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Communication between cells

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R I1I1 Biology Electrical equivalent I2I2 I = I 1 + I 2 I

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Nernst Equation The Nernst equation relates the potential difference to the concentration difference in equilibrium: C i / C o = exp { - Z e (V i -V o )/kT } or V i - V o = (kT/Ze) ln (C o /C i ) with e = charge electron Z = valence of ion k = Boltzmann constant T = temperature C i (C o ) = concentration inside (outside) membrane

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Example of Nernst equation t=0 R + 50 N a + 50 Cl - 100 N a + 100 Cl - 100 t >> 0 R + 50 N a + 64 Cl - 114 N a + 86 Cl - 86 ΔV = 0ΔV = - 10 If both N a and Cl, but not R + can migrate through the membrane.

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Nernst equation for squid axon

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Problems with Nernst equation considers only a single ion. If multiple ions are involved, it assumes equal permeability for all ions Applies only to passive transport ions migrate independently of each other

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Goldman/Hodgkin/Katz equation The Nernst equation relates the potential difference to the concentration difference in equilibrium for a single neuron. When several ions are involved, we obtain for equilibrium : P k [K] o +P Na [Na] o +P Cl [Cl] i ΔV = (RT/F) ln --------------------------------- P k [K] i +P Na [Na] i +P Cl [Cl] o with Pi = permeability of ion i [K] i/o = concentration inside/outside F = Faraday constant T = temperature ΔV = potential difference across membrane

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Current through an ion channel

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Schematic overview of the active membrane

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Hodgkin & Huxley: Current through an ion-channel Ohm’s law Conductance G=1/R Conductance G is a product of maximal conductance g Ca and the fraction of open channels m 3 h R V ion outside inside 0 mV V mV I 0 mV V mV I

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State: Gating kinetics

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Open State: Gating kinetics

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Open Closed mm mm m Probability: State: (1-m) mm mm Gating kinetics

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V (mV) mm mm Open Closed mm mm m Probability: State: (1-m) mm mm Channel Open Probability: Gating kinetics

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-150-100-50050100150 0 0.2 0.4 0.6 0.8 1 m (V) -150-100-50050100150 0 2 4 6 8 x 10 -3 m (V) m (s) -150-100-50050100150 0.2 0.4 0.6 0.8 1 1.2 h (V) V clamp (mV) -100-50050100 0 2 4 6 8 10 h (V) V clamp (mV) h (s) Parameter fitting (2)

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-150-100-50050100150 0 0.2 0.4 0.6 0.8 1 m (V) -150-100-50050100150 0 2 4 6 8 x 10 -3 m (V) m (s) -150-100-50050100150 0.2 0.4 0.6 0.8 1 1.2 h (V) V clamp (mV) -100-50050100 0 2 4 6 8 10 h (V) V clamp (mV) h (s) Parameter fitting (2)

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V (mV) mm mm Open Closed mm mm m Probability: State: (1-m) mm mm Channel Open Probability: Gating kinetics g m (t) = g m, max (m ∞ - m 0 )(1 - e -t/τ ) g h (t) = g h, max (h ∞ - h 0 )e -t/τ m3hm3h time g gmgm ghgh g Na

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V mV 0 mV V mV 0 mV ICIC I Na Kirchoff’s law: Membrane voltage equation I Na = g max, Na m 3 h(V- V Na ) -C m dV/dt = g max, Na m 3 h(V-V na ) + g max, K n 4 (V-V K ) + g leak (V-V na )

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Actionpotential

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