Stability and Oscillations in Mathematical Models of Neural Systems

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The brain is made up of cells called neurons which communicate with each other through electrical signals. It is well known that 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. This proposal comprises a study of mathematical models of several different neural networks, some corresponding to specific brain regions and others incorporating properties common to many brain networks. The study will use mathematics to understand how networks transform the periodic activity in a single neuron into the collective electrical activity of the network. One particular focus will be on how the time delay in the communication between neurons affects the activity that is produced. One project in this study will contribute to our understanding of human balance control. A second project will contribute to our understanding of how normal brain rhythms are produced in the hippocampus region of the brain. This in turn may help understand how destructive rhythms, such as epilepsy, occur.