text: A B C 35 35 3 30 30 25 25 2.5 20 20 2 15 15 10 10 1.5 5 5 0 0 1 -5 -5 -10 -10 0.5 0 50 100 150 0 50 100 150 0 50 100 Actual laser current (mA) Actual laser current (mA) Number of ibuprofen samples Figure 7. Sensed nociception decrease due to treatments. (A) Diﬀerences of the inferred laser current between the control and the ibuprofen datasets, ∆ . Errors represent standard Iibuprofen deviations, estimated with bootstrapping. (B) Similar diﬀerences for the mutant worm, ∆ . Error Imutant bars are estimated by bootstrapping. (C) Dependence of the statistical signiﬁcance of the nociception level reduction (measured by the score) for ibuprofen at 110 mA on the number of ibuprofen trials. Z the response. Thus this drug application likely reduced the perceived nociception level in the worm. In contrast, a TRPV mutation changes locomotion in a way that is not as well captured by the template model. Thus we can be objectively critical about any nociception inference made with this strain. The model was also useful in determining key experimental parameters for future measurements. After veriﬁcation that the model works well with the native and ibuprofen treated stimulus-response data, we quantiﬁed the changes in thermal nociception perception due to ibuprofen treatment. Our modeling and experimental assessment of thermal nociception identiﬁed the optimal stimulus range and required number of trials to determine statistically signiﬁcant diﬀerences between the inferred current of N2 in the untreated and treated conditions. In conclusion, we have built a general model that connects stereotyped behavior to stimulus. With a language to describe this relationship, it is now possible to study quantitatively the eﬀects of genetics and chemicals on this sensorimotor behavior. We believe that the utility of the model is quite general and could be applied to diﬀerent model system. However, we particularly hope that this work helps further establish C. elegans as a model for nociceptive research. Materials and Methods Worm preparation and experiment design All worms were grown and maintained under standard conditions [38], incubated with food at 20 ◦C. Well fed worms were washed twice then gently spun down for 1 minute and the supernatant discarded by aspiration. For the drug application, 100 L of ibuprofen in M9 at 100 M was added to the eppendorf µ µ tube. For the wild-type and mutant data set, M9 was used instead of the drug solution. Worms were then placed in an incubator for 30 minutes at 20◦C. After that worms were poured onto a seeded agar plate and transferred to agar assay plates by a platinum wire pick. These assay plates were incubated at 20◦C for 30 minutes, and then the experimental trials were done within the next 30 minutes. In total = 201 worms for the control group, = 441 worms for the ibuprofen group, and = 100 worms for N N N the mutant group (ocr-2(ak47) osm-9(ky10) IV; ocr-1(ak46)) group were tested. The mutant strain was obtained from the Caenorhabditis Genetics Center. The thermal nociceptive stimulation instrument has been described previously [30]. In summary, an ∼ infrared laser is directed to heat the head of a freely crawling worm ( 0 5mm FWHM) on an agar plate. . 12 )Am( tnerruc resal derrefni fo esaerceD erocs-Z ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: B

text: C

text: Actual

text: laser

text: current

text: (mA)

text: Actual

text: laser

text: current

text: (mA)

text: Number

text: of

text: ibuprofen

text: samples

text: Figure

text: Sensed

text: nociception

text: decrease

text: due

text: to

text: treatments.

text: (A)

text: Diﬀerences

text: of

text: the

text: inferred

text: laser

text: current

text: between

text: the

text: control

text: and

text: the

text: ibuprofen

text: datasets,

text: Errors

text: represent

text: standard

text: Iibuprofen

text: deviations,

text: estimated

text: with

text: bootstrapping.

text: (B)

text: Similar

text: diﬀerences

text: for

text: the

text: mutant

text: worm,

text: Error

text: Imutant

text: bars

text: are

text: estimated

text: by

text: bootstrapping.

text: (C)

text: Dependence

text: of

text: the

text: statistical

text: signiﬁcance

text: of

text: the

text: nociception

text: level

text: reduction

text: (measured

text: by

text: the

text: score)

text: for

text: ibuprofen

text: at

text: mA

text: on

text: the

text: number

text: of

text: ibuprofen

text: trials.

text: Z

text: the

text: response.

text: Thus

text: this

text: drug

text: application

text: likely

text: reduced

text: the

text: perceived

text: nociception

text: level

text: in

text: the

text: worm.

text: In

text: contrast,

text: a

text: TRPV

text: mutation

text: changes

text: locomotion

text: in

text: a

text: way

text: that

text: is

text: not

text: as

text: well

text: captured

text: by

text: the

text: template

text: model.

text: Thus

text: we

text: can

text: be

text: objectively

text: critical

text: about

text: any

text: nociception

text: inference

text: made

text: with

text: this

text: strain.

text: The

text: model

text: was

text: also

text: useful

text: in

text: determining

text: key

text: experimental

text: parameters

text: for

text: future

text: measurements.

text: After

text: veriﬁcation

text: that

text: the

text: model

text: works

text: well

text: with

text: the

text: native

text: and

text: ibuprofen

text: treated

text: stimulus-response

text: data,

text: we

text: quantiﬁed

text: the

text: changes

text: in

text: thermal

text: nociception

text: perception

text: due

text: to

text: ibuprofen

text: treatment.

text: Our

text: modeling

text: and

text: experimental

text: assessment

text: of

text: thermal

text: nociception

text: identiﬁed

text: the

text: optimal

text: stimulus

text: range

text: and

text: required

text: number

text: of

text: trials

text: to

text: determine

text: statistically

text: signiﬁcant

text: diﬀerences

text: between

text: the

text: inferred

text: current

text: of

text: in

text: the

text: untreated

text: and

text: treated

text: conditions.

text: In

text: conclusion,

text: we

text: have

text: built

text: a

text: general

text: model

text: that

text: connects

text: stereotyped

text: behavior

text: to

text: stimulus.

text: With

text: a

text: language

text: to

text: describe

text: this

text: relationship,

text: it

text: is

text: now

text: possible

text: to

text: study

text: quantitatively

text: the

text: eﬀects

text: of

text: genetics

text: and

text: chemicals

text: on

text: this

text: sensorimotor

text: behavior.

text: We

text: believe

text: that

text: the

text: utility

text: of

text: the

text: model

text: is

text: quite

text: general

text: and

text: could

text: be

text: applied

text: to

text: diﬀerent

text: model

text: system.

text: However,

text: we

text: particularly

text: hope

text: that

text: this

text: work

text: helps

text: further

text: establish

text: C.

text: elegans

text: as

text: a

text: model

text: for

text: nociceptive

text: research.

