BME 5742 Biosystems Modeling and Control

Transcription

1 BME 5742 Biosystems Modeling and Control Hodgkin-Huxley Model for Nerve Cell Action Potential Part 1 Dr. Zvi Roth (FAU) 1

2 References Hoppensteadt-Peskin Ch. 3 for all the mathematics. Cooper s The Cell : Nice illustrations of the sodium and potassium channels. Dr. Zvi Roth (FAU) 2

3 The Neuron Dr. Zvi Roth (FAU) 3

4 The Neuron V m change propagates to cell body, axon hillock. Axon hillock integrates all inputs; generates spike if V m > threshold. Spikes = Action Potentials. Spikes travel along axons. Dendrites collect inputs. Input either or membrane potential, Vm. Spike propagates to synapse. Dr. Zvi Roth (FAU) 4

5 Connection to the Next Neuron(s) Dr. Zvi Roth (FAU) 5

6 Qualitative Facts: A given dendritic input either raises or lowers membrane potential. Elevated membrane potential raises the rate of spike generation. Lowered membrane potential lowers the rate of spike generation. Spikes are largely transparent in the process. Spikes may be all-or-none, but information is transferred as a continuous signal. Dr. Zvi Roth (FAU) 6

7 Pulses Information Axons are highly specialized wires that conduct the neurone's output signal to the target cells - up to 10,000 in some cases. The majority of nerve cells encode their output as a series of brief voltage pulses, also referred to as action potentials or spikes. The pulses originate at or close to the cell body of nerve cells and propagate down the axon at constant velocity and amplitude. Hodgkin and Huxley modelled the initiation and propagation of these action potentials in the giant axon of a squid. Dr. Zvi Roth (FAU) 7

8 Circuit Model of Cell Membrane Channels Dr. Zvi Roth (FAU) 8

9 Circuit Description The membrane potential is determined by three conductances: voltage-independent leak conductance g Cl. voltage dependent sodium conductance g Na. voltage dependent potassium conductance g K. The conductances are in series with voltage sources, the values of which correspond to the respective reverse potentials of the ionic currents, E Cl, E Na, E K. Dr. Zvi Roth (FAU) 9

10 Circuit Description (cont d) The potential of the ICF is taken as a zero voltage reference. Before stimulation of a membrane patch, the membrane resting voltage is about -65mV and g Na and g K are almost fully inactive, g K is still >> g Na. At this point in time, the membrane is largely dominated by the leak current and residual potassium current. Dr. Zvi Roth (FAU) 10

11 Simulation Results The applied current slowly depolarizes the membrane by charging up the capacitance. As V m approaches the threshold -50mV, sodium channels begin to open, the resultant flow of N a ions depolarizes the membrane further. The state variables m, h, and n describe the action of the grating particles. Dr. Zvi Roth (FAU) 11

12 Simulation Results Two events occur to reverse V m about 1ms later. V m The sodium channels conductance deactivates. Potassium channels open causing an outward current. V m The potassium current pulls V m to below minus 65mV, (hyperpolarisation), but it also deactivates allowing g L to return it to rest. Dr. Zvi Roth (FAU) 12

13 Now let s delve into the mathematics: We shall build up on the electrical-chemicalosmotic cell volume control model presented in Ion Movement in Cells lecture, with some changes: Due to the fast electrical events (in the time scale of milliseconds), we can no longer neglect membrane capacitance effect. There is no steady-state Ionic currents do not go to zero. Dr. Zvi Roth (FAU) 13

14 Cv Cell Membrane Capacitance Effect = ( Vq[ Na dv C = dt dv C dt d dt = I Recall: + ] Na i ( Vq[ Na + Vq[ K + I ] K i I + Cl ] i + Vq[ K Vq[ Cl + KT [ Na ] o I Na = g Na ( v ln( )) + + q [ Na ] I K I + KT [ K ] = g K ( v ln( + q [ K ] Cl = g Cl ( v + KT q o i i + ] i )) [ Cl ln( [ Cl ] ] ] i xzq) + V ( q)[ Cl pq o i pq )) ] i xzq) Electro-neutrality assumption is to take C 0. Dr. Zvi Roth (FAU) 14

15 Cell Membrane Capacitance Model dv dt Where: C + g Na Na K K Cl Cl KT q ( v E ) + g ( v E ) + g ( v E ) = [ Na ln( [ Na + E Na = + ] ] o i ) 0 KT q [ K ln( [ K + E K = + E Cl = KT q [ Cl ln( [ Cl ] ] o i ] ] ) o i ) Pump currents cancelled out due to 1:1 approximation Dr. Zvi Roth (FAU) 15

16 Simplifying Assumption Cell is large enough, and ion fluxes are small enough, so that ICF ion concentrations change only slightly during an action potential. There are no short-time-scale large swings in ions concentration We can treat the potentials E K < E Cl < 0 < E Na as given constants. Dr. Zvi Roth (FAU) 16

17 Action Potential Model C g E dv + g( v E) = 0 dt = g + g + g = g Na Na E g Na K Na Cl + g K EK + g + g + g K Cl Cl E Cl We see from equation that v(t) always wants to approach E. Cell adjusts E simply by changing membrane conductance values. Dr. Zvi Roth (FAU) 17

18 Sequence of E Changes When cell is at rest v=e is close to E Cl. Early in an action potential, Na + channels open up, dramatically increasing g Na. This brings E closer to E Na, with v following. Later, Na + channels close while K + channels open. E comes close to E K, at a more negative level than the resting potential. Finally, K + channels close, and v goes back to the resting voltage. Dr. Zvi Roth (FAU) 18

19 The resulting pulse: Dr. Zvi Roth (FAU) 19

20 Ion Conductance Change Via Controlled Opening and Closing of Ion Channels Transport of ions through channels is extremely rapid. More than 10 6 ions per second flow through open channels flow rate which is thousands of times larger than for ions transport through carrier proteins. Ion channels are highly selective. Narrow pores restrict passage to ions of appropriate size and charge. Dr. Zvi Roth (FAU) 20

21 More about Ion Channels Most ion channels are not permanently open. Some are though. Channel gates open in response to some stimuli. Ligand-gated channels open in response to the binding of neuro-transmitters or other signaling molecules. Voltage-gated channel open in response to changes in electric potential across the membrane. Dr. Zvi Roth (FAU) 21

22 Phases 1 and 2 Dr. Zvi Roth (FAU) 22

23 Phases 2 and 3 Dr. Zvi Roth (FAU) 23

24 Phases 3 and back to 1 Dr. Zvi Roth (FAU) 24

25 Ion Selectivity of Sodium Channels Narrow pore acts as a size filter. Ionic radius of Na + is smaller than that of K Dr. Zvi Roth (FAU) 25

26 Ion Selectivity of Sodium Channels Narrow pore acts as a size filter. Ionic radius of Na + is smaller than that of K +. But now the question is how does a K + channel work?? - 2 Dr. Zvi Roth (FAU) 26

27 Ion Selectivity of Potassium Here, the selectivity of the pore is thought to be related to interaction between K + and polar amino acid side chains lining the pore. Channels - 1 Dr. Zvi Roth (FAU) 27

28 Ion Selectivity of Potassium Here, the selectivity of the pore is thought to be related to interaction between K + and polar amino acid side chains lining the pore. Water molecule is displaced, and K + goes through. Channels - 2 Dr. Zvi Roth (FAU) 28

29 Ion Selectivity of Potassium In contrast, dehydrated Na + is too small to interact with the polar side chains. Na + remain bound to water molecules, and the complex is too large to pass through the pore. Channels - 3 Dr. Zvi Roth (FAU) 29