The limit equation will then be

$$ egin{cases} 
\frac{\partial n}{\partial t} + \frac{\partial}{\partial v} \left[ \left( -g_L v + (V_E - v) \int_0^{+\infty} g M(v, N(t)) dg \right) n(v, t) \right] + \phi_F n(v, t) = 0, \\ 
N(t) = \int_0^{V_E} \phi_F(v) n(v, t) dv, \\ 
V_E \int_0^{+\infty} g M(v, N(t)) dg n(v, t) = N(t), \quad n(V_E, t) = 0, \\ 
n(v, 0) = n^0(v) := \int_0^{+\infty} \rho^0(v, g) dg. 
\end{cases} 
\quad (7.13) $$

Note that it is possible to compute explicitly

$$ \int_0^{+\infty} g M(g, N(t)) dg = \frac{\sqrt{a} e^{-\frac{N(t)^2}{2a}}}{\sqrt{\frac{\pi}{2}} \left( 1 + \operatorname{erf} \left( \frac{s_0 + N(t)}{\sqrt{2a}} \right) \right)}. $$

**Theorem 7.13** (Perthame, Salort [212]). Assume for all $k \in \mathbb{N}$, $\int_0^{+\infty} \int_0^{V_E} g^k \rho^0 dg dv < +\infty$, $\phi_F \in W_+^{1, +\infty}(0, V_E)$. Then, there exists $C(k)$, $C > 0$ such that for all $\varepsilon > 0$, for all $t \ge 0$,

$$ \int_0^{+\infty} \int_0^{V_E} g^k \rho^\varepsilon(v, g, t) dg dv \le C(k), \quad \|\mathcal{N}\|_\infty + \|\mathcal{N}'\|_\infty \le C. $$

In the weak topology of bounded measures, $\rho^\varepsilon \to n(v, t)M(g, \mathcal{N}(t))$, where $n$ is solution to (7.13)

For the solutions to the limit equation (7.13), [212] also proves the $L_{\text{loc}}^\infty$ bound

$$ \sup_{t \in [0, T]} \|n(\cdot, t)\|_{\infty} \le C(T) \|n^0\|_{\infty}. $$

Note that in the formal low-noise limit $a \to 0$, the Maxwellian converges to a Dirac mass and its average to an affine function of the firing rate:

$$ M(g, N(t)) \to \delta_{s_0 + S_E N(t)}, \quad \int_0^{+\infty} g M(g, N(t)) dg \to s_0 + S_E N(t). $$

The main equation of (7.13) then becomes

$$ \frac{\partial n}{\partial t} + \frac{\partial}{\partial v} \left[ \left( -g_L v + (s_0 + S_E N(t))(V_E - v) \right) n(v, t) \right] + \phi_F(v) n(v, t) = 0, $$

which is almost identical to (6.2) studied in [70, 71, 212] and obtained as a mean-field limit in [113, 78].

### 7.7.2 Fast voltage limit

If we now consider the case where it is not the conductance which evolves way faster but the voltage, then the solution should concentrate in $v$ around the curve of solutions to $J_v = 0$, which means that $v$ will converge to

$$ V^*(g) = \frac{gV_E}{g + g_L}. $$

Consider for simplicity $\sigma_E = 0$. The fast firing regime consists in replacing the first equation in (7.11) by

$$ \frac{\partial \rho}{\partial t} + \frac{1}{\varepsilon} \frac{\partial}{\partial v} \left[ \left( -g_L v + g(V_E - v) \right) \rho \right] + \frac{\partial}{\partial g} \left[ (s_0 + S_E \mathcal{N}(t) - g) \rho \right] - a_0 \frac{\partial^2 \rho}{\partial v^2} + \phi_F \rho = 0, $$
with $\epsilon > 0$ the scaling parameter. The article [146] thus proposes that in the infinitely fast voltage regime, the equation for

$$n(g, t) = \int_{0}^{V_E} \rho(v, g, t)dv$$

is the parabolic equation

$$\left\{
\begin{array}{l}
\displaystyle \frac{\partial n}{\partial t} + \frac{\partial}{\partial g} [(s_0 + S_E N(t) - g)n] - a_0 \frac{\partial^2 n}{\partial g^2} = 0, \\
\\
\displaystyle N(t) = \int_{0}^{+\infty} \phi_F(V^*(g))n(g, t)dg, \\
\\
\displaystyle J_g(0, N(t))n(0, t) - a_0 \frac{\partial n}{\partial g}(0, t) = 0, \quad n(g, 0) = n^0(g) := \int_{0}^{V_E} \rho^0(v, g)dv.
\end{array}
\right.
\qquad (7.14)$$

For technical reasons, the proof in [146] is performed on a version of the equation with flux $J_g$ bounded from above uniformly. Using careful estimates on (7.11), first for the hyperbolic case $a = 0$ and then for the case $a > 0$, they prove rigorously that the limit equation in the fast firing regime is an equation of the form (7.14).

## 7.8 Other variants of the voltage-conductance model

Besides the random firing voltage-conductance model we have described, some other simplifications have been proposed in order to find which features are key for (V-C) to exhibit complex behaviours. Although we will not describe them in detail, note the 2-conductances model [92] and the kinetic Fokker-Planck equation with jumps [237] which both share many similarities with (V-C).

### 7.8.1 Full understanding of a toy model and the importance of the voltage drift

In order to understand self-organized oscillations in (V-C), one can make two simplifications to isolate the difficulty of the firing rate being computed as a moment evaluated at a pointwise value of the voltage. One difficulty stems from the velocity field in $v$, i.e., $J_v(v, g) = -g_L v + g(V_E - v)$ as in (V-C). The velocity field can not be written in a separable form $J_v(v, g) = f(v)h(g)$, in contrast to the velocity field for the position variable in classical kinetic models. This non-separable velocity field prohibits the use of many analysis tools. To understand the consequences of a non-separable velocity field in $v$ and $g$, we consider a simpler velocity field in $v$ neglecting the leaky mechanism, $J_v(v, g) = g(V_E - v)$, which is further reduced to $J_v(v, g) = g$ for the sake of simplicity, see [47] for further generalizations. Moreover, it leads to the complicated boundary condition (7.3), which makes it difficult to analyse even in the steady state case [89]. Second, we extend the domain of $g$ from $\mathbb{R}^+$ to $\mathbb{R}$. Then we get the equation for the probability density function $p(t, v, g)$, denoted by $p$ in order to make the distinction to (V-C) apparent, as

$$\partial_t p + g \partial_v p + \partial_g ((g_{\text{in}}(t) - g)p - a(t) \partial_g p) = 0, \quad v \in (0, V_F), \ g \in \mathbb{R}, \ t > 0, \qquad (7.15)$$

and the boundary condition in voltage $v$ is simplified to

$$p(t, 0, g) = p(t, V_F, g), \quad \forall t > 0, g \in \mathbb{R}.$$

For the conductance $g$, we no longer need a boundary condition at $g = 0$ thanks to the extension of the domain. For the initial data, we assume it is a probability density function on $(0, V_F) \times (-\infty, +\infty)$, denoted as

$$p(0, v, g) = p^0(v, g), \quad v \in (0, V_F), g \in \mathbb{R}.$$

The expressions of $g_{\text{in}}(t)$ and $a(t)$ depend on the firing rate $N(t)$ in the same way as in the original model given by

$$g_{\text{in}}(t) = g_0 + g_1 N(t), \quad a(t) = a_0 + a_1 N(t), \quad g_0, g_1, a_0, a_1 > 0. \qquad (7.16)$$
We define the firing rate $N(t)$ in a similar manner as the integration of flux at voltage $v = V_F$, over $g > 0$, i.e.,

$$N(t) := \int_{0}^{\infty} gp(t, V_F, g)dg. \qquad (7.17)$$

Though simplified, this model (7.15)-(7.17) inherits the nonlinear mechanism of the original model (V-C) – dependence of $g_{in}(t)$, $a(t)$ on $N(t)$ as the firing rate $N(t)$ still only depends on the flux at one voltage value $v = V_F$.

