Model for Transition from Waves to Synchrony in the Olfactory Lobe of Limax

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浙江省杭州市2022-2023学年高三上学期教学质量检测(一模)英语试题

一、听力选择题二、听力选择题1.A ．Mess up the buffet.B ．Embarrass himself in public.C ．Jump the queue.D ．Consume too much.2. What are the speakers mainly talking about?A ．The hot dog.B ．The beerC ．The tickets.3.A ．The hunting is to blame for the disappearance of the birds.B ．The cause of the decline in the ducks’ number is uncertain.C ．She is unhappy with the climate change throughout the world.D ．The man should find more scientific evidence for the birds’ extinction.4. How many people on the bus escaped unhurt in the accident?A ．6.B ．5.C ．4.5.A ．He is quiet.B ．He was hard-working.C ．He is a musician.D ．He couldn’t wake up.6. 听下面一段较长对话，回答以下小题。

1. What are the speakers going to do?A ．Have a class.B ．Have a hike.C ．Have a snack.2. What is probably the man?A ．A foreign teacher.B ．At our guide.C ．An international student 3. Why is go used to replace zou ?A ．Zou and Go rhyme.B ．Go is inaccurate.C ．Zou is easy to remember.7. 听下面一段较长材料，回答以下小题。

有关绿色发展的英语作文理解当代中国

有关绿色发展的英语作文理解当代中国全文共6篇示例，供读者参考篇1Green Growth: The Path to China's Bright FutureHi there! My name is Xiaoming and I'm a 4th grader here in Beijing. Today I want to talk to you about something really important happening in my country - green development. I know it might sound a bit boring, but I promise it's actually super cool and will help make the future much better for all of us!You see, for a long time, China was focused on growing its economy as quickly as possible. We built a ton of factories, roads, cities and other things to become a modern, powerful nation. But during that time, we also created a lot of pollution that hurt the environment. The air in many cities became really smoky and hard to breathe. Factories dumped toxic chemicals into rivers. It was not a very green way of developing.Luckily, China's leaders realized that continuing down that old path wasn't sustainable. If we kept polluting and damaging the planet, eventually there would be no clean air to breathe, water to drink or healthy environment for plants and animals tolive in. So a few years ago, they made a smart decision to put more emphasis on green growth instead of just growing as fast as possible.Green growth means developing the economy in a way that protects the environment and natural resources rather than destroying them. It involves using clean energy like solar and wind power instead of dirty coal and oil. It means companies have to follow stricter rules about pollution and be environmentally responsible. Individuals like you and me are encouraged to live greener lifestyles by reducing waste, reusing items, and recycling as much as possible.I see examples of green growth happening all around my city every day. My school installed a bunch of solar panels on the roof to generate renewable energy from the sun's rays. We have a vegetable garden where we grow our own food without chemical pesticides. We even have a class where we learn about climate change and how to be better environmental stewards.When I'm out around Beijing, I notice more and more electric vehicles on the roads instead of gasoline-powered cars that spew out fumes. The city government has planted millions of new trees to help clean the air and capture carbon dioxide. They are also expanding the metro and bus systems so fewerpeople need to drive. Many buildings now have green roofs covered in plants and grasses.On trips to the countryside, I've seen enormous wind farms sprouting up with their tall white turbines slowly spinning in the breeze. These wind farms generate pollution-free electricity. There are also huge solar panel arrays harnessing energy from the sun's rays. Farmers are shifting to more sustainable methods that avoid overusing synthetic fertilizers and pesticides that can pollute soil and groundwater.China is doing a lot because we have to act now to protect the environment for the future. If we keep going the old way by burning tons of coal and gas while destroying forests and polluting the rivers and air, the planet will continue getting hotter and hotter due to climate change. Weather could become more extreme with worse droughts, storms and heat waves that ruin farms and threaten food supplies. Rising sea levels could flood many coastal cities where millions of people live. Plants and animals could go extinct. It's a scary thought!That's why green growth that prevents those bad things from happening is so important. The old economic model of growth at any cost is no longer sustainable or wise. We have to find a better balance between developing a strong economywhile also protecting the natural environment that we all depend on to survive and thrive. It's the only way China and the world can keep developing in a healthy, safe way for many generations to come.I'm really glad China's leaders are embracing this green growth model for the future. It makes me excited and hopeful that the air will keep getting cleaner, the food and water will be purer, and we'll have beautiful forests and landscapes to enjoy. More renewable energy from the sun, wind and other green sources will power our homes and businesses. Maybe future cities will evenbe designed to operate in a fully sustainable loop with Zero waste!So that's my take on why green growth is so vital for China's future and the world's future too. I hope this has helped explain what it's all about and why we should all embrace it wholeheartedly. It's up to my generation to build on the green growth happening today to make it even bigger and better tomorrow. With hard work and care for the environment, I'm confident we can create a brilliant green future for China and the entire globe. What篇2Green Development and Understanding Contemporary ChinaHi there! My name is Lily, and I'm a fourth-grader at Sunshine Elementary School. Today, I want to talk to you about something really important – green development and understanding contemporary China.China is a country that's changing rapidly, and one of the biggest changes happening is a shift towards a more environmentally friendly way of living and developing. This is known as "green development," and it's super important for the future of our planet!You see, for a long time, many countries (including China) focused on industrialization and economic growth without really thinking about the impact it had on the environment. Factories spewed out smoke and pollution, forests were cut down to make way for new buildings, and natural resources were used up without much thought for the future.But over time, people started to realize that this wasn't sustainable. If we keep damaging the environment at this rate, we'll run out of clean air, water, and natural resources – and our planet will become a really unpleasant place to live.That's why the concept of green development emerged. It's all about finding ways to grow our economies and improve people's lives without harming the environment. It's about using renewable energy sources like solar and wind power instead of burning fossil fuels. It's about protecting forests, rivers, and wildlife habitats. It's about reducing waste and recycling as much as possible.And you know what? China has become a leader in green development! The government has set ambitious goals to reduce carbon emissions, promote clean energy, and protect natural resources. They've invested heavily in things like solar and wind farms, electric vehicles, and energy-efficient buildings.One really cool example is the city of Shenzhen, which used to be a small fishing village but is now a major metropolis. Shenzhen has transformed itself into a hub for green technology and sustainable living. They have electric buses, bike-sharing programs, and tons of parks and green spaces.But green development isn't just about big cities and fancy technology. It's also about everyday people like you and me making small changes in our lives. We can reduce our energy use by turning off lights and unplugging devices when we're not using them. We can recycle paper, plastic, and glass instead ofthrowing them away. We can walk, bike, or take public transportation instead of driving everywhere.And you know what else is really important? Understanding contemporary China! China is a fascinating country with a rich history and culture, but it's also a rapidly changing and modernizing society. By learning about China's past and present, we can better appreciate the challenges and opportunities it faces.For example, did you know that China has made incredible progress in reducing poverty over the past few decades? Hundreds of millions of people have been lifted out of extreme poverty, thanks to economic reforms and development programs. That's amazing!At the same time, China is grappling with issues like urbanization, an aging population, and the need to transition to a more sustainable growth model. It's a complex andmulti-faceted country, and there's always more to learn and understand.So, let's all do our part to support green development and learn more about contemporary China! It's not just good for the environment and our planet – it's also really interesting and fun to explore new ideas and perspectives.That's all from me for now, but I encourage you to keep an open mind, ask questions, and never stop learning. Together, we can build a greener, more sustainable, and more understanding world!篇3My Understanding of Green Development in Today's ChinaHi there! My name is Xiaoming, and I'm a 10-year-old student from Beijing. Today, I want to share with you my thoughts on green development and how China is working towards a more sustainable future.First of all, let me explain what green development means. It's all about finding ways to grow our economy and improve people's lives without harming the environment. We need to use natural resources carefully, reduce pollution, and protect the planet for future generations.In China, green development is a really big deal. Our country has experienced rapid economic growth over the past few decades, but this has also led to some environmental problems like air pollution, water pollution, and deforestation. The good news is that our government and people are now taking action to address these issues.One of the main things China is doing is promoting renewable energy sources like solar power, wind power, and hydropower. You've probably seen solar panels on the roofs of buildings or huge wind turbines spinning in the countryside. These renewable energy sources don't produce as much pollution as burning coal or oil, which is better for the environment.China is also working hard to improve energy efficiency. This means finding ways to use less energy to do the same tasks. For example, many new buildings in China are designed to be more energy-efficient, with better insulation and more efficient heating and cooling systems. Some factories are also upgrading their equipment to use less energy.Another important part of green development in China is protecting and restoring natural habitats. Our country has created many nature reserves and national parks to protect endangered species and their habitats. There are also programs to plant trees and restore forests, which help to improve air quality and prevent soil erosion.In my city, Beijing, there are lots of efforts to promote green development too. The government has taken steps to reduce air pollution by encouraging people to use public transportation,bicycles, or electric vehicles instead of cars that produce a lot of emissions. There are also campaigns to encourage people to recycle and reduce waste.At my school, we learn about the importance of protecting the environment and practicing sustainable habits. We have recycling bins for paper, plastic, and metal, and we're encouraged to bring reusable water bottles and lunch boxes instead of using disposable ones. We also have a school garden where we grow fruits and vegetables, and we learn about composting.While there's still a long way to go, I'm hopeful that with continued efforts and cooperation, we can create a world where economic development and environmental protection go hand in hand. As a kid growing up in today's China, I'm excited to be part of this journey towards a more sustainable future.So, that's my understanding of green development in contemporary China. It's an important issue that affects all of us, and I believe that by working together and making eco-friendly choices in our daily lives, we can make a real difference.Thanks for listening, and remember: every little action counts when it comes to protecting our planet!篇4Certainly! Here's an essay of around 2,000 words on the topic of green development and understanding contemporary China, written in English with a tone suitable for elementary school students.Title: China's Green Journey: Protecting Our Beautiful HomeHello, friends! Today, I want to talk to you about something really important – China's efforts to protect our environment and achieve green development. You see, our country is like a big, beautiful garden, and it's our responsibility to take care of it.Have you ever heard about air pollution? It's like a thick, smoggy blanket that covers our cities, making it hard to breathe clean air. Or what about those towering piles of trash that seem to grow taller and taller every day? These are just some of the challenges we face when it comes to protecting our environment.But did you know that China is working really hard to solve these problems? Our leaders understand how important it is to take care of our planet, and they've come up with some amazing plans to make our country greener and cleaner.One of the coolest things China is doing is investing in renewable energy. You know how we get electricity from burning coal or natural gas? Well, those fossil fuels release a lot of harmful gases into the air, which is not good for our environment. That's why China is building more and more wind turbines and solar panels to generate electricity from the wind and the sun –sources of energy that are clean and renewable!Another exciting thing happening in China is the development of electric vehicles. Have you ever seen those cool, futuristic-looking cars that don't have any exhaust pipes? Those are electric cars, and they don't release any harmful gases into the air like regular gasoline-powered cars do. China is encouraging more people to use electric cars, buses, and even bicycles to help reduce air pollution in our cities.But that's not all! China is also working hard to protect our forests and natural habitats. You see, trees and plants are like the lungs of our planet – they take in carbon dioxide (that's the gas that contributes to climate change) and release oxygen for us to breathe. That's why China has launched massive tree-planting campaigns and is creating more national parks and nature reserves to protect our precious forests and wildlife.And do you know what else China is doing? They're encouraging people to reduce, reuse, and recycle! That's right, instead of just throwing away our trash, we can sort it and recycle things like plastic, paper, and glass so that they can be turned into new products. This helps to reduce the amount of waste we produce and conserve our natural resources.Now, I know all of this might sound a bit complicated, but trust me, it's really important stuff. By working together to promote green development, we can make our country – and our whole planet – a cleaner, healthier, and more beautiful place to live.So, what can we do to help? Well, there are lots of small things we can do in our daily lives. We can turn off lights and electronics when we're not using them to save energy. We can bring our own reusable bags when we go shopping instead of using plastic bags. We can even start our own little recycling projects at home or at school!Remember, every little bit counts, and together, we can make a big difference. Let's all do our part to support China's green journey and protect our beautiful home!篇5Green Growth in Today's ChinaHi there! My name is Xiaoming and I'm 10 years old. Today I want to tell you all about the really cool things China is doing to grow in an environmentally friendly way. It's called "green development" and it means developing our economy and society while also protecting the natural environment.In China, we've realized how important it is to take care of the earth as we build new cities, factories, and infrastructure. If we're not careful, pollution and environmental damage can happen, which is bad for people's health and the plants and animals. That's why green development has become such a big priority.One major part of green development is using clean energy sources like solar, wind, and nuclear power instead of dirty fossil fuels like coal that create a lot of pollution. China is now a world leader in renewable energy. We have the biggest solar farm on earth and lead the world in installing new solar and wind power capacity each year. The renewable energy sector also creates millions of new jobs which is awesome.In my hometown, I see more and more solar panels going up on rooftops and in big solar farms out in the countryside. There are also huge wind turbines towering over the fields and hillscapturing the wind's energy. My dad says that pretty soon, a big portion of our electricity will come from these clean energy sources rather than coal power plants. How cool is that?Green transportation is another key part of China's green development plan. This means things like high-speed rail networks, electric vehicles, and better public transit to reduce pollution from cars and trucks. China already has the longest high-speed rail network in the world with over 25,000 miles of tracks! That allows people and goods to get around quickly without driving and creating emissions.The cities are also working hard to build more subway and bus systems to get people out of their personal cars. Electric vehicles are becoming mainstream too. My uncle just bought an electric car and he can drive all around the city without any emissions coming from the tailpipe. In the future, more and more of the cars, trucks, and buses will be powered by electricity rather than gasoline and diesel.China is also a leader in green finance and green tech innovation. This means investing lots of money into developing new clean technologies like better solar panels, wind turbines, batteries for electric cars, and eco-friendly construction materials.Lots of new companies are being created in these high-tech green industries. Maybe I'll work at one of them when I grow up!Another really important part of green development is improving resource efficiency and recycling to create a more "circular economy." This means designing products that use fewer raw materials, last longer, and can be repaired, reused, or recycled at the end of their life instead of just being thrown away. It also means finding ways to reuse and recycle water as much as possible since fresh water can be scarce.My school has started teaching us how to sort our trash properly so that paper, plastics, metals, and other materials can be recycled. We're also learning to avoid single-use plastics and to buy products with less packaging waste. At home, we now have different bins for separating out the recyclables which then get picked up and processed. It feels good to do our part!On school field trips, we've visited some of these national parks and seen pandas, golden monkeys, cranes, and other amazing wildlife up close and in their natural habitats. It's incredible getting to experience China's rich biodiversity firsthand. I'm glad efforts are being made to protect these special places and animals.China is also on the forefront of eco-city development, where entire cities are being designed and built from the ground up with environmental sustainability in mind. Things like energy-efficient buildings, green spaces, excellent public transit, clean heating systems, recycling programs, and renewable power generation are all incorporated into the city planning.I have some relatives living in one of these new eco-cities and they tell me it's like living in a green utopia! The air is fresh, there are parks and trees everywhere, getting around is a breeze with all the subway and bike lines, and they even capture and reuse their wastewater. I can't wait to visit and see it myself. Maybe that's what cities of the future will be like!As you can see, China has been taking huge strides in green development in recent years. Of course, there's still more work to be done and challenges to overcome, but I'm really proud of the progress my country has made. By growing the economy through clean industries, investing in green innovation, protecting ecosystems, and building sustainable cities, we are ensuring a higher quality of life for today's people while also preserving the environment for future generations like me.I hope this gives you a glimpse into how green development works and why it's so important, especially for a big country likeChina. If we want to keep developing and building prosperous societies, we have to do it in an environmentally-responsible way. That's the path to a greener, healthier, more sustainable future. Thanks for reading!篇6Green Growth: China's Path to a Sustainable FutureHi there! My name is Lily and I'm a 10-year-old student from Beijing. Today, I want to tell you all about the amazing changes happening in my country when it comes to green development. It's such an important topic that impacts the whole world!You see, China is the world's most populous nation with over 1.4 billion people. That's a lot of mouths to feed and a huge demand for energy, homes, transportation and consumer goods. For many years, China's rapid economic growth came at a big cost to the environment through pollution, deforestation, and heavy use of fossil fuels like coal.However, the Chinese government realized that this model wasn't sustainable in the long run. Pollution was making people sick, climate change was a looming threat, and our natural resources were being used up rapidly. So about 10 years ago,China made a huge commitment to green development and building an "ecological civilization."What does that mean exactly? Well, it means shifting from just chasing economic growth at all costs, to pursuing growth in a way that protects the environment, conserves natural resources, and promotes harmony between humans and nature. It's all about finding that perfect balance!One of the biggest areas of focus has been clean energy. China used to rely heavily on dirty coal power, but now it's going full speed ahead on renewable energy like solar, wind, nuclear and hydropower. My dad works for a big solar panel company and he told me China is installing a new solar farm every single week! How cool is that?There are also a lot more electric vehicles on the roads now to cut down on pollution from gasoline cars. The government offers people incentives to buy electric and they are building tons of new charging stations. I really want an electric car when I'm older!In addition to clean energy, China is a world leader inhigh-speed rail now. Bullet trains allow people to travel long distances while producing way less emissions than planes or cars. Just last month, my class took a high-speed rail trip to Xi'an tosee the Terracotta Warriors. It only took a few hours to get there instead of a whole day's drive!China is also making a massive effort to increase forest cover and green spaces. The government has started huge tree planting projects across the country, especially in areas that have suffered from deserts expanding due to deforestation. President Xi has said that "lucid waters and lush mountains are invaluable assets." I couldn't agree more!My school has started lots of neat green initiatives too. We have a vegetable garden that the students help tend to. We also have recycling bins and energy-saving lights throughout the building. For our field trips, we prefer taking public transit or visiting parks and nature reserves to learn about ecosystems. My favorite was when we went to Wuyuan and saw those amazing bright yellow fields of rapeseed flowers!Sometimes I still see people improperly disposing of waste or I smell factory smoke in the air. But I know we are making great progress. More and more people recognize the importance of environmental protection with each passing year.In my hopeful vision for the future, I see China as a true ecological civilization. One with booming clean industries, pristine rivers and lakes, and flourishing natural wonders. A placewith modernization in harmony with nature, not destroying it. A nation that is prosperous but also sustainable for generations to come.With continued efforts on renewable energy, reforestation, conservation, and green transportation, I'm confident we can make this dream a reality. China is showing the world that an economy and the environment don't have to be in conflict. With creativity, willpower and the right priorities, green development is possible even for the most populous。

