Figure 1

(a) Narrow straits formed by a cusp between two reflecting circles. The domain delimited by the big reflecting circle and the cusped neck is $\Omega$. A Brownian particle can escape $\Omega$ only through the segment AB. (b) The image ${\Omega }_w=w\left(\Omega \right)$ under the conformal mapping (2). The narrow neck leading to AB in $\Omega$ is mapped into the semi-ring-shaped domain in ${\Omega }_w$. The remaining part of $\Omega$ is mapped into the red domain. The segment AB (of length $\varepsilon$) in the left panel is mapped into the thick black segment AB in the right panel [of length $\sqrt{\varepsilon }+O\left(\varepsilon \right)$].Reuse & Permissions

Figure 2

(a) The solution $v\left(e^{i\theta }\right)$ for $\varepsilon ={10}^{-1},{10}^{-2},{10}^{-3}$. (b) The NET $v\left(e^{ic\sqrt{\epsilon }}\right)$ for $c=0.5$.Reuse & Permissions

Figure 3

NET from the domain (a) with $D=1$, $L=1$. Statistics were obtained from 1000 exit times of simulated Brownian trajectories (dashed line). (b) NET vs. obstacle scaled radius $r=a/L=\frac{1}{2}(1-\varepsilon )$. The analytical approximation is [Eq. (13), continuous curve] with $0<r=r_1=0.2$, $r_1<r<r_2=0.45$, and $0.45<r<0.5$.Reuse & Permissions

Figure 4

Organization of the neuronal membrane. (a) Schematic representation of a Brownian particle diffusing in a crowded microdomain. (b) Mean-square displacement (MSD) of the particle in a domain paved with microdomains. The MSD is linear, showing that crowding does not affect the nature of diffusion. The effective diffusion coefficient is computed from $⟨\mathrm{MSD}\left(t\right)/4t⟩$ ($D=1$). (c) Effective diffusion coefficient computed from the MSD for different radiuses of the obstacles. Brownian simulations (continuous curve): there are three regions (separated by the dashed lines). While there is no crowding for $a<0.2$, the decreasing of the effective diffusion coefficient for $0.2<a<0.4$ is logarithmic and like the square root for $a>0.4$. (d) Effective diffusion coefficient of a particle diffusing in a domain as a function of the fraction of the occupied surface. An $\alpha$-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPAR) receptor has a diffusion coefficient of 0.2 $\mu$m${}^2$/s in a free membrane [27].Reuse & Permissions