  Recurrent neural networks (RNNs) with random, but sufficiently strong and
balanced coupling display a well known high-dimensional chaotic dynamics. Here,
we investigate if externally applied inputs to these RNNs can stabilize
globally synchronous, input-dependent solutions, in spite of the strong
chaos-inducing coupling. We find that when the balance between excitation and
inhibition is exact, that is when the row-sum of the weights is constant and 0,
a globally applied input can readily synchronize all neurons onto a synchronous
solution. The stability of the synchronous solution is analytically explored in
this work with a master stability function. For any synchronous solution to the
network dynamics, the conditional Lyapunov spectrum can be readily determined,
with the stability of the synchronous solution critically dependent on the
largest real eigenvalue component of the RNN weight matrix. We find that the
smaller the maximum real component of the weight matrix eigenvalues, the more
readily the network synchronizes. Further, the conditional Lyapunov exponents
are easily computed numerically for any synchronization signal without
simulating the RNN. Finally, for certain oscillatory synchronization signals,
the conditional Lyapunov exponents can be determined analytically.