text: Materials

text: and

text: Methods

text: Worm

text: preparation

text: and

text: experiment

text: design

text: All

text: worms

text: were

text: grown

text: and

text: maintained

text: under

text: standard

text: conditions

text: incubated

text: with

text: food

text: at

text: ◦C.

text: Well

text: fed

text: worms

text: were

text: washed

text: twice

text: then

text: gently

text: spun

text: down

text: for

text: minute

text: and

text: the

text: supernatant

text: discarded

text: by

text: aspiration.

text: For

text: the

text: drug

text: application,

text: L

text: of

text: ibuprofen

text: in

text: at

text: M

text: was

text: added

text: to

text: the

text: eppendorf

text: µ

text: µ

text: tube.

text: For

text: the

text: wild-type

text: and

text: mutant

text: data

text: set,

text: was

text: used

text: instead

text: of

text: the

text: drug

text: solution.

text: Worms

text: were

text: then

text: placed

text: in

text: an

text: incubator

text: for

text: minutes

text: at

text: After

text: that

text: worms

text: were

text: poured

text: onto

text: a

text: seeded

text: agar

text: plate

text: and

text: transferred

text: to

text: agar

text: assay

text: plates

text: by

text: a

text: platinum

text: wire

text: pick.

text: These

text: assay

text: plates

text: were

text: incubated

text: at

text: for

text: minutes,

text: and

text: then

text: the

text: experimental

text: trials

text: were

text: done

text: within

text: the

text: next

text: minutes.

text: In

text: total

text: worms

text: for

text: the

text: control

text: group,

text: worms

text: for

text: the

text: ibuprofen

text: group,

text: and

text: worms

text: for

text: N

text: N

text: N

text: the

text: mutant

text: group

text: IV;

text: group

text: were

text: tested.

text: The

text: mutant

text: strain

text: was

text: obtained

text: from

text: the

text: Caenorhabditis

text: Genetics

text: Center.

text: The

text: thermal

text: nociceptive

text: stimulation

text: instrument

text: has

text: been

text: described

text: previously

text: In

text: summary,

text: an

text: infrared

text: laser

text: is

text: directed

text: to

text: heat

text: the

text: head

text: of

text: a

text: freely

text: crawling

text: worm

text: FWHM)

text: on

text: an

text: agar

text: plate.

text: )Am(

text: tnerruc

text: resal

text: derrefni

text: fo

text: esaerceD

text: erocs-Z

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text: 39. Efron B, Tibshirani R. An introduction to the bootstrap. Boca Raton, FL: Chapman & Hall/CRC; 1994. 40. Nemenman I, Bialek W. Occam factors and model independent Bayesian learning of continuous distributions. Phys Rev E. 2002;65:026137. 17

label: reference

text: Efron

text: B,

text: Tibshirani

text: R.

text: An

text: introduction

text: to

text: the

text: bootstrap.

text: Boca

text: Raton,

text: FL:

text: Chapman

text: Hall/CRC;

text: Nemenman

text: I,

text: Bialek

text: W.

text: Occam

text: factors

text: and

text: model

text: independent

text: Bayesian

text: learning

text: of

text: continuous

text: distributions.

text: Phys

text: Rev

text: E.

label: paragraph


text: level following pharmacological or genetic treatments from the behavior of an individual worm. This quantiﬁcation requires only a handful of worms (about 60) to show statistically signiﬁcant nociception reduction for a common analgesic, and its statistical power quickly improves with an increasing number of subjects. The paper is organized as follows. First we discuss the structure of the dataset and the model. Then we evaluate the performance of the model and discuss the stereotypical behavior we discovered. Further, we use the model to infer the nociceptive stimulus level of worms in three diﬀerent conditions: wild-type untreated, wild-type treated with ibuprofen, and a mutant with defects in thermal nociception. We argue that, while eﬀects of the ibuprofen treatment can be attributed largely to a reduced nociception, the mutant’s response shows changes to the behavior beyond nociceptive eﬀects. Finally, we use the statistical model to discuss how the nociception experiments should be designed to achieve the highest statistical power to quantify analgesic eﬀects. Results We aim to infer the perceived worm nociception level from the temporal dynamics of the worm response. The noxious signal is administered by thermal stimulation using an infrared laser while the worm crawls on an agar plate. The worm motion is captured by video microscopy and analyzed using custom image analysis software. The worm postures are very stereotypical, adding up to simple sinusoidal motions forwards or backwards, and to turns [24]. Thus without much loss to the statistical power, we characterize the entire nociceptive behavior of the animal by a time series of its center of mass velocity (see Materials and Methods). Our task is then to verify if such responses are stereotypical, scaling in frequency and amplitude with the applied laser current. If they are stereotypical and thus can be used to characterize the perceived nociception, the next task is to infer the applied laser current from the velocity data. For each noxious stimulus trial a random worm is selected on the plate and its motion is sampled at 60 Hz for 15 s. An infrared laser pulse with a randomly chosen current between 0 to 200 mA and a duration of 0.1 s is then directed to the head of the worm 1 s after the start of the video recording. The worm’s center of mass motion in a typical trial consists of a forward motion before the stimulus, a stop and/or backwards motion after the stimulus, then followed by an “omega” turn, after which the worm emerges with a forward motion in a diﬀerent direction, cf. Fig. 1. This is a typical response to many noxious stimuli in C. elegans, and not just to thermal noxious stimulus [30]. Such nociceptive response, at least at large stimuli levels, is diﬀerent from the more commonly studied thermotactic behavior, and is mediated by diﬀerent neural and genetic pathways [29, 30]. To understand the eﬀects of pharmacological and genetic interventions on nociception transduction, we collected three distinct datasets. In the ﬁrst, the stimulus was applied to wild-type worms (N2). In the second, the wild-type worms were pre-treated with an ibuprofen solution (see Materials and Methods). In the third, we applied the stimulus to an untreated triple mutant (ocr-2(ak47) osm-9(ky10) IV; ocr- 1(ak46)). Hereafter we refer to these datasets as “control”, “ibuprofen”, and “mutant”, respectively. We collectively refer to ibuprofen and mutant worms as “treated” worms. Forward motion in all three ± ± ± data sets was similar (typical forward velocity of 13 9, 9 7, 13 10 pixel/s respectively, where the error denoted the standard deviation of the velocity distribution), so that there are no drastic defects in motility. Statistical model of the nociception We bin the laser current of the nociceptive stimulus into ﬁve distinct levels (bins), deﬁned to have an equal number of control worms in each bin (40 per bin). The maximum reverse escape velocity is indistinguishable among all three worm types for the largest stimulus level, indicating again no gross defects to motility or noxious response. At the same time, the velocity at smaller stimuli levels shows substantial diﬀerences ( scores of up to 3.8) between the control and the treated worms, especially in Z 3

text: level

text: following

text: pharmacological

text: or

text: genetic

text: treatments

text: from

text: the

text: behavior

text: of

text: an

text: individual

text: worm.

