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Dilating blow-up time: A generalized solution of the NNLIF neuron model and its global well-posedness

The nonlinear noisy leaky integrate-and-fire (NNLIF) model is a popular mean-field description of a large number of interacting neurons, which has attracted mathematicians to study from various aspects. A core property of this model is the finite time blow-up of the firing rate, which scientifically corresponds to the synchronization of a neuron network, and mathematically prevents the existence of a global classical solution. In this work, we propose a new generalized solution based on reformulating the PDE model with a specific change of variable in time. A firing rate dependent timescale is introduced, in which the transformed equation can be shown to be globally well-posed for any connectivity parameter even in the event of the blow-up. The generalized solution is then defined via the backward change of timescale, and it may have a jump when the firing rate blows up. We establish properties of the generalized solution including the characterization of blow-ups and the global well-posedness in the original timescale. The generalized solution provides a new perspective to understand the dynamics when the firing rate blows up as well as the continuation of the solution after a blow-up.


Introduction
To describe a large number of interacting neurons, mean-field approaches have been successfully used in computational neuroscience [4,9,33,37], which lead to various novel models including nonlinear partial differential equations with unique structures. Recently such equations have attracted many mathematicians to study both from the PDE point of view [7,19,31,34] and from the corresponding SDE point of view [14,16,17,29]. One of the common features shared by these models is the synchronization, which means a positive fraction of neurons spike at the same time. Mathematically this leads to the finite-time blow-up of the firing rate which obstructs the existence of a global classical solution. Our goal of this work, is to propose a new generalized solution which admits blow-ups, and can be shown to be globally well-posed.

Model description
We consider the nonlinear noisy leaky integrate-and-fire (NNLIF for short) model proposed in [1,3,4], which has become a standard model in computational neuroscience (see e.g. [5,6,24]). Here each neuron is characterized by its voltage (membrane potential) v, which is no greater than a threshold VF . And the ensemble of neurons is described by p(v, t), the probability density at each time t to find a neuron with voltage v. The density function p(v, t) is governed by the following PDE ∂tp + ∂v (−v + VL + µ(t))p = a(t)∂vvp + N (t)δv=V R , t > 0, v ∈ (−∞, VF ). (1.1) The term −v + VL in the drift models the leaky mechanism which drives the voltage to the equilibrium potential VL < VF . Each neuron is also subject to a fluctuating input current, of which the mean is µ(t) and the noise level is described by a(t). Such an input can drive the voltage of a neuron away from VL and may exceed VF at a certain time.
The unique feature of (1.1) comes from the firing mechanism. A neuron spikes when its voltage arrives at VF , the firing potential. In this macroscopic model, the spike activity is described by the mean firing rate N (t), whose physical meaning is the number of spikes in expectation per unit time.
A spike has two consequences. First, after the spike, the voltage of the neuron is immediately reset to VR, the reset potential. Such a reset gives the rise to the Dirac source term N (t)δv=V R in (1.1), where we use δv=V R to denote the Dirac measure supported at VR. Physically, we have VL ≤ VR < VF . Also the instantaneous reset gives an absorbing boundary condition at VF p(VF , t) = 0, t > 0, (1.2) since no neuron is allowed to stay at VF . Second, when a neuron spikes, it influences other neurons by contributing to their input currents. In this way, the input current consists of two parts: an external part and an intrinsic part from the spikes of other neurons within this network. Precisely, its mean µ(t) and noise level a(t) are given by µ(t) = µ0 + bN (t), a(t) = a0 + a1N (t). (1.3) Here µ0 ∈ R and a0 > 0 represents the mean and the noise level of the external part, which are assumed to be constant in time. The connectivity parameter b ∈ R determines the influence of one spike. When b > 0, the spike of a neuron increases the voltage of other neurons, which is called the excitatory case. When b < 0, the spike of a neuron decreases the voltage of other neurons, which is called the inhibitory case. Last but not least, a1 ≥ 0 is the scaling parameter of the noise level induced by spikes. Finally, the firing rate N (t) is determined in a self-consistent way N (t) := −a(t)∂vp(VF , t) = −(a0 + a1N (t))∂vp(VF , t). In view of (1.4), the dependence of N (t) in (1.3) makes (1.1) nonlinear. The system is completed with the boundary condition at the infinity p(−∞, t) = 0, t > 0, and an initial condition p(v, 0) = pI (v), v ≤ VF . For more detailed derivations of this model, we refer to [4,7,30]. The PDE (1.1) can be viewed as a Fokker-Planck equation of a corresponding Mckean-Vlasov SDE [16,17]. Similar mathematical structures have also been investigated in a wide range of modeling context, including portfolio or bank systems in mathematical finance [25,32] and the supercooled Stefan problem [18].

Literature Review
Mathematical analysis on (1.1) started relatively recently albeit its modeling success. A remarkable phenomenon of (1.1) is the possible finite-time blow-up of the firing rate N (t), first discovered in [7]. Such a blow-up not only becomes a core concept in the mathematical studies of (1.1) but is indeed scientifically relevant.
Mathematically, the blow-up of N (t) is exactly the obstacle for the existence of a global solution in the classical theory, while [13] has established the global well-posedness of a classical solution as long as N (t) remains finite. Many studies have been devoted to the emergence of the blow-up and the existence of a global solution in the absence of the blow-up. In [13] (and via a different method in [12]), the blow-up of N (t) is excluded when b ≤ 0, a1 = 0, and thus in this case the global well-posedness is established. However, as long as b > 0, there exists smooth initial data such that the blow-up occurs in finite time [7,38]. Conditions on the initial data and the connectivity parameter b > 0 to exclude or ensure the blow-up have been studied from a stochastic perspective in [16,25] and via deterministic methods in [38]. Besides, if a time-delayed effect is added to (1.1), then the resulting modified model has been shown to be globally well-posed [8,17].
Scientifically, a blow-up of the firing rate N (t) corresponds to synchronization of a neuron network. Here synchronization refers to the scenario when a portion of the neurons within the network fire simultaneously, a phenomenon ubiquitous in a brain and of fundamental importance in neuroscience. We remark that this definition of synchronization shares many similarities to the so-called multi-firing event, a relatively recently proposed and less understood concept in computational neuroscience [35,36,41].
Quite interesting it is to consider generalized solutions of (1.1), which allows the blow-up of N (t) to be part of the dynamics. Such a generalized solution might yield global well-posedness for a much larger parameter regime than the classical counterpart, and it can describe the dynamics in the presence of synchronization. A pioneering work in this direction is [17] which defines a generalized solution called the "physical solution" for the corresponding SDE of (1.1). In their definition, the blow-up of N (t) gives a Dirac measure in time, which induces a jump of the solution. The jump size, which corresponds to the proportion of neurons synchronized at that time, is characterized by a "physical" condition. They proved the global existence of this "physical solution" but its uniqueness was left open. Such a physical solution has been studied subsequently in [25,28,32]. More recently, a detailed characterization for the regularity of N (t) around the blow-up time is given and the global uniqueness is finally established in [18]. Actually, [18,25,28,32] study different but mathematically closely related models rather than (1.1) itself. For those models blow-up is also important. Up to now, most theoretical contributions to generalized solutions of (1.1) works with the SDE formulation with probability techniques, see also a recent numerical study [27].
Last but not least, all global existence results aforementioned deals with the case a1 = 0. By (1.4) the firing rate is given by −a0∂vp(VF , t) 1 + a1∂vp(VF , t) . (1.5) When a1 = 0, N (t) < +∞ is equivalent to −∂vp(VF , t) < +∞. But when a1 > 0, to have a finite and non-negative N (t) we need to require additionally −∂vp(VF , t) < 1/a1. Thus in terms of the classical solution, the case a1 > 0 is more "ill-posed". Indeed, [12, Appendix A] has shown that when a1 > 0 for any b ∈ R, including the inhibitory case (b ≤ 0), there is no global classical solution if the initial data is concentrated enough near VF .

