We employ optimal control theory to design an event-based, minimum energy, desynchronizing control stimulus for a network of pathologically synchronized, heterogeneously coupled neurons. This works by optimally driving the neurons to their phaseless sets, switching the control off, and letting the phases of the neurons randomize under intrinsic background noise. An event-based minimum energy input may be clinically desirable for deep brain stimulation treatment of neurological diseases, like Parkinson's disease. The event-based nature of the input results in its administration only when it is necessary, which, in general, amounts to fewer applications, and hence, less charge transfer to and from the tissue. The minimum energy nature of the input may also help prolong battery life for implanted stimulus generators. For the example considered, it is shown that the proposed control causes a considerable amount of randomization in the timing of each neuron's next spike, leading to desynchronization for the network.

We consider the problem of desynchronizing a network of synchronized, globally (all-to-all) coupled neurons using an input to a single neuron. This is done by applying the discrete time dynamic programming method to reduced phase models for neural populations. This technique numerically minimizes a certain cost function over the whole state space, and is applied to a Kuramoto model and a reduced phase model for Hodgkin-Huxley neurons with electrotonic coupling. We evaluate the effectiveness of control inputs obtained by averaging over results obtained for different coupling strengths. We also investigate the applicability of this method to Hodgkin-Huxley models driven by multiplicative stimuli.

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The optimal input current for a reduced neuron model and a specific target spiking time is obtained. The objective of optimization is to minimize the total input energy to the system subject to a zero net input integral over the time horizon. This “charge-balance constraint” ensures that no net external charge is injected into the neuron. The results are compared to optimal currents for which the charge-balance constraint is not imposed.