# Appendix: Numerical methods

We now present the numerical schemes used in the simulations of previous chapters. We focus on adaptations of the well-known upwind scheme, which is widely applied in advection problems. For further reference on numerical methods for hyperbolic equations, see Bouchut [4], Godlewski et al. [32] and LeVeque [39].

## Classical elapsed time model

We start with the numerical scheme corresponding to the classical elapsed time Equation (1). We truncate the domain for the elapsed time s to the interval $[0, L]$ with $L > 0$ and we consider a uniformly spaced grid

$$s_i = i\Delta s, \quad \text{for } i = 1, \dots, M.$$

The initial data is given by the array $(n^0)_{i=0}^M$. The iterations in time are given by

$$t_k = k\Delta t, \quad k > 0$$

and we define $n_i^k \approx n(t_k, s_i)$ as the numerical solution, with $n_0^k = N^k \approx N(t_k)$ the approximate activity. From the explicit upwind scheme we get the discretization

$$\frac{n_i^{k+1} - n_i^k}{\Delta t} + \frac{n_i^k - n_{i-1}^k}{\Delta s} + p(s_i, N^{k-1})n_i^k = 0. \quad (67)$$

so that the numerical scheme is as follows.

<caption>**Algorithm 1** Explicit upwind scheme for classical elapsed time equation</caption>

N^{-1} \leftarrow \mathbf{solve} \left( y = \Delta s \sum_{i=1}^{M} p(s_i, y)n_i^0 \right) \quad \triangleright \text{Initial guess for } N(0)
\text{for } k = 0: k_{\max} \text{ do}
\quad n_i^{k+1} \leftarrow n_i^k - \frac{\Delta t}{\Delta s}(n_i^k - n_{i-1}^k) - \Delta t p(s_i, N^{k-1})n_i^k \quad \triangleright \text{Update } n
\quad N^k \leftarrow \Delta s \sum_{i=1}^{M} p(s_i, N^{k-1})n_i^k + n_M^k \quad \triangleright \text{Update } N
\text{end for}

An important property of this numerical scheme, which is easily verified, is the following.

**Lemma 12.** *The numerical Scheme 1 is mass-conservative i.e.*

$$\Delta s \sum_{i=1}^{M} (n_i^{k+1} - n_i^k) = 0.$$
In order to have this property, we added the term $n_M^k$ in the formula for computing $N^k$. Thus we need to assure that $n_M^k$ is small enough, which means that we need $L$ large enough and assume that solution $n$ vanishes at infinity.

Moreover, note that for all $t \ge 0$ the activity $N(t)$ satisfies the fixed point equation

$$y = \int_{0}^{\infty} p(s, y)n(t, s) ds. \qquad (68)$$

The fixed point Equation (68) can be solved with Newton's method or gradient descend and this allows to compute in particular the initial guess for $N(0)$. In particular, the non linear equation for $N(0)$ may have multiple solutions that determine different branches of solutions for the elapsed time Equation (1). The argument of fixed point may be extended for each iteration in time, which inspires the following variant of the numerical scheme

<caption>**Algorithm 2** Variant upwind scheme for classical elapsed time equation</caption>

N⁻¹ ← **solve** (y = Δs Σ<sub>i=1</sub><sup>M</sup> p(sᵢ, y)nᵢ⁰)
    ▷ Calculate N(0)
    **for** k = 0: k<sub>max</sub> **do**
        nᵢ<sup>k+1</sup> ← nᵢ⁰ - Δt/Δs(nᵢ⁰ - nᵢ<sup>k-1</sup>) - Δt p(sᵢ, N<sup>k</sup>)nᵢ⁰
        ▷ Update n
        nᵢ<sup>k+1</sup> ← (nᵢ<sup>k+1</sup>/Σ<sub>j=1</sub><sup>M</sup> n<sub>j</sub><sup>k+1</sup>)
        ▷ Normalization
    N<sup>k+1</sup> ← **solve** (y = Δs Σ<sub>i=1</sub><sup>M</sup> p(sᵢ, y)nᵢ<sup>k+1</sup>)
    ▷ Update N
    **end for**

This scheme is more accurate in computing $N(t)$ and no condition on $n_M^k$ is required, but we have to normalize the solution for $n$ at each iteration, so that we still need $L$ large enough. Moreover, we may consider Scheme 2 with different types of discretizations for the corresponding transport equation, but we are still forced to normalize at each iteration.

