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2024-03-07 20:27:54

PAY中文(简体)翻译：剑桥词典

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pay 在英语-中文（简体）词典中的翻译

payverb uk

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/peɪ/ us

Your browser doesn't support HTML5 audio

/peɪ/ paid | paid

pay verb

(BUY)

Add to word list

A1 [ I or T ] to give money to someone for something you want to buy or for services provided

付费;付酬

How much did you pay for the tickets?

你买那些票花了多少钱？

I pay my taxes.

我缴付税款。

[ + two objects ] I'll pay you the fiver back tomorrow.

我明天还你那5英镑。

I paid the driver (in/with) cash.

我付给司机现金。

Would you prefer to pay with/by cash, cheque, or credit card?

你喜欢用现金、支票还是信用卡支付？

[ + obj + to infinitive ] I think we'll need to pay a builder to take this wall down.

我想我们得雇个建筑工人来把这面墙推倒。

Did Linda pay you for looking after her cats while she was away?

琳达出门时让你替她照看她的猫，有没有付钱给你？

I paid (out) a lot of money to get the washing machine fixed and it still doesn't work!

我花了一大笔钱修洗衣机，结果还是不能用！

pay for itself

If something pays for itself, it works so well that it saves the same amount of money that it cost.

使损益相当;够本

The advertising should pay for itself.

这则广告所带来的收益应该能够本。

更多范例减少例句I pay my electricity bill by direct debit.Very few people can afford to pay those prices.How much did you pay for your glasses?We agreed to pay for the car by instalments.I've been saving all year to pay for our holiday.

pay verb

(WORK)

B1 [ I or T ] to give money to someone for work that they have done

付钱（给…），支付（…）报酬

The company pays its interns $4,000 a month.

这家公司给实习生每个月4千美元的报酬。

We pay €200 a day for this kind of work.

这样的工作我们每天付200欧元。

Accountancy may be boring but at least it pays well.

当会计可能很乏味，但至少收入可观。

Most of these women are very poorly paid and work in terrible conditions.

这些妇女大多工资很低，且工作条件恶劣。

更多范例减少例句I'll pay you double if you get the work finished by Friday.This magazine has considerable financial muscle and can afford to pay top journalists.They pay me next to nothing but I really enjoy the work.The law obliges companies to pay decent wages to their employees.You'll be paid on completion of the project.

pay verb

(PROFIT)

[ I ] to give a profit or advantage to someone or something

有收益，有利可图，有好处

It never pays to take risks where human safety is concerned.

凡是涉及人身安全时，冒险决不会有什么好处。

更多范例减少例句Crime really doesn't pay.It always pays to keep on top of your work.It would pay you to be more cautious about future investments.It pays to get some professional advice first.It never pays to rush into things.

pay verb

(GIVE)

C2 [ T ] to give or do something

给予，致以;进行

The commander paid tribute to the courage of his troops.

司令员高度赞扬了部队官兵的勇敢无畏。

It's always nice to be paid a compliment.

被人赞扬总是件好事。

A crowd of mourners gathered to pay their respects to the dead man.

一群哀悼者聚集在一起向死者告别。

pay attention (to something)

B1 to watch, listen to, or think about something carefully

（通过仔细看、听或思考）关注（某事）

You weren't paying attention to what I was saying.

你没有注意听我的话。

pay (someone/something) a call/visit

B2 to visit a person or place, usually for a short time

拜访（某人）

I'll pay you a call when I'm in the area.

我到这一带来时会去拜访你的。

We thought we'd pay a visit to the museum while we were in Lisbon.

我以为在里斯本时我们会去参观那家博物馆。

If you leave your address, I'll pay a call on you when I'm in the area.

如果你留下地址，我到这一带来时会去拜访你的。

更多范例减少例句She complained that her husband never paid her any compliments any more.He never paid attention in class and seemed to be in a permanent daydream.On this occasion we pay homage to him for his achievements.The teacher gently reproved the boys for not paying attention.You'd do well to pay heed to what your grandmother says.

习语

he who pays the piper calls the tune.

pay your dues

pay your way

pay dividends

pay the price

pay the ultimate price

pay through the nose

pay top dollar

put paid to something

you pays your money and you takes your choice/chance短语动词

pay someone/something back

pay someone back

pay down something

pay for something

pay something in

pay off

pay something off

pay someone off

pay (something) out

pay something out

更多短语动词

pay up

paynoun [ U ] uk

Your browser doesn't support HTML5 audio

/peɪ/ us

Your browser doesn't support HTML5 audio

/peɪ/

B1 the money you receive for doing a job

工资，薪金

It's a nice job but the pay is appalling.

这份工作不错，但工资低得不像话。

be in the pay of someone

to work for someone, especially secretly

（尤指秘密地）受雇于（某人）

更多范例减少例句The unions are in dispute with management over pay.Many employees have had to take drastic cuts in pay.Management has/have offered staff a 3% pay increase.When you reckon in all my overtime, my total pay is quite good.If it's a choice between higher pay and job security, I'd prefer to keep my job.

(pay在剑桥英语-中文（简体）词典的翻译 © Cambridge University Press)

pay的例句

pay

They found that, overall, listeners of these languages paid less attention to stress than to intrasentential cues in deciding agent - patient relations.

来自 Cambridge English Corpus

It also remains unclear who paid for them.

来自 Cambridge English Corpus

To this end it is disappointing that the book has paid only scant attention to the psychological processes in the disorder.

来自 Cambridge English Corpus

Willingness to pay for ultrasound in normal pregnancy.

来自 Cambridge English Corpus

It holds that it is never permitted to pay for an object x, if x could have been obtained for free.

来自 Cambridge English Corpus

In any case, now that the big cosmetics companies were paying for research into alternatives, they were no longer available as targets.

来自 Cambridge English Corpus

Have consumers pay the difference for costlier plans, if they think they provide better value.

来自 Cambridge English Corpus

People preferred to be paid in bracelets, rings, glass necklaces, and other items.

来自 Cambridge English Corpus

示例中的观点不代表剑桥词典编辑、剑桥大学出版社和其许可证颁发者的观点。

A1,B1,C2,B1,B2,B1

pay的翻译

中文（繁体）

購買, 付費, 付酬…

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西班牙语

pagar, salario, sueldo…

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葡萄牙语

pagar, salário, salário [masculine]…

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韩语

意大利语

पैसे देणे, मोबदला देणे, वाहणे…

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（勘定や代金）を払う, ～に給料を支払う, 給料…

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ödemek, vermek, yapılan iş karşılığı ücret ödemek/vermek…

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paie [feminine], paye [feminine], salaire [masculine]…

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pagar, paga, sou…

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betalen, lonend zijn, betuigen…

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நீங்கள் வாங்க விரும்பும் ஒன்றுக்கு அல்லது வழங்கப்பட்ட சேவைக்காக ஒருவருக்கு பணம் செலுத்த…

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भुगतान करना, पैसे देना, वेतन या मज़दूरी देना…

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ચૂકવવું, કોઈ સેવા લીધા બદ્દલ કે કંઈ ખરીદવા બદ્દલ કોઈને પૈસા આપવા, કોઈને તેમણે કરેલ કામ માટે નાણાં…

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betale, betale tilbage, betale sig…

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betala, löna sig, höra [upp]…

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membayar, menerima padah, ada faedahnya…

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bezahlen, sich auszahlen, zollen…

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lønn [masculine], betale, gi lønn…

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رقم ادا کرنا…

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платити, сплачувати, поплатитися…

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платить, платить (за работу), быть выгодным…

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చెల్లించు, ఎవరికైనా మీరు కొనదలచుకున్న వస్తువుకు గానీ, పొందదలచుకున్న సేవలకు గానీ డబ్బు ఇచ్చు…

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يَدْفَع, يَدْفَع (الأجر), أجْر…

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অর্থ প্রদান করা, কিছু কেনা বা প্রদত্ত পরিষেবার জন্য কাউকে টাকা দেওয়া, বেতন…

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(za)platit, splatit, platit…

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membayar, mengembalikan uang pinjaman, menerima hukuman…

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จ่าย, ชำระหนี้, ชดใช้…

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trả tiền, trả nợ, trả giá…

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płacić (za), zapłacić (za), opłacić…

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돈을 지불하다, 급료, 보수를 주다…

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pagare, retribuire, retribuzione…

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pay的发音是什么？

在英语词典中查看 pay 的释义

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pawnbroker

pawnshop

pawpaw

Pax

pay

pay (something) out

pay a forfeit phrase

pay channel

pay claim

pay更多的中文(简体)翻译

全部

pay TV

prepay

base pay

pay dirt

pay rise

sick pay

pay claim

查看全部意思»

词组动词

pay up

pay down something

pay off

pay (something) out

pay something in

pay someone/something back

pay someone back

查看全部动词词组意思»

惯用语

pay dividends idiom

pay the price idiom

pay your dues idiom

pay your way idiom

pay top dollar idiom

crime doesn't pay idiom

the devil to pay idiom

查看全部惯用语意思»

“每日一词”

veggie burger

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/ˈvedʒ.i ˌbɜː.ɡər/

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/ˈvedʒ.i ˌbɝː.ɡɚ/

a type of food similar to a hamburger but made without meat, by pressing together small pieces of vegetables, seeds, etc. into a flat, round shape

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Three dimensional path planning using Grey wolf optimizer for UAVs | Applied Intelligence

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Applied Intelligence

Article

Three dimensional path planning using Grey wolf optimizer for UAVs

Published: 07 January 2019

Volume 49, pages 2201–2217, (2019)

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Applied Intelligence

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Ram Kishan Dewangan1, Anupam Shukla1 & W. Wilfred Godfrey1

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AbstractRobot path planning is essential to identify the most feasible path between a start point and goal point by avoiding any collision in the given environment. This task is an NP-hard problem and can be modeled as an optimization problem. Many researchers have proposed various deterministic and meta-heuristic algorithm to obtain better results for the path planning problem. The path planning for 3D multi-Unmanned Aerial Vehicle (UAV) is very difficult as the UAV has to find a viable path between start point and goal point with minimum complexity. This work utilizes a newly proposed methodology named ‘grey wolf optimization (GWO)’ to solve the path planning problem of three Dimensional UAV, whose task is to find the feasible trajectory while avoiding collision among obstacles and other UAVs. The performance of GWO algorithm is compared with deterministic algorithms such as Dijkstra, A* and D*, and meta-heuristic algorithms such as Intelligent BAT Algorithm (IBA), Biogeography Based Optimization (BBO), Particle Swarm Optimization (PSO), Glowworm Swarm Optimization (GSO), Whale Optimization Algorithm (WOA) and Sine Cosine Algorithm (SCA), so as to find the optimal method. The results show that GWO algorithm outperforms the other deterministic and meta-heuristic algorithms in path planning for 3D multi-UAV.