text: This

text: quantiﬁcation

text: requires

text: only

text: a

text: handful

text: of

text: worms

text: (about

text: to

text: show

text: statistically

text: signiﬁcant

text: nociception

text: reduction

text: for

text: a

text: common

text: analgesic,

text: and

text: its

text: statistical

text: power

text: quickly

text: improves

text: with

text: an

text: increasing

text: number

text: of

text: subjects.

text: The

text: paper

text: is

text: organized

text: as

text: follows.

text: First

text: we

text: discuss

text: the

text: structure

text: of

text: the

text: dataset

text: and

text: the

text: model.

text: Then

text: we

text: evaluate

text: the

text: performance

text: of

text: the

text: model

text: and

text: discuss

text: the

text: stereotypical

text: behavior

text: we

text: discovered.

text: Further,

text: we

text: use

text: the

text: model

text: to

text: infer

text: the

text: nociceptive

text: stimulus

text: level

text: of

text: worms

text: in

text: three

text: diﬀerent

text: conditions:

text: wild-type

text: untreated,

text: wild-type

text: treated

text: with

text: ibuprofen,

text: and

text: a

text: mutant

text: with

text: defects

text: in

text: thermal

text: nociception.

text: We

text: argue

text: that,

text: while

text: eﬀects

text: of

text: the

text: ibuprofen

text: treatment

text: can

text: be

text: attributed

text: largely

text: to

text: a

text: reduced

text: nociception,

text: the

text: mutant’s

text: response

text: shows

text: changes

text: to

text: the

text: behavior

text: beyond

text: nociceptive

text: eﬀects.

text: Finally,

text: we

text: use

text: the

text: statistical

text: model

text: to

text: discuss

text: how

text: the

text: nociception

text: experiments

text: should

text: be

text: designed

text: to

text: achieve

text: the

text: highest

text: statistical

text: power

text: to

text: quantify

text: analgesic

text: eﬀects.

text: Results

text: We

text: aim

text: to

text: infer

text: the

text: perceived

text: worm

text: nociception

text: level

text: from

text: the

text: temporal

text: dynamics

text: of

text: the

text: worm

text: response.

text: The

text: noxious

text: signal

text: is

text: administered

text: by

text: thermal

text: stimulation

text: using

text: an

text: infrared

text: laser

text: while

text: the

text: worm

text: crawls

text: on

text: an

text: agar

text: plate.

text: The

text: worm

text: motion

text: is

text: captured

text: by

text: video

text: microscopy

text: and

text: analyzed

text: using

text: custom

text: image

text: analysis

text: software.

text: The

text: worm

text: postures

text: are

text: very

text: stereotypical,

text: adding

text: up

text: to

text: simple

text: sinusoidal

text: motions

text: forwards

text: or

text: backwards,

text: and

text: to

text: turns

text: Thus

text: without

text: much

text: loss

text: to

text: the

text: statistical

text: power,

text: we

text: characterize

text: the

text: entire

text: nociceptive

text: behavior

text: of

text: the

text: animal

text: by

text: a

text: time

text: series

text: of

text: its

text: center

text: of

text: mass

text: velocity

text: (see

text: Materials

text: and

text: Methods).

text: Our

text: task

text: is

text: then

text: to

text: verify

text: if

text: such

text: responses

text: are

text: stereotypical,

text: scaling

text: in

text: frequency

text: and

text: amplitude

text: with

text: the

text: applied

text: laser

text: current.

text: If

text: they

text: are

text: stereotypical

text: and

text: thus

text: can

text: be

text: used

text: to

text: characterize

text: the

text: perceived

text: nociception,

text: the

text: next

text: task

text: is

text: to

text: infer

text: the

text: applied

text: laser

text: current

text: from

text: the

text: velocity

text: data.

text: For

text: each

text: noxious

text: stimulus

text: trial

text: a

text: random

text: worm

text: is

text: selected

text: on

text: the

text: plate

text: and

text: its

text: motion

text: is

text: sampled

text: at

text: Hz

text: for

text: s.

text: An

text: infrared

text: laser

text: pulse

text: with

text: a

text: randomly

text: chosen

text: current

text: between

text: to

text: mA

text: and

text: a

text: duration

text: of

text: s

text: is

text: then

text: directed

text: to

text: the

text: head

text: of

text: the

text: worm

text: s

text: after

text: the

text: start

text: of

text: the

text: video

text: recording.

text: The

text: worm’s

text: center

text: of

text: mass

text: motion

text: in

text: a

text: typical

text: trial

text: consists

text: of

text: a

text: forward

text: motion

text: before

text: the

text: stimulus,

text: a

text: stop

text: and/or

text: backwards

text: motion

text: after

text: the

text: stimulus,

text: then

text: followed

text: by

text: an

text: “omega”

text: turn,

text: after

text: which

text: the

text: worm

text: emerges

text: with

text: a

text: forward

text: motion

text: in

text: a

text: diﬀerent

text: direction,

text: cf.

text: Fig.

text: This

text: is

text: a

text: typical

text: response

text: to

text: many

text: noxious

text: stimuli

text: in

text: C.

text: elegans,

text: and

text: not

text: just

text: to

text: thermal

text: noxious

text: stimulus

text: Such

text: nociceptive

text: response,

text: at

text: least

text: at

text: large

text: stimuli

text: levels,

text: is

text: diﬀerent

text: from

text: the

text: more

text: commonly

text: studied

text: thermotactic

text: behavior,

text: and

text: is

text: mediated

text: by

text: diﬀerent

text: neural

text: and

text: genetic

text: pathways

text: To

text: understand

text: the

text: eﬀects

text: of

text: pharmacological

text: and

text: genetic

text: interventions

text: on

text: nociception

text: transduction,

text: we

text: collected

text: three

text: distinct

text: datasets.

text: In

text: the

text: ﬁrst,

text: the

text: stimulus

text: was

text: applied

text: to

text: wild-type

text: worms

text: In

text: the

text: second,

text: the

text: wild-type

text: worms

text: were

text: pre-treated

text: with

text: an

text: ibuprofen

text: solution

text: (see

text: Materials

text: and

text: Methods).

text: In

text: the

text: third,

text: we

text: applied

text: the

text: stimulus

text: to

text: an

text: untreated

text: triple

text: mutant

text: IV;

text: ocr-

text: Hereafter

text: we

text: refer

text: to

text: these

text: datasets

text: as

text: “control”,

text: “ibuprofen”,

text: and

text: “mutant”,

text: respectively.