## Elapsed time model with two times

We now present the numerical methods for the multiple time renewal Equation (2.1) by extending the ideas of the classical elapsed time model. We truncate the values of $(s, a)$ to the set $[0, L] \times [0, L]$ with $L > 0$ and we consider a uniformly spaced grid

$$(s_i, a_j) = (i\Delta s, j\Delta a) \quad \text{for } 1 \le i \le j \le M, \quad \Delta s = \Delta a.$$

The initial data is given by the matrix $(n^0)_{i,j}$ of size $M \times M$. From the domain $\mathcal{D}$ we impose the following support conditions

$$\begin{cases} n_{i,j}^0 = 0 & \text{if } j \le i - 1, \\ n_{i,i-1}^0 = n_{i,M}^0 & \text{for } i = 1, \dots, M. \end{cases} \qquad (69)$$

The iterations in time are given by

$$t_k = k\Delta t, \quad k > 0,$$

and we define $n_{i,j}^k \approx n(t_k, s_i, a_j)$ as the numerical solution, with $n_{0,j}^k = N_j^k \approx N(t_k, a_j)$ the approximate discharging flux and $X^k \approx X(t_k)$ the approximate activity.
From the explicit upwind scheme we get the discretization

$$ \frac{n_{i,j}^{k+1} - n_{i,j}^{k}}{\Delta t} + \frac{n_{i,j}^{k} - n_{i-1,j}^{k}}{\Delta s} + \frac{n_{i,j}^{k} - n_{i,j-1}^{k}}{\Delta a} + p(s_i, N^{k-1}) n_i^k = 0, \quad (70) $$

so that the numerical scheme is as follows.

<caption>**Algorithm 3** Explicit upwind scheme for elapsed time equation with two times</caption>

X⁻¹ ← **solve** (y = ΔsΔa ∑<sub>i=1</sub><sup>M</sup> ∑<sub>j=i</sub><sup>M</sup> p(sᵢ, aⱼ, y)nᵢⱼ⁰) ▷ Initial guess for X(0)
N⁻¹ ← Δs ∑<sub>l=j</sub><sup>M</sup> p(aⱼ, s_l, X⁻¹)nⱼl⁰ ▷ Initial guess for N(0, a)
**for** k = 0: k<sub>max</sub> **do**
    **if** i ≤ j **then**
        nᵢⱼ<sup>k+1</sup> ← nᵢⱼ<sup>k</sup> - Δt/Δs(nᵢⱼ<sup>k</sup> - nᵢ⁻₁ⱼ<sup>k</sup>) - Δt/Δa(nᵢⱼ<sup>k</sup> - nᵢⱼ⁻₁<sup>k</sup>) - Δt p(sᵢ, X<sup>k-1</sup>)nᵢⱼ<sup>k</sup> ▷ Update n
    **else**
        nᵢⱼ<sup>k+1</sup> ← 0 ▷ Support condition
    **end if**
    nᵢᵢ⁻₁<sup>k+1</sup> ← nᵢ,ₘ<sup>k+1</sup>
    Nⱼ<sup>k</sup> ← Δs ∑<sub>l=j</sub><sup>M</sup> p(aⱼ, s_l, X<sup>k-1</sup>)nⱼl<sup>k</sup> + nⱼ,ⱼ<sup>k</sup> ▷ Update N
    X<sup>k</sup> ← Δa ∑<sub>j=1</sub><sup>M</sup> Nⱼ<sup>k</sup> ▷ Update X
**end for**

Analogously to Scheme 1, after a direct calculation we have following remarkable property.

**Lemma 13.** *The numerical Scheme 3 is mass-conservative i.e.*

$$ \Delta s \Delta a \sum_{i=1}^{M} \sum_{j=i}^{M} (n_{i,j}^{k+1} - n_{i,j}^{k}) = 0. $$

Like Scheme 1, we need to assure that $n_{i,j}^k$ is small enough i.e. the solution $n$ vanishes at the diagonal $\{(s, a) \in \mathcal{D}: s = a\}$.

In the particular case when the firing rate $p$ does not depend on the second elapsed time $a$, we can establish a relation between Schemes 1 and 3.

**Proposition 3.** Assume that $p = p(s, X)$ and let $n_{i,j}^k$ be a numerical solution for Scheme 3, then $\bar{n}_i^k := \Delta a \sum_{j=i}^M n_{i,j}^k$ is a numerical solution for Scheme 1.

As in the classical elapsed time equation, we note that for all $t \ge 0$ the total activity $X(t)$ satisfies the fixed point equation

$$ y = \int_{0}^{\infty} \int_{s}^{\infty} p(s, a, y) n(t, s, a) da ds. \quad (71) $$

Like Equation (68), this equation may have different solutions for $X(0)$, which determines different branches of solution for the extended system. Similarly to Scheme 2, we may adapt the fixed point argument to the extended model to get the numerical Scheme 4.