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Download referencesAuthor informationAuthors and AffiliationsABV - IIITM Gwalior, Gwalior, IndiaRam Kishan Dewangan, Anupam Shukla & W. Wilfred GodfreyAuthorsRam Kishan DewanganView author publicationsYou can also search for this author in

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Appl Intell 49, 2201–2217 (2019). https://doi.org/10.1007/s10489-018-1384-yDownload citationPublished: 07 January 2019Issue Date: 15 June 2019DOI: https://doi.org/10.1007/s10489-018-1384-yShare this articleAnyone you share the following link with will be able to read this content:Get shareable linkSorry, a shareable link is not currently available for this article.Copy to clipboard

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Predicting Stock Price Trend Using MACD Optimized by Historical Volatility

icting Stock Price Trend Using MACD Optimized by Historical VolatilityJournalsPublish with usPublishing partnershipsAbout usBlogMathematical Problems in EngineeringJournal overviewFor authorsFor reviewersFor editorsTable of ContentsSpecial IssuesMathematical Problems in Engineering/2018/Article/On this pageAbstractIntroductionConclusionData AvailabilityConflicts of InterestAcknowledgmentsReferencesCopyrightRelated ArticlesResearch Article | Open AccessVolume 2018 | Article ID 9280590 | https://doi.org/10.1155/2018/9280590Show citationPredicting Stock Price Trend Using MACD Optimized by Historical VolatilityJian Wang1and Junseok Kim1Show moreAcademic Editor: Luis MartínezReceived18 Sept 2018Revised13 Nov 2018Accepted21 Nov 2018Published25 Dec 2018AbstractWith the rapid development of the financial market, many professional traders use technical indicators to analyze the stock market. As one of these technical indicators, moving average convergence divergence (MACD) is widely applied by many investors. MACD is a momentum indicator derived from the exponential moving average (EMA) or exponentially weighted moving average (EWMA), which reacts more significantly to recent price changes than the simple moving average (SMA). Traders find the analysis of 12- and 26-day EMA very useful and insightful for determining buy-and-sell points. The purpose of this study is to develop an effective method for predicting the stock price trend. Typically, the traditional EMA is calculated using a fixed weight; however, in this study, we use a changing weight based on the historical volatility. We denote the historical volatility index as HVIX and the new MACD as MACD-HVIX. We test the stability of MACD-HVIX and compare it with that of MACD. Furthermore, the validity of the MACD-HVIX index is tested by using the trend recognition accuracy. We compare the accuracy between a MACD histogram and a MACD-HVIX histogram and find that the accuracy of using MACD-HVIX histogram is 55.55% higher than that of the MACD histogram when we use the buy-and-sell strategy. When we use the buy-and-hold strategy for 5 and 10 days, the prediction accuracy of MACD-HVIX is 33.33% and 12% higher than that of the traditional MACD strategy, respectively. We found that the new indicator is more stable. Therefore, the improved stock price forecasting model can predict the trend of stock prices and help investors augment their return in the stock market.1. IntroductionSecurities investment is a financial activity influenced by many factors such as politics, economy, and psychology of investors. Its process of change is nonlinear and multifractal [1]. The stock market has high-risk characteristics; i.e., if the stock price volatility is excessive or the stability is low, the risk is uncontrollable. Financial asset returns in the short term are persistent; however, those in the long term will be reversed [2].Asness [3] reported that the stock, foreign exchange, and commodity markets have a trend. Hassan [4] noted that complex calculations are not particularly effective for predicting stock markets. Many trend analysis indicators and prediction methods for financial markets have been proposed. Pai [5] used Internet search trends and historical trading data to predict stock markets using the least squares support vector regression model. Lahmiri [6] accurately predicted the minute-ahead stock price by using singular spectrum analysis and support vector regression. Researchers have also used other methods to forecast stock markets. Singh et al. [7] designed a forecasting model consisting of fuzzy theory and particle swarm optimization to predict stock markets using historical data from the State Bank of India. Lahmiri et al. [8] proposed an intelligent ensemble forecasting system for stock market fluctuations based on symmetric and asymmetric wavelet functions. Das et al. [9] proposed a hybridized machine-learning framework using a self-adaptive multipopulation-based Jaya algorithm for forecasting the currency exchange value. Laboissiere et al. [10] developed a model involving correlation analysis and artificial neural networks (NNs) to predict the stock prices of Brazilian electric companies. Lei [11] proposed a wavelet NN prediction method for the stock price trend based on rough set attribute reduction. Lahmiri [12] used variational mode decomposition to forecast the intraday stock price. Lahmiri [13] addressed the problem of technical analysis information fusion and reported that technical information fusion in an NN ensemble architecture improves the prediction accuracy. In [14], the authors argued that time series of stock prices are nonstationary and highly noisy. This led the authors to propose the use of a wavelet denoising-based backpropagation (WDBP) NN for predicting the monthly closing price of the Shanghai composite index. Shynkevich et al. [15] investigated the impact of varying the input window length and the highest prediction performance was observed when the input window length was approximately equal to the forecast horizon. In [16], a prediction model based on the input/output data plan was developed by means of the adaptive neurofuzzy inference system method representing the fuzzy inference system. Zhou et al. [17] proposed a stock market prediction model based on high-frequency data using generative adversarial nets. Wang et al. [18] used a bimodal algorithm with a data-divider to predict the stock index. In [19], the author used multiresolution analysis techniques to predict the interest rate next-day variation. Using K-line patterns’ predictive power analysis, Tao et al. [20] found that their proposed approach can effectively improve prediction accuracy for stock price direction and reduce forecast error.We will introduce the concept of moving average convergence divergence (MACD) and help the readers understand its principle and application in Section 2. Although the MACD oscillator is one of the most popular technical indicators, it is a lagging indicator. In Section 3, we propose an improved model called MACD-HVIX to deal with the lag factor. In Section 4, data for empirical research are described. Finally, in Section 5, we develop a trading strategy using MACD-HVIX and employ actual market data to verify its validity and reliability. We also compare the prediction accuracy and cumulative return of the MACD-HVIX histogram with those of the MACD histogram. The performance of MACD-HVIX exceeds that of MACD. Therefore, the trading strategy based on the MACD-HVIX index is useful for trading. Section 6 presents our conclusion.2. MACD and Its StrategyMACD evolved from the exponential moving average (EMA), which was proposed by Gerald Appel in the 1970s. It is a common indicator in stock analysis. The standard MACD is the 12-day EMA subtracted by the 26-day EMA, which is also called the DIF. The MACD histogram, which was developed by T. Aspray in 1986, measures the signed distance between the MACD and its signal line calculated using the 9-day EMA of the MACD, which is called the DEA. Similar to the MACD, the MACD histogram is an oscillator that fluctuates above and below the zero line. The construction formula is as follows: where , , and . The weight number is a fixed value equal to . The number of the MACD histogram is usually called the MACD bar or OSC. The analysis process of the cross and deviation strategy of DIF and DEA includes the following three steps.(i) Calculate the values of DIF and DEA.(ii) When DIF and DEA are positive, the MACD line cuts the signal line in the uptrend, and the divergence is positive, there is a buy signal confirmation.(iii) When DIF and DEA are negative, the signal line cuts the MACD line in the downtrend, and the divergence is negative, there is a sell signal confirmation.3. MACD-HVIX Weighted by Historical Volatility and Its StrategyThe essence of a good technical indicator is a smooth trading strategy; i.e., the constructed index must be a stationary process. We present an empirical study in Section 5. The validity and sensitivity of MACD have a strong relationship with the choice of parameters. Different investors choose different parameters to achieve the best return for different stocks. In this study, the weight is based on the historical volatility. It is expected that the accuracy and stability of MACD can be improved. The construction formula is as follows:Here, the weight changes over time; HVIX is the change index of the historical volatility of a stock. The HVIX in this paper is the change index of the volatility in the past days. It is similar to the market volatility index VIX used by the Chicago options exchange. It reflects the panic of the market to a certain extent; thus, it is also called the panic index. The above process is expressed by the code shown in Algorithm 1.Require: Set up parameters The stock closing price is , the historical volatility index is , the length of the closingstock price data is , the weight of is , the weight of is ,and the time parameters are and . for j=1 to () do sum=0; for i=j to (j+) do Generate return series Sum the return in the past days sum=sum+ Calculate the mean return in the past days R_mean= sum=0 for i=j to (j+) do Sum the variance in the past days sum=sum+; Calculate the standard deviation in the past days for j=1 to () do sum1=0 for i=j to (j+) do sum1=sum1+; sum2=0; for i=j to (j+) do sum2=sum2+; Calculate for i=2 to () do Calculate and Calculate for i=2 to length() do Calculate Algorithm 1 General algorithm for HVIX and EMA-HVIX.The analysis process of the cross and deviation strategy of DIF-HVIX and DEA-HVIX includes the following three steps.(i)Calculate the values of DIF-HVIX and DEA-HVIX.(ii)When DIF-HVIX and DEA-HVIX are positive, the MACD-HVIX line cuts the signal line of HVIX in the uptrend, and the divergence is positive, there is a buy signal confirmation.(iii)When DIF-HVIX and DEA-HVIX are negative, the signal line of HVIX cuts the MACD-HVIX line in the downtrend, and the divergence is negative, there is a sell signal confirmation.4. Data DescriptionWe first perform an empirical study on the buy-and-sell strategy, which involves buying today and selling tomorrow. We use the historical data for the stock “-zgrs-” from November 2, 2015, to September 21, 2017, from the Shanghai stock market. First, we develop the strategy for the new index and calculate the prediction accuracy and cumulative return of the stock with two different indicators. Then, we compare the accuracy rate and cumulative return. The accuracy here is calculated according to whether the stock price rises on the second day. Furthermore, we test a buy-and-hold strategy for the proposed model. The buy-and-hold strategy is a trading strategy in which the traders hold the stock for a while instead of selling it on the next trading day. We use the historical data for the stock “-dggf-” from July 27, 2009, to November 3, 2017, from the Shanghai stock market to test a 5 d buy-and-hold strategy and use the historical data for the stock “-payh-” from June 22, 1993, to May 10, 2010, from the Shanghai stock market to test a 10 d buy-and-hold strategy. The detailed trading strategy is similar to the buy-and-sell strategy. Here, we use , , and 5. Empirical Results5.1. Empirical Results of Buy-and-Sell StrategyFrom the “-zgrs-” stock data chosen in Section 4, we calculate the HVIX of the past m days and the past n days. A higher stock index means that investors feel anxiety regarding the stock market, and a lower stock index means that the rate of change of the stock price will decrease. Figure 1 shows the HVIX index.Figure 1 Historical volatility of “-zgrs-” with the buy-and-sell strategy.Next, using the calculated volatility index, we calculate the weight of the EMA formula in Section 3 and obtain the values of