The separability of the voltage flux makes possible an analysis via Fourier modes for classical solutions since the boundary conditions in $v$ now simplify to periodic boundary conditions. The main outcome is showing that the long time asymptotics are dictated by the zeroth-order model and then by homogeneous in $v$ solutions to (7.15)-(7.17). Let us consider the Fourier expansion in $v$ direction

$$\begin{align} p(t, v, g) &= \frac{1}{V_F} \sum_{k=-\infty}^{+\infty} p_k(t, g) e^{ikv \frac{2\pi}{V_F}}, \quad \text{where} \tag{7.18} \\ p_k(t, g) &:= \int_{0}^{V_F} p(t, v, g) e^{-ikv \frac{2\pi}{V_F} d} dv, \quad k \in \mathbb{Z}. \nonumber \end{align}$$

**Theorem 7.14** (Carrillo, Dou, Zhou [47]). Suppose $p(t, v, g)$ is a classical solution of (7.15) and $p_0(t, g)$ is the zeroth mode in $v$ defined in (7.18). Then $p(t, v, g)$ converges to the $v$-homogeneous mode $\frac{1}{V_F}p_0(t, g)$ exponentially. Precisely, we have

$$\left\| p(t, v, g) - \frac{1}{V_F} p_0(t, g) \right\|_{L^1((0, V_F) \times \mathbb{R})} \le 2 \frac{e^{-\left(\frac{2\pi}{V_F}\right)^2 d(t)}}{1 - e^{-\left(\frac{2\pi}{V_F}\right)^2 d(t)}}$$

where $d(t)$ is given by

$$d(t) := a_0 \left( t - 2 \frac{e^t - 1}{e^t + 1} \right) > 0, \quad t > 0.$$

The second simplification extending the conductance to the whole line allows for a deep analysis of the $v$-homogeneous solutions to (7.15)-(7.17). In fact, explicit computations show the existence of gaussian solutions with means and variances in $g$ depending only on time. The $v$-homogeneous solutions to (7.15)-(7.17) satisfy

$$\partial_t p_0 = \partial_g ((-g_{in}(t) + g)p_0 + a(t)\partial_g p_0), \quad g \in \mathbb{R}, t > 0, \qquad (7.19)$$

where $g_{in}(t)$ and $a(t)$ still given by (7.16). The $v$-homogeneous equation (7.19) admits the following Gaussian type solution

$$p(t, g) = \frac{1}{\sqrt{2\pi c(t)}} \exp \left( - \frac{(g-b(t))^2}{2c(t)} \right),$$

where the mean $b(t)$ and the variance $c(t)$ satisfy the following autonomous ODE

$$\begin{align} \frac{db(t)}{dt} &= g_0 + g_1 N(b(t), c(t)) - b(t), \tag{7.20} \\ \frac{dc(t)}{dt} &= 2a_0 + 2a_1 N(b(t), c(t)) - 2c(t). \nonumber \end{align}$$

Here $N(b, c)$ is a function of $(b, c)$ which denotes the firing rate of such a Gaussian type solution:

$$N(b, c) := \frac{1}{V_F} \int_{0}^{+\infty} g \frac{1}{\sqrt{2\pi c}} \exp \left( -\frac{(g-b)^2}{2c} \right) dg \ge 0, \quad b \in \mathbb{R}, c > 0.$$

This combined with Theorem 7.14 leads to the following final long time asymptotics result for solutions to (7.15)-(7.17).
**Theorem 7.15** (Carrillo, Dou, Zhou [47]). For initial data $p^0(v, g)$ with $\int_{\mathbb{R}} \int_0^{V_F} |g|p^0(v, g)dvdg < \infty$, the classical solution of the simplified voltage-conductance model (7.15)-(7.17) satisfy:

1. When $g_1/V_F \ge 1$, the firing rate $N(t)$ goes to infinity as time evolves, i.e.,

$$N(t) \to +\infty, \quad \text{as } t \to +\infty.$$

2. When $0 < g_1/V_F < 1$, as time evolves the density function $p(t, v, g)$ converges to the unique steady state

$$p^*(v, g) := \frac{1}{V_F} \frac{1}{\sqrt{2\pi c^*}} \exp \left( - \frac{(g - b^*)^2}{2c^*} \right), \quad (7.21)$$

in $L^1((0, V_F) \times \mathbb{R})$. Here $(b^*, c^*)$ is the unique steady state of the ODE system (7.20).

3. When $g_1/V_F \to 1^-$, we have $b^* \to +\infty$, which means in $g$ direction the center of the steady state (7.21) moves towards infinity.

By introducing a parameter $\varepsilon > 0$, modelling the timescale ratio between the conductance $g$ and the voltage $v$, we can consider the fast conductance limit $\varepsilon \to 0^+$. One can derive a limit model governing the $v$ marginal density, whose long time behavior can also be characterized clearly as in the previous theorem, see [47] for further details. Depending on $g_1$ and the $L^\infty$ norm of initial data, the solution either blows up in finite time or globally exists in the form of periodic solutions. The authors only find periodic solutions in the fast conductance limit showing that the other neglected difficulties might be relevant for the appearance of periodic solutions in the full model (V-C).

### 7.8.2 Wassertein contraction for a Moris-Lecar model

In [135], a kinetic Moris-Lecar [188] model is proposed for the modelling of a population of neurons described by voltage and conductance. Instead of letting the neurons fire at a threshold $V_F$ and resetting them, the voltage domain is extended from $[V_L, V_F]$ to $[V_L, V_E]$, where $V_L, V_E$ are again the leak and the excitatory reversal potential, satisfying here $0 < V_L < V_E$. The model reads

$$\left\{
\begin{array}{l}
\displaystyle \frac{\partial \rho}{\partial t} + \frac{\partial \rho}{\partial v} [(g_L(V_L - v) + g(V_E - v))\rho] + \gamma \frac{\partial \rho}{\partial g} [(G(v) - g)\rho] - a \frac{\partial^2 \rho}{\partial g^2} = 0, \\
\rho(V_L, g, t) = \rho(V_E, g, t) = 0, \\
(G(v) - g)\rho(v, 0, t) - a \frac{\partial \rho}{\partial g}(v, 0, t) = 0, \\
\rho(v, g, 0) = \rho^0(v, g) \ge 0, \quad \int_{V_L}^{V_E} \int_0^{+\infty} \rho^0(v, g)dvdg = 1,
\end{array}
\right.
\quad (7.22)$$

with $g_L > 0$ the leak conductance, $\gamma > 0$ a timescale parameter for the conductance, $a > 0$ the strength of noise on the conductance and $G \in C^2([V_L, V_E]) > 0$ a bounded positive function. Contrarily to (V-C), this PDE system is linear, and, most important, it does not have the issue of a complex nonlocal boundary condition, which was a technical issue even in linear (V-C).

The method [135] proposes for the study of (7.22) is to focus on the underlying stochastic process

$$\left\{
\begin{array}{l}
dV_t = [g_L(V_L - V_t) + g_E(V_E - V_t)]dt, \\
dG_t = \gamma(G(V_t) - G_t)dt + \sqrt{2\alpha d}B_t + dL_t,
\end{array}
\right.
\quad (7.23)$$

with $(B_t)_{t \ge 0}$ a standard Brownian motion and $(L_t)_{t \ge 0}$ the local time preventing the conductance $G_t$ from becoming negative (see [227, Chapter 6] or the discussion on a similar issue in Section 2).

