Mathematical Methods–Homework Assignment 1

Mathematical Methods–Homework Assignment 1 November 30, 2015

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Introduction on the models Hodgkin-Huxley

In this exercise we consider the Hodgkin-Huxley model for the generation of an action potential in the axon of giant squid. It describes how the membrane potential of a small patch varies over time as a function of the current state and the fluxes of sodium and potassium currents. You may find more background information in the course Dynamical Behavior of Neuronal Networks or on the internet. These exercises consider the mathematical part, i.e. analysis of fixed points and simulation. The equations are given by  0 V = −gN a m3 h(V − EN a ) − gK n4 (V − EK ) − gL (V − EL ) + I(t),    0 m = (m∞ (V ) − m)/τm (V ), (1) h0 = (h∞ (V ) − h)/τh (V ),    0 n = (n∞ (V ) − n)/τn (V ). In the sequel we write X 0 = f (t, X), even if f would not depend explicitly on t. The vector X has the components X = (V, m, h, n). All constants are fixed, except for the input I that will be varied. gN a = 120; gK = 36; gL = .003; EN a = 50; EK = −90; EL = −70; The functions xi,∞ (V ) and τi (V ) for i = m, h, n are defined by xi,∞ (V ) = αi /(αi + βi ) and τi (V ) = 1/(αi + βi ) with αm = 0.1 ∗ (V + 35)/(1 − exp(−(V + 35)/10)); βm = 4 ∗ exp(−(V + 60)/18); αh = 0.07 ∗ exp(−(V + 60)/20); βh = 1/(1 + exp(−(V + 30)/10)); αn = 0.01 ∗ (V + 50)/(1 − exp(−(V + 50)/10)); βn = 0.125 ∗ exp(−(V + 60)/80);

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Wilson-Cowan

Here we consider the average dynamics of a neural population consisting of many cells and two types, i.e. excitatory E and inhibitory I. We have 0 ≤ E, I ≤ 1. The model is given by E˙ I˙

= −E + (1 − E)Se (c1 E − c2 I + P ) = −I + (1 − I)Si (c3 E − c4 I + Q)

where the function Sx with x = E, I is defined by Sx (u) =

1 1 − . 1 + exp(−bx (u − θx )) 1 + exp(bx θx ))

This means that u = c1 E − c2 I + P for the first equation. The parameters are given by θe = 4, be = 1.3, θi = 3.7, bi = 2, Q = 1, c1 = 20, c2 = 11.5, c3 = 15, c4 = 1, P = 2. 1

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Exercises • Write a matlab function HH.m so that the call f=HH(t,X,I) is the evaluation of (1). • Write a matlab function JACHH.m so that result of the call A=JACHH(t,X,I) is the Jacobian DfX . • For every value of I there is a unique fixed point. Implement a method to approximate this fixed point for a given value of I. Make an (I, V )-diagram where I is plotted horizontally, while vertically you plot the value V of the fixed point. Start at I = 0 and vary I up to a sufficiently large value, e.g. I ≥ 100. • There is a critical value Ic such that two eigenvalues of the fixed point have real part zero. Determine this critical value with 4 digits correct. At this instance only, you may use eig. • For which values of I is the fixed point stable? • Simulate the model using Euler-Backward without adapting the stepsize. Use the initial condition x0 = [−65, .1, .1, .5] with I = 30 constant and 0 ≤ t ≤ 60. As a reference solution, use the matlab-solver ode45. Plot the two solutions within the same figure. The large and short deflections in V are called action potentials. Choose a (maximal) stepsize for EB such that the norm difference kV (t = 60)ode45 − V (t = 60)EB k is smaller than 0.1mV. Determine the number of steps needed. Compare these numbers and timing and comment. Pre-allocation of variables may decrease the required computational time. • Use your EB-solver (or RK4, see slides/notes if you like) to find the range of I for which you have periodic orbits. For each I determine Vmax and Vmin of this periodic orbit. Plot these values (I, Vmax (I)) and (I, Vmin (I)) in your plot of the equilibria. • Show that for I = 10 there are different initial conditions x0 such that the solution X(t) either converges to either a fixed point or to a periodic orbit. Hence, for this value of I, the system is here bistable. • Consider the effect of additive white noise. Take the same initial condition x0 as above and I = 2 + σdW , i.e. noise is added only to the V component. Implement the Euler-Maruyama scheme and show that if σ is large enough, the noise can induce action potentials that would not be present without noise. • Determine for σ from 0 to 4 in appropriate steps the firing rate, i.e. the number of action potential during a sufficiently long interval. • Find the three fixed points of the Wilson-Cowan model. Consider to visualize the curves where E 0 = 0 and I 0 = 0. • Determine the linearization for each fixed point. Classify the fixed point based on the eigenvalues. Make local phase portraits of the linearization x0 = DF (E0 , I0 )x. • Make a global phase portrait by simulating the model with the Heun method (or RK4) with appropriate stepsizes for various (but not too many) initial conditions in the plane. Also indicate the directions of the eigenvectors of the saddle in this plot.

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Report

Write a report about the methods you have used with a motivation of your choice. Comment briefly on the results obtained. Also, hand in the working code (in electronic form) that you have used to obtain your results. There should be a single main-script that reproduces all your results. Call it main with your initials as suffix, e.g. for me this would be main HGEM. The code should be

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readable, i.e. contain documentation. It should be clear what you do at every step, by choosing correct names for your variables and referencing to algorithms in the book. If parts of your solution use the symbolic toolbox or commands such as fsolve, eig or equivalent performing the numerical work in a ”black box”, this part will not be given credit.

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