Lecture 10 : Neuronal Dynamics. Eileen Nugent

Transcription

1 Lecture 10 : Neuronal Dynamics Eileen Nugent

2 Origin of the Cells Resting Membrane Potential: Nernst Equation, Donnan Equilbrium Action Potentials in the Nervous System Equivalent Electrical Circuits and the Derivation of the Cable Equation Voltage-Gating Hypothesis, Hodgkin-Huxley Measurements

3 Osmotic Pressure Osmosis flow of water across semi-permeable membrane due to concentration difference of non permeable solutes Osmotic Pressure Pressure difference needed across membrane to stop osmotic flow Equilibrium (c) Fluid pressure constant in the pore Osmosis (d - solid line) Pressure equal on both sides pressure drop in channel cancels osmotic pressure jump Reverse osmosis (d - dashed line) Pressure generated >c 0 k B T on right hand side. Reverse osmosis

4 Osmotic Pressure and the Cell Globular Protein Concentration Corresponding Surface tension Rp/2 = 1.5 x 10-3 N/m Enough to rupture cell Salt Concentration Lipid Bilayer Membranes almost Impermeable Concentration 1027 ions per m-3 Red blood cells can t manage this in pure water How can the cell maintain this in the body?

5 Why do cells not burst? The cytoplasm has a very different composition from the extra-cellular environment Why doesn t osmotic flow through the membrane burst or shrink the cell Predictions for equilibrium state don t quite fit with real cell behaviour

6 Membranes and Ions Membrane permeable to K+ but not Cl- Higher concentration of K+ inside than outside. Same for negative charges (neutrality) K+ ions could increase entropy of system by crossing membrane but electrostatic attraction pulls them back Measuring membrane potential

7 K+ ions cross the membrane to try to erase the difference in concentration up to a point. They are still attracted to Cl- ions They form a diffuser layer at the outer membrane boundary Similarly there is a diffuse negative layer at the inner boundary This gives rise to a potential difference called the Nernst Potential In equilibrium the Nernst relation gives the Nernst potential of the permanent species (K+ ions) Nernst Potentials

8 Nernst Relation and Ion Flow

9 Donnan Potential (1) (2) (3)

10 Ion Pumps

11 Origin of the Cells Resting Membrane Potential: Nernst Equation, Donnan Equilbrium Action Potentials in the Nervous System Equivalent Electrical Circuits and the Derivation of the Cable Equation Voltage-Gating Hypothesis, Hodgkin-Huxley Measurements

12 Neurons Synapse Image : Axon cross-section Input = dendrites (one of many) Output = axons (one can split into two) Computation : performed in soma sometimes called cell body Connections : synapse Signal insulation = myelin

13 Nervous System : How are electrical signals transmitted? Axon length scales : mm to m Sciatic nerve longest in human body with axons ~ 1m long How are electrical signals transported over long distances despite : * loss of ions through cell membrane * ionic fluid resistance REGENERATION mechanism needed

14 Impulse Response :Graded Potentials Membrane potential actively maintained at -60 mv Apply a depolarizing impulse (e.g. inject some positive charges) Small depolarizing impulse (membrane potential varies by couple mv) Graded potential - The strength of the depolarization diminishes over time and distance (as it travels down axon) and spreads. - Strength of signal depends on stimulus strength - Passive propagation : signal travels only a short distance down axon.

15 Impulse Response : Action Potentials Large depolarizing impulse (membrane potential crosses a threshold value) Action potential - Large signal which does not diminish propagates down axon at constant speed ( m/s) - Propagating impulse has constant form ( action potential ) - Peak value of response to impulse independent of stimulus strength - All or nothing pulse generation

16 Action Potential Measurements ~ 500 um 2 ms Hodgkin Huxley 1939 Giant axon of squid Features of action potential : Change sign, overshoot

17 Origin of the Cells Resting Membrane Potential: Nernst Equation, Donnan Equilbrium Action Potentials in the Nervous System Equivalent Electrical Circuits and the Derivation of the Cable Equation Voltage-Gating Hypothesis, Hodgkin-Huxley Measurements

18 Cell Membrane Modeling : Electrical Networks fluid inside Membrane Fluid outside DISCRETE-ELEMENT MODELS Model for small patch of membrane of area A Battery : Nernst potential Resistance : Ion channels Capacitance : Membrane

19 Ionic Conduction in a Nerve Fibre DISTRIBUTED-ELEMENT MODEL OF AXON Each cylindrical segment is discrete circuit element Resistance R R for ions moving across membrane (passive diffusion) dr x, dr x resistances to current flowing from one axial element to the next in cell interior/exterior Assume dr x negligible

20 Cable Equation dx ionic conductivity inside axon ionic flux across membrane C is capacitance per unit area CABLE EQN

21 Linear Cable Equation Assuming Ohmic conductance through membrane gives : LINEAR CABLE EQN With response to localized impulse: Axon space Constant Axon time Constant

22 Membrane Resting potential Resting potential electrical potential across plasma membrane of a cell that is not conducting an impulse Resting potential is dynamically maintained by active transport Na + concentration maintained far from equilibrium Rapid increase in sodium conductance would rapidly depolarize the membrane

23 Sodium Concentration and Voltage Gating Shape of action potential depends on external sodium concentration Peak of action potential tracks sodium Nernst potential

24 Origin of the Cells Resting Membrane Potential: Nernst Equation, Donnan Equilbrium Action Potentials in the Nervous System Equivalent Electrical Circuits and the Derivation of the Cable Equation Voltage-Gating Hypothesis, Hodgkin-Huxley Measurements

25 Voltage Gating Hypothesis Assume conductance is voltage dependent

26 Nonlinear Travelling Wave Na + channels closed Na + channels open Non-linear term gives travelling wave of fixed amplitude Shape Only possible to send one action potential

27 Hodgkin - Huxley ELECTROPHYSIOLOGY EXPT: EVIDENCE FOR VOLTAGE GATED ION-SELECTIVE CONDUCTANCES Space clamping Voltage clamping Separation of ion currents

28 Na + channels open Na + channels start to close Delayed opening of K + channels K + channels close K + diffusion