大学英语四级词汇前后缀

第一部分前缀a,an①无，不，非astable不稳定的acentric无中心的②含有in，on，at，by，with，to等意义asleep在熟睡中ahead向前ab脱离abnormal不正常的abaxial离开轴心的anti反抗antiwar反战的anti—imperialist反帝的auto自动，自己autobiography自传auto—stable自动稳定be使…加强belittle使缩小befriend友好相待bi二,双biweekly双周刊biligual两种语言的co，col，com，con，cor共同cooperation协作collaboration协作，勾结combine联合；correlation相互关系counter，contra反，对应counteraction反作用contrast对比，对照di，dif，dis否定，相反diffident不自信的dislike不喜欢en，em使…enlarge扩大enable使…能empower使…有权力e，ex外,出external外部的erupt喷出extra以外，超过extraordinary格外的extra-solar太阳系以外的hyper在上，超hyper—frequency超高频hypersonic超声的il,im，in，ir否定illogical不合逻辑的impossible不可能的invisible不可见irrational不合理的inter互相interchange交换interlock连锁intra，intro在内，内部intrapersonal个人内心的introspect内省mal恶,不良maltreat虐待malfunction机能失常micro微microscrope显微镜microware微波mid中，中间mid-air半空中midstream中流mini小minibus小公共汽车ministate小国mis错，坏mistake错误misspell拼错multi多multiparty多党的multilingual多种语言的non否定nonsmoker不抽烟的人nonexistent不存在的out超过,过度，外，出，除去outgo走得比…远outgrow长得太大outdoor户外的outroot除根over上,过度overwork工作过度overbridge天桥post后postwar战后postnatal诞生后的pre前，领先prewar战前的prefix前缀re回，再rsemi半semicircle半圆semiconductor半导体sub，suc，suf，sup次,亚，在下，低于substandard低于标准规格的succeed继承suffix后缀supplement增补trans转换，变换，横过，越过transmit传送transatlantic横渡大西洋的tri三tricar三轮车triangle三角（形）un否定unstable不稳定的unknown未知的under下，内（指衣服)，不足underground地下的underskirt衬裙underpay付资不足uni单一uniform均匀的unipolar单极的后缀able，ible可…的,能…的readable可读的sensible可觉察的age表状态，性质,行为breakage破损shortage缺乏al(adj）具有…性质的，如…的personal个人的regional地区的al（n）表动作,人，事物proposal提案professional专业人员ance，ence表状态，行为，性质importance重要性confidence自信ancy，ency表状态，行为，性质constancy一惯性urgency紧迫性ant，ent表人，物，行为applicant申请人correspondent通信者arian表各种年龄，派别,主义，职业的人humanitarian人道主义的utilitarian功利主义者crat支持,参与者democrat民主人士bureaucrat官僚dom表性质，状态，行为freedom自由wisdom智慧ee表动物的承受者employee雇员trainee受训练的人eer从事…的人pioneer开拓者volunteer志愿者en（v）使变成harden使硬shorten使短en（adj）有…质的，似…的golden金色的woolen毛的er，or表人,物singer歌唱家survivor幸存者ery表状态，性质，行为,活动地点nursery托儿所robbery掠夺slavery奴隶身份ese某国（地)的，某国（地）的人及语言Chinese中国人（汉语）Portuguese葡萄牙人（语）ess表女性，雌性hostess女主人actress女演员ful充满…的，具有…性质的cupful一满杯fearful可怕的hood表身份,状况，性质neighborhood邻里manhood男子气概ic,ical类似…的，具有…的heroic英雄般的logical合逻辑的〈Bify使成…，使…化magnify放大purify提纯ion，tion表情况，状态,性质，行为fashion时髦decision决定ish似…的，有…的selfish自私的childish儿童般的ism表示……主义,行为，…学，…派realism现实主义tourism旅游fatalism宿命；论modernism现代派ist从事…工作的人socialist社会主义者dentist牙科医生ity表行为，性质,状态unity团结一致maturity成熟性ive有…性质的，有…作用的，属于…protective保护的productive生产的；active积极主动的ize,ise,yze，yse以…方式对待，…化realize实现analyze分析modernize现代化less没有…的fearless无畏的useless无用的let小booklet小册子leaflet传单like有…性质的，像…的dreamlike梦一般的humanlike像人类的logy论，研究biology生物学technology技术ly如…的,有…特性的manly男子气的lovely可爱的ment表行为，状态,性质movement运动management管理ness表状态，性质darkness黑暗kindness仁慈ous，eous,ious充满…的dangerous危险的famous著名的ship表状况，状态，身份，技艺，技能hardship苦难doctorship博士学位；workmanship手艺,工艺some像…的，引起…的troublesome讨厌的lonesome孤独的ster表示人youngster年轻人gangster歹徒ward(s）表方向upward（s）向上outward（s）向外y①（n）表状态，性质，行为difficulty困难victory胜利②（adj）充满，包含hairy多毛发的irony钢铁般的第二部分词缀1．否定前缀un- uncomfortable, unacceptable，unavoidablein- incomplete, incapable，indispensable, inevitableim— imperfect, impartial, immortaldis— disable，disadvantagemis- misplace, mislead, misfortuneab- abnormalde- deactivate, degradeil— illegal，illiterateir- irregular, irrelativemal- malfunction, malnutrition2。

HP 3326A 双通道信号合成器说明书

FREQUENCY, FUNCTION & WAVEFORM SYNTHESIZERSTwo-Channel Synthesizer, DC to 13 MHzModel 3326AHP 3326AThe HP 3326A Two-Channel Synthesizer combines two indepen-dent synthesizers, flexible modulation, and control circuitry into a sin-gle, powerful package. This single instrument can provide precisephase offset, two-tone sweep, fast frequency switching, internal mod-ulation, and pulse signals for bench or systems use.Complete Two-Phase SolutionThe HP 3326A can provide two signals whose phase is adjustableand calibrated anywhere in its 13 MHz frequency range without anexternal phasemeter.Self-calibration can be performed internally or externally andyields accuracy of ±0.2 degrees below 100 kHz. Phase can be set with0.01 degree resolution at all frequencies. Using its unique phase-cali-bration circuitry, calibrated multi-phase signals are easily achievedwith two or more HP 3326As.Powerful Two-Tone CapabilityThe HP 3326A is the single-source answer for producing a widevariety of broadband two-tone signals. It's two channels can be offsetup to ±100 kHz, either in the CW mode or while sweeping.Channel amplitudes and functions (sine or square) can be selectedindependently and provided from separate outputs or through thebuilt-in signal combiner. Low sinewave distortion (harmonics are atleast -70 dBc below 100 kHz) makes low-distortion intermodulationmeasurements a simple task.Versatile ATE SourceWith two complete synthesizers in a single instrument, rack spaceand power are conserved. Features like internal amplitude and phasemodulation, two-tone, and pulse modes allow this one instrument todo the job of several sources.All functions, modes, and parameters of the HP 3326A are com-pletely programmable over the HP-IB. Maximum accuracy is en-sured with amplitude/phase calibrations that can be enabled,disabled and initiated under remote control.High Performance Modulation and PulsesPrecise amplitude and phase modulation is easy with both channelsof the HP 3326A. Each channel can be used with simultaneous AMand PM, or one channel can modulate the other. Amplitude modula-tion frequency is dc to 100 kHz and envelope distortion is better than-46 dB.In the pulse mode both pulse and pulse-complement outputs areprovided. Symmetry range is 1 % to 99% and is settable in 0.1 % incre-ments. In addition, both pulse amplitudes and their offsets are inde-pendently controllable.Other FeaturesThe HP 3326A has a host of convenience features to speed andsimplify signal generation. Nine complete setup states can be storedin nonvolatile memory, along with automatic storage of the power-offstate.A discrete sweep mode is available to generate from 2 to 63frequency pairs with dwell times individually selectable for each fre-quency and each channel. Several flexible triggering modes allowhardware or software triggers to initiate frequency, amplitude, orphase changes, and sweeps.DC offset is available in all modes, and all outputs are floating.Frequency resolution is I1 digits, and all.sweeps and frequencychanges are phase-continuous.SpecificationsFor complete specifications refer to the HP 3326A data sheet.Operating ModesTwo Channel:Channels A and B are independentTwo-Phase-,Channels A and B are the same frequency, with cali-brated phase offset between the two signalsTwo-Tons:Channel B frequency offset 0 to 100 kHz from channel AfrequencyPulse:Channel B is the complement of Channel AFrequency (Waveforms are Sine, Square, Pulse, and DC)Range:0 Hz to 13 MHzResolution:1MHz below 100 kHz, 1 mHz at and above 100 kHzStability:±5x10-6/year, 20° to 30°C. See also option001, High Sta-bility Frequency Reference.Accuracy:±5x10-6of selected value, 20° to 30°C, at time of calibra-tion with standard frequency referenceSinewave Spectral PurityHarmonics:Harmonically related signals will be less than the follow-ing levels relative to the fundamental, or <-90 dBm, whichever isgreater:10 Hz50 kHz100 kHz1MHz13 MHz+23.98 dBm -80 dBc-70 dBc-35 dBc-30 dBc+13.98 dBm -80 dBc-80 dBc-65 dBc-30 dBc-56.02 dBmIntegrated Phase noise:-66 dBc (Option 001 only, for a 30kHzband centered on a 10 MHz carrier excluding ±1 Hz about the carri-er)Main Signal Outputs (Channels A d B, All Waveforms Un-less Noted)Connectors:Front panel BNC femaleI mpedance:50 0; output may be floated to ±42 V peakSync A:TTL level squarewave at Channel A frequency.Output Amplitude (Sine Mode)Range:1mVpp to 10 Vpp in 8 ranges without DC offset. See alsooption 002 High Voltage OutputUnits:Volts peak-peak, Volts rms, dBm (50i2), dBV1981Resolution:0.1% of full range for peak-peak entry0.3% of full range for rms entry0.01 dB for dBm or dBV entry Accuracy: Relative to programmed value after self-calibration0.001 Hz100 kHz1MHz13 MHz +23.98 dBm±0.1 dB±0.3 dB±0.6 dB+3.98 dBm ±0.8 dB-36.02 dBm±0.2 dB±0.5 dB +1.0 dB-56.02 dBmSquarewave and Pulse CharacteristicsRise/fall time::515 ns, 10% to 90% at full output Overshoot:< 5% of peak-to-peak amplitude at full output Pulse width range:1% to 99% of period or 20 ns, whichever is greaterPulse width resolution:0.01% of periodPulse width accuracy:<±l% of period ±20 ns Amplitude accuracy:±2%, 0.001 Hz to 100 kHzDC OffsetRange:(See also option 002, high voltage output)DC only:0 to ±5 VDC+AC:DC+AC peak <5V; Max. DC offset is affected by AC range,Maximum is 4.5 V decreasing to 4.5 mV on lowest range Resolution:3 digitsAccuracy:(After self-calibration)DC only.±75 mVDC+AC: (Sinewave) 10 Hz to 1 MHz: *2% of rangeI MHz to 13 MHz: ±5% of rangePhase Offset(Channel A vs B in Two-Phase mode)Range:±720 degreesResolution:0.01 degreeAccuracy: After self-calibration, for equal-level sinewaves 1 V to 10 V peak-peak0.1Hz to 10 Hz±0.5 degrees10 Hz to 100 kHz±0.2 degrees100 kHz to 1 MHz±0.3 degrees1MHz to 13 MHz±2.0 degreesAmplitude ModulationSpecifications apply to Channel A and Channel B with external modulation or to Channel A internal modulation with Channel B as the modulation source. External modulation is allowed in all modes; internal modulation is allowed only in the two-channel mode. Waveforms:Sine, square, or (external only) pulse, DC, etc. Frequency Range:Carrier: DC to 13 MHzModulation: DC to 100 kHz Modulation Depth:0 to 100%Phase ModulationSpecifications apply to Channel A and Channel B with external modulation or to Channel A internal modulation with Channel B as the modulation source. External modulation is allowed in all modes; internal modulation is allowed only in the two-channel mode. Waveforms:Sine, square, or (external only) pulse Frequency Range:Carrier: DC to 13 MHzModulation: DC to 5 kHzPhase Deviation: 0° to 360°Frequency SweepSweep Types:Linear, discreteSweep Forms:Triangle, rampSweep Time:5 ms to 1000 sSweep Elements(Discrete): 2 to 63 frequency pairs and dwell times, user defined; dwell times = 5 ms to 1000 s/element Maximum Sweep Width:13 MHz Output CombinerChannel A and B are combined on the Channel A output. B output is off. Combiner may be used in the two-channel, two-phase, and two-tone modes. DC offset is automatically set to 0 V. Frequency Range:DC to 13 MHzReturn Loss:>20 dBAuxiliary Outputs (All Connectors are Rear-Panel BNC) 10 MHz reference-.+3 dBm output to phase lock other instruments to the HP 3326A10 MHz oven output:+3 dBm oven-stabilized frequency reference (option 001 only)X-axis drive:Linear ramp proportional to sweep timeZ-axis blank:TTL low during sweepSweep Marker TTL low at selected marker frequency in sweep 20-33 MHz LO:>I 00 mV square wave output offset 20 MHz from Channel B outputAuxiliary Inputs (All Connectors are Rear-Panel BNC) Reference Input:For phase-locking to an external frequency refer-ence. Signal of 1,2,5, or 10 MHz, ±10 ppm, 0 to +20 dBm External Trigger Input:TTL level to initiate linear or discrete sweep on high to low transitionChannel A and B external phase calibration inputs Channel A and B external amplitude modulation inputs Channel A and B external phase modulation inputsHP-IB Remote ControlCompatible with IEEE Standard 488-1978Interface Functions:SH I ,AH I ,T6,L4,SRI,RLI,PPO,DC 1,DTI,CO,EIOption 001 High Stability Frequency Reference Stability:±5X10-°/week after 72 hours continuous operation ±1X10-7 /month after 15 days continuous operation Option 002High Voltage OutputMultiplies the output level by 4 and expands the allowable DC off-set range. Specifications apply to both channels in all modes with the internal combiner off.Frequency range:DC to 1 MHzOutput Impedance:<2 0, DC to 50 kHz; < 10 0, 50 kHz to 1 MHz Amplitude range:4 mV to 40 Vpp into > 1 k 0, <200 pF load without DC offset (must be entered in peak-to-peak units only)DC offset:±20 V, independent of amplitude range. DC + AC peak must not exceed 20 VOption 003 Rear Terminal OutputsProvides Channel A and B main outputs only on rear panel BNC's. Front panel main outputs are removed. Specifications unchanged. GeneralPower.100/ 120/220/240 V, +5%, -10%,48 to 66 Hz; 120 VA, 150 VA with all options, 10 VA standbyWeight: 27 kg (60 lb) net, 37 kg (81 lb) shipping Dimensions:177 mm H x 425.5 mm W x 497.8 mm D (7 x 163/x 195/8")Accessories AvailableHP 15507A Isolator.For isolation of signal ground between frequen-cy reference and instrument input/outputHP 9211-2656 transit case for protection in transportation and stor-ageOrdering Information Price HP 3326A Two-Channel Synthesizer$9,760 Option 001High Stability Frequency Reference$665 Option 002High Voltage Output$305 Option 003 Rear Terminal Outputs (Rear only)N/C Option 907Front Handle Kit S61 Option 908Rack Flange Kit$36 Option 909Rack Flange and Handle Combination Kit$92 Option910 Extra Operating Manual$102 Option 914Delete Service Manual less $115 Option W30 Ext. Warranty$190。

