Dynamical Systems in Neuroscience:
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Dynamical Systems in Neuroscience:

Dynamical Systems in Neuroscience: Dynamical Systems in Neuroscience:

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12.07.2015 Views

32 Electrophysiology of NeuronsFigure 2.7: To tease out neuronal currents, biologists employ an arsenal of sophisticated“clamp” methods, such as current-, voltage-, conductance-, and dynamic-clamp.current needed to stabilize the potential at the new value is a function of time, the prestepholding potential V c , and the step potential V s . First, the current jumps to a newvalue to accommodate the instantaneous voltage change from V c to V s . From (2.5) wefind that the amplitude of the jump is g inp (V s −V c ). Then, time- and voltage-dependentprocesses start to occur and the current decreases and then increases. The value at thenegative peak, marked by circle “o” in Fig. 2.6, depends only on V c and V s and it iscalled the instantaneous current-voltage (I-V) relation, or I 0 (V c , V s ). The asymptotic(t → ∞) value depends only on V s and it is called the steady-state current-voltage (I-V)relation, or I ∞ (V s ).Both relations, depicted in Fig. 2.6b, can be found experimentally (black dots) ortheoretically (curves). The instantaneous I-V relation usually has a non-monotone N-shape reflecting non-linear auto-catalytic (positive feedback) transmembrane processes,which are fast enough on the time scale of the action potential so that they could beassumed to have instantaneous kinetics. The steady-state I-V relation measures theasymptotic values of all transmembrane processes, and it may be monotone, as inthe figure, or not, depending on the properties of the membrane currents. Both I-Vrelations provide an invaluable quantitative information about the currents operatingon fast and slow time scale, and both are useful in building mathematical models ofneurons. Finally, when I ∞ (V ) = 0, the net membrane current is zero, and the potentialis at rest or equilibrium, which may still be unstable as we discuss in the next chapter.2.2 ConductancesIonic channels are large transmembrane proteins having aqueous pores through whichions can flow down their electrochemical gradients. The electrical conductance of indi-

Electrophysiology of Neurons 33ExtracellularNa +voltagesensorselectivityfilterchannelproteinmembrane+ + + +++ ++ ++ +++ + ++ +activationgateIntracellularinactivationgateClosed(not activated)Open(activated)Closed(inactivated)Figure 2.8: Structure of voltage-gated ion channels. Voltage sensors open activationgate and allow selected ions to flow through the channel according to their electrochemicalgradients. The inactivation gate blocks the channel (modified from Armstrong andHille 1998).vidual channels may be controlled by gating particles (gates), which switch the channelsbetween open and closed states. The gates are sensitive to one or more of the followingfactors:• Membrane potential. Example: voltage-gated Na + or K + channels.• Intracellular agents (second-messengers). Example: Ca 2+ -gated K + channels.• Extracellular agents (neurotransmitters and neuromodulators). Example: AMPA,NMDA, or GABA receptors.Despite the stochastic nature of transitions between open and closed states in individualchannels, the net current generated by a large population or ensemble of identicalchannels can reasonably be described by the equationI = ḡ p (V − E) (2.7)where p is the average proportion of channels in the open state, ḡ is the maximalconductance of the population, and E is the reverse potential of the current, i.e., thepotential at which the current reverses its direction. If the channels are selective for asingle ionic species, then the reverse potential E equals the Nernst equilibrium potential(2.1) for that ionic species, see Ex. 2.2.2.1 Voltage-gated channelsWhen the gating particles are sensitive to the membrane potential, the channels aresaid to be voltage-gated. The gates are divided into two types: Those that activate or

Electrophysiology of Neurons 33ExtracellularNa +voltagesensorselectivityfilterchannelprote<strong>in</strong>membrane+ + + +++ ++ ++ +++ + ++ +activationgateIntracellular<strong>in</strong>activationgateClosed(not activated)Open(activated)Closed(<strong>in</strong>activated)Figure 2.8: Structure of voltage-gated ion channels. Voltage sensors open activationgate and allow selected ions to flow through the channel accord<strong>in</strong>g to their electrochemicalgradients. The <strong>in</strong>activation gate blocks the channel (modified from Armstrong andHille 1998).vidual channels may be controlled by gat<strong>in</strong>g particles (gates), which switch the channelsbetween open and closed states. The gates are sensitive to one or more of the follow<strong>in</strong>gfactors:• Membrane potential. Example: voltage-gated Na + or K + channels.• Intracellular agents (second-messengers). Example: Ca 2+ -gated K + channels.• Extracellular agents (neurotransmitters and neuromodulators). Example: AMPA,NMDA, or GABA receptors.Despite the stochastic nature of transitions between open and closed states <strong>in</strong> <strong>in</strong>dividualchannels, the net current generated by a large population or ensemble of identicalchannels can reasonably be described by the equationI = ḡ p (V − E) (2.7)where p is the average proportion of channels <strong>in</strong> the open state, ḡ is the maximalconductance of the population, and E is the reverse potential of the current, i.e., thepotential at which the current reverses its direction. If the channels are selective for as<strong>in</strong>gle ionic species, then the reverse potential E equals the Nernst equilibrium potential(2.1) for that ionic species, see Ex. 2.2.2.1 Voltage-gated channelsWhen the gat<strong>in</strong>g particles are sensitive to the membrane potential, the channels aresaid to be voltage-gated. The gates are divided <strong>in</strong>to two types: Those that activate or

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