We derive analytical solutions for macroscopic regimes in populations of identical and nonidentical quadratic integrate-and-fire neurons (QIFs), where the irregularity of synaptic input is represented by a continuous Gaussian noise. This amounts to the diffusion approximation well established in mathematical neuroscience. The obtained results rely on the first-passage time problem and allow understanding of the statistical physics properties of parabolic bursting systems, from quantum physics over intermittency theory to mathematical neuroscience. Within the developed theoretical framework, we study the effect of fluctuations on the interplay between electrical and chemical synapses in networks of QIF neurons.

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• Collective behavior in networks
• Spiking neurons
• Stochastic resetting
• Complex networks
• First passage problems
• Fokker–Planck equation
• Integrate-and-fire model
• Mean field theory