Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models

Figure 3.  Firing rates and probability densities for $b_E^E=3$, $b_I^E=0.75$, $b_E ^I=0.5$, $b_I^I=0.25$, in case of a normalized Maxwellian initial condition with mean 0 and variance 0.5 (see (39)). $N_E$ blows-up because of the large value of $b_E ^E$

Figure 4.  Firing rates and probability densities for $b_E^E=0.5$, $b_I^E=0.25$, $b_E ^I=0.25$, $b_I^I=1$, in case of a normalized concentrated Maxwellian initial condition with mean 1.83 and variance 0.003 (see (39)).The initial condition concentrated close to $V_F$ provokes the blow-up of $N_E$

Figure 5.  Firing rates and probability densities for $b_E^E=3$, $b_I^E=0.75$, $b_E ^I=0.5$, $b_I^I=3$, in case of a normalized Maxwellian initial condition with mean 0 and variance 0.5 (see (39)).The blow-up of $N_E$ cannot be avoided by a large value of $b_I^I$

Figure 1.  The function $I_2$ in terms of $N_I$, for different values of $N_E$ fixed. $I_2$ is an increasing function on $N_I$ and decreasing on $N_E$

Figure 6.  $\mathcal{F}(N_E)$ for different parameter values corresponding to the first case of Theorem 4.1: there are no steady states (left) or there is an even number of steady states (right). Left figure: $b_E^E=3$, $b_I^E=0.75$, $b_E^I=0.5$ and $b_I^I=5$. Right figure: $b_E^E=1.8$, $b_I^E=0.75$, $b_E^I=0.5$ and $b_I^I=0.25$

Figure 7.  $\mathcal{F}(N_E)$ for different parameter values corresponding to the second case of Theorem 4.1: there is an odd number of steady states

Figure 8.  Comparison between an uncoupled excitatory-inhibitory network ($b_E^I=b_I^E=0$) and a coupled network with small $b_E^I$ and $b_I^E$. The qualitative behavior is the same in both cases

Figure 9.  Analysis of the number of steady states for $b_E^E=3$, $b_E^I=0.5$, $b_I^I=0.25$ and different values for $b_I^E$

Figure 2.  Study of the limits of $\mathcal{F}(N_E)$ and $N_I^2(N_E)I(N_E)$ (see (29)) when it is finite

Figure 10.  Firing rates for the case of two steady states (right in Fig. 6), for different initial conditions: $\rho_{\alpha}^0-1,2$ are given by the profile (40) with $(N_E,N_I)$ stationary values and $\rho_{\alpha}^0-3$ is a normalized Maxwellian with mean 0 and variance 0.25 (see (39)). For both firing rates, the lower steady state seems to be asymptotically stable whereas the higher one seems to be unstable

Figure 11.  Stability analysis for the case of three steady states (right in Fig. 7)