Dynamical Systems in Neuroscience:

Simple Models 3278. (Another theta neuron) Show that the quadratic integrate-and-fire neuron (8.2)is equivalent to˙ϑ = ϑ 2 + (1 − |ϑ|) 2 r .where ϑ ∈ [−1, 1] and r have the same meaning as in the previous exercise. Arethere any other “theta neurons”?9. When is the linear version of (8.3, 8.4),˙v = I − v − u if v = 1, then˙u = a(bv − u) v ← 0, u ← u + d,equivalent to the integrate-and-fire or resonate-and-fire model?10. Show that the simple model (8.3, 8.4) with b < 0 is equivalent to the quadraticintegrate-and-fire neuron with a passive dendritic compartment.11. All membrane potential responses in Fig. 8.8 were obtained using model (8.3,8.4) with appropriate values of the parameters. Use MATLAB to experimentwith the model and reproduce the figure.12. Simulate the FS spiking neuron in Fig. 8.27 using simple model (8.5, 8.6) withlinear equation for u. What can you say about its possible bifurcation structure?13. Fit the recordings of RS neuron in Fig. 8.12 using the following modelC ˙v = I − g(v − v r ) + p(v − v t ) 2 + − u if v = v peak , then˙u = a(b(v − v r ) − u) v ← c, u ← u + dwhere x + = x when x > 0 and x + = 0 when otherwise. This model better fitsthe upstroke of the action potential.14. Explore numerically the model (8.3, 8.4) with a nonlinear after-spike reset v ←f(u), u ← g(u), where f and g are some functions.15. [M.S.] Analyze the generalization of the system (8.3, 8.4)where e is another parameter.˙v = I + v 2 + evu − u if v = 1, then˙u = a(bv − u) v ← c, u ← u + d16. [M.S.] Analyze the generalization of the following system, related to the exponentialintegrate-and-fire modelwhere k is another parameter.˙v = I − v + ke v − u if v = 1, then˙u = a(bv − u) v ← c, u ← u + d