Dynamical Systems in Neuroscience:

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34 Electrophysiology of Neuronsm (V)1.00.80.60.40.20.0-80 -40 0 40 80V (mV)τ(V) (ms)1.61.41.21.00.80.60.40.2-120 -80 -40 0 40 80V (mV)Figure 2.9: The activation function m ∞ (V ) and the time constant τ(V ) of the fasttransient K + current in layer 5 neocortical pyramidal neurons (modified from Korngreenand Sakmann 2000).open the channels, and those that inactivate or close them; see Fig. 2.8. Accordingto the tradition initiated in the middle of 20th century by Hodgkin and Huxley, theprobability of an activation gate to be in the open state is denoted by the variable m(sometimes the variable n is used for K + and Cl − channels). The probability of aninactivation gate to be in the open state is denoted by the variable h. The proportionof open channels in a large population isp = m a h b , (2.8)where a is the number of activation gates and b is the number of inactivation gates perchannel. The channels can be partially (0 < m < 1) or completely activated (m = 1);not activated or deactivated (m = 0); inactivated (h = 0); released from inactivationor deinactivated (h = 1). Some channels do not have inactivation gates (b = 0), hencep = m a . Such channels do not inactivate, and they result in persistent currents. Incontrast, channels that do inactivate result in transient currents.Below we describe voltage- and time-dependent kinetics of gates. This descriptionis often referred to as the Hodgkin-Huxley gate model of membrane channels.2.2.2 Activation of persistent currentsThe dynamics of the activation variable m is described by the first-order differentialequationṁ = (m ∞ (V ) − m)/τ(V ) (2.9)where the voltage-sensitive steady-state activation function m ∞ (V ) and the time constantτ(V ) can be measured experimentally: They have sigmoid and unimodal shapes,respectively, as in Fig. 2.9; see also Fig. 2.20. The steady-state activation functionm ∞ (V ) gives the asymptotic value of m when the potential is fixed (voltage-clamp).Smaller values of τ(V ) result in faster dynamics of m.
Electrophysiology of Neurons 35-30 mV(a)-60 mVpresteppotential V 0-100 mVstep potentialsV scurrent, I (pA)0-20-40-60I s(c)I(V)-80(b)I(t)-5 pAV s-100-80 -60 -40 -20 0membrane voltage, V (mV)10.5(e)activationactivationI s-85 pAm sgating varitable, m10.80.60.40.2m s(d)m (V)m(t)00 2 4 6 8 10time (ms)V s0-80 -60 -40 -20 0membrane voltage, V (mV)Figure 2.10: An experiment to determine m ∞ (V ). Shown are simulations of the persistentNa + current in Purkinje cells (see Sect. 2.3.5).
Page 1: Eugene M. IzhikevichThe Neuroscienc
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128 Two-Dimensional SystemsabcdFigu
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130 Two-Dimensional SystemsFigure 4
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134 Conductance-Based Models• If
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136 Conductance-Based Modelsinward(
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148 Conductance-Based Models1I=651I
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150 Conductance-Based Modelshave co
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166 Conductance-Based Models
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168 Bifurcationssaddle-node bifurca
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172 Bifurcations0.5v 2V-nullclineK
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186 Bifurcationsmembrane potential,
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224 Bifurcations19. [M.S.] A leaky
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226 Excitabilityspike?spikerestrest
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274 ExcitabilityK + activation gate
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280 Simple Modelsspikemembrane pote
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408 Solutions to Exercises, Chap. 3
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442 Referencesterneurons mediated b
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444 ReferencesDickson C.T., Magistr
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448 Referencestional Journal of Bif
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450 ReferencesMarkram H, Toledo-Rod
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