Results and contributions
In this work, we propose a generalized solution of (1.1) from a new perspective, and throughout this paper we always take a1 > 0. We formally derive from (1.1) an equation in a new timescale linked with the firing rate N (t). This equation can be shown to be globally well-posed for any b ∈ R, even in the event of the blow-up of N (t). The generalized solution of (1.1) is then defined via the backward change of timescale, and in particular it may have a jump when the firing rate blows up. We establish properties of the generalized solution including the characterization of blow-ups (Proposition 2.5) and the global well-posedness in the original timescale (Theorem 2.2). The idea can be described simply as dividing everything by N (t) + 1 in the equation. More precisely, we introduce a new timescale τ called the dilated timescale given by We stress that a1 > 0 is crucial for the well-posedness of our generalized solution. It ensures the equation in timescale τ is uniformly parabolic. Hence, we view a1 > 0 as an essential parameter that yields the well-posedness of the generalized solution. This is in contrast to the fact that a1 > 0 deteriorates the well-posedness in the classical theory [12, Appendix A].
As a new perspective, this generalized solution gives several immediate implications on understanding the NNLIF dynamics as well as opens new questions. First, it provides a way to understand the dynamics when the firing rate blows up as well as the continuation of the solution after a blow-up: first solve the equation in τ and then obtain the solution in t by the inverse change of timescale. Second, the equation in τ may be a suitable platform to study the long time behavior of the NNLIF dynamics (1.1). It is globally well-posed in the classical sense and has at least one steady state for all b ∈ R. Third, the idea of this generalized solution, namely introducing the dilated timescale, might be applied to other integrate-and-fire models, including the model with a refractory state [15], population density models of integrate-and-fire neurons with jumps [19][20][21], and a model derived via a fast conductance limit [10].
There are also other possible ways to define generalized solutions and it is interesting to investigate relations between various generalized solutions. In particular, in some recent works [39,40] the authors have studied a similar model to (1.1), and also used the idea of introducing a new timescale related with the firing rate N (t). However, there are some subtle but essential differences between their generalized solution and ours. We shall give a discussion on issues in defining a generalized solution in Section 6.
The rest of this paper is arranged as follows. Section 2 is a detailed introduction to our generalized solution and main results. Proofs are given in Section 3, 4 and 5. Precisely, in Section 3 we prove the global well-posedness in timescale τ . And the global well-posedness in timescale t is treated in Section 4 and 5 for the excitatory and the inhibitory case, respectively. Finally, additional discussions are given in Section 6.
We recall the NNLIF model (1.1). Substituting the definitions of µ(t) and a(t) (1.3), we have where the constant drift b0 is a sum of the leaky potential and the external input b0 := VL + µ0.
The boundary and initial conditions of (2.1) are given by And the firing rate N (t) is determined by (1.5) We assume the parameters satisfy It is worth emphasizing that we work with the case a1 > 0, which imposes a finite upper bound −p(VF , t) < 1/a1 to get a finite and non-negative firing rate from (2.3). We might interpret the solution of (2.3) as Note that the lower bound −∂vp(VF , t) ≥ 0 is expected since the solution is non-negative with p(VF , t) = 0 (2.2). However, in general there is no guarantee of the upper bound −∂vp(VF , t) < 1/a1. The possible blow-up of N (t) is the main obstacle towards a global solution of (2.1) in the classical sense [13].

Notion of the generalized solution
For heuristic purposes, we explain the notion of the generalized solution in an informal way in Section 2.
The key idea, is to divide both sides of (2.5) by N (t). But to avoid the degeneracy from N (t) = 0, we divide by N + c instead with some c > 0, whose specific value is unessential, as we shall see in Proposition 2.1.
Precisely, we introduce another timescale τ , called the dilated timescale, given by the following change of variable in time with some fixed constant c > 0, e.g., c = 1. Then we divide both sides of (2.5) by N (t) + c to derive, at least formally, an equation with the same boundary and initial conditions as (2.2) In (2.7),Ñ is a shorthand forÑ (τ ), which is defined as And the coefficients bc, ac are just constants given by 1 bc := b0 − cb, ac := a0 − ca1. (2.9) In principle, the dilated timescale τ , and the solution (p,Ñ ) for (2.7) also depend on the parameter c > 0 in (2.6). However, as we shall see, the specific choice of c > 0 won't influence the resulting generalized solution (Proposition 2.1). Therefore for convenience, we do not write out the dependence of c for τ,Ñ andp(v, τ ) explicitly unless necessary. We refer to (2.7)-(2.8) as the equation in the dilated timescale, in contrast to (2.1), the equation in the original timescale. Several observations indicate that (2.7) is more favorable than (2.1) for the well-posedness. From (2.8) we knowÑ (τ ) ∈ [0, 1/c], (2.10) thanks to −∂vp(VF , τ ) ≥ 0 since p ≥ 0 andp(VF , τ ) = 0. Therefore in (2.7) the source of nonlinearityÑ (τ ) is uniformly bounded, in contrast to N (t) in the equation in the original timescale t (2.1). Moreover, (2.7) is uniformly parabolic since the diffusion coefficient is between a1 > 0 and 1 c a0 > 0. In fact, we can establish the global well-posedness for classical solutions of the equation in the dilated timescale τ (2.7), for any b ∈ R. Precise statements are postponed to Section 2.1.2.
In the classical sense, the above derivation from (2.1) to (2.7) is rigorous only if N < +∞, since otherwise when N (t) = +∞ the change of variable (2.6) may not be well-defined. Recall that the blow up of the firing rate N (t) is exactly the obstacle for the existence of a global classical solution of (2.1) [13]. However, in the dilated timescale τ , N = +∞ is allowed since it just corresponds toÑ = 0 by (2.8), which does not hinder the evolution of (2.7). In view of its global well-posedness, (2.7) may evolve through a period when N = +∞ (which corresponds toÑ = 0) and go back to the non-blow up regime when N < +∞ (which corresponds toÑ > 0). Hence, if we change back from τ to the original timescale t, we may be able to define a generalized solution of (2.1) which goes beyond the blow-up time, thus having a longer lifespan. We shall call this "change-back process" the inverse time-dilation transform (see the remark below (2.15) or Definition 2.2 for a precise definition), while the process from (2.5) to obtain (2.7) by dividing by N + c is called a time-dilation transform. In the next, we proceed to examine the inverse time-dilation transform in details to define a generalized solution.
To remove the ambiguity, we consider the following generalized inverse of (2.11) In principle, picking another τ with τ 0Ñ (ω)dω = t shall not make an essential difference, but to fix the choice (2.12) is convenient, which makes the map τ (t) càdlàg. An illustration is given in Figure 1.
We define the generalized solution of (2.1) at time t by assigning the value of the classical solution of (2.7) at τ = τ (t). Precisely, we define the generalized solution at time t to be p(v, t) :=p(v, τ (t)), (2.13) and its associated firing rate is given by (2.14) which is from inverting the definition ofÑ (2.8). Recalling the definition ofÑ (2.8), we see that (2.13) and (2.14) give the same firing rate as (2.4) In (2.13) and (2.14), we shall call (p,Ñ ) the time-dilation transform of (p, N ), and call (p, N ) the inverse timedilation transform of (p,Ñ ).
The jump of p comes from the fact that the whole interval [τ1, τ2] is compressed to one point by (2.12), due toÑ = 0. In timescale τ , the blow up time t * is "dilated" to an interval [τ1, τ2]. This is why we call τ a "dilated timescale". In this "dilated" interval [τ1, τ2], while N ≡ +∞, or equivalentlyÑ ≡ 0, the density functionp evolves continuously. A more detailed discussion on the relationship between p(·, t * ) and p(·, t − * ) is given in Section 2.3.
Lifespan While the solution in the dilated timescale τ is always global, the generalized solution in the original timescale t may be not. By definition, we can define a generalized solution at time t if and only if the map in (2.12) τ (t) < +∞, or equivalently Therefore, the maximal lifespan of a generalized solution is exactly given by And then the global well-posedness for the generalized solution in the original timescale t is equivalent to T * = +∞. Of course, the integral in (2.18) does not depend on the choice of c > 0 either (Proposition 2.1).
Remark 2.1. The reformulation from (2.1) to (2.5) is indeed essential for the construction of the generalized solution.
It turns out that although (2.1) and (2.5) are equivalent when N (t) < +∞, they give different time dilation transforms when N = +∞. We will discuss more on this in Section 6.

