A Complex-Valued Firing-Rate Model That Approximates the Dynamics of Spiking Networks

Firing-rate models provide an attractive approach for studying large neural networks because they can be simulated rapidly and are amenable to mathematical analysis. Traditional firing-rate models assume a simple form in which the dynamics are governed by a single time constant. These models fail to replicate certain dynamic features of populations of spiking neurons, especially those involving synchronization. We present a complex-valued firing-rate model derived from an eigenfunction expansion of the Fokker-Planck equation and apply it to the linear, quadratic and exponential integrate-and-fire models. Despite being almost as simple as a traditional firing-rate description, this model can reproduce firing-rate dynamics due to partial synchronization of the action potentials in a spiking model, and it successfully predicts the transition to spike synchronization in networks of coupled excitatory and inhibitory neurons.Author Summary: Neuronal responses are often characterized by the rate at which action potentials are generated rather than by the timing of individual spikes. Firing-rate descriptions of neural activity are appealing because of their comparative simplicity, but it is important to develop models that faithfully approximate dynamic features arising from spiking. In particular, synchronization or partial synchronization of spikes is an important feature that cannot be described by typical firing-rate models. Here we develop a model that is nearly as simple as the simplest firing-rate models and yet can account for a number of aspects of spiking dynamics, including partial synchrony. The model matches the dynamic activity of networks of spiking neurons with surprising accuracy. By expanding the range of dynamic phenomena that can be described by simple firing-rate equations, this model should be useful in guiding intuition about and understanding of neural circuit function.