Paradoxical phase response of gamma rhythms facilitates their entrainment in heterogeneous networks
Authors: Xu Xize , Riecke Hermann 
Published: December 2020

The synchronization of differentγ-rhythms arising in different brain areas has been impli-cated in various cognitive functions. Here, we focus on the effect of the ubiquitous neuronal heterogeneity on the synchronization of PING (pyramidal-interneuronal network gamma) and ING (interneuronal network gamma) rhythms. The synchronization properties of rhythms depends on the response of their collective phase to external input. We therefore determined the macroscopic phase-response curve for finite-amplitude perturbations (fmPRC), using numerical simulation of all-to-all coupled networks of integrate-and-fire (IF) neurons exhibiting either PING or ING rhythms. We show that the intrinsic neuronal heterogeneity can qualitatively modify the fmPRC. While the phase-response curve for the individual IF-neurons is strictly positive (type I), the fmPRC can be biphasic and exhibit both signs (type II). Thus, for PING rhythms, an external excitation to the excitatory cells can, in fact, delay the collective oscillation of the network, even though the same excitation would lead to an advance when applied to uncoupled neurons. This paradoxical delay arises when the external excitation modifies the internal dynamics of the network by causing additional spikes of inhibitory neurons, whose delaying within-network inhibition outweighs the immediate advance caused by the external excitation. These results explain how intrinsic heterogeneity allows the PING rhythm to become synchronized with a periodic forcing or another PING rhythm for a wider range in the mismatch of their frequencies. We demonstrate a similar mechanism for the synchronization of ING rhythms. Our results identify a potential function of neuronal heterogeneity in the synchronization of coupledγ-rhythms, which may play a role in neural information transfer via communication through coherence.

Author Summary

The interaction of a large number of oscillating units can lead to the emergence of a collective, macroscopic oscillation in which many units oscillate in near-unison or near-synchrony. This has been exploited technologically, e.g., to combine many coherently interacting, individual lasers to form a single powerful laser. Collective oscillations are also important in biology. For instance, the circadian rhythm of animals is controlled by the near-synchronous dynamics of a large number of individually oscillating cells. In animals and humans brain rhythms reflect the coherent dynamics of a large number of neurons and are surmised to play an important role in the communication between different brain areas. To be functionally relevant, these rhythms have to respond to external inputs and have to be able to synchronize with each other. We show that the ubiquitous heterogeneity in the properties of the individual neurons in a network can contribute to that ability. It can allow the external inputs to modify the internal network dynamics such that the network can follow these inputs over a wider range of frequencies. Paradoxically, while an external perturbation may delay individual neurons, their ensuing within-network interaction can overcompensate this delay, leading to an overall advance of the rhythm.