Denote $(P_t)_{t \ge 0}$ the semigroup associated to (7.23), which acts left on measures and right on test functions as seen in other duality arguments in this document. The article [135] uses the following
coupling method: choose two $C^2$ Lipschitz non-negative functions $\alpha, \beta$ such that $\alpha^2 + \beta^2 = 1$, and consider the coupled system

$$
\left\{
\begin{array}{l}
dv_t = [g_L(V_L - v_t) + g_E(V_E - v_t)]dt, \\
dg_t = \gamma(G(v_t) - g_t)dt + \sqrt{2a}[\alpha_t dB_t + \beta_t dB'_t] + dL_t, \\
dv'_t = [g_L(V_L - v'_t) + g_E(V_E - v'_t)]dt, \\
dg'_t = \gamma(G(v'_t) - g'_t)dt + \sqrt{2a}[\alpha_t dB_t - \beta_t dB'_t] + dL'_t,
\end{array}
\right.
$$

where $(B_t)_{t \ge 0}$, $(B'_t)_{t \ge 0}$ are standard Brownian motions,

$$
\alpha_t = \alpha(|g_t - g'_t|^2), \quad \beta_t = \beta(|g_t - g'_t|^2),
$$

and $L, L'$ are the local times at 0 of $g_t, g'_t$. Since by Levy's characterisation the processes

$$
W_t = \int_0^t [\alpha_t dB_t + \beta_t dB'_t], \quad W'_t = \int_0^t [\alpha_t dB_t - \beta_t dB'_t]
$$

are Brownian motions, $(v_t, g_t)$ and $(v'_t, g'_t)$ are also solutions to (7.23). The regularity involved allows to write Itô formulae for the coupling system.

When $\alpha = 1, \beta = 0$ both equations are subject to the same noise source and the coupling is called synchronous. Using this coupling, [135] proves that if $(V_E - V_L) \|G'\|_\infty < g_L$, then $(v_t, g_t)$ and $(v'_t, g'_t)$ subject to the same Brownian noise are subject to a deterministic contraction: there exist $C \ge 1, \lambda > 0$ such that

$$
|(v_t, g_t) - (v'_t, g'_t)| \le C e^{-\lambda t} |(v_0, g_0) - (v'_0, g'_0)|
$$

This allows to prove both that there exists a unique measure stationary state $\rho_\infty \in \mathcal{P}([V_L, V_E] \times \mathbb{R}_+)$ to (7.22) when $a > 0$ and that when $a = 0$ the deterministic system converges exponentially fast towards a unique equilibirum $(V_\infty, G_\infty)$.

By constructing a mirror coupling, [135] proves that there is also a stochastic contraction for (7.23). Notwithstanding the regularity assumptions not being satisfied by this choice, the idea of a mirror coupling is to choose $\beta(s) = 1_{s>0}$ and the corresponding $\alpha$ in order to have $\alpha^2 + \beta^2 = 1$. A regularised mirror coupling is crafted in [135] and used to prove a very general Wasserstein contraction for (7.23) and hence for (7.22). More precisely, denote for any $p \ge 1$, $\mathcal{P}_p([V_L, V_E] \times \mathbb{R}_+)$ the set of probability measures on $[V_L, V_E] \times \mathbb{R}_+$ with finite $p^{th}$ moment and $\mathcal{W}_p$ the $L^p$-Wasserstein distance on this space. Then,

**Theorem 7.16** (Herda, Monmarché, Perthame [135]). There are constants $\lambda > 0, C \ge 1$ such that for any initial distributions $\nu, \mu \in \mathcal{P}_1([V_L, V_E] \times \mathbb{R}_+)$,

$$
\forall t \ge 0, \quad \mathcal{W}_1(\nu P_t, \mu P_t) \le C e^{-\lambda t} \mathcal{W}_1(\nu, \mu).
$$

There exists a unique invariant measure $\rho_\infty$ and for all $\nu \in P([V_L, V_E] \times \mathbb{R}_+)$, $\nu P_t$ converges weakly to $\rho_\infty$. For all $p > q \ge 1$, if $\rho_\infty, \nu \in \mathcal{P}_p([V_L, V_E] \times \mathbb{R}_+)$, then $\mathcal{W}_q(\nu P_t, \rho_\infty)$ converges to 0 as $t \to +\infty$.

The method can be extended to a system of finitely many neurons of potentials and conductances $V_t^1, G_t^1, \dots, V_t^N, G_t^N$ interacting through their voltage via the quantities $G_i(V_t^1, \dots, V_t^N)$, with $G_i \in C^1([V_L, V_E]) > 0$. The system of $N$ neurons is then

$$
\left\{
\begin{array}{l}
dV_t^i = [g_L(V_L - V_t^i) + g_E(V_E - V_t^i)]dt, \\
dG_t^i = \gamma(G_i(V_t^1, \dots, V_t^N) - G_t^i)dt + \sqrt{2a}dB_t^i + dL_t^i.
\end{array}
\right.
$$

Denote $\mathcal{W}_{1,l_1}$ the Wasserstein-1 distance associated to

$$
\|(v, g) + (w, h)\|_{1} = \sum_{i=1}^{N} (|v^{i} - w^{i}| + |g^{i} - h^{i}|).
$$
**Theorem 7.17** (Herda, Monmarché, Perthame [135]). Assume there exists $K > 0$ such that

$$ \max_{1 \le i, j \le N} \| \partial_{v_j} G_i \|_{\infty} \le K. $$

There are constants $\lambda, \eta > 0, C \ge 1$ such that for any initial distributions $\nu, \mu \in \mathcal{P}_1([V_L, V_E] \times \mathbb{R}_+)$,

$$ \forall t \ge 0, \quad W_{1,l_1}(\nu P_t, \mu P_t) \le C e^{-(\lambda-\eta)t} W_{1,l_1}(\nu, \mu). $$

If $\lambda > \eta$, there exists a unique invariant measure $\rho_{\infty}^N$ whose moments are all finite and for all $\nu \in P([V_L, V_E]^N \times \mathbb{R}_+^N)$, $W_p(\nu P_t, \rho_{\infty}^N)$ converges to 0 as $t \to +\infty$.

The constants $C, \lambda, \eta$ being constructed explicitly in [135], it is in fact possible to determine when $\lambda > \eta$ and to play on the functions and parameters to ensure it is the case.

If the control $K > 0$ over the norms $\|\partial_{v_j} G_i\|$ is uniform when $N \to +\infty$, the work done on the linear model and the system of $N$ neurons allow to pass to the mean-field limit and prove propagation of chaos. The article [135] considers the choice

$$ G_i(V^1, \dots, V^N) = H_0(V^i) + \frac{1}{N-1} \sum_{j \ne i} H_1(V^i, V^j). $$

with $H_0 \in C^1([V_L, V_E], (0, +\infty))$ and $H_1 \in C^1([V_L, V_E]^2, (0, +\infty))$. Then $|\partial_{v_i} G_i| \le \|H_0'\|_{\infty} + \|\nabla H_1\|_{\infty}$ and for $j \ne i$, $|\partial_{v_j} G_i| \le \frac{1}{N-1} \|\nabla H_1\|_{\infty}$.

