# Hindmarsh rose model with delay (dde)

## Main Question or Discussion Point

hi all,

I have a problem using the MATLAB dde23 solver to evaluate the delayed HR model:

dx/dt = y(t) - a*x(t)^3 + b*x(t)^2 - z(t) + e (x(t-tau) - x(t))

dy/dt = c - d*x(t)^2 - y(t)

dz/dt = r [ s*(x(t) - xo) - z(t)]

how could I set the history values (-tau<t<0)?

it is possible use the dde23 solver or I must use some approximation? if so what's the best approximation method?

tnx

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hi all,

I have a problem using the MATLAB dde23 solver to evaluate the delayed HR model:

dx/dt = y(t) - a*x(t)^3 + b*x(t)^2 - z(t) + e (x(t-tau) - x(t))
dy/dt = c - d*x(t)^2 - y(t)
dz/dt = r [ s*(x(t) - xo) - z(t)]

how could I set the history values (-tau<t<0)?

it is possible use the dde23 solver or I must use some approximation? if so what's the best approximation method?

tnx
Hi,

I should have thought that the simplest thing to do would be just to put x(t) = x(0) for all t<0. An alternative would be x(t)=0 for t<0: this would imply that the delayed variable simply 'doesn't exist' before t=tau. Finally, you could put x(t-tau)=x(t)for t<0: this is tantamount to solving the undelayed ODE up to t-tau, and putting the delay in for t>tau. You could always experiment.

That said, most of the DDEs I have worked with have come out integrating a PDE w.r.t. its 'non-time' variable (e.g. Gurney WSC, Nisbet RM, Lawton, JH. 1983. The systematic formulation of tractable single-species population models incorporating age structure. Journal of Animal Ecology 52: 479-495.) and so the history of the delayed variable comes out naturally. Maybe just putting t-tau for t in the right-hand side of an ODE is not a good way of getting a DDE without extra thought.

The dde23 solve in Matlab should also work with this problem. There is also Simon Wood's solv95 programme (http://www.maths.bath.ac.uk/~sw283/simon/dde.html [Broken]).

Ashley

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