Precise definitions and properties
In this section we give the precise definition of the generalized solution of (2.1) and its basic properties. We start with the definition of the classical solution for the equation in the dilated timescale τ (2.7), and the statement of its global well-posedness.
Following definitions of classical solutions for the NNLIF model in time variable t (2.1) [13,38], we give the definition of a classical solution for the NNLIF model in τ variable (2.7)-(2.8).
To state the global well-posedness for the classical solution of (2.7), we need a natural assumption on the initial data as in [13,38]. Assumption 1. For the initial data pI (v) we impose the following assumptions.
The proof of Theorem 2.1, given in Section 3, is an adaption of [13]. As discussed in Section 2.1.1, the boundedness ofÑ and the uniform parabolicity thanks to a1 > 0 play important roles in the proof.
We define generalized solutions for the equation in the original timescale t (2.1), by applying the transform (2.12) to classical solutions in the dilated timescale τ (2.7), as discussed in Section 2.1.1. The precise definition of our generalized solution is given as follows.
Here τ (t) is the change of variable defined in (2.12).
We refer to (p,Ñ ) as the time-dilation transform of (p, N ), and correspondingly, (p, N ) as the inverse timedilation transform of (p,Ñ ). The generalized solution defined in Definition 2.2 may be called a time-dilated solution of (2.1). Remark 2.3. Definition 2.2 implies that the density function p and the firing rate N satisfies the relation (2.15).
By the uniqueness of (2.7) (Theorem 2.1), we shall show that the generalized solution does not depend on the value of c > 0 and its uniqueness. Moreover, we can always chooseτ = +∞ in the definition thanks to the global well-posedness of (2.7) (Theorem 2.1), and get the maximal lifespan of the generalized solution as in (2.18). These are summarized in the following proposition. Proof of Proposition 2.1. Suppose (q1,Ñ1) and (q2,Ñ2) are two solutions for (2.7) with different c > 0, denoted as c1, c2 > 0 respectively, but sharing the same initial data. Let τ1(t) and τ2(t) be the corresponding maps (2.12), then for (i) it suffices to show q1(·, τ1(t)) = q2(·, τ2(t)), ∀t ∈ [0,T ). (2.19) Indeed, this follows from a change of timescale from τ1 to τ2, which is well-defined, and the uniqueness of (2.7) for a fixed c. To give details, starting from q1(τ1, ·), we consider the change of timescale which is well-defined since N+c 1 N+c 2 is between 1 and c 1 c 2 > 0. Define the corresponding map h(τ1) := Then the new density function induced from q1 by the change of timescale (2.20) is given bŷ q1(·, h(τ1)) := q1(·, τ1).
Finally, noting that h is a strictly increasing function, one directly checks τ2(t) = h(τ1(t)), therefore (2.19) holds. (Here with abuse of notation, we use τi to denote the time variables and τi(t) for the maps (2.12).) Hence we have shown that the generalized solution does not depend on the choice of c > 0, its uniqueness also follows from the uniqueness of (2.7) (or taking c1 = c2 in the above proof). By Definition 2.2 and the global well-posedness of (2.7), the maximal lifespan is given by T * = +∞ 0Ñ (τ )dτ as in (2.18 Next, we examine the connection between the generalized solution and the classical solution of (2.1). Here the definition of a classical solution for (2.1) is similar to Definition 2.1 with an additional requirement that the corresponding N (t) < +∞, or equivalently −a1∂vp(VF , t) < 1. For a generalized solution (p, N ) on [0,T ) to be a classical solution, clearly it has to satisfy that N (t) < +∞ for t ∈ [0,T ). On the other hand, suppose a generalized solution indeed satisfy N (t) < +∞ for all t ∈ [0,T ). This implies N (τ ) > 0 for all τ ∈ [0,τ ) in the dilated timescale. Hence, the map t(τ ) (2.11), and thus its inverse "change-backmap" τ (t) (2.12) are both C 1 strictly increasing function. Therefore, by definition the inverse time-dilation transform preserves all regularities required for (p, N ) to be a classical solution.
In general, a generalized solution p always has the same spatial regularity as a classical solution, but in time we only expect it to be càdlàg when N (t) = +∞. We summarize the regularity of a generalized solution as follows.  In particular, if N (0) < +∞, then p(·, t = 0) coincides with the initial data pI (·). Otherwise the initial data pI (v) shall be interpreted as pI (·) = p(·, 0 − ).
Proof of Proposition 2.3. These are all direct consequences of Definition 2.2. Note that the càdlàg property of p follows from the càdlàg property of the map τ (t) defined in (2.12).
For a generalized solution in the original timescale, the density function p can has a jump in time when N (t) = +∞, i.e., the firing rate blows up. We postpone the discussion on this scenario to Section 2.3. We shall discuss the global well-posedness first, along which we introduce the limit equation at the blow-up, which is an important notion for the description of the blow-up as well as the global well-posedness.

Main results
Now we state the global well-posedness in the original timescale t. Depending on the connectivity parameter b, we identify three scenarios: 1. b ≥ VF − VR, the strongly excitatory case.
In short, the global well-posedness for the generalized solution holds if and only if b < VF − VR.
Precisely, we have the following theorem.
Theorem 2.2 (Global well-posedness in the original timescale t). Let the initial data pI(v) satisfy Assumption 1, there exists a unique generalized solution for the equation in the original timescale t (2.1) on the time interval [0, T * ). Here T * ∈ [0, +∞] is the maximal lifespan given by (2.18) and we have 1. When b ≥ VF − VR, the strongly excitatory case, there exists an initial data satisfying Assumption 1 but giving T * = 0, that is, there is no generalized solution on any time interval [0, t) with t > 0.
2. When b < VF − VR we have T * = +∞, that is, there exists a unique global generalized solution, provided the following additional assumptions on the drift and the initial data hold.
(a) When 0 < b < VF − VR, the mildly excitatory case, we additionally assume The proof of Theorem 2.2 is given in Section 4 for the excitatory part and in Section 5 for the inhibitory part.
Remark 2.4. Assumptions on the drift (2.22) and (2.23) are not restrictive. A range of µ0 is allowed since VL ≤ VR < VF . Besides, we conjecture that they are technical rather than essential for the global well-posedness (at least based on some preliminary numerics).
Let's explain more on the non-existence of a global solution when b ≥ VF − VR. By Proposition 2.1, for a generalized solution, its maximal lifespan T * is given by (2.18) In the strongly excitatory case, we can give examples with T * < +∞ which deny the global well-posedness. Such examples are due to a phenomenon which we refer to as the eternal blow-up. It means in the dilated timescale there exists some time τ0 ≥ 0 such thatÑ (τ ) ≡ 0, for all τ ≥ τ0. (2.24) We call it an eternal blow-up sinceÑ = 0 is equivalent to N = +∞ (2.16). Clearly, (2.24) implies that in (2.18) sinceÑ ≤ 1/c by (2.8). Thus then a global solution does not exist when there is an eternal blow-up. A numerical illustration is given in Figure 2. The critical threshold b = VF − VR is the same as that of the "physical solution" in [17] (corresponds to α = 1 in their notations), which is a generalized solution from the SDE perspective in the regime a1 = 0. They also give an intuition that when b ≥ VF − VR, a neuron can spike infinite times in an instant. It is interesting to note that our eternal blow-up (2.24) has a similar meaning, see Remark 2.7.
In the next section we give the proof strategy for Theorem 2.2. There, in particular in Proposition 2.4, we can see how to identify the threshold b = VF − VR.
Remark 2.5 (When the initial data blows up). In Theorem 2.2 we do not require that the initial data satisfies −a1∂vpI (VF ) < 1, therefore the firing rate N (t) is allowed to blow up at t = 0. As described in Proposition 2.3, when N (0) = +∞ the initial data pI shall be interpreted as pI (·) = p(·, 0 − ). Let's understand Theorem 2.1 in this special case.
In the dilated timescale, an initial blow-up of the firing rate meansÑ = 0 for τ = 0. Even so, we have the global well-posedness for generalized solutions when b < VF − VR. In view of the formula for T * (2.18), this implies that in the dilated timescale,Ñ (τ ) ≡ 0 (or equivalently N ≡ +∞). In other words, the solution would get out of the blow-up regime eventually when b < VF − VR.
But when b ≥ VF − VR there exist examples such thatÑ (τ ) ≡ 0 (or equivalently N ≡ +∞) for all τ ≥ 0, which is the eternal blow-up defined in (2.24) with τ0 = 0. It implies that the lifespan in the original time scale T * = 0 and hence denies the global well-posedness of the generalized solution in the strongly excitatory case. An investigation into the eternal blow-up is also a motivation towards the proof strategy for Theorem 2.2.
0Ñ (τ )dτ by (2.18). Right: The corresponding N (t) in the original timescale. The firing rate blows up as the time t approaches the maximal lifespan T * . In this plot, the firing rate is truncated at N = 100. Parameters: The initial data is a localized Gaussian with variance 0.17 around v = −1. The numerical scheme is a variant version from the one in [26].

