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The idealisation of neuronal pulses as $\delta$-spikes is a convenientapproach in neuroscience but can sometimes lead to erroneous conclusions. Weinvestigate the effect of a finite pulse-width on the dynamics of balancedneuronal networks. In particular, we study two populations of identicalexcitatory and inhibitory neurons in a random network of phase oscillatorscoupled through exponential pulses with different widths. We consider threecoupling functions, inspired by leaky integrate-and-fire neurons with delay andtype-I phase-response curves. By exploring the role of the pulse-widths fordifferent coupling strengths we find a robust collective irregular dynamics,which collapses onto a fully synchronous regime if the inhibitory pulses aresufficiently wider than the excitatory ones. The transition to synchrony isaccompanied by hysteretic phenomena (i.e. the co-existence of collectiveirregular and synchronous dynamics). Our numerical results are supported by adetailed scaling and stability analysis of the fully synchronous solution. Aconjectured first-order phase transition emerging for $\delta$-spikes issmoothed out for finite-width pulses.

Collective dynamics in the presence of finite-width pulses