# Homework Solution: The file ActionPotential.DAT contains two columns of data – time and voltage – sho…

The file ActionPotential.DAT contains two columns of data – time and voltage – showing a typical action potential in a sensory neuron. (The MATLAB command load ActionPotential.DAT will import the data into the workspace with the variable name ActionPotential.) We are interested not only in the voltage, but also in total membrane current over the course of the action potential. We have not measured current directly, but can estimate it using the relationship I =−C dV/dt where C is membrane capacitance. C can be estimated from cell surface area using the approximation that 1 cm2 of membrane has a capacitance of 1µF. Using a central difference approximation of order h2 to estimate the derivative of V, calculate the current I using the equation above when the cell diameter is 1 micron (that is, 1µm, or 10-6 m). Indicate the time points at which the central difference approximation cannot be applied, and use a forward or backward difference approximation (also of order h2), as appropriate, to calculate the derivative at those points. (Your result, therefore, should produce a value of the derivative at EVERY time point represented in the data set.) Make two plots: one will be the voltage (y-axis) as a function of time (x-axis), and the other will be the estimated total membrane current (y-axis) as a function of voltage (xaxis). This first plot is a time-domain plot, and the second is a state-space plot. On both plots, label (for example, by adding symbols or text labels to the plot) the peaks (maximum and minimum voltage) of the action potential. On the state-space plot, also label the peak inward (negative) current and the peak outward (positive) current. Because the membrane returns to the same voltage from which it started, the statespace plot forms a loop. If we were to draw arrows representing how we proceed around the loop with time, would those arrows proceed in a clockwise or a counterclockwise direction? How can you tell? What does this tell us about the current involved in creating an action potential – does the early part of the action potential start with mostly inward (negative) or mostly outward (positive) current? (Action Potential data) -1.92   -57.75 -1.905 -57.7 -1.89   -57.75 -1.875 -57.75 -1.86   -57.75 -1.845 -57.7 -1.83   -57.7 -1.815 -57.8 -1.8     -57.75 -1.785 -57.75 -1.77   -57.75 -1.755 -57.8 -1.74   -57.8 -1.725 -57.75 -1.71   -57.7 -1.695 -57.75 -1.68   -57.8 -1.665 -57.75 -1.65   -57.75 -1.635 -57.75 -1.62   -57.75 -1.605 -57.75 -1.59   -57.7 -1.575 -57.7 -1.56   -57.75 -1.545 -57.75 -1.53   -57.7 -1.515 -57.7 -1.5     -57.7

The polish ExercisePotential.DAT contains span columns of axioms – span and voltage – showing a customary exercise virtual in a sensory neuron. (The MATLAB command

obtain tenor the axioms into the workspace with the capricious spectry ExercisePotential.)

We are ardent referable simply in the voltage, beside so in aggregate membrane vulgar balance the series of the exercise virtual. We bear referable measured vulgar at-once, beside can admire it using the relationship

I =−C dV/dt

where C is membrane capacitance. C can be admired from cell demeanor area using the mode that 1 cm2 of membrane has a capacitance of 1µF.

Using a convenient dissonance mode of adjust h2 to admire the derivative of V, reckon the vulgar I using the equation overhead when the cell transection is 1 micron (that is, 1µm, or 10-6 m). Indicate the span aims at which the convenient dissonance mode canreferable be applied, and conservation a impertinent or inconversant dissonance mode (so of adjust h2), as divert, to reckon the derivative at those aims. (Your remainder, hence, should effect a appreciate of the derivative at EVERY span aim represented in the axioms established.)

Make span contrives: individual obtain be the voltage (y-axis) as a office of span (x-axis), and the other obtain be the admired aggregate membrane vulgar (y-axis) as a office of voltage (xaxis). This highest contrive is a span-domain contrive, and the promote is a state-space contrive. On twain contrives, label (control specimen, by adding symbols or passage labels to the contrive) the peaks (consummation and minimum voltage) of the exercise virtual. On the state-space contrive, so label the peak internal (negative) vulgar and the peak external (positive) vulgar.

Becaconservation the membrane produce to the similar voltage from which it agoing, the statespace contrive controlms a loop. If we were to pull arrows representing how we income encircling the loop with span, would those arrows income in a clockwise or a counterclockwise course? How can you utter? What does this utter us environing the vulgar confused in creating an exercise virtual – does the forthcoming distribute of the exercise virtual set-out with for-the-most-part internal (negative) or for-the-most-part external (positive) vulgar?

(Exercise Virtual axioms)

-1.92   -57.75

-1.905 -57.7

-1.89   -57.75

-1.875 -57.75

-1.86   -57.75

-1.845 -57.7

-1.83   -57.7

-1.815 -57.8

-1.8     -57.75

-1.785 -57.75

-1.77   -57.75

-1.755 -57.8

-1.74   -57.8

-1.725 -57.75

-1.71   -57.7

-1.695 -57.75

-1.68   -57.8

-1.665 -57.75

-1.65   -57.75

-1.635 -57.75

-1.62   -57.75

-1.605 -57.75

-1.59   -57.7

-1.575 -57.7

-1.56   -57.75

-1.545 -57.75

-1.53   -57.7

-1.515 -57.7

-1.5     -57.7

## Expert Exculpation  