Oscillatory behavior has been found in different specialized genetic networks. Previous work has demonstrated nonsynchronous, erratic single-cell oscillations in a genetic network composed of nonspecialized regulatory components and based entirely on negative feedback. Here, we present the construction of a more robust, hysteresis-based genetic relaxation oscillator and provide a theoretical analysis of the conditions necessary for single-cell and population synchronized oscillations. The oscillator is constructed by coupling two subsystems that have previously been implemented experimentally. The first subsystem is the toggle switch, which consists of two mutually repressive genes and can display robust switching between bistable expression states and hysteresis. The second subsystem is an intercell communication system involved in quorum-sensing. This subsystem drives the toggle switch through a hysteresis loop in single cells and acts as a coupling between individual cellular oscillators in a cell population. We demonstrate the possibility of both population synchronization and suppression of oscillations (cluster formation), depending on diffusion strength and other parameters of the system. We also propose the optimal choice of the parameters and small variations in the architecture of the gene regulatory network that substantially expand the oscillatory region and improve the likelihood of observing oscillations experimentally.

Dopaminergic neurons of the midbrain fire spontaneously at rates <10/s and ordinarily will not exceed this range even when driven with somatic current injection. When driven at higher rates, these cells undergo spike failure through depolarization block. During spontaneous bursting of dopaminergic neurons in vivo, bursts related to reward expectation in behaving animals, and bursts generated by dendritic application of N-methyl-d-aspartate (NMDA) agonists, transient firing attains rates well above this range. We suggest a way such high-frequency firing may occur in response to dendritic NMDA receptor activation. We have extended the coupled oscillator model of the dopaminergic neuron, which represents the soma and dendrites as electrically coupled compartments with different natural spiking frequencies, by addition of dendritic AMPA (voltage-independent) or NMDA (voltage-dependent) synaptic conductance. Both soma and dendrites contain a simplified version of the calcium-potassium mechanism known to be the mechanism for slow spontaneous oscillation and background firing in dopaminergic cells. The compartments differ only in diameter, and this difference is responsible for the difference in natural frequencies. We show that because of its voltage dependence, NMDA receptor activation acts to amplify the effect on the soma of the high-frequency oscillation of the dendrites, which is normally too weak to exert a large influence on the overall oscillation frequency of the neuron. During the high-frequency oscillations that result, sodium inactivation in the soma is removed rapidly after each action potential by the hyperpolarizing influence of the dendritic calcium-dependent potassium current, preventing depolarization block of the spike mechanism, and allowing high-frequency spiking.

Dopaminergic (DA) neurons of the mammalian midbrain exhibit unusually low firing frequencies in vitro. Furthermore, injection of depolarizing current induces depolarization block before high frequencies are achieved. The maximum steady and transient rates are about 10 and 20 Hz, respectively, despite the ability of these neurons to generate bursts at higher frequencies in vivo. We use a three-compartment model calibrated to reproduce DA neuron responses to several pharmacological manipulations to uncover mechanisms of frequency limitation. The model exhibits a slow oscillatory potential (SOP) dependent on the interplay between the L-type Ca2+ current and the small conductance K+ (SK) current that is unmasked by fast Na+ current block. Contrary to previous theoretical work, the SOP does not pace the steady spiking frequency in our model. The main currents that determine the spontaneous firing frequency are the subthreshold L-type Ca2+ and the A-type K+ currents. The model identifies the channel densities for the fast Na+ and the delayed rectifier K+ currents as critical parameters limiting the maximal steady frequency evoked by a depolarizing pulse. We hypothesize that the low maximal steady frequencies result from a low safety factor for action potential generation. In the model, the rate of Ca2+ accumulation in the distal dendrites controls the transient initial frequency in response to a depolarizing pulse. Similar results are obtained when the same model parameters are used in a multi-compartmental model with a realistic reconstructed morphology, indicating that the salient contributions of the dendritic architecture have been captured by the simpler model.

Oscillatory regulatory networks have been discovered in many regulatory pathways. Due to their enormous complexity, it is necessary to study their dynamics by means of highly simplified models. These models have received particular value because artificial regulatory networks can be engineered experimentally. In this paper, we study dynamical properties of an artificial regulatory oscillator called repressilator. We have shown that oscillations arise from the existence of an absorbing toruslike region in the phase space of the model. This geometric structure requires monotonic repression at all promoters and the absence of any regulatory connections apart from a cyclic repression loop. We show that oscillations collapse as only weak extra connections are introduced if there is imbalance between the attended concentrations and those sufficient for saturation of the promoters. We found that a pair of diffusively coupled repressilators displays synchronization properties similar to those of relaxation oscillators if the regulatory connections in the cyclic repression loop are strong. Thus, the role of strengthening these connections can be viewed as introducing time scale separation among variables. This may explain controversial synchronization properties reported for repressilators in earlier studies.