Biophysical Foundations

Transcription

1 Biophysical Foundations BENG/BGGN 260 Neurodynamics University of California, San Diego Week 1 BENG/BGGN 260 Neurodynamics (UCSD Biophysical Foundations Week 1 1 / 15

2 Reading Material B. Hille, Ion Channels of Excitable Membranes, Sinauer, 2001, Ch. 1 and 10, pp and C. Koch, Biophysics of Computation, Oxford Univ. Press, 1999, Ch. 1, pp P. Dayan and. Abbott, Theoretical Neuroscience, MIT Press, 2001, Ch. 5, pp E.M. Izhikevich, Dynamical Systems in Neuroscience, MIT Press, 2007, Ch. 2, pp BENG/BGGN 260 Neurodynamics (UCSD Biophysical Foundations Week 1 2 / 15

3 Neurodynamics: Overview Dendrite Cell body Node of Ranvier Axon Terminal Axon Nucleus Myelin sheath Membrane Dynamics Temporal Coding Signal Propagation Adaptation and earning Action Potential Generation Axon Dynamics STDP, reinforcement,... 1 ms ms 1 s years BENG/BGGN 260 Neurodynamics (UCSD Biophysical Foundations Week 1 3 / 15

4 Nernst Potential intracellular extracellular [K + ] i [K + ] o V i V o V m = V i V o membrane potential density ratio [K + ] o [K + ] i = n o n i energies per molecule = e Eo Ei Boltzman constant = e Vo V i / absolute temperature energies per mol ( = e Uo U i RT gas constant charge per molecule = C per valence index E rest = V m euilibrium = room T {}}{ ln [K+ ] o [K + ] i = ln 10 }{{} [K + ] o log 10 [K + ] i room T E energy Fermi level Boltzman tail T n density BENG/BGGN 260 Neurodynamics (UCSD Biophysical Foundations Week 1 4 / 15

5 Nernst Potential TABE 1.3 Free Ion Concentrations and Euilibrium Potentials for Mammalian Skeletal Muscle Intracellular concentration Ion o (mm Extracellular concentration (mm Euilibrium potential a (mv Ion Ion i Na K Ca nm 15, Cl b 29 b 90 b a Calculated from Euation 1.11 at 37 C (E rest = [Ion]o ln [Ion] i. b Calculated assuming a 90-mV resting potential for the muscle membrane and that Cl ions are a euilibrium at rest. Hille 2001, p. 17 BENG/BGGN 260 Neurodynamics (UCSD Biophysical Foundations Week 1 5 / 15

6 Nernst-Planck i inside o outside x 0 J [K + ] D i [K + ] o Euilibrium between diffusion and Diffusion: (chemical ( J D = D [K + ] drift: (electrical x V o V m J E V i 0 x Drift: J E = [K + ]µ ( E = µ[k + ] V J K + = J D + J E = 0 euilibrium D d[k+ ] dx µ[k + ] dv dx = 0 D µ d[k + ] [K + ] = dv d ( ln [K + ] D µ ln [K+ ] o [K + ] i = (V o V i = V m = E rest with D µ = indeed, D = µ Dissipation-fluctuation (Einstein BENG/BGGN 260 Neurodynamics (UCSD Biophysical Foundations Week 1 6 / 15

7 Membrane Capacitance Charge Density ρ E +Q Q + V i + Q A Q A V o Electric 1 Q Field E ε A x V i Potential V V m = E V o x x capacitance = differential charge accumulation with voltage across insulator (membrane Q ε A E = ρ ε, or de dx = ρ ε V = E, or dv dx = E Q = C m V m with C m = ε A V m ε 0.01 F/m2 Dynamics: V i V o I or I C m : C m dv m dt = I (V m,... BENG/BGGN 260 Neurodynamics (UCSD voltage Biophysical dependent Foundations Week 1 7 / 15

8 Membrane Dynamics K + : current density ( J K = µ[k] ln[k] + V current I K µ[k] A approximate arguments... more rigorous later ( ln [K]o g K E K V m C m average concentration membrane thickness Ohmic approximation effective cross-section [K] i + V o V i = g K (V m E K { gk = µ[k] A E K = [K]o ln [K] i K +, Na + : V m g K g Na C m Euilibrium: E rest = V m = g KE K + g Na E Na g K + g Na E K E Na Not uite... BENG/BGGN 260 Neurodynamics (UCSD Biophysical Foundations Week 1 8 / 15

