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Ch5 Exercises.py

@@ -16,43 +16,139 @@
 #----------------------------Exercises------------------------------------
 def Ex1():
     '''
+    The Nernst equation (equation 5.4) was derived in chapter 5 under
+    the assumption that the membrane potential was negative and the
+    ion being consideredwas positively charged. Rederive this result for
+    a negatively charged ion and for the case when E is positive to verify
+    that it applies in all these cases.
     '''
 
 def Ex2():
     '''
+    Verify that equation 5.47 is a solution of equation 5.46 when Vinfinity is
+    independent of time. Then, solve equation 5.46 for the casewhen Vinfinity
+    is an arbitrary function of time. In this solution, V(t) is expressed in
+    terms of integrals involving Vinfinity(t).
     '''
 
 
 def Ex3():
     '''
+    Build a model integrate-and-fire neuron from equation 5.8. Use
+    Vrest = -70 mV, Rm = 10 M
+    , and taum = 10 ms. Initially set V = Vrest.
+    When the membrane potential reaches Vth = -54 mV, make the neuron
+    fire a spike and reset the potential to Vreset = -80 mV. Show
+    sample voltage traces (with spikes) for a 300-ms-long current pulse
+    (choose a reasonable current Ie) centered in a 500-ms-long simulation.
+    Determine the firing rate of the model for various magnitudes
+    of constant Ie and compare the results with equation 5.11.
     '''
 
 def Ex4():
     '''
+    Include an extra current in the integrate-and-firemodel to introduce
+    spike-rate adaptation, as described in equations 5.13 and 5.14, and in
+    the caption to figure 5.6.
     '''
 
 def Ex5():
     '''
+    Add an excitatory synaptic conductance to the integrate-and-fire neuron
+    of exercise 3 by adding the extra synaptic conductance term in
+    equation 5.43 with Es = 0. Set the external current to zero, Ie = 0, in
+    this example, and assume that the probability of release on receipt
+    of a presynaptic spike is 1. Use rmgs = 0.5 and describe Ps using the
+    alpha function of equation 5.35 with taus = 10 ms and Pmax = 0.5. To
+    incorporatemultiple presynaptic spikes, Ps should be described by a
+    pair of differential equations,
+    
+    ts (dPs/dt) = e*Pmax*Z - Ps
+    
+    with e = exp(1), and
+    
+    taus (dz/dt) = -z
+    
+    with the additional rule that z is set to 1 whenever a presynaptic spike
+    arrives. Plot V(t) in one graph and the synaptic current in another.
+    Trigger synaptic events at times 50, 150, 190, 300, 320, 400, and 410
+    ms. Explain what you see.
     '''
 
 def Ex6():
     '''
+    The equations in exercise 5 generate an alpha function response to a single
+    input spike, but they do not prevent Ps from growing greater than 1
+    when themodel synapse is driven bymultiple spikes at a sufficiently
+    high frequency. In otherwords, this model synapse does not saturate.
+    To introduce saturation, modify the equations of exercise 5 to
+    
+    ts (dPs/dt) = e*Pmax*Z(1 -Ps) - Ps
+    
+    with e = exp(1), and
+    
+    taus (dz/dt) = -z
+    
+    with the additional rule that z is set to 1 whenever a presynaptic
+    spike arrives. Compare Ps(t) computed using these equations with
+    Ps(t) computed using the equations of exercise 5 for constant rate,
+    regular (periodic) presynaptic spike trains with frequencies ranging
+    from 1 to 100 Hz. In both cases, use taus = 10 ms and Pmax = 0.5.
     '''
 
 def Ex7():
     '''
+    Construct a model of two coupled integrate-and-fire neurons similar
+    to that of figure 5.20. Both model neurons obey equation 5.43
+    with EL = -70 mV, Vth = -54 mV, Vreset = -80 mV, taum = 20 ms,
+    rmgs = 0.15, and RmIe = 18 mV. Both synapses should be described
+    as in exercise 5 with Pmax = 0.5 and taus = 10 ms. Consider cases
+    where both synapses are excitatory, with Es = 0 mV, and both are
+    inhibitory, with Es = -80 mV. Show how the pattern of firing for
+    the two neurons depends on the type (excitatory or inhibitory) of the
+    reciprocal synaptic connections. For these simulations, set the initial
+    membrane voltages of the two neurons to slightly different values,
+    randomly, and run the simulation until an equilibrium situation has
+    been reached, which may take a few seconds of simulated run time.
+    Start froma fewdifferent randominitial conditions to study whether
+    the results are consistent. Investigate what happens if you change
+    the strengths and time constants of the reciprocal synapses.
     '''
+
 
 def Ex8():
     '''
+    Build a Hodgkin-Huxley model neuron by numerically integrating
+    the equations for V, m, h, and n given in chapter 5 (see, in particular
+    equations 5.6, 5.17-5.19, 5.22, 5.24, and 5.25). Take cm = 10 nF/mm2,
+    and as initial values take: V = -65 mV, m = 0.0529, h = 0.5961, and
+    n = 0.3177. Use an integration time step of 0.1 ms. Use an external
+    current with Ie/A = 200 nA/mm2 and plot V, m, h, and n as functions
+    of time for a suitable interval. Also, plot the firing rate of the model
+    as a function of Ie/A over the range from 0 to 500 nA/mm2. Show
+    that the firing rate jumps discontinuously from zero to a finite value
+    when the current passes through the minimum value required to
+    produce sustained firing. Finally, apply a pulse of negative current
+    with Ie/A = -50 nA/mm2 for 5 ms followed by Ie/A = 0 and show
+    what happens.
     '''
 
 def Ex9():
     '''
+    Construct and simulate the K+ channel model of figure 5.12. Plot the
+    mean squared deviation between the current produced by N such
+    model channels and the Hodgkin-Huxley current as a function of N
+    over the range 1 =< N =< 100, matching the amplitude of theHodgkin-
+    Huxley model so that the mean currents are the same.
     '''
 
 def Ex10():
     '''
+    Compute analytically the value of the release probability Prel just before
+    the time of each presynaptic spike for a regular (periodic rather
+    than Poisson), constant-frequency presynaptic spike train as a function
+    of the presynaptic firing rate. Do this for both the depression
+    and facilitation models described by equation 5.37.
     '''