Proof strategy: The limit equation at the blow-up
This section is devoted to the proof strategy of Theorem 2.2, concerning whether a global generalized solution exists in the original timescale t. Recall the definition ofÑ (2.8)Ñ Condition (2.25) implies that the firing rate firing rate N , or the boundary flux −a1∂vp(VF , τ ) can not be always too large. Hence, the global well-posedness in the original timescale t is a question on the long time behavior of (2.7) in the dilated timescale τ . First, we consider the extreme case whenÑ (τ ) ≡ 0 for all τ ≥ 0, which simply gives T * = 0 < +∞ in (2.18). This is the eternal blow-up defined in (2.24) with τ0 = 0. In this case, the dynamics of (2.7) simplifies to To find a solution of (2.26) satisfying (2.27), it is natural to look into the steady state. When b > 0, i.e., the excitatory case, we have a unique probability density steady state with an explicit formula for its boundary flux as follows.
Proposition 2.4 (Part of Proposition 4.1). When b > 0, for the limit equation (2.26), there exists a unique probability density steady statep∞(v). Moreover, its boundary flux is given by The critical threshold b = VF − VR emerges in the formula (2.28), which implies that −a1∂vp∞(VF ) ≥ 1 if and only if b ≥ VF − VR. Therefore, when b ≥ VF − VR, the steady state of (2.26), which givesÑ = 0, is also a steady state of (2.7). If we set this steady statep∞(v) as the initial data, we shall haveÑ (τ ) ≡ 0, which implies T * = 0 in (2.18). Hence, we prove the first part of Theorem 2.2 -an example of non-existence in the strongly excitatory case. Now we describe the proof strategy for the global well-posedness when b < VF − VR. The key intuition also comes from the limit equation (2.26). We first consider the mildly excitatory case, i.e., the case when 0 < b < VF − VR.
Again we start with examining an extreme case -the eternal blow-up. Precisely, we ask whether it is possible that N = 0 holds for all τ ≥ 0 for a solution of (2.7) when 0 < b < VF − VR. That is equivalent to ask whether there exists a solution of the limit equation (2.26) with −a1∂vp(VF , τ ) ≥ 1 for all τ ≥ 0, a question concerning only the linear equation (2.26). Intuitively the answer is no, because entropy methods [7] tell us that a solution of (2.26) converges in the long time to the steady state. But the boundary flux of the steady state −a1∂vp∞(VF ) is strictly less than 1 when 0 < b < VF − VR by Proposition 2.4.
The long time behavior of (2.26) provides the cornerstone for analysis of general cases. We argue by contradiction, if on the contrary +∞ 0Ñ (τ )dτ < +∞, (2.29) then +∞ LÑ (τ )dτ → 0 as L → +∞, which implies a smallness ofÑ (τ ). Hence, we can view the nonlinear equation (2.7) as a perturbation of the linear limit equation (2.26), where theÑ (τ ) terms are regarded as non-autonomous perturbations. By (2.29) the total influence of such perturbations shall be finite and therefore eventually the dynamics would be dominated by the linear equation (2.26) in the long time. If so, however, by the entropy estimate for the linear equation, the boundary flux of the solution shall be near to that of the steady state, which would contradict (2.29).
To turn these intuitions into a rigorous proof, we adopt various techniques developed in literature for the NNLIF model (2.1), including the entropy estimate [7], the control of the boundary flux [12,15], and the L ∞ bounds by constructing super solutions [12]. The proof for the excitatory case is given in Section 4.
For the inhibitory case, i.e., b ≤ 0, the general strategy is the same. Certain adjustments are needed, which are mainly due to a different behavior of the limit equation (2.26). The proof for the inhibitory case is given in Section 5.

Characterization of the blow-up scenario
This section is devoted to characterizing the scenario when the firing rate blows up. We start by revisiting the argument for (2.17) in a rigorous way as now we have the precise definition of the generalized solution.
When τ1 < τ2, the blow-up time t * is dilated to an interval [τ1, τ2]. Thus in the original timescale p is discontinuous in time at t * since by Definition 2.2 . We can further characterize the relationship between p(·, t + * ) and p(·, t − * ). Thanks to (2.30), on "the blow-up interval" [τ1, τ2] the dynamics (2.7) is reduced to the linear limit equation (2.26). This observation leads to the following proposition.
In other words, suppose we solve the linear limit equation (2.26) with the initial data p(·, t − * ) and denote the solution as w(v, τ ), i.e., Then we have Here ∆τ ∈ [0, +∞) is given by Proof of Proposition 2.5. Following the discussion above the statement of this proposition, it remains to show that τ2 − τ1 coincides with ∆τ defined in (2.32), or equivalently (2.33). Note that by (2.8), the boundary flux condition in (2.33) is equivalent to that the correspondingÑ = 0. Hence, by (2.30) we have τ2 − τ1 ≤ ∆τ . Suppose on the contrary ∆τ > τ2 − τ1. We can construct another solution of (2.7) on [τ1, τ1 + ∆τ ] withÑ ≡ 0, by copying w from the time interval [0, ∆τ ]. Note that the definition of τ2 ensures thatÑ ≡ 0 on any (right) neighborhood of its. Hence, we have two different solutions of (2.7) with a same initial data at τ = τ1. This contradicts the uniqueness of (2.7), which is ensured by Theorem 2.1.
Remark 2.7. Formally, the eternal blow-up (2.24) when b ≥ VF − VR can be viewed as ∆τ = +∞ (or even ∆τ ≥ +∞). In this case a solution can never return to the non-blowup regime (N < +∞) and can not continue to evolve in the original timescale. The neurons keep actively spiking with N = +∞ in the dilated timescale, but this happens within an instant in the original timescale.
To summarize, when the firing rate blows up, the density function evolves according to the limit equation (2.26) for a time ∆τ (in the dilated timescale), which is by (2.32) the minimal time for the boundary flux to become below the critical value, or equivalently to get out of the blow-up regime. All this happens in an infinitesimal time in the original timescale but typically ∆τ > 0, which leads to a jump of the density function. A numerical illustration is given in Figure 3.
The following two remarks discuss the dynamics of the limit equation (2.26).
Remark 2.8. We identify two groups of terms in equation (2.5). The first group is which does not depend on N (t). The second group is, on the other hand, the terms containing a factor N (t) From a modeling point of view, (2.34) describes the dynamics of a single and independent neuron, while (2.35) is the contribution of the interaction within a neuronal network, which is proportion to N (t). If the firing rate N (t) is very small, then (2.35) is small and (2.34) dominates. While in the (nearly) blow-up regime when N (t) is very large, (2.35) is also very large, which motivates us to study the dynamics in the dilated timescale τ .
In the limit equation (2.26), (2.34) disappears since it is divided by N = +∞. In particular, the parameters b0, a0 for the single-neuron dynamics -the leaky mechanism and the external input, does not influence (2.26). What matters then is the network connectivity parameter b ∈ R and the scaling parameter of the intrinsic noise a1 > 0. Remark 2.9. We might view (2.26) as a drift-diffusion equation with a unique fire-reset mechanism. The direction of the drift is determined by b ∈ R: rightwards for b > 0 (the excitatory case) and leftwards for b < 0 (the inhibitory case). And we see again a1 > 0 is crucial to make (2.26) a well-posed parabolic problem.
3 Global well-posedness in the dilated timescale τ In this section, we prove Theorem 2.1, the global well-posedness for classical solutions of the equation in the dilated timescale τ (2.7). For notation simplicity, we drop the tilde inp since in this section we work exclusively in the dilated timescale. First we recall (2.7) with p(VF , τ ) = 0, τ ≥ 0 whereÑ is given by (2.8) Here c > 0 is the parameter to avoid the degeneracy from N = 0, and bc, ac are constants defined in (2.9). Classical solutions for the equation in the original timescale t (2.1) have been analysed in [13], where they have established the local well-posedness for any b ∈ R and the global well-posedness for b ≤ 0, for (2.1) with a1 = 0. Our treatment for (2.7) is an adaption of [13], since (2.7), which is formally derived from (2.1), shares similar structures with it. However, there are two differences. We shall prove the global well-posedness for classical solutions of (2.7), regardless of the connectivity parameter b ∈ R, i.e., for both the excitatory and the inhibitory case. Moreover, we work in the scenario a1 > 0 rather than a1 = 0.
An important feature of (2.7), the equation in the dilated timescale τ , is that the source of nonlinearityÑ ∈ [0, 1/c] is always bounded as in (2.10). This is in contrast to N (t) in (2.1) for the original timescale t, which is allowed to take any value among [0, +∞) in principle. Our proof shall show that the boundedness ofÑ indeed plays a key role for the existence of a global solution.
Besides, when a1 > 0, together with a0 > 0, the equation possesses uniform parabolicity in the dilated timescale τ , in the sense that the diffusion coefficient in (2.7) satisfies thanks to the definition of ac (2.9) and thatÑ ∈ [0, 1/c] by (2.8). Hence, by providing enough diffusion, a1 > 0 gives crucial benefits in the dilated timescale τ , while it is an obstacle for the well-posedness of classical solutions in the original timescale t [12, Appendix A].
Although the global well-posedness of (2.7) is the foundation for our generalized solution, its proof is independent of the reasoning in Section 4 and 5, and thus may be skipped at a first reading.
The proof strategy of Theorem 2.1 is to first transform (2.7) to a Stefan-like free boundary problem, which is given in Section 3.1. Then we can reduce the PDE problem to an integral equation for M (s), the boundary flux, which is an analogy of the firing rate in the free boundary formulation. In this way, with further analysis, the global well-posedness for the equivalent free boundary problem is established in Section 3.2. This proof strategy closely follows [13].

