neural_models

neural_models - 1 Single-Cell and Mean Field Neural Models...

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Unformatted text preview: 1 Single-Cell and Mean Field Neural Models Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 2 The neuron is the basic unit of the nervous system. It is a normal cell that has been adapted morphologically and in terms of protein expression for direct communication with other neurons, with various receptors (e.g., photoreceptors), and with muscle tissue. Figure 1: Stained single neuron 3 Figure 2: Population of interconnected stained neurons 4 Figure 3: Structure of a neuron Dendrites : input pathways, from afferent neurons Soma or cell body : integration center Axon : output pathway Synapses : structures where electrical signals are converted to chemical signals and transmitted to efferent neurons Myelin : insulating cells on vertebrate axons 5 Neurons often encode information in the frequency of spik- ing, or electrical impulse firing rate . How are electricity and the neuron related? Answer: ion channels. Figure 4: Atomic model of the bacterial KcsA K + channel A concentration gradient is maintained across the plasma membrane by ion pumps , which hydrolyze ATP to provide the energy to pump ions upstream. An ion channel is a portal that allows ions of a speci c type (e.g., potassium ions, K + ) to 6 move through the membrane. The channel is like a gate; when it is open the ions move through it in a downstream fashion, powered by the concentration gradient. plasma membrane channel KCl KCl K + Figure 5: An ion channel allows speci c types of ions to ow through As ions move through the channels an electrical potential develops, which opposes the concentration gradient. The total ion ux across the membrane is described by the Nernst-Planck equation : J =- D dC dx + zCF RT d dx (1) where C is ion concentration and is the electrical poten- tial. ( D is the di usion coe cient, R is the gas constant, T is temperature, and F is the Faraday constant.) The rst term describes the concentration gradient, while the second describes the electrical potential. Eventually an equlibrium 7 is reached ( J = 0). The equilibrium potential is called the Nernst potential : V ion = RT zF ln [ion] out [ion] in (2) where V ion = in- out and z is the ion valence. Typically, [ K + ] in > [ K + ] out (3) so V K < 0, [ Na + ] in < [ Na + ] out (4) and [ Ca 2+ ] in < [ Ca 2+ ] out (5) so V Na , V Ca > 0. In fact, typical values are: V K - 70 mV (6) V Na 50 mV (7) V Ca 100 mV (8) The resting potential is the weighted average of the Nernst potentials, with the weights being the macroscopic con- ductance ( g ) of the channels in the membrane: 8 V rest = g Na V Na + g Ca V Ca + g K V K g Na + g Ca + g K (9) Here, macroscopic conductance re ects the permeability of an ion type through all open channels permeable to that ion type. For example, g Ca is the total permeability of Ca 2+ through all open Ca 2+ channels....
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neural_models - 1 Single-Cell and Mean Field Neural Models...

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