It is then possible to take the constants as independent of $N$ in Theorem 7.17, allowing the negative part $\lambda$ of the decay rate to be independent from $N$ as well. Using this strategy, [135] proves rigorous propagation of chaos: if we consider the initial conditions distributed according to $(\rho^0)^{\otimes N}$ and denoting for $k \le N$ by $\rho_t^{(k,N)}$ the law of $(V_t^1, G_t^1, \dots, V_t^k, G_t^k)$, then

$$ W_2(\rho_t^{(k,N)}, \rho(t)^{\otimes k}) \le \sqrt{\frac{k}{N}} C_0 e^{ct}, $$

for some constants $C_0, c > 0$ and where $\rho$ is the solution to the mean-field system

$$ \left\{ \begin{array}{l} \displaystyle \frac{\partial \rho}{\partial t} + \frac{\partial \rho}{\partial v} [J_v(v, g)\rho] + \gamma \frac{\partial \rho}{\partial g} [(H_0(v) + H_1 \star \rho(v) - g)\rho] - a \frac{\partial^2 \rho}{\partial g^2} = 0, \\[1.5em] \rho(V_L, g, t) = \rho(V_E, g, t) = 0, \\[1.5em] (H_0(v) + H_1 \star \rho(v) - g)\rho(v, 0, t) - a \frac{\partial \rho}{\partial g}(v, 0, t) = 0, \\[1.5em] \rho(v, g, 0) = \rho^0(v, g) \ge 0, \quad \int_{V_L}^{V_E} \int_0^{+\infty} \rho^0(v, g) dv dg = 1, \end{array} \right. \qquad (7.24) $$

where $J_v(v, g) = g_L(V_L - v) + g(V_E - v)$ and

$$ H_1 \star \rho(v) = \int_{V_L}^{V_E} \int_0^{+\infty} H_1(v, w) \rho(w, g) dg dw. $$

As proved in [135], the coupling method still applies to this now nonlinear PDE problem: there exist $C, \theta, \lambda > 0$ such that if $\rho_1^0, \rho_2^0 \in \mathcal{P}([V_L, V_E] \times \mathbb{R}_+)$ yield two solutions $\rho_1, \rho_2$ of (7.24), then

$$ W_1(\rho_1(t), \rho_2(t)) \le C e^{-(\lambda-\theta)t} W_1(\rho_1^0, \rho_2^0). $$

Again, $\lambda$ and $\theta$ are constructed explicitly and can be compared, providing when $\lambda > \theta$ the existence and uniqueness of a stationary state and a weakly-nonlinear stability result for (7.24).
# 8 Kinetic FitzHugh-Nagumo models

One of the most comprehensive descriptions of individual neurons and the ionic exchanges leading to action potentials is the celebrated Hodgkin-Huxley model [137]. Due to its complexity and high computational cost, the FitzHugh-Nagumo model has been proposed [109, 194] as a simplification which preserves many important features while remaining mathematically simple and numerically tractable at the network scale.

Consider a neuron of voltage $v(t)$ and adaptation variable $w(t)$, whose voltage evolution responds to an external current $I_{\text{ext}}(t)$, an intrinsic voltage dynamics $N(v)$ and is dampened by the adaptation variable $w$. The FitzHugh Nagumo system writes

$$ egin{cases} \frac{dv}{dt} = N(v) - w + I_{\text{ext}}(t), \\ \frac{dw}{dt} = bv - aw + c, \end{cases} \qquad (8.1) $$

with $a, b$ positive parameters and $c \in \mathbb{R}$ that we will often set to 0. A widespread choice [194, 107, 3, 258] for the nonlinearity $N$ is a cubic function like $N(v) = v - v^3$.

We first explain how this model for one neuron can be used to derive a mean-field equation for a whole network. Then we discuss existence of solutions, asymptotic convergence to stationary states and the arising of self-sustained oscillations. Last, we cover some results about spatial extensions of the model and the reduction towards a macroscopic limit.

## 8.1 From individual FitzHugh-Nagumo neurons to a mesoscopic system

Consider a network of $K$ interconnected neurons each described by the system (8.1) and subject to a standard Brownian noise $(B_t^i)_{t \ge 0}$:

$$ egin{cases} dV_t^i = (N(V_t^i) - W_t^i + \sum_{j=1}^N J_{ij}(V_t^i - V_t^j) + I_{\text{ext}}(t)) dt + \sigma dB_t^i, \\ dW_t^i = (bV_t^i - aW_t^i) dt, \end{cases} \qquad (8.2) $$

where we set here the cubic nonlinearity $N(v) = v(v - \lambda)(1 - v)$, $\lambda > 0$ and assume for clarity $\sigma^2 = 2$, $I(t) \equiv I_0 \in \mathbb{R}$. The coefficients $J_{ij}$ represent the connections between neurons $i$ and $j$.

Propagation of chaos and description of the mean-field limit for system (8.2) have been rigorously obtained in [3] and slightly improved in [183, Appendix A]; see also [11, 64, 170, 73] for other approaches on mean field limits for systems of FitzHugh-Nagumo neurons. In the limit $K \to +\infty$, the density $\rho(v, w)$ of the limiting process is solution to the nonlinear kinetic equation

$$ egin{cases} \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial w}((bv - aw)\rho) + \frac{\partial}{\partial v}(B_{\varepsilon}(v, w, \mathcal{J}(t))\rho) - \frac{\partial^2 \rho}{\partial v^2} = 0, \\ \mathcal{J}(t) = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} v\rho(v, w, t)dv dw, \\ B_{\varepsilon}(v, w, \mathcal{J}) = v(v - \lambda)(1 - v) + w - \varepsilon(v - \mathcal{J}) + I_0, \\ \rho(v, w, 0) = \rho^0(v, w) \ge 0, \quad \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \rho^0(v, w)dv dw = 1. \end{cases} \qquad (\text{FhN}) $$

The parameter $\varepsilon$ represents the average connectivity and depends upon scaling assumptions on the $J_{ij}$ in the mean-field limit. When it is small enough, we consider the model to be weakly-nonlinear.

Consider the weights $\omega(v, w) = 1 + \frac{v^2}{2} + \frac{w^2}{2}$ and the spaces $L^1(\omega)$ and $L^1(\omega^{1/2})$ of functions which are integrable when multiplied by $\omega$ or $\omega^{1/2}$. We take from [183] the following notion of weak-solution.

**Definition 8.1.** We say that $\rho \in C(\mathbb{R}_+, L^1(\omega^{1/2}))$ is a global-in-time weak solution to (FhN) if
• for almost every $t \ge 0$, $\rho(\cdot, \cdot, t) \ge 0$ and

$$ \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \rho(v, w, t) \mathrm{d}v \mathrm{d}w = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \rho^0(v, w) \mathrm{d}v \mathrm{d}w = 1; $$

• for all $\phi \in C^1(\mathbb{R}_+, C_c^\infty(\mathbb{R}^2))$ and all $t \ge 0$,

$$ \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \phi(v, w, t) \rho(v, w, t) \mathrm{d}v \mathrm{d}w = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \phi(v, w, t) \rho^0(v, w) \mathrm{d}v \mathrm{d}w \\ + \int_{0}^{t} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \left[ \frac{\partial \phi}{\partial t} + (bv - aw) \frac{\partial \phi}{\partial w} + B_{\varepsilon}(v, w, \mathcal{J}(s)) \frac{\partial \phi}{\partial v} + \frac{\partial^2 \phi}{\partial v^2} \right] \rho(v, w, s) \mathrm{d}v \mathrm{d}w \mathrm{d}s. $$

## 8.2 Existence of weak solutions

Let us introduce the space

$$ L^1 \log L^1 = \{ \rho \in L^1(\mathbb{R}^2) \mid \rho \ge 0, \mathcal{H}(\rho) < +\infty \}, \quad \mathcal{H}(\rho) = \int_{\mathbb{R}^2} \rho(v, w) \log \rho(v, w) \mathrm{d}v \mathrm{d}w, $$

and for some $\kappa > 0$, the weight function $m(v, w) = e^{\kappa(\omega-1)} = e^{\kappa(\frac{v^2}{2} + \frac{w^2}{2})}$. Define also the space with additional $v$-derivatives

$$ H_v^2 = \left\{ f \in H^1(\mathbb{R}^2) \mid \frac{\partial^2 f}{\partial v^2} \in L^2(\mathbb{R}^2) \right\}, \quad \|f\|_{H_v^2} = \|f\|_{H^1} + \left\| \frac{\partial^2 f}{\partial v^2} \right\|_{L^2}. $$

and its natural weighted version $H_v^2(m)$ where all norms are taken with the weight $m$.

