Parameter estimation of a two state delay differential equation modeling the human respiratory system

We study parameter estimation for the two state model which describes the balance equation for carbon dioxide and oxygen in human respiratory system. These are nonlinear parameter dependent and because of the transport delay in the respiratory control system, they are modeled with delay differential equation. Numerically simulated noisy data are generated and several examples are studied with Levenberg-Marquardt and Trust-region algorithms to determine the values of unknown parameters.Comment: 17 page

Automated analyses of innate olfactory behaviors in rodents.

Olfaction based behavioral experiments are important for the investigation of sensory coding, perception, decision making and memory formation. The predominant experimental paradigms employ forced choice operant assays, which require associative learning and reinforced training. Animal performance in these assays not only reflects odor perception but also the confidence in decision making and memory. In this study, we describe a versatile and automated setup, "Poking-Registered Olfactory Behavior Evaluation System" (PROBES), which can be adapted to perform multiple olfactory assays. In addition to forced choice assays, we employ this system to examine animal's innate ability for odor detection, discrimination and preference without elaborate training procedures. These assays provide quantitative measurements of odor discrimination and robust readouts of odor preference. Using PROBES, we find odor detection thresholds are at lower concentrations in naïve animals than those determined by forced choice assays. PROBES-based automated assays provide an efficient way of analyzing innate odor-triggered behaviors

Behavior Box Design.

<p><b>A</b>) Single port design with one pair IR LED emitter and detector that can be used to trigger odor delivery. <b>B</b>) A triple port design that combines the odor port in <b>A</b> with two water ports. A water spout is included in each water port. <b>C</b>) A single port design that combines water and odor delivery. Two sets of paired IR LED emitter and detector are implemented to detect nose poking and licking activity.</p

Odor Delivery Controlled by the Olfactometer.

<p><b>A</b>) Sample traces of PID measurement of AA at various concentrations. <b>B</b>) Dose response curve of four different odors at seven concentrations. <b>C&D</b>) PID measurements of repeated odor applications over multiple sessions using single odor vials (<b>C</b>) and syringe top filters (<b>D</b>). Flow speed and application duration are as indicated. <b>E</b>) Normalized PID measurement values as a function of time after odor delivery. Odor concentration: 2.5×10⁻³ s.v. <b>F</b>) Sample PID traces of odor mixture delivery using olfactometer. The black bars indicate the delivery of HXH. The red bars indicate the delivery of HXO.</p

PROBES based habituation/Dis-habituation Analysis of Innate Odor Preference.

<p><b>A&B</b>) Bar plots of odor port investigation by a single animal (upper panel) and the average of <i>NPI</i> value for FU (<b>A</b>, 10 animals) or 2-MBA (<b>B</b>, 11 animals). Dashed lines indicate 100% of average <i>NPI</i> to mineral oil. Calculation of attraction and aversion indexes is indicated. <b>C&D</b>) Comparison of attraction(<b>C</b>) and aversion (<b>D</b>) indexes obtained by PROBES assay and 3-chamber assays. The <i>p</i> values of the tests are indicated. <b>E</b>) Comparison table of PROBES and 3-chamber methods for innate preference assays.</p

Reinforced Two-Choice Assay and Go/No Go Assay using PROBES.

<p><b>A</b>) The learning curve in a two-choice assay to discriminate AA and HPH (1∶100 concentration, v/v) in 9 animals. Individual animals’ training curves are shown in grey lines. Error bars, S.E.M. <b>B</b>) Success rate (<i>SR</i>) in two-choice testing with decreasing odor concentration using the same set of animals as <b>A</b>. Chance level of discrimination is 50%, indicated by a dashed line. <b>C</b>) Training curve in a Go/No Go training using AA as reward odor (CS+, 1∶100, v/v) and MO (CS−). Individual animals training curves are shown in grey. <b>D</b>) Success rate in Go/No Go test with increasing odor concentration. The average <i>SR</i> from eight animals are indicated in black. The psychometric performances of three individual animals are plotted in grey. The results are fit with a Weibull psychometric function. Threshold values (T) calculated from the fitting are indicated in the figure.</p

The Design of PROBES.

