Dynamical Systems Tools and the Study of Neural Oscillations

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The brain is made up of cells called neurons which communicate with each other through electrical signals. The signal produced by a single neuron is a periodic train of pulses, with properties that depend on the characteristics of the individual neuron. Groups of neurons form networks which perform the functions of the brain such as storing and retrieving memories. The electrical activity of each network is generally also periodic, with properties that may change depending on the function the network performs and the state of the individual, for example sleeping vs waking. This proposal comprises studies of mathematical models ranging from detailed models of particular neurons which incorporate experimental data, to models of specific brain regions and models incorporating properties common to many brain networks. The research will use mathematics to understand how networks transform the periodic activity in a single neuron into the collective electrical activity of the network, often called a brain rhythm. One particular focus will be on how the time delay in the communication between neurons affects the activity that is produced. The results of this work may help understand how normal brain rhythms, such as those that occur during sleep, and destructive brain rhythms, such as epilepsy, occur.