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

The authors acknowledge support from projects MTM2011-27739-C04-02 and MTM2014-52056-P of Spanish Ministerio de Economía y Competitividad and the European Regional Development Fund (ERDF/FEDER). The second author was also sponsored by the grant BES-2012-057704

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  • Excitatory and inhibitory nonlinear noisy leaky integrate and fire models are often used to describe neural networks. Recently, new mathematical results have provided a better understanding of them. It has been proved that a fully excitatory network can blow-up in finite time, while a fully inhibitory network has a global in time solution for any initial data. A general description of the steady states of a purely excitatory or inhibitory network has been also given. We extend this study to the system composed of an excitatory population and an inhibitory one. We prove that this system can also blow-up in finite time and analyse its steady states and long time behaviour. Besides, we illustrate our analytical description with some numerical results. The main tools used to reach our aims are: the control of an exponential moment for the blow-up results, a more complicate strategy than that considered in [5] for studying the number of steady states, entropy methods combined with Poincaré inequalities for the long time behaviour and, finally, high order numerical schemes together with parallel computation techniques in order to obtain our numerical results.

    Mathematics Subject Classification: 35K60, 35Q92, 82C31, 82C32, 92B20.


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  • 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)

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