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In this work we study the synchronization of Kuramoto oscillators driven byexternal forces in complex modular networks. The motivation is the neuronaldynamics that takes place during information processing in the neural cortex.The neuron organization is modular, with clusters associated to differentfunctions and brain structures, and need to constantly respond to externalstimuli. We study the behavior of forced Kuramoto oscillators where only afraction of them is subjected to a periodic external force. When alloscillators receive the external drive the system always synchronize with theperiodic force if its intensity is sufficiently large. We show that theconditions for global synchronization depend on the fraction of nodes beingforced and on network topology, strength of internal couplings and intensity ofexternal forcing. We develop numerical and analytical calculations for thecritical force for global synchronization as a function of the fraction offorced oscillators. As an application we study the response of the electricjunction \textit{C. elegans} network to external stimuli using the partiallyforced Kuramoto model. Stimuli were applied to topological modules, to ganglia,specified by their anatomical localization, and to the functional groups(sensory and motoneurons). We found that topological modules do not containpurely anamotical groups or functional classes, and that stimulating differentclasses of neurons lead to very different responses, measured in terms ofsynchronization and phase velocity correlations. In all cases the modularstructure hindered full synchronization, protecting the system from seizures.The responses to stimuli applied to topological and functional modules showedpronounced patterns of correlation or anti-correlation with other modules thatwere not observed when the stimulus was applied to a ganglion with mixedfunctional neurons.

Synchronization induced by external forces in modular networks