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Ch6 Exercises.py

@@ -16,43 +16,140 @@
 #----------------------------Exercises------------------------------------
 def Ex1():
     '''
+    Build aConnor-Stevensmodel neuron by numerically integrating the
+    equations for V, m, h, n, a, and b given in chapter 6 (see, in particular,
+    equations 6.1, 6.4, and appendix A). Use cm = 10 nF/mm2, and as
+    initial values take: V = -68 mV, m = 0.0101, h = 0.9659, n = 0.1559,
+    a = 0.5404, and b = 0.2887. Use an integration time step of 0.1 ms.
+    Use an external current with Ie/A = 200 nA/mm2 and plot V, m, h, n,
+    a, and b as functions of time over a suitable interval. Plot the firing
+    rate of the model as a function of Ie/A over the range from 0 to 500
+    nA/mm2. How does this differ from what you got for the Hodgkin-
+    Huxley model in exercise 8 of chapter 5. Finally, apply a pulse of
+    negative current with Ie/A = -500 nA/mm2 for 5 ms followed by
+    Ie/A = 200 nA/mm2and show what happens.
     '''
 
 def Ex2():
     '''
+    Construct aMorris-Lecarmodel neuron (Morris, C & Lecar, H (1981)
+    Voltage oscillations in the barnacle giant muscle fiber. Biophysical
+    Journal 35:193–213). Instead of simulating the fast sodium spikes
+    of an action potential, this model describes slower calcium spikes.
+    The model has just two active currents, an instantaneous voltagedependent
+    Ca2+ current and a persistent K+ current, described by a
+    single dynamical gating variable N:
+    
+    im = gL(V - EL) + gCaMinfinity(V)(V - ECa) + gKN(V - EK)
+    
+    with gL = 0.005mS/mm2, gCa = 0.01mS/mm2 and gK = 0.02mS/mm2,
+    EL = -50mV, ECa = 100mV and EK = -70mV. Use cm = 10 nF/mm2.
+    The function Minfinity(V) is given by
+    
+    Minfinity(V) = 1 / (1 + e**[-0.133(V+1)]
+    
+    and the gating variable N is given by
+    
+    tauN(V) (dN/dt) = Ninfinity(V) - N
+    
+    with
+    
+    tauN(V) = 3 / (cosh[0.345(V-10)]
+    
+    and
+    
+    Ninfinity(V) = 1 / (1 + e**[-0.138(V - 10)]
+    
+    Here, V is understood to be in mV units, and tauN is expressed in ms
+    units. Determine the firing rate as a function of injected current and
+    plot themembrane potential andN as a functions of time. Also, show
+    a phase-plane trajectory, which is a plot of that path taken by these
+    variables in the two-dimensional space described by the points (V,N), while the model is firing. In the phase plane, plot the nullclines
+    for the V and N equations. These are lines in the V-N plane along
+    which either dV/dt = 0 or dN/dt = 0. (Phase-plane descriptions and
+    nullclines are described in chapter 7.)
     '''
 
 
 def Ex3():
     '''
+    The FitzHugh-Nagumo equations (see FitzHugh, R (1961) Impulses
+    and physiological states in models of nerve membrane. Biophysical
+    Journal 1:445–466) are given by
+    
+    dv/dt =  v(1 - v**2) - u + Ie
+    
+    and
+    
+    du/dt = epsilon * (v - 0.5u)
+    
+    Draw the nullclines for these equations for Ie = 0 and Ie = -1. These
+    are the lines in the v-u plane where the right side of one or the other
+    of these two equations is zero. In which case or cases do you think
+    the model will produce oscillations? Next simulate the model to
+    see what happens when these equations are integrated over time.
+    Determine what happens for Ie = 0 with epsilon = 0.3, 0.1, and 1 and for
+    Ie = -1 with epsilon = 0.3. (Phase-plane descriptions and nullclines are
+    described in chapter 7.)
     '''
 
 def Ex4():
     '''
+    Show that solution of equation 6.19 satisfies the cable equation
+    along an infinite cable in response to the injected current ie =
+    Ie*taum*d(x)d(t)/(2pia).
     '''
 
 def Ex5():
     '''
+    Verify that the solution for an isolated junction given by equations
+    6.21 and 6.22 satisfies the correct boundary conditions at the junction
+    point: v1(0) = v2(0) = v3(0) and
+    
+    3[sigma]i=1 (ai**2 * (dvi/dx)x=0) = 0
     '''
 
 def Ex6():
     '''
+    Generalize the solution for an isolated junction of equation 6.21 to
+    the time-dependent case when the injected current on segment 2 is
+    ie = Ie*taum*delta(x2 - y)delta(t)/(2pia).
     '''
 
 def Ex7():
     '''
+    Show that the expression for v(x) given in figure 6.10,with R1 and R2
+    given by equations 6.23 and 6.24, satisfies the cable equation and the
+    boundary conditions, v(0) = vsoma and dv/dx = 0 when x = L.
     '''
 
 def Ex8():
     '''
+    Show that the expression for v(x) given in figure 6.12,with R3 and R4
+    given by equations 6.26 and 6.27, satisfies the cable equation and the
+    boundary conditions, v(0) = 0 and dv/dx = 0 when x = L.
     '''
 
 def Ex9():
     '''
+    Construct a non-branching axonal cable with conductances in each
+    compartment described by the Connor-Stevens model (as in exercise
+    1). Solve for the membrane potential using the methods of
+    appendix B of chapter 6. Initiate action potential propagation at one
+    end of the cable by injecting current into the terminal compartment
+    of the cable. Plot the action potential propagation velocity as a function
+    of the axon radius. Inject current into the middle of the cable
+    to generate two, opposite-moving action potentials. Generate action
+    potentials from each end of the cable and show that they annihilate
+    each other when they collide.
     '''
 
 def Ex10():
     '''
+    Determine the numerical solution for a multi-compartment cable
+    with a single branching node (where a single cable splits into two
+    branches) analogous to the solution for a non-branching cable (equations
+    6.53–6.56) given in appendix B of chapter 6.
     '''