nc 02 Neurons
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Membranes, Neurons, and Action Potentials Some Organizational Stuff... For credit points: At the end of the semester you will have a written test. 28.07.2014: 8.15-8.45; Waldweg 26, Altbau – 1.201 Repeated test: 15.09.2014: 8.15-8.45; Waldweg 26, Altbau – 1.201 2-3 times during the semester you will get some exercises. They are voluntary, but useful for everybody. Lecture for basics of (experimental) neuroscience: Neuro- und Verhaltensbiologie – Prof. Fiala; Wed. 10-12; Mikrobiologie; MN06 Slides can be downloaded from http://www.physik3.gwdg.de/cns/... The Interdisciplinary Nature of Computational Neuroscience The Applications Artificial Neural Networks (Problem Solving) Chip Design new approaches questions Computer Science Non-Linear Dynamics Special Robotics Medicine Psychology Computational Neuroscience tools Social Networks answers predictions ElectroBiophysiology chemistry The World (Problems) Marketing The Methods Information Theory Computer Vision Psychophysics facts CNS Neuroanatomy Neurophysiology The Brain (Substrate) Systems Areas Local Nets Neurons Synapses Mathematics Physics Chemistry Biology The Sciences (Fundament) Molecules Levels of Information Processing in the Nervous System 1m CNS 10cm Sub-Systems 1cm Areas / „Maps“ 1mm Local Networks 100mm Neurons 1mm Synapses 0.01mm Molecules Structure of a Neuron: At the dendrite the incoming signals arrive (incoming currents) At the soma current are finally integrated. At the axon hillock action potential are generated if the potential crosses the membrane threshold The axon transmits (transports) the action potential to distant sites CNS At the synapses are the outgoing signals transmitted onto the dendrites of the target neurons Systems Areas Local Nets Neurons Synapses Molecules The Underlying Structure of Neurons: The Membrane CNS Systems Areas Local Nets Neurons • Potassium (K+) • Sodium (Na+) • Chloride (Cl-) • Calcium (Ca++) • Protein Anions (A-) Synapses Molecules Selective Ion Channels How complicated are Ion Channels? For instance, a sodium (Na+) channel looks (schematically!) something like this: How complicated are Ion Channels? For instance, a sodium (Na+) channel looks (schematically!) something like this: Different Types of Ion Channels membrane + out - in always open opening depends on membrane potential transports ions against concentration gradient What is the influence of ion channels on the neuron’s dynamics? At synapses: neurotransmitter + out fast slow opening depends on neurotransmitters - in The Influence of Passive Channels The Influence of Passive Channels + Outside of cell Although the inside of the cell is negatively charged, amounts of positive potassium stays outside. But why? Inside of cell - Potassium diffuses via the ion channels to the outside → Equilibrium between Force of Diffusion and Electrostatic Force What are the effects of this equilibrium? What are the effects of the equlibrium at a membrane? Difficult question Let’s make a detour and think about our atmosphere (perhaps this helps) Our Atmosphere and the Distribution of Particles ππ πππ = πππππ Equilibrium: Sink current ππ πππ air particles Earth’s surface π: Boltzmann’s constant π: Temperature Height β Diffusion current πππππ ππ(β) μππ β π β = −π· β πβ β π μππ 1 ππ − π π· πβ = β=0 ? Number of particles π(β) Einstein relation: π· = μππ π(π): number of particles in height π μ: mobility π: mass π: gravity constant π·: diffusion