Course description
As with modeling any complex system, detailed mathematical modeling of neural networks can quickly become too complicated to allow analysis, or even simulation, of the resulting systems of equations. In this course, we will explore methods of simplifying neuron and neural network models to better understand their behavior. We will examine how these models, coupled with experimental data, can yield insights into the functions of neural networks in the brain.
We will start with examining the behavior of networks of simplified firing rate neuron models, exploring how network structure can influence the dynamics of these networks. We will discuss approaches to developing effective population equations that describe the dynamics of neurons, especially in the case with correlations within the network structure
We will derive spiking neuron models using the Hodgkin-Huxley formalism and investigate methods of simplifying these equations while retaining their essential behavior.
The course will explore standard mathematical techniques of taming equations, such as time/space scale separation, asymptotic analysis, ensemble averaging, ad hoc approximation, and neglect of unpleasant details. We may study simplified neuron models such as the integrate-and-fire neuron, the theta-neuron, the Poisson spiking model, and the population density approach. We will show how these models can be applied to neuroscience questions such as feature selectivity in local corticalcircuits, working memory, analysis of auditory or visual signals, and/or the estimation of connectivity patterns in neuronal networks.
No previous experience in neuroscience required. Moderate mathematical sophistication (e.g. basic familiarity with differential equations) will be assumed.