Transform to a Stefan free boundary problem
By a translation, without loss of generality we set VF = 0 and therefore VR < 0. We introduce the following notations for the drift and diffusion coefficients, recalling definitions of bc, ac (2.9) Then we rewrite (2.7) as In this section, we transform (3.3) into an equivalent free boundary problem, which resembles the classical Stefan problem. Then in the next section, we do a further reduction to an integral equation of the boundary flux and prove the global well-posedness.

Changes of variables
First, we utilize a transform which is sightly modified from a classical change of variable [11]. In our case, the reformulation involvesÑ (τ ), due to the dependence of the coefficients in (3.3) on it.
Introduce a new space variable y = β(τ )v and a new time variable s = S(τ ), where Denote the solution in the new variables as q(y, s), given by Then one derives from (3.3) via lengthy but direct calculations, whose details are presented in Appendix A.1.1,

The free boundary problem
In view of the characteristics of (3.12), we consider a further change of variable u(x, s) = q(y, s), where Then we get the following free boundary problem, which is equivalent to (3.3).   Hence, to prove the global well-posedness of (2.7), it is equivalent to study (3.13). We shall establish the global well-posedness for classical solutions of (3.13). The definition of a classical solution of (3.13) and the assumption on its initial data can be directly deduced from Definition 2.1 and Assumption 1 via the change of variable discussed above. Thus they are omitted here.
The nonlinearity of (3.13) comes from the fact that ℓ(s) and ℓR(s) depend on the solution through its boundary flux M (s). Compared to the free boundary problem in [13], the only but important difference is how ℓ(s) and ℓR(s) move. Here we have a uniform a priori control on the moving speeds of the free boundaries ℓ(s), ℓR(s), which is crucial for getting a global solution. In this way we can show that β(s; M ) and D(s; M ) are well-defined, formulated as the following lemma.

Well-posedness of the equivalent free boundary problem
In this section we establish the global well-posedness of the free boundary problem (3.13), which is equivalent to (2.7), therefore proving Theorem 2.1. This step is also a modification from the approach in [13] (see also classical treatments on the Stefan problem [22,23]). We first establish the local well-posedness of (3.13), with an existence time controlled by an upper bound of the derivative of the solution. Then we shall show that the derivative of a solution can not blow up in finite time, which leads to the global well-posedness. Here the uniform bounds on the boundary moving speeds ℓ ′ (s), ℓ ′ R (t), given in Lemma 3.2, play a key role for the proof.

Local well-posedness
This section is devoted to proving the following local well-posedness result for (3.13). ∂u(x, s0) ∂x .
The proof strategy is to reduce (3.13) to an integral equation for the boundary flux M (s) on which a fixed point argument can be applied.
We start with deriving integral equations from (3.13) via the heat kernel The derivations are identical to those in [13, Section 3.1], which do not depend on the specific forms of ℓ(s) and ℓR(s). In this way, the free boundary problem (3.13) is reduced to an integral equation for M (s) (3.26). This integral formulation provides a platform to perform a fixed point argument for local-wellposedness.

Global well-posedness
In view of the local result Theorem 3.1, towards a global well-posedness we need to control sup x≤ℓ(s) |ux(x, s)|.

U0 < ∞ is ensured by the definition of a classical solution.
Such a uniform estimate on ux is obtained through the integral formula (3.25). The proof is omitted here since it is identical to the proof of [13, Proposition 4.1], thanks to the Lipschitz continuity for ℓ, ℓR in Lemma 3.2.
Then, one can extend the lifespan of a solution as long as M (s) is bounded. Indeed, if (3.28) holds by Proposition 3.1 we get bounds on sup x≤ℓ(s) |ux(x, s)|. Then applying Theorem 3.1 for s ∈ [0, S), we can extend the solution to [s, s + σ] with a uniform σ > 0. For a detailed proof, one can see [13,Theorem 4.2] Then, the following proposition rules out the possibility that M (s) blows up in a finite time, which allows the solution to be extended step by step towards infinity. |ux(x, s0 − ǫ)| < ∞.
Here M0 < +∞ is implied by the definition of a classical solution. And we assume ǫ < s0, otherwise we can choose a smaller ǫ which does not matter for the analysis of global well-posedness.
We postpone the proof of Proposition 3.3 to Appendix A. In this case, the key to the proof is the uniform Lipschitz continuity of ℓ(s) (Lemma 3. we see a longer lifespan is closely related with upper controls on the boundary flux −a1∂vp(VF , τ ). Therefore, the global well-posedness for generalized solutions of (2.1), which is equivalent to T * = +∞, is a problem concerning the long time behavior of (2.7) in the dilated timescale τ . In this section we prove Theorem 2.2 for the excitatory case, i.e., the case b > 0. We shall show T * = +∞ for 0 < b < VF − VR and give examples with T * = 0 when b ≥ VF − VR. The inhibitory case when b ≤ 0 is treated in the next section. The general strategy, as sketched in Section 2.2.2, is the same.
First in Section 4.1 we study the limit equation (2.26). In the strongly excitatory case b ≥ VF − VR, the steady state of (2.26) provides an example of the eternal blow-up defined in (2.24) therefore denying the global well-posedness in that case. Then in Section 4.2, a contradiction argument allows us to treat (2.7) as a perturbation of (2.26). And we prove the global well-posedness of the generalized solution of (2.1) in the mildly excitatory case, i.e., the case 0 < b < VF −VR.
For notation convenience, we shall drop the tilde inp in this section and the next section, as we will work in timescale τ only to prove T * = +∞.
We start with the steady state of (2.26), which gives the counter-example against the global well-posedness in the case b ≥ VF − VR. Then the long time behavior of (2.26) is analyzed via entropy estimates, which serves as the foundation for the global well-posedness proof for the case 0 < b < VF − VR.