MACD-HVIX, DEA-HVIX, and OSC.Figure 2 shows the candlestick chart and MACD histogram. In the candlestick chart, the blue line represents the 12-d EMA, and the red line represents the 26-d EMA. Candlesticks are usually composed of a red and green body, as well as an upper wick and a lower wick. The area between the opening and the closing prices is called the body, and price excursions above or below the real body are called the wick. The body indicates the opening and closing prices, and the wick indicates the highest and lowest traded prices of a stock during the time interval represented. For a red body, the opening price is at the bottom, and the closing price is at the top. For a green body, the opening price is at the top, and the closing price is at the bottom.Figure 2 Candlestick chart and MACD histogram for “-zgrs-” with the buy-and-sell strategy. (For interpretation of the references to color in the figure, the reader is referred to the web version of the article.)In the MACD histogram, the solid line represents the DIF, the dotted line represents the DEA, and the histogram represents the MACD bar. According to the strategy described in Section 3, we buy the stock when the DIF and DEA are positive, the DIF cuts the DEA in an uptrend, and the divergence is positive. We sell the stock when the DEA cuts the DIF in a downtrend, and the divergence is negative. As shown in Figure 2, we sell the stock on days 155 and 355 and buy the stock on days 212, 290, 310, 381, and 393. The buy-and-sell signals in the candlestick chart and the MACD histogram are shown in Figure 3.Figure 3 Buy-and-sell signals in the candlestick chart and MACD histogram for the buy-and-sell strategy.Figure 4 shows the candlestick chart and MACD histogram of HVIX. In the candlestick chart, the blue line represents the 12-d EMA-HVIX, and the red line represents the 26-d EMA-HVIX. In the MACD-HVIX histogram, the solid line represents the DIF-HVIX, the dotted line represents the DEA-HVIX, and the histogram represents the MACD-HVIX bar. According to the strategy described in Section 3, we buy the stock when the DIF-HVIX and DEA-HVIX are positive, the DIF-HVIX cuts the DEA-HVIX in an uptrend, and the divergence is positive, and we sell the stock when the DEA-HVIX cuts the DIF-HVIX in a downtrend, and the divergence is negative. As shown in Figure 4, we sell the stock on days 118 and 187 and buy the stock on days 222, 231, 241, 243, 292, 415, and 447. The buy-and-sell signals in the candlestick chart and the MACD histogram are shown in Figure 5.Figure 4 Candlestick chart and MACD-HVIX histogram for “-zgrs-” with the buy-and-sell strategy. (For interpretation of the references to color in this figure, the reader is referred to the web version of the article.)Figure 5 Buy-and-sell signals in the candlestick chart and the MACD-HVIX histogram for the buy-and-sell strategy.To verify the stability of the new indicator, we compare the MACD and MACD-HVIX in Figure 6. The MACD and MACD-HVIX have basically the same trend and the stability of the MACD-HVIX is better than that of the MACD.Figure 6 Comparison of the two indicators for the buy-and-sell strategy.Table 1 shows a comparison of the specific values of the buying-selling points for the MACD index and MACD-HVIX index, as well as a comparison of the predicted and actual trends. Here, we see that the prediction accuracy of MACD-HVIX is 0.667 and that of MACD is 0.4286. By using the proposed indicator, we can improve the prediction accuracy by 55.55% compared with the traditional MACD. The “-Price-” in the table represents the closing price of stock. Next, we compare the cumulative returns for the two indicators. According to the trading points shown in the table, we perform a simulation test. We assume that the initial fund is 1 million. The cumulative returns under the two indexes are 1.1136 million and 1.3365 million, for MACD and MACD-HVIX indices, respectively.Table 1 Comparison of the specific values of the buying-selling points for the buy-and-sell strategy.5.2. Empirical Results of Buy-and-Hold StrategyUsing the “-dggf-” stock data chosen in Section 4, we first investigate the buy or sell points for both the indicators with the buy-and-hold strategy applied for 5 d. Then, we compare the prediction accuracy between the two indicators. The MACD histogram shown in Figure 7 indicates the buy-and-sell points; we should buy the stock at a buy point on days 391, 1,071, 1,181, 1,326, and 1,481, and sell the stock at a sell point on days 791, 881, and 911. The prediction situation is shown in Table 2.Table 2 Comparison of the specific values of the buying-selling points with the buy-and-hold strategy applied for 5 d.Figure 7 Candlestick chart and MACD histogram for “-dggf-” with the buy-and-hold strategy applied for 5 d. (For interpretation of the references to color in this figure, the reader is referred to the web version of the article.)The MACD-HVIX histogram in Figure 8 indicates the buy-and-sell points. We should buy the stock at a buy point on days 371, 1,201, 1,331, and 1,561 and sell the stock at a sell point on days 751 and 771. The prediction situation is shown in Table 2. A comparison between MACD and MACD-HVIX is shown in Figure 9.Figure 8 Candlestick chart and MACD-HVIX histogram for “-dggf-” with the buy-and-hold strategy for 5 d. (For interpretation of the references to color in this figure, the reader is referred to the web version of the article.)Figure 9 Comparison of the two indicators with the buy-and-hold strategy applied for 5 d.Table 2 shows a comparison of the specific values of the buying-selling points for the MACD and MACD-HVIX indices with the buy-and-hold strategy for 5 d, as well as a comparison of the predicted and actual trends. Here, we observe that the prediction accuracy of MACD-HVIX is 0.8333 and that of MACD is 0.6250. By using the proposed indicator, we can improve the prediction accuracy by 33.33% compared with the traditional MACD. The “-Price-” in the table represents the closing price of the stock.Next, using the “-payh-” stock data chosen in Section 4, we investigate the buy or sell points for both the indicators with the buy-and-hold strategy applied for 10d. Then, we compare the prediction accuracy between the two indicators. The MACD histogram in Figure 10 indicates the buy-and-sell points. We should buy the stock at a buy point on day 901, and sell the stock at a sell point on days 621, 1,971, 2,071, 2,291, 2,431, and 2,661. The prediction situation is shown in Table 3.Table 3 Comparison of the specific values of the buying-selling points with the buy-and-hold strategy applied for 10 d.Figure 10 Candlestick chart and MACD histogram for “-payh-” with the buy-and-hold strategy applied for 10 d. (For interpretation of the references to color in this figure, the reader is referred to the web version of the article.)The MACD-HVIX histogram in Figure 11 indicates the buy-and-sell points. We should buy the stock at a buy point on day 901 and sell the stock at a sell point on days 2,071, 2,421, 2,661, and 2,741. The prediction situation is shown in Table 3. A comparison of MACD and MACD-HVIX is presented in Figure 12.Figure 11 Candlestick chart and MACD-HVIX histogram for “-payh-” with the buy-and-hold strategy applied for 10 d. (For interpretation of the references to color in this figure, the reader is referred to the web version of the article.)Figure 12 Comparison of the two indicators with the buy-and-hold strategy applied for 10 d.Table 3 shows the comparison of the specific values of the buying-selling points for the MACD and MACD-HVIX indices with the buy-and-hold strategy applied for 5 d, as well as a comparison of the predicted and actual trends. Here, we observe that the prediction accuracy of MACD-HVIX is 0.8 and that of MACD is 0.7143. By using the proposed indicator, we can improve the prediction accuracy by 12% compared with the traditional MACD. The “-Price-” in the table represents the closing price of the stock.5.3. Computational ComplexityThe computational complexity of the MACD and MACD-HVIX for a stock which has a length of n are and , respectively. In terms of trend prediction processing time, the average time required to process a buy-and-sell strategy, a buy-and-hold strategy for 5 days, and a buy-and-hold strategy for 10 days with the MACD approach (MACD-HVIX) are, respectively, 1.25 (1.51), 1.12 (1.35), and 1.41 seconds (1.58) using Matlab R2017b on an Intel(R) Core(TM) i5-6200 CPU @ 2.30GHz processor.6. ConclusionAs indicated by Tables 1, 2, and 3, we buy-and-sell stock based on improved MACD; then we found all the accuracy is higher than that before the improvement. Therefore, the improved model has higher maneuverability in securities investment and allows investors to capture every buy-and-sell points in the market. For both the buy-and-sell strategy and the buy-and-hold strategy, the empirical results indicated that the proposed model can make more precise predictions than the traditional model. The proposed model was tested with three different stocks and it generated the high prediction accuracy for all the cases. In addition, while a smoothing index is used to construct the MACD index and the impact of the past price declines exponentially, the MACD-HVIX does not have this property. Although the MACD-HVIX index is improved compared with the MACD index, the stationarity of the MACD-HVIX index is difficult prove theoretically. Test shows that it is stable; however, in the ever-changing market, an abnormal situation can cause incalculable losses to investors. In future research, we will investigate other factors for the model by constantly updating the data and the training model to obtain a better prediction effect.Data AvailabilityThe data used to support the findings of this study are available from the corresponding author upon request.Conflicts of InterestThe authors declare that there are no conflicts of interest regarding the publication of this paper.AcknowledgmentsThe first author (Jian Wang) was supported by the China Scholarship Council (201808260026). The corresponding author (J.S. Kim) expresses thanks for the support from the BK21 PLUS program.ReferencesF. Schmitt, D. Schertzer, and S. Lovejoy, “Multifractal Fluctuations in Finance,” International Journal of Theoretical and Applied Finance, vol. 03, no. 03, pp. 361–364, 2000.View at: Publisher Site | Google ScholarT. J. Moskowitz, Y. H. Ooi, and L. H. 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Guo, “Forecasting stock indices with back propagation neural network,” Expert Systems with Applications, vol. 38, no. 11, pp. 14346–14355, 2011.View at: Publisher Site | Google ScholarY. Shynkevich, T. M. McGinnity, S. A. Coleman, A. Belatreche, and Y. Li, “Forecasting price movements using technical indicators: Investigating the impact of varying input window length,” Neurocomputing, vol. 264, pp. 71–88, 2017.View at: Publisher Site | Google ScholarI. Svalina, V. Galzina, R. Lujić, and G. Šimunović, “An adaptive network-based fuzzy inference system (ANFIS) for the forecasting: the case of close price indices,” Expert Systems with Applications, vol. 40, no. 15, pp. 6055–6063, 2013.View at: Publisher Site | Google ScholarX. Zhou, Z. Pan, G. Hu, S. Tang, and C. Zhao, “Stock Market Prediction on High-Frequency Data Using Generative Adversarial Nets,” Mathematical Problems in Engineering, vol. 2018, Article ID 4907423, 11 pages, 2018.View at: Publisher Site | Google ScholarZ. Wang, J. Hu, and Y. Wu, “A Bimodel Algorithm with Data-Divider to Predict Stock Index,” Mathematical Problems in Engineering, vol. 2018, Article ID 3967525, 14 pages, 2018.View at: Publisher Site | Google ScholarS. Lahmiri, “Interest rate next-day variation prediction based on hybrid feedforward neural network, particle swarm optimization, and multiresolution techniques,” Physica A: Statistical Mechanics and its Applications, vol. 444, pp. 388–396, 2016.View at: Publisher Site | Google ScholarL. Tao, Y. Hao, H. Yijie, and S. Chunfeng, “K-Line Patterns' Predictive Power Analysis Using the Methods of Similarity Match and Clustering,” Mathematical Problems in Engineering, vol. 2017, Article ID 3096917, 11 pages, 2017.View at: Publisher Site | Google ScholarCopyrightCopyright © 2018 Jian Wang and Junseok Kim. 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Curr Biol. Author manuscript; available in PMC 2019 Apr 14.Published in final edited form as:Curr Biol. 2018 Sep 10; 28(17): R1043–R1058. doi: 10.1016/j.cub.2018.04.058PMCID: PMC6462409EMSID: EMS82581PMID: 30205054Principles of insect path integrationStanley Heinze,*,1 Ajay Narendra,2 and Allen Cheung3Stanley Heinze

