Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: blow-up and global existence
The aim of the article is to review, explain and, to some extent, refine some up-to-date results on Nonlinear Noisy Leaky Integrate \& Fire model, NNLIF. One refers to the model introduced in the papers of \textit{N. Brunel} [J. Comput. Neurosci. 8, No. 3, 183--208 (2000; Zbl 1036.92008)], and \textit{N. Brunel} and \textit{V. Hakim} [``Fast global oscillations in networks of integrate-and-fire neurons with long firing rates'', Neural Comput. 11, No. 7, 1621--1671 (1999; \url{doi:10.1162/089976699300016179})]. This model is described by an initial boundary value problem for a PDE and it writes: \[ \frac {\partial p} {\partial t} (v,t) + \frac{\partial}{\partial v}[(-v + bN(t))p(v,t)]- a \frac{\partial^2 {p}}{\partial {v^2}}(v,t) = N(t)\delta(v-V_R),\text{ for }v\leq V_F, \] \[ N(t) = -a\frac {\partial p} {\partial v}(V_F,t)\geq0,\ a>0 \] \[ p(v,0) =p^0(v)\geq 0,\quad p(V_F,t) =0,\quad p(-\infty,t)=0, \] \[ \int_{-\infty}^{V_F}p^0(v)dv =1 \] where \(p(., t)\) is the probability density of the electric potential of a randomly chosen neuron at time $t, v$ describes the membrane potential, $V_F$ is a threshold value, $\delta$ is the Dirac mass at point 0, $N$ is the firing rate of network, a is a diffusion coefficient and b is the connectivity parameter. One mainly investigates the problem of finite-time blow-up, the possibility to extend the solutions after a blow-up event and the global-in-time existence of classical solutions in the excitatory case, $b>0$. In the second section basics and assumptions are introduced. Blow-up for all initial conditions when b is large enough is discussed in the third section. The global-in-time existence in an excitatory network is largely presented and covers the fourth section. Conclusions are to be found in the fifth section and some technical results in the fifth section of the article.
scientific article; zbMATH DE number 7451609
scientific article