For fixed external input $\mathcal{J}$, (FhN) is the evolution equation $\partial_t \rho = Q_{\mathcal{J},\varepsilon} \rho$ where the operator $Q_{\mathcal{J},\varepsilon}$ is defined by

$$ Q_{\mathcal{J},\varepsilon} \rho = \frac{\partial}{\partial w} [(aw - bv)\rho] + \frac{\partial}{\partial v} [B_{\varepsilon}(v, w, \mathcal{J})\rho] + \frac{\partial^2 \rho}{\partial v^2}. $$

First, [183] proves by direct calculations an estimate of the form

$$ \frac{d}{dt} \int_{\mathbb{R}^2} \rho(v, w, t) \omega(v, w) \mathrm{d}v \mathrm{d}w \le K_1 - K_2 \int_{\mathbb{R}^2} \rho(v, w, t) \omega(v, w) \mathrm{d}v \mathrm{d}w $$

which by application of Grönwall lemma and use of the control $|\mathcal{J}(t)| \le 2 \|\rho(t)\|_{L^1(\omega)}$ ensures that any solution of (FhN) satisfies, for appropriate initial conditions, the a priori bounds

$$ \|\rho(\cdot, \cdot, t)\|_{L^1(\omega)} \le \max(C, \|\rho^0\|_{L^1(\omega)}), \quad \sup_{t \ge 0} |\mathcal{J}(t)| \le C, $$

for some constant $C$ depending on $\rho^0$ and the parameters. Then, for fixed $\mathcal{J}$, they derive a bound the operator $Q_{\mathcal{J},\varepsilon}$ and deduce a similar bound for $\rho$ in the space $L^1(m)$. This first step allows to move to the stronger spaces $H^1(m)$ and $H_v^2(m)$. The core idea is to take advantage of the hypodissipativity of the equation by use of the twisted space approach and the Nash-Villani technique [267]. More precisely, choosing two positive constants $\kappa_1 < \kappa_2$ and associated $m_1 = e^{\kappa_1(\omega-1)}$, $m_2 = e^{\kappa_2(\omega-1)}$, they prove (not without long and tedious technicalities) that for all $\mathcal{J}$ fixed, there exist constants $K_1, K_2$ and $\delta \in (0, 1)$ such that

$$ \|\rho\|_{\mathcal{H}^1}^2 = \|\rho\|_{L^2(m_2)}^2 + \|\partial_w \rho\|_{L^2(m_2)}^2 + \langle \partial_w \rho | \partial_v \rho \rangle_{L^2(m_1)} + \|\partial_v \rho\|_{L^2(m_2)}^2 $$

defines a Hilbert norm equivalent to the usual norm on $H^1(m_2)$ and satisfying

$$ \forall \rho \in H^1(m_2), \quad \langle Q_{\mathcal{J},\varepsilon} \rho | \rho \rangle_{\mathcal{H}^1} \le K_1 \|\rho\|_{L^2(\mathbb{R}^2)}^2 - K_2 \|\rho\|_{\mathcal{H}^1}^2. $$
From there, choosing $m = m_2$, Nash's inequality and the equivalence between $\|\cdot\|_{H^1}$ and $\|\cdot\|_{H^1(m)}$ allow to derive the differential inequality

$$ \frac{d}{dt} \|\rho(\cdot, \cdot, t)\|_{H^1}^2 \le \langle Q_{\mathcal{J}(t),\varepsilon} \rho(\cdot, \cdot, t) | \rho(\cdot, \cdot, t) \rangle \le K_3 - K_4 \|\rho(\cdot, \cdot, t)\|_{H^1}^2, $$

which proves a priori control of $\|\rho(\cdot, \cdot, t)\|_{H^1}$ for any solution of (FhN). Then, [183] proves, using the partial Fischer information

$$ I_v(\rho) = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \frac{|\partial_v \rho(v, w)|^2}{f(v, w)} dv dw, $$

that solutions to (FhN) with finite initial entropy remain in the space $L^1 \log L^1$ of finite entropy functions.

**Lemma 8.2.** For all $\rho^0 \in L^1(\omega) \cap L^1 \log L^1 \cap \mathcal{P}(\mathbb{R}^2)$, for any solution to (FhN) and any time $T > 0$, there exists $C(T) > 0$ such that

$$ \sup_{t \in [0, T]} \mathcal{H}(\rho(\cdot, \cdot, t)) + \int_0^t I_v(\rho(\cdot, \cdot, s)) ds \le C(T). $$

This entropy bound yields the well-posedness bound

$$ \sup_{t \in [0, T]} \| \rho_1(\cdot, \cdot, t) - \rho_2(\cdot, \cdot, t) \|_{L^1(\omega^{1/2})} \le C_2(T) \| \rho_1^0 - \rho_2^0 \|_{L^1(\omega^{1/2})}, $$

which ensures uniqueness of any prospective solution. Although standard results don't apply and the equation has a degenerate structure and unbounded drifts, [183] finds its way to the existence of solutions using the a priori bounds above along with a cutoff and a tailored coercive bilinear form. An Aubin-Lions compactness argument yields a solution for any given external input $t \mapsto \mathcal{J}(t) \in L^\infty(\mathbb{R}_+)$, and then the full solution to (FhN) can be recovered by a fixed point argument on the functional that maps $\mathcal{J} \in L^\infty([0, T])$ to the internal input $\mathcal{J}_{\rho\mathcal{J}}$ associated to the solution $\rho_{\mathcal{J}}$ obtained from the linear equation with external input $\mathcal{J}$.

**Theorem 8.3** (Mischler, Quiñinao, Touboul, [183]). For any $\rho^0 \in L^1(\omega) \cap L^1 \log L^1 \cap \mathcal{P}(\mathbb{R}^2)$, there exists a unique global-in-time weak solution $\rho$ to (FhN). It depends continuously on $\rho^0$ in $L^1(\omega^{1/2})$ and satisfies for some constant $C_0 > 0$,

$$ \|\rho(\cdot, \cdot, t)\|_{L^1(\omega)} \le \max(C_0, \|\rho^0\|_{L^1(\omega)}). $$

If the initial condition enjoys the higher decay rate $\rho^0 \in L^1(m)$, then for some constant $C_1 > 0$ depending on $\rho^0$,

$$ \|\rho(\cdot, \cdot, t)\|_{L^1(m)} \le C_1. $$

There exist norms $\|\cdot\|_{H^1}$ and $\|\cdot\|_{H_v^2}$ equivalent respectively to $\|\cdot\|_{H^1(m)}$ and $\|\cdot\|_{H_v^2(m)}$ such that if the initial condition enjoys the higher regularity $\rho^0 \in H^1(m)$ or $\rho^0 \in H_v^2(m)$, then for some constants $C_2, C_3 > 0$ depending on $\rho^0$,

$$ \|\rho(\cdot, \cdot, t)\|_{H^1(\omega)} \le C_2, \quad \text{or} \quad \|\rho(\cdot, \cdot, t)\|_{H_v^2(\omega)} \le C_3. $$

## 8.3 Long time behaviour of the mesoscopic system

Consider the following variant of the Brouwer fixed point theorem:

**Theorem 8.4** ([102], [119]). Let $\mathcal{Z}$ a convex and compact subset of a Banach space $\mathcal{X}$ and $(S_t)_{t \ge 0}$ a continuous semigroup on $\mathcal{Z}$. Assume $\mathcal{Z}$ is invariant under the action of $(S_t)_{t \ge 0}$. Then, there exists $z_0 \in \mathcal{Z}$ such that for any $t \ge 0$, $S_t z_0 = z_0$.
The article [183] uses this result with the space

$$ \mathcal{X} = H_v^2(m) \cap L^1 \log L^1 \cap \mathcal{P}(\mathbb{R}^2), $$

an appropriate $\mathcal{Z}$ and the semigroup generating the solutions of (FhN) to prove existence of at least one stationary solution to the nonlinear system.