<p><b>A</b>) Schematic illustration of the system. A single PROBES software interface gives commands and integrates readout from the olfactometers and behaviors boxes. Four separate olfactometers and behavioral boxes are independently controlled, and each olfactometer controls eight odor channels. Odor from each olfactometer is delivered to the corresponding behavior box. Readout is indicated by purple lines, commands are indicated by dark blue lines and air flow paths are indicated by green lines. <b>B</b>) Photograph of PROBES setup with four olfactometers. Each box contains two independent olfactometers. <b>C</b>) Detailed layout of a box containing two separate olfactometers. <b>D</b>) A typical behavior box with the triple port configuration for reinforced two-choice assays. Odors are delivered through the middle nose cone while the water reward ports are controlled by two water valves.</p

Cross Habituation Assay of Odor Discrimination.

<p><b>A</b>) Schematic illustration of odor application sequence in a cross habituation assay. <b>B</b>) Bar plots of odor port investigation duration of two animals. Dashed lines indicate the average activity during eight sessions of mineral oil presentation. Odor dilution, 1∶1000. <b>C</b>) Bar plots of <i>NPI</i> from <b>B</b>. <i>ΔNPI</i> calculation is indicated. MO: mineral oil. <b>D</b>) Bar plots of average <i>NPI</i> in one cross habituation experiment. The number of animals tested is indicated; N.S., not significant. **: <i>p</i><0.01. <b>E</b>) Bar plot of average <i>ΔNPI</i> for five odor pairs. Odor structures and pairwise <i>t-</i>test <i>p</i> values are shown. Error bars: S.E.M.</p

Behavior Assay Using PROBES.

<p><b>A</b>) Schematic illustration of odor delivery sequence in dis-habituation and cross habituation assays. S1, S2, S3 indicate odor stimulus or air control. <b>B</b>) Event log from software package. (<b>i</b>) List of event types and corresponding device locations used in the log sheet. (<b>ii</b>) An example event log listing time, type, location and magnitude. Five pairs of events are labelled with different colors. The “on” and “off” events of each pair are indicated. Time elapsed between “off” and “on” indicates the duration of an event. <b>C</b>) Raster plots showing odor port investigation of two female animals (<b>i</b> and <b>ii</b>) in response to male urine registered by PROBES and with video recording. Enlarged views of activities during the last trial of mineral oil delivery and the first trial of male urine delivery are shown to illustrate the differences in investigation measurement between PROBES and video recording. TTL square pulses are marked to indicate odor delivery. Shaded columns indicate odor delivery of mineral oil (grey) and mouse urine (blue). Arrows 1–5 mark the five event pairs in <b>B</b>. <b>D</b>) Bar plots of the duration of odor investigation shown in <b>C</b>. The total time of odor port investigation in each trial was plotted. <b>E</b>) Comparison table for these two methods.</p

Three-chamber Test for Odor Preference.

<p><b>A</b>) Schematic illustration of a classic 3-chamber design. The positions of odor vials are indicated. Shaded areas in each arm of the chamber indicate the odor zone for behavior analysis (<b>i</b>). Tracking traces of four animals in a control experiment with mineral oil vials in both chambers are shown (<b>ii</b>). <b>B</b>) Scatter plots of time spent in two zones in a control session with mineral oil. Each test session was 10 minutes. Twelve animals were examined. Pair wised <i>t</i>-test <i>p</i> value is indicated. <b>C</b>) Odor preference test for female urine (FU, <b>i</b>) and 2-MBA (<b>ii</b>). The upper panels show the tracking trace of one animal in the behavior chamber. Bar graphs indicate time spent in each zone. The number of animals (n) and <i>p</i> values of pairwise <i>t</i>-test are shown. <b>D</b>) Box plot of the ratio between test odor and MO. Dashed line indicates no difference between two zones. One-way ANOVA <i>p</i> values are indicated.</p