1Lund University, Lund, SwedenFind articles by Stanley HeinzeAjay Narendra

2Macquarie University, Sydney, AustraliaFind articles by Ajay NarendraAllen Cheung

3The University of Queensland, Queensland Brain Institute, Upland Road, St. Lucia, Queensland, AustraliaFind articles by Allen CheungAuthor information Copyright and License information PMC Disclaimer

1Lund University, Lund, Sweden

2Macquarie University, Sydney, Australia

3The University of Queensland, Queensland Brain Institute, Upland Road, St. Lucia, Queensland, Australia*Email: es.ul.loib@eznieh.yelnatsPMC Copyright notice The publisher's final edited version of this article is available free at Curr BiolAssociated DataSupplementary MaterialsSupplemental Information.NIHMS82581-supplement-Supplemental_Information.docx (100K)GUID: 973C078F-4061-4BC1-A06B-B5E8060A14FAAbstractContinuously monitoring its position in space relative to a goal is one of the most essential tasks for an animal that moves through its environment. Species as diverse as rats, bees, and crabs achieve this by integrating all changes of direction with the distance covered during their foraging trips, a process called path integration. They generate an estimate of their current position relative to a starting point, enabling a straight-line return (home vector). While in theory path integration always leads the animal precisely back home, in the real world noise limits the usefulness of this strategy when operating in isolation. Noise results from stochastic processes in the nervous system and from unreliable sensory information, particularly when obtaining heading estimates. Path integration, during which angular self-motion provides the sole input for encoding heading (idiothetic path integration), results in accumulating errors that render this strategy useless over long distances. In contrast, when using an external compass this limitation is avoided (allothetic path integration). Many navigating insects indeed rely on external compass cues for estimating body orientation, whereas they obtain distance information by integration of steps or optic-flow-based speed signals. In the insect brain, a region called the central complex plays a key role for path integration. Not only does it house a ring-attractor network that encodes head directions, neurons responding to optic flow also converge with this circuit. A neural substrate for integrating direction and distance into a memorized home vector has therefore been proposed in the central complex. We discuss how behavioral data and the theoretical framework of path integration can be aligned with these neural data.IntroductionWell before the advent of GPS, humans have had a need to revisit specific sites, for instance, the home cave or a specific tree that is abundant in fruit. While a map would be useful in these instances to avoid getting lost, creating a map requires information of where one is relative to the navigational goal or the starting point of a trip. This is particularly difficult if there are no conspicuous features of the environment, for example in the desert or at sea. Early sailors, when venturing out into the sea, found a solution to this problem. They regularly updated their position relative to the point of departure by measuring traveling speed and direction, a strategy called dead reckoning. In animals this behavior is termed path integration [1]. Many arthropods, including a wide variety of insects (Figure 1A, shield bugs [2,3], field crickets [4], cockroaches [5], Drosophila [6], honeybees [7]and ants [8]) are known to use this strategy to return to their nest, hive or burrow by the shortest possible route after convoluted foraging trips. This is critical for survival, since it reduces predatory pressure and allows the animal to minimize exposure to hostile weather conditions. In principle, during path integration animals continuously keep track of the distance and the directions traveled and then integrate this information to produce a single vector that takes them directly back to the point of origin (Figure 1B). While path integration in a strict sense does not include the homeward journey, i.e. it only refers to the update of the internal position estimate with respect to a point of origin, without homing behavior it is very difficult to obtain an experimental readout of the animal’s internal position estimate. This raises the question whether homing insects are the only species with the capability for path integration. If it is not used for homing, which behaviors would a path integration based position estimate be used for? In more general terms this leads to the question of how useful path integration is as an independent navigation strategy and how it interacts with other behavioral modules. We propose that the answer might be found by examining the neural circuits underlying path integration behavior.Open in a separate windowFigure 1If the circuits underlying path integration are unique to homing insects, this might suggest that path integration can be understood as an isolated behavioral module. In contrast, if path integration results from neural ensembles participating in many functions, all insects with those circuits could potentially have this capability and one can expect that path integration interacts strongly with other behaviors also driven by the shared circuits. Intuitively, one might think that any neural circuit for path integration must comprise an angular integrator that keeps track of the animal’s turns, and an odometer that integrates the current speed of the animal over time, i.e. keeps track of distance using dedicated odometer cells. Together both would encode the animal’s position relative to a point of origin. As laid out in the following sections, this simplistic view of the neural basis of path integration is not in agreement with behavioral data or theoretical considerations, and recent insights into the neural substrates of path integration indeed point to a much more integrated internal representation of the insect’s position.Path integration behaviorRequisites of the path integratorThe strongest evidence for the ability to path integrate comes from displacing animals to an unfamiliar location while they attempt to return home (Figure 1C). If these displaced animals travel in the direction where home would have been and ignore the passive displacement, one can conclude that the animal has a path integrator. When these experiments were done in desert ants, it was revealed that one key input required for the path integrator is a reliable external compass. The main compass cue that these and other insects use for path integration is the pattern of polarized light present in the blue sky [9]. It allows to infer the Sun’s position without the need to see the Sun directly. As with all celestial cues, the polarization pattern is reliable because it is located at an infinite distance from the animal, so that the orientation of the celestial cue changes only when animals carry out rotational motion, but not during translational motion. Specialized sets of ommatidia located in the dorsal rim area of the insect’s compound eyes detect changes in the angle of polarization (E-vector) [10,11]. Many insects in addition possess simple eyes (ocelli) on the dorsal surface of the head that are often also sensitive to polarized light [12–15], but their contribution to path integration behavior is unclear. Similarly, compass information derived from other global cues such as the position of the Sun [16], the Moon [17] spectral and intensity cues [18,19], magnetic cues [20–22], and the Milky Way [23,24], are also used for insect navigation, yet whether they contribute to path integration is still unclear.The second input required for the path integrator is an odometer. Which sensory information insects use for measuring distances depends on their mode of locomotion. Walking insects, such as ants, rely on a pedometer, that is, their distance estimation is based on integrating the number of steps [25,26]. This was elegantly shown by shortening or lengthening the legs of ants that had arrived at a food source after travelling a specific distance. Ants with modified longer legs overshot on their home journey, whereas those with modified shorter legs underestimated the homeward distance.As ants and other insects also walk over undulating terrain, the path integrator takes into account the angular upward and downward slopes that the animal travels. It encodes not the entire distance traveled, but the sum of the horizontal projections it traversed during the outbound journey [27]. This efficiently compensates for differences in terrain along the outbound and inbound trip. While behaviorally well-described, it is still unknown whether step integration is driven by the proprioceptors located in body joints or by monitoring activity of a pattern generator for walking.Insects can also use another sensory cue to gauge distance traveled: optic flow, i.e., the rate at which visual information moves across their retina when they navigate their environment [28,29]. While the pedestrian ants generally appear to ignore lateral optic flow and respond only weakly to changes in the ventral optic flow [30,31], a convincing case for the use of optic flow in ants comes from studying social carrying behavior [32]. During social carrying a transporter ant carries one of its nestmates to a new nest, which is unfamiliar to the carried individual. If the pair is separated and the passenger is released within a channel placed in an unfamiliar location, it travels the entire homeward distance, demonstrating that the ant has obtained distance information despite not having been able to use its step integrator. If the ventral region of the eye was occluded during the passive transport, i.e., preventing optic flow detection, the ant appears to be lost when released in the test channel, suggesting that optic flow information is indeed used by passengers to estimate the distance home. Interestingly, distance information obtained from optic flow during an outward journey could not be transferred to a stride integrator during the homeward journey, suggesting distinct neural correlates for each process.Different from ants, flying insects appear to estimate distances exclusively by integrating optic flow [28,29]. Optic flow based distance estimation has been largely studied in honeybees. Individual honeybees that return to the hive after a successful foraging trip convey information about the location of the food source to their nestmates through the waggle dance [33,34]. During the dance the bees move along a straight line while they waggle their abdomen, loop to the left to return to the starting point, repeat the waggling along the straight line, before making a loop to the right to repeat the entire sequence of movements. The duration of the waggle phase is proportional to the distance between the hive and the food source, and the angle between the axis of the waggle and the vertical direction provides the direction to the food source with respect to the Sun. Importantly, observing the dance allows to gain direct insight into the knowledge the bee has of its foraging trip and thus view the result of its path integration computations.A functional celestial compass is also required for distance estimation. In fact, ants walking in channels only integrate distances when they have had access to polarized skylight and ignore the distance they walk in conditions where the sky is occluded [35]. This indicates that insects do not record distance and direction information separately, but store both in an integrated manner, refuting the notion of explicit odometer cells. Honeybees do something slightly different, giving rise to the possibility that that they store multiple