Simplifiedwavemodelling

Simplified wave modellingJohn C. BancroftABSTRACTWave motion is modelled using the acoustic wave equation and implemented using MATLAB. This method requires two initial conditions that are introduced using a simple wavelet on a one dimensional propagator such as a string, spring, or wire. This model is expanded to two dimensions that illustrate plane-wave propagation, boundary effects, and Huygen’s wavelets.WAVE ON A STRING.Assume we want to model transverse wave motion on a string (spring, wire, etc). We will compute the motion of a wavelet at equal time increments that simulate photos to evaluate the motion.We start with a Gaussian-shaped wavelet illustrated below in Figure 1 that represents a photograph of the transverse displacement. We have chosen the width of the wavelet to be large, relative to the sample interval so that linear interpolation provides an adequate description, i.e. the sample rate is approximately ten times the maximum frequency to minimize grid dispersion. Given this information:•What is the direction of motion?•What will be the shape in the next photograph?FIG 1: A representation of a Gaussian shaped wavelet on a string.Is the wavelet in Figure 1 going to move to the left, the right or bounce up and down? We simply don’t know because we don’t have enough information. How do we provide that additional information? Let’s examine the wave equation to find out.The string has one distance dimension, x, and we define the displacement p(x), at different time intervals such as p(x, t=1), p(x, t=2), etc. It therefore becomes convenient to define a 2D array, p(x, t), that defines the amplitude of the strings displacement at agiven location, x , and time, t . The movement of energy on the string is governed by the one dimensional acoustic wave-equation()()22222,,1p x t p x t x v t∂∂=∂∂. (1) We will use the finite difference method to approximate the wave-equation. A second derivative of a function, f(x), is approximated in a discrete form of f at position, n , i.e. f n , and is approximated by()211222n n n f x f f f x x δ−+∂−+=∂, (2) where the increments of f n are d x . The finite difference equation for the wave equation becomes: 1,,1,,1,,1222221i j i j i j i j i j i j p p p p p p x v tδδ−+−+−+−+=. (3) This equation can be represented in graphical form in the following figure, with the increment, i , representing distance to the left, and j , the time increment that increases vertically.FIG 2: Finite difference model of full wave-equation.We wish to compute the position of the string at a new time, given some initial condition. We will choose p i, j+1 as the unknown value in the finite-difference equation because it is a single value at a new time, giving()22,1,,11,,1,222i j i j i j i j i j i j v t p p p p p p x δδ+−−+=−−+. (4) According to this equation, we need to know the position of the string at the two previous times, j , and j-1; .i.e. the two initial conditions of the string. Shown below is thefinite-difference operator positioned at the at i th spatial location to calculate the amplitude on the string for the third time, at j=3.i-1 i+1iFIG 3: The first two initial conditions at j =1 and 2 to compute the first line of samples at j=3. If we desire to model a wave travelling down the string, then we need to start with a wavelet on the string at time j=1, and the same wavelet repeated on the string at time j=2 but moved to a new spatial location. The location on the j=2 string is critical and must have a spatial increment defined by the wave velocity, v , and the time increment, d t . These two initial conditions would also match any two sequential photos of the wave moving down the string. They are shown above, along with the finite-difference operator, illustrating the computation of the amplitudes on the string at time, j=3. (The above figure has used a very large d x on the operator for the purposes of illustration only).Computer simulations are illustrated in Figure 4, with (a) showing the amplitude of a Gaussian wavelet on the string at the first two times and the resulting calculation of the wavelet at the third time, k=3. The amplitudes at succeeding times may be calculated from the previous two times as illustrated in (b) that shows the wavelet at a time increment, k=50. The initial two wavelets are also shown to demonstrate how well the amplitude and shape of the wavelet has been preserved after 48 iterations.The complete MATLAB code for producing Figures 4 and 5 is listed in Appendix A. The part of the code that propagates the wavefield is six lines that are encircled to illustrate the simplicity. The coding method is similar to FORTRAN and does not take advantage of vector math that is available to a programmer.a)b)FIG 4: The first two initial conditions and the computation of a) the next third time increment, and b) at 50 time increments later. Note the preservation of the shape and amplitude.The full set of iterations are shown below in a three-dimensional view that shows the amplitude of the wavelet at the incremental times. This is a trivial task in MATLAB that simply displays 2D array, p(t, x), in Figure 5 with the call, “figure (3); mesh(p);”, highlighted in the program listing in appendix A. This figure has zoom androtation features that allow the programmer to view the image from any position. Thesimple coding of this program, along with the powerful MATLAB graphics, provide anexcellent analysis tool.FIG 5: A full set of time images of the displacementTo illustrate the importance of the initial conditions, various configurations of the first two wavelets are shown in the following figures. Note the effect on the subsequent propagations.•Figure 6a and b show the first two wavelets at the same amplitude and location at the centre of the string. The result is two wavelets moving away from eachother. (These two images demonstrate two different perspective views of thesame figure).•Figure 6c shows the two initiating wavelets at the same location, but the second is twice the amplitude, producing an interesting deformation of thestring. A similar type of deformation is obtained by zeroing the amplitude ofthe second wavelet as viewed in Figure 6d.We refer to the setting of the two initial conditions or the defining the deformation ofthe string at the first two time intervals as “exciting” the string.a) b)c) d) FIG 6: Various wave propagations that result when the initial conditions are varied, a) when the location and amplitude are the same, b) the same as (a) but a different view, c) with the same location but the amplitude of the second initial condition has twice the amplitude, and d) when the second initial condition is zero.2D WAVE PROPAGATIONThe propagation of energy on a 2D plane is also quite simple to program by extending the concepts of the 1D program. The two-dimensional wave equation,()()()2222222,,,,,,1p x z t p x z t p x z t x z v t∂∂∂+=∂∂∂, (5) becomes the finite difference equation, 1,,,,1,,,1,,,,1,,,1,,,,122222221i j k i j k i j k i j k i j k i j k i j k i j k i j k p p p p p p p p p x z v t δδδ−+−+−+−+−+−++=. (6) Solving for the single time sample at k+1 we get:1,,,,1,,,1,,,,1,22,,1,,,,122222i j k i j k i j k i j k i j k i j k i j k i j k i j k p p p p p p p p p v t x z δδδ−+−++−−+−+ =−++. (7) This difference equation is illustrated in Figure 7. A plane through the central points represents time at the k th level and the one sample point below this plane is a sample on the previous time layer at k-1. The single point to be computed (the circle) lies on the upper plane at time k+1.FIG 7: Schematic representation of the three-dimensional (x, z, t ) operator.We therefore need two planes of the wavefield (at k-1, and k ) to start the propagation. As in the 1D case, it is critical that the first two planes represent the desired initial conditions. In Figure 8a below, the initial wavefield starts at the centre of the plane and then, after 80 time iteration of computing the wavefield, we get the wavefield shifted to the left as evident in part (b). Note that the amplitude and shape of the wave field has been preserved.a) b) FIG 8: A plane wave designed to propagate to the left, a) at time zero, and b) at after 80 time increments.Figure 9a show the an initialization of a plane-wave that only extend part way across the surface. The resulting wavefield after 80 time iterations is shown in various perspectiveviews in panels b), c), and d). Note the unusual effect of this result as energy has propagated opposite to the direction to the main part of the wavefront.a) b)c) d)FIG 9: A portion of a plane-wave is propagated 40 time increments with: a) the initial location with a rounded edge, b) a perspective view after 40 time lags, c) side view, and d) a plan view.Figure 10 contains two images that result from a 3-D Gaussian-shaped wavelet that is circular in (x, z) as displayed in Figure 11a. The circular wavefront in part (a) in Figure 10 was excited by keeping the two excitation wavelets at the same location, while part (b) was excited by moving the second wavelet to the left with the propagation velocity. Note that even though the wavelet was initially propagated to the left, some energy also moves in the opposite direction.Figure 11 displays three additional wavelets in (b), (c), and (d) that are truncated to widths of five, three, and one sample. The intent of these wavelets is to represent some form of decomposition of a plane wave: i.e. part (d) is just one slice from the wavefront. This slice is definitely aliased in the direction parallel to the wavefront and we should expect some form of dispersion during propagation.Each wavelet is propagated in the direction of the wavefront and creates the circular images on the left side of the corresponding parts of Figure 12. These circular imagesrepresent some form of Huygen’s wavelets that are used to propagate wavefronts. The right side of these figures are the wavefront that are reconstructed from the Huygen’s wavelets. This was achieved by assuming source points were positioned at the original location of the wavefront (i.e. a continuous line of samples) and the resulting Huygen’s wavelets summed.a) b) FIG 10: Two wavefront responses from a Gaussian shaped wavelet at the centre with different initial conditions; with a) the initial wavelets at the same location, and b) when the second initial wavelet is shifted to the left with the medium’s velocity.a) b) c) d) FIG 11: Wavelet sources with a) the full circular wavelet, b) truncated to five points wide, c) three points wide and d) one point wide.a)b)c)d)FIG 12: Huygen’s wavefronts on the left side of each image when the source wavelet is: a) circular; b) 5 points wide; c) 3 points wide; and d) 1 sample wide. The right side is the corresponding reconstruction of the wavefront from an array of source points.FIG 13: Reconstructed wavefront and Huygen’s wavelet approximation when a single sample spike is propagated as part of a wavefront.The results of using a single sample for excitation is illustrated in Figure 13. Aliasing in the direction of wavefield propagation causes excessive dispersion of the energy.The non-aliased circular/Gaussian wavelet of Figure 11a was used to demonstrate the formation of the Huygen’s wavelet. This is illustrated in Figure 14, which shows the propagating wavefields at time intervals of 5, 10, 20, 30, 40, 50, 60, and 80. Note the formation of the phase-shift as the amplitudes become negative, and that some energy is propagated in the reverse direction.The Huygen’s wavelet energy lies on a circular path that extends in all directions around the excitation point. The amplitude tapers to zero in the direction opposite the excitation direction, and then to a maximum in the direction of excitation or propagation. Figure 15 is a side view of the Huygen’s wavelet that shows the peak amplitude has a linear slope that rises in the direction of propagation. This amplitude is described in theory as 1+cosine(q), where q is the angle from the direction of propagation. Also note that the phase-shift appears to be 45 degrees.COMMENTSThe images in Figure 10 illustrate the difference in the energy radiated from an isotropic point source and that of an element on a wavefront.The small portion of MATLAB code that propagates the 2D energy is included asAppendix B.FIG 14: Formation of a Huygen’s wavelet at 5,10, 20, 30, 40, 50, 60, 80 time units.FIG 15: Side view of Huygen’s wavelet displaying the amplitude.CONCLUSIONSWave propagation can be illustrated using simple programs written and displayed in MATLAB.ACKNOWLEDGEMENTSWe acknowledge NSERC and the CREWES sponsors for their continued support.APPENDIX AMATLAB code for modelling 2D data. The two parts encircled contain the wave propagation code and the function call to plot the view of the full movement of the waveform.% Wave on stringclearv = 1000.0; % Velocitydx = 1.0; % x incrementdt = 0.001; % time incrementnx = 100; %Number of x samplesnt = 100; %Number of z samplesp = zeros(nt,nx); % Matrix for strings1 = zeros(nx); % String at time 1s2 = zeros(nx); % String at time 2s40= zeros(nx); % String at time 50xary=zeros(nx); % Plot axis% Define the position of the string at the first two times for ix = -10:10p(1, ix+20 ) = 100*exp(-(ix^2 )/16.0 );p(2, ix+21 ) = 100*exp(-(ix^2 )/16.0 );end% Loop for each time increment: limit time to prevent distortion due to boundary reflectionfor it = 3: nt-55for ix = 2:99p(it,ix) = 2*p(it-1,ix) - p( it-2, ix) +((v*dt/dx)^2)*(p(it-1,ix-1)-2*p(it-1,ix)+p(it-1,ix+1)); endend% Get singel copies of the two excitation arrays and one latter array.for ix = 1:nxxary(ix)=ix;s1(ix) = p(1, ix);s2(ix) = p(2, ix);s40(ix) = p(40, ix);end% Plot the datafigure (1); plot(xary,s2,'r--',xary,s1,'b','LineWidth',3); xlabel('x','FontSize',20), ylabel('Amp','FontSize',20)title(' \it{ Wave on string}', 'FontSize',20)figure (2); plot(xary,s40,'g:',xary,s2,'r--',xary,s1,'b','LineWidth',3);xlabel('x','FontSize',20), ylabel('Amp','FontSize',20)title(' \it{ Wave on string}', 'FontSize',20)figure(3); mesh( p);xlabel('x','FontSize',20), ylabel('t','FontSize',20),zlabel('Amp','FontSize',20)title(' \it{ Wave on string}', 'FontSize',20)a) b)c)FIG A1: MATLAB windows with results of the 1D modelling code showing: a) the initial conditions on a string; b) initial conditions on a string and the wave after 40 time increments; and c) all 40 time increments on the string.APPENDIX BPortion of MATLAB code for modelling 3D data that propagates the wave.%********************************************************* % Compute each time layerfor it = 3:ntdit% Compute each x tracefor ix = 2:nx-1%Compute each sample in tracefor iz = 2:nz-1p = vol(ix, iz, it-1);ptm1 = vol(ix, iz, it-2);pzm1 = vol(ix, iz-1, it-1);pzp1 = vol(ix, iz+1, it-1);pxm1 = vol(ix-1, iz, it-1);pxp1 = vol(ix+1, iz, it-1);%solve wave-equationptp1 = 2.0*p - ptm1 + V^2*dt^2*( (pxm1 -2*p+pxp1)/dx^2 + (pzm1 -2*p + pzp1)/dz^2 );vol(ix, iz, it) = vol(ix, iz, it) + ptp1;endend%********************************************************。

