Improving the Analysis of Coupled Neuronal Networks

Robert Allen, 3rd year PhD, University of Nottingham

BACKGROUND:

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While I am technically a Mathematical Neuroscientist, the word mathematical is doing a lot of the heavy lifting there. My project is based on the concept of coupled oscillator networks. We can mathematically model a neuron as a system of voltage equations with an oscillating pattern and then model the brain as a network of these oscillators coupled together and affecting each other.

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The standard technique for analysing these systems for the last 4 decades has been to approximate each neuron with a single phase coordinate and then see how the network structure affects the stability of certain brain activity states.

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The problem with this method is that the approximation is incredibly inaccurate if the interactions between neurons are strong compared to the size of the oscillations of the uncoupled neuron, which in a brain, they are. This is where our project comes in, we introduce a second coordinate, known as an isostable coordinate, to give a sense of the amplitude of the oscillations in the hope that we can make a more accurate approximation, and hence do better analysis of the full coupled oscillator network.

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METHODOLOGY and RESULTS:

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So far, I’ve taken the methods that already exist for single neurons described with phase-isostable coordinates and extended them to use in networks. I then derived the conditions that describe when different brain activity states exist in our phase-isostable network and when these states are stable against perturbations. We’ve investigated what this looks like for networks of very simple neuronal models. The figure shows the phase-isostable coordinates for the Morris-Lecar model.

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FUTURE WORK:

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At the moment I’m working on repeating the analysis we’ve already done but for systems where the oscillations are induced by delay differential equations, not ordinary differential equations. I also want to properly incorporate synaptic coupling into my methods, it’s possible in theory, I just need to sit down and do the math!

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FUNDED BY:

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CONTACT:

or you could try to find me on Facebook (Robert Allen, I’m stroking a grey cat in my profile picture).