If asymptotic stability holds for any initial condition, the equilibrium point is said to be globally asymptotically stable. Note that for linear networks, asymptotic stability and global asymptotic stability are equivalent, so the distinction between local and global stability is not necessary. We saw already that the stability of linear networks can be characterized in terms of eigenvalues of the weight matrix. But this doesn't generalize to nonlinear systems. Here is another way of deriving sufficient conditions for stability.
Equality holds only at an equilibrium point.
As long as E is concave up, we get convergence to minimum. Unfortunately not, as we can check by equality of mixed partials. Then the set of minima of L is globally asymptotically stable, except for the set of initial conditions that are unstable equilibria. This can be used to show that the network dynamics converges to fixed points. This means that L is nonincreasing.