vectors; a personal vector that the bee individually relies on and a communal vector to convey the distance information to nestmates [36]. This was discovered by training honeybees to a feeder at a known distance and depriving the bees of celestial cues along some segments of the outbound trip. The personal vector ignored the distance travelled without celestial compass, whereas the communal vector included these segments.Path integration interacts with other navigational strategiesIn visually poor landscapes, such as saltpans or mudflats, the visual world for an animal remains fairly similar over 100s of meters. Hence in such landscapes, irrespective of how far animals are displaced from their typical foraging area, they rely primarily on their path integrator and travel most of the distance indicated by their home vector [37,38]. In contrast, ants that live in slightly cluttered landscapes learn visual landmark information and establish individualistic routes that lead them to their food sources and back to the nest [39]. If they are displaced close to the familiar foraging corridor, they initially orient using visual landmarks (Figure 2A, 0 m) or chose an intermediate direction between the real home and the direction indicated by the path integrator (Figure 2A) [40,41], and subsequently find home by relying on visual landmarks along the route [40]. In contrast, when ants are displaced to previously unvisited locations but within their foraging range, they ignore compass information from the path integrator and exclusively use visual landmark information to directly head home [42,43] (Figure 2B). Finally, ants occupying landmark-rich habitats follow their home vector only for a short distance when displaced to unfamiliar locations [44,45] (Figure 1C). Overall, the degree to which ants rely on their path integrator when placed in an unfamiliar location thus inversely correlates with the availability of landmarks in their habitat [45].Open in a separate windowFigure 2Optimally, path integration and other spatial estimates should be combined by weighting each estimate’s level of certainty, which for path integration is inversely proportional to path length, i.e. the longer the path the less reliable the position estimate. Intuitively, when in conflict, a less certain estimate should be weighted less. However, [41] showed that positional certainty cannot explain behavior resulting from conflicts between path integration and visual cues (Figure 2A). Instead, the data matched better with a strategy during which the ants ignore the distance to the goal and only account for the angular component of the uncertainty distribution, i.e. weighting directional certainty. These results could be explained by averaging path integration vectors of various length in memory, suggesting that the working memory resulting from path integration is transferred to long term memory, which then is optimized over the course of multiple homing runs.The extent to which insects use polarized skylight, a key input to the path integrator, is also context dependent. Foraging ants that leave the nest weight both celestial and terrestrial compass cues equally and hence respond to a change in the pattern of polarized skylight by about half of the manipulation [46]. When ants are returning home, they weight their polarization compass more strongly the further they are away from the nest [47].Path integration also serves as a reference system for learning visual landmarks. Naïve eusocial insects emerging from their nest for the first time learn the nest-associated visual landmarks through carefully choreographed learning walks or flights [48–50]. Information learnt during this phase is used to pinpoint home during their return journeys. After leaving the nest, flying insects turn back and move in a series of arcs centered around the nest, gradually gaining distance and elevation, while ensuring the nest is always seen in the left or right visual field. Walking insects, such as ants, engage in learning walks where they carry out multiple loops that begin and end at the nest [20,49,51]. During these loops, ants systematically turn back and look towards the nest. The only strategy that allows them to repeatedly acquire nest-oriented views is by relying on the path integrator [52]. This is because the ants cannot see the nest entrance during their learning walks, which is typically an inconspicuous hole in the ground. Thus, the path integrator is the main framework on which visual representation of the nest environment is generated.In conclusion, the insect path integrator does not work in isolation but interacts with other navigational information present in the environment, resulting in robust as well as flexible orientation behavior. Moreover, path integration appears to be the fallback strategy used if no other options are available. To illuminate why this is the case, the theoretical underpinnings of path integration have to be examined.Theory of path integrationMathematically, path integration is perfect. Even though it can be described using different references frames and coordinate systems, these implementations all have the same outcome: the animal exactly pinpoints its origin. In the real world, and as indicated by the behavioral data above, this is clearly not the case. This discrepancy results from the fact that errors occur, which causes uncertainty in the animal’s estimate of its position. Where do these errors originate? They result from noise. Noise is a ubiquitous part of the world, manifest at all physical scales [53–55]. It ranges from inherent quantum uncertainty, randomness of molecular ensembles, the stochasticity of neural spikes, to environmental obstruction or distortion of sensory cues. Noise is considered here as any random perturbation on a path integration input, update or output, which cannot be predicted or detected by the insect. Consequently, noise reduces path integration accuracy. To survive, insects have evolved corrective and compensatory strategies for mitigating the impact of path integration errors.Path integration errors and reference framesActive displacement is the canonical input for all path integration computations. Unavoidably, noise impacts on displacement estimates, resulting in angular and linear input errors that affect the positional uncertainty of the animal. While linear errors from distance estimates always accumulate with increasing path length, angular errors must be subdivided based on their impact on path integration accuracy: First, direction errors occur when using a compass or stable landmark, and second, rotation errors occur when integrating angular velocity to estimate direction. Of those, only rotational errors accumulate with increasing foraging time.Path integration updates and memories are also susceptible to error. Both information transfer and the changing neural code during path integration computations are impacted by noise, causing some discrepancy between path integration updates and true displacement. The maintenance of path integration memory during a journey, and across multiple path integration tasks are also susceptible. As it is assumed that only those movement and steering errors (path integration output) that are not detected or updated will affect path integration, these output errors are considered together with sensory and update errors.As mentioned above, only rotational angular errors accumulate. Thus, the impact of input noise depends on whether rotations or directions are used to estimate displacement (Figure 3A; [56–58]. Compass or compass-like information is used to estimate displacement in allothetic path integration. By definition, compass cues provide a stable direction input, i.e., displacement estimates will not accumulate directional error over time. Using idiothetic (e.g. self-motion) cues alone, displacement must use rotational estimates cumulatively, resulting in idiothetic path integration.Open in a separate windowFigure 3The properties of allothetic path integration and idiothetic path integration differ substantially (for mathematical details see [56–58]). Given identical stepwise errors, idiothetic path integration accumulates positional uncertainty much more rapidly, and is much less accurate than allothetic path integration (Figure 3B,C). This is because the accumulation of angular errors in idiothetic path integration is without bound, thereby progressively increasing the input error. Using allothetic path integration, angular errors are bounded by the compass, and path integration positional uncertainty increases linearly with journey length.Additionally, idiothetic path integration systematically underestimates the net distance travelled, capping the maximal distance that can be represented. The combination of large positional uncertainty and systematic distance underestimation makes idiothetic path integration untenable for all but the shortest of paths.Spatial coordinate systems for path integrationIn principle, path integration information needs to be stored temporarily and updated regularly, a process that results in some error with each update. This process can be implemented in various coordinate systems differing in two characteristics: (1) egocentric versus allocentric reference frame, and (2) static versus dynamic vectorial basis [59]. The reference frame defines the origin and reference vectors of the coordinate system. Egocentric means the self is at the origin and all other locations are defined with respect to body axes. Allocentric means an external location (e.g., nest) is at the origin, and all other locations, including the current location, are represented with respect to that external reference (Figure 3D,E; [59,60]). Coordinates may either be projected along pre-defined directions (static vectorial basis) or be free to rotate (dynamic vectorial basis). There is no maximum number of reference directions, but as the animal’s heading in the real world will only rarely align with any pre-defined direction, an accurate heading representation always requires an additional projection computation onto at least two static reference vectors at each step (Figure 3D,E).Irrespective of the type of directional input, the type of coordinate system that is internally used for storing and updating the animal’s position determines whether path integration update requires a rotation estimate, therefore affecting path integration accuracy (Figure 3F, [59,61]). Any dynamic vectorial basis or egocentric reference frame requires rotation estimates and thus cause cumulative angular errors similar to idiothetic path integration (subtle differences described in [59]), resulting in large, nonlinear positional uncertainty. An allocentric static vectorial representation is therefore expected to be most resistant to update noise, since it is the only coordinate system class not requiring rotation estimates. Moreover, if a compass is used, positional uncertainty from allocentric static vectorial representations accumulates linearly like allothetic path integration. Therefore, the variance of positional uncertainty should increase linearly with path length, which is indeed consistent with insect data [62].When distance estimates during path integration build up over time in ‘allocentric static vectorial representations’, this is functionally equivalent to a ring of static reference vectors pointing into all azimuth directions, upon which the animal’s movements continuously accumulate (Figure 3). In other implementations of allocentric static vectorial representations, even distance components are static, so that each vector represents an external location or ‘place’. Path integration updates would entail shifting the neural representation along a 2D array of places. If the neural representation is