Analyzing the competition of gamma rhythms with delayed pulse-coupled oscillators in phase representation

Networks of neurons can generate oscillatory activity as result of various types of coupling that lead to synchronization. A prominent type of oscillatory activity is gamma (30-80 Hz) rhythms, which may play an important role in neuronal information processing. Two mechanisms have mainly been proposed for their generation: (1) interneuron network gamma (ING) and (2) pyramidal-interneuron network gamma (PING). In vitro and in vivo experiments have shown that both mechanisms can exist in the same cortical circuits. This raises the questions: How do ING and PING interact when both can in principle occur? Are the network dynamics a superposition, or do ING and PING interact in a nonlinear way and if so, how? In this article, we first generalize the phase representation for nonlinear one-dimensional pulse coupled oscillators as introduced by Mirollo and Strogatz to type II oscillators whose phase response curve (PRC) has zero crossings. We then give a full theoretical analysis for the regular gamma-like oscillations of simple networks consisting of two neural oscillators, an "E neuron" mimicking a synchronized group of pyramidal cells, and an "I neuron" representing such a group of interneurons. Motivated by experimental findings, we choose the E neuron to have a type I PRC [leaky integrate-and-fire (LIF) neuron], while the I neuron has either a type I or type II PRC (LIF or "sine" neuron). The phase representation allows us to define in a simple manner scenarios of interaction between the two neurons, which are independent of the types and the details of the neuron models. The presence of delay in the couplings leads to an increased number of scenarios relevant for gamma-like oscillatory patterns. We analytically derive the set of such scenarios and describe their occurrence in terms of parameter values such as synaptic connectivity and drive to the E and I neurons. The networks can be tuned to oscillate in an ING or PING mode. We focus particularly on the transition region where both rhythms compete to govern the network dynamics and compare with oscillations in reduced networks, which can only generate either ING or PING. Our analytically derived oscillation frequency diagrams indicate that except for small coexistence regions, the networks generate ING if the oscillation frequency of the reduced ING network exceeds that of the reduced PING network, and vice versa. For networks with the LIF I neuron, the network oscillation frequency slightly exceeds the frequencies of corresponding reduced networks, while it lies between them for networks with the sine I neuron. In networks oscillating in ING (PING) mode, the oscillation frequency responds faster to changes in the drive to the I (E) neuron than to changes in the drive to the E (I) neuron. This finding suggests a method to analyze which mechanism governs an observed network oscillation. Notably, also when the network operates in ING mode, the E neuron can spike before the I neuron such that relative spike times of the pyramidal cells and the interneurons alone are not conclusive for distinguishing ING and PING.