text: We

text: collectively

text: refer

text: to

text: ibuprofen

text: and

text: mutant

text: worms

text: as

text: “treated”

text: worms.

text: Forward

text: motion

text: in

text: all

text: three

text: data

text: sets

text: was

text: similar

text: (typical

text: forward

text: velocity

text: of

text: pixel/s

text: respectively,

text: where

text: the

text: error

text: denoted

text: the

text: standard

text: deviation

text: of

text: the

text: velocity

text: distribution),

text: so

text: that

text: there

text: are

text: no

text: drastic

text: defects

text: in

text: motility.

text: Statistical

text: model

text: of

text: the

text: nociception

text: We

text: bin

text: the

text: laser

text: current

text: of

text: the

text: nociceptive

text: stimulus

text: into

text: ﬁve

text: distinct

text: levels

text: (bins),

text: deﬁned

text: to

text: have

text: an

text: equal

text: number

text: of

text: control

text: worms

text: in

text: each

text: bin

text: per

text: bin).

text: The

text: maximum

text: reverse

text: escape

text: velocity

text: is

text: indistinguishable

text: among

text: all

text: three

text: worm

text: types

text: for

text: the

text: largest

text: stimulus

text: level,

text: indicating

text: again

text: no

text: gross

text: defects

text: to

text: motility

text: or

text: noxious

text: response.

text: At

text: the

text: same

text: time,

text: the

text: velocity

text: at

text: smaller

text: stimuli

text: levels

text: shows

text: substantial

text: diﬀerences

text: scores

text: of

text: up

text: to

text: between

text: the

text: control

text: and

text: the

text: treated

text: worms,

text: especially

text: in

text: Z


text: A B C 800 1 1 700 control control control ibuprofen 0.8 ibuprofen 0.8 ibuprofen 600 mutant mutant mutant 500 0.6 0.6 400 300 0.4 0.4 200 0.2 0.2 100 0 0 0 1 1.5 2 2.5 3 3.5 1 1.5 2 2.5 3 3.5 1 1.5 2 2.5 3 3.5 Time (s) Time (s) Time (s) Figure 5. Variability of the active nociceptive response. For the three worm types, in (A) we plot the total variance of velocity proﬁles 2 . Note that these numbers depend strongly on the preprocessing σtotal of the data – in particular, smoothing of the velocity (with the 500 ms ﬁlter, see Materials and Methods) decreases the total variance. In (B) we show the fraction of potentially explainable variance 2 2 . Finally, (C) shows the part of the explainable variance that was not explained by our σstIa/tiσsttoictaall model 2 2. See the main text for the discussion of the diﬀerences between the three σres/σ worm types on this plot.I treated worms is smaller than that for the control worms at the same applied current, the nociception level perceived by the treated worms is smaller. Figure 6 shows the overall structure of the inference done with the model. In the ﬁrst row, we plot the conditional distribution of the inferred laser current given the actual applied current for the three I | worm types, ( type). For presentation purposes, we again bin the trials using into ﬁve bins , P Iinf I, Ii Iµ = 1 5 as before, and plot µ , . . . , X Ntype | | | ( type) = ( v ) (v ) (12) P Iinf Iµ, Pcontrol Iinf i Ptype i Iµ . i | Here (v ) is 1 if the stimulus on the ’th trial for this worm type was in the bin, and zero Ptype i Iµ | i Iµ otherwise. Further, ( v ) is given by the full model, Eq. (8), with the parameters inferred for Pcontrol Iinf i the control worm, and with the empirically observed velocities v in trial for each worm type. We see that i there is more probability concentrated at small for the ibuprofen and the mutant worms, suggesting Iinf a reduction in the perceived stimulus level. Similarly, in the second row in Fig. 6, we plot the expected value, ¯ , of the distribution of the current inferred using the control model, ( |v ), for each of Ii Pcontrol Iinf i the individual trials in each of the three worm types. To the extent that the dots for the ibuprofen and the mutant worms are again somewhat lower than for the control worms, there is some reduction in the perceived current by this measure as well. However, these population averaged results wash out important diﬀerences in the structure of the nociception response. To quantify these small eﬀects more accurately, we now look at the perceived stimulus changes for individual worms in the datasets. Speciﬁcally, for each trial in the control dataset, i a trial ( ) with the closest value of the applied laser current is found in the ibuprofen / mutant dataset j i (the mean magnitude of the current mismatch is 1 mA for both the ibuprofen and the mutant worms). < We then use the control model to calculate the expected value of the inferred current for the th trial j in the ibuprofen / mutant datasets. This expectation is subtracted from the expectation value of the inferred current for the matched trial for the control dataset. The diﬀerence of the expectation values, i averaged over all control worms, is our measure of the reduction in the perceived nociception level 9 )2s/2lexip( ecnairav latoT ecnairav elbanialpxe fo noitcarF ecnairav denialpxenu fo noitcarF ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

font: default

text: B

text: C

text: control

text: control

text: control

text: ibuprofen

text: ibuprofen

text: ibuprofen

text: mutant

text: mutant

text: mutant

text: Time

text: (s)

text: Time

text: (s)

text: Time

text: (s)

text: Figure

text: Variability

text: of

text: the

text: active

text: nociceptive

text: response.

text: For

text: the

text: three

text: worm

text: types,

text: in

text: (A)

text: we

text: plot

text: the

text: total

text: variance

text: of

text: velocity

text: proﬁles

text: Note

text: that

text: these

text: numbers

text: depend

text: strongly

text: on

text: the

text: preprocessing

text: σtotal

text: of

text: the

text: data

text: in

text: particular,

text: smoothing

text: of

text: the

text: velocity

text: (with

text: the

text: ms

text: ﬁlter,

text: see

text: Materials

text: and

text: Methods)

text: decreases

text: the

text: total

text: variance.

text: In

text: (B)

text: we

text: show

text: the

text: fraction

text: of

text: potentially

text: explainable

text: variance

text: Finally,

text: (C)

text: shows

text: the

text: part

text: of

text: the

text: explainable

text: variance

text: that

text: was

text: not

text: explained

text: by

text: our

text: σstIa/tiσsttoictaall

text: model

text: See

text: the

text: main

text: text

text: for

text: the

text: discussion

text: of

text: the

text: diﬀerences

text: between

text: the

text: three

text: σres/σ

text: worm

text: types

text: on

text: this

text: plot.I

text: treated

text: worms

text: is

text: smaller

text: than

text: that

text: for

text: the

text: control

text: worms

text: at

text: the

text: same

text: applied

text: current,

text: the

text: nociception

text: level

text: perceived

text: by

text: the

text: treated

text: worms

text: is

text: smaller.

text: Figure

text: shows

text: the

text: overall

text: structure

text: of

text: the

text: inference

text: done

text: with

text: the

text: model.