encoded by spike rate, neurons may behave like ‘place cells’ of the mammalian hippocampus [59].The presence of anatomical substrates for a ring-like path integration system in the insect central complex (see below), the absence of definitive ‘place cell’ evidence, and the apparent match between theoretical and behavioral data suggests that insect path integration is best described by a ring-like ‘allocentric static vectorial representation’ model [59,63,64].Systematic search and path integrationAs all path integrators accumulate errors they lead animals only to the vicinity of the goal, rather than to the goal itself. To overcome this uncertainty, insect path integration must always operate together with a search strategy to ensure the target is found [37,65–67]. An insect’s uncertainty about the precise location of its goal increases with foraging distance. During path integration this means that the longer the outbound path becomes, the greater the uncertainty of the animal’s position estimate is. Insect search therefore has likely evolved to complement path integration to find a target as efficiently as possible and, given this intimate link, both behaviors potentially result from shared neural substrates [64].Insects begin searching behavior when they have not found their goal after traveling their entire vector towards home or towards a food source. Food-specific searches have been mostly described in honeybees [67] and are needed as the waggle dance that recruits them to the food source will not direct them with pinpoint accuracy [68]. In contrast, we know most about home-specific searches in ants and isopods. In all cases, and irrespective whether the animals are walking or flying, individuals search in ever increasing loops, returning close to the start of their search at the end of each loop [65,69] (Figure 2C). As longer traveling distances decrease the accuracy of the path integrator, the search is adjusted accordingly. Ants adapt by searching in wider loops when the accuracy of the path integrator is low, but focus their search to a narrow area when the accuracy of the path integrator is high [37]. Irrespective of the state of the path integrator, some ant species also search when they detect that their familiar visual landmarks are absent during homing. In these cases, they generate a progressive search where loops get larger but drift towards the nest [70].Like path integration, search is imperfect. For each increment of search effort, there is a chance that an insect may miss the target. The total probability of successfully locating the target additionally takes into account knowledge of the target’s location (target uncertainty function), which, during path integration, is equal to the positional uncertainty when search is initiated. If this target uncertainty function follows a circular Gaussian distribution, success is optimized when the search distribution follows an inverted parabola [71–73] (Box 1(I); for mathematical details see Supplemental information). The shape of this curve dictates that the optimal search radius should increase slowly over time, proportional to the fourth root of the total allocated search time. Crucially, there is no need to plan the total search effort before beginning to search, as later search effort can simply add to earlier search effort without affecting optimality. Optimal search can be implemented by adding widening discs of search area, similar to the above described ant search strategy [65,66]. Note that until an insect finds its target or other familiar cue, its uncertainty is expected to continue increasing.How do these considerations apply to path integration? For path integration using a compass-driven, allocentric static vectorial representation, optimal search theory makes a number of testable predictions. For example, [74] compared two ant populations of which one covered double the distances prior to search initiation (14 vs 28 m). Optimal search theory predicts a very small difference in the search radii between the two groups, consistent with observations (Supplemental Information), but contrary to the authors’ original interpretation that small differences meant ants did not adapt their search widths. Similarly small differences in search radius were reported in another species of ant [75]. In contrast, the uncertainty distribution itself should differ substantially under the same conditions, again in close agreement with data (Supplemental Information). Furthermore, the maximum search radius should be proportional to t0.5, almost identical to Cataglyphis search data from [65] (Box 1(II), Supplemental Information).In contrast to ants in open fields, bees are often tested in one-dimensional tunnels. Does optimal search still hold in these situations? If an insect simply takes a slice through the optimal 2D search distribution, initial search width should be proportional to d04 (Supplemental Information). However, 1D search width increases linearly with foraging distance (d0) in honeybees [76]. This result can still be explained by optimal search theory, if bees use a 2D path integration-search heuristic, but are forced to execute search in 1D. Estimated using the bees’ U-turn positions, 1D search radius should then be proportional to training distance, as reported in honeybees (Supplemental information for details; Box 1(III)).Despite the near perfect matches between predictions and observations described so far, there remains a potentially serious gap in our understanding of path integration-related search. Two parameters of optimal search are properties of the insect - the effective search width W and the uncertainty constant kσ, which is the total rate of increase of positional variance during path integration. Using data from Cataglyphis [65] yields a search width of W = 3.8 m (Supplemental Information), i.e., the ant must recognize its nest perfectly at 1.9 m away either side – which is in stark contrast to the observed 2-3 cm reported elsewhere [62]. One possibility to resolve this discrepancy is that insects could actually search for a relatively large familiar region from which they can find home, rather than home itself. Supporting this view, ants which inhabit landmark-rich environments seem to initiate search early to find a familiar route, rather than the nest per se [45].Importantly, the uncertainty constant kσ allows insight into the magnitude of path integration-related positional uncertainty. Using the above data, and assuming a circular Gaussian uncertainty distribution, path integration positional uncertainty in ants increases at a seemingly inexplicable rate, i.e., >310 cm2 (95% CI, ~ half an A4 page) per 1 cm walked (Box 1(IV)). Comparable estimates are found using the median radius of search initiation points [37]. It is unclear what could cause such large path integration errors, or whether current neurocomputational models can cope with them [64,77]. As even unrealistically large sensory errors cannot come close to such uncertainty (Box 1(IV)), path integration error may be largely due to internal path integration update noise rather than input/output noise. A possible explanation for such large errors is that insects which navigate long distances could have evolved a smaller gain per unit distance travelled in their path integration accumulator, i.e. they sacrifice accuracy for range given finite neural resources. Insects with smaller foraging ranges would be predicted to have smaller errors per unit distance (larger gain). In contrast, a larger foraging range would require learning an extended familiar nest region, with correspondingly large search width.A duo accumulator model for path integrationAs outlined above, honeybee dance provides a vectorial readout of recent navigation journeys and appears to depend on the same information used for path integration [78], thus offering a window to the underlying mechanisms. Additional to informing us about the nature of the bee’s odometer, dance observations suggest that a bee must remember a net path integration vector between nest and reward, which is reset to zero at the nest. Yet, at the same time, the bee cannot forget the reward vector which it uses to produce the dance. This suggests that separate accumulators for outbound and inbound journeys may be required (Figure 4A). A consequence of such a duo-accumulator is that inbound and outbound path integration behaviors become dissociable. If, for instance, the outbound and inbound paths differ, dance may only show the outbound (i.e., reward-bound) information, and it is left to the recruit to find home after foraging, consistent with experimental data [36,79] and difficult to explain by other models. Another consequence is that dance distance and travel distance may differ, as was shown when honeybees are guided along a partially occluded path to reward during training [36,79] (Figure 4C). A similar reduction in path integration distance along unoccluded channels was also reported in ants [35].Open in a separate windowFigure 4Path integration may be used directly to control dance behavior or indirectly via stored memories. Theoretically, a honeybee could more accurately estimate a reward location by combining noisy estimates from multiple visits, particularly if path integration uncertainty is large. That requires a longer-term memory which averages multiple path integration estimates, potentially explaining the multimodal honeybee dances reported by [80] (Figure 4D).Overall, dance observations suggest that the path integrator of the insect brain likely interacts with long-term memory to allow vector averaging, in line with the above mentioned behavioral observations in ants [41]. Additionally, the dissociable inbound and outbound vectors discovered via dance observations are easily explainable if the bee’s path integration circuit comprises two separate accumulators. To gauge how realistic this model is, we have to ask what is known about how insect brains control path integration?The neural basis of path integrationA brain region called the central complex (CX) has emerged as likely site in the insect brain to serve as the neural substrate for path integration. This highly conserved, midline-spanning group of neuropils in the center of the brain consists of the upper and lower divisions of the central body (CBU, CBL; ellipsoid and fan-shaped body in flies), the protocerebral bridge (PB) and the paired noduli (Figure 5A) [81–83]. It is characterized by a regular, repetitive neuroarchitechture of 16-18 vertical columns and several horizontal layers [84,85], both of which are formed by repeating sets of columnar neurons, combined with tangential input neurons innervating entire layers. Most cell types of the CX are conserved at least anatomically across all insect species investigated to date, suggesting that the principal circuit outline has evolved more than 350 million years ago [84–87].Open in a separate windowFigure 5Sensing directionsEstimating the insect’s current heading requires information either from global compass cues, or, if these are unavailable, from rotational motion cues. The most prominent pathway for sending compass information to the CX comprises a set of neurons indirectly linking the optic lobe to the CBL (Figure 5A) [10]. In many insects, these cells carry global compass information derived from the skylight polarization pattern and other celestial cues. This compass-pathway has been most extensively studied in locusts (reviewed in [10,88]), but has also been identified in the Monarch butterfly [86,89], dung beetles [87,90], ants [91], and bees [64]. When presenting a rotating polarizer to the animal from above, polarization sensitive neurons (POL-neurons) of this pathway respond with sinusoidal changes in their action potential frequency [88]. Each cell is tuned to a particular plane of polarization (E-vector), at which it exhibits maximal activity. As a given E-vector correlates with the solar azimuth, at least when viewed in the zenith, each POL cell encodes the direction of the animal relative to solar azimuth. Across the population of POL neurons all tuning angles exist, thus representing all possible compass directions. After the information is transmitted to the CX, an ordered array of these E-vector tunings emerges in the columns of the PB [92]. Here, a POL neuron’s tuning corresponds to its anatomical location along the width of the PB, systematically shifting from left to right until all possible E-vectors are covered. Activity across the PB therefore is predicted to form a localized bump of maximal spike-rate, depending on the direction the animal faces with respect to the Sun.In Drosophila, this prediction based on global compass cues has been confirmed directly by imaging entire populations of neurons homologous to the locust POL-neurons [93–96]. These experiments were carried out in closed-loop conditions with flies walking on an air-suspended ball or during tethered flight. The flies used were genetically engineered to express a calcium indicator in a set of neurons called the E-PG cells (homologous to CL1 cells) (Figure 5B). These neurons connect single CBL-columns (ellipsoid body in flies) to single PB-columns in a highly stereotypical projection pattern (Figure 5C). When imaged during behavior, the population of E-PG cells generated a single bump of high activity across the width of the CBL [93]. According to the projection pattern of these cells, this bump is also observable in each hemisphere of the PB. When the fly turned clockwise, the bump moved anti-clockwise to a new position and vice versa when the fly moved anti-clockwise. The activity bump thus tracked the angular movements of the fly, generating a distribution of activity across the CX that encodes head direction. The principal input to these cells are the Drosophila ring neurons (the homologous counterparts of TL neurons from other insects) [97]. These carry information about the visual environment in flies [97], while they are the final element of the POL pathway in other insects [10]. This suggests that the TL neuron pathway has been rededicated to relay the directional information to the CX that is most relevant in each particular species. Whatever the source of information, the result of the computations in the CX always appears to be a bump of activity encoding the current heading of the insect.The Drosophila head direction cells are not only active in the presence of visual input, but function also in darkness [93]. This indicates that proprioceptive input is used by these cells and suggests that rotation estimates converge with visual input. If both pathways are conserved in all insects, allothetic compass signals should converge with idiothetic rotation estimates in the head-direction circuit. As demonstrated by path integration theory reviewed above, a heading estimate based only on idiothetic information should accumulate significant error [56,58,59]. Indeed, the head-direction activity bump drifted out of sync with the real heading of the fly in darkness. By measuring the activity of individual neurons with extracellular recordings rather than imaging an entire neuronal population, head-direction cells resembling those in the fly were found also in cockroaches [98].Considering the data from compass encoding in locusts, bees, butterflies and beetles together with head-direction encoding in flies and cockroaches, it appears likely that the same circuitry is implemented in the CX of all insects. While shifting emphasis on different inputs according to ecological need, the head-direction circuit can be expected to use a combination of global compass cues, reliable landmark information and rotation estimates (rotational optic flow and proprioceptive input) to encode the current heading of the insect in the context of any navigation strategy, including for path integration. At least some neural substrates required for path integration, i.e. the mentioned head-direction code, are therefore most likely shared between all behaviors that require an oriented response within either an internal or external reference frame.Recently, the nature of the circuit that generates and maintains the head-direction activity bump in Drosophila has been illuminated in detail. As predicted by theoretical considerations more than two decades ago, a circuit motif called a ‘ring attractor’ can account for the observed activity in the CX [99]. This hypothetical circuit comprises, in principle, a circular array of neurons that are linked by mutual inhibitory connections, complemented by local excitatory connection between neighboring members of the ring. Any activity injected into that circuit will consolidate itself into a single bump and will be maintained even in the absence of new input. If each neuron of the ring resembles a different azimuth angle (i.e., a fixed reference vector), then the combined activity in all neurons in the ring encodes the current heading of the animal and, when combined with an accumulator, can be used directly as input for a path integrator, creating a static vectorial representation of a position estimate (the number of cells being equivalent to the number of reference vectors). The eight columnar Drosophila E-PG neurons per hemisphere form the core of such a ring attractor circuit [96]. Which neurons mediate the postulated local excitation between neighboring cells and the ring-wide inhibition needed for the attractor circuit?First, E-PG neurons are connected to another type of columnar neuron of the CX, the P-EN neurons. While the E-PG cells receive input in the CBL and project to the PB, the P-EN cells possess the opposite polarity (Figure 5B,D) [94,95]. Using these morphological features, both types of cells form reciprocal excitatory connections. Crucially, the projection patterns of both cell types are not identical, but are offset by one column to the right (in one hemisphere) or to the left (in the other hemisphere). This offset means that any individual E-PG neuron indirectly excites its neighbor via a P-EN intermediary. To prevent this excitation to spread across the entire circuit, lateral inhibition is required [94–96,100]. Whereas direct evidence for the cell-type providing this inhibition is lacking, local interneurons of the PB (closely resembling TB1 neurons from other insects [92,101]) appear to be the most promising candidates [100]. These cells have complex branching patterns [84,102] covering the entire PB and could connect all E-PG cells to one another in ways compatible with ring attractor circuits. In the context of a path integration circuit, TB1 neurons in bees, differing slightly in their morphology from their fly counterparts, were used to construct a ring attractor circuit model without the need for recurrent excitation [64], indicating that there might be several possible implementations of ring attractors in the insect CX with subtle differences reflecting different needs of each species.How does the movement of the animal update the location of the bump within the CX? Two potential input pathways to the fly ring attractor circuit exist: The most prominent one (mediated by TL-neurons/ring-neurons) directly feeds onto the E-PG neurons in the CBL and carries visual feature information [97,103], and, in other insects, information about compass cues [10]. Unfortunately it has not been resolved how individual neurons of that pathway connect to the elements of the head-direction network. Therefore, it is still elusive how the disordered compass information from several parallel pathways is used to update the bump position in the CBL/PB. The second input pathway is less well described, but, in contrast, its effects on the circuit have been clearly revealed [94,95,104]. It carries information about body rotations (rotational optic flow and likely proprioceptive information) and feeds onto the P-EN neurons. Via an intricate mechanism rooted in subtle anatomical features of the CX (Figure 5D), the head-direction bump in the E-PG cells will always be pulled either clockwise or counterclockwise along the ring by the P-EN cells, as long as the fly performs counterclockwise or clockwise angular movements, respectively. While the specific role of this circuit for path integration, and its link to compass input remains to be resolved, the head-direction signal it generates in the PB-columns is suited to be one principal input to any possible downstream path integrator.Sensing distancesDirection signals are a crucial input for path integration, but have to be combined with information about the distances covered to enable returning to a point of origin. As mentioned above, bees use translational optic flow to estimate distances, while ants do the same by integrating their steps. Recently, neurons responding to translational optic flow in a speed dependent way have been shown to terminate in the bee CX [64]. Specifically, they relay information from the ventromedial and lateral protocerebrum to the noduli, small regions of previously unknown function present in all flying insects. Two of these cells (TN1 and TN2 cells; Figure 5B) exist on either side of the midline, yielding a set of four neurons per brain. Each cell is tuned to translational optic flow originating from a particular azimuth located ca. 45˚ on either side of the midline, both in front and behind the bee. These neurons thus tile the animal’s movement space into four cardinal directions and together can robustly encode all translational movements of the bee [64]. This yields a reliable result even during periods of holonomic motion, i.e. when the body axis of the bee is not aligned with its movement direction and the bee performs sideways movements while hovering in front of flowers. Whether these cells are also involved in mediating information from legs for stride integration in ants remains to be explored, particularly in the light of behavioral results in ants suggesting that distance information based on optic flow cannot be transferred to measuring distance by step integration during walking [32].Path integration memoryUnlike using visual landmark information, path integration as such does not involve long term memory. The information required to return home is solely based on the integrated version of their preceding foraging journey, stored in working memory. The memories of the path integrator, i.e., both the abilities to estimate heading direction and distances, lasts only for 24h and declines rapidly thereafter [105,106]. In stark contrast, memories of nest-associated visual landmarks last for the entire lifetime, at least in desert ants [106,107]. To establish a working memory of a foraging trip, translational speed has to be integrated over time to obtain a measure for the flown distance. Whereas the mechanism for this integration has not been identified in vivo, a computational model of the bee CX has combined optic flow sensitive TN neurons from the noduli with compass neurons in the PB to generate a distance memory [64]. This model results in an accumulation of neural activity in each of the eight directions represented by the head-direction cell population. As each accumulator unit is a separate, direction-locked odometer, no dedicated odometer cell representing the total distance traveled is required. Rather, the combined activity of all eight integrators represents the distance to the point of origin in conjunction with its direction at any given moment in time during foraging. The neural substrate for this proposed path integration memory are columnar neurons of the CBU (CPU4 neurons; P-FN in Drosophila;