Steady state
The steady state of (2.26) satisfies Solving (4.1) for b > 0, we obtain the following proposition.
Proposition 4.1. When b > 0, the steady state of (2.26) is unique up a multiplying constant. Moreover, we can pick a constant such that the steady state is a probability density on (−∞, VF ), given by V F − 1)c2. Therefore the steady state is unique up to a multiplying constant. We can choose an appropriate Z such that is a probability density. Such a normalization constant Z is given by Then, it is direct to compute the diffusion flux at v = VF In the strongly excitatory case, i.e., b ≥ VF −VR, by Proposition 4.1 the steady state of (2.26) satisfies −a1∂vp∞(VF ) ≥ 1. Therefore, p∞ is also a steady state of the nonlinear equation in the dilated variable (2.7), withÑ = 0. Taking this steady state as the initial data, and then we clearly get a solution to the time-evolution problem withÑ (τ ) ≡ 0 for all τ ≥ 0. Hence, the eternal blow-up (2.24) happens and the corresponding lifespan of the generalized solution in the original time scale is zero as given in (2.18).
Therefore, the lifespan of the corresponding generalized solution of (2.1) T * is zero, as given in (2.18).
Using the steady state of (2.26) as a counter-example, Corollary 4.1 denies the global well-posedness of the generalized solution when b ≥ VF − VR. The first part of Theorem 2.2 is then proved.

Relative entropy estimate
The steady state (4.2) allows us to study the long time behavior of (2.26), which is a linear equation, using the relative entropy method as in [7,Section 4] for the NNLIF equation. We aim to derive a control on the boundary flux, by modifying the treatments in [15, Appendix A] and [12,Theorem 2.1]. Estimates in this section serve as a basis for the global well-posedness proof in Section 4.2.
Let p(v, τ ) be a solution of the limit equation (2.26). We denote its boundary flux as Let p∞(v) be the steady state (4.2) and M∞ be its boundary flux (4.3). We use the following notations for the ratios as in [12] h For a C 2 convex function G : R → R, we denote the relative entropy Calculating d dτ SG[p](τ ), we have the entropy dissipation as summarized in the following. where the entropy dissipation DG[p](τ ) is given by Proof of Proposition 4.2. We start from checking the boundary conditions of h. At v = VF since 0 = p(VF , τ ) = p∞(VF , τ ), by L.Hôpital's rule we get At v = VR we rewrite the flux jump condition for p as Then we multiply (4.9) by p∞, and integrate by parts. Noting that ∂vv is understood in the distributional sense, we deduce (4.10) Finally, by the boundary conditions for h and p∞, we obtain Plugging this to (4.10), we get (4.7). The two terms in (4.7) are non-negative because G is convex. (4.11) The first term in this dissipation can be estimated via the following Poincáre inequality.
where α > 0 is a constant independent of h.
The proof of this Poincáre inequality is similar to that of [7,Proposition 4.3] and is postponed to Appendix B. Together with the entropy dissipation (4.11), Proposition 4.3 implies the exponential convergence to the steady state, in the norm induced by the entropy SG[p](τ ) with G = (x − 1) 2 .
Corollary 4.2. When b > 0, for a solution p(v, τ ) of (2.26) with the initial data satisfying Then p(·, τ ) converges to the steady state p∞ exponentially in the following sense (4.5). Thanks to the conservation of mass of (2.26) and the first condition in (4.12), we have This convergence result suggests that in the long time the boundary flux M (τ ) = −a1∂vp(VF , τ ) as defined in (4.4) shall also be close to that of the steady state M∞. However, the convergence in L 2 ((−∞, VF ), dv p∞(v) ) given by Corollary 4.2 is not strong enough to imply that M (τ ) also converges to M∞. Instead, using the treatments in [12,15], we control ν(τ ) = M (τ )/M∞ using entropy dissipation DG[p](τ ) as follows.

By Proposition 4.3 we have
Next, we apply the Sobolev injection H 1 (I) ⊂ L ∞ (I) to h(v) − 1 on a small neighbourhood I of VR = 0. The interval I is chosen such that p∞(v) as in (4.2) is bounded from below on I. Then noting ∂v(h − 1) = ∂vh, from (4.13) we obtain with some C > 0. Thanks to the expression of DG[p](t) in (4.11), choosing ǫ such that 2ǫM∞C < a1, and ǫ < 1/2, we get the desired results.
Then we can show M (τ ) is close to M∞ in the long term as follows.

Global well-posedness in t in the mildly excitatory case
With the results on the linear limit equation (2.26) as preparations, this section is devoted to the global well-posedness of the generalized solution in the original timescale t (2.1) when 0 < b < VF − VR. In other words, we aim to prove the mildly excitatory case of Theorem 2.2.
As shown in Proposition 2.1, the lifespan of a generalized solution is independent of the parameter c > 0 in the change of variable dτ = (N + c)dt. Hence, without loss of generality we take c = a 0 a 1 to set ac = 0 in view of (2.9), therefore simplifying the diffusion coefficient. With this choice c = a 0 a 1 , the equation in τ (2.7) becomes with the boundary condition p(VF , τ ) = 0, τ > 0. Here in place of bc in (2.9), b * is given by Then the drift condition (2.22) is equivalent to b * ≤ VF .
Hence, under the assumption (4.17), the long time behavior of (4.14) is expected to be similar to that of (2.26). In particular in the long term the boundary flux M (τ ) should also be close to M∞ (4.3), which is less than one when 0 < b < VF − VR. However, this would imply some positive lower bound ofÑ thanks to (4.16), which contradicts the assumption (4.17). Our proof is to fulfill this intuitive plan in a rigorous way.
First we note that it is sufficient to get an estimate similar to Corollary 4.3. By this lemma, we thus aim to prove (4.18) by carrying out the relative entropy estimate to (4.14). We need to control the extra terms involvingÑ (τ ). To this aim, we first give the L ∞ bounds of h = p/p∞, using the method of super-solutions developed in [12]. Then, we do the entropy estimate. Under the assumption (4.17) we prove (4.18), which in turn contradicts (4.17) and therefore the proof is completed.

L ∞ bound
We establish the L ∞ bounds in this section as preparations for the entropy estimate.
First, we construct super-solutions of (4.14) similar to [12, Defintion 4.2 and Lemma 4.4]. The super-solution consists of two ingredients: the steady state (4.2) of the limit equation (2.26) and a time factor, where the former is to tackle the terms withoutÑ and the latter is to deal with the terms involvingÑ .
Proposition 4.4. When b > 0 and b * ≤ VF , there exists a γ > 0 such that for any given solution p andÑ of (4.14), is a super solution of (4.14) in the following sense: q(VF , τ ) = 0 and on (−∞, VF ) × (0, +∞) in the sense of distributions. Here p∞ is the steady state of the limit equation (2.26) given in (4.2).
To check (4.19) in the sense of distribution, as in Remark 2.2 it suffices to check that q satisfies (4.19) in the classical sense on ((−∞, VR) ∩ (VR, VF )) × (0, +∞) and a flux jump inequality at VR Since p∞(v) is the steady state of (2.26), the flux jump condition is satisfied (it becomes an equality) and the terms withoutÑ in (4.19) cancel. Therefore, we only need to check the extra terms related withÑ V R , which is zero at VF . Since in this case ∂vp∞(v) ≤ 0, the last term in (4.20) is positive if v ≥ max(VR, b * ). Therefore it suffices to choose Noting that the sup term above is finite when b * ≤ VF , we conclude the proof.
We remark that γ > 0 in Proposition 4.4 does not depend on a particular solution, while the super-solution q(v, τ ) depends on a specificÑ .
By comparing a solution with a super-solution, we can derive L ∞ bounds as in [12,Lemma 4.3].
Proposition 4.5. When b > 0 and b * ≤ VF , suppose the initial data pI (v) of (4.14) satisfy for some CI > 0. Then the following L ∞ bound holds Here γ is the constant for the super-solution in Proposition 4.4.
Corollary 4.4. When b > 0 and b * ≤ VF , suppose the initial data of (4.14) satisfy pI (v) ≤ CI p∞(v) for some CI > 0. If additionally, (4.17) holds, then for the solution p of (4.14) we have for some constant C independent of time.

