Today we begin treating systems with 2 state variables, which we will analyze in 2D state space also called the phase plane. In Chapter 5 of Strogatz we will study types of fixed points such as stable nodes, unstable nodes, neutral stability, saddle points, and we will discuss Liapunov stability and the notion of globally attracting as they pertain to the characteristics of the fixed points. The concept of nullclines will also be introduced. With respect to saddle points we will discuss the stable manifold (approached as t --> -∞) and the unstable manifold (approached as t --> +∞).

We will study Linear differential equations and linear algebra to analyze 2D systems. It is important that you be comfortable with the characteristic equation and how we can determine the eigenvalues (λ_1,2) for a typical 2D ODE using the trace and determinant from the matrix of coefficients. We will also determine the eigenvectors (v_1,2) for 2D linear systems. Knowing λ_1,2 and v_1,2, we can draw the phase portrait for 2D systems to show all the behaviors exhibited by the system.

You will need to use the Hodgkin-Huxley equations in a working *.ode file for XPPAUT for this week's homework

(1) Plot h_∞(V) and τ_h(V). Use a voltage range that includes (at a minimum) -90 to +10 mV and be sure to include proper units.

(2) Run a 40 ms simulation of the HH equations using a step command that begins at t₁ = 5 ms and ends at t₂ = 15 ms. Make the amplitude of step command netative, varying from -10 μA/cm² to zero. Explain why in some cases, the negative step commands, which actually hyperpolarize the membrane potential, cause spike discharge at the termination of the pulse (i.e., after t₂)? To answer this, you may find it useful to plot not just the V_M, but to also plot how the state variables m, n, and h change during the simulation in response to different amplitude step commands. This type of excitation following a negative current pulse (of sufficient amplitude and duration) is called anode-break excitation. Reproducing anode-break excitation was a major achievement of the HH model and provided a good explanation for this phenomenon that was observable in squid axons in vitro.

(3) Use the same *.ODE file as above, but now make the step command positive from slighly above zero to +5 μA/cm². Compare anode-break action potentials (from #2) to the action potentials evoked by positive step commands here. Explain why the action potentials evoked by the negative commands (in #2) have a higher amplitude (approximately +42 mV peak) compared to the spikes evoked by positive pulses (approximately +37 mV peak) here in #3.

(4) Create another *.ODE file with the HH equations. Set I_app to +3 μA/cm² and run the simulation for 20 ms. Choose a very small Dt in the nUmerics window. Now set m equal to m_∞(V) and instead of using h in the equation for I_Na, use the quantity (1-n) in its place. A small Dt is essential to keep this simulation stable (i.e., no numerical errors). Plot and print the two types of spike responses and comment on their similarities and differences in terms of gating properties of m, n, and h.

(5) Solve the change equation for m in closed form, which is given by:

m′ = α_m(V)*(1 - m) - β_m(V)*m

Show that the definitions m_∞ = α/(α+β) and τ = 1/(α+β) allows you to re-write the ODE as:

m′ = (m_∞(V) - m)/τ(V).