Figure 5B), which connect the PB and the noduli to project to the CBU in a regular pattern [84,85,108]. Per column there are a minimum of 18 cells in the examined bee species [64], allowing the existence of recurrent microcircuits within each columnar circuit (i.e., the accumulator for one compass direction). While remaining hypothetical, this implementation of path integration uses only biologically plausible connections between existing CX neurons and predicts specific activity patterns across the population of CPU4 cells after defined foraging trips. These patterns represent a distributed neural code for the home vector, which can be experimentally verified in the future.SteeringThe distributed home vector representation of the bee path integration model essentially encodes the goal of the insect’s behavior during the homing trip. Likely downstream neurons have been suggested to compare this information to the current compass heading and initialize compensatory steering movements to align intended and actual headings [64]. The CX has been implicated in motor planning for nearly three decades, based on deficits in walking behavior, turning ability, and leg coordination in structural CX mutants of Drosophila [109,110]. More recently, extracellular recordings from freely moving cockroaches have revealed that activity of CX neurons predict the imminent movements of the animal [111]. These cells encode specific combinations of rotational and translational movements and thereby encode future trajectories of the cockroach. All recordings combined constitute a population of cells that mapped the possible movement space of the animal. Current injection through the same electrodes that were used for recording CX activity during free movement caused the insect to turn in a predicted direction, presumably by activating neurons near the electrode terminals, thereby demonstration a causal relation between the neural activity and the motor action [111]. Unfortunately, the morphological identity of these cells is not known. However, virtual current injections into the modeled path integration circuit of the bee showed that similarly predictable turning behavior can be initiated when artificially driving the output cells of the circuit [64]. In the model, these neurons (CPU1 columnar cells; Figure 5B) compare the neural activity of the PB compass cells with the output of the memory units. If the CPU1 population in one hemisphere exceeds the activity level of that in the other hemisphere, steering is initiated to turn the insect towards the target. These cells possibly correspond to the movement predicting neurons in cockroaches.CPU1 cells are highly conserved across many insects and are considered the major output cells of the CX [64,85,86,102]. They converge in a brain region called the lateral accessory lobe, which is known to contain premotor neurons in silk moths (Bombyx mori) [112,113]. These moth neurons show so-called flip flop activity that is characterized by sustained periods of alternating high and low firing rates [114]. The transition between the two states is initiated by sensory information, most commonly short pheromone pulses [113], but can also be triggered by light flashes [115]. As neck motor neurons show flip-flop activity as well and head-turns often precede body-turns, the flip flop circuit is considered a control network for turning movements during moth odor plume tracking [116]. Mirroring the neural activity, this tracking behavior features typical zig-zagging movements, during which the moth passes through the odor plume multiple times and reverses flight direction after each exposure to the plume. Consistent with the idea that this circuit fundamentally underlies steering in insects [112], the path integration circuit model of bees yielded similar steering behavior when implemented on a robot [64].Overall, the insect CX appears to contain all elements for a path integration circuit, ranging from the representation of the animal’s current heading, input from speed sensing cells, a possible neural substrate for a distributed home vector representation, and output neurons that link to known steering centers of the insect brain.Towards unified neural control of navigationMany questions about how insect brains control path integration remain unanswered. Arguably the most fundamental question is how path integration is embedded into the general behavioral repertoire of insects, especially as it has become clear from behavioral data that path integration is essentially never used in isolation. How does the brain switch between explorative navigation, during which the path integrator accumulates information, and homing, during which this information is used to drive the turning movements of the animal? Locating the motivational switches responsible for these transitions between strategies might also resolve how other navigation strategies, e.g. route following and landmark navigation, are integrated with path integration. One region of the CX appears to be anatomically suited to serve as integration site for different strategies: The CBU (fan-shaped body) receives input from many different brain areas via a diverse set of input neurons [86,117,118]. These regions include areas that receive projections from the output cells of the mushroom body and thus could carry information controlling navigation strategies that require long-term memory (route following, landmark navigation, retrieval of averaged path integration vectors). The axonal terminals of these cells in the CBU could provide input to the dendrites of CPU1 neurons and would thus converge onto the same steering circuit as path integration output [64,119]. Using the dendritic trees of CPU1 cells as integration site for working-memory based output from the path integrator and for long-term memory based output generated by other strategies would also allow efficient weighting of these navigation modes. The input with the best signal to noise ratio would dominate CPU1 neuron activity and all inputs combined would generate an integrated activity pattern indicating the insect’s desired heading, which then could be compared to the PB-based representation of the current compass heading to guide the animal’s next movement decision.Finally, if the path integration memory is based on ongoing activity in the CBU, information flow from the CBU towards long-term storage must exist, e.g., to allow averaging of vectors and association of vectors with landmarks, that are suggested for instance by honeybee dance observations. The mushroom body is the most likely location for long term storage, yet there is no known direct connection between the CX and this area, highlighting the need for more comprehensive analysis of the neural connections of the CX.Open conceptual questionsIn summary, insects have evolved strategies in the context of path integration that approximate optimal computations. However, some aspects of path integration and search behavior remain elusive, such as the drifting search patterns observed in some ants [70], or how search paths themselves are generated [62,65,66]. Similarly, it is unclear whether search is an emergent property of the path integration system itself [59,64,120] or requires additional neural control. A reward-based motivational system may be able to orchestrate largely independent functional modules, of which path integration, search, route following, and landmark navigation are examples [77,121]. If so, path integration can be understood independently of the rest of an insect’s navigational system. However, if path integration is interwoven with other strategies such as search, dance, and landmark memories, then a reductionist approach may not suffice. While a definite answer is still to be found, several lines of evidence seem to favor the latter possibility. The fact that a highly conserved brain region like the central complex appears to be at the heart of path integration is striking, as the basic neural circuits proposed to underlie path integration are also present in non-homing insects. Clearly this brain region is involved in a stunningly rich variety of behaviors beyond path integration [81,89,109,111,122–124], and it will be a major task to pin these to specific components or computations of the neural circuitry of this brain center. Whether these path integration circuits have been modified from an ancestral version to endow homing insects with a capable path integrator, or if all insects have a basic ability to path integrate is a yet to be illuminated. Given the intimate relation between systematic search and path integration behaviors, it appears promising to use search that can be centered around a random, non-salient position in space as an indication for the existence of an underlying path integrator, allowing to also access path integration control circuits in non-homing insects. On the other hand, it may be questioned whether behavioral experiments in artificial, simplified settings make full use of the animal’s mental abilities. If more complex strategies are used by the insect brain during natural foraging (e.g. information fusion, cognitive maps), these could produce behavior indistinguishable from path integration during simplified tasks. While the natural ecology of some insect species may indeed favor path integration as the most parsimonious interpretation of their navigational behaviors, it remains unclear whether this is true of insects in general.Supplementary MaterialSupplemental Information includes a mathematical description of the relation between optimal search theory and path integration uncertainty estimates and can be found with this article online at *bxs.Supplemental InformationClick here to view.(100K, docx)AcknowledgementsWe would like to thank Dr. Marie Dacke for helpful comments on the manuscript. The authors are grateful for financial support from the following organizations: the Swedish Research Council (VR, 621-2012-2213, to S.H.), the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement no. 714599, to S.H.), the Australian Research Council (FT140100221, DP150101172, to A.N.) and the Hermon Slade Foundation (HSF17/08, to A.N.).FootnotesDeclaration of InterestsThe authors declare no competing interests.References1. Mittelstaedt H, Mittelstaedt ML. Avian Navigation. Berlin, Heidelberg: Springer, Berlin, Heidelberg; 1982. Homing by path integration; pp. 290–297. [Google Scholar]2. Hironaka M, Nomakuchi S, Filippi L, Tojo S, Horiguchi H, Hariyama T. The directional homing behaviour of the subsocial shield bug, Parastrachia japonensis (Heteroptera: Cydnidae), under different photic conditions. Zool Sci. 2003;20:423–428. [PubMed] [Google Scholar]3. Hironaka M, Filippi L, Nomakuchi S, H H, Hariyama T. 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to not do anything about something or someone when you should try to change or correct that thing or person: I knew he wasn’t telling me everything, but I decided to let it slide. It’s easy to let exercise slide in the suburbs where you have to drive your car all the time.

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Alternatives to the Six-Minute Walk Test in Pulmonary Arterial Hypertension - PMC

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PLoS One. 2014; 9(8): e103626. Published online 2014 Aug 11. doi: 10.1371/journal.pone.0103626PMCID: PMC4128819PMID: 25111294Alternatives to the Six-Minute Walk Test in Pulmonary Arterial HypertensionVincent Mainguy, Simon Malenfant, Anne-Sophie Neyron, Didier Saey, François Maltais, Sébastien Bonnet, and Steeve Provencher

Vincent Mainguy