In　this　paper,　we　investigate　the　dynamic　behavior　of　a　system　of　two　coupled　Hindmarsh-Rose　(HR)　neurons,　based　on　bifurcation　analysis　of　its　fast　subsystem.　The　individual　HR　neuron　has　chaotic　behavior,　but　they　can　become　regularized　when　coupled　through　synaptic　coupling　or　joint　electrical-synaptic　coupling.　Through　numerical　methods　we　first　investigate　the　bifurcation　structure　of　its　fast　subsystem.　We　show　that　the　emerging　of　periodic　patterns　of　neurons　is　related　to　topological　changes　of　its　underlying　bifurcations.　The　Lyaponov　exponent　calculations　further　reveal　the　pathway　from　chaotic　bursting　behavior　to　regular　bursting　of　HR　neurons.　Finally,　we　include　both　electrical　and　synaptic　coupling　in　the　system,　and　numerically　calculate　the　time　dynamics.　Even　though　electrical　couplings　(or　gap　junctions)　usually　does　not　regularize　chaotic　trajectories,　but　joint　coupling　has　been　more　effective　than　synaptic　coupling　alone　in　producing　stable　rhythms.　The　main　contribution　of　this　paper　is　that　we　provide　a　mathematical　description　for　transitions　of　neuron　dynamics　from　chaotic　trajectories　to　regular　bursting　when　synaptic　and　electrical-synaptic　coupling　strengthens,　using　bifurcation　analysis.