**Theorem 8.5** (Mischler, Quiñinao, Touboul, [183]). For all $\varepsilon \ge 0$, there exists at least one stationary state $\rho_\infty \in H_v^2(m)$ to (FhN). When $\varepsilon \to 0$, it converges strongly in $L^2(m)$ towards the linear stationary state obtained for $\varepsilon = 0$.

**The linearised problem** Let $\rho_\infty$ and $\rho$ a stationary state and a solution of (FhN) and consider the variation $\rho = \rho_\infty + h$. Then the linearised operator for $h$ is

$$ \mathcal{L}_\varepsilon h = Q_{\mathcal{J}_\infty, \varepsilon} h + \varepsilon \mathcal{J}_h(t) \frac{\partial \rho_\infty}{\partial v}, $$

with $\mathcal{J}_h(t) = \int_{\mathbb{R}^2} v h(v, w, t) dv dw$. Similar to the method described<fnref n="9" /> in Subsection 4.6.1, the idea is to split the operator in two:

$$ A = N\chi_R, \quad B_\varepsilon = \mathcal{L}_\varepsilon - A, $$

with $N > 0$ a constant and $\chi$ a smooth cutoff function equal to 1 in $B(0, R)$ and supported in $B(0, 2R)$. The goal is to take advantage of the regularity of $A$ and the dissipativity of $B_\varepsilon$ to conclude that for small $\varepsilon$ the spectrum of $\mathcal{L}_\varepsilon$ is close to the spectrum of $B_\varepsilon$; [183] emphasises that this procedures corresponds to Weyl's theorem or to a generalised Krein-Rutman [185] (see also the survey [238]).

Recall the definition of hypodissipativity above (Definition 4.25).

**Lemma 8.6** (Mischler, Quiñinao, Touboul, [183]). For $N, R > 0$ large enough, $B_\varepsilon$ is (-1)-hypodissipative in $H_v^2(m)$.

This allows to perform a detailed spectral analysis in the disconnected case and to extend it to $\varepsilon \simeq 0$.

**Weakly-nonlinear case** When $\varepsilon$ is small enough, [183] proves that the stationary state is unique. Then, building upon the study of the linear spectrum, they proceed to characterise $\Sigma(\mathcal{L}_\varepsilon)$.

Coming back to the perturbation $h = \rho - \rho_\infty$, they write the following Duhamel formula:

$$ h = S_{\mathcal{L}_\varepsilon}(t)h^0 + \int_0^t S_{\mathcal{L}_\varepsilon}(t-s) \left( \varepsilon \mathcal{J}_h(t) \frac{\partial h}{\partial v} \right) ds. $$

Combining it with the spectral information, it is possible to prove the decay

$$ \|h\|_{L^2(m)} \le C e^{-\alpha t} \|h^0\|_{L^2(m)}, $$

for some $C \ge 1, \alpha > 0$.

**Theorem 8.7** (Mischler, Quiñinao, Touboul, [183]). There exists $\varepsilon_0 > 0$ such that for all $\varepsilon \in (0, \varepsilon_0)$, there exists a unique stationary state $\rho_\infty$ to (FhN) and $\eta(\varepsilon) > 0$ such that for all $\rho^0 \in H^1(m) \cap \mathcal{P}(\mathbb{R}^2)$, if $\|\rho^0 - \rho_\infty\|_{L^2(m)} \le \eta(\varepsilon)$, there exist $C \ge 1$ and $\nu > 0$ such that the solution to (FhN) satisfies

$$ \forall t \ge 0, \quad \|\rho(\cdot, \cdot, t) - \rho_\infty\|_{L^2(m)} \le C e^{-\nu t}. $$

Moreover, $\lim_{\varepsilon \to 0} \eta(\varepsilon) = +\infty$.

Numerical solutions on the particle system ($N = 2000$) in [183] confirm that for small $\varepsilon$ the unique stationary state is stable. For large $\varepsilon$, the system can have two stationary states and there is convergence towards one of them. In the intermediate connectivity regime, stable self-sustained oscillations are observed.

<footnote n="9">⁹Note that [187] was published two years later than [183] and precedes it only in our exposition.</footnote>
## 8.4 Periodic solutions arising through noise and interaction

As we just mentioned, numerical simulations in [183] displayed convergence towards a periodic solution in the intermediate connectivity regime. This behaviour was rigorously investigated in [169, 168] using probabilistic methods and noise-driven bifurcations. They study, very similar to the mean-field system (FhN), the system of stochastic differential equations

$$ \begin{cases} dV_t = \delta \left( N(V_t) - W_t - \frac{K}{\delta} (V_t - \mathbb{E}[V_t]) \right) dt + \sqrt{2\sigma} dB_t, \\ dW_t = \frac{\delta}{c} (V_t + a - bW_t) dt, \end{cases} \qquad (8.3) $$

with $b, c, \delta, K$ positive parameters, $a \in \mathbb{R}$ and $N(v) = v - \frac{v^3}{3}$. The main idea is to look at the small $\delta$ range, where the term $V_t - \mathbb{E}[V_t]$ is close to 0 and thus the density in $v$ is close to a Gaussian. It allows [169] to perform a slow-fast perturbative analysis. In a parameter range where the system without noise ($\sigma = 0$) nor interactions ($K = 0$) would converge to equilibrium, the noise-alone could send the individual neuron dynamics (no interaction) towards a periodic Carnard-like dynamics. Strong interaction ($K \neq 0$ and $\delta$ small) then allow these individual oscillations to happen collectively.

The slow-fast dynamics approach proceeds by replacing, for small enough $\delta$, the voltage variable by a random variable of Gaussian distribution $N(\mathbb{E}[V_t], \frac{\sigma^2}{K})$. The dynamics of $W_t$ being linear in $V_t$, it will also be Gaussian at first order. The couple $V_t, W_t$ will then behave like $N(m_t, \Sigma_\delta)$, and an evolution equation can be written for $m_t$ in the form

$$ \frac{d}{dt} m_t = \delta F_{\frac{\sigma^2}{K}}(m_t), \quad F_u(v, w) = \begin{pmatrix} (1-u)v - \frac{v^3}{3} - w \\ \frac{1}{c}(v+a-bw) \end{pmatrix}, $$

which is also of FitzHug-Nagumo type. The covariance matrix $\Gamma_\delta$ can be characterized as

$$ \Gamma_{\delta} = \frac{\sigma^2}{K} \begin{pmatrix} 1 & \frac{\delta}{Kc+b\delta} \\ \frac{\delta}{Kc+b\delta} & \frac{Kc+b\delta}{b(Kc+b\delta)} \end{pmatrix}. $$

Making this analysis rigorous, [169] proves that the nonlinear mean-field system (8.3) behaves close enough to the system $(v, w) = F_u(v, w)$, which, for a suitable choice of parameters, has a stable fixed point for $u = 0$ and a stable periodic solution for some range of the parameter $u$. Because $u = \frac{\sigma^2}{K}$ in the equation for $m_t$, the mean-field oscillations are induced by both noise and interaction.