constant π(β=0) − μππβ π = ln π· π0 π β = π0 ππ₯π − π β = π0 ππ₯π − π0 β π β = 0 μππβ π· ππβ ππ πΈππππ£ π β = π0 ππ₯π − ππ Gravitational energy πΈππππ£ Barometric formula Our Atmosphere and the Distribution of Particles Barometric formula π β = π0 ππ₯π − Sink current ππ πππ Height β Diffusion current πππππ π1 β π β1 π(β): number of particles in height β πΈππππ£ ππ π: Boltzmann’s constant π: Temperature πΈ1 = π0 ππ₯π − ππ β1 β2 air particles Earth’s surface Number of particles π(β) π2 β π β2 = π0 ππ₯π − πΈ2 ππ πΈ1 π1 ππ = ππ₯π − πΈ1 − πΈ2 = π2 π ππ₯π − πΈ2 ππ 0 ππ π0 ππ₯π − π1 ΔπΈ = ππ₯π − π2 ππ Boltzmann distribution Our Atmosphere and the Distribution of Particles = The Membrane with Channels and Ions Barometric formula π β = π0 ππ₯π − Sink current ππ πππ Height β Diffusion current πππππ channel membrane potassium ions air particles Earth’s surface π1 β1 β2 π2 Number of particles π(β) π1 β π β1 π(β): number of particles in height β πΈππππ£ ππ π: Boltzmann’s constant π: Temperature πΈ1 = π0 ππ₯π − ππ π2 β π β2 = π0 ππ₯π − πΈ2 ππ πΈ1 π1 ππ = ππ₯π − πΈ1 − πΈ2 = π2 π ππ₯π − πΈ2 ππ 0 ππ π0 ππ₯π − π1 ΔπΈ = ππ₯π − π2 ππ Boltzmann distribution Our Atmosphere and the Distribution of Particles = The Membrane with Channels and Ions Electric current π: Boltzmann’s constant π: Temperature Diffusion current πππππ Sink current ππ πππ channel π1 πππ’π‘ π2 πππ membrane potassium ions Now, we take as energy difference ΔπΈ the difference in electric energy ΔπΈππππ = π§ππ. πππ [πΎ + ]ππ π§ππ = = ππ₯π − πππ’π‘ ππ [πΎ + ]ππ’π‘ π1 ΔπΈ = ππ₯π − π2 ππ π: potential π§: moles of charges π: elementary charge Boltzmann distribution The Membrane with Channels and Ions: Nernst Equation + πππ [πΎ ]ππ π§ππ = = ππ₯π − πππ’π‘ ππ [πΎ + ]ππ’π‘ Force of Diffusion Electrostatic Force π=− channel ππ πππ ππ π§π πππ’π‘ πππ’π‘ membrane potassium ions π: Boltzmann’s constant π: Temperature π: potential π§: moles of charges π: elementary charge πππ π π πππ π=− ππ π§πΉ πππ’π‘ Nernst Equation π π = π πΉ π : universal gas constant πΉ: Faraday constant For π = 310πΎ (36.85 °πΆ) π=− 26.7ππ πππ ππ π§ πππ’π‘ A membrane with open channels (semipermeable) induces an electric potential: the membrane potential The Influence of Passive Channels + Outside of cell Although the inside of the cell is negatively charged, amounts of positive potassium stays outside. But why? Inside of cell - Potassium diffuses via the ion channels to the outside → Equilibrium between Force of Diffusion and Electrostatic Force What are the effects of this equilibrium? A membrane with open channels (semipermeable) induces an electric potential: the membrane potential Different Types of Ion Channels membrane + out - in always open opening depends on membrane potential transports ions against concentration gradient membrane potential At synapses: neurotransmitter + out fast slow opening depends on neurotransmitters - in The Role of Ion Pumps in General For instance the sodium(Na+)-potassium(K+) pump: Nernst Equation π π πππ π=− ππ π§πΉ πππ’π‘ in general Goldman Equation π π π= ππ πΉ π + πΎ [π΄π− ]ππ π πππ+ [ππ ]ππ’π‘ + π ππ΄− π + π πΎ − [π΄ ]ππ’π‘ π πππ+ [ππ ]ππ + π ππ΄− π π π: permeability The Role of Ion Pumps in Neurons K+ Na+ Herleitung Goldman Equation π π π= ππ πΉ π + πΎ [π΄π− ]ππ π πππ+ [ππ ]ππ’π‘ + π ππ΄− π + π πΎ − [π΄ ]ππ’π‘ π πππ+ [ππ ]ππ + π ππ΄− π π simulation Goldman-Hodgkin-Katz Equation neuron π π ππΎ+ [πΎ + ]ππ’π‘ +πππ+ [ππ+ ]ππ’π‘ +ππΆπ− [πΆπ − ]ππ π= ππ πΉ ππΎ+ [πΎ + ]ππ +πππ+ [ππ+ ]ππ +ππΆπ− [πΆπ− ]ππ’π‘ Different Types of Ion Channels membrane + out - in always open opening depends on membrane potential transports ions against concentration gradient At synapses: neurotransmitter + out fast slow opening depends on neurotransmitters - in Schematic Diagram of a Synapse: Receptor ≈ Channel Transmitter Vesicle CNS Systems Dendrite Areas Local Nets Neurons Synapses Molecules What happens at a chemical synapse during signal transmission: Pre-synaptic action potential The pre-synaptic action potential depolarises the axon terminals and Ca2+-channels open. Ca2+ enters the pre-synaptic cell by which the transmitter vesicles are forced to open and release the transmitter. Concentration of transmitter in the synaptic cleft Post-synaptic action potential Thereby the concentration of transmitter increases in the synaptic cleft and transmitter diffuses to the postsynaptic membrane. Transmitter sensitive channels at the postsyaptic membrane open. Na+ and Ca2+ enter, K+ leaves the cell. An excitatory postsynaptic current (EPSC) is thereby generated which leads to an excitatory postsynaptic potential (EPSP). The Effect of Synaptic Inputs on the Membrane Potential Goldman-Hodgkin-Katz Equation: π π ππΎ+ [πΎ + ]ππ’π‘ +πππ+ [ππ+ ]ππ’π‘ +ππΆπ− [πΆπ − ]ππ π= ππ πΉ ππΎ+ [πΎ + ]ππ +πππ+ [ππ+ ]ππ +ππΆπ− [πΆπ− ]ππ’π‘ Task for everybody: Discuss with your neighbors possible problems using the GoldmanHodgkin-Katz Equation to describe the membrane potential influenced by synaptic inputs! • Permeability depends on Voltage The Effect of Synaptic Inputs on the Membrane Potential Goldman-Hodgkin-Katz Equation: π π ππΎ+ [πΎ + ]ππ’π‘ +πππ+ [ππ+ ]ππ’π‘ +ππΆπ− [πΆπ − ]ππ π= ππ πΉ ππΎ+ [πΎ + ]ππ +πππ+ [ππ+ ]ππ +ππΆπ− [πΆπ− ]ππ’π‘ At synapses, the ion channels open and close dependent on time-dependent signals. This means that the permeabilities change over time: ππ₯ → ππ₯ (π‘). really complicated… The Goldman-Hodgkin-Katz Equation holds only for the equilibrium. Problem? The Effect of Synaptic Inputs on the Membrane Potential: the problem of equilibrium Membrane Potential π π ππΎ+ [πΎ + ]ππ’π‘ +πππ+ [ππ+ ]ππ’π‘ +ππΆπ− [πΆπ − ]ππ π= ππ πΉ ππΎ+ [πΎ + ]ππ +πππ+ [ππ+ ]ππ +ππΆπ− [πΆπ− ]ππ’π‘ Transient dynamic Equilibrium Transient dynamic Equilibrium Equilibrium π½=? π½=? Synaptic Input time π½=? time For fast changing inputs the GoldmannHodkin-Katz Equation is not valid. The Effect of Synaptic Inputs on the Membrane Potential Goldman-Hodgkin-Katz Equation: π π ππΎ+ [πΎ + ]ππ’π‘ +πππ+ [ππ+ ]ππ’π‘ +ππΆπ− [πΆπ − ]ππ π= ππ πΉ ππΎ+ [πΎ + ]ππ +πππ+ [ππ+ ]ππ +ππΆπ− [πΆπ− ]ππ’π‘ At synapses, the ion channels open and close dependent on time-dependent signals. This means that the permeabilities change over time: ππ₯ → ππ₯ (π‘). really complicated… The Goldman-Hodgkin-Katz Equation holds only for the equilibrium. Problem? YES!!! Different Types of Ion Channels membrane + out - in always open opening depends on membrane potential transports ions against concentration gradient At synapses: The chemical model of a membrane becomes neurotransmitter too complicated + out fast slow - in We need another model opening depends on neurotransmitters Different Types of Ion Channels: Second Model membrane + out - in always open opening depends on membrane potential transports ions against concentration gradient At synapses: neurotransmitter + out fast slow opening depends on neurotransmitters - in The Influence of Ion Channels membrane without channels + out - in acts like a capacitor + Kirchhoff’s law: πΆ= π π π π β πΆ β π = πβ ππ‘ ππ‘ const. ππ πΆ β ππ ππ‘ = ππ‘ πΆ β ππ ππ‘ = −πΌ πΆ: capacity π: charge π: potential πΌ: current The Influence of Ion Channels membrane without channels leakage channels + out - in acts like a capacitor + Kirchhoff’s law: acts like a resistance + πΆ= π π π π β πΆ β π = πβ ππ‘ ππ‘ const. ππ πΆ β ππ ππ‘ = ππ‘ πΆ β ππ ππ‘ = −πΌ Ohm’s law: π = π β πΌ with π = 1 π πΌ =πβπ π: conductance π : resistance πΆ: capacity π: charge π: potential πΌ: current The Influence of Ion Channels membrane without channels membrane with channels leakage channels + out - in acts like a capacitor acts like a resistance + Kirchhoff’s law: + πΆ= π π π π β πΆ β π = πβ ππ‘ ππ‘ const. ππ πΆ β ππ ππ‘ = ππ‘ πΆ β ππ ππ‘ = −πΌ Ohm’s law: π = π β πΌ with π = 1 π πΆ β ππ ππ‘ = −π β π πΌ =πβπ π: conductance π : resistance πΆ: capacity π: charge π: potential πΌ: current The Influence of Ion Channels: Resting Potential Fixed Points: ππ ! =0 ππ‘ πΆ β ππ ππ‘ = −π β π π(π‘) π(π‘0 ) 1 π ππ = − π π‘ ln π π‘0 π π‘ πΆ π‘0 constant =− π π0 π‘0 = 0 The membrane potential becomes equal zero. π − π0 β π(π‘0 ) π π‘ ππ‘ π →π=0 πΆ (π‘ − π‘0 ) πβπ‘ π π‘ = π0 β ππ₯π − πΆ ππ ππ π =− πΆβπ =0 ππ‘ ππ‘ π‘ πΆ π→0 π‘→∞ π π πππ π=− ππ ≠0 π§πΉ πππ’π‘ The Influence of Ion Channels: Resting Potential membrane with channels A battery with ππππ π‘ constant potential ππππ π‘ πΆ β ππ ππ‘ = −π β π lim π = 0 π‘→∞ πΆ β ππ ππ‘ = −π β (π − ππππ π‘ ) lim π = ππππ π‘ π‘→∞ ππππ π‘ : resting potential π: conductance πΆ: capacity π: potential Different Types of Ion Channels membrane + out - in always open opening depends on membrane potential transports ions against concentration gradient At synapses: neurotransmitter + out fast slow opening depends on neurotransmitters - in The Influence of Synapses ππππ π‘ : resting potential π: conductance πΆ: capacity π: potential time-varying conductance ππππ π‘ ππππ π‘ πΆ β ππ ππ‘ = −ππ π¦π (π‘) β (π − ππππ π‘ ) πΆ β ππ ππ‘ = −π β (π − ππππ π‘ ) lim π = ππππ π‘ π‘→∞ τπ π¦π β πππ π¦π 1 0 δ(π‘ − π‘ππππ’π‘ ) ππ‘ = −ππ π¦π π‘ + πΏ(π‘ π‘ππππ’π‘ synaptic input t Different Types of Ion Channels membrane + out - in always open opening depends on membrane potential transports ions against concentration gradient At synapses: neurotransmitter + out fast slow opening depends on neurotransmitters - in The Influence of Ion Channels ππππ π‘ πΆ β ππ ππ‘ = −π β (π − ππππ π‘ ) Hodgkin and Huxley Squid Hodgkin-Huxley Model: dVm C ο½ ο g Na m 3 h(Vm ο VNa ) ο g K n 4 (Vm ο VK ) ο g L (Vm ο VL ) ο« I inj dt • voltage dependent gating variables mο¦ ο½ ο‘ m (V )(1 ο m) ο ο’ m (V )m nο¦ ο½ ο‘ n (V )(1 ο n ) ο ο’ n (V )n hο¦ ο½ ο‘ (V )(1 ο h ) ο ο’ (V )h h h (for the giant squid axon) V mV Hodgkin-Huxley Model: Action Potential / Threshold 40 20 0 20 40 60 80 Iinj = 0.42 nA V mV 0 40 20 0 20 40 60 80 10 t ms 15 20 Iinj = 0.43 nA 0 V mV 5 Short, weak current pulses depolarize the cell only a little. 5 40 20 0 20 40 60 80 10 t ms 15 20 Iinj = 0.44 nA An action potential is elicited when crossing the threshold. 0 5 10 t ms 15 20 Action Potential Action Potential dVm C ο½ ο g Na m 3 h(Vm ο VNa ) ο g K n 4 (Vm ο VK ) ο g L (Vm ο VL ) ο« I inj dt action potential • If u (orV) increases, m increases -> Na+ ions flow into the cell • at high u, Na+ conductance shuts off because of h • h reacts slower than m to the voltage increase • K+ conductance, determined by n, slowly increases with increased u With a mathematical model we can now test the influences of different induction protocols
