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Contents Lectures in Mathematical Neuroscience Paul C. Bressloff 1 Lectures in Mathematical Neuroscience 3 Lecture 1.1. 1.2. 1.3. 1. Single Neuron Models Conductance–based models Periodically forced neural oscillator Integrate–and–fire models 5 5 8 15 Lecture 2.1. 2.2. 2.3. 2.4. 2. Synaptic and Dendritic Processing Excitatory and inhibitory synapses Kinetic model of a synapse Dendritic filtering of synaptic inputs Synaptic plasticity 27 27 31 35 38 Lecture 3.1. 3.2. 3.3. 3.4. 3. Firing Rates, Spike Statistics and the Neural Code The neural code Spike statistics and the Poisson process Stochastically driven IF neuron Homogeneous population of IF neurons 49 49 53 56 60 Lecture 4.1. 4.2. 4.3. 4. Network Oscillations and Synchrony Phase reduction for synaptically coupled neural oscillators Phase-locked solutions Oscillations in large homogeneous networks 65 66 69 76 Lecture 5.1. 5.2. 5.3. 5.4. 5. Neural Pattern Formation Reduction to rate models Turing mechanism for cortical pattern formation Persistent localized states Traveling waves 81 81 83 89 95 Bibliography 103 i Lectures in Mathematical Neuroscience Paul C. Bressloff IAS/Park City Mathematics Series Volume XV, 2005 Lectures in Mathematical Neuroscience Paul C. Bressloff Neurons in the brain communicate with each other by transmitting electrical spikes known as action potentials (see Figure 1). An action potential propagates along the axon of a neuron until it reaches a terminal that forms ...
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