Relative entropy estimate. Proof of Theorem 2.2-(2)-(a)
In this part, we extend the relative entropy estimate of the limit equation (2.26) to (4.14). Then we fulfill the aforementioned contradiction argument to establish the global well-posedness for generalized solutions of (2.1) when 0 < b < VF − VR, proving Theorem 2.2-(2)-(a), For a solution p(v, τ ) of (4.14) with its boundary flux M (τ ) (4.4), as in (4.5) we use the notations Here p∞ is the steady state of the limit equation (2.26) and M∞ is its boundary flux as in Proposition 4.1. And as in (4.6) we consider the relative entropy for p against p∞ Similar to Proposition 4.2 we calculate d dτ SG[p](τ ). The difference is that now p(v, τ ) is governed by (4.14) instead of (2.26), thus terms involvingÑ appear. where the dissipation DG[p](τ ) is still given by (4.7) And the "perturbation" EG[p](τ ) is given by Proof of Proposition 4.6. We rewrite (4.14) as a perturbation from (2.26).
Note that p for (4.23) has the same boundary condition and the flux jump condition as (2.26). Thus here h(v, τ ) has the same boundary conditions as in Proposition 4.2. Therefore terms withoutÑ can be manipulated in the exact same way. As in Proposition 4.2, we multiply (4.23) by G ′ (h) and integrate to obtain where the dissipation DG[p] is the same as given in (4.7). And the "extra"Ñ terms give rise to to EG[p](τ ), which is given by Thanks to the zero boundary condition at VF , we integrate by parts Noting that hG ′′ (h)∂vh = ∂v(G ′ (h)h − G(h)), we do another integration by parts to transfer the derivative to p∞ where the boundary term vanishes due to that p∞(VF ) = 0. Now we give the proof of Theorem 2.2 in the mildly excitatory case, i.e., when 0 < b < VF − VR.
Proof of Theorem 2.2. mildly excitatory case. First we check that for initial data pI (v), there exists a constant CI such that Then we control the "perturbation" EG[p](τ ) (4.22) with G = (x − 1) 2 . Thanks to the uniform bound of h, hG ′ (h), G(h) are also bounded uniformly. Hence we derive In the last line, to conclude the integrability of these two terms, we use the formula for p∞ (4.2), in particular its exponential decay at infinity. Therefore by the relative entropy calculation in Proposition 4.6, we have Thus we get where SG[p](0) < ∞ thanks to (4.24) and +∞ 0Ñ (τ )dτ < ∞ is the contradiction assumption. Then, using the entropy dissipation DG[p] to control the boundary flux by Lemma 4.1, we deduce Finally, by Lemma 4.2 this leads to a contradiction with the assumption +∞ 0Ñ (τ )dτ < ∞. Then the proof is completed.

Inhibitory case: Global well-posedness in t
In this section we prove the global well-posedness for the generalized solution of (2.1) in the inhibitory case, i.e., when b ≤ 0, which is the last part of Theorem 2.2. The proof strategy is similar to the mildly excitatory case in Section 4 but some adjustments are needed. A main difference lies in the long time behavior of the limit equation (2.26).
As in Section 4, we first study the limit equation (2.26). Then we are able to treat the equation in timescale τ as a perturbation of (2.26).

Non-L 1 positive steady state
As in the excitatory case, the steady state of (2.26) satisfies (4.1) However, in the inhibitory case when b ≤ 0, the steady equation (4.1) does not have a solution, unless we drop the requirement on the decay when v → −∞. Nevertheless, if we relax this condition, we can still find positive steady states, summarized as the following.
Proposition 5.1. When b ≤ 0, the limit equation (2.26) does not have steady states in L 1 (−∞, VF ). Nevertheless, if we do not require the decay at infinity, we have the following "steady state" which is positive for v < VF . When b < 0 it is given by While when b = 0 the steady state is given by We can not normalize these "steady states" p∞(v) when b ≤ 0 since they are not in L 1 . Nevertheless, for later references we fix the expressions of p∞ as in Proposition 5.1 and denote its boundary flux as M∞ = −a1∂vp∞(v) > 0, using the same notation as in (4.3).

Relative entropy estimate
Though the "steady state" p∞ in Proposition 5.1 is not in L 1 , we can still use it to derive the relative entropy estimate. By the relative entropy estimate we can see the long time behavior of (2.26) and control the boundary flux, as in Section 4.1.2 and [7,12,15].
The entropy dissipation calculation in Proposition 4.2, which does not depend on the sign of b or any specific property of p∞, still holds when b ≤ 0 with p∞ defined in Proposition 5.1. We recall the result here as follows.
Let p(v, τ ) be a solution of the limit equation ( For a C 2 convex function G : R → R, let the relative entropy associated with G be as in (4.6) The difference is in the choice of the convex function G. Now p∞ is not in L 1 , if we still choose G(x) = (x − 1) 2 , then the entropy SG[p](τ ) will not be a finite number in general. Instead, we shall choose G(x) = x 2 which gives .
And when b = 0 we have We can see SG[p] as a weighed integral of p 2 with the weight 1/p∞(v) by (5.4). Intuitively, the decreasing of SG[p](τ ) may indicate that the mass of p tends to move from where the weight is large to where the weight is small. Note that when b < 0, p∞ goes to infinity as v → −∞ and thus 1/p∞(v) goes to zero when v goes to −∞. Hence the dissipation of this entropy may tell that the mass shall move to −∞ as time evolves. For the case b = 0 in (5.6), the weight at v = VF is also the highest. Therefore when b ≤ 0, we shall expect that the mass near VF becomes small in the long time, which intuitively implies the boundary flux M (τ ) shall be close to zero. Indeed, using the entropy dissipation DG[p](τ ), we can control the boundary flux M (τ ) via an inhibitory counterpart of Lemma 4.1. We can control the distance between ν(τ ) = M (τ )/M∞ and a δK (τ ) which can be made small. Precisely, we have the following lemma.
Lemma 5.1. When b ≤ 0, for G = x 2 , for any K > 0 there exists ǫ = ǫ(K) > 0 such that with a constant C independent of p(v, τ ) or K.

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For some 0 < ǫ < 1 2 to be determined, by the elementary inequality a 2 + 2ǫb 2 ≥ ǫ(a − b) 2 , we deduce in the same way as for Lemma 4.1 By definition δK (τ ) is the average of h on [VR − K, VR]. Thus, by Poincaré-Wirtinger inequality on [VR − K, VR] and noting that p∞(v) is bounded from below on [VR − K, VR], we have Together with the Sobolev injection from H 1 (I) to L ∞ (I) applying to h(·, τ ) − δK (τ ) on the interval I = [VR − K, VR], we get for some CK > 0 Finally, note that the entropy dissipation DG[p] for G = x 2 is the same as that for G = (x − 1) 2 , as in (4.11) Thus by choosing ǫ > 0 small enough such that 2ǫM∞CK < a1, and ǫ < 1/2 we get the desired results.  Thus we obtain the desired result by taking δ(τ ) = M∞δK (τ ).

Global well-posedness in t in the inhibitory case
With preparations on the limit equation (2.26), now we start to analyze the equation in the dilated timescale (2.7). As in Section 4.2, we set c = a 0 a 1 WLOG and thus work with with the boundary condition p(VF , τ ) = 0, τ > 0. We still use the notation M (τ ) = −a1∂vp(VF , τ ) (4.4) for the boundary flux, and recall in this caseÑ is determined by M in (4.16) A difference lies in the assumption on the drift b * , which is given by (4.15) In this inhibitory case, instead of b * ≤ VF (2.22), we assume (2.23) b0 ≤ VF , which implies in (4.14) thanks to (4.15) andÑ ∈ [0, a 1 a 0 ] (4.16).
We argue by contradiction as in the excitatory case. Suppose the solution does not globally exists, in view of the lifespan formula (2.18), we have (4.17) +∞ 0Ñ (τ )dτ < ∞, which gives a smallness forÑ (τ ). Therefore we may see (4.14) as a perturbation of the limit equation (2.26). However, as indicated in Section 5.1, when b ≤ 0 a solution of (2.26) moves away from VF in the long time, which would lead to small boundary flux M (τ ). In view of (4.16), this would result in some positive lower bound ofÑ (4.16), which contradicts (4.17). As in the mildly excitatory case, our proof is a justification of this intuition.
First, we note that it suffices to get an estimate similar to Corollary 5.1.
Lemma 5.2 is the inhibitory counterpart of Lemma 4.2, and its proof, which we omit here, is essentially the same as Lemma 4.2.
To derive (5.7), we first establish L ∞ bounds and then use the relative entropy estimate.

