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# Stability of synchronization under stochastic perturbations in leaky integrate and fire neural networks of finite size

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• We study the synchronization of fully-connected and totally excitatory integrate and fire neural networks in presence of Gaussian white noises. Using a large deviation principle, we prove the stability of the synchronized state under stochastic perturbations. Then, we give a lower bound on the probability of synchronization for networks which are not initially synchronized. This bound shows the robustness of the emergence of synchronization in presence of small stochastic perturbations.

Mathematics Subject Classification: Primary: 92B25, 92B20; Secondary: 60F10, 60J75.

 Citation: • • Figure 1.  Simulation of the model for $N = 3$, $\gamma = 100$ $\text{s}^{-1}$, $\beta = -52$ mV, $\theta = -55$ mV, $V_r = -70$ mV, $\varepsilon = 2.25\times10^{-4}$ $\text{V}^2/\text{s}$ and $H_{ji} = 1.5$ mV for all $i\neq j\in\{1,2,3\}$

Figure 2.  $\mathbb{P}\left(S_{n+1}^\varepsilon\middle|S_{n}^\varepsilon\right)$ versus $\varepsilon$ for $N = 1599$, $\gamma = 100$ $\text{s}^{-1}$, $\theta = -55$ mV, $V_r = -70$ mV, $m = 0.75$ mV. Red plot $\beta = -52$ mV, green plot $\beta = -54.7$ mV and blue plot $\beta = -55.3$ mV. $\varepsilon\in[0, 2.5\times 10^{-3}\text{V}^2/\text{s}]$

Figure 3.  Zoom on Fig.2 for $\varepsilon\in[0,2.25\times 10^{-4}\text{V}^2/\text{s}]$.

Figure 4.  $\mathbb{P}\left(BS_{n}^\varepsilon\right)$ versus $n$. The values of the parameters are: $N = 1599$, $\gamma = 100$ $\text{s}^{-1}$, $\theta = -55$ mV, $V_r = -70$ mV, $\beta = -52$ mV and $\varepsilon = 0.225\times 10^{-4}\text{V}^2/\text{s}$. Magenta $m = 0.0150$ mV, green $m = 0.0225$ mV, blue $m = 0.0270$ mV and orange $m = 0.0300$ mV. The initial distribution of the membrane potentials are i.i.d. with uniform law on $[-100\text{ mV}, -55\text{ mV}]$

Figure 5.  $\mathbb{P}\left(BS_{n}^\varepsilon \setminus BS^\varepsilon_{n-1}\right)$ probability of first synchronization at the $n$-th firing instant versus $n$. The values of the parameters are: $N = 1599$, $\gamma = 100$ $\text{s}^{-1}$, $\theta = -55$ mV, $V_r = -70$ mV, $\beta = -52$ mV and $\varepsilon = 0.225\times 10^{-4}\text{V}^2/\text{s}$. Magenta $m = 0.0150$ mV, green $m = 0.0225$ mV, blue $m = 0.0270$ mV and orange $m = 0.0300$ mV. The initial distribution of the membrane potentials are i.i.d. with uniform law on $[-100\text{ mV}, -55\text{ mV}]$

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