text: In

text: the

text: ﬁrst

text: row,

text: we

text: plot

text: the

text: conditional

text: distribution

text: of

text: the

text: inferred

text: laser

text: current

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text: actual

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text: three

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text: worm

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text: type).

text: For

text: presentation

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text: we

text: again

text: bin

text: the

text: trials

text: using

text: into

text: ﬁve

text: bins

text: P

text: Iinf

text: I,

text: Ii

text: Iµ

text: as

text: before,

text: and

text: plot

text: µ

text: X

text: Ntype

text: type)

text: v

text: (v

text: P

text: Iinf

text: Iµ,

text: Pcontrol

text: Iinf

text: i

text: Ptype

text: i

text: Iµ

text: i

text: Here

text: (v

text: is

text: if

text: the

text: stimulus

text: on

text: the

text: ’th

text: trial

text: for

text: this

text: worm

text: type

text: was

text: in

text: the

text: bin,

text: and

text: zero

text: Ptype

text: i

text: Iµ

text: i

text: Iµ

text: otherwise.

text: Further,

text: v

text: is

text: given

text: by

text: the

text: full

text: model,

text: Eq.

text: with

text: the

text: parameters

text: inferred

text: for

text: Pcontrol

text: Iinf

text: i

text: the

text: control

text: worm,

text: and

text: with

text: the

text: empirically

text: observed

text: velocities

text: v

text: in

text: trial

text: for

text: each

text: worm

text: type.

text: We

text: see

text: that

text: i

text: there

text: is

text: more

text: probability

text: concentrated

text: at

text: small

text: for

text: the

text: ibuprofen

text: and

text: the

text: mutant

text: worms,

text: suggesting

text: Iinf

text: a

text: reduction

text: in

text: the

text: perceived

text: stimulus

text: level.

text: Similarly,

text: in

text: the

text: second

text: row

text: in

text: Fig.

text: we

text: plot

text: the

text: expected

text: value,

text: of

text: the

text: distribution

text: of

text: the

text: current

text: inferred

text: using

text: the

text: control

text: model,

text: |v

text: for

text: each

text: of

text: Ii

text: Pcontrol

text: Iinf

text: i

text: the

text: individual

text: trials

text: in

text: each

text: of

text: the

text: three

text: worm

text: types.

text: To

text: the

text: extent

text: that

text: the

text: dots

text: for

text: the

text: ibuprofen

text: and

text: the

text: mutant

text: worms

text: are

text: again

text: somewhat

text: lower

text: than

text: for

text: the

text: control

text: worms,

text: there

text: is

text: some

text: reduction

text: in

text: the

text: perceived

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text: by

text: this

text: measure

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text: well.

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text: population

text: averaged

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text: wash

text: out

text: important

text: diﬀerences

text: in

text: the

text: structure

text: of

text: the

text: nociception

text: response.

text: To

text: quantify

text: these

text: small

text: eﬀects

text: more

text: accurately,

text: we

text: now

text: look

text: at

text: the

text: perceived

text: stimulus

text: changes

text: for

text: individual

text: worms

text: in

text: the

text: datasets.

text: Speciﬁcally,

text: for

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text: the

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text: the

text: applied

text: laser

text: current

text: is

text: found

text: in

text: the

text: ibuprofen

text: mutant

text: dataset

text: j

text: i

text: (the

text: mean

text: magnitude

text: of

text: the

text: current

text: mismatch

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text: mA

text: for

text: both

text: the

text: ibuprofen

text: and

text: the

text: mutant

text: worms).