Like the variant Scheme 2, this scheme is more accurate in computing $X(t)$ and no condition on the diagonal and sub-diagonal are required. However, this is at the cost of solving the fixed point Equation 71, which is more expensive computationally, and normalizing the solution for $n$ at each iteration.
<caption>**Algorithm 4** Variant upwind scheme for elapsed time equation with two times</caption>

```
X^0 ← solve (y = ΔsΔa ∑∑_{i=1}^M ∑_{j=i}^M p(s_i, a_j, y)n_{i,j}^0)
▷ Calculate X(0)
N_j^0 ← Δs ∑_{l=j}^M p(a_j, s_l, X^0)n_{jl}^0
▷ Calculate N(0, a)
for k = 0: k_max do
    if i ≤ j then
        n_{i,j}^{k+1} ← n_{i,j}^k - Δt/(Δs)(n_{i,j}^k - n_{i-1,j}^k) - Δt/(Δa)(n_{i,j}^k - n_{i,j-1}^k) - Δt p(s_i, X^k)n_{i,j}^k
        ▷ Update n
    else
        n_{i,j}^{k+1} ← 0
        ▷ Support condition
end if
n_{i,j}^{k+1} ← (n_{i,j}^{k+1} / ΔsΔa ∑_{i=1}^M ∑_{j=i}^M n_{j,i}^{k+1})
▷ Normalization
X^{k+1} ← solve (y = ΔsΔa ∑_{i=1}^M ∑_{j=i}^M p(s_i, a_j, y)n_{i,j}^{k+1})
▷ Update X
N_j^{k+1} ← Δs ∑_{l=j}^M p(a_j, s_l, X^{k+1})n_{jl}^{k+1}
▷ Update N
end for
```

## Elapsed time model with learning

We now present the numerical methods for the elapsed time system with spatial dependence and learning (3.2). We truncate the domain for the elapsed time s to the interval $[0, L]$ with $L > 0$ and the domain for the position x is $\Omega = [0, a]$. We consider the respective uniformly spaced grids for $[0, L] \times \Omega$ and $\Omega \times \Omega$

$$ 
\begin{align*}
(s_i, x_j) &= (i\Delta s, j\Delta x) & \text{for } i = 1, \dots, M, j = 1, \dots, P. \\
(x_j, y_l) &= (i\Delta x, l\Delta x) & \text{for } i, j = 1, \dots, P.
\end{align*}
$$

The initial probability density is $(n_0)_{i,j}$, which a matrix of size $M \times P$, the initial connectivity kernel is $(w_0)_{j,l}$, which is a matrix of size $M \times M$, and $I_j := I(x_j)$ is the given external input. The iterations in time are given by

$$ t_k = k\Delta t, \quad k > 0. $$

and we define $n_{i,j}^k \approx n(t_k, s_i, x_j)$ as the numerical solution, $n_{0,j}^k = N_j^k \approx N(t_k, x_j)$ the approximate activity, $S_j^k \approx S(t_k, x_j)$ the amplitude of stimulation and $w_{j,l}^k \approx w(t_k, x_j, y_l)$ the connectivity kernel.

The idea is to solve the transport equation for each position $x_j$, adapting Scheme 2. From the explicit upwind scheme we get the following discretization for $n$

$$ \frac{n_{i,j}^{k+1} - n_{i,j}^{k}}{\Delta t} + \frac{n_{i,j}^{k} - n_{i-1,j}^{k}}{\Delta s} + p(s_i, S_j^k) n_{i,j}^{k} = 0 \quad (72) $$

and we consider the following discretization for $w$

$$ \frac{w_{j,l}^{k+1} - w_{j,l}^{k}}{\Delta t} = -w_{j,l}^{k+1} + \gamma G(N_j^k, N_l^k), \quad (73) $$

so that we get the numerical Scheme 5.

In order to compute the initial guess for $S(0, x)$, we use the fact that for all $t \ge 0$ the amplitude of stimulation satisfies the following integral equation
<caption>**Algorithm 5** Upwind scheme for elapsed time equation with learning</caption>

(S⁻¹)ₚ =1 ← **solve** (Yₗ = ΔsΔx ∑ₘ₁ =1 ∑ₙ₁ =1 p(sᵢ, Yₙ) wᵣₗ,₁ nᵢ,₁ =1 + Iₗ) ▷ Initial guess for S(0, x)
Nⱗ⁻¹ ← Δs ∑ₘ₁ =1 p(sᵢ, Sⱗ₀) nᵢ,₁ =1
**for** k = 0: kmax **do**
&nbsp;&nbsp;&nbsp;&nbsp;nᵢ,₁ =1 ← nᵢ,₁ =1 − Δt/Δs (nᵢ,₁ =1 − nᵢ−₁,₁) − Δt p(sᵢ, Sⱗᵏ⁻¹) nᵢ,₁ =1 ▷ Update n
&nbsp;&nbsp;&nbsp;&nbsp;Nⱗ =1 ← Δs ∑ₘ₁ =1 p(sᵢ, Sⱗᵏ⁻¹) nᵢ,₁ =1 + nᵢ,₁ =1 ▷ Update N
&nbsp;&nbsp;&nbsp;&nbsp;Sⱗ =1 ← Δx ∑ₙ₁ =1 wᵣₗ,₁ Nᵣ =1 + Iₗ ▷ Update S
&nbsp;&nbsp;&nbsp;&nbsp;wᵣ =1 ← 1/(1+Δt) (wᵣ =1,₁ + ΔtγG(Nⱗᵏ, Nᵣ =1)) ▷ Update w
**end for**