L ∞ bound
To derive the L ∞ bounds, first we construct super-solutions similar to [12] and Proposition 4.4.
Proposition 5.2. When b ≤ 0 and b0 ≤ VF , there exists a γ > 0 such that for a given solution p andÑ of (4.14), is a super solution of (4.14) in the following sense: q(VF , τ ) = 0 and . It is direct to check that the flux jump condition is satisfied. Indeed for a fixed τ , q(·, τ ) is a steady state of (2.26) with b = 0. Therefore, we only need to check the following inequality And it suffices to choose γ ≥ 1. On (VR, VF ), in this case ∂vq < 0. We rewrite the drift coefficient by plugging in the definition of b * (4.15) (4.16). Therefore, in this case owing to ∂vq ≤ 0, it suffices to ensure The last term is positive if v ≥ max(VR, b0) thanks to ∂vq ≤ 0. Therefore it is sufficient to choose Note that the sup term above is finite thanks to the assumption b0 ≤ VF . We conclude the proof.
Remark 5.1 (Comparison to the excitatory case). In Proposition 4.4, we construct super-solutions for b > 0 via the steady state of the associated limit equation (2.26). In this inhibitory case, we construct super-solutions for all b ≤ 0 by the steady state of (2.26) when b = 0, which is closer to the setting of "universal super-solution" in [12,Lemma 4.4]. This choice in some sense uses the benefits of b ≤ 0.
With super-solutions, we give L ∞ bounds as in Proposition 4.5 and [12, Lemma 4.3].
Proposition 5.3. When b ≤ 0 and b0 ≤ VF , if the initial data pI (v) of (4.14) is bounded and satisfies lim sup Then the following L ∞ bound holds Here γ is the constant for the super-solution in Proposition 5.2 and CI > 0 is a constant depending on the initial data.
Proof of Proposition 5.3. Noting that by the condition (5.11) and boundedness of pI (v), we can choose CI large enough, such that Then the rest of the proof is identical to that of Proposition 4.5.
Under the contradiction assumption (4.17), Proposition 5.3 directly implies uniform-in-time L ∞ bounds similar to Corollary 4.4 in the excitatory case. For later purposes, we give bounds on both p and h = p/p∞. Corollary 5.2. When b ≤ 0 and b0 ≤ VF , for (4.14) with bounded initial data satisfying (5.11). If additionally, (4.17) holds then we have the uniform bounds for some constant C independent of time, where p∞ is the "steady state" of (2.26) given in Proposition 5.1.

Relative entropy estimate. Proof of Theorem 2.2-(2)-(b)
Now we prove the global well-poseness for generalized solutions of (2.1) in the inhibitory case b ≤ 0, completing the proof of Theorem 2.2-(2)-(b). We start with recalling the relative entropy calculations. For a solution p(v, τ ) of (4.14) with its boundary flux M (τ ) (4.4), as before we use the notations (4.5) Here p∞ is the "steady state" of the linear equation (2.26), whose expression is given in Proposition 5.1, and M∞ is its boundary flux.
In the inhibitory case, we choose the convex function G(x) = x 2 and consider the associated relative entropy for p against p∞ (4.6) Now we start the proof, whose general strategy is identical to the mildly excitatory case.
Integrating in τ , if (5.14) holds we get since pI (v) is a probability density by Assumption 1. It remains to show the second term Using the contradiction assumption +∞ 0Ñ (τ )dτ < +∞ (4.17), it suffices to derive a uniform-in-τ bound By the triangle inequality, we have Thanks to (5.16), by Corollary 5.2 both p and h = p/p∞ are uniformly bounded. Therefore for the second term, we have And for the first term, we split the integral into two parts as follows Finally, note that when v ≤ VR, |(−v + b * )∂vp∞| ≤ Cp 2 ∞ by the expression in Proposition 5.1. Thus we obtain thanks to the uniform bound of p. Then the proof is complete.
The only difference is the coefficient of the Dirac source term at v = VR. In (2.1) it is N (t) while in (2.5) it is (a0 + a1N (t))(−∂vp(VF , t)). These two coefficients are the equal by (2.3) for classical solutions and thus the two formulations are equivalent in this case.
The flux-limited version (6.1) also deserves further understanding. If we naively integrate in space, then formally the reset at VR can be less than the loss at VF , which gives an apparent violation of mass conservation. Therefore the precise formulation of (6.1) might be more sophisticated, but it might be a worthy challenge.
Another issue on defining a generalized solution is the refractory period. Physically, after a neuron spikes it enters a short refractory period, during which it does not respond to any stimulus. From a modeling perspective, one can impose a finite or an infinitesimal refractory period. The latter case can be viewed as a limit of the former. If we add a finite refractory period, the model as well as its solution structure is substantially changed (see e.g. [15]). But an infinitesimal refractory period might be added to modify the definition of a generalized solution without changing the classical solution, since it only makes a difference when the firing rate blows up. A refractory period ensures that one neuron can not spike more than once in an instant of the original timescale t. Thus it might exclude the eternal blowup defined in (2.24) and make it possible to have global well-posedness in timescale t for arbitrary large connectivity parameter b.
To sum up, we have discussed two issues on defining a generalized solutions. We observe that the existing definitions in [17,39,40] and this work are different in those two aspects. Thus, it is appealing to study the uniqueness criteria in finding a scientifically relevant solution within a certain context.

Other further directions
We discuss possible future directions to explore the NNLIF dynamics (1.1) in the framework of the generalized solution in this paper.
The equation in the dilated timescale τ (2.7) might serve as an ideal platform to study the long time behavior of the original system, since the time-dilated transform (2.7) is globally well-posed in the classical sense for all b ∈ R (Theorem 2.2).
Moreover, (2.7) enriches the structure of the steady states of (1.1). It not only has classical steady states with N < +∞ but also allows steady states with N = +∞, which is equivalent toÑ = 0. These additional steady states appear if and only if b ≥ VF − VR, and are exactly those given by Proposition 4.1 since they are also steady states to the limit equation (2.26).
Interestingly, different solution behaviors in timescale t might be unified as the long time convergence to a steady state in timescale τ : If a solution converges to a steady state withÑ > 0 (equivalent to N < +∞), then it is global in timescale t. But if a solution converges to a steady state withÑ = 0 (equivalent to N = ∞), it might have an eternal blow-up in timescale t as defined in (2.24). Besides, the additional steady state might provide a way to understand the various phenomena including bistability observed in [27].
To obtain the generalized solution, we first solve the equation in timescale τ , then switch to the dynamics in timescale t via the inverse change of time. This procedure also facilitates efficient numerical simulations. Indeed, by simulating the equation in τ , the numerical scheme in [26] naturally becomes a computational approach for the generalized solution, which works for the dynamics with or without blow-ups.
Other interesting directions include to investigate how the blow-up times are distributed in the original timescale t (for similar models this has been studied in [18,39]), and to apply the idea of such time-dilation transforms to other integrate-and-fire models [10,15,21].

A.1.2 Proofs of Lemma 3.3 and Lemma 3.4
Here we present the proof of Lemma 3.3 and Lemma 3.4. From (3.14) we know a priori that β(s) ≥ 0 thusÑ ∈ [0, 1/c] by (3.15). Hence, in turn by (3.14) we know β(s) ≥ 1 and is C 1 . Therefore γ = β 2 ≥ 1 and is in C 1 . Then we differentiate (A.6) with respect to s to derive an ODE whereÑi is the shorthand forÑ (γi, Mi) for i = 1, 2. Using the mean value theorem to estimate the above two terms, and noting that β

A.2 Proof of Proposition 3.3
Here we present the proof of Proposition 3.3.