text: We

text: then

text: use

text: the

text: control

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text: ibuprofen

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text: diﬀerence

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text: Tidal Disruption Events Prefer Unusual Host Galaxies 9 K. Decker French 1, Iair Arcavi 2,3 Ann Zabludoff 1 Subject headings: galaxies: evolution, galaxies: nuclei 5. CONCLUSIONS special and may suggest an answer. Many have had a recent galaxy-galaxy merger (Zabludoﬀ et al. 1996), in- We demonstrate the preference of tidal disruption creasing the possibility of a black hole binary, perturbed events (TDEs) to occur in quiescent galaxies with strong stellar orbits, a spatially-concentrated population of A Balmer line absorption. Quiescent galaxies with the − stars, and/or an evolved population of easily-disrupted strongest Balmer absorption, Hδ σ(Hδ ) > 4 ˚A, make A A A giants. The high incidence of LINER-like emission up only 0.2% of local galaxies, yet host 3 of 8 optical/UV (Yan et al. 2006; Y. Yang et al. 2006) may also play a TDE candidates. A softer cut, Hδ > 1.31˚A, includes A role. In future work, we will explore these connections. only 2.3% of local galaxies, but 6 of 8 optical/UV TDE We thank A. Levan and A. de Ugarte Postigo for pro- host galaxies. The optical/UV TDE rates are thus en- × viding the digital version of their Swift J1644 host galaxy hanced by 190+−115 in the strongest Hδ galaxies and × 100 A spectrum. We thank Yujin Yang for his contributions by 33+−711 in galaxies with HδA > 1.31˚A. Because of this to the stellar population ﬁtting routines. KDF acknowl- preference, the corresponding rates of optical/UV TDEs edges support from NSF grant DGE-1143953, P.E.O., − × − − − × − − are 1 3 10 3 yr 1 per galaxy, and 2 4 10 4 yr 1 and the ARCS Phoenix Chapter and Burton Family. IA per galaxy, respectively. As a result, we predict a lower is grateful for support from the Karp Discovery Award. optical/UV rate in normal star-forming and early-type AIZ and IA thank the Center for Cosmology and Parti- − × − − galaxies, of 1 5 10 6 yr 1 per galaxy. Even the one cle Physics at NYU for its support. AIZ acknowledges high energy TDE with a full measured optical spectrum, funding from NSF grant AST-0908280 and NASA grants Swift J1644, lies in a galaxy with strong Balmer absorp- ADP-NNX10AD476/ADP-NNX10AE88G. AIZ thanks tion and weak nebular line emission, which implies a the John Simon Guggenheim Foundation for its support. × > 80 enhancement in such hosts, and a link between Funding for SDSS-III has been provided by the Alfred the optical/UV and high energy TDE classes. P. Sloan Foundation, the Participating Institutions, the Why do TDEs prefer quiescent, Balmer-strong hosts? National Science Foundation, and the U.S. Department This type of galaxy has several properties that make it of Energy Oﬃce of Science. The SDSS-III web site is http://www.sdss3.org/. REFERENCES Aihara, H., Allende Prieto, C., An, D., et al. 2011, The Conroy, C., & Gunn, J. E. 2010, The Astrophysical Journal, 712, Astrophysical Journal Supplement Series, 193, 29 833 Arcavi, I., Gal-Yam, A., Sullivan, M., et al. 2014, The Conroy, C., Gunn, J. E., & White, M. 2009, The Astrophysical Astrophysical Journal, 793, 38 Journal, 699, 486 Bloom, J. S., Giannios, D., Metzger, B. D., et al. 2011, Science Couch, W. J., & Sharples, R. M. 1987, Monthly Notices of the (New York, N.Y.), 333, 203 Royal Astronomical Society (ISSN 0035-8711), 229, 423 Brinchmann, J., Charlot, S., White, S. D. M., et al. 2004, Dressler, A., & Gunn, J. E. 1983, The Astrophysical Journal, 270, Monthly Notices of the Royal Astronomical Society, 351, 1151 7 Brown, G. C., Levan, A. J., Stanway, E. R., et al. 2015, Monthly Dressler, A., Smail, I., Poggianti, B. M., et al. 1999, The Notices of the Royal Astronomical Society, 452, 4297 Astrophysical Journal Supplement Series, 122, 51 Burrows, D. N., Kennea, J. A., Ghisellini, G., et al. 2011, Nature, Elbaz, D., Dickinson, M., Hwang, H. S., et al. 2011, Astronomy & 476, 421 Astrophysics, 533, 119 Casey, C. M., Narayanan, D., & Cooray, A. 2014, Physics Evans, C. R., & Kochanek, C. S. 1989, The Astrophysical Reports, 541, 45 Journal, 346, L13 Cenko, S. B., Bloom, J. S., Kulkarni, S. R., et al. 2012a, Monthly Gehrels, N. 1986, The Astrophysical Journal, 303, 336 Notices of the Royal Astronomical Society, 420, 2684 Gezari, S., Chornock, R., Rest, A., et al. 2012, Nature, 485, 217 Cenko, S. B., Krimm, H. A., Horesh, A., et al. 2012b, The Hills, J. G. 1975, Nature, 254, 295 Astrophysical Journal, 753, 77 Holoien, T. W.-S., Prieto, J. L., Bersier, D., et al. 2014, Monthly Chen, X., Madau, P., Sesana, A., & Liu, F. K. 2009, The Notices of the Royal Astronomical Society, 445, 3263 Astrophysical Journal, 697, L149 Holoien, T. W.-S., Kochanek, C. S., Prieto, J. L., et al. 2016, Chen, X., Sesana, A., Madau, P., & Liu, F. K. 2011, The Monthly Notices of the Royal Astronomical Society, 455, 2918 Astrophysical Journal, 729, 13 Levan, A. J., Tanvir, N. R., Cenko, S. B., et al. 2011, Science Chornock, R., Berger, E., Gezari, S., et al. 2014, The (New York, N.Y.), 333, 199 Astrophysical Journal, 780, 44 Li, S., Liu, F. K., Berczik, P., & Spurzem, R. 2015, arXiv:1509.00158, 14 MacLeod, M., Guillochon, J., & Ramirez-Ruiz, E. 2012, The 1 Steward Observatory, University of Arizona, 933 North Astrophysical Journal, 757, 134 Cherry Avenue, Tucson AZ 85721 MacLeod, M., Ramirez-Ruiz, E., Grady, S., & Guillochon, J. 2 Las Cumbres Observatory Global Telescope, 6740 Cortona 2013, The Astrophysical Journal, 777, 133 Dr, Suite 102, Goleta, CA 93111 Magorrian, J., & Tremaine, S. 1999, Monthly Notices of the Royal 3 Kavli Institute for Theoretical Physics, University of Cali- Astronomical Society, 309, 447 fornia, Santa Barbara, CA 93106 1 Negative values indicate emission. Hα EW values are cor- Phinney, 1989, The Center of the Galaxy: Proceedings of the 136th Symposium of the International Astronomical Union rected for stellar absorption. 2 Absolute magnitudes (r band, no extinction correction) from Prentice, , Ashall, , Inserra, , et al. 2015, The Astronomer’s SDSS (model mag) for optical/UV TDE hosts, and from Levan Telegram et al−. (2011) for Swift J1644. We assume H0 = 70 km s−1 Rees, M. J. 1988, Nature, 333, 523 Mpc 1, ΩΛ = 0.7, and Ωm = 0.3. ##LTLine##

text: Tidal

text: Disruption

text: Events

text: Prefer

text: Unusual

text: Host

text: Galaxies

text: K.

label: author

text: Decker

text: French

text: Iair

text: Arcavi

text: Ann

text: Zabludoff

text: Subject

label: section

text: headings:

text: galaxies:

label: abstract

text: evolution,

text: galaxies:

text: nuclei

text: CONCLUSIONS

text: special

text: and

text: may

text: suggest

text: an

text: answer.

text: Many

text: have

text: had

text: a

text: recent

text: galaxy-galaxy

text: merger

text: (Zabludoﬀ

text: et

text: al.

text: in-

text: We

text: demonstrate

text: the

text: preference

text: of

text: tidal

text: disruption

text: creasing

text: the

text: possibility

text: of

text: a

text: black

text: hole

text: binary,

text: perturbed

text: events

text: (TDEs)

text: to

text: occur

text: in

text: quiescent

text: galaxies

text: with

text: strong

text: stellar

text: orbits,

text: a

text: spatially-concentrated

text: population

text: of

text: A

text: Balmer

text: line

text: absorption.

text: Quiescent

text: galaxies

text: with

text: the

text: stars,

text: and/or

text: an

text: evolved

text: population

text: of

text: easily-disrupted

text: strongest

text: Balmer

text: absorption,

text: Hδ

text: σ(Hδ

text: ˚A,

text: make

text: A

text: A

text: A

text: giants.

text: The

text: high

text: incidence

text: of

text: LINER-like

text: emission

text: up

text: only

text: of

text: local

text: galaxies,

text: yet

text: host

text: of

text: optical/UV

text: (Yan

text: et

text: al.

text: Y.

text: Yang

text: et

text: al.

text: may

text: also

text: play

text: a

text: TDE

text: candidates.

text: A

text: softer

text: cut,

text: Hδ

text: includes

text: A

text: role.

text: In

text: future

text: work,

text: we

text: will

text: explore

text: these

text: connections.

text: only

text: of

text: local

text: galaxies,

text: but

text: of

text: optical/UV

text: TDE

text: We

text: thank

text: A.

text: Levan

text: and

text: A.

text: de

text: Ugarte

text: Postigo

text: for

text: pro-

text: host

text: galaxies.

