solving 2D FitzHugh-Nagumo system

Question

Kate Heinzman 2020-3-5

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/509275-solving-2d-fitzhugh-nagumo-system

评论： darova 2020-3-7

function [U,V,t] = fhn1d(tend,dx,stimMask,stimStr,stimDur,U0,V0,t0)

% function [U,V,t] = fhn1d(tend,dx,stimMask,stimStr,stimDur,U0,V0,t0)

% simulates propagation of a FitxHugh-Nagumo cable

%

% INPUT: tend time to integrate to

% dx spatial time step in nm

% stimMask length of this vector specifies how many nodes are in

% the cable and specifies which nodes will receive a

% stimulus current

% stimStr strength of current stimulus

% stimDur duration of current stimulus

% U0 vector the same length of stimMask that contains the

% initial condition for u on each node

% V0 vector the same length of stimMask that contains the

% initial condition for v on each node

% t0 time at which to begin integrating

%

% OUTPUT: U contains solution where it has a column for each node on

% the cable and a row for each timestep

% V contains solution where it has a column for each node on

% the cable and a row for each timestep

% t vector with the time of each timestep

% Set argument defaults

% The first three arguments are required (others are optional)

if nargin < 8 || isempty(t0)

t0 = 0;

end

if nargin < 7 || isempty(V0)

V0 = zeros(size(stimMask));

end

if nargin < 6 || isempty(U0)

U0 = zeros(size(stimMask));

end

if nargin < 5 || isempty(stimDur)

stimDur = 0;

end

if nargin < 4 || isempty(stimStr)

stimStr = 0;

end

% Set membrane parameters and cable conductivity

a = 0.13;

b = 0.013;

c1 = 0.26;

c2 = 0.1;

D = 5;

% stimMask is a column vector

stimMask = stimMask(:);

% Number of nodes

nx = length(stimMask);

% Stack up initial values for u and v into a long column vector

% All u values are first and then the v values

UV0 = [U0(:);V0(:)];

% Set time at which stimulus turns off

stimOff = stimDur + t0;

% Make a handle to an anonymous function that will evaluate the

% right-hand-sides of all the ODEs in the system

fhnHandle = @(t,UV) fhn1(t,UV,stimStr,stimOff,stimMask,dx,a,b,c1,c2,D);

% Call ode 45 to do the integration

% On output, t is a column vector of times

% UV has one column for each ODE in the system and one row per timestep

% fhn1 is written so that the first nx columns in UV contain u values going

% down the cable and the next nx columns contain the v values

[t,UV] = ode45(fhnHandle,[t0, tend],UV0);

% Unpack UV into seperate U and V arrays

U = UV(:,1:nx);

V = UV(:,nx+1:end);

plotFiber(U,V,t)

end

function dUVdt = fhn1(t,UV,stimStr,stimOff,stimMask,dx,a,b,c1,c2,D)

% function dUVdt = fhn1(t,UV,stimStr,stimOff,stimMask,dx,a,b,c1,c2,D)

% evaluates the right-hand-sides of a system of ODEs that stimulate

% propagation on an FHN cable; called by ode45()

%

% INPUT: t current time

% UV column vector with the current values of the dependent

% variables

% stimStr strength of current stimulus

% stimOff time at which the stimulus switches off

% stimMask vector that specifies which nodes receive the stimulus

% dx spatial time step in nm

% a,b,c1,c2,D model parameters

%

% OUTPUT: dUVdt column vector containing solutions

% Get number of nodes

nx = length(stimMask);

% Unpack u and v

u = UV(1:nx);

v = UV(nx+1:end);

% Compute the diffusive term for each node, d2u/dx2, and store in column

% vector L

% Preallocate L

L = zeros(nx,1);

% Special term for first node that imposes no-flux boundary

L(1) = D*(u(2)-u(1))/(dx*dx);

% Diffusive term for all interior nodes

for i = 2:nx-1

L(i) = D*(u(i-1) + u(i+1) - 2*u(i))/(dx*dx);

end

% No-flux boundary condition for last node

L(nx) = D*(u(nx-1)-u(nx))/(dx*dx);

% Make a vector that contains the current stimulus for each node

if t < stimOff

IStim = stimMask*stimStr;

else

IStim = stimMask*0;

end

% Get fhn current for each node

I = c1.*u.*(u-a).*(1-u)-c2*u.*v;

% Add all the terms together to give dUdt on each node

dUdt = L + I + IStim;

% Compute dVdt

dVdt = b*(u-v);

% Stack all of our dUdt and dVdt values into one column vector

dUVdt = [dUdt;dVdt];

end

How do I take my code for the solving the fitzhugh-nagumo system in one dimension and solve it two dimensions according to the equations below?

The membrane parameters (a,b,c1,c2) are the same. D11 and D22 are conductivity constants in the x and y directions respectively; both can be set to five. How do you specify the number of nodes in the sheet in both directions? The spatial steps dx and dy should be input arguments to the function. How do you do the no flux boundary conditions for d2u/dx2 on the left and right edges of the sheet and the top and bottom edges for d2u/dy2? Then once the diffusive terms are figured out for this system, how do you pass them into ode45? Overall, I want to know how to convert my 1d version into what needs to be solved for the 2d version.

5 个评论
显示 3更早的评论隐藏 3更早的评论

darova 2020-3-7

I already have answered you about your difference scheme. Did you see that?

LINK

Kate Heinzman 2020-3-7

Yes, but this solution did not help me. Can you please read over this current problem that I am asking about? I am not concerned with the difference scheme; I am confused on how to pass the data to ode45 such that it returns a 3D array where the first layer is time and second and third layers are the two spatial dimensions. What I am asking is how to take the correct 1 dimensional code I already have posted above and update it to a 2 dimensional version.

请先登录，再进行评论。

请先登录，再回答此问题。

Answer 1

darova 2020-3-7

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/509275-solving-2d-fitzhugh-nagumo-system#answer_419045

FDM_central_diff.m

How do you do the no flux boundary conditions for d2u/dx2 on the left and right edges of the sheet and the top and bottom edges for d2u/dy2?

It's question to you. What are boundary conditions (temperature or heat flux) ?

Then once the diffusive terms are figured out for this system, how do you pass them into ode45? Overall, I want to know how to convert my 1d version into what needs to be solved for the 2d version.

Once you have "2D version" you can't use ode45

I attach similar problem solved using difference scheme. See if it helps

2 个评论
显示无隐藏无

Kate Heinzman 2020-3-7

I've been tasked with using ode45 to solve it; I can't use the difference scheme. Boundary conditions are the same as in the above code except they have to be applied for d2u/dx2 on the left and right columns of the sheet, d2u/dy2 on the top and bottomo row of sheet, and both d2u/dx2 and d2u/dy2 on the corners since this is where they overlap.

darova 2020-3-7

ode45 (Ordinary Differencial Equation) is for ordinary diff equations

You have partial differential equations (function depends on 2 or more variables)

You can't solve your equations using ode45

请先登录，再进行评论。

solving 2D FitzHugh-Nagumo system

5 个评论
显示 3更早的评论隐藏 3更早的评论

回答（1 个）

2 个评论
显示无隐藏无

另请参阅

类别

标签

Community Treasure Hunt

solving 2D FitzHugh-Nagumo system

5 个评论 显示 3更早的评论隐藏 3更早的评论

回答（1 个）

2 个评论 显示 无隐藏 无

另请参阅

类别

标签

Community Treasure Hunt

5 个评论
显示 3更早的评论隐藏 3更早的评论

2 个评论
显示无隐藏无