(S_{2})$.

$(ii)$ We now exploit the asymptotic exponential behavior of the hazard
function associated to $1-H(\cdot,\Lambda)$ to show that $\pi_{2}$, the size
of the next full blowup triggered in $S_{2}$, can be made arbitrary close to
$1-P(S_{2})$. In this perspective, let us define the finite time

$\displaystyle\sigma^{\sharp}=\sup\left\\{\sigma\geq\sigma^{\star}\,\bigg{|}\,\partial_{\sigma}\ln\left(1-H(\sigma,\Lambda)\right)<-\frac{1}{4}\right\\}<\infty\,.$

For all $\sigma>\sigma^{\sharp}$, integrating the hazard rate inequality
$\partial_{\sigma}\ln\left(1-H(\sigma,\Lambda)\right)<-1/4$ yields

$\displaystyle\mathbb{P}\left[\xi\geq\sigma\,\Big{|}\,\xi\geq\sigma^{\sharp}\right]=\frac{1-H(\sigma)}{1-H(\sigma^{\sharp})}\leq
e^{-\frac{\sigma-\sigma^{\sharp}}{4}}\,,$ (45)

showing that the survival probability of active processes decays at least
exponentially past time $S_{2}+\sigma^{\sharp}$. Moreover, the fraction of
processes $\Delta P_{2}$ that inactivates during the full blowup but before
$S_{2}+\sigma^{\sharp}$ satisfies

$\displaystyle\Delta
P_{2}=P(S_{2}+\sigma^{\sharp})-P(S_{2})\geq\pi_{1}\int_{\sigma^{\dagger}}^{\sigma^{\star}}h(\sigma,\Lambda)\,\mathrm{d}\sigma\geq\Delta
H_{\Lambda}/2>0\,.$

where we utilize the assumption that $\pi_{1}>1-C_{\Lambda}/\lambda>1/2$ and
where we define the constant $\Delta
H_{\Lambda}=\int_{\sigma^{\dagger}}^{\sigma^{\star}}h(\sigma,\Lambda)\,\mathrm{d}\sigma>0$,
which only depends on $\Lambda$. The above lower bound follows from the fact
that we choose $\lambda$ and $\pi_{1}$ so that
$S_{2}-R_{1}\leq\sigma^{\dagger}<\sigma^{\star}\leq\sigma^{\sharp}$. By (45),
the lower bound $\Delta H_{\Lambda}/2\leq\Delta P_{2}\leq\pi_{2}$ implies that

$\displaystyle\mathbb{P}\left[\xi\geq\lambda\pi_{2}\,\Big{|}\,\xi\geq\sigma^{\sharp}\right]\leq
e^{-\frac{\lambda\pi_{2}-\sigma^{\sharp}}{4}}\leq e^{-\frac{\lambda\Delta
H_{\Lambda}-2\sigma^{\sharp}}{8}}\,.$

Thus, the fraction of inactivated process during blowup can be made
arbitrarily close to $1-P(S_{2})$ at the cost of choosing larger $\lambda$, as
shown by:

$\displaystyle\pi_{2}$ $\displaystyle=$ $\displaystyle\Delta
P_{2}+\big{(}1-P(S_{2}+\sigma^{\sharp})\big{)}\mathbb{P}\left[\xi\leq\lambda\pi_{2}\,\Big{|}\,\xi\geq\sigma^{\sharp}\right]\,,$
$\displaystyle\geq$ $\displaystyle\Delta
P_{2}+\big{(}1-P(S_{2}+\sigma^{\sharp})\big{)}\left(1-e^{-\frac{\lambda\Delta
H_{\Lambda}-2\sigma^{\sharp}}{8}}\right)\,,$ $\displaystyle\geq$
$\displaystyle
1-P(S_{2})-\big{(}1-P(S_{2}+\sigma^{\sharp})\big{)}e^{-\frac{\lambda\Delta
H_{\Lambda}-2\sigma^{\sharp}}{8}}\xrightarrow{\lambda\to\infty}1-P(S_{2})\,.$

$(iii)$ It remains to exhibit a criterion for $\lambda$ ensuring that the
blowup size $\pi_{2}$ is larger than $1-C_{\Lambda}/2$. Such a criterion can
be obtained by first observing that

