In　this　chapter,　we　analyze　the　transition　mechanisms　between　periodic　and　chaotic　behavior　in　a　synaptically　coupled　system,　which　consists　of　bursting　neurons　(Hindmarsh-Rose　model).　We　observe　that　chaotic　bursting　neurons　can　turn　to　periodic　neurons　or　vice　versa,　by　changes　in　coupling　strength.　The　Lyaponov　exponent　calculations　show　the　fine　structure　of　a　pathway　to　chaotic　behavior　when　the　synaptic　coupling　of　neurons　gets　weaken　in　strength.　Further　we　use　Poincare　maps　to　gain　more　insights　of　the　transitions.