**Theorem 8.8 (Luçon, Poquet [169]).** There exist a choice for $a, b, c, K, \sigma$ such that for all $\delta$, the dynamical system $\frac{1}{\delta}(v'(t), w'(t)) = F_{\frac{\sigma^2}{K}}(v(t), w(t))$ admits a stable limit cycle. There exists $\delta^*$ such that for all $\delta \in (0, \delta^*)$, there exists a periodic solution $\rho^{\text{per}}$ to (8.3). This solution is associated to an invariant limit cycle $C_\delta$ and there exist $C_1, C_2$ independent of $\delta$, $C(\delta)$, $\lambda(\delta)$ such that if $\rho^0 \in \mathcal{P}_2(\mathbb{R}^2)$, $\int_{\mathbb{R}^2} \|(v, w)\|^6 d\rho^0 \le C_1$ and $W_2(\rho^0, C_\delta) \le C_2\delta$, then the solution $\rho$ to (8.3) satisfies

$$ W_2(\rho_t, C_\delta) \le C(\delta)e^{-\lambda(\delta)t}W_2(\rho^0, C_\delta). $$

They obtain in the process that $C(\delta) \to +\infty$ and $\lambda(\delta) \to \min(\lambda(1), \frac{b}{c})$ when $\delta \to 0$.

## 8.5 A spatially extended kinetic FitzHugh-Nagumo model and its macroscopic limit

From the same base used in the model (FhN), it is possible to provide a mesoscopic description of a brain area while also taking into account the spatial structure. As we shall see, this allows to rigorously construct macroscopic equations.
The article [10] considers a system of $n$ interacting FitzHug-Nagumo neurons $((V_t^i, W_t^i, x_i))_{1\le i\le n}$ subject to independent Brownian noises at the voltage level,

$$ egin{cases} 
dV_t^i = (N(V_t^i) - W_t^i + \frac{1}{n}\sum_{j=1}^n \Phi(x_i, x_j)(V_t^i - V_t^j))dt + \sigma dB_t^i, \\ 
dW_t^i = (aV_t^i - bW_t^i + c)dt, 
\end{cases} $$

with $c \in \mathbb{R}$, $a, b > 0$ and again for simplicity $\sigma^2 = 2$. Note that the case $\sigma = 0$ has also been thoroughly investigated [75, 73, 74]. Each neuron is described at time $t$ by a voltage $V_t^i$, an adaptation variable $W_t^i$ and a position $x_i \in K$ lying in a compact set $K$. The coupling function $\Phi(x, y)$ describes the strength of interactions between points $x$ and $y$. Following [12, 170, 219], the coupling function is composed of strong local connections and weaker long range interactions

$$ \Phi(x, y) = \frac{1}{\varepsilon} \delta_{x=y} + \Psi(x, y). \qquad (8.4) $$

In the formal limit $n \to +\infty$, the system is described by the probability density $\rho(x, v, w, t)$ that we will often write as $\rho(x, u, t)$ with $u = (v, w)$ for clarity.

$$ egin{cases} 
\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial w}((av - bw + c)\rho) + \frac{\partial}{\partial v}((N(v) - w - \mathcal{J}(x, v, t))\rho) - \frac{\partial^2 \rho}{\partial v^2} = 0, \\ 
\mathcal{J}(x, v, t) = \int_K \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \Phi(x, x')(v - v')\rho(x', v', w', t)dv'dw'dx', \\ 
\rho(x, v, w, 0) = \rho^0(x, v, w) \ge 0, \quad \int_K \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \rho^0(x, v, w)dv dw dx = 1. 
\end{cases} 
\qquad (\text{se-FhN}) $$

Given the choice (8.4), consider the following parameter assumptions.

**Assumption 8.9 (Regularity of parameters).** Assume $N \in C^2(\mathbb{R})$, $\Psi \in C(K, L^1(K))$ and there exists $p \ge 2$ and $r \in (1, +\infty]$ such that

$$ \limsup_{|v| \to +\infty} \frac{N(v)}{v} = -\infty, \quad \sup_{|v| \ge 1} \frac{|N(v)|}{|v|^p} + \sup_{x' \in K} \int_K \Psi(x, x) dx + \sup_{x \in K} \int_K |\Psi(x, x')|^r dx' < +\infty. $$

The articles [10, 8, 9] investigate the limit of strong local interactions $\varepsilon \to 0$ in (8.4), the main idea being that the density should concentrate on a particular time-dependent voltage $V(t, x)$ solution to a macroscopic model. In view of this mesoscopic to macroscopic limit, consider the density $\rho^\varepsilon$ obtained as a solution to (se-FhN) from an initial condition $\rho^{0,\varepsilon}$ with the interaction function as in (8.4). Then, renormalise $\rho$ by the quantity of neurons $m^\varepsilon(x)$ at location $x$ and define the macroscopic voltage and adaptation variables $V^\varepsilon, W^\varepsilon$:

$$ egin{aligned} m^\varepsilon(x) &= \int_{\mathbb{R}^2} \rho^{0,\varepsilon}(x, u) du, \quad m^\varepsilon(x) \mu_{t,x}^\varepsilon = \rho(x, \cdot, t), \quad \bar{\mu}_{t,x}^\varepsilon = \int_{-\infty}^{+\infty} \mu_{t,x}^\varepsilon(v, \cdot) dv \\ m^\varepsilon(x) V^\varepsilon(t, x) &= \int_{\mathbb{R}^2} v \rho^\varepsilon(x, u, t) du, \quad m^\varepsilon(x) W^\varepsilon(t, x) = \int_{\mathbb{R}^2} w \rho^\varepsilon(x, u, t) du. \end{aligned} $$

The system (se-FhN) can then be rewritten as

$$ egin{cases} 
\frac{\partial \mu^\varepsilon}{\partial t} + \nabla_u \cdot \left[ \begin{pmatrix} N(v) - w - \mathcal{J} \\ av - bw + c \end{pmatrix} \mu^\varepsilon \right] - \frac{\partial^2 \mu^\varepsilon}{\partial v^2} = \frac{1}{\varepsilon} m^\varepsilon(x) \frac{\partial}{\partial v} \left[ (v - V^\varepsilon) \mu^\varepsilon \right], \\ 
\mathcal{J}(x, v, t) = \int_K \Psi(x, x') m^\varepsilon(x) (v - V^\varepsilon(t, x)) dx'. 
\end{cases} 
\qquad (8.5) $$

Because of the leading order term in $1/\varepsilon$, it is natural to expect that in the strong local interaction limit $(v - V^\varepsilon)\mu^\varepsilon \to 0$. To better quantity this concentration, [10] uses the $W_2$ Wasserstein distance over
the space $\mathcal{P}_2(\mathbb{R}^2)$ of probability measures with finite second moment and they find that the measure $\mu^\epsilon$ concentrates in the $v$ variable as a Dirac mass centred at the macroscopic variable $\mathcal{V}(x, t)$ and in the $w$ variable as the continuous marginal $\bar{\mu}$. More precisely, they obtain an estimate of the form

$$W_2(\mu, \delta_{\mathcal{V}^\epsilon} \otimes \bar{\mu}) \sim \sqrt{\epsilon}.$$

The dynamics is then asymptotically reduced to a macroscopic model with a voltage $\mathcal{V}(t, x)$ and an adaptation variable $W(t, x) = \int w d\bar{\mu}$ defined through the average of the limit marginal $\bar{\mu}$. The long-range interaction through the kernel $\Psi$ becomes the macroscopic convolution operator

$$\mathcal{L}_m[\mathcal{V}] = \int_K \Psi(x, x')m(x')(\mathcal{V}(x) - \mathcal{V}(x'))dx.$$

The macroscopic system is then of the form

$$\left\{
\begin{array}{l}
\displaystyle \frac{\partial \mathcal{V}}{\partial t} = N(\mathcal{V}(t, x)) - W(t, x) - \mathcal{L}_m[\mathcal{V}], \\[1.5ex]
\displaystyle \frac{\partial \bar{\mu}}{\partial t} + \frac{\partial}{\partial w}[(a\mathcal{V}(t, x) - bW(t, x) + c)\bar{\mu}] = 0, \\[1.5ex]
\displaystyle W(t, x) = \int_{-\infty}^{+\infty} w d\bar{\mu}_{t,x}(w).
\end{array}
\right.
\qquad (8.6)$$

Consider the following assumptions on initial data.