2024-2025学年北京版小学六年级英语上册暑假预习试题及答案

2024-2025学年北京版小学六年级英语上册暑假预习试题班级：____________________ 学号：____________________ 姓名：____________________一、听力题（每题3分）Question 1:Recording:“Good morning, everyone. Today is a beautiful day. The weather is sunny and warm. How about your day?”Question: What is the weather like today?A. Cloudy.B. Sunny and warm.C. Rainy.D. Windy.Answer: BQuestion 2:Recording:“Look at that girl. She is my cousin, Alice. She’s very good at playing the piano.”Question: Who is Alice?A. The speaker’s sister.B. The s peaker’s cousin.C. The speaker’s friend.D. The speaker’s teacher.Answer: BQuestion 3:Recording:“I usually go to school by bus, but today I’m going to ride my bike. It’s a nice day for a bike ride.”Question: How is the speaker going to school today?A. By car.B. By bus.C. On foot.D. By bike.Answer: DQuestion 4:Recording:“In the park, you can see many flowers. They are red, yellow, and purple. They are very beautiful.”Question: What colors are the flowers in the park?A. Red, green, and blue.B. Red, yellow, and purple.C. Blue, yellow, and pink.D. Orange, white, and purple.Answer: BQuestion 5:Recording:“My favorite subject is science. I like experiments and learning about new things.”Question:What is the speaker’s favorite subject?A. Math.B. Science.C. English.D. Art.Answer: B二、单选题（每题3分）1._______is my brother’s favorite sport.A. SwimB. SwimsC. SwimmingD. SwamAnswer: C2.She_______her homework yesterday evening.A. doesB. didC. doingD. doAnswer: B3.We have_______art lesson every week.A. aB. anC. theD. /Answer: B4.Can you_______the word “happy” in Chinese?A. speakB. tellC. sayD. talkAnswer: C5.What_______your sister_______last Sunday?A. does, doB. did, doesC. did, doD. do, didAnswer: CNote: These questions are designed to help students review and preview the content of the upcoming Grade 6 English textbook. They cover various grammar points and vocabulary usage.三、多选题（每题4分）Question 1:Which of the following are correct ways to express “I enjoy reading books”? (Choose two.)A. I love to read books.B. I like reading books.C. I am enjoying read books.D. I enjoy to read books.E. Reading books is my hobby.Answer: A, BExplanation: Options A and B are both correct ways to express “I enjoy reading books.” Option C has a grammatical error, “read” should be in the gerund form “reading.” Option D uses the incorrect phrase “enjoy to,” as “enjoy” is followed by a gerund or noun. Option E is also correct, but it is expressed as a statement rather than a direct expression of enjoyment.Question 2:Select the two sentences that contain an adjective clause.A. The book that I borrowed from the library is very interesting.B. I saw the movie last night.C. The girl who is standing over there is my friend.D. He plays basketball every day.E. The cat which is black and white is mine.Answer: A, CExplanation:Options A and C contain adjective clauses. In A, “that I borrowed from the library” modifies “the book” and describes which book is being referenced. In C, “who is standing over there” modifies “the girl” and provides additional information about her. Options B, D, and E do not contain adjective clauses.Question 3:Which two phrases have the same meaning as “at the end of the day”?A. FinallyB. In conclusionC. To begin withD. EventuallyE. At the startAnswer: A, DExplanation:Phrases “Finally” and “Eventually” have the same meaning as “at the end of the day,” which signifies the conclusion or ultimate result of something. Option B, “In conclusion,” also means to sum up, but it is more formal and often used in writing. Options C and E refer to the beginning, not the end.Question 4:Choose the two sentences that use the present perfect tense correctly.A. I have studied English for five years.B. She studies English every day.C. They have been living in this city since 2010.D. He is studying for his exam now.E. We have finished our homework.Answer: A, CExplanation:Options A and C use the present perfect tense correctly. In A, “I have studied English for five years” expresses a completed action that started in the past and continues to the present. In C, “They have been living in this city since 2010” expresses a continuous action that began in the past and continues to the present. Options B, D, and E use different tenses: present simple, present continuous, and present perfect simple, respectively.Question 5:Identify the two sentences that are passive voice.A. The teacher explained the lesson clearly.B. The book was given to me by my friend.C. He wrote a letter to his mother.D. The cat was playing with the ball.E. The report has been completed.Answer: B, EExplanation:Options B and E are in passive voice. In B, “The book was given to me” has the s ubject “the book” receiving the action of “was given.” In E, “The report has been completed” has the subject “the report” being acted upon by the verb “has been completed.” Options A, C, and D are in active voice, with the subjects performing the actions.四、阅读理解（每题4分）Title: The Power of ReadingReading is a powerful tool that can open up a world of knowledge and imagination. It has the ability to transport us to faraway places, introduce us to new ideas, and help us understand ourselves and others better. As children grow from elementary school to middle school, the importance of reading becomes even more apparent.In elementary school, reading is often focused on building basic literacy skills like phonics, vocabulary, and comprehension. However, as students enter middle school, the demands on their reading skills increase. They are expected to read more complex texts, analyze information critically, and make inferences based on what they read.One of the best ways to improve reading skills is through regular practice. By reading a variety of materials, students can expand their vocabulary, learn new grammar structures, and develop a deeper understanding of the world. Additionally, reading for pleasure can also be beneficial as it fosters a lifelong love of learning and encourages students to seek out new information.Not only does reading improve academic performance, but it also has numerous benefits for personal development. It can help build empathy, enhance critical thinking skills, and promote self-awareness. By reading about different cultures and perspectives, students can gain a broader understanding of the world and learn to appreciate diversity.Questions:1.What is the main idea of the passage? (4 points)Answer: The passage discusses the power of reading and its importance for students as they transition from elementary school to middle school.2.What are some of the basic literacy skills that are focused on in elementary school? (4points)Answer: Phonics, vocabulary, and comprehension.3.How do the demands on reading skills change in middle school? (4 points)Answer: The demands increase, expecting students to read more complex texts, analyze information critically, and make inferences.4.What are some of the benefits of reading for pleasure? (4 points)Answer: It fosters a lifelong love of learning and encourages students to seek out new information.5.How can reading help with personal development? (4 points)Answer: It can help build empathy, enhance critical thinking skills, and promote self-awareness. By reading about different cultures and perspectives, students can gain a broader understanding of the world and learn to appreciate diversity.五、作文（30分）Title: My Favourite TeacherInstructions: Write a composition about your favorite teacher. Describe their appearance, personality, and why they are your favorite. Use your imagination and creativity to make your composition interesting and engaging.My Favourite TeacherIn the vast ocean of knowledge, I have encountered many teachers who have guided me through the waves of learning. Among them all, Mrs. Smith stands out as my favorite teacher.Mrs. Smith is a woman of elegance and wisdom. Her tall figure is always dressed in simple yet elegant attire, giving her an air of authority yet warmth. Her eyes sparkle with intelligence and kindness, always ready to lend an ear to our queries and concerns.Her personality is as captivating as her appearance. She is strict yet fair, demanding excellence from us but understanding our limitations. She encourages us to think critically and challenges us to go beyond our comfort zones. Her sense of humor often breaks the monotony of the classroom, turning even the most mundane topics into interesting conversations.The reason Mrs. Smith is my favorite teacher is not just because of her teaching skills or her personality. It is her dedication to her students that truly sets her apart. She goes beyond the syllabus, introducing us to new ideas and perspectives. She cares deeply about our progress, not just academically but also personally. She is always there to offer advice, guidance, and a shoulder to lean on.In conclusion, Mrs. Smith is not just a teacher; she is a mentor, a friend, and a role model. Her influence on me has been immense, shaping me into the person I am today. I am grateful to have had such a wonderful teacher in my life and I aspire to be as dedicated and passionate about my work as she is.Note: This is a sample composition for students to refer to while writing their own compositions. Students are encouraged to use their own imagination and creativity to write unique compositions.。

自动化专业英语复习资料

Translate the following sentences into Chinese。

(2)Although current and voltage are the two basic variables in an electric circuit，they are not sufficient by themselves。

For practical purposes，we need to know how muchpower an electric device can handle。

虽然电流和电压是电路中的两个基本变量，但是它们本身是不足够的。

出于实用的目的，我们需要知道电气设备可以处理多大的功率。

(3)In any linear circuit containing multiple independent sources，the current or voltage at any point in the network may be calculated as the algebraic sum of the individual contributions of each source acting alone。

任一含有多个独立电源的线性电路，某处电压或电流都是电路中各个独立电源单独作用时，在该处分别产生的电压或电流的叠加。

(5)Since an 8-bit code can represent 256 segments，its codes for the same analogvalue are shown with the maximum analog signal of 1.5V equal to 255 。

Notice thatthe 8-bit code is two groups of 4-bit codes，which are also expressed inhexadecimal form。

山东省临沂市2023-2024高一下学期7月期末考试英语试题

山东省临沂市2023-2024高一下学期7月期末考试英语试题Located in the heart of Europe, France offers bike riders its beautiful landscapes, rich history, and delicious food. Here are four of the most popular bike paths in France that show the best of what France has to offer to cycling lovers.The Loire Valley RouteFrom Blois to Angers Length: 800 km Time: 7-10 daysThis bike path is a journey through the royal history of France. It passes by the splendid castles of the Loire Valley, such as Chambord and Chenonceau. The route is mostly flat, making a relaxing ride.The Canal des Deux MersFrom Bordeaux to Toulouse Length: 360 km Time: 4-6 daysThis path connects the Atlantic Ocean with the Mediterranean Sea. Bike riders will experience a mix of city landscapes, vineyards and the peaceful countryside. The path, relatively flat, with a few gentle inclines (斜坡), is well-suited for cyclists of all abilities.The Côte d' Azur RouteFrom Hyères to Menton Length: 230 km Time: 3-4 daysFor those seeking coastal beauty and a touch of attraction, this route offers a ride along the French Riviera. It’s a fairly challenging route with some steep climbs but provides views of the Mediterranean.The Alsace Wine RouteFrom Selestat to Thann Length: 170 km Time: 3-5 daysThis route is a perfect for food lovers, offering a-scenic tour through the vineyards of Alsace. The flat ground provides plenty of opportunities to enjoy local wines and cooking.1. Which route lasts the longest time?A．The Loire Valley Route. B．The Canal des Deux Mers.C．The Côte d' Azur Route. D．The Alsace Wine Route.2. What is unique about the Canal des Deux Mers route?A．It is the most challenging route.B．It satisfies riders of different levels.C．Riders can learn the royal history of France.D．It connects the cities around the Mediterranean.3. If a rider is fond of food, which route will he prefer?A．The Canal des Deux Mers. B．The Loire Valley Route.C．The Côte d' Azur Route. D．The Alsace Wine Route.Within 60 seconds of hearing a bell ring, firefighters are dressed and rushing toward a high-rise on fire. They have to head inside, knowing that floors could fall down under them and roofs could fall on their heads. One wrong decision might mean death.Who would sign up for a job like that? Ask firefighter Sam Telfer. He used to sit in a high-rise office of a software company in San Diego, California. Now he is trained to break into rooms full of blinding smoke and overwhelming heat and come up with a rescue strategy.“I wanted to learn new and practical skills and have a job that varied day by day.” That’s quite an understatement to describe a career that requires you to keep a cool head while attacking a fire. But Telfer says that fires and tension can be conquered in the same way: ongoing physical and mental training.Telfer was a runner in college, but nowadays he builds up his lung ability for life-threatening conditions instead of races. He might climb hundreds of steps, wearing 34 kilograms of gear(装备), including oxygen tanks. He may carry sufferers to safety. No matter what he does, he must control his breathing.His survival will depend on more than body fitness. Firefighters must have sufficient mental preparation. To respond to disasters effectively, they study a wide range of subjects such as emergency (紧急) medicine, structural(结构的) design and chemical reactions. They practice quickly judging the changing conditions of a fire. “It is why we love the job,” Telfer says. “We are continuo usly learning.”Going from being a software specialist to a first responder was quite a transition for Telfer. Never could he have imagined the challenges he would face. But he says, “Not once have I ever felt that it was not the right job for me.” Danger might be waiting as he heads to work each day. But his body and mind are trained to stay calm, even in the roar of a great fire.4. Why did Telfer change his job?A．He would like to challenge himself.B．He wanted to get a high paying job.C．He had been dreaming of being a firefighter.D．He was unfit for working in a software company.5. What do firefighters need to survive the fire?A．They need strong lung ability for races.B．They need to study how to design buildings.C．They need more body health than mental preparation.D．They need mental preparation in addition to body health.6. What can we learn about Telfer from the last paragraph?A．He regrets leaving the software company.B．He has faced many difficulties in his former job.C．He finds his job challenging but still a right choice.D．He once thought of giving up the job as a firefighter.7. What can be a suitable title for the text?A．A Firefighting HeroB．Firefighters To The RescueC．Firefighter: A Challenging JobD．From A Software Specialist To A First ResponderOut of a total of 308, 000 entries in the Oxford English Dictionary, just 366 words begin with the letter “x”, and very few of those make it into most people’s vocabulary. However, the letter is welcomed in popular culture and by brands, such as X-Files, X-Factor, X-man, X-rays and Planet X.The letter “x” has had somewhat puzzling –history, perhaps contributing to its malleable(可塑的) meanings. It first came from a Phoenician letter called “samekh”, meaning fish. The Creeksborr owed it, called it “Chi”, used it for the “kh” sound and wrote it as a cross. The Romans later adopted the “x” sound from Chalcidian alphabet (字母表), a Western variant of the Greek alphabet, and combined it with the “Chi” ( x) symbol and “x” as we know it t oday was born.The tech industry seems to have a particular preference for the letter. This week, the 24th letter of the alphabet was the focus of the spotlight once again, as Elon Musk announced an unexpected rename of Twitter — the social media company h e bought in October 2022 for $44 bn. The site’s iconic(图标的) blue bird logo is no more, replaced by a black letter “X”, part of Musk’s long-term plan to be a “platform that can deliver, well ... everything.” And he also named an early Tesla car Model X, and his space exploration programme Space X.Besides Musk’s X , there’s Microsoft’s gaming console Xbox, Sony’s Xperia smartphone, Apple’s iPhone X and operating system OS X. In 2010, Google chose the letter for the name of its secret idea-testing laboratory, first known as Google x and now simply X.So why do the tech giants prefer “x”, this one single letter, when there are 25 others they could use instead? Tony Thorne, Director of the Slang and New Language Archive at King’s College, London, thought its shape can signify everything from love ( the kiss we used to end our emails and texts), to absence, and even death ( as with the skull and crossbones ). “It’s impressive and people see it as an attempt to appear enigmatic- a symbol of unknowns and possibilities,” he tells BBC Culture.8. What does paragraph 2 mainly talk about?A．How the letter “x” changed the history.B．How the letter “x” was created by the Greeks.C．How the letter “x” developed and came into being.D．How the letter “x” was originally used by Phoenicians.9. Why are other tech companies mentioned in paragraph 4?A．To stress the importance of the letter.B．To list the reasons why they prefer the letter.C．To further show tech companies favor the letter.D．To provide the historical background of the letter.10. What does the unde rlined word “enigmatic” mean in the last paragraph?A．Attractive. B．Mysterious. C．Typical. D．Intelligent.11. What can we know about the letter “x”?A．Its shape seldom appears in our daily life.B．It’s powerful and charming in the tech industry.C．Words that begin with the letter are widely used.D．The sounds of the letter are the same through history.At times we all wish that we knew what other people were thinking. Fortunately, people’s facial expressions, movements and positions can communicate a lot about what is going on in their minds.12 , but Joe Navarro’s book What Everybody is Saying can help develop that ability.According to Navarro, establishing a baseline for how a person behaves is important. 13 . Most facial expressions and positions can mean several different things, so figure out when people use certain expressions. Recognizing this baseline can help you understand whether expressions and positions indicate a reaction to something you said or not.14 . Even when people don’t lie dire ctly, they sometimes try to hide their feelings. Whether you are communicating with people in business or in your personal life, recognizing these feelings can help you look out for their interests and your own.When people think of body language, the first thing that comes to mind is the facial expressions. Some people have become good at controlling their facial expressions to hide their emotions. When dealing with someone like that, look at the rest of their bodies, especially their feet and legs. 15 . However, if someone feels uncomfortable, they may point their feet away from the person they are speaking to.Nearness to other people is part of body language. People stand closer to people they like or feel comfortable with than to strangers or people the y don’t like. 16 . Sitting or standing up straight canshow that someone feels confident, while sitting with crossed arms indicates that they feel defensive or uncomfortable.Body language can communicate a lot about people’s thoughts and feelings, and thu s, help you understand them.A Human Chain, as they called themselves, risking their lives to rescue two boys from drowning(溺水), is still remembered.One sunny afternoon, on the Sunny Beach, two boys, Mike and Jim, walked into the waves without the adults ______ When about 80 yards from the shore, they realized the sea________them further and further. Having tried but________to go back, they waved and screamed for help________, the life guards were off duty.The brothers had been________desperately(拼命地) when a couple, Jane and Alex, wandered by. They didn’t see the boys at first, but th ey_________their screams.Without hesitation, the couple________into the water and soon reached the boys. Jane________the scared boys, telling them they would be safe, only to find they, too, were now________. They couldn’t go back to the _____and could ha rdly hit the sandy bottom with their________.Minutes later, Jane decided they were________in a scaring whirlpool(旋涡).Just then, 42 tourists were walking to them, hand in hand, forming a( an)________. Soon the first person held Jane’s hand. By turns, her h usband and the two boys________her. The long line moved steadily back to the beach.The rescuers called it the Human Chain. And it was the deeply “human” aspect of the rescuers’ team work that made it so amazing and_________.17.A．noticing B．helping C．inspiring D．training18.A．drove B．prevented C．sailed D．pulled19.A．feared B．failed C．chose D．pretended20.A．Definitely B．Unfortunately C．Normally D．Eventually21.A．struggling B．complaining C．surfing D．rushing22.A．described B．copied C．heard D．controlled23.A．limped B．tripped C．jumped D．looked24.A．greeted B．calmed C．blamed D．forgave25.A．in danger B．at ease C．in tears D．on board26.A．dock B．ship C．tent D．shore27.A．arms B．hands C．feet D．sticks28.A．trapped B．gathered C．ignored D．spotted29.A．access B．ladder C．circle D．line30.A．replaced B．hugged C．joined D．assisted31.A．reliable B．joyful C．satisfying D．memorable阅读下面短文，在空白处填入1个适当的单词或括号内单词的正确形式。