$$
S(t, x) = \int_0^\infty p(s, S(t, y))w(t, x, y)n(t, s, y) ds dy + I(x), \quad (74)
$$

so that we have to solve the corresponding system of equations to compute the vector S⁻¹ in Scheme 5. Furthermore, Equation (74) may have multiple solutions, which determine different branches of solutions. We may modify Scheme 5 by solving Equation (74) at each time iteration for computing S(t, x) and normalizing the solution n. However, solving a large nonlinear system of equations at each iteration is computationally expensive.

Another property of Scheme 5 is the conservation of mass like Scheme 1.

**Lemma 14.** The numerical scheme 5 is mass-conservative i.e.

$$
\Delta s \sum_{i=1}^{M} (n_{i,j}^{k+1} - n_{i,j}^{k}) = 0, \quad \forall j = 1, \dots, P.
$$

Like Scheme 1, we need to assure that $n_{M,j}^k$ is small enough for all $j = 1, \dots, P$. We may avoid this condition in the following numerical scheme at the cost normalizing the solution $n$ with respect to elapsed time variable $s$ at each time iteration, as we see in Scheme 6.

<caption>**Algorithm 6** Variant upwind scheme for elapsed time equation with learning</caption>

g<sub>j</sub> &larr; &Delta;s &sum;<sub>i=1</sub><sup>M</sup> n<sub>i,j</sub><sup>0</sup> &larr; Calculate g(x) = &int; n<sup>0</sup>(s, x) ds
(S<sup>0</sup>)<sub>j=1</sub><sup>P</sup> &larr; solve (Y<sub>j</sub> = &Delta;s&Delta;x &sum;<sub>i=1</sub><sup>M</sup> &sum;<sub>l=1</sub><sup>P</sup> p(s<sub>i</sub>, Y<sub>l</sub>) w<sub>j,l</sub><sup>0</sup> n<sub>i,l</sub><sup>0</sup> + I<sub>j</sub>) &larr; Calculate S(0, x)
N<sub>j</sub><sup>0</sup> &larr; &Delta;s &sum;<sub>i=1</sub><sup>M</sup> p(s<sub>i</sub>, S<sub>j</sub><sup>0</sup>)n<sub>i,j</sub><sup>0</sup> &larr; Calculate N(0, x)
for k = 0: k<sub>max</sub> do
n<sub>i,j</sub><sup>k+1</sup> &larr; n<sub>i,j</sub><sup>k</sup> - Δt/Δs (n<sub>i,j</sub><sup>k</sup> - n<sub>i-1,j</sub><sup>k</sup>) - Δt p(s<sub>i</sub>, S<sub>j</sub><sup>k</sup>)n<sub>i,j</sub><sup>k</sup> &larr; Update n
n<sub>i,j</sub><sup>k+1</sup> &larr; Δs &sum;<sub>i=1</sub><sup>M</sup> n<sub>i,j</sub><sup>k+1</sup> n<sub>i,j</sub><sup>k+1</sup> &larr; Normalization
w<sub>j,l</sub><sup>k+1</sup> &larr; 1/(1+Δt) (w<sub>j,l</sub><sup>k</sup> + ΔtγG(N<sub>j</sub><sup>k</sup>, N<sub>l</sub><sup>k</sup>)) &larr; Update w
N<sub>j</sub><sup>k+1</sup> &larr; &Delta;s &sum;<sub>i=1</sub><sup>M</sup> p(s<sub>i</sub>, S<sub>j</sub><sup>k</sup>)n<sub>i,j</sub><sup>k+1</sup> &larr; Update N
S<sub>j</sub><sup>k+1</sup> &larr; Δx &sum;<sub>l=1</sub><sup>M</sup> w<sub>j,l</sub><sup>k+1</sup> N<sub>l</sub><sup>k+1</sup> + I<sub>j</sub> &larr; Update S
end for

Alternatively, we may also consider Scheme 6 with different explicit and implicit numerical method to solve the transport equation for n.
Concerning the limit System (3.43) of slow learning dynamics with $\epsilon = 0$, we can get explicitly the solution for probability density $n$ in terms of the activity and the amplitude of stimulation since there is no transport equation to solve. Hence, we may adapt the numerical Method 6 to get the following scheme for the limit System (3.43).

