A mean-field model of Integrate-and-Fire neurons: non-linear stability of the stationary solutions

Abstract

We investigate a stochastic network composed of Integrate-and-Fire spiking neurons, focusing on its mean-field asymptotics. We consider an invariant probability measure of the McKean-Vlasov equation and establish an explicit sufficient condition to ensure the local stability of this invariant distribution. Furthermore, we prove a conjecture proposed initially by J. Touboul and P. Robert regarding the bistable nature of a specific instance of this neuronal model.

Figures (2)

• Figure 1: Plot of the function $\alpha \mapsto J(\alpha) := \frac{\alpha}{\gamma(\alpha)}$, for $b(x) = -x$ and $f(x) =x^2$. We prove in Proposition \ref{['prop:toy-model-multi-stab']} that this function is decreasing on $(0, \alpha_*]$ and increasing on $[\alpha_*, \infty)$.
• Figure 2: Let $f(x) = x^2$ and $b(x) = -x$. For $J = 2.12$, the invariant probability measures of \ref{['eq:McKeanVlasov']} are $\{\delta_0, \nu^\infty_{\alpha_1}, \nu^\infty_{\alpha_2} \}$ with $\alpha_1 \approx 1.108$ and $\alpha_2 \approx 1.7383$. The shape of the non-trivial invariant probability measures $\nu^\infty_{\alpha_1}$ and $\nu^\infty_{\alpha_2}$ are reported in Figures \ref{['fig:sub-third']} and \ref{['fig:sub-fourth']} respectively. Figure \ref{['fig:sub-first']}, we plot the curves $\Re F(\alpha_1, z) = 0$ (in blue) and $\Im F(\alpha_1, z)$ (in red). The two curves intersect at a zero of $F(\alpha_1, \cdot)$. We find numerically that $F(\alpha_1, 0.3065) \approx 0$. This suggests that $\nu^\infty_{\alpha_1}$ is unstable. For $\alpha = \alpha_2$, we find in Figure \ref{['fig:sub-second']} that the zeros of $z \mapsto F(\alpha_2, z)$ have negative real part, suggesting that $\nu^\infty_{\alpha_2}$ is stable.

Theorems & Definitions (46)

• theorem 1
• remark 1
• proposition 1
• remark 2
• theorem 2
• proposition 2
• lemma 1
• proof
• lemma 2
• proof