$\displaystyle\pi_{2}\geq
1-P(S_{2})-\big{(}1-P(S_{2}+\sigma^{\sharp})\big{)}e^{-\frac{\lambda\Delta
H_{\Lambda}-2\sigma^{\sharp}}{8}}\geq 1-P(S_{2})-e^{-\frac{\lambda\Delta
H_{\Lambda}-2\sigma^{\sharp}}{8}}\,.$

Then let us define $\lambda^{\sharp}>0$ as

$\displaystyle\lambda^{\sharp}=\sup\left\\{\lambda\geq
0\,\bigg{|}\,\lambda\leq\frac{2}{\Delta
H_{\Lambda}}\left(\sigma^{\sharp}-4\ln{\left(\frac{C_{\Lambda}}{2\lambda}\right)}\right)\right\\}<\frac{C_{\Lambda}}{2}e^{-\sigma^{\sharp}/4}<\infty\,,$

such that for all $\lambda\geq\lambda^{\sharp}$, we have

$\displaystyle e^{-\frac{\lambda\Delta
H_{\Lambda}-2\sigma^{\sharp}}{8}}\leq\frac{C_{\Lambda}}{2\lambda}\,.$

Utilizing the fact that $P(S_{2})<C_{\Lambda}/(2\lambda)$ by Proposition 3.4,
we have

$\displaystyle\pi_{2}\geq 1-P(S_{2})-\frac{C_{\Lambda}}{2\lambda}\geq
1-\frac{C_{\Lambda}}{\lambda}\,.$

Thus choosing

$\displaystyle\lambda\geq
L_{\Lambda}=\max\left(2l_{\Lambda},2/C_{\Lambda},2\sqrt{2\pi\Lambda}e^{\Lambda^{2}/2C_{\Lambda}},\lambda^{\sharp}\right)$

suffices to ensure that the next full blowup has size at least
$1-C_{\Lambda}/\lambda$. ∎

The above proposition directly implies the existence and uniqueness of global
explosive solutions under the following assumptions:

######  Assumption 4.

Consider a dPMF dynamics with refractory period
$0\leq\epsilon<C_{\Lambda}/\nu$, with interaction parameter $\lambda\geq
L_{\Lambda}$, and with natural initial conditions such that $q_{0}$ contains
an Dirac-delta mass $\pi_{0}\delta_{\Lambda}$ with $\pi_{0}\geq
1-C_{\Lambda}/\lambda$.

######  Theorem 3.2.

Under Assumption 4 , the fixed-point problem 1.3 admits a unique global
solution $\Psi$ defined over the whole half-line $\mathbbm{R}^{+}$. Moreover,
this solution presents an infinite but discrete set of full blowups with
trigger times $\\{S_{k}\\}_{k\in\mathbbm{N}}$ and with sizes
$\\{\pi_{k}\\}_{k\in\mathbbm{N}}$ such that for all $k\geq 1$, $\pi_{k}\geq
1-C_{\Lambda}/\lambda$ and $S_{k+1}-U_{k}\geq\nu\epsilon$ where
$U_{k}=S_{k}+\lambda\pi_{k}$.

###### Proof.

By Theorem 2.6, the fixed-point problem 1.3 admits a local smooth solution up
to time $S_{1}=\inf\left\\{\sigma>0\,|\,\Psi^{\prime}(\sigma)\geq 0\right\\}$.
By the same arguments as in the proof of Proposition 3.3, the blowup condition
must be satisfied in finite time: $S_{1}<\infty$. By the same arguments as in
the proof of Proposition 3.5, the blowup size satisfies
$\pi_{1}>1-C_{\Lambda}/\lambda$. From there on, one can iteratively apply
Proposition 3.5 to define a global solution to the fixed-point problem 1.3
with an infinite number of blowups. The uniqueness of the global solution
follows from the uniqueness of the local solutions up to blowups and the
uniqueness of the continuation process at the exit of a blowup. Finally, this
unique global solution $\Phi$ is defined over the whole half-line
$\mathbbm{R}^{+}$ since all blowups are lower bounded by
$1-C_{\Lambda}/\lambda$, so that all local solutions are defined over a domain
of duration at least $\lambda(1-C_{\Lambda}/\lambda)$ during blowup episodes.
Finally, a blowup can only happen after resetting the processes that
inactivated during the previous blowup, which must happens after a duration
$\nu\epsilon$ by the proof of Proposition 3.3. ∎