<caption>**Algorithm 7** Variant upwind scheme for the limit system with $\epsilon = 0$</caption>

```
g_j ← Δs ∑<sub>i=1</sub><sup>M</sup> n<sup>0</sup><sub>i,j</sub> \quad ▷ Calculate g(s) = ∫ n<sup>0</sup>(s, x) ds
(S<sup>0</sup>)<sup>P</sup><sub>j=1</sub> ← solve (Y_j = ΔsΔx ∑<sub>i=1</sub><sup>M</sup> ∑<sub>l=1</sub><sup>P</sup> p(s<sub>i</sub>, Y_l) w<sup>0</sup><sub>j,l</sub> n<sup>0</sup><sub>i,l</sub> + I_j) \quad ▷ Calculate S(0, x)
N<sup>0</sup><sub>j</sub> ← Δs ∑<sub>i=1</sub><sup>M</sup> p(s<sub>i</sub>, S<sup>0</sup><sub>j</sub>)n<sup>0</sup><sub>i,j</sub> \quad ▷ Calculate N(0, x)
for k = 1: k<sub>max</sub> do
    n<sup>k+1</sup><sub>i,j</sub> ← N<sup>k</sup><sub>j</sub> exp(-Δs ∑<sub>l=1</sub><sup>i</sup> p(s<sub>l</sub>, S<sup>k</sup><sub>j</sub>)) \quad ▷ Update n
    n<sup>k+1</sup><sub>i,j</sub> ← (g<sub>j</sub> / Δs ∑<sub>i=1</sub><sup>M</sup> n<sup>k+1</sup><sub>i,j</sub>) n<sup>k+1</sup><sub>i,j</sub> \quad ▷ Normalization
    w<sup>k+1</sup><sub>j,l</sub> ← (1 / (1+Δt)) (w<sup>k</sup><sub>j,l</sub> + ΔtγG(N<sup>k</sup><sub>j</sub>, N<sup>k</sup><sub>l</sub>)) \quad ▷ Update w
    N<sup>k+1</sup><sub>j</sub> ← Δs ∑<sub>i=1</sub><sup>M</sup> p(s<sub>i</sub>, S<sup>k</sup><sub>j</sub>)n<sup>k+1</sup><sub>i,j</sub> \quad ▷ Update N
    S<sup>k+1</sup><sub>j</sub> ← Δx ∑<sub>l=1</sub><sup>M</sup> w<sup>k+1</sup><sub>j,l</sub> N<sup>k+1</sup><sub>l</sub> + I_j \quad ▷ Update S
end for
```
Bibliography

[1] A. Abbassian, M. Fotouhi, and M. Heidari. "Neural fields with fast learning dynamic kernel". In: *Biological cybernetics* 106.1 (2012), pp. 15–26.

[2] S.-i. Amari. "Dynamics of pattern formation in lateral-inhibition type neural fields". In: *Biological cybernetics* 27.2 (1977), pp. 77–87.

[3] V. Bansaye, B. Cloez, and P. Gabriel. "Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions". In: *Acta Applicandae Mathematicae* 166.1 (2020), pp. 29–72.

[4] F. Bouchut. *Nonlinear stability of finite Volume Methods for hyperbolic conservation laws: And Well-Balanced schemes for sources*. Springer Science & Business Media, 2004.

[5] H. Brezis. *Functional analysis, Sobolev spaces and partial differential equations*. Springer Science & Business Media, 2010.

[6] N. Brunel. "Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons". In: *Journal of computational neuroscience* 8.3 (2000), pp. 183–208.

[7] M. J. Cáceres, J. A. Carrillo, and B. Perthame. "Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states". In: *The Journal of Mathematical Neuroscience* 1.1 (2011), p. 7.

[8] J. A. Cañizo, P. Gabriel, and H. Yoldaş. "Spectral gap for the growth-fragmentation equation via Harris's Theorem". In: *arXiv preprint arXiv:2004.08343* (2020).

[9] J. A. Cañizo and H. Yoldaş. "Asymptotic behaviour of neuron population models structured by elapsed-time". In: *Nonlinearity* 32.2 (2019), p. 464.

[10] J. A. Carrillo, B. Perthame, D. Salort, and D. Smets. "Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience". In: *Nonlinearity* 28.9 (2015), p. 3365.

[11] J. Chevallier. "Mean-field limit of generalized Hawkes processes". In: *Stochastic Processes and their Applications* 127.12 (2017), pp. 3870–3912.

[12] J. Chevallier, M. J. Cáceres, M. Doumic, and P. Reynaud-Bouret. "Microscopic approach of a time elapsed neural model". In: *Mathematical Models and Methods in Applied Sciences* 25.14 (2015), pp. 2669–2719.

[13] J. Crevat. "Diffusive limit of a spatially-extended kinetic FitzHugh-Nagumo model". In: *arXiv preprint arXiv:1906.08073* (2019).

[14] J. Crevat. "Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network". In: (2019).
[15] J. Crevat, G. Faye, and F. Filbet. "Rigorous Derivation of the Nonlocal Reaction-Diffusion Fitzhugh-Nagumo System". In: *SIAM Journal on Mathematical Analysis* 51.1 (2019), pp. 346-373.

