Stability of two cluster solutions in pulse coupled networks of neural oscillators

Abstract

Phase response curves (PRCs) have been widely used to study synchronization in neural circuits comprised of pacemaking neurons. They describe how the timing of the next spike in a given spontaneously firing neuron is affected by the phase at which an input from another neuron is received. Here we study two reciprocally coupled clusters of pulse coupled oscillatory neurons. The neurons within each cluster are presumed to be identical and identically pulse coupled, but not necessarily identical to those in the other cluster. We investigate a two cluster solution in which all oscillators are synchronized within each cluster, but in which the two clusters are phase locked at nonzero phase with each other. Intuitively, one might expect this solution to be stable only when synchrony within each isolated cluster is stable, but this is not the case. We prove rigorously the stability of the two cluster solution and show how reciprocal coupling can stabilize synchrony within clusters that cannot synchronize in isolation. These stability results for the two cluster solution suggest a mechanism by which reciprocal coupling between brain regions can induce local synchronization via the network feedback loop.

Coupling of the cell cycle and metabolism in yeast cell-cycle-related oscillations via resource criticality and checkpoint gating

“…By clustering we do not mean spatial clustering (cultures that exhibit YMO occur in well-mixed bioreactors), but temporal clustering -cohorts of cells traversing the CDC in near synchrony (see Figure 1). The mathematical results in Boczko et al (2010) are similar to those in Chandrasekaran, Achuthan, and Canavier (2011) where, in a different context, it was observed that reciprocal coupling can stabilize a 2-cluster solution by enforcing synchrony within clusters that would not synchronize in isolation.…”

Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators

“…Synchronization was not evident at weak values for the connectivity between modules, likely due to the inability of weak coupling strengths to overcome the effects of noise. The applicability of the methods presented herein is limited only by the ability to measure the appropriate PRCs, by the requirement that the coupling be effectively pulsatile, and the restriction to only two oscillators unless the dynamics of the population are analogous to those of a component oscillator (Achuthan and Canavier 2009; Chandrasekaran et al 2011). For example, absence epileptic seizures have been postulated to arise from phase-locking with a small but noticeable phase lag between thalamic and cortical sites (Perez Velasquez et al 2007).…”

“…If the appropriate PRCs for neural oscillators can be measured, then we do not require any additional assumptions regarding the intrinsic dynamics of the oscillator or the form of the coupling. These results can be extended to apply to two populations of reciprocally coupled oscillators (Achuthan and Canavier 2009; Chandrasekaran et al 2011) under additional assumptions regarding the dynamics of each population.…”

Globally attracting synchrony in a network of oscillators with all-to-all inhibitory pulse coupling

“…Stability of the two cluster mode can be calculated by treating the two clusters as two oscillators, provided synchrony within each cluster is stable [37], otherwise a more complicated analysis applies [38]. Since we are only interested in cases in which synchrony is stable, this caveat is not relevant.…”