A minimally invasive neurostimulation method for controlling abnormal synchronisation in the neuronal activity

Many collective phenomena in Nature emerge from the -partial- synchronisation of the units comprising a system. In the case of the brain, this self-organised process allows groups of neurons to fire in highly intricate partially synchronised patterns and eventually lead to high level cognitive outputs and control over the human body. However, when the synchronisation patterns are altered and hypersynchronisation occurs, undesirable effects can occur. This is particularly striking and well documented in the case of epileptic seizures and tremors in neurodegenerative diseases such as Parkinson’s disease. In this paper, we propose an innovative, minimally invasive, control method that can effectively desynchronise misfiring brain regions and thus mitigate and even eliminate the symptoms of the diseases. The control strategy, grounded in the Hamiltonian control theory, is applied to ensembles of neurons modelled via the Kuramoto or the Stuart-Landau models and allows for heterogeneous coupling among the interacting unities. The theory has been complemented with dedicated numerical simulations performed using the small-world Newman-Watts network and the random Erdős-Rényi network. Finally the method has been compared with the gold-standard Proportional-Differential Feedback control technique. Our method is shown to achieve equivalent levels of desynchronisation using lesser control strength and/or fewer controllers, being thus minimally invasive.Author summary: Synchronisation plays an important role in most of the neuronal activities and in particular in the control of the motor system. However, due to biochemical dysfunction in the brain activity, an abnormal and excessive synchronisation may occur being responsible for severe symptoms of several neurological diseases. For the case of Parkinson’s disease, for instance, an insufficient dopamine production in the basal ganglia causes rigidity or continuous tremors. In the case of epilepsy instead, imbalance between excitation and inhibition causes strong unpredictable seizures. Several neurostimulation techniques have been developed with the aim to control and relieve the symptoms as alternatives to oral medication. In this line of research, we propose a new method which has the property of being as little invasive as possible, in the number of electrodes needed and the strength of the current applied, while still controlling the symptoms. It is based on the consideration that neuronal patches resemble a set of phase-coupled oscillators which dynamics can be described by the celebrated Kuramoto model. The control technique we employ is inspired by a Hamiltonian formulation of the Kuramoto model. To verify the effectiveness of our method, we test it in a more realistic model of coupled neuronal patches described by the Stuart-Landau equations. Numerical simulations validate our approach.