[16] O. Diekmann, S. A. Van Gils, S. M. V. Lunel, and H.-O. Walther. *Delay equations: functional-, complex-, and nonlinear analysis*. Vol. 110. Springer Science & Business Media, 2012.

[17] R. J. DiPerna and P.-L. Lions. "Ordinary differential equations, transport theory and Sobolev spaces". In: *Inventiones mathematicae* 98.3 (1989), pp. 511-547.

[18] W. Doblin. "Éléments d'une théorie générale des chaînes simples constantes de Markoff". In: *Annales Scientifiques de l'École Normale Supérieure*. Vol. 57. 1940, pp. 61-111.

[19] W. Doeblin. "Sur deux problèmes de M. Kolmogoroff concernant les chaînes dénombrables". In: *Bulletin de la Société Mathématique de France* 66 (1938), pp. 210-220.

[20] G. Dumont and P. Gabriel. "The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states". In: *Nonlinearity* 33.12 (2020), p. 6381.

[21] G. Dumont, J. Henry, and C. O. Tarniceriu. "A theoretical connection between the noisy leaky integrate-and-fire and the escape rate models: the non-autonomous case". In: *Math. Model. Nat. Phenom*. 15 (2020), Paper No. 59, 20.

[22] G. Dumont, J. Henry, and C. O. Tarniceriu. "Noisy threshold in neuronal models: connections with the noisy leaky integrate-and-fire model". In: *J. Math. Biol.* 73.6-7 (2016), pp. 1413-1436.

[23] O. Faugeras, T. Papadopoulo, J. Touboul, D. Talay, E. Tanre, and M. Bossy. "The statistics of spikes trains for some simple types of neuron models". PhD thesis. INRIA, 2007.

[24] G. Faye. "Existence and stability of traveling pulses in a neural field equation with synaptic depression". In: *SIAM Journal on Applied Dynamical Systems* 12.4 (2013), pp. 2032-2067.

[25] G. Faye and O. Faugeras. "Some theoretical and numerical results for delayed neural field equations". In: *Physica D: Nonlinear Phenomena* 239.9 (2010), pp. 561-578.

[26] G. Faye, J. Rankin, and P. Chossat. "Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis". In: *Journal of mathematical biology* 66.6 (2013), pp. 1303-1338.

[27] L. Ferretti et al. "Quantifying SARS-CoV-2 transmission suggests epidemic control with digital contact tracing". In: *Science* 368.6491 (2020).

[28] C. Fonte and V. Schmutz. "Long time behavior of an age and leaky memory-structured neuronal population equation". In: *arXiv preprint arXiv:2106.11110* (2021).

[29] N. Fournier and B. Perthame. "A non-expanding transport distance for some structured equations". In: *arXiv preprint arXiv:2102.04092* (2021).

[30] P. Gabriel. "Measure solutions to the conservative renewal equation". In: *ESAIM: Proceedings and Surveys* 62 (2018), pp. 68-78.

[31] W. Gerstner and W. M. Kistler. *Spiking neuron models: Single neurons, populations, plasticity*. Cambridge university press, 2002.

[32] E. Godlewski and P.-A. Raviart. *Numerical approximation of hyperbolic systems of conservation laws*. Vol. 118. Springer, 1996.

[33] P. Gwiazda and E. Wiedemann. "Generalized entropy method for the renewal equation with measure data". In: *Communications in Mathematical Sciences* 15.2 (2017), pp. 577-586.
[34] K.-P. Hadeler and J. Tomiuk. "Periodic solutions of difference-differential equations". In: *Arch. Rational Mech. Anal.* 65.1 (1977), pp. 87-95.

[35] M. Hairer and J. C. Mattingly. "Yet another look at Harris' ergodic theorem for Markov chains". In: *Seminar on Stochastic Analysis, Random Fields and Applications VI*. Springer. 2011, pp. 109-117.

[36] T. Harris. *The existence of stationary measures for certain Markov processes Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability 1954-1955* 2 Univ. 1956.

[37] D. Hebb. *The Organization of Behavior: a neuropsychological approach.* (1949).

[38] M.-J. Kang, B. Perthame, and D. Salort. "Dynamics of time elapsed inhomogeneous neuron network model". In: *Comptes Rendus Mathematique* 353.12 (2015), pp. 1111-1115.

[39] R. J. LeVeque and R. J. Leveque. *Numerical methods for conservation laws*. Vol. 132. Springer, 1992.

[40] J.-g. Liu, Z. Wang, Y. Zhang, and Z. Zhou. "Rigorous justification of the Fokker-Planck equations of neural networks based on an iteration perspective". In: *arXiv preprint arXiv:2005.08285* (2020).

