Length-Independent Stability and $L_2$-Gain Certificates for Chains of Delayed Neural Modules

Abstract

Networks of repeated neural modules with delayed neighbour coupling need stability guarantees for every number of modules, but delay-dependent criteria certify one network at a time and grow with its size. This paper derives stability and $L_2$-gain certificates of length-independent size for open chains and rings of modules of two alternating types with slope-restricted activations. A local and a transmission delay act independently and are only bounded and continuous. A Lyapunov-Krasovskii functional with fixed integration limits and module-local variables that repeat with the module type reduces the certificate of every chain to one cell of two modules. Local conditions and a single linear matrix inequality of order 96 on this cell, or 104 for the gain, certify global asymptotic stability and a common $L_2$-gain bound for all open chains and all rings with an even number of at least four modules. Up to a margin, the stability inequality is also necessary for the repeated variables to certify every length. All uniform certificates, including certified coupling regions, are verified in exact rational arithmetic. In the numerical study, the certified coupling reaches 92% of an outer bound given by unstable members of the model class for short delays and 79% at the nominal delay bounds. For 16 modules at unit coupling strength, certificates of the same functional computed for that length alone raise the certified delay scale by 2% to 7% in four cases. A single stability problem with 1298 scalar variables thus replaces per-length certification of long chains at little cost.