text: The

text: optical/UV

text: TDE

text: rates

text: are

text: thus

text: en-

text: viding

text: the

text: digital

text: version

text: of

text: their

text: Swift

text: host

text: galaxy

text: hanced

text: by

text: in

text: the

text: strongest

text: Hδ

text: galaxies

text: and

text: A

text: spectrum.

text: We

text: thank

text: Yujin

text: Yang

text: for

text: his

text: contributions

text: by

text: in

text: galaxies

text: with

text: HδA

text: Because

text: of

text: this

text: to

text: the

text: stellar

text: population

text: ﬁtting

text: routines.

text: KDF

text: acknowl-

text: preference,

text: the

text: corresponding

text: rates

text: of

text: optical/UV

text: TDEs

text: edges

text: support

text: from

text: NSF

text: grant

text: P.E.O.,

text: are

text: yr

text: per

text: galaxy,

text: and

text: yr

text: and

text: the

text: ARCS

text: Phoenix

text: Chapter

text: and

text: Burton

text: Family.

text: IA

text: per

text: galaxy,

text: respectively.

text: As

text: a

text: result,

text: we

text: predict

text: a

text: lower

text: is

text: grateful

text: for

text: support

text: from

text: the

text: Karp

text: Discovery

text: Award.

text: optical/UV

text: rate

text: in

text: normal

text: star-forming

text: and

text: early-type

text: AIZ

text: and

text: IA

text: thank

text: the

text: Center

text: for

text: Cosmology

text: and

text: Parti-

text: galaxies,

text: of

text: yr

text: per

text: galaxy.

text: Even

text: the

text: one

text: cle

text: Physics

text: at

text: NYU

text: for

text: its

text: support.

text: AIZ

text: acknowledges

text: high

text: energy

text: TDE

text: with

text: a

text: full

text: measured

text: optical

text: spectrum,

text: funding

text: from

text: NSF

text: grant

text: and

text: NASA

text: grants

text: Swift

text: lies

text: in

text: a

text: galaxy

text: with

text: strong

text: Balmer

text: absorp-

text: AIZ

text: thanks

text: tion

text: and

text: weak

text: nebular

text: line

text: emission,

text: which

text: implies

text: a

text: the

text: John

text: Simon

text: Guggenheim

text: Foundation

text: for

text: its

text: support.

text: enhancement

text: in

text: such

text: hosts,

text: and

text: a

text: link

text: between

text: Funding

text: for

text: SDSS-III

text: has

text: been

text: provided

text: by

text: the

text: Alfred

text: the

text: optical/UV

text: and

text: high

text: energy

text: TDE

text: classes.

text: P.

text: Sloan

text: Foundation,

text: the

text: Participating

text: Institutions,

text: the

text: Why

text: do

text: TDEs

text: prefer

text: quiescent,

text: Balmer-strong

text: hosts?

text: National

text: Science

text: Foundation,

text: and

text: the

text: U.S.

text: Department

text: This

text: type

text: of

text: galaxy

text: has

text: several

text: properties

text: that

text: make

text: it

text: of

text: Energy

text: Oﬃce

text: of

text: Science.

text: The

text: SDSS-III

text: web

text: site

text: is

text: REFERENCES

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text: University

text: of

text: Cali-

text: Astronomical

text: Society,

text: fornia,

text: Santa

text: Barbara,

text: CA

text: Negative

text: values

text: indicate

text: emission.

text: Hα

text: EW

text: values

text: are

text: cor-

text: Phinney,

text: The

text: Center

text: of

text: the

text: Galaxy:

text: Proceedings

text: of

text: the

text: Symposium

text: of

text: the

text: International

text: Astronomical

text: Union

text: rected

text: for

text: stellar

text: absorption.

text: Absolute

text: magnitudes

text: (r

text: band,

text: no

text: extinction

text: correction)

text: from

text: Prentice,

text: Ashall,

text: Inserra,

text: et

text: al.

text: The

text: Astronomer’s

text: SDSS

text: (model

text: mag)

text: for

text: optical/UV

text: TDE

text: hosts,

text: and

text: from

text: Levan

text: Telegram

text: et

text: al−.

text: for

text: Swift

text: We

text: assume

text: km

text: Rees,

text: M.

text: J.

text: Nature,

text: Mpc

text: ΩΛ

text: and

text: Ωm

text: ##LTLine##


text: • The velocity modulus distribution proposed by Mao et al. [33]: | | ∝ | | −| | − | | − | | f( v ) v 2 exp[ v /v ](v2 v 2)p Θ(v v ) , (4.3) 0 esc esc with free parameters v and p. 0 • The velocity modulus distribution proposed by Lisanti et al. [52]: | | ∝ | | − | | − − | | f( v ) v 2 exp[(v2 v 2)/(k v2) 1]k Θ(v v ) , (4.4) esc 0 esc with free parameters v and k. 0 The normalisation of all the ﬁtting functions is such that the 1D integral in velocity | | space of f( v ) is equal to 1 (integrated from 0 to v ). esc We perform the ﬁt to the velocity modulus distributions of the simulated haloes with the above ﬁtting functions, by minimisin(cid:88)g the χ2 function: − ≡ N (y f(v , p))2 χ2(p) i i , (4.5) σ2 i i where y is value of the velocity modulus distribution at the velocity v , σ is the corresponding i i i 1σ Poisson error, N is the number of data points in the speed distribution (determined by the number of bins in speed which is 24 – 32 depending on the halo in the hydrodynamic simulations), and p are the free parameters determined by the minimisation procedure. The local Galactic escape speed, v , is a ﬁxed parameter and as explained above it is determined esc for each simulated halo from the total mass enclosed in a sphere of radius 7 kpc, thus representing an upper limit on the escape speed at the Solar circle. As can be seen from tables 2 and 3 in Appendix , the Maxwellian distribution with a free peak (α = 1 in eq. (4.2)) provides a better ﬁt to most haloes in the hydrodynamic simulations compared to their DMO counterparts. However, the other ﬁtting functions which have one extra free parameter in general provide a slightly better ﬁt to haloes in both the hydrodynamic and DMO simulations. In particular, the Mao et al. function [33] provides the best ﬁt for almost all haloes (except a few) for both the hydrodynamic and DMO cases. In ﬁgure 3, we show the distribution of the best-ﬁt parameters and compare the good- ness of ﬁt of the three ﬁtting functions presented in eqs. (4.2), (4.3), and (4.4) for both hydrodynamic (left panels) and DMO simulations (right panels). In each panel the results are shown for the 12 eagle HR and 2 apostle IR MW-like haloes. The function proposed by Mao et al. [33] (centre panels) ﬁts both the hydrodynamic and DMO simulated haloes slightly better than the other ﬁtting functions. There is no clear distinction in the goodness of ﬁt values for DMO and hydrodynamic haloes. However, lower values for both the parameters p and v are preferred by the DMO haloes, while the ﬁt to the hydrodynamic counterparts 0 spans a wider range in the parameter space. The generalized Maxwellian distribution (top panels) provides a better ﬁt to the DMO simulated haloes compared to the hydrodynamic haloes. This is due to the fact that the local DM speed distributions of the DMO haloes are in general more symmetric around the peak than those obtained from the hydrodynamic haloes. Hence a generalized Maxwellian distribution can parametrize the speed distribution of the DMO haloes rather well. However, – 11 – ##LTLine##

label: list

text: The

text: velocity

text: modulus

text: distribution

text: proposed

text: by

text: Mao

text: et

text: al.