KeyTek EMCPro PLUS 高级EMC测试系统用户指南说明书

Analyze •Detect •Measure •Control TMKeyTek EMCPro ®PLUSAdvanced EMC test system forcompliance testing to 6 IEC/EN standardsThe newly configured KeyTek EMCPro ®PLUS test system features resident capabilities for EMC CE Mark compliance testing to 6 IEC/EN standards,and fully addresses new requirements for a 100 kHz burst rate per IEC 61000-4-4, Edition 2 (EFT) and 80%dip per IEC 61000-4-11, Edition 2 (PQF ™).Portable and low cost, the KeyTek EMCPro PLUS is the answer to manufacturers’ demand for a mid-range, multi-capability EMC immunity tester. It’s ideal for companies who require flexibility, versatility, and the highest test level-to-cost ratio instrument on the market.Portable, mid-range EMC test systemResident capabilities for compliance testing to 6 IEC/EN standards Addresses ANSI/IEEE, ITU, ETSI & UL standardsSurge testing to 6.6kV with the combination, telecom, & ring wavesMonitors surge voltage & current at the output terminals Monitors output of the coupling unit & automatically switches connections according to coupling modeHighest test levels, widest selection of tests & lowest in-use costsUpgradable as standards changeTechnical Specifications Model PRO-BASEEMCPro PLUS Base UnitSystem Voltage90-240VAC, 50/60HzINTEGRATED EUT MAINS COUPLER/DECOUPLER AC Voltage 1 phase, 50 - 250VAC. 50/60Hz AC Current 16A max.**DC Voltage 100VDC max.DC Current:10A max.Frequency 50/60Hz EUT Connectors Nema, British, SchukoCONTROL INTERFACE Interface RS232 Fiber-optic SAFETY FEATURESExternal Interlock for usersInterlock for CCL connectorExternal stop input ENVIRONMENTAL OPERATING CONDITIONS Temperature 15°- 40°C Humidity 10-75%, non-condensing Altitude 8000 ft. max.PHYSICAL Height 22.9cm (8.7 in)Width 43.4cm (17.1 in)Depth 64.8cm (25.5 in)Weight 39kg (85 lbs.)CE MARKINGSafety and EMC Directives 1981Model PRO-ESDESD per IEC 61000-4-2 and EN 61000-4-2Trigger Modes One shot manual, multi-shot tripodRepetition Rate Single shot, 1pps or 20ppsAir Discharge Voltage500V - 8.8kV ±10%Contact Discharge Voltage500V - 4.4kV ±10%Discharge Capacitor150pF ±10%Discharge Resistance330Ω±10%Charging Resistance50MΩ- 100MΩPolarity Front panel or software controlledShot Counter 1 - 999 dischargesEnergy Storage***********Model PRO-EFTEFT per IEC 61000-4-4 Edition 2, EN 61000-4-4 and ANSI C62.41Voltage Waveform5/50ns ±30%Peak Voltage250V - 4.4kV ±5%Burst Period300ms ±10%Burst Duration15ms ±20%, for pulse frequencies uo to 5kHz, 0.75msabove 5kHzFrequency1-100kHz, in 0.5kHz steps, ±10%DC Blocking Capacitor10nF (internal)Options Model CM-3CD-16/32:16 or 32 Amp, 3 phase EFT &surge coupler/decouplerModel CM-CCL: Capacitive coupling clampModel CM-CCLC: Coupling clamp coverModel EFT-ATTN:EFT attenuator for oscilloscopemonitoringModel PRO-SURGESurge for compliant testing per IEC 61000-4-5, EN 61000-4-5, ANSI C62.41 Category B and UL 1449Voltage Waveform 1.2/50µsPeak Voltage250V - 6.6kV ±5%, 12Ωmode250V - 6.0kV ±5%, 2ΩmodePeak Current125A - 3.3kA ±10%Additional 10ΩResistor Software selectableRepetition Rate Up to 4 per minuteOpen-circuit Voltage Front time: 1.2µs ±30%Duration: 50 µs ±20%1Undershoot: ≤30%Short-circuit Current Front time: 8.0µs ±20%Duration*50µs ±20%Undershoot≤30%Line sync accuracy±15%, 50 - 277VACOptions ModelCM-3CD-16/32:16 or 32 Amp, 3 phase EFT& surge coupler/decouplerModel CM-I/OCD: External 8 line coupler/decouplerfor I/O signal linesModel CM-I/OCD-HS:High speed I/OCD option fortesting data rates to >100kHz*Durations are reduced in 12Ωmode and when coupling multiple lines to PE Model PRO-RING**Ring Wave Surge per ANSI C62.41 Cat. A, B, and UL 864Voltage Waveform100kHz damped cosinePeak Voltage250 - 6.6kV ±5%Repetition Rate<4/minute at 6kV, faster at lower voltagesOpen-circuit Voltage Rise Time: 0.5µs ±30%Short-circuit Current Vp/Ip: 12Ω±3Ωor 30Ω±8Ωsoftware selectable Options ModelCM-3CD-16/32:16 or 32 Amp, 3 phase EFT& surge coupler/decouplerModel PRO-TELECOM**Surge Telecom compliant testing per IEC 61000-4-5, EN 61000-4-5, FCC Part 68, ITU K.17, K.20, K.21 and ETSIVoltage Waveform10/700µs (9/720µs FCC Part 68)Peak Voltage250V - 6.6kV ±5%Peak Current 6.25 - 165A +10/-0%, 40ΩmodeRepetition Rate Up to 4 per minuteOpen-circuit Voltage Front time: 7.0µs to 11.7µsDuration: 576µs to 840µsShort-circuit Current Front time: 3.5µs to 6.5µsDuration: 256µs to 384µsOptions Model CM-TELCD: External coupler for telecom linesSurge Waveform MonitoringLines Monitored Monitors are automatically switched to matchgenerator coupling modeOpen-circuit Voltage1000:1 ±10%Short-circuit 200:1 ±7%Current AttenuationModel PRO-HPOWERPower Frequency Magnetic Field for compliant testing per IEC 61000-4-8 and EN 61000-4-8 Field Frequency50Hz/60HzField Amplitude0.5 - 4A/m, in 0.25A steps, ±10% (with CM-HCOIL)up to 100A/m with optional external HPOWER-EXT AC Source InternalResolution0.25A minimumCoil Factor0.65 to 1.00Coil Resistance0.05ΩmaximumOptions ModelCM-HMON:Measurement probe for powerfrequency magnetic fieldsModel CM-HCOIL: 1m x1m magnetic field coilModel HPOWER - EXT: External generator for powerfrequency magnetic field to 30A/m**PRO-TELECOM and PRO-RING can not be installed in same unit.Model PRO-HPULSEPulse Magnetic Field for compliant testing per IEC 61000-4-9 and EN 61000-4-9Field Pulse8/20µsField Amplitude100A/m - 1000A/m, ±10%Resolution5A/mCoil Factor0.65 to 1.00Options ModelCM-HMON:Measurement probe for powerfrequency magnetic fieldsModel CM-HCOIL: 1m x 1m magnetic field coilModel PRO-PQFDips and Interrupts for compliant testing IEC 61000-4-11 Edition 2, and EN 61000-4-11 Dips40%, 70%, 80%Interrupts0% (short and open)Transition Time1µs - 5µsInrush Minimum 250Amps @ 100 - 120V,Minimum 500Amps @ 220 - 240VAC Voltage50 - 250VAC, 50/60HzAC Current16A max.**PQF Sync Output5V signal occurs at each dip or interrupt transition Options Model PQF-QUAL:Circuit per IEC 61000-4-11 fortesting PQF generator inrush capabilityPQF Waveform MonitoringVoltage Input Connection Fixed, L1 to L2Voltage Attenuation100:1 ±5%Current Input Connection Fixed, L1Peak Current Minimum 500A inrush into 1700µFCurrent Attenuation200:1 ±5%Model CM-3CD-16 & CM-3CD-32*Semi-automatic, stand alone, three-phase AC/DC mains coupler/decouplers for EFT & Surge per IEC 61000-4-4, Edition 2 and IEC 61000-4-5ELECTRICALWaveforms EFT: 5/50ns, per IEC 61000-4-4Surge: Combination wave: 1.2/50µs open-circuitvoltage, 8/20µs short-circuit current, per IEC 61000-4-5 Maximum Surge 6.6kV, 3.3kAVoltage & CurrentMaximum EFT Voltage 4.4kVCoupling Modes EFT: L1, L2, L3, N or PESurge Hi: L1, L2, L3 or NSurge Lo: L1, L2, L3, N or PE* Not available for delivery until October 2004COUPLER/DECOUPLERSAC Voltage50 to 250V, 50/60Hz line to ground, 50 to 433Vline to lineAC Current CM-3CD-16:16A/phase continuousCM-3CD-32:32A/phase continuousDC Current CM-3CD-16:16A up to 48V8A up to 110V1.2A up to 220V0.3A up to 440VCM-3CD-32:25A up to 48V8A up to 220V1.2A up to 220V0.3A up to 440VEUT Mains Safety SocketsOutput ConnectorsPOWER REQUIREMENTSInput Voltage90-250VAC, 50/60HzInput Current1A at 120VAC; 0.5A at 240VACModel CM-I/OCDI/O coupler/decoupler - provides the ability to couple surges from EMCPro PLUS or any surge simulator, to I/O or data lines per IEC 61000-4-5ELECTRICALWaveforms Designed to couple combination waves of 1.2/50µsopen-circuit voltage, 8/20µs short-circuit currentsupplied by option PRO-SURGE with the KeyTekEMCPro PLUSRepetition Rate Up to 5 per minute at 4.4kVData Line Frequency To greater than 100kHz without significant degradationwhen CM-I/OCD-HS is installed. Option CM-I/OCD-HSis recommended for data line frequencies greater than1kHzNumber of Lines Eight lines - any line can be surged to any other line orgroundMaximum Surge Voltage 4.4kVMaximum Signal Line Voltage200VMaximum Signal Line Current1A AC or DCClamping Selectable built-in clamps of 20V and 220V; externalbias input for other clamp levelsAvailable Options CM-I/OCD-HS:Internally-installed option providesselectable parallel resistors (400s, 200s, 100s) - highlyrecommended for data line frequencies greater than1kHz.Model CM-TELCDTelecom line coupler/decoupler - provides the ability to couple both the telecom wave and combination wave per IEC 61000-4-5ELECTRICALWaveforms Designed to couple 1.2/50µs combination or 10/700µstelecom wavesTelecom Line Frequency To 100kHz without significant degradationNumber of Lines Up to four lines - one or two pairs of balanced TelecomlinesMaximum Surge Voltage 4.4kVMaximum Signal Line Voltage200VMaximum Signal Line Current1A AC or DCClamping Selectable built-in clamps of 20V and 225V: externalbias input for other clamp levelsExperience the many benefits of working with recognized experts in the field of EMC (Electromagnetic Compatibility)testing. Our commitment to the discipline is wide ranging; we actively participate on global standards committees, andhave helped define test methodologies to achieve regulatory standards such as CE Mark requirements, as well ascompany and market-driven product quality objectives,.Our goal is to support you with lifelong service — from applications support, calibration services and preventativemaintenance scheduling to full tactical field support.Thermo can help you reach the next level of success.Please see the KeyTek EMC Test System Options & Accessories data sheet for additional KeyTek EMCPro PLUS testsystem options and accessories.Specialists who understand the challenges you face. Innovative ideas. Leading technologies. Breadth of EMC test equipment.Thermo–your EMC test solutions partner. Contact us today for details.This sheet is for informational purpose only and is subject to change without notice.© 2004 Thermo Electron Corporation. All rights reserved. Thermo Electron Corporation, Question everything, and Analyze. Detect. Measure. Control.are trademarks of Thermo Electron Corporation.Control Technologies Division One Lowell Research Center(978) /esdLowell, MA, 01852(978) 275-0850 faxFormerly doing business as Thermo KeyTek。

