Strongly nonlinear age structured equation,time-elapsed model and large delays

TL;DR

The paper tackles a strongly nonlinear time-elapsed age-structured PDE for neural assemblies, addressing whether inhibitory dynamics without delay converge to a unique steady state and how large delays can induce periodic behavior. The authors develop a global contraction (non-expansion) framework that yields exponential convergence to a stationary state under wide conditions, including a new non-degeneracy requirement tied to strict nonlinearity. When delays are added, they first show linear and weakly nonlinear regimes still relax to a steady state, and for large delays the dynamics can be described by iterates of a nonlinear map, potentially leading to periodic solutions with period 2d in rescaled time. They also extend the non-expansion principle to distributed birth and simple systems, and provide a rigorous formalism to handle large delays. The results offer a robust mechanism for desynchronization in inhibitory networks and contribute broadly to the theory of nonlinear age-structured equations with renewal terms.

Abstract

The time-elapsed model for neural networks is a nonlinear age structured equationwhere the renewal term describes the network activity and influences the dischargerate, possibly with a delay due to the length of connections.We solve a long standing question, namely that an inhibitory network withoutdelay will converge to a steady state and thus the network is desynchonised. Ourapproach is based on the observation that a non-expansion property holds true.However a non-degeneracy condition is needed and, besides the standard one, weintroduce a new condition based on strict nonlinearity.When a delay is included, and following previous works for Fokker-Planck models,we prove that the network may generate periodic solutions. We introduce a newformalism to establish rigorously this property for large delays.The fundamental contraction property also holds for some other age structuredequations and systems.

Figures (3)

• Figure 1: Graph of $\Phi$ and $\Phi\circ\Phi$. Here $\Phi'(\overline{I}) <-1$ and a periodic solution arises oscillating between the two extreme fixed points of $\Phi\circ\Phi$.
• Figure 2: The solution $\tilde{I}_d(\tau)$ in blue and the function $\tilde{I}_\infty(\tau)$ in red. Here $d$ is large and the solution is periodic of period $2$ in the variable $\tau = \frac{t}{d}$.
• Figure 3: The solution $I(\tau)$ in blue and the function $\tilde{I}_\infty(\tau)$ in red. Here $d$ is moderate and a chaotic behaviour of $I(t)$ is observed while the iterates converge to $\overline{I}=0.75$.

Theorems & Definitions (25)

• Theorem 1.1: Long term convergence for inhibitory connections
• Theorem 1.2: Long time delays
• Proposition 2.1: Contraction property
• proof
• Proposition 2.2: A priori bounds
• proof
• Proposition 2.3
• proof
• Theorem 2.4: Convergence to the steady state
• proof