Ponente
Descripción
In this work, we study a nonlinear structured population model motivated by neuronal dynamics. The system couples an elapsed-time transport equation for the neuronal density with a nonlocal evolution equation for the voltage field:
\begin{equation}
\label{eq:PVI}
\begin{cases}
\displaystyle
\partial_t u + \partial_s u
= -p(v(t,x),s)\,u(t,s,x)
+ \int_{s}^{\infty}!\int_{\Omega}
k(s,r,y,x)\, p(v(t,y),r)\, u(t,r,y)\, dy\, dr,
& t \ge 0,\ s>0,\ x \in \Omega, \[0.8em]
\displaystyle
\partial_t v
= -v(t,x) + \int_{\Omega} w(t,x,y)\, v(t,y)\, Nu,v\, dy,
& t \ge 0,\ x \in \Omega, \[0.8em]
u(0,s,x) = u_0(s,x) \ge 0, \quad v(0,x) = v_0(x),
& (s,x) \in \mathbb{R}^+ \times \Omega.
\end{cases}
\end{equation}
where (\Omega\subset\mathbb R^3) is an open bounded domain
supplemented with the renewal boundary condition
[
u(t,0,x)=Nu,v.
]
Here (u(t,s,x)) represents the density of neurons whose last discharge occurred (s) units of time before (t), while (v(t,x)) denotes the voltage field. The elapsed-time variable encodes the recent firing history of the population, including refractory effects and activity-dependent responses, whereas the voltage field provides a macroscopic description of the electrical state influencing neuronal firing.
The nonlocal operators (N[u,v]), (G_0[u,v]), and (G_1[u,v]) describe renewal at zero elapsed time, redistribution of firing activity across elapsed-time states, and voltage-mediated coupling, respectively. Through these terms, the voltage-dependent firing rate affects the evolution of the neuronal density, while the resulting renewal and redistribution mechanisms feed back into the voltage dynamics. This produces a nonlinear nonlocal feedback structure between firing activity, density evolution, and voltage dynamics, which constitutes the main analytical difficulty of the model.
The mathematical model couples a transport equation for the elapsed-time neuronal density with a nonlocal evolution equation for the voltage field. The interaction between these two components produces a nonlinear feedback mechanism: the voltage affects the firing rate, the firing rate determines renewal and redistribution mechanisms in the elapsed-time density, and the resulting firing activity feeds back into the voltage dynamics through a nonlocal coupling operator. This structure leads to a nonlinear and nonlocal evolution system that is naturally analyzed in a mixed functional framework.
The main analytical contribution of the paper is the proof of local well-posedness for the coupled system in a mixed $(L^1)-(L^\infty)$ setting. The neuronal density is controlled in ($L^1$), which is appropriate for a population density, while the voltage field is controlled in $(L^\infty)$, which allows us to handle the nonlinear dependence of the firing rate on the voltage. The proof combines several tools: characteristic representations for the elapsed-time transport equation, weak formulations adapted to low-regularity data, estimates for the nonlinear nonlocal operators, and a fixed-point argument for the full coupled system.