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Periodic pulse train stimulation is generically used to study the function ofthe nervous system and to counteract disease-related neuronal activity, e.g.,collective periodic neuronal oscillations. The efficient control of neuronaldynamics without compromising brain tissue is key to research and clinicalpurposes. We here adapt the minimum charge control theory, recently developedfor a single neuron, to a network of interacting neurons exhibiting collectiveperiodic oscillations. We present a general expression for the optimalwaveform, which provides an entrainment of a neural network to the stimulationfrequency with a minimum absolute value of the stimulating current. As in thecase of a single neuron, the optimal waveform is of bang-off-bang type, but itsparameters are now determined by the parameters of the effective phase responsecurve of the entire network, rather than of a single neuron. The theoreticalresults are confirmed by three specific examples: two small-scale networks ofFitzHugh-Nagumo neurons with synaptic and electric couplings, as well as alarge-scale network of synaptically coupled quadratic integrate-and-fireneurons.

Entrainment of a network of interacting neurons with minimum stimulating charge