  Periodic pulse train stimulation is generically used to study the function of
the nervous system and to counteract disease-related neuronal activity, e.g.,
collective periodic neuronal oscillations. The efficient control of neuronal
dynamics without compromising brain tissue is key to research and clinical
purposes. We here adapt the minimum charge control theory, recently developed
for a single neuron, to a network of interacting neurons exhibiting collective
periodic oscillations. We present a general expression for the optimal
waveform, which provides an entrainment of a neural network to the stimulation
frequency with a minimum absolute value of the stimulating current. As in the
case of a single neuron, the optimal waveform is of bang-off-bang type, but its
parameters are now determined by the parameters of the effective phase response
curve of the entire network, rather than of a single neuron. The theoretical
results are confirmed by three specific examples: two small-scale networks of
FitzHugh-Nagumo neurons with synaptic and electric couplings, as well as a
large-scale network of synaptically coupled quadratic integrate-and-fire
neurons.
