a. Run the simulation with the constants given. Use Euler integration with interval = 0.001. Plot yo, psNT, Vm, Nbar, Ns, and NT for 0 9 t 9 80 ms. Use a vertical scale of -0.15 to 0 .6. Note what happens to bNT(t) as the presynaptic input frequency is raised by increasing Vin.

b. Now try varying system rate constants. Note that in the real world, concentrations and densities are non-negative quantities. What does it mean regarding the simulation if non- negativity is not preserved?

1.8. A linear transfer function of the form: H(s) = an/(s + a)n = (Yn/X)(s) can be used to model the dynamics of certain neurophysiological systems. When n = 2, the dynamics of the so-called alpha function that is often used to model excitatory postsynaptic potentials is obtained. Let the input to the transfer function, x, be a unit impulse. (In the simulation, make the pulse rectangular with width 0.001 ms, and its height 1000.) The system can be simulated by writing 10, simple, linear concatenated ODEs of the form:

yk = ayk + ayk-i where k = 1, 2, ... 10, and y0 = x, the system input.

a. Simulate and plot the 10 yk(t). Note what happens to the response as k ^ 10.

b. What mathematical form does yk(t) take as k ^ x?

1.9. A passive, cylindrical dendrite is to be modeled by an RCG transmission line model, as shown in Figure P1.9. The dendrite is 250 |im long, and has a 1-|im radius. A synapse at the far end introduces an epsp that can be modeled as an ideal voltage source of the form:

where a = 200 r/s. This pulse propagates passively down the dendrite to the cell body, where it is measured by a glass micropipette electrode as vm(200, t). The dendrite membrane has a capacitance of CM = 1 |F/cm2, and a net passive conductance of GM = 2 x 10-4 S/cm2. The axoplasm resistivity is pi = 630 ohm cm. The dendrite is to be modeled by a lumped-parameter RCG transmission line with five sections (one for every 50 |im of dendrite length). The soma is known to terminate the dendrite with a parallel CG of Cs = 78.54 pF, and Gs = 1.571 x 10-8 S. vm(200, t) is measured across CsGs.

a. Calculate the appropriate ri, cm , and gm for each RCG section.

b. Use a circuit simulation software package such as PSPICE™ or Micro-Cap™ to observe the vm(200, t) transient, given the vs above. (Note that the simulation of this linear system can also be done with Simnon or Matlab® after writing node equations for the circuit of Figure P1.9.)

c. Plot the sinusoidal frequency response 20 log[Vm/Vs(jra)] of the den-drite model.

d. Now examine the result when only two (instead of five) RCG transmission line sections are used. It is necessary to multiply ri, gm , and cm each by (5/2) to scale the two, 125-|im sections correctly. Use the same Rs and Cs to terminate the line model.

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