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Compromising: a lesson from coupled oscillators

January 4, 2012

In a recent paper about the neural control of breathing, we looked at the phenomenon of phase-locking in bursting neurons. We used a model (available on ModelDB) that consists of a network of bursting cells, each with an independent natural burst frequency. When the oscillators are coupled, their bursts can become phase-locked at a frequency that is somewhere between the smallest and largest natural burst frequencies of the individuals. This is known as the “compromise frequency”. We are currently working to understand this for bursting cells, and an obvious starting point is to consider the simpler case of complete synchronization in a pair of non-bursting oscillators.

The synchronization of two coupled oscillators is well understood. For example, see Chapter 8 in Synchronization: a universal concept in nonlinear sciences by Pikovsky, Rosenblum, and Kurths. The perspective from dynamical systems theory can be found in this nice article by Josic et al and in greater detail in Chapter 10 in Dynamical Systems in Neuroscience by Izhikevich (available free online). While looking for an explanation of the frequency of a coupled system, I was pleased to find a nice example in Nonlinear dynamics and chaos by Strogatz (section 8.6). In typical fashion, Strogatz makes everything crystal clear by combining a simple example with just the right amount of analysis. It is commonly known that two coupled oscillators will have a compromise frequency that lies somewhere between their respective natural frequencies. In his simple example, Strogatz shows how the resonant frequency is a function of the coupling strengths between the two oscillators and their natural frequencies. His derivation seems quite elegant to me.

My challenge now is to understand this for nonlinear oscillators, as discussed by Heath and Wiesenfeld in 1998. Bigger challenges lie in analyzing bursting neurons as well as different coupling architectures. The experimental results of our paper showed large phase differences (80 ms) between pairs of cells. To replicate this in our model, we defined clusters of cells such that connections within any given cluster were stronger than connections between clusters. The result is that all cells in the network become phase-locked, but the phase differences are significantly smaller between cells in the same cluster as compared to cells across two different clusters. In this case, each cluster becomes an oscillator, and even these populations find a compromise frequency between their own natural frequencies.

Just imagine if human society could master the science of the “compromise frequency”. If people are oscillators within clusters, what coupling strengths and architectures are necessary for synchronization? I wonder.

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