For $\epsilon>0$, Theorem 3.2 directly implies the existence of global time
change $\Phi$ parametrizing the original dPMF dynamics over the the half-line
$\mathbbm{R}^{+}$ with the properties listed in Theorem 1.8. However, Theorem
3.2 only implies that
$T_{k+1}-T_{k}=\Psi(S_{k+1})-\Psi(S_{k})=\Psi(S_{k+1})-\Psi(U_{k})\geq\epsilon$,
where $\\{T_{k}=\Psi(S_{k})\\}_{k\in\mathrm{N}}$ is the sequence of blowup
times for the original dPMF dynamics. Thus, in agreement with the remark
following Proposition 2.5, Theorem 3.2 does not allow us to conclude about the
existence of a global time change $\Phi$ in the limit $\epsilon\to 0^{+}$.
Indeed, it could be that $\lim_{\epsilon\to 0^{+}}T_{k+1}-T_{k}=0$ for all
large enough $k$, which is compatible with accumulating the blowup times so
that

$\displaystyle
T_{\infty}=\lim_{k\to\infty}T_{k}=\lim_{k\to\infty}\Psi(S_{k})=\lim_{S\to\infty}\Psi(S)<\infty\,.$
(46)

As a result, $\Phi=\Psi^{-1}$ would only define a solution time change $\Phi$
up the finite time $T_{\infty}$. We address this point in the following
section, where we characterize PMF dynamics (with zero refractory period) as
limits of dPMF dynamics when $\epsilon\to 0^{+}$.

##  4 Limit of vanishing refractory periods

In this section, we show that explosive PMF dynamics can be defined
consistently over the whole half-line $\mathbbm{R}^{+}$ for zero refractory
period, i.e., with instantaneous reset. First, we define PMF dynamics for zero
refractory period $\epsilon=0$, which by contrast dPMF dynamics with
$\epsilon>0$, admit an explicit iterative construction. Second, we introduce a
series of continuity results showing that PMF dynamics are “physical” in the
sense that they are recovered from dPMF dynamics in the limit $\epsilon\to
0^{+}$. Finally, we provide the proofs for these continuity results, which
essentially amounts to show that at all blowup trigger, exit, and reset times,
the spatial densities of dPMF dynamics converge toward their PMF counterpart
in $L_{1}$ norm.

###  4.1 Solutions with zero refractory periods

Theorem 3.2 applies to the case of zero refractory period $\epsilon=0$, that
is for PMF dynamics, assuming natural initial condition with normalized
spatial component, $\int_{0}^{\infty}q_{0}(x)\,\mathrm{d}x=1$, and no inactive
processes. In particular, for $\epsilon=0$, the fixed-point problem 1.3 admits
a unique global solution for large enough interaction parameter $\lambda$ and
for sufficiently large initial mass concentrated at $\Lambda$. Let us denote
this solution by $\Psi$ for simplicity. As an inverse time change, $\Psi$
parametrizes a PMF dynamics with an infinite but discrete set of well ordered
blowups with successive trigger times $\\{S_{k}\\}_{k\in\mathbbm{N}}$ and
corresponding size $\\{\pi_{k}\\}_{k\in\mathbbm{N}}$. A major benefit of
considering PMF dynamics is that the corresponding inverse time changes $\Phi$
admit an explicit iterative construction. Specifically, the sequences
$\\{S_{k}\\}_{k\in\mathbbm{N}}$ and $\\{\pi_{k}\\}_{k\in\mathbbm{N}}$ can be
defined explicitely via the adjunction of the sequence of initial spatial
conditions $\\{q_{0,k}(\cdot)=q(S_{k},\cdot)\\}_{k\in\mathbbm{N}}$. This
explicit definition is made possible by the fact that for zero refractory
period $\epsilon=0$, the renewal problems at stake loose their delayed
character and can be solved analytically. In particular, in between blowup
episodes, we have the fundamental solution

$\displaystyle g(\sigma,x)=\sum_{k=0}^{\infty}h(\sigma,x+k\Lambda)\,,$

where $g(\sigma,x)$ is the instantaneous flux of a process started in $x>0$
with zero delay reset. This explicit fundamental solution allows one to define
the announced sequence $\\{S_{k},\pi_{k},q_{0,k}\\}_{k\in\mathbbm{N}}$ in
$\mathbbm{R}^{+}\times(0,1]\times\mathcal{M}((0,\infty))$ as follows:

######  Definition 4.1.

With the convention that $S_{0}=0$ and for a natural initial condition $q_{0}$
with Dirac-delta mass $\pi_{0}\delta_{\Lambda}$ satisfying Assumption 4, let
us define the sequence $\\{S_{k},\pi_{k},q_{0,k}\\}_{k\in\mathbbm{N}}$ by
setting $q_{0,0}=q_{0}$ and iterating for all $k\geq 1$:

  1. 1.

