Firing rate models for gamma oscillations in I-I and E-I networks.

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, even when extended to include an explicit dynamic synaptic activation variable, are found unable to capture some dynamics such as Interneuronal Network Gamma oscillations (ING). Use of an explicit delay is helpful in simulations at the expense of complicating mathematical analysis. We resolve this issue by introducing a dynamic variable, u, that acts as an effective delay in the negative feedback loop between firing rate (r) and synaptic gating of inhibition (s). In effect, u endows synaptic activation with second order dynamics. With linear stability analysis, numerical branch-tracking and simulations, we show that our r-u-s rate model captures some key qualitative features of spiking network models for ING. We also propose an alternative formulation, a v-u-s model, in which mean membrane potential v satisfies an averaged current-balance equation. Furthermore, we extend the framework to E-I networks. With our six-variable v-u-s model, we demonstrate 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.

Synaptic homeostasis transiently leverages Hebbian mechanisms for a multiphasic response to inactivity.

Homeostatic regulation of synapses is vital for nervous system function and key to understanding a range of neurological conditions. Synaptic homeostasis is proposed to operate over hours to counteract the destabilizing influence of long-term potentiation (LTP) and long-term depression (LTD). The prevailing view holds that synaptic scaling is a slow first-order process that regulates postsynaptic glutamate receptors and fundamentally differs from LTP or LTD. Surprisingly, we find that the dynamics of scaling induced by neuronal inactivity are not exponential or monotonic, and the mechanism requires calcineurin and CaMKII, molecules dominant in LTD and LTP. Our quantitative model of these enzymes reconstructs the unexpected dynamics of homeostatic scaling and reveals how synapses can efficiently safeguard future capacity for synaptic plasticity. This mechanism of synaptic adaptation supports a broader set of homeostatic changes, including action potential autoregulation, and invites further inquiry into how such a mechanism varies in health and disease.