text: f(

text: v

text: v

text: exp[

text: v

text: /v

text: v

text: Θ(v

text: v

text: esc

text: esc

text: with

text: free

text: parameters

text: v

text: and

text: p.

text: The

text: velocity

text: modulus

text: distribution

text: proposed

text: by

text: Lisanti

text: et

text: al.

text: f(

text: v

text: v

text: v

text: Θ(v

text: v

text: esc

text: esc

text: with

text: free

text: parameters

text: v

text: and

text: k.

text: The

text: normalisation

text: of

text: all

text: the

text: ﬁtting

text: functions

text: is

text: such

text: that

text: the

text: integral

text: in

text: velocity

text: space

text: of

text: f(

text: v

text: is

text: equal

text: to

text: (integrated

text: from

text: to

text: v

text: esc

text: We

text: perform

text: the

text: ﬁt

text: to

text: the

text: velocity

text: modulus

text: distributions

text: of

text: the

text: simulated

text: haloes

text: with

text: the

text: above

text: ﬁtting

text: functions,

text: by

text: the

text: χ2

text: function:

text: N

text: (y

text: f(v

text: i

text: i

text: σ2

text: i

text: i

text: where

text: y

text: is

text: value

text: of

text: the

text: velocity

text: modulus

text: distribution

text: at

text: the

text: velocity

text: v

text: σ

text: is

text: the

text: corresponding

text: i

text: i

text: i

text: 1σ

text: Poisson

text: error,

text: N

text: is

text: the

text: number

text: of

text: data

text: points

text: in

text: the

text: speed

text: distribution

text: (determined

text: by

text: the

text: number

text: of

text: bins

text: in

text: speed

text: which

text: is

text: depending

text: on

text: the

text: halo

text: in

text: the

text: hydrodynamic

text: simulations),

text: and

text: p

text: are

text: the

text: free

text: parameters

text: determined

text: by

text: the

text: minimisation

text: procedure.

text: The

text: local

text: Galactic

text: escape

text: speed,

text: v

text: is

text: a

text: ﬁxed

text: parameter

text: and

text: as

text: explained

text: above

text: it

text: is

text: determined

text: esc

text: for

text: each

text: simulated

text: halo

text: from

text: the

text: total

text: mass

text: enclosed

text: in

text: a

text: sphere

text: of

text: radius

text: kpc,

text: thus

text: representing

text: an

text: upper

text: limit

text: on

text: the

text: escape

text: speed

text: at

text: the

text: Solar

text: circle.

text: As

text: can

text: be

text: seen

text: from

text: tables

text: and

text: in

text: Appendix

text: the

text: Maxwellian

text: distribution

text: with

text: a

text: free

text: peak

text: (α

text: in

text: eq.

text: provides

text: a

text: better

text: ﬁt

text: to

text: most

text: haloes

text: in

text: the

text: hydrodynamic

text: simulations

text: compared

text: to

text: their

text: DMO

text: counterparts.

text: However,

text: the

text: other

text: ﬁtting

text: functions

text: which

text: have

text: one

text: extra

text: free

text: parameter

text: in

text: general

text: provide

text: a

text: slightly

text: better

text: ﬁt

text: to

text: haloes

text: in

text: both

text: the

text: hydrodynamic

text: and

text: DMO

text: simulations.

text: In

text: particular,

text: the

text: Mao

text: et

text: al.

text: function

text: provides

text: the

text: best

text: ﬁt

text: for

text: almost

text: all

text: haloes

text: (except

text: a

text: few)

text: for

text: both

text: the

text: hydrodynamic

text: and

text: DMO

text: cases.

text: In

text: ﬁgure

text: we

text: show

text: the

text: distribution

text: of

text: the

text: best-ﬁt

text: parameters

text: and

text: compare

text: the

text: good-

text: ness

text: of

text: ﬁt

text: of

text: the

text: three

text: ﬁtting

text: functions

text: presented

text: in

text: eqs.

text: and

text: for

text: both

text: hydrodynamic

text: (left

text: panels)

text: and

text: DMO

text: simulations

text: (right

text: panels).

text: In

text: each

text: panel

text: the

text: results

text: are

text: shown

text: for

text: the

text: eagle

text: HR

text: and

text: apostle

text: IR

text: MW-like

text: haloes.

text: The

text: function

text: proposed

text: by

text: Mao

text: et

text: al.

text: (centre

text: panels)

text: ﬁts

text: both

text: the

text: hydrodynamic

text: and

text: DMO

text: simulated

text: haloes

text: slightly

text: better

text: than

text: the

text: other

text: ﬁtting

text: functions.

text: There

text: is

text: no

text: clear

text: distinction

text: in

text: the

text: goodness

text: of

text: ﬁt

text: values

text: for

text: DMO

text: and

text: hydrodynamic

text: haloes.

text: However,

text: lower

text: values

text: for

text: both

text: the

text: parameters

text: p

text: and

text: v

text: are

text: preferred

text: by

text: the

text: DMO

text: haloes,

text: while

text: the

text: ﬁt

text: to

text: the

text: hydrodynamic

text: counterparts

text: spans

text: a

text: wider

text: range

text: in

text: the

text: parameter

text: space.

text: The

text: generalized

text: Maxwellian

text: distribution

text: (top

text: panels)

text: provides

text: a

text: better

text: ﬁt

text: to

text: the

text: DMO

text: simulated

text: haloes

text: compared

text: to

text: the

text: hydrodynamic

text: haloes.

text: This

text: is

text: due

text: to

text: the

text: fact

text: that

text: the

text: local

text: DM

text: speed

text: distributions

text: of

text: the

text: DMO

text: haloes

text: are

text: in

text: general

text: more

text: symmetric

text: around

text: the

text: peak

text: than

text: those

text: obtained

text: from

text: the

text: hydrodynamic

text: haloes.

text: Hence

text: a

text: generalized

text: Maxwellian

text: distribution

text: can

text: parametrize

text: the

text: speed

text: distribution

text: of

text: the

text: DMO

text: haloes

text: rather

text: well.

text: However,

text: ##LTLine##