Rotating waves in the Theta model for a ring of synaptically connected neurons

a r X i v:n lin/04529v3[nli n.PS]3J un24ROTATING W A VES IN THE THETA MODEL FOR A RING OF SYNAPTICALLY CONNECTED NEURONS GUY KATRIEL Abstract.We study rotating waves in the Theta model for a ring of synaptically-interacting neurons.We prove that when the neurons are oscillatory,at least one rotating wave always exists.In the case of excitable neurons,we prove that no travelling waves exist when the synaptic coupling is weak,and at least two rotating waves,a ‘fast’one and a ‘slow’one,exist when the synaptic coupling is suﬃciently strong.We derive explicit upper and lower bounds for the ‘critical’coupling strength as well as for the wave velocities.We also study the special case of uniform coupling,for which complete analytical results on the rotating waves can be achieved.1.introduction In this work we study rotating waves in rings of neurons described by the Theta model.The Theta model [2,3,5,6],which is derived as a canonical model for neurons near a ‘saddle-node on a limit cycle’bifurcation,assumes the state of the neuron is given by an angle θ,with θ=(2l +1)π,l ∈Z corresponding to the ‘ﬁring’state,and the dynamics described by (1)dθ∂t =1−cos(θ(x,t ))+(1+cos(θ(x,t ))) β+g ΩJ (x −y )s (y,t )dy,(3)∂s (x,t )Partially supported by the Edmund Landau Center for Research in Mathematical Anal-ysis and Related Areas,sponsored by the Minerva Foundation (Germany).12GUY KATRIELwhere J is a positive function and P is deﬁned by∞ l=−∞δ(θ−(2l+1)π).(4)P(θ)=Here s(x,t)(x∈Ω,t∈R)measures the synaptic transmission from the neuron located at x,and according to(3),(4)it decays exponentially,except when the neuronﬁres(i.e.whenθ(x,t)=(2l+1)π,l∈Z),when it experiences a jump.(2)says that the neurons are modelled as Theta-neurons,where the input I(x,t)to the neuron at x,as in(1),is given byI(x,t)=g ΩJ(x−y)s(y,t)dy.J(x−y)(here assumed to be positive)describes the relative strength of the synaptic coupling from the neuron at x to the neuron at y,while g>0is a parameter measuring the overall coupling strength.The above model,in the case c>0,is the one presented in[2,7].In the case c=0this model is the one presented in[6](Remark2)and[9].We always assume c≥0.When the geometry is linear,Ω=R,it is natural to seek travelling waves of activity along the line in which each neuron makes one or more oscillations and then approaches rest.In[8]it was proven that for suﬃciently strong synaptic coupling g,at least two such waves,a slow and a fast one,exist,and also that they always involve each neuronﬁring more than one time before it approaches rest,while for suﬃciently small g such waves do not exist.It was not determined how many times each neuronﬁres before coming to rest,and it may even be that each neuronﬁres inﬁnitely many times.Some numerical results in the case of a one and a two-dimensional geometry were obtained in [7].In this work we consider a diﬀerent possibility for the spacial geometry:Ω=S1,so the neurons are placed on a ring and our equations are(3)and∂θ(x,t)(5)ROTATING WA VES IN THETA NEURONS3(10)s(x+2π,t)=s(x,t)∀x,t∈RThe integer m(the‘winding number’)is determined by the initial condition θ(x,0),and will be preserved as long as the solution remains continuous.In this geometry,a diﬀerent kind of wave of activity is possible:a wave that rotates around the ring repeatedly.Such waves,that is solutions of the form:(11)θ(x,t)=φ(x+vt)(12)s(x,t)=r(x+vt)where v is the wave velocity,are the focus of our investigation.In section2we show that in the case that the winding number m=0, there can exist only trivial rotating waves.Thus the interesting cases are when m>0.Here we study the case m=1,the case m>1being beyond our reach.Thus,this work concentrates on theﬁrst non-trivial case.Our central results about existence,nonexistence and multiplicity of ro-tating waves can be summarized as follows(seeﬁgures1,2for the simplest diagrams consistent with these results):Theorem 1.Consider the equations(5),(3)with conditions(9),(10),and m=1.(I)In the oscillatory caseβ>0:for all g>0there exists a rotating wave, with velocity going to+∞as g→+∞.(II)In the excitable caseβ<0:(i)For g>0suﬃciently small there exist no rotating waves.(ii)for g suﬃciently large there exist at least two rotating waves,a‘fast’and a ‘slow’one,in the sense that their velocities approach+∞and0,respectively, as g→+∞.Therefore our results bear resemblance to those obtained in[8]for the case of a linear geometry.We note that although for the rotating waves found here each neuronﬁres inﬁnitely many times,the reason for this is that it is re-excited each time,because of the periodic geometry.During each revolution of the rotating wave,each neuronﬁres once,so naively one could think that the analogous phenomenon in a linear geometry would be a travelling wave with each neuronﬁring once-but this was shown to be impossible in[8]. It is interesting to note that while in[8]some restrictions were made on the coupling function J,like being decreasing with distance,here no such restrictions are imposed beyond(7),(8).We would expect however that some restriction would need to be imposed on J in order to obtain stability of travelling waves.The whole issue of stability remains quite open and awaits future investigation.In the caseΩ=R,both numerical evidence in[7,8]4GUY KATRIELand results obtained in other models[1]indicate that the fast wave is stable while the slow wave is unstable,so we might conjecture that this is true for the case investigated here as well-at least under some natural assumptions on J.Some analytical progress on the stability question in the caseΩ=R has recently been achieved in[9].Let us note that the model considered here,in the caseβ<0,describes waves in an excitable medium,about which an extensive literature exists(see [10]and references therein).However,most models consider diﬀusive rather than synaptic coupling.In the case of the Theta model on a ring,with diﬀusive coupling,and m=1,it is proven in[4]that a rotating wave exists regardless of the strength of coupling(i.e.the diﬀusion coeﬃcient),so that our results highlight the diﬀerence between diﬀusive and synaptic coupling.In section3we reduce the study of rotating waves to the investigation of the zeroes of a function of one variable.In section4we investigate the special case in which the coupling is uniform(J(x)is a constant function), which,although artiﬁcial from a biological point of view,allows us to obtain closed analytic expressions for the wave-velocity vs.coupling-strength curves in an elementary fashion.We can thus gain some intuition for the general case,and obtain information which is unavailable in the case of general J, like precise multiplicity results.It is interesting to investigate to what extent the more precise results obtained in the uniform-coupling case extend to the general case,and we shall indicate several questions,which remain open,in this direction.In section5we turn to the case of general coupling functions J,and prove the results of theorem1above,obtaining also some quantitative estimates:lower and upper bounds for the critical values of synaptic coupling coupling strength g,as well as for the wave velocities.2.preliminariesWe begin with an elementary calculus lemma which is useful in several of our arguments below.Lemma2.Let f:R→R be a diﬀerentiable function,and let b,c∈R,b=0, be constants such that we have the following property:(13)f(z)=c⇒f′(z)=b.Then the equation f(z)=c has at most one solution.proof:Assume by way of contradiction that the equation f(z)=c has at least two solutions z0<z1.Deﬁne S⊂R byS={z>z0|f(z)=c}.S is nonempty because z1∈S.Let z)=c.We have either z=z0,and we shall show that both ofROTATING WA VES IN THETA NEURONS5 these possibilities lead to contradictions.If z)=csign(f′(z0))=sign(f′(z)with f(z2)=c,contradicting the deﬁnition of z=z0then z0is a limit-point of S,which implies that f′(z0)=0,contradicting(13).These contradictions conclude our proof.Turning now to our investigation,we note a few properties of the functions h(θ)and w(θ)deﬁned by(6)which will be used often in our arguments: (14)h((2l+1)π)=2∀l∈Z,(15)h(2lπ)=2β,∀l∈Z,(16)w((2l+1)π)=0∀l∈Z,(17)w(2lπ)=2∀l∈Z,Plugging(11),(12)into(5),(3),and setting z=x+vt we obtain the following equations forφ(z),r(z):(18)vφ′(z)=h(φ(z))+gw(φ(z)) π−πJ(z−y)r(y)dy,(19)vr′(z)+r(z)=P(φ(z))(1−cr(z)).In order to satisfy the boundary conditions(9),(10),φand r have to satisfy(20)φ(z+2π)=φ(z)+2πm∀z∈R,(21)r(z+2π)=r(z)∀z∈R.Let usﬁrst dispose of the case of zero-velocity waves,v=0.We get the equations(22)h(φ(z))+gw(φ(z)) π−πJ(z−y)r(y)dy=0,(23)r(z)=P(φ(z))(1−cr(z)).If there exists some z0∈R withφ(z0)=(2l+1)π,l∈Z,then,substituting z=z0into(22)and using(14),(16),we obtain2=0,a contradiction.Hence we must have(24)φ(z)=(2l+1)π∀z∈R,l∈Z6GUY KATRIELwhich implies that P(φ(z))≡0,so that(23)gives r(z)≡0,and(22)reduces to h(φ(z))≡0,and thusφ(z)is a constant function,the constant being a root of h(θ).This implies,ﬁrst of all,that the winding number m is0,since a constantφ(z)cannot satisfy(20)otherwise.In addition the function h(θ) must vanish somewhere,which is equivalent to the conditionβ≤0.We have thus provenLemma3.Zero-velocity waves exist if and only if m=0andβ≤0,and in this case they are just the stationary solutionsr(z)≡0φ(z)≡±cos−1 β+1.vThus the assumptions of lemma2,with f=φ,c=(2l+1)π,b=2ROTATING WA VES IN THETA NEURONS7 Having found all possible rotating waves in the case m=0,we can now turn to the case m>0.In fact,as was mentioned in the introduction,we shall treat the case m=1,the cases m>1being harder.By lemma3we know that there are no zero-velocity waves,so we can assume v=0and deﬁne1(27)λ=8GUY KATRIELBy lemma2,(35)implies that the equationφ(z)=2l+1has at most one solution for each l∈Z.In particular,sinceφ(0)=π,φ(2π)=3πwe have φ(z)=π,3πfor z∈(0,2π),and by continuity ofφ(z)this implies(33).Let us note that if we knew that for rotating waves the functionφ(z)must be monotone,then lemma5would follow immediately from(32).Question6.Is it true in general that rotating wave solutions are monotone (for m¿0)?Lemma7.Assume(λ,φ,r)satisfy(28),(29)with conditions(30),(31),(32). Thenλ>0.In other words,v>0for all rotating waves with m=1,so the waves rotate clockwise.Of course in the symmetric case m=−1the waves will rotate counter-clockwise.proof:By(32)and(35)we haveφ′(0)=2λ.We have already noted that λ=0.Ifλwere negative,thenφwould be decreasing near z=0,so for small z>0we would haveφ(z)<0,contradicting(33).Our next step is to solve(29),(31)for r(z),in terms ofφ(z).We will use the following important consequence of lemma5:Lemma8.P(φ(z))|(−2π,2π)=12λδ(z).Letχ(z)∈C∞0(R)be a test ing lemma5again we have (37) π−πχ(u)δ(φ(u)−π)du= ǫ−ǫχ(u)δ(φ(u)−π)du,whereǫ>0is arbitrary.In particular,sinceφ′(0)=2λ>0,we may choose ǫ>0suﬃciently small so thatφ′(z)>0for z∈(−ǫ,ǫ),so that we can make a change of variablesϕ=φ(u),obtainingǫ−ǫχ(u)δ(φ(u)−π)du= φ(ǫ)φ(−ǫ)χ(φ−1(ϕ))δ(ϕ−π)dϕφ′(φ−1(π))=χ(0)2λ.