Low dimensional dynamics of spiking neuron networks

The collective dynamics of neuronal networks can be complex even when simplified one-compartment spiking (integrate-and-fire, IF) neurons are considered for modeling, due to the high-dimensionality of the system and the quenched randomness in the synaptic couplings. Despite several decades of efforts, a general expression for the dynamics of the population firing rate \(\nu(t)\) independent from neuronal models and dynamical regimes expressed is still lacking. Here, we contribute to solve this issue by deriving a low-dimensional mean-field dynamics of \(\nu(t)\) (i.e., the network activity) valid for a wide class of IF neurons and outside equilibrium. To this purpose we focused on the evolution of the population density of neuron membrane potentials which in these networks is fully captured by a Fokker-Plank (FP) equation. Relying on the first modes in the state-dependent spectral expansion of the FP operator, we derive the 2nd-order ODE for the firing rate. In this framework the \(\nu(t)\) dynamics naturally incorporates the strength of the synaptic couplings and synaptic delays. The remarkable agreement between theory and simulations is shown both under noise- and drift-dominate regimes in networks of leaky IF neurons