9 Goldman-Hodgkin-Katz i inside [K] i Vi 0 o outside x [K](x [K] o V m V o x 0 x J K J K = µ ( [K] + [K] V CURRENT DENSITY, µa/cm 2 DIFFUSION DRIFT 1-D, along x perpendicular to membrane: ( d[k] J K = µ dx + [K]dV dx J K = constant (flow conservation dv dx = constant (assumption! = Vm d[k] dx = J K µ + V m [K] BENG/BGGN 260 Neurodynamics (UCSD Biophysical Foundations Week 1 9 / 15

10 Goldman-Hodgkin-Katz Continued d[k] dx = J K µ + V m [K] [ ( d ln J K V m µ + V ] m [K] J K (1 e Vm/ J K = µv m Ohm s limit: [K] i = [K] o = [K] J K = µv m = dx = µv m [K] d[k] J K µ + Vm dx = [K] V m ([K] i e Vm/ [K] o [K] i e Vm/ [K] o 1 e Vm/ = ln J K + µvm [K] o J K + µvm [K] i GHK current euation BENG/BGGN 260 Neurodynamics (UCSD Biophysical Foundations Week 1 10 / 15

11 Goldman-Hodgkin-Katz Current aw Hille 2001, Fig. 14.2, p. 447 BENG/BGGN 260 Neurodynamics (UCSD Biophysical Foundations Week 1 11 / 15

12 Goldman-Hodgkin-Katz imits J K = µ KV m [K] i e Vm/Vt [K] o V t = 1 e Vm/Vt 25.9mV 1 J K = 0 for [K] i = e Vm/Vt [K] o, or V m = V t ln [K]o [K] i = E K (OK! µ K 2 J [K] i V m = P K [K] i V m µ P K = K K µ K [K] ov m = P K [K] o V m permeability V m V t V m V t Current asymptotes to a linear conductance limited by the sourcing concentration at active terminal. 3 J K g K (V m E K for J K 0 (V m E K? BENG/BGGN 260 Neurodynamics (UCSD Biophysical Foundations Week 1 12 / 15

13 GHK Ohmic Approximation V m E K = ln [K] o [K] i e Vm = e E K e Vm E K J K µe [K] K i [K] i [K] o ( 1 Vm E K [K] o 1 [K] i V [K] o (1 m E K or: J K g K (V m E K with [K] i [K] o µ g }{{} K = ln [K] o [K] i [K] o }{{} [K] i [K] o [K] channel }{{} i conductance permeability average concentration ( 1 V m E K BENG/BGGN 260 Neurodynamics (UCSD Biophysical Foundations Week 1 13 / 15

14 Goldman-Hodgkin-Katz Euilibrium J K = µ KV m J Na = µ NaV m [K] i e Vm / [K] o 1 e Vm / [Na] i e Vm / [Na] o 1 e Vm / Euilibrium : J K + J Na 0, neglecting common terms : µ K ( [K] i e Vm / [K] o + µ Na ( [Na] i e Vm / [Na] o 0 E rest = V m = ln µ K [K] o + µ Na [Na] o µ K [K] i + µ Na [Na] i usually expressed in terms of permeabilities: P K = µ K and P Na = µ Na BENG/BGGN 260 Neurodynamics (UCSD Biophysical Foundations Week 1 14 / 15

15 GHK Neuromorphism (for the silicon enthusiast { (K + [K] i [K] o J K P K [K] i for V m V t = P K [K] o for V m V t G gate SiO 2 J channel p J 0 e κ Vg Vt e Vs Vt for V ds V t = S (holes D J 0 e κ Vg Vt e V d Vt for V ds V t source drain Si P K J 0 e κ Vg Vt controlled by V g, gate voltage IPID BIAYER : SIICON MOSFET : (pmos* neuromorphisms : (imperfect [K] i e Vs Vt controlled by V s, source voltage [K] o e V d Vt controlled by V d, drain voltage V m V ds = V d V s, i.e. : V i V s V o V d * C.A. Mead, Analog VSI and Neural Systems, Addison-Wesley, 1989 BENG/BGGN 260 Neurodynamics (UCSD Biophysical Foundations Week 1 15 / 15