Smooth dynamics:

$\displaystyle
g_{k}(\sigma)=\int_{0}^{\infty}g(\sigma,x)q_{0,k}(x)\,\mathrm{d}x\,.$
$\displaystyle
q_{k}(\sigma,y)=\int_{0}^{\sigma}\kappa(\sigma-\tau,y,\Lambda)g_{k}(\tau)\,\mathrm{d}\tau+\int_{0}^{\infty}\kappa(\sigma,y,x)q_{0,k}(x)\,\mathrm{d}x\,.$

  2. 2.

Blowup trigger time:

$\displaystyle
S_{k+1}-S_{k}=\inf\left\\{\sigma>0\,\Big{|}\,g_{k}(\sigma)>1/\lambda\right\\}<\infty\,.$

  3. 3.

Blowup size:

$\displaystyle\pi_{k+1}=\inf\left\\{p\geq
0\,\bigg{|}\,p>\int_{0^{+}}^{\infty}H(\lambda
p,x)q_{k}(S_{k+1},x)\,\mathrm{d}x\right\\}>0\,.$

  4. 4.

Blowup exit/reset distribution:

$\displaystyle
q_{0,k+1}(y)=\int_{0}^{\infty}\kappa(\lambda\pi_{k+1},y,x)q_{k}(S_{k+1},x)\,\mathrm{d}x+\pi_{k+1}\delta_{\Lambda}(x)\,.$

The distributions $\\{q_{0,k}\\}_{k\in\mathbbm{N}}$ defined above are in fact
the reset distributions for PMF dynamics, $q_{0,k}=q(R_{k},\cdot)$. Moreover,
with $\epsilon=0$, post-blowup reset is instantaneous so that $U_{k}=R_{k}$
and exit distributions are recovered by excluding the reset mass $\pi_{k}$:
$q(R_{k}^{-},\cdot)=q_{0,k}-\pi_{k}\delta_{\Lambda}$, . The sequence
$\\{S_{k},\pi_{k},q_{0,k}\\}_{k\in\mathbbm{N}}$ allows one to give an explicit
piecewise representation of the global solution $\Psi$. This representation
makes use of the functions

$\displaystyle\sigma\mapsto\Psi_{k}(\sigma)=\mathbbm{1}_{\\{\sigma>0\\}}\left(\sigma-\lambda\int_{0}^{x}G(\sigma,x)q_{k}(x)\,\mathrm{d}x\right)/\nu\,,$

where $G(\cdot,x)$ denotes the cumulative flux function associated to
$g(\cdot,x)$. Together with the trigger time $S_{k}$, the functions $\Psi_{k}$
allows one to specify the sequence of blowup time
$\\{T_{k}\\}_{k\in\mathbbm{N}}$ for the original dynamics as:

$\displaystyle T_{0}=0\,,\quad T_{k+1}=T_{k}+\Psi_{k}\big{(}S_{k+1}\big{)}\,.$

We then obtain the following piecewise representation:

######  Definition 4.2.

For zero refractory period $\epsilon=0$, the global solution $\Psi$ is given
by

$\displaystyle\Psi(\sigma)=\sum_{k=0}^{\infty}\mathbbm{1}_{\\{S_{k}<\sigma\leq
S_{k+1}\\}}\left[T_{k}+\Psi_{k+1}\big{(}\sigma-(S_{k}+\pi_{k})\big{)}\right]\,.$

Observe that at this stage, although $\Psi$ is defined over the whole half-
line $\mathbbm{R}^{+}$ by Theorem 3.2, we cannot invoke Proposition 2.5 to
show that the corresponding time change $\Phi=\Psi^{-1}$ is defined over the
whole half-line $\mathbbm{R}^{+}$. However, we can establish this point by
using the following estimate for inter-blowup times:

######  Proposition 4.1.

For natural initial condition $q_{0}$ with Dirac-delta mass
$\pi_{0}\delta_{\Lambda}$ satisfying Assumption 4, we have:

$\displaystyle\Psi(S_{k+1})-\Psi(S_{k})>\frac{1}{4\nu}\inf\left\\{\sigma>0\,\bigg{|}\,h(\sigma,\Lambda)>\frac{a_{\Lambda}}{\lambda}\right\\}>0\,,$

where $a_{\Lambda}$ is a positive constant that only depends on $\Lambda$.
Thus, $T_{\infty}=\lim_{\sigma\to\infty}\Psi(\sigma)=\infty$ and
$\Phi=\Psi^{-1}$ is defined over the whole half-line $\mathbbm{R}^{+}$.

