An E-I Firing Rate Model for Gamma Oscillations

Firing rate models for describing the mean-field activities of neuronal ensembles can be used effectively to study network function and dynamics, including synchronization and rhythmicity of excitatory-inhibitory populations. However, traditional Wilson-Cowan-like models, being amenable to mathematical analysis, are found unable to capture some dynamics such as Interneuronal Network Gamma oscillations (ING) although use of an explicit delay can help. We resolve this issue by introducing a mean-voltage variable ( v ) that considers the subthreshold integration of inputs and works as an effective delay in the negative feedback loop between firing rate ( r ) and synaptic gating of inhibition ( s ). Here we describe a three-variable r-s-v firing rate model for I-I networks, which is biophysically interpretable and capable of generating ING-like oscillations. Linear stability analysis, numerical branch-tracking and simulations show that the rate model captures many of the common features of spiking network models for ING. We also propose an alternative r-u-s model for ING which considers an implicit delay by a pre-synaptic variable u . Furthermore, we extend the framework to E-I networks. With our six-variable r-s-v model, we describe for the first time in firing rate models the transition from Pyramidal-Interneuronal Network Gamma (PING) to ING by increasing the external drive to the inhibitory population without adjusting synaptic weights. Having PING and ING available in a single network, without invoking synaptic blockers, is plausible and natural for explaining the emergence and transition of two different types of gamma oscillations. Author Summary Gamma rhythm (30-90 Hz) is a neuronal oscillation broadly observed across various animal species. It is associated with cognitive functions and gamma oscillation dysfunction has been found in neurological diseases. Traditional studies of gamma oscillations are primarily conducted in detailed but computational demanding spiking network models. Motivated by the growing size of accessible datasets and large-scale modeling, we develop mean-field representations of gamma oscillations that are computationally lightweight and mathematically analyzable. We resolve the incapability of generating oscillation in inhibitory network with traditional Wilson-Cowan firing rate model by introducing implicit delay with a mean membrane potential variable and generalize the model for excitatory-inhibitory network to capture richer dynamics. Our work provides a potential building block for multi-area large-scale modeling to help understanding the functional roles of circuits at different levels.