(38)ROTATING WA VES IN THETA NEURONS9 This proves(36),completing the proof of the lemma.By lemma8we can rewrite equation(29)on the interval(−2π,2π)as(39)r′(z)+(λ+c2δ(z),The solution of which is given by(40)r(z)= 12H(z)),where H is the Heaviside function:H(z)=0for z<0,H(z)=1for z>0. Substituting z=πinto(40)and using(31),we obtain an equation for r(−π)whose solution isr(−π)=12−e−πλ)−1,and substituting this back into(40),we obtain that the solution of(29),(31) which we denote by rλ(z)in order to emphasize the dependence on the pa-rameterλ,is given on the interval(−2π,2π)by(41)rλ(z)=12H(z)) H(z)+(e2πλ+c2λρc(λ),whereρc(λ)=e2πλ−12−1.We note that(43)ρ0(λ)≡1,A fact that considerably simpliﬁes the formulas in the case c=0.The rotating waves correspond to solutions(λ,φ)of the equation (44)φ′(z)=λh(φ(z))+λgw(φ(z)) π−πJ(z−y)rλ(y)dy,withφ(z)satisfying(32)and(45)φ(π)=φ(−π)+2π.To simplify notation,we deﬁne(46)Rλ(z)= π−πJ(z−y)rλ(y)dy,10GUY KATRIELso that (44)is rewritten as (47)φ′(z)=λh (φ(z ))+λgR λ(z )w (φ(z )).We note that (47)is a nonautonomous diﬀerential equation for φ(z ),and since the nonlinearities are bounded and Lipschitzian,the initial value problem(47),(32)has a unique solution,which we denote by φλ.Rotating waves thus correspond to solutions λ>0of the equation(48)φλ(π)−φλ(−π)=2π.Rewriting (47)and (50)we have(49)φ′λ(z )=λh (φλ(z ))+λgR λ(z )w (φλ(z )),(50)φλ(0)=π,and deﬁning(51)Ψ(λ)=12λρc (λ),so that (49)reduces to(53)φ′λ(z )=λh (φλ(z ))+g 2ππ−πφ′(z )dz 2ρc (λ)w (φ(z ))=1λh (ϕ)+g 2π π−πdϕ2ρc (λ)w (ϕ).(55)ROTATING WA VES IN THETA NEURONS11 Substituting the explicit expressions for h and w from(6),and using the formula(56)1A+Bcos(φ)=1A2−B2(|A|>|B|),(55)becomes(57)1=14λ2β+2gλρc(λ),so that rotating waves correspond to solutions of(57),with their velocities given by v=12λρc(λ).In the following lemma we collect some properties of the functions f c,β(λ), which are obtained by elementary calculus:Lemma9.(i)Whenβ<0,f c,βis positive and convex on(0,∞),and (60)limλ→0f c,β(λ)=∞,(61)limλ→∞f c,β(λ)=∞.(ii)Whenβ≥0,f c,βis decreasing on(0,∞),and(60)holds.Ifβ>0it has a zero atλ=1β,ifβ=0it is positive on(0,∞)and limλ→∞f c,0(λ)=0.From lemma9we conclude that whenβ<0(58)has exactly two solutions if g>Ω(c,β),where(62)Ω(c,β)=minλ>0f c,β(λ),which we will denote byλλc,β(g),no solution if g<Ω(c,β),and a unique solution when g=Ω(c,β).Whenβ≥0,part(ii)of lemma9implies that(58)has a unique solution for any g>0,which we denote byλc,β(g).An elementary asymptotic analysis of the equation(58)yieldsLemma10.(i)Whenβ<0we have the following asymptotics as g→∞(63)2g as g→∞.12GUY KATRIELFor λc,β(g )=112−1)1g +O10,β(g )=1g+O1c,β(g ),given in (64)for c >0and (65)for c =0.We thus obtainTheorem 11.When J ≡1:(I)In the excitable case β<0:(i)If g >Ω(g,c )there exist two rotating waves with velocities given by (66)vv c,β(g )=1c,β(g ),and we have,for the slow wave (67)v21g 3as g →∞,for the fast wave when c >0:(68)π2−1√v 0,β(g )=2g +O1c,β(g )=λc,β(g ),and for large g it has the same asymptotics as in (68),(69)in the cases c >0,c =0,respectively.In the excitable case we thus have two rotating waves born at a supercritical saddle-node bifurcation as the coupling strength g crosses Ω(c,β).ROTATING WA VES IN THETA NEURONS 13246810v12345678gFigure 1.Velocity of waves (v )vs.coupling strength (g )for the case J ≡1,c =0,β=−0.5.We now note that in the special case c =0(the model introduced in [6])we can obtain more explicit ing (43)we havef 0,β(λ)=1−4βλ2|β|.We can also solve (58)explicitly,and obtain the velocities of the rotating waves.When β<0,g >Ω(0,β)vg 2+4β,g 2+4β.When β≥0,for all g >0v 0,β(g )=14GUY KATRIEL246810v12345678gFigure 2.Velocity of waves (v )vs.coupling strength (g )for the case J ≡1,c =0,β=0.5.246810v12345678gFigure 3.Velocity of waves (v )vs.coupling strength (g )for the case J ≡1,c =1,β=−0.5.ROTATING WA VES IN THETA NEURONS 15246810v12345678gFigure 4.Velocity of waves (v )vs.coupling strength (g )for the case J ≡1,c =1,β=0.5.5.The general caseWe now return to the case when J is a general continuous positive 2π-periodic function,and prove that several of the results about rotating waves obtained above for the special case J ≡1remain valid,though the proofs are necessarily less direct.Lemma 12.lim λ→0Ψ(λ)=0.proof:We shall prove that (72)φλ(z )=π+O (λ)as λ→0uniformly in z ∈[−π,π].The lemma follows immediately from this and from (51).When c >0,the claim (72)is immediate,since,using (41),lim λ→0r λ(z )=12H (z ) (e c16GUY KATRIELFor c=0,rλ(z)becomes singular asλ→0,so we need a more reﬁned argument.Forλrλ(z)we have(74)λrλ(z)=12J=1J2λρc(λ)minx∈RJ(x)≤Rλ(z)≤1ρc(λ)≥gρc(λ)≥gROTATING WA VES IN THETA NEURONS17(79)φλ(z)>0∀z∈[−π,0].Together with(51),these imply the result of our lemma.To prove our claim we note that,using(49),(15),(17),part(ii)of lemma13(which is why we need the assumption that J is non-constant)and(77)φλ(z)=0or2π⇒φ′λ(z)=2λβ+2λgRλ(z)<2λβ+gρc(λ)maxx∈R J(x)≤2λβ+2λ|β|=0.(80)We now show that(80)implies(78).If(78)fails to hold,then we setz0=min{z∈[0,π]|φλ(z)=2π}.This number is well-deﬁned by continuity and by the fact thatφλ(0)=π, which implies also that z0>0.By(80)we haveφ′λ(z0)<0,but this implies thatφλ(z)is decreasing in a neighborhood of z0,and in particular that there exist z∈(0,z0)satisfyingφλ(z)=2π.But this contradicts the deﬁnition of z0,and this contradiction proves(78).Similarly,assuming(79)does not hold and deﬁningz1=max{z∈[−π,0]|φλ(z)=0},we conclude that z1<0andφ′λ(z1)<0,so thatφλ(z)is decreasing in a neighborhood of z1,and this implies a contradiction to the deﬁnition of z1 and proves that(79)holds.This concludes the proof of the lemma.Sincelimλ→∞ρc(λ)=e−c2∀λ>0,we conclude from lemma14thatLemma15.In the excitable caseβ<0,we haveλ≥ge−c2|β|maxx∈RJ(x)⇒Ψ(λ)<1,Let us note that sinceΨ(λ)<1implies that(52)doesn’t hold,and since v=12g.18GUY KATRIELThe following theorem shows that,in the excitable case and for suﬃciently weak synaptic coupling,there are no rotating waves(so it implies part(II)(i) of theorem1).Theorem17.In the excitable caseβ<0,if g∈(0,g0),whereg0=Ω(c,β)2|β|ρc(λ)maxx∈RJ(x).We deﬁneµ=β+g2λρc(λ)maxx∈RJ(x) (1+cos(φλ(z)))=λ[(µ+1)+(µ−1)cos(φλ(z))].(85)which implies(note that the integral below is well-deﬁned because of(84))π−πφ′λ(z)dz(µ+1)+(µ−1)cos(ϕ)≤2πλ.If we assume,by way of contradiction,that(82)does not hold,i.e.that φλ(π)−φλ(−π)≥2π,then,using(56),φλ(π)φλ(−π)dϕ(µ+1)+(µ−1)cos(ϕ)=πµ,ROTATING WA VES IN THETA NEURONS19so together with(86)we obtain1µ≤2λ,which is equivalent tog≥12λρc(λ),which contradicts g<g0.This contradiction proves(82),concluding the proof of the theorem.We now proceed to prove that in the excitable case when the synaptic coupling is suﬃciently large we have at least two rotating waves(see theorem 20below).Lemma18.In the excitable caseβ<0,if there exists someλ0>0withΨ(λ0)>1,then there exist at least two solutionsλ1,λ2of(52)with0<λ2<λ0<λ1, hence two rotating waves,with velocities satisfyingv1=1λ0<12λfor allλ≥λ′1,and in particular it follows thatλ′1>λ0.Wethus haveλ′2<λ0<λ′1withΨ(λ′2)<1,Ψ(λ0)>1,Ψ(λ′1)<1.Thus by the intermediate value theorem,the equation(52)has a solution λ2∈(λ′2,λ0)and a solutionλ1∈(λ0,λ′1),corresponding to two rotating waves.The following lemma is valid for all values ofβ:Lemma19.Assume thatλ>0satisﬁes the inequality(87)f c,β(λ)<g minx∈RJ(x),where f c,βis deﬁned by(59).ThenΨ(λ)>1.20GUY KATRIELproof:By(51),our claim is equivalent to(88)φλ(π)−φλ(−π)>2π.We deﬁnegη=β+.4λ2Using(49)and lemma13we haveφ′λ(z)=λ[h(φλ(z))+gRλ(z)w(φλ(z))]≥λ 1−cos(φλ(x))+ β+g(η+1)+(η−1)cos(φλ(z))≥2πλ.Making the change of variablesϕ=φλ(z),we obtain (91) φλ(π)φλ(−π)dϕ2π0dϕ√(η+1)+(η−1)cos(ϕ)≤√,min x∈R J(x)whereΩ(c,β)is deﬁned by(62).Then when g>g1,there exist at least two rotating waves.In fact,we have a‘slow’wave with velocity v s bounded from above by(93)v s≤vROTATING WA VES IN THETA NEURONS21 and a‘fast wave’with velocity v f bounded from below by(94)v f≥c,β,2g+O 1πmin x∈R J(x)2−1√g as g→∞.proof:g>g1and(62)imply the existence ofλ>0satisfying(87),hence by lemma19Ψ(λ)>1,so that lemma18implies the existence of two rotating waves.To prove(93),(94),we note that,assuming g>g1,the range of values of λ0for which(87)holds is the intervalλλc,β(g minx∈RJ(x)),where the functionsλλc,βare deﬁned in section4.Thus,applying lemma 18withλ0=λc,β(g min x∈R J(x))−ǫ.Sinceǫ>0is arbitrary,we have a solutionλof(52)withλ≥c,β(g min x∈R J(x))+ǫ,we obtain the existence of a wave with velocity v fsatisfying(94).The estimates(95)-(97)follow from(93),(94)and lemma10.We note that along with the upper bound(95),we have a lower bound for the velocity of the slow wave,given by lemma16.Question21.Derive an upper bound for the velocities of the fast waves(note that(94)gives a lower bound).Question22.Theorems17and20show that several of the qualitative features that we saw explicitly in the case of uniform coupling(section4)remain valid in the general case.It is natural to ask whether more can be said,e.g.,whether22GUY KATRIELthe following conjecture,or some weakened form of it,is true:for any J,there exists a value g crit such that:(i)For g<g crit there exist no travelling waves.(ii)For g=g crit there exists a unique travelling wave.(iii)For g>g crit there exist precisely two travelling waves.The next theorem deals with the oscillatory caseβ>0,as well as the borderline caseβ=0,and in particular proves part(I)of theorem1. Theorem23.Ifβ≥0,there exists a rotating wave solution for any value of g>0,with velocity v bounded from below byJ(x)),(98)v≥v(g minx∈Rwhere v is the function deﬁned by(71),and the asymptotic formulas(96),(97) hold with v f replaced by v.proof:Ifβ>0,then for any g>0the equationJ(x)f c,β(λ)=g minx∈Rhas the unique solutionλc,β(g min x∈R J(x)).Hence,anyJ(x))λ0>λc,β(g minx∈Rsatisﬁes(87),so that by lemma19Ψ(λ0)>1.On the other hand forλsuﬃciently small we have,by lemma12,Ψ(λ)<1.Hence there exists a solutionλ∈(0,λ0)of(52).Sinceλ0>λc,β(g min x∈R J(x))is arbitrary,we conclude that there exists a solutionλ≤λc,β(g min x∈R J(x))of(52).Hence a rotating wave with velocity satisfying(98).Question24.Is it true that in the oscillatory caseβ≥0the rotating wave is always unique?We saw that this is the case when J≡1.Finally,we stress the important question of stability of the rotating waves, which remains open:Question25.Investigate the question of stability of the rotating waves,i.e., do arbitrary solutions of(5),(3)approach one of the rotating waves in large time?We conjecture that,at least under some restrictions on J,the rotating wave is stable in the caseβ>0,while in the caseβ<0the fast rotating wave is stable and the slow one is unstable.References[1]P.C.Bressloﬀ,Travelling waves and pulses in a one-dimensional network of excitableintegrate-and-ﬁre neurons,J.Math.Biol.40(2000),169-198.[2]G.B.Ermentrout,Type I membranes,phase resetting curves and synchrony,NeuralComput.8(1996),979-1001.[3]G.B.Ermentrout&N.Kopell,Parabolic bursting in an excitable system coupled witha slow oscillation,SIAM J.Appl.Math.46(1986),233-253.ROTATING WA VES IN THETA NEURONS23 [4]G.B.Ermentrout&J.Rinzel,Waves in a simple,excitable or oscillatory,reaction-diﬀusion model,J.Math.Biology11(1981),269-294.[5]F.C.Hoppensteadt&E.M.Izhikevich,‘Weakly Connected Neural Networks’,Springer-Verlag(New-York),1997.[6]E.M.Izhikevich,Class1neural excitability,conventional synapses,weakly connectednetworks,and mathematical foundations of pulse-coupled models,IEEE Trans.Neural Networks10(1999),499-507.[7]R.Osan&B.Ermentrout,Two dimensional synaptically generated travelling waves ina theta-neuron neuronal network,Neurocomputing38-40(2001),789-795.[8]R.Osan,J.Rubin&B.Ermentrout,Regular travelling waves in a network of Thetaneurons,SIAM J.Appl.Math.62(2002),1197-1221.[9]J.E.Rubin,A nonlocal eigenvalue problem for the stability of a travelling wave in aneuronal medium,Discrete&Continuous Dynamical Systems10,(2004),925-940.[10]A.T.Winfree,‘The Geometry of Biological Time’,Springer-Verlag(New-York),2001.Einstein Institute of Mathematics,The Hebrew University of Jerusalem,Jerusalem, 91904,IsraelE-mail address:haggaik@。