###### Proof.

Let us first observe that for $\epsilon=0$, we have $U_{k}=R_{k}$ and

$\displaystyle\nu\big{(}\Psi(S_{k+1})-\Psi(S_{k})\big{)}$ $\displaystyle=$
$\displaystyle\nu\big{(}\Psi(S_{k+1})-\Psi(R_{k})\big{)}\,,$ $\displaystyle=$
$\displaystyle S_{k+1}-R_{k}-\lambda\big{(}G(S_{k+1})-G(R_{k})\big{)}\,.$

Then the proposed lower bound will follow from bounding above
$G(S_{k+1})-G(R_{k})$, the cumulative flux of inactivation between the $k$-th
blowup and the $(k\\!+\\!1)$-th blowup. In this view, we split that cumulative
flux as $G(S_{k+1})-G(R_{k})=\Delta G_{\cancel{\pi_{k}},k}+\Delta
G_{\pi_{k},k}$, where $\Delta G_{\cancel{\pi_{k}},k}$ and $\Delta
G_{\pi_{k},k}$ represent the contributions of the processes that survive or
inactivate during the last blowup, respectively. As Assumption 4 ensures that
$\|g_{\cancel{\pi_{k}}}\|_{R_{k},S_{k+1}}\leq 1/(4\lambda)$, we have

$\displaystyle\Delta
G_{s,k}\leq\|g_{\cancel{\pi_{k}}}\|_{R_{k},S_{k+1}}(S_{k+1}-R_{k})\leq\frac{S_{k+1}-R_{k}}{4\lambda}\,.$

Moreover, Assumption 4 ensures that $g_{\pi_{k}}$ remains a convex function up
to the next blowup so that $g_{\pi_{k}}(R_{k})=0$ and
$g_{\pi_{k}}(S_{k+1})\leq 1/\lambda$ implies that

$\displaystyle\Delta
G_{\pi_{k},k}\leq\frac{\|g_{\pi_{k}}\|_{R_{k},S_{k+1}}(S_{k+1}-R_{k})}{2}\leq\frac{S_{k+1}-R_{k}}{2\lambda}\,.$

Considering the above bounds together leads to

$\displaystyle\nu\big{(}\Psi(S_{k+1})-\Psi(S_{k})\big{)}=S_{k+1}-R_{k}-\lambda\big{(}\Delta
G_{{\cancel{\pi_{k}}},k}+\Delta G_{\pi_{k},k}\big{)}$ $\displaystyle\geq$
$\displaystyle(S_{k+1}-R_{k})/4\,.$

The announced lower bound follows from the lower bound $s_{1}$ defined in (40)
which satisfies

$\displaystyle S_{k+1}-R_{k}\geq
s_{1}\geq\inf\left\\{\sigma>0\,\bigg{|}\,h(\sigma,\Lambda)>\frac{a_{\Lambda}}{\lambda}\right\\}=\frac{\Lambda^{2}}{2\ln\left(\lambda\right)}+o(1/\ln\lambda)\,.$

∎

Proposition 4.1 directly implies that
$T_{\infty}=\lim_{S\to\infty}\Phi(S)=\infty$, so that $\Phi$ parametrizing a
PMF dynamics over the whole half-line $\mathbbm{R}^{+}$. We conclude by giving
the explicit, iterative construction of explosive PMF dynamics with zero
refractory period $\epsilon=0$.

######  Theorem 4.3.

Under Assumption 4, PMF dynamics admits the density function
$p(t,y)=\mathrm{d}\mathbb{P}\left[X_{t}\leq x|X_{t}>0\right]/\mathrm{d}x$

$\displaystyle p(t,x)=\sum_{k=0}^{\infty}\mathbbm{1}_{\\{T_{k}\leq
t<T_{k+1}\\}}q_{k}\big{(}\Phi(t)-T_{k},x\big{)}\,,$

where the sequence $(T_{k},q_{k})$ is specified in Definition 4.1 and the time
change $\Phi=\Psi^{-1}$ is specified in Definition 4.2.

