It is a biological modeling assignment to be written using the Xppaut software. There will need to be postscript graphs.

1) A modified Hodgkin-Huxley model

One considers a neuron model where voltage evolution is given by

C ddVt = Gl(Vl - V ) + GNam8(V )3h(ENa - V ) + GKn(EK - V ) + I

Sodium current activation is assumed to be instantaneous m

and the gating variables h and n satisfy the equations th ddht = h8(V ) - h and tn ddnt = n8(V ) - n

where th and tn are both equal to 1 ms,

h n

Other parameters are C = 0.8nF, Gl = 0.3µS, GNa = 40µS, GK = 3.5µS, Vl = - 66mV, ENa = 50mV, EK = - 90mV.

Draw the voltage response of the model to a constant injected current for several values of this current.

Draw the bifurcation parameter using the injected current as the bifurcation parameter.

Interpret the results.

2) Effect of an AHP current

One adds to the model a afterhyperpolarization (AHP) current IAHP = GAHPz(EK - V ) where GAHP = 0.3µS.

The gating variable z satisfies

tz(V ) ddzt = z8(V ) - z

where z8(V ) = 1 and tz = 0.1ms above -20 mV (spike voltage threshold) and z8(V ) = 0 and tz = 10ms below -20 mV.

Show how the discharge of the neuron is modified by this AHP current.

Show that small oscillations are observed before action potentials and characterize these oscillations. What is their typical frequency? How do they vary with the injected current? Show that these oscillations disappear if the injected current is too much increased.

Show that the firing frequency of the neuron displays a series of plateaus above the firing threshold and explain why. Explain why the firing frequency smoothly increases for higher values of the injected current.

3) Reduction to a three-dimensional model

Reduce the above four-dimensional model to a three-dimensional model in a way similar to what is done for the standard Hodgkin-Huxley model. Explain how you do that and write down the resulting equations.

What is the discharge of this three-dimensional model? How does it vary with the injected current? Does it display small oscillations between action potentials.

4) Origin of the small oscillations

Compute the bifurcation diagram of the reduced model. How does the resting state become unstable when the current is increased?

Draw the trajectory of the model in three dimensions when the injected current is just above the firing threshold. Describe this trajectory. Why does it explain the small oscillations between action potentials?

Draw the the trajectory of the model in three dimensions when the injected current is larger and one does not observe small oscillations. How does it differ for the previous case? Why are small oscillations no longer observed?

Conclude on the role of the AHP in the emergence of small oscillations between action potentials. 5) Excitability

One increases the excitability of the reduced model by adding a persistent sodium current, which does not inactivate. Its activation is instantaneous, and its activation curve is the same as the transient sodium current above but shifted downward by 5 mV.

Draw the two parameter bifurcation diagram of the model using the injected curent and the conductance GNap of the persistent sodium current as bifurcation parameters. Interpret the results. Show that the small oscillations preceding action potentials then disappear as GNap is increased.

1) A modified Hodgkin-Huxley model

One considers a neuron model where voltage evolution is given by

C ddVt = Gl(Vl - V ) + GNam8(V )3h(ENa - V ) + GKn(EK - V ) + I

Sodium current activation is assumed to be instantaneous m

and the gating variables h and n satisfy the equations th ddht = h8(V ) - h and tn ddnt = n8(V ) - n

where th and tn are both equal to 1 ms,

h n

Other parameters are C = 0.8nF, Gl = 0.3µS, GNa = 40µS, GK = 3.5µS, Vl = - 66mV, ENa = 50mV, EK = - 90mV.

Draw the voltage response of the model to a constant injected current for several values of this current.

Draw the bifurcation parameter using the injected current as the bifurcation parameter.

Interpret the results.

2) Effect of an AHP current

One adds to the model a afterhyperpolarization (AHP) current IAHP = GAHPz(EK - V ) where GAHP = 0.3µS.

The gating variable z satisfies

tz(V ) ddzt = z8(V ) - z

where z8(V ) = 1 and tz = 0.1ms above -20 mV (spike voltage threshold) and z8(V ) = 0 and tz = 10ms below -20 mV.

Show how the discharge of the neuron is modified by this AHP current.

Show that small oscillations are observed before action potentials and characterize these oscillations. What is their typical frequency? How do they vary with the injected current? Show that these oscillations disappear if the injected current is too much increased.

Show that the firing frequency of the neuron displays a series of plateaus above the firing threshold and explain why. Explain why the firing frequency smoothly increases for higher values of the injected current.

3) Reduction to a three-dimensional model

Reduce the above four-dimensional model to a three-dimensional model in a way similar to what is done for the standard Hodgkin-Huxley model. Explain how you do that and write down the resulting equations.

What is the discharge of this three-dimensional model? How does it vary with the injected current? Does it display small oscillations between action potentials.

4) Origin of the small oscillations

Compute the bifurcation diagram of the reduced model. How does the resting state become unstable when the current is increased?

Draw the trajectory of the model in three dimensions when the injected current is just above the firing threshold. Describe this trajectory. Why does it explain the small oscillations between action potentials?

Draw the the trajectory of the model in three dimensions when the injected current is larger and one does not observe small oscillations. How does it differ for the previous case? Why are small oscillations no longer observed?

Conclude on the role of the AHP in the emergence of small oscillations between action potentials. 5) Excitability

One increases the excitability of the reduced model by adding a persistent sodium current, which does not inactivate. Its activation is instantaneous, and its activation curve is the same as the transient sodium current above but shifted downward by 5 mV.

Draw the two parameter bifurcation diagram of the model using the injected curent and the conductance GNap of the persistent sodium current as bifurcation parameters. Interpret the results. Show that the small oscillations preceding action potentials then disappear as GNap is increased.

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