Synchronous rhythmic spiking in neuronal networks can be brought about by the interaction between E-cells and Icells (excitatory and inhibitory cells). The I-cells gate and synchronize the E-cells, and the E-cells drive and synchronize the I-cells. We refer to rhythms generated in this way as PING (pyramidal-interneuronal gamma) rhythms. The PING mechanism requires that the drive I I to the I-cells be sufficiently low; the rhythm is lost when I I gets too large. This can happen in at least two ways. In the first mechanism, the I-cells spike in synchrony, but get ahead of the E-cells, spiking without being prompted by the E-cells. We call this phase walkthrough of the I-cells . In the second mechanism, the I-cells fail to synchronize, and their activity leads to complete suppression of the E-cells. Noisy spiking in the E-cells, generated by noisy external drive, adds excitatory drive to the I-cells and may lead to phase walkthrough. Noisy spiking in the I-cells adds inhibition to the E-cells and may lead to suppression of the E-cells. An analysis of the conditions under which noise leads to phase walkthrough of the I-cells or suppression of the E-cells shows that PING rhythms at frequencies far below the gamma range are robust to noise only if network parameter values are tuned very carefully. Together with an argument explaining why the PING mechanism does not work far above the gamma range in the presence of heterogeneity, this justifies the “G” in “PING.”

In model networks of E-cells and I-cells (excitatory and inhibitory neurons, respectively), synchronous rhythmic spiking often comes about from the interplay between the two cell groups: the E-cells synchronize the I-cells and vice versa. Under ideal conditions—homogeneity in relevant network parameters and all-to-all connectivity, for instance—this mechanism can yield perfect synchronization. We find that approximate, imperfect synchronization is possible even with very sparse, random connectivity. The crucial quantity is the expected number of inputs per cell. As long as it is large enough (more precisely, as long as the variance of the total number of synaptic inputs per cell is small enough), tight synchronization is possible. The desynchronizing effect of random connectivity can be reduced by strengthening the E→I synapses. More surprising, it cannot be reduced by strengthening the I→E synapses. However, the decay time constant of inhibition plays an important role. Faster decay yields tighter synchrony. In particular, in models in which the inhibitory synapses are assumed to be instantaneous, the effects of sparse, random connectivity cannot be seen.

We analyze the existence and stability of phase-locked states of neurons coupled electrically with gap junctions. We show that spike shape and size, along with driving current (which affects network frequency), play a large role in which phase-locked modes exist and are stable. Our theory makes predictions about biophysical models using spikes of different shapes, and we present simulations to confirm the predictions. We also analyze a large system of all-to-all coupled neurons and show that the splay-phase state can exist only for a certain range of frequencies.

Coherent rhythms in the gamma frequency range are ubiquitous in the nervous system and thought to be important in a variety of cognitive activities. Such rhythms are known to be able to synchronize with millisecond precision across distances with significant conduction delay; it is mysterious how this can operate in a setting in which cells receive many inputs over a range of time. Here we analyze a version of mechanism, previously proposed, that the synchronization in the CA1 region of the hippocampus depends on the firing of “doublets” by the interneurons. Using a network of local circuits that are arranged in a possibly disordered lattice, we determine the conditions on parameters for existence and stability of synchronous solutions in which the inhibitory interneurons fire single spikes, doublets, or triplets per cycle. We show that the synchronous solution is only marginally stable if the interneurons fire singlets. If they fire doublets, the synchronous state is asymptotically stable in a larger subset of parameter space than if they fire triplets. An unexpected finding is that a small amount of disorder in the lattice structure enlarges the parameter regime in which the doublet solution is stable. Synaptic noise reduces the regime in which the doublet configuration is stable, but only weakly.

If an oscillating neural circuit is forced by another such circuit via a composite signal, the phase lag induced by the forcing can be changed by changing the relative strengths of components of the coupling. We consider such circuits, with the forced and forcing oscillators receiving signals with some given phase lag. We show how such signals can be transformed into an algorithm that yields connection strengths needed to produce that lag. The algorithm reduces the problem of producing a given phase lag to one of producing a kind of synchrony with a “teaching” signal; the algorithm can be interpreted as maximizing the correlation between voltages of a cell and the teaching signal. We apply these ideas to regulation of phase lags in chains of oscillators associated with undulatory locomotion.

We study the properties of a network consisting of two model neurons that are coupled by reciprocal inhibition. The study was motivated by data from a pair of cells in the crustacean stomatogastric ganglion. One of the model neurons is an endogenous burster; the other is excitable but not bursting in the absence of phasic input. We show that the presence of a hyperpolarization activated inward current ( i h ) in the excitable neuron allows these neurons to fire in integer subharmonics, with the excitable cell firing once for every N ≥ 1 bursts of the oscillator. The value of N depends on the amount of hyperpolarizing current injected into the excitable cell as well as the voltage activation curve of i h . For a fast synapse, these parameter changes do not affect the characteristic point in the oscillator cycle at which the excitable cell bursts; for slower synapses, such a relationship is maintained within small windows for each N . The network behavior in the current work contrasts with the activity of a pair of coupled oscillators for which the interaction is through phase differences; in the latter case, subharmonics exist if the uncoupled oscillators have near integral frequency relationships, but the phase relationships of the oscillators in general change significantly with parameters. The mechanism of this paper provides a potential means of coordinating subnetworks acting on different time scales but maintaining fixed relationships between characteristic points of the cycles.