## Introduction

This document describes construction of delay lines in JSim MML. A delay line is a mechanism for delaying the delivery of a signal. Mathematically, this may be thought of as querying the value of a variable v(t) at an earlier time v(t-delay). While MML does not have an explicit "delay()" function currently (one is being planned), delay lines may be created in MML using several alternative methods which are described below.

Prerequisites:

- Introductory Survey of MML (required)
- Introduction to the JSim GUI (recommended)

Contents:

- Variable Function (VF) Delay Lines
- PDE Delay Lines
- ODE Series Delay Lines
- Comments or Questions?

## Variable Function (VF) Delay Lines

This form of delay line uses MML VF syntax, which treats a dynamic variable v as a function v(expression). For a simple delay line, the appropriate VF call is v(t-delay) which is described below (those interested if the full generality of VFs, should see Using Variable Functions in MML). Consider the following example:

// simple variable function delay line math vf1 { realDomain t; t.min=0; t.max=10; t.delta=0.25; real delay = 1; // user-specified time delay real u(t); // the normal state variable real udelayed(t); // the delayed state variable when (t=t.min) u=1; u:t = -udelayed; udelayed = if (t<delay) 0 else u(t-delay); }

Note that the definition of udelayed uses an "if" clause to prevent u() queries for t less than 0. Such queries would cause a out-of-range error when the model is run.

While the above delay implementation is adequate in most circumstances, there are some subtleties that can sometimes cause problems:

- JSim calculates u(t-delay) by linearly interpolating u(t) values calculated from previous time points, that is, u's values on the defined time grid. This means the accuraccy of u(t-delay) depends upon the pseudo-linear behaviour of u. Normally, specifying an appropriately small time step (t.delta) will address this problem.
- The above interpolation behaviour means that if delay is less than t.delta, then u(t-delay) will return a NaN because the interpolated values will include u values not yet calculated. This means you must specify a delay value greater that t.delta to get acceptable results. In any case, delay must normally be large compared to t.delta in order to achieve acceptable interpolation accuraccy.
- Third, the "if" clause above is, strictly speaking, correct only for t.min=0. If t.min is not necessarily zero in your model, the correct "if" clause should be "if (t-t.min < delay) ..."

The following model incorporates two refinements of the previous one's "if" clause. First, udelayed is set equal to u if the delay is too small, allowing the case of the delay=0 to be handled correctly. Second, the case of non-zero t.min is handled correctly.

// refined variable function delay line math vf2 { realDomain t; t.min=0; t.max=10; t.delta=0.25; real delay = 1; // user-specified time delay real u(t); // the normal state variable real udelayed(t); // the delayed state variable when (t=t.min) u=1; u:t = -udelayed; udelayed = if (delay < t.delta) u else ( if (t-t.min < delay) 0 else u(t-delay)); }

## PDE Delay Lines

To understand this section, you should first read Using PDEs in MML.

*This section is currently under development.*

## ODE Series Delay Lines

To understand this section, you should first read Using ODEs in MML.

*This section is currently under development.*

**Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants:** NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.