Quantifying Dynamics of Synchronization

My research looks at the dynamical behavior of biological neural networks and the model I’m using partially simulated the Pre-Botzinger complex of the brainstem found in mammals that regulates breathing. The neurons in the network rhythmically activate and deactivate together to form a synchronized pattern. The model I created in python plots the 3 different state variables of the neurons in the network but I needed to create a different measure to more quantitatively indicate if the network was synchronized or not.

My solution to this was to take the minimum and maximum values for the voltage across all neurons during the simulation and transform it into an interval between 0 and pi. Then using the formula e^(ix) I could turn each neuron’s voltage into a vector in the complex plane. So if the neurons had similar voltages their vectors would be pointing in the same direction, I could then sum them and divide by the number of neurons to give a number between 0 and 1, 0 if the neurons all had very different values and 1 if they all had similar values. Then I could plot this synchronization variable over time as the simulation ran to give me and indication of when the network was synchronized. This has provided useful information to the classification and analysis of the different structures neural networks can have and will help my research continue into exploring this area of neuroscience.


A network of six neurons all connected and their voltage.


The associated synchronization variable for the network

The next step in my research will be to find what local characteristics of the topological structure of this networks are necessary to their dynamics. Once we can find certain properties of biological networks that dictate the complicated behavior, we can move closer to finding out how the inherent architecture of neural networks plays a role in their behavior.