Tonic neuron position and network topology determine synchronization dynamics

Understanding how collective neuronal rhythms emerge from the interplay between network structure and single-neuron dynamics remains a fundamental challenge in neurodynamics. In this work, we show that in small heterogeneous networks, the network topology and the position of a single tonic neuron govern synchronization dynamics. We study minimal four-neuron Huber-Braun motifs composed of one tonic neuron and three bursting neurons coupled via electrical synapses. By systematically varying the tonic neuron's centrality-quantified through degree and a quasi-Laplacian measure-we identify two critical coupling thresholds: the critical coupling for global phase synchronization and the coupling at which the network returns to synchronized periodic bursting. Remarkably, centrality determines the nature of the synchronized attractor: motifs in which the tonic neuron has low centrality synchronize into tonic spiking, whereas highly central tonic neurons drive the network toward synchronized bursting. Moreover, the networks exhibit nontrivial collective behaviors, including large-ISI "stairway-to-heaven" structures, partial firing death characterized by the coexistence of spiking and subthreshold neurons, and remote synchronization in the star configuration with the tonic neuron at the hub. While the present study focuses on the Huber-Braun model in four-neuron motifs, excitability and electrical coupling are features shared by many conductance-based neuronal models. This motivates the hypothesis that analogous structure-dynamics relationships may also arise in other neuronal models and small network motifs, highlighting the potential role of local structural control in shaping collective neural rhythms.