An Introduction to Modeling Neuronal Dynamics

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Börgers, Christoph. An Introduction to Modeling Neuronal Dynamics by Christoph Börgers. -
XIII, 457 p. 356 illus., 186 illus. in color. online resource. - (Texts in Applied Mathematics, 66) Content notes : Vocabulary and Notation -- Modeling a Single Neuron -- The Nernst Equilibrium -- The Classical Hodgkin-Huxley ODEs -- Numerical Solution of the Hodgkin-Huxley ODEs -- Three Simple Models of Neurons in Rodent Brains -- The Classical Hodgkin-Huxley PDEs -- Linear Integrate-and-fire (LIF) Neurons -- Quadratic Integrate-and-fire (QIF) and Theta Neurons -- Spike Frequency Adaptation -- Dynamics of Single Neuron Models -- The Slow-fast Phase Plane -- Saddle-node Collisions -- Model Neurons of Bifurcation Type 1 -- Hopf Bifurcations -- Model Neurons of Bifurcation Type 2 -- Canard Explosions -- Model Neurons of Bifurcation Type 3 -- Frequency-current Curves -- Bistability Resulting from Rebound Firing -- Bursting -- Modeling Nuronal Communication -- Chemical Synapses -- Gap Junctions -- A Wilson-Cowan Model of an Oscillatory E-I Network -- Entertainment, Synchronization, and Oscillations -- Entertainment by Excitatory Input Pulses -- Synchronization by Fast Recurrent Excitation -- Phase Response Curves (PRCs) -- Synchronization of Two Pulse-coupled Oscillators -- Oscillators Coupled by Delayed Pulses -- Weakly Coupled Oscillators -- Approximate Synchronization by a Single Inhibitory Pulse -- The PING Model of Gamma Rhythms -- ING Rhythms -- Weak PING Rhythms -- Beta Rhythms -- Nested Gamma-theta Rhythms -- Functional Significance of Synchrony and Oscillations -- Rhythmic vs. Tonic Inhibition -- Rhythmic vs. Tonic Excitation -- Gamma Rhythms and Cell Assemblies -- Gamma Rhythms and Communication -- Synaptic Plasticity -- Short-term Depression and Facilitation -- Spike Timing-dependent Plasticity (STDP) -- Appendices -- A. The Bisection Method -- Fixed Point Iteration -- Elementary Probability Theory -- Smooth Approximations of Non-smooth Functions -- Solutions to Selected Homework Problems. 9783319511719 * Neurosciences. Statistical physics. Vibration. Mathematical Models of Cognitive Processes and Neural Networks. Mathematical and Computational Biology. Neurosciences. Statistical Physics and Dynamical Systems. Vibration, Dynamical Systems, Control. * Series