[41] E. Löcherbach, V. Schmutz, and T. Schwalger. "On a finite-size neuronal population equation". In: *arXiv preprint arXiv:2106.14721* (2021).

[42] C. Ly and D. Tranchina. "Spike train statistics and dynamics with synaptic input from any renewal process: a population density approach". In: *Neural Computation* 21.2 (2009), pp. 360-396.

[43] P. Michel, S. Mischler, and B. Perthame. "General relative entropy inequality: an illustration on growth models". In: *Journal de mathématiques pures et appliquées* 84.9 (2005), pp. 1235-1260.

[44] S. Mischler, C. Quiñinao, and Q. Weng. "Weak and strong connectivity regimes for a general time elapsed neuron network model". In: *Journal of Statistical Physics* 173.1 (2018), pp. 77-98.

[45] S. Mischler and Q. Weng. "Relaxation in time elapsed neuron network models in the weak connectivity regime". In: *Acta Applicandae Mathematicae* 157.1 (2018), pp. 45-74.

[46] J. D. Murray. *Mathematical biology: I. An introduction*. Vol. 17. Springer Science & Business Media, 2007.

[47] K. Pakdaman, B. Perthame, and D. Salort. "Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation". In: *J. Math. Neurosci.* 4 (2014), Art. 14, 26.

[48] K. Pakdaman, B. Perthame, and D. Salort. "Dynamics of a structured neuron population". In: *Nonlinearity* 23.1 (2010), pp. 55-75.

[49] K. Pakdaman, B. Perthame, and D. Salort. "Relaxation and self-sustained oscillations in the time elapsed neuron network model". In: *SIAM J. Appl. Math.* 73.3 (2013), pp. 1260-1279.

[50] B. Perthame and D. Salort. "On a voltage-conductance kinetic system for integrate & fire neural networks". In: *Kinetic & Related Models* 6.4 (2013), pp. 841-864.

[51] B. Perthame. *Transport equations in biology*. Springer Science & Business Media, 2006.
[52] B. Perthame and D. Salort. "Derivation of a voltage density equation from a voltage-conductance kinetic model for networks of integrate-and-fire neurons". In: *Communications in Mathematical Sciences* 17.5 (2019), pp. 1193-1211.

[53] B. Perthame, D. Salort, and G. Wainrib. "Distributed synaptic weights in a LIF neural network and learning rules". In: *Physica D: Nonlinear Phenomena* 353 (2017), pp. 20-30.

[54] J. Pham, K. Pakdaman, J. Champagnat, and J.-F. Vibert. "Activity in sparsely connected excitatory neural networks: effect of connectivity". In: *Neural Networks* 11.3 (1998), pp. 415-434.

[55] R. Potthast. "The Amari Model in Neural Field Theory". In: *Encyclopedia of Computational Neuroscience*. Ed. by D. Jaeger and R. Jung. New York, NY: Springer New York, 2013, pp. 1-6.

[56] T. Schwalger and A. V. Chizhov. "Mind the last spike—firing rate models for mesoscopic populations of spiking neurons". In: *Current opinion in neurobiology* 58 (2019), pp. 155-166.

[57] H. L. Smith. *An introduction to delay differential equations with applications to the life sciences*. Vol. 57. Springer New York, 2011.

[58] N. Torres and D. Salort. "Dynamics of neural networks with elapsed time model and learning processes". In: *Acta Appl. Math.* 170 (2020), pp. 1065-1099.

[59] N. Torres, M. J. Cáceres, B. Perthame, and D. Salort. "An elapsed time model for strongly coupled inhibitory and excitatory neural networks". In: *Physica D: Nonlinear Phenomena* (2021), p. 132977.

[60] N. Torres, B. Perthame, and D. Salort. "A multiple time renewal equation for neural assemblies with elapsed time model". In: *arXiv preprint arXiv:2108.10577* (2021).

[61] S. K. Tumuluri. "Steady state analysis of a nonlinear renewal equation". In: *Mathematical and computer modelling* 53.7-8 (2011), pp. 1420-1435.

[62] H. R. Wilson and J. D. Cowan. "Excitatory and inhibitory interactions in localized populations of model neurons". In: *Biophysical journal* 12.1 (1972), pp. 1-24.
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ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF THE ELAPSED TIME MODEL FOR NEURAL ASSEMBLIES

Abstract

The elapsed time model has been widely studied in the context of mathematical neuroscience with many open questions left. The model consists of an age-structured equation that describes the dynamics of interacting neurons described by their refractory period, i.e. the elapsed time since their last discharge. Our interest lies in weakly connected networks where mathematical tools such as the Entropy method and Doeblin's theory have proved to be useful methods in proving convergence to steady state, and highly connected networks leading to strong nonlinearities where classical perturbation methods do not apply. Moreover, we are interested in extensions of the elapsed time model by incorporating new variables such as the elapsed time since penultimate discharge and the spatial dependence with learning processes via the change in the interconnections.

