Some theoretical and numerical results for delayed neural field equations

Robust stabilization of delayed neural fields with partial measurement and actuation

“…We stress that, reasoning as in the proof of (Faye and Faugeras, 2010, Theorem 3.2.1), it can be seen that the solutions of (14b) exist, are unique, and are continuously differentiable over [0; +∞), provided that the kernels w ij and initial states are in L 2 , delays d j and time constants τ j are continuous, and inputs I j are continuous.…”

“…As observed in e.g. Faye and Faugeras (2010), such systems can actually be casted into the more general setup of a delayed state-space representation whose state is a function of the space variable r. More precisely, (2)-(3) in closed loop with either (4) or (5) can be seen as a particular case of the more general dynamics…”

“…In line with Faye and Faugeras (2010), it relies on the upper right Dini derivative of a functional V ∈ C(C, R ≥0 ) along the solutions of (6), as defined byV…”

“…This class of uncontrolled delayed neural fields was extensively studied in Faye and Faugeras (2010) and Veltz and Faugeras (2011), with the only difference that, in those works, the time constant τ and the activation function S were assumed homogeneous, meaning independent of r. A bifurcation analysis was also conducted in Atay and Hutt (2004), under the additional requirement that the kernel w depends only on the distance |r − r |, but allowing for higher order dynamics. From all these works, it is a known fact that the product of the Lipschitz constant of the activation function S by the square of the L 2 -norm of the kernel w regulates the stability of the origin of (17) in the absence of inputs ρ: if this product is smaller than 1, then the origin is asymptotically stable: see (Atay and Hutt, 2004, Theorem 2.1), (Faye and Faugeras, 2010, Theorem 4.2.3) or (Veltz and Faugeras, 2011, Proposition 3.15).…”

Special issue: Mathematical neuroscience

“…The final paper by Faye and Faugeras [25] treats models with space-dependent delays representing axonal communication lags and makes use of techniques from functional analysis to establish existence and uniqueness of solutions for smooth nonlinearities, as well performing a Lyapunov analysis to determine asymptotic stability.…”