A Microscopic Spiking Neuronal Network for the Age-Structured Model

Abstract

We introduce a microscopic spiking network consistent with the age-structured/renewal equation proposed by Pakdaman, Perthame and Salort. It is a jump process interacting through a set of global activity variables with random delays. We show the well-posedness of the particle system and the mean-field equation. Moreover, by studying the tightness of the empirical measure, we prove the propagation of chaos property. Eventually, we quantify the rate of convergence by using the coupling method.

Appendix:  General Theorems for Stochastic Processes

1.1 Remarks on the Proof of Lemma 13

Our first task is to prove that $(X_{t},M_{t})$ is a semi martingale. To do so, recall that at the end of the proof of Lemma 13, we got that for any $\varphi \in C_{b} ^{2}({\mathbb {R}}_{+}^{2})$, the process

$$\begin{aligned} \begin{aligned}[b] &\varphi (X_{t},M_{t})-\varphi (X_{0},M_{0}) \\ &\quad {}- \int_{0}^{t}\partial_{m} \varphi (X_{s},M_{s}) \biggl[-\alpha M_{s}+\alpha J \int_{0}^{s}{\mathbb {E}}\bigl[a(X _{s-s'},M_{s-s'}) \bigr]b\bigl(ds'\bigr) \biggr]\,ds \\ &\quad {}- \int_{0}^{t}\partial_{x}\varphi (X_{s},M_{s})\,ds- \int_{0}^{t} a(X _{s},M_{s}) \bigl(\varphi (0,M_{s})-\varphi (X_{s},M_{s}) \bigr)\,ds, \end{aligned} \end{aligned}$$

(31)

is a local martingale. Then, by using Jacod-Shiryaev [7, Theorem II.2.42, p. 86] adequately we can conclude. Indeed, the Theorem writes

Theorem 14

There is equivalence between:

• $(X_{t},M_{t})$ is a semimartingale, and it admits the characteristics $(B,0,\nu )$; i.e., $(X_{t},M_{t})$ writes

  $$\begin{aligned} (X_{t},M_{t})= (X_{0},M_{0})+{\mathcal {M}}^{c}+B, \end{aligned}$$

  where ${\mathcal {M}}^{c}$ is the continuous local martingale of the canonical decomposition, $B$ is predictable and $\nu$ is a predictable random measure on ${\mathbb {R}}_{+}\times {\mathbb {R}}_{+}^{2}$, namely the compensator of the random measure associated to the jumps of $X$.

• For each bounded function $\varphi \in C^{2}({\mathbb {R}}_{+}^{2})$, the process

  $$\begin{aligned} &\varphi (X_{t},M_{t})-\varphi (X_{0},M_{0}) - \int_{0}^{t}\partial _{x}\varphi (X_{s-},M_{s-})\, dB^{X}_{s}- \int_{0}^{t}\partial_{m} \varphi (X_{s-},M_{s-})\, dB^{M}_{s} \\ &\quad {}- \int_{0}^{t} \int_{{\mathbb {R}}_{+}^{2}} \bigl\{ \varphi (X_{t-}+x,M_{t-}+m)- \varphi (X_{t-},M_{t-}) \\ &\quad {} -x \partial_{x}\varphi (X_{t-},M_{t-})-m \partial_{m}\varphi (X_{t-},M_{t-}) \bigr\} \nu (ds,dx,dm) \end{aligned}$$

  is a local martingale.

Then, in our case of study, by choosing the characteristics

$$\begin{aligned} B^{X}_{t}&= \int_{0}^{t} \bigl[1+X_{s} a(X_{s-},M_{s-}) \bigr]\,ds, \\ B^{M}_{t} &= \int_{0}^{t} \biggl[-\alpha M_{s}+\alpha J \int_{0}^{s}{\mathbb {E}}\bigl[a(X _{s-s'},M_{s-s'}) \bigr] b\bigl(ds'\bigr) \biggr]\,ds, \end{aligned}$$

and

$$ \nu (ds,dx,dm)= a(X_{s-},M_{s-})\,ds \delta_{-X_{s-}}(dx)\delta _{0}(dm), $$

we get that $(X_{t},M_{t})$ is indeed a semi martingale.

Now, we have the necessary conditions to prove that the process $(X_{t},M_{t})$ solves (8)–(9). This is actually an application of Jacod-Shiryaev [7, Theorem III.2.26, p. 157]. Consider the stochastic differential equation

$$ \textstyle\begin{cases} (X_{0},M_{0}) = (\xi_{x},\xi_{m}) \\ d(X_{t},M_{t}) = \beta (t,X_{t},M_{t})\,dt \\ \qquad\qquad \quad {}+\delta (t,X_{t-},M_{t-},u) ({\mathcal {N}}(du,dt)-q(du,dt) ), \end{cases} $$

(32)

where ${\mathcal {N}}$ is a standard Poisson random measure with intensity measure

$$\begin{aligned} q(du,dt)=du\otimes dt. \end{aligned}$$

Hence, if ${\mathcal {N}}$ has a jump at $(t,u)$, then $\Delta (X_{t},M _{t})=\delta (t,X_{t-},M_{t-},u)$.

Theorem 15

Let $\eta$ be a suitable initial condition (i.e., a probability on ${\mathbb {R}}_{+}^{2}$), and $\beta ,\delta$ be

$$ \textstyle\begin{cases} \beta =\bigl(\beta^{1},\beta^{2}\bigr),\ \textit{a Borel function:}\ {\mathbb {R}}_{+} \times {\mathbb {R}}_{+}^{2}\rightarrow {\mathbb {R}}_{+}^{2}, \\ \delta =\bigl(\delta^{1},\delta^{2}\bigr),\ \textit{a Borel function:}\ {\mathbb {R}}_{+}\times {\mathbb {R}}_{+}^{2}\times {\mathbb {R}}_{+}\rightarrow {\mathbb {R}}_{+}^{2}. \end{cases} $$

The set of all solution-measures (or weak solutions) to (32) with initial condition $\eta$ is the set of all solutions to a martingale problem on the canonical space where the characteristics $(B,0,\nu )$ are given by

$$ B^{i}_{t}(w)= \int_{0}^{t} \beta^{i} \bigl(s,X_{s}(w),M_{s}(w)\bigr)\,ds, $$

and $\nu$ is namely the compensator of the random measure associated to the jumps of $(X_{t},M_{t})$.

We notice that $(X_{t},M_{t})$ indeed solves the Martingale problem given by (31), therefore it is a solution-measure to (32), i.e., there exists a Poisson measure ${\mathcal {N}}(ds, du)$ on ${\mathbb {R}}_{+} \times {\mathbb {R}}_{+}$ with intensity $ds\,du$ such that $(X_{t},M_{t})$ solves (8)–(9), and Lemma 13 is proved.