Existence of a global energy function for symmetric multiregion RNNs

Determine whether the multiregion recurrent neural network with symmetric effective interactions—specifically, with the effective-interaction matrix \hat{T}^{\mu\nu,\rho\sigma} symmetric so that T^{\mu\nu\rho} = \delta^{\mu\rho} c^{\mu\nu} and c^{\mu\nu} = c^{\nu\mu}—admits a global Lyapunov energy function that guarantees convergence of the current variables S^{\mu\nu}(t) to fixed points from any initial condition.

Background

The paper analyzes a multiregion recurrent neural network with low-rank, structured inter-region connectivity and disordered within-region connectivity. Under a symmetry constraint on the effective-interaction tensor (implemented via symmetry of the matrix \hat{T}^{\mu\nu,\rho\sigma}), the current dynamics reduce to fixed points whose stability is studied via a local energy-like quantity.

While the authors establish conditions under which fixed points are (marginally) stable by showing a nonpositive time derivative of a constructed local energy for appropriate perturbations, they note that this does not constitute a proof of global convergence. The existence of a global Lyapunov function, akin to that in classical symmetric-coupling neural networks, remains unresolved.