## Resumo

We design and analyze the dynamics of a large network of theta neurons, which are idealized type I neurons. The network is heterogeneous in that it includes both inherently spiking and excitable neurons. The coupling is global, via pulselike synapses of adjustable sharpness. Using recently developed analytical methods, we identify all possible asymptotic states that can be exhibited by a mean field variable that captures the network's macroscopic state. These consist of two equilibrium states that reflect partial synchronization in the network and a limit cycle state in which the degree of network synchronization oscillates in time. Our approach also permits a complete bifurcation analysis, which we carry out with respect to parameters that capture the degree of excitability of the neurons, the heterogeneity in the population, and the coupling strength (which can be excitatory or inhibitory). We find that the network typically tends toward the two macroscopic equilibrium states when the neuron's intrinsic dynamics and the network interactions reinforce one another. In contrast, the limit cycle state, bifurcations, and multistability tend to occur when there is competition among these network features. Finally, we show that our results are exhibited by finite network realizations of reasonable size.

## Resumo Limpo

design analyz dynam larg network theta neuron ideal type neuron network heterogen includ inher spike excit neuron coupl global via pulselik synaps adjust sharp use recent develop analyt method identifi possibl asymptot state can exhibit mean field variabl captur network macroscop state consist two equilibrium state reflect partial synchron network limit cycl state degre network synchron oscil time approach also permit complet bifurc analysi carri respect paramet captur degre excit neuron heterogen popul coupl strength can excitatori inhibitori find network typic tend toward two macroscop equilibrium state neuron intrins dynam network interact reinforc one anoth contrast limit cycl state bifurc multist tend occur competit among network featur final show result exhibit finit network realiz reason size