**Assumption 8.10 (Initial conditions).** For all $\epsilon > 0$, assume $x \mapsto \mu_{0,x}^\epsilon \in C(K, \mathcal{P}_2(\mathbb{R}^2))$, $m^\epsilon \in C(K)$ and there exist $p \ge 2$, $r \in (1, +\infty]$ of conjugate exponent $r'$ and $C_p, \bar{C}_p > 0$ independent of $\epsilon$ such that

$$m^* \le m^\epsilon(x) \le \frac{1}{m^*}, \quad \sup_{x \in K} \int_{\mathbb{R}^2} |u|^{2p} d\mu_{0,x}^\epsilon(u) \le C_p, \quad \int_{K \times \mathbb{R}^2} |u|^{2pr'} m^\epsilon(x) d\mu_{0,x}^\epsilon(u) \le \bar{C}_p,$$

and that there exist $c_1^\epsilon, c_2^\epsilon > 0$ such that

$$\sup_{x \in K} \int_{\mathbb{R}^2} e^{\frac{|u|^2}{2}} d\mu_{0,x}^{\epsilon}(u) \le c_1^{\epsilon}, \quad \sup_{x \in K} \int_{\mathbb{R}^2} \log(\mu_{0,x}^{\epsilon}) d\mu_{0,x}^{\epsilon}(u) \le c_2^{\epsilon}, \quad \sup_{x \in K} \left\| \nabla_u \sqrt{\mu_{0,x}^{\epsilon}} \right\|_{L^2(\mathbb{R}^2)}^2 \le c_2^{\epsilon}.$$

Then both the mesoscopic and macroscopic problems are well-posed.

**Theorem 8.11 (Blaustein, Filbet, [10]).** *Grant assumptions 8.9 and 8.10. There exists a unique weak solution*

$$\mu^{\epsilon} \in C(\mathbb{R}_{+} \times K, L^{1}(\mathbb{R}^{2})) \cap L_{\text{loc}}^{\infty}(\mathbb{R}_{+} \times K, \mathcal{P}_{2}(\mathbb{R}^{2}))$$

to (8.5) and the macroscopic quantities satisfy $\mathcal{V}^\epsilon, \mathcal{V}^\epsilon \in C(\mathbb{R}_+ \times K)$. There exists a unique weak solution $\mathcal{V} \in C(\mathbb{R}_+ \times K)$, $\bar{m}u \in C(\mathbb{R}_+ \times K, L^1(\mathbb{R}))$ to (8.6).

Using this setting, it is possible to make the asymptotic reduction above rigorous. Rescale the densities $\mu^\epsilon$ and $\bar{\mu}^\epsilon$ as

$$\mu_{t,x}^{\epsilon}(v, w) = \frac{1}{\epsilon^{\alpha}} \nu_{t,x}^{\epsilon} \left( \frac{1}{\epsilon^{\alpha}}(v - \mathcal{V}^{\epsilon}), w - \mathcal{W}^{\epsilon} \right), \quad \bar{\nu}_{t,x}^{\epsilon}(w) = \int_{-\infty}^{+\infty} \nu_{t,x}^{\epsilon}(v, w) dv,$$

with $\epsilon^\alpha$ a concentration profile. Inputing this change of variables in (8.5) allows to find that the correct profile should be with $\alpha = \frac{1}{2}$. Then it can be proved that

$$\nu^{\epsilon} = \mathcal{M}_{m^{\epsilon}, 0} \otimes \bar{\nu}^{\epsilon} + O(\sqrt{\epsilon}), \quad \mathcal{M}_{m^{\epsilon}(x), \mathcal{V}(x)}(v) = \sqrt{\frac{m^{\epsilon}(x)}{2\pi}} e^{-\frac{m^{\epsilon}(x)|v-\mathcal{V}(x)|^2}{2}}.$$
**Theorem 8.12** (Blaustein, Filbet, [10]). Grant assumptions 8.9 and 8.10. Consider the solutions $\mu^\epsilon$ to (8.5) and $(\mathcal{V}, \bar{\mu})$ to (8.6). Define the macroscopic and mesoscopic errors

$$ \mathcal{E}_{\text{mac}} = \|(\mathcal{V}^0, \mathcal{W}^0) - (\mathcal{V}^{0,\epsilon}, \mathcal{W}^{0,\epsilon})\|_{L^\infty(K)} + \|m - m^\epsilon\|_{L^\infty(K)}, \quad \mathcal{E}_{\text{mes}} = \sup_{x \in K} W_2(\bar{\nu}_{0,x}^\epsilon, \bar{\nu}_{0,x}). $$

Then there exists $C, \epsilon_0 > 0$ such that for all $0 < \epsilon \le \epsilon_0$ and all $t, x \in \mathbb{R}_+ \times K$,

$$ W_2(\mu^\epsilon, \mathcal{M}_{\frac{1}{\epsilon}m, \mathcal{V}} \otimes \bar{\mu}) \le C(\min((\mathcal{E}_{\text{mac}} + \epsilon)e^{Ct}, 1) + \mathcal{E}_{\text{mes}}e^{-bt} + e^{-\frac{t}{\epsilon}\rho}). \quad (8.7) $$

If $\mathcal{E}_{\text{mac}} + \mathcal{E}_{\text{mes}} = O(\epsilon)$, the right-hand side in (8.7) can be improved into $C(\min(\epsilon e^{Ct}, 1) + e^{-\frac{t}{\epsilon}\rho})$ and uniformly on all time compacts,

$$ \lim_{\epsilon \to 0} \sup_{x \in K} (|\mathcal{V}^\epsilon(t, x) - \mathcal{V}(t, x) + W_2(\bar{\mu}_{t,x}^\epsilon, \bar{\mu}_{t,x})) = 0, \quad \lim_{\epsilon \to 0} \sup_{x \in K} W_2(\mu_{t,x}^\epsilon, \delta_{\mathcal{V}(t,x)} \otimes \bar{\mu}_{t,x}) = 0, $$

with convergence rates explicitly given in terms of $t, m^*, \epsilon$ and universal constants not depending on $t, \epsilon$. Refined estimates on the Gaussian profile during the $\epsilon \to 0$ blow-up are provided in [8, 9], the former proving estimates in a $L^1$ and a weighted $L^2$ framework, the latter using a Hopf-Cole approach to derive $L^\infty$ bounds.

## Acknowledgements

JAC was supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 883363). JAC and PR want to thank Westlake University, Hangzhou, China, where a significant part of this review article was written, for its warm hospitality. JAC and PR thank Zhennan Zhou and Alejandro Ramos-Lora for valuable feedback on the manuscript.

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