To deal with this problem of strong interconnections in the elapsed time model, we choose a particular case which can be reduced to delay differential equations and we prove a general convergence result to a stationary state in the inhibitory and the weakly excitatory cases. In addition, we prove the existence of particular periodic solutions with jump discontinuities in the strongly excitatory case.

In the first extension of the classical elapsed time model we incorporate the elapsed since the penultimate discharge, obtaining a more complex system of integro-differential equations. For this new system we prove convergence to stationary state by means of Doeblin's theory in the case of weak nonlinearities in an appropriate functional setting, inspired by the case of the classical elapsed time equation. We also aim to study how the incorporation of this new elapsed time may change the general dynamics of the system.

In the second extension we focus on incorporating the spatial dependence (e.g. position of neurons across the cortex) and some learning processes to the classical elapsed time model. The learning rules that we consider are essentially inspired from the Hebbian rule, which is a classical principle in modeling the process of learning. We then obtain a new system of integro-differential equations, from which we analyze the convergence to stationary states by the means of entropy method and Doeblin's theory in the case of weak interconnections. We also consider the situation where neural activity is faster than the learning process and give conditions where one can approximate the dynamics by a solution with a similar profile of a steady state.

For each model we present some numerical simulations to observe how the parameters of the system can give different behaviors and pattern formations in order to contrast them with the theoretical results.

**Keywords:** mathematical neuroscience, elapsed time, structured equations, neural networks, periodic solutions, delay differential equations, weak interconnections, learning rule, connectivity kernel, convergence to equilibrium, entropy method, doeblin theory.
Résumé

Le modèle du temps écoulé a été largement étudié dans le contexte des neurosciences mathématiques, mais de nombreuses questions restent ouvertes. Le modèle consiste en une équation structurée par âge qui décrit la dynamique des neurones en interaction décrite par leur période réfractaire, c'est-à-dire le temps écoulé depuis leur dernière décharge. Nous nous intéressons aux réseaux faiblement connectés pour lesquels des outils mathématiques tels que la méthode de l'entropie et la théorie de Doeblin se sont avérés utiles pour prouver la convergence vers un état stable, et aux réseaux hautement connectés conduisant à de fortes non-linéarités pour lesquels les méthodes classiques de perturbation ne s'appliquent pas. De plus, nous nous intéressons aux extensions du modèle de temps écoulé en incorporant de nouvelles variables telles que le temps écoulé depuis la pénultième décharge et la dépendance spatiale avec les processus d'apprentissage via le changement des interconnexions.

Pour traiter le problème des interconnexions fortes dans le modèle du temps écoulé, nous choisissons un cas particulier qui peut être réduit à des équations différentielles à retard et nous prouvons un résultat général de convergence vers un état stationnaire dans les cas inhibiteurs et faiblement excitateurs. En outre, nous prouvons l'existence de solutions périodiques particulières avec des discontinuités de saut dans le cas fortement excitateur.

Dans la première extension du modèle classique du temps écoulé, nous incorporons le temps écoulé depuis l'avant-dernière décharge, obtenant ainsi un système plus complexe d'équations intégro-différentielles. Pour ce nouveau système, nous prouvons la convergence vers l'état stationnaire au moyen de la théorie de Doeblin dans le cas de non-linéarités faibles dans un cadre fonctionnel approprié, inspiré du cas de l'équation classique du temps écoulé. Nous cherchons également à étudier comment l'incorporation de ce nouveau temps écoulé peut modifier la dynamique générale du système.

Dans la deuxième extension, nous nous concentrons sur l'incorporation de la dépendance spatiale (par exemple, la position des neurones dans le cortex) et de certains processus d'apprentissage au modèle classique du temps écoulé. Les règles d'apprentissage que nous considérons sont essentiellement inspirées de la règle de Hebb, qui est un principe classique dans la modélisation du processus d'apprentissage. Nous obtenons alors un nouveau système d'équations intégro-différentielles, dont nous analysons la convergence vers des états stationnaires au moyen de la méthode de l'entropie et de la théorie de Doeblin dans le cas d'interconnexions faibles. Nous considérons également la situation où l'activité neuronale est plus rapide que le processus d'apprentissage et nous donnons les conditions dans lesquelles on peut approximer la dynamique par une solution avec un profil similaire à un état stationnaire.

Pour chaque modèle, nous présentons quelques simulations numériques afin d'observer comment les paramètres du système peuvent donner des comportements et des formations de modèles différents afin de les contraster avec les résultats théoriques.

**Mots clés :** neurosciences mathématiques, temps écoulé, équations structurées, réseaux de neurones, solutions périodiques, équations différentielles à retard, interconnexions faibles, règle d'apprentissage, noyau de connectivité, convergence à l'équilibre, méthode d'entropie, théorie de doeblin.

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