###  4.2 Global continuity in the limit of vanishing refractory periods

Considering Theorem 3.2 for zero refractory period $\epsilon=0$ allows for the
explicit construction of explosive PMF dynamics in the large interaction
regime. To justify that the thus-constructed dynamics are “physical”, we need
to check that these non-delayed solutions can be recovered from dPMF dynamics
in the limit of vanishing refractory period $\epsilon\to 0^{+}$. In this
perspective, let us consider some natural initial conditions for explosive PMF
dynamics that satisfy Assumption 4. Such initial conditions are entirely
specified by the spatial component $q_{0}$, which includes an Dirac-delta mass
of size $\pi_{0}$ at $\Lambda$. By Theorem 3.2, the corresponding explosive
PMF dynamics is fully parametrized by the inverse time-changed function $\Psi$
specified in Definition 4.1. Given the same natural initial condition, Theorem
3.2 also guarantees the existence and uniqueness of explosive dPMF dynamics
for small enough refractory period $\epsilon$ under Assumption 4. Let us
denote the corresponding inverse time change by $\Psi_{\epsilon}$. By Theorem
3.2, $\Psi_{\epsilon}$ specifies a countable infinity of jumps with trigger
times $\\{S_{\epsilon,k}\\}_{k\in\mathbbm{N}}$, exit times
$\\{U_{\epsilon,k}\\}_{k\in\mathbbm{N}}$, and reset times
$\\{R_{\epsilon,k}\\}_{k\in\mathbbm{N}}$ such that
$0=R_{\epsilon,0}<S_{\epsilon,1}<U_{\epsilon,1}<R_{\epsilon,1}<S_{\epsilon,2}<U_{\epsilon,2}<R_{\epsilon,2}<\dots$.
Our goal is to justify the following continuity result:

######  Theorem 4.4.

Given the same purely spatial natural initial condition satisfying Assumption
4, the dPMF solution $\Psi_{\epsilon}$ converges compactly toward the PMF
solution $\Psi$ on $\mathbbm{R}^{+}$ and the dPMF blowup times
$S_{\epsilon,k}$, $U_{\epsilon,k}$, $R_{\epsilon,k}$ converge toward their PMF
counterparts $S_{k}$ and $U_{k}=R_{k}$ as well.

The proof of the above Proposition relies on a simple recurrence argument that
makes use of four continuity results, which we first give without proof. The
first set of results, comprising Proposition 4.2 and Proposition 4.3, deal
with the continuity of dPMF dynamics in between blowup episodes in the limit
of vanishing refractory period $\epsilon\to 0^{+}$. These results essentially
follow from the uniform boundedness of the various functions involved in the
integral representation of the density of the dPMF dynamics. Proposition 4.2
bears on the continuity of the cumulative fluxes, which boils down to the
continuity of blowup trigger times with respect to the initial conditions for
the $L_{1}$ norm.

######  Proposition 4.2.

Consider some natural initial conditions $(q_{\epsilon,0},g_{\epsilon,0})$ in
$\mathcal{M}((0,\infty))\times\mathcal{M}([-\xi_{\epsilon,0},0))$ such that
$q_{\epsilon,0}\to q_{0}$ in $L_{1}$ norm when $\epsilon\to 0^{+}$, where
$q_{0}$ is a probability density with Dirac-delta mass
$\pi_{0}\delta_{\Lambda}$. Then under Assumption 4, we have:

$(1)$ $\lim_{\epsilon\to 0^{+}}S_{\epsilon,1}=S_{1}$ where $S_{\epsilon,1}$
and $S_{1}$ denotes the next blowup trigger times of the time changes
$\Psi_{\epsilon}$ and $\Psi$, respectively.

$(2)$ for all $k$ in $\mathbbm{N}$, the $k$-iterated derivatives of
$G_{\epsilon}$ and $G$ satisfy $G^{(k)}_{\epsilon}\to G^{(k)}$ uniformly on
$[0,S_{1}+\lambda/3]$.

Proposition 4.2 establishes that PMF dynamics are locally recovered from dPMF
dynamics in the limit of vanishing refractory period, at least up to the first
blowup episode. Then, a natural strategy to prove Proposition 4.4 is to apply
Proposition 4.2 iteratively after each blowup episodes in conjunction with
some continuity result about the blowup episodes. It turns out that these