From b6e7c5717a3afe901f6a95209be4b37c9d55f134 Mon Sep 17 00:00:00 2001 From: Jordan Breffle Date: Wed, 30 Aug 2017 09:41:02 -0500 Subject: [PATCH] Add files via upload --- Ch5 Exercises.py | 96 ++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 96 insertions(+) diff --git a/Ch5 Exercises.py b/Ch5 Exercises.py index ecaa0a5..ef600af 100644 --- a/Ch5 Exercises.py +++ b/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. '''