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Journal of Computational Neuroscience17,365–383,2004c 2004Kluwer Academic Publishers.Manufactured in The Netherlands.Model for Transition from Waves to Synchrony in the OlfactoryLobe of LimaxBARD ERMENTROUT∗Department of Mathematics,University of Pittsburgh,Pittsburgh,PA15260,USAJING W.WANGDivision of Biological Sciences,University of California,San Diego,2218Bonner Hall9500Gilman Drive,La Jolla,CA92093-0368,USAJORGE FLORESCenter for Neurobiology and Behavior,Columbia University,New York,NY10032,USAALAN GELPERINMonell Chemical Senses Center,3500Market St.,Philadelphia,PA19104,USAReceived March19,2004;Revised July7,2004;Accepted July14,2004Action Editor:Christiane LinsterAbstract.A biophysical model for the interactions between bursting(B)cells and nonbursting(NB)cells in the procerebral lobe of Limax is developed and tested.Phase-sensitivity of the NB cells is exhibited due to the strong inhibition from the rhythmically bursting B cells.Electrical and chemical junctions coupled with a parameter gradient lead to sustained periodic waves in the lobe.Excitatory interactions between the NB cells,which rarely ﬁre,lead to stimulus evoked synchrony in the lobe oscillations.A novel calcium current is suggested to explain the effects of nitric oxide(NO)on the lobe.Gap junctions are shown both experimentally and through simulations to be required for the oscillatingﬁeld potentials.Keywords:oscillations,waves,olfaction,limax,nitric oxide,gap junctions1.IntroductionInformation processing in the olfactory system typ-ically involves oscillatory dynamics of ongoing or odor-elicited activity in early odor processing centers (Dorries and Kauer,2000;Gelperin,1999;Lam et al., 2000;Laurent et al.,2001;Martin et al.,2004).The computational role of oscillatory dynamics in olfaction is not yet known in detail,although two recent manipu-∗To whom correspondence should be tions of oscillatory dynamics during odor processing suggest a role for oscillations in discriminating be-tween closely related odors(Stopfer et al.,1997;Teyke and Gelperin,1999).Detailed knowledge as to how input from olfactory receptors affects the cooperative dynamics of mitral cells in olfactory bulb and mitral cell analogs in invertebrates would help clarify the computational role of oscillatory dynamics in olfaction. The early olfactory system of the terrestrial mollusc Limax maximus is well suited to studies of oscilla-tory dynamics and olfactory information processing.366Ermentrout et al.The structure which receives direct olfactory input,the procerebral(PC)lobe of the cerebral ganglion,dis-plays0.7Hz oscillations in localﬁeld potential(LFP) which are modulated by olfactory input(Gelperin and Tank,1990;Gervais et al.,1996;Kimura et al.,1998c; Toda et al.,2000).The PC lobe also propagates activity waves from its apical to basal pole at1.1mm/sec in vitro (Delaney et al.,1994;Kimura et al.,1998b;Kleinfeld et al.,1994).The activity waves serve to separate odor representations set up in the PC lobe by associative conditioning when odors are used as conditioned stim-uli(Ermentrout et al.,2001;Gelperin,1999;Kimura et al.,1998a).The traveling activity wave in the Limax PC lobe results from interactions among a group of bursting (B)neurons which have a gradient of excitability from apex to base of the PC lobe(Ermentrout et al.,1998). Figure4A of this paper shows slices of the procere-bral lobe from the apical region of the PC lobe os-cillate at a higher frequency than slices taken from the basal region.This is interpreted to show that B neurons in the apical region are more excitable,i.e., have a higher frequency of bursting,than B neurons in the basal region.The B neurons are coupled local inhibitory neurons which produce5–7mV IPSPs in nonbursting(NB)neurons,ﬁrst in the apical region and then in successively more basal NB cells as the activ-ity wave propagates from apex to base in the B neuron network(Wang et al.,2001).The LFP arises mainly from the coherent postsynaptic currents in the local population of NB neurons elicited by coherent burst-ing in the local B cells(Watanabe et al.,1998,1999, 2001).At rest the PC lobe produces oscillatory waves.Ei-ther spontaneously or after odor input the PC can switch from waves to brief bouts of apical-basal synchrony. Odor inputs make afferent synapses onto excitatory interneurons,the NB cells.Shocking olfactory input ﬁbers results in phase-dependent modulation of the LFP(Gelperin and Tank,1990).This is a consequence of phase dependent activation of NB cells and their ex-citatory synapses onto B cells(Inoue et al.,2000).In addition to direct fast excitation from NB cells onto B cells,NB cell activity may induce the release of nitric oxide(NO)onto B cells,which causes an in-crease in B cell burst frequency and an increase in LFP frequency(Gelperin,1994).In an earlier paper (Ermentrout et al.,1998)we used an abstract model of coupled phase oscillators to explain the origin of the phase lags of the LFP as well as the transient synchrony due to odor inputs.In order to explain several recent experimental results,we now introduce simpliﬁed bio-physical models for B and NB cells and couple them into networks.The coupled networks show waves,tran-sient synchrony to inputs and phase-dependence of the long-lasting LFP frequency increase to inputs.Addi-tionally,we describe several new physiological exper-iments and use the model to explain these.We show that uncaged NO can lead to a change in the fre-quency of the rhythm as well as a change in the di-rection of propagation and that this alteration is lo-cation dependent.We furthermore assess the role of electrical coupling in the PC lobe and its importance in maintaining both the rhythm and the structure of the wave.2.Methods2.1.Experimental2.1.1.Distribution of Bursting Neurons.Anatomi-cal studies of the number and distribution of burster neurons in the Limax PC lobe used desheathed PC lobes from newly hatched slugs incubated at room tem-perature for20min in slug saline containing5ng/ml sulforhodamine101(Molecular Probes)(Cina and Hochman,2000)or0.005%Hoechst33258(Sigma). The latter is normally used to stain nuclear DNA(Bale et al.,2001)but we found that Hoechst33258labels burster neurons with the same speciﬁcity as sulforho-damine101(Wang et al.,1999).Excess dye was re-moved by rinsing the PC lobe with saline containing 1%BSA for one hour.Stained neurons were visual-ized with a custom-built two-photon laser-scanning mi-croscope(TPLSM)(Denk and Svoboda,1997).The excitation wavelength was830nm.Imageﬁles from TPLSM were analyzed using NIH image v.1.57.Some preparations were examined in150µm thick vibratome sections made after embedding the desheathed PC lobe in low melting point agar.2.1.2.Electrophysiology of the PC Lobe.The PC lobe was isolated byﬁne dissection after anesthetizing slugs(Limax maximus)by exposure to5◦C and injec-tion of cold isotonic(55mM)MgCl2.Dissection of the CNS to isolate the PC lobe is done in high magne-sium(5X)cold saline.After return to normal saline at room temperature the LFP oscillation and apical-basal wave propagation are very regular.Dualﬁeld potentialModel for Transition from Waves to Synchrony in the Olfactory Lobe of Limax 367recordings from the PC lobe used saline-ﬁlled patch electrodes (3–5µm tip)inserted into the cell body layer at the apex and base of the lobe.Field potential signals were recorded with HEKA EPC-7ampliﬁers,digitized at 500Hz and recorded on computer (Apple Mac II)using custom software.The olfactory nerve (ON)was severed at its peripheral origin in the digitate ganglion and placed in a tight ﬁtting suction electrode for stim-ulation.The phase of the ON shock was calculated as the time from the LFP peak to the shock divided by the duration of the last complete LFP cycle prior to the shock.To measure the effects of gap junctional blockers such as 18αglycyrrhetinic acid (GA)(Bou-Flores and Berger,2001;Davidson and Baumgarten,1988;Wong et al.,1998)and octanol (OCT)(Mills and Massey,2000;Spray et al.,1985)we recorded intra-cellularly from buccal cells B7and SB (Copeland and Gelperin,1983;Prior and Gelperin,1977)and mea-sured the decrease in their electrical coupling caused by addition of the gap junctional blocker.Effects of uncoupling agents and molluscan glutamatergic modi-ﬁers like picrotoxin (PTX)(Cleland,1996)were as-sessed by dual recordings (apical and basal)of the LFP in the isolated PC lobe to measure the phase de-lay due to wave propagation in the B cell network.When drug treatments suppressed spontaneous activ-ity in the B cell network,we activated B cells either by adding an NO donor (diethylamine/NO,DEA/NO)(Hrabie et al.,1993)to the bath or by treating the prepa-ration with caged NO (nitrosylpentachlororuthenate,NPR)(Carter et al.,1993;Makings and Tsien,1994)and applying uncaging ﬂashes to the apical or basal region of the PC lobe by shutter-controlled localizedepiillumination.Figure 1.Schematic of the two-layer model.Bursting neurons,B 0,...,B 20are coupled with local electrical junctions and longer-range inhibitory coupling (black arrows).They have a T-type calcium current,potassium current,leak,and modulation through nitric oxide (NO)produced from the non-bursting cells (red arrow).Nonbursting neurons,N B 0,...,N B 20are coupled with excitatory chemical synapses (open arrows)and receive local inhibition from the bursting cells.They have fast sodium,potassium,leak,and a slower potassium current.2.2.Model DescriptionThe model consists of a spatially distributed network of bursting and nonbursting cells.Since the wave on the PC lobe appears to vary only along one spatial di-mension,we have modeled the spatial distribution as a one-dimensional line of cells—21cells of each type (see Fig.1).2.2.1.Bursting Cells.These are responsible for the local ﬁeld potential and generate the oscillations.Since little is known of the currents responsible for the gen-esis of the spontaneous oscillations,we have made a fairly simple model with only three currents,a leak,a potassium current,and a T-type calcium current.Wang et al.(2001)have shown that calcium is the dominant inward current responsible for the activity of bursting neurons.There are two reasons we chose the T-type current instead of a simpler calcium cur-rent.First,the oscillations occur at very hyperpolar-ized potentials suggesting a low threshold calcium current.Secondly,the oscillation is very sensitive to NO.In a recent paper,Kawai and Miyachi (2001)showed that T-type calcium channels in olfactory re-ceptor neurons are modulated by cGMP.NO is a pow-erful signalling molecule and has been shown to ac-tivate cGMP through several pathways (Ahern et al.,2002).Thus,we introduce a T-type calcium chan-nel which is modulated by NO.The potassium cur-rent aids in the repolarization and keeps the magni-tude of the bursts low.The leak plays an important role as follows.Watanabe et al.(2003)have recently shown that intracellular chloride is important in deter-mining the direction of propagation of the waves.We368Ermentrout et al.thus suppose that the leak consists of a chloride and a potassium leak.The gradient in intracellular chloride then acts to alter the reversal potential of the leak.At the apical end,the reversal potential is more depolarized than an the basal end.This results in an increase in the local frequency of the oscillation.The model equations for the voltage are:C dVdt=−g L(V−E L)−g k n4(V−E K)−g Ca m2hV(V−E Ca)−I couplewhere h,n are dynamical variables satisfying the usual gating equations,τx(V)dx/dt=x∞(V)−x.The func-tions and parameters for these are given in the Ap-pendix.The activation of the T-current satisﬁes:m=1/(1+exp(−(V−V th)/6.2))where V th=−58−f[NO].f=2is a factor that con-verts concentration of nitric oxide[NO]into a millivolt shift.We do not include separate equations for the ac-tivation of cGMP through nitric oxide although such equations could easily be incorporated in the present formulation.Nitric oxide concentration obeys the fol-lowing equations:d[NO]dt=([NO]back−[NO])/τNO+Swhere we take[NO]back=1µM andτNO=5000 msec.The term S includes sources such as through the ﬁring of the local NB cells.We have not included dif-fusion of[NO];simulations(not shown)indicate that it makes little difference.Clearly,if S=0then diffusion will make no difference because of the uniform back-ground concentration.We remark that a recent study by Jacklet and Tieman(2004)has shown that NO causes an increase in excitability of Aplysia neurons by sup-pressing a voltage-dependent potassium channel.This mechanism for action of NO on a model cell would re-sult in similar behavior—by depolarizing the neuron, the frequency of oscillations would increase.For the network results that we present below,the details of how NO works are unimportant as long as it effects an increase in the burster frequency.There are three terms in I couple consisting of(i) nearest neighbor gap-junctional coupling;(ii)equal coupling through inhibition to the5neighbors on ei-ther side;and(iii)synaptic excitation from the local nonbursting cell.Inhibitory synaptic currents have the form:I syn=g ii s(V−E i)wheredsdt=αi(V)(1−s)−s/τi.Excitatory synapses have a similar form but with a dif-ferent reversal potential and with the gate,s dependent on the local NB cell potential.2.2.2.Non-Bursting Cells.Since these cells do not burst,only spiking currents were included:CdVdt=−g L(V−E L)−(g K n4+g W w)(V−E K)−g Na m3h(V−E Na)−I syn+I stim.We included a slow voltage-dependent potassium cur-rent,g W for spike-frequency adaptation.The synaptic currents consisted of local inhibition from the bursting cell and recurrent excitatory connections from three neighbors on either side.Stimulation was meant to mimic the application of odorant,localized at discrete spatial points in the network and had the form:I stim(t)=−g stim te−t/40(V−E e)where E e=0is the reversal potential of the excitatory input stimulus.2.2.3.Local Field Potential.“Localﬁeld potential”is obtained by adding the inhibitory synaptic currents of the11cells surrounding a given spatial location, and then passing through a low-passﬁlter with a time constant of100milliseconds.3.Results3.1.Distribution of Bursting NeuronsThe distribution of bursting neurons within the PC lobe was determined by applying the activity-dependent vi-tal dyes sulforhodamine101or Hoechst33258to the intact desheathed lobe and observing the pattern of stained neurons with TPSLM.Burster neurons have characteristic oval shaped somata and multiple pro-cesses projecting within the cell body layer(Fig.2D),Model for Transition from Waves to Synchrony in the Olfactory Lobe of Limax369Figure2.Distribution of burster neurons in the Limax procerebral lobe.Cells labeled by Hoechst33258are located in the cell body layer of the procerebral lobe immediately adjacent to the zone of termination of the olfactory afferentﬁbers.The PC lobe from a newly hatched slug was optically sectioned by two-photon laser scanning microscopy(Wang et al.,2001).Optical sections of the whole lobe with the cell layer facing the microscope objective are shown(A):for the surface layer,(B):50µm deep and(C):100µm deep.(D):A high resolution micrograph of labeled cells after sectioning the PC lobe at150µm thickness shows the oval nuclei and processes projecting within the cell layer,showing that the labeled cells are bursting neurons.(E):A projection of150optical sections at1µm intervals onto one image plane(NIH Image v.1.57) reveals all the labeled cells in this image volume,suggesting that there are about300bursting neurons in this PC lobe.CM cell mass,TM terminal mass,IM internal mass.as determined by sharp electrode recording and intra-cellular dye injection(cf.Fig.1in Wang et al.,2001). Nonburster neurons have spherical somata and a single process projecting directly to the neuropil of the PC. Both sulforhodamine101and Hoechst33258stained burster neurons and revealed that the buster neuron so-mata are located in a20µm layer adjacent to the ter-minal neuropil(TM).Optical sections of the whole PC lobe with the cell layer facing the microscope objective also show that the bursting neurons are predominately located adjacent to the TM neuropil(Fig.2A,B and C).A projection of150optical sections at1µm intervals onto one image plane reveals all the labeledB cells in the image volume(Fig.2E).There are about300burst-ing neurons in this PC lobe from a newly hatched slug. The PC lobe is a site of active neurogenesis posthatch-ing(Zakharov et al.,1998)so the adult PC lobe will have a considerably larger population of B cells.Since we do not know the mechanism of selective Hoechst dye uptake into burster neurons and did not attempt an independent veriﬁcation that all of the burster neurons were indeed stained,it is possible that the evidence of burster neuron density and distribution is an underesti-mate of the true density and distribution.3.2.Response to ON Shock is Phase Dependent The phase-dependence of responses of the B and NB cell networks in the Limax PC lobe is demonstrated most clearly by measuring responses to stimulation of the ON as a function of the phase of the sponta-neous LFP oscillation at which the stimulus is given. Stimulation of the ON elicits a short-latency response370Ermentrout et al.comprised of two components in the LFP record fromthe PC lobe cell body layer.Theﬁrst peak in the LFPresponse to ON shock is due to currents set up by thevolley of action potentials invading the PC lobe fromthe ON.This peak is not phase sensitive and is not sen-sitive to the calcium concentration in the bathing solu-tion.The second peak in the LFP response to ON shockis phase sensitive and is reduced in low calcium saline(Gelperin and Tank,1990).The size of the synapticcomponent of the LFP response to ON shock is min-imal when the ON is shocked at a phase of0.1andincreases linearly to an ON shock phase of0.9(cf.Fig.3B in Gelperin and Tank,1990).The inputﬁbersfrom the ON synapse on NB cells(Inoue et al.,2000)so the synaptic component of the LFP response to ONshock arises from the summed EPSP in the populationof NB cells contributing to the potential measured bythe LFP electrode.Since the NB cell membrane poten-tial is modulated by the5–7mV IPSP from the B cellswith each cycle of the oscillation,the EPSP evoked inNB cells by the ON shock varies in amplitude duringeach cycle of the oscillation.Single shocks of the ON also can elicit longer lastingchanges in the frequency of the LFP oscillation,partic-ularly at higher stimulus intensities(data not shown).As the ON shock is delivered at later and later phases ofthe ongoing LFP oscillation the NB cells are more andmore likely to spike in response to the ON shock.NBspikes have two effects.They provide direct fast excita-tion of B cells and may trigger increased NO productionby olfactory inputﬁbers,which increases the burst rateof B cells.As shown in Fig.2,the NO-sensitive B cellsomata and processes are localized in the cell layer im-mediately adjacent to the neuropil where NO is likelyto be released from inputﬁber neurites.3.3.Single Cell Behavior in ModelA singleB cell is capable of spontaneously oscillat-ing.We have adjusted parameters so that the frequencyvaries from about0.8to1.5Hz.The frequency of oscil-lations depends on the two main parameters in the sim-ulations:(i)the leak potential,E L and(ii)the shift ofthe calcium current activation threshold,V th.In Fig.3A,we illustrate the potential of a single B cell for two val-ues of the leak potential when V th=−60mV.We have included an inhibitory autapse since in the full spatialmodel,there is inhibition from neighboring neurons.For high and low values of the leak,there is a stableﬁxed point.Decreasing from a high leak potential,there is a subcritical Hopf bifurcation(Fig.3B)which turnsaround leading to a stable periodic orbit.Lower valuesof the leak restabilize theﬁxed point through anotherHopf bifurcation.For−83<E L<−80,theﬁxed point is unstable and there is a stable periodic solu-tion.The frequency of the limit cycles ranges from1to1.5Hz with a large linear region.Figures3D and Eshow similar diagrams for variation of the calcium acti-vation parameter,V th with the leakﬁxed at−82mV.Aswith the leak,there are two Hopf bifurcations.The fre-quency can range from less than1Hz up to3Hz.Thisrange of frequencies matches the frequency range ofthe LFP oscillation measured in150µm thick slices ofthe PC lobe cut along the apical-basal axis(cf.Fig.4Ain Ermentrout et al.,1998,and unpublished data).Forvalues of leak below around−83mV,the cells are toohyperpolarized for the voltage to cross the activation ofthe calcium current.On the other hand,when the leakpotential is too positive,the cell is depolarized enoughso that the T-current is inactivated.Next,we consider a pair of cells,one a burster(B)andthe second,a nonbursting(NB)cell.The B cell sponta-neously oscillates and thus provides a phasic inhibitionto the NB cell(Fig.4).In our model,the inputs fromthe receptor neurons target the NB cells which dom-inate the lobe.As shown in Fig.3B in Gelperin andTank(1990),there is a consistent phase-dependence ofthe response of NB cells to inputs.This is shown forthe model in Fig.4.In Fig.4A the input fails to makethe NB cell spike and thus has no effect on the localB cell.The magnitude of the EPSPs evoked on the NBcell depend on the time of the input.Figure4C showsthe relative height of the EPSPs as a function of thetime of the B-cell.Figure4B shows that if the inputcomes late enough in the phase of the rhythm,then theNB cell can spike.This in turn increases the local Bcell frequency,an effect which lasts for several cycles,as observed in experiments.3.4.Baseline NetworkFigure5shows a space-time picture of the potentialof the B cells when there is a gradient in the leak(E L=−80mV at the apical end and−83mV at the basal end)(cf.Fig.3A).Nearest neighbor gap junc-tional coupling has strength0.03as does the inhibitory coupling.The network is active over about60%of the cycle.Figure5B shows the localﬁeld potential(LFP, see methods)at3sites(0is most apical and20is most basal).We point out that in experiments,the apical LFPModel for Transition from Waves to Synchrony in the Olfactory Lobe of Limax371Figure 3.Properties of an individual B-cell.(A)V oltage of the B-cell for two different leaks E L =−83,−80.The more depolarized leak has a higher frequency and a lower amplitude.(B)Bifurcation diagram showing the voltage as a function of the leak for V th =−60mV .(C)Frequency as a function of the leak.(D)Bifurcation diagram as a function of V th ,the activation threshold for the T-current for E L =−82mV .(E)Frequency as a function of the threshold.has a larger magnitude than the basal end.This could be due to the fact that the cell layer is thicker at the api-cal end,so that the LFP may sum more currents.In our model,the faster cells have smaller amplitude action potentials.3.5.Electrical and Chemical Coupling Among B Cells;ExperimentsTo guide further modeling,we explored the nature of neuronal coupling among B cells,both electrical and chemical.To characterize the effects of agents known to uncouple gap junctions,we used a Limax prepa-ration that allowed direct assessment of gap junctional coupling between buccal neurons SB and B7(Prior and Gelperin,1977),as shown in Fig.6.A single electrode voltage clamp maintains neuron SB at a constant mem-brane voltage while current pulses of varying amplitude are injected into neuron B7.The change in membranepotential of neuron B7was measured vs.the clamp current in neuron SB with and without 2mM octanol or 100µM GA.Octanol reduced the conductance be-tween buccal cells B7and SB by 28%,while 100µM GA reduced B7-SB conductance by 17%(3experi-ments,2preparations).Having characterized the effects of GA and octanol on electrically coupled neurons in Limax,we then ap-plied these gap junction blockers to the PC lobe and assessed their effects on both wave conduction and the local ﬁeld potential in the B cell network.The prepa-ration was bathed in 50µM NPR (caged NO donor)and uncaging ﬂashes of 250msec were applied to a 100µm diameter region either just apical to the api-cal LFP recording electrodes or just basal to the basal LFP recording electrode.Before the stimulus is given,the LFP is clearly visible in the PC lobe treated with saline but is completely absent in the GA-treated lobe.Thus,gap junctions are required for the existence of a LFP .The effects of the localized apical and basal NO372Ermentrout etal.Figure 4.Phase dependence of the evoked EPSP onto NB cells.(A)Bursting cell (black)and nonbursting cell (colored)potentials for three different stimuli at t =1200,1400,1600milliseconds.(B)Stimulus at t =1800milliseconds is sufﬁcient to cause ﬁring of the NB cell (blue),and subsequent speeding up of the B cell rhythm (black)compared to the case of no stimulus (red).(C)Dependence of the relative EPSP size measured from base to peak (in Fig.A)as a function of the timing relative to the burst.For stimuli occurring later than about 70%of the cycle,the stimulus evokes a spike in the NB cell.uncaging ﬂashes when the lobe is in normal saline are shown in the upper part of Fig.7.The apical ﬂash in-creased activity at both apical and basal recording sites while the basal ﬂash produced a strong effect at the basal site but a weaker effect at the apical site.Addi-tion of 100µM GA blocked the basal to apical propa-gation (except for a single peak occurring 2.3seconds after the ﬂash)but left intact the apical to basal prop-agation (Fig.7,middle panels).This is in contrast to the control response to a basal ﬂash recorded in normal saline,where small amplitude activity is evident in the apical record immediately after the basal ﬂash and the ﬁrst of two large amplitude apical events occurs with a latency of 0.5sec after the ﬂash.One possible ex-planation for the reduced efﬁcacy of GA in blocking apical-to-basal wave propagation versus the efﬁcacy of GA in blocking basal-to-apical wave propagation seen in Fig.7is due to the asymmetry in the effectiveness of apical and basal uncaging ﬂashes in normal saline.However,the simulations described below and illus-trated in Fig.8have no such asymmetries and are ex-plained below.Further addition of 5mM PTX,which blocks glutamate-mediated chloride conductances in molluscs (Cleland,1996),suppressed the remaining apical-to-basal propagating activity and rhythmic re-sponding at the recording sites adjacent to the uncaging ﬂash (Fig.7,lower panels).Similar results were seen with 2mM OCT.Both electrical and chemical synapticModel for Transition from Waves to Synchrony in the Olfactory Lobe of Limax373Figure 5.Normal wave behavior.(A)B-cell potentials in a space-time plot.Cell index (0–20)is horizontal and time runs downward.(B)Local ﬁeld potentials of cells 1,10,19.Figure 6.Buccal ganglion preparation used to measure the ef-fects of gap junction uncoupling agents.B7,B9=identiﬁed buccal motoneurons,SB =salivary burster neuron,B1=identiﬁed buccal cell,BC =buccobuccal connective,CBC =cerebrobuccal connec-tive,GN =gastric nerve,N1,N2,N3=buccal nerve roots 1,2,3,SN =salivary nerve.Single electrode voltage clamp maintains SB at a constant Vm.Current pulses of varying amplitude are injected into B7.The change in Vm of neuron B7is then plotted vs the clamp current ﬂow in neuron SB.transmission clearly play a role in propagation within the B cell network,which is also highly NO sensitive.3.6.Effects of Local NO Depend on LocationFigure 8illustrates a simulation in which the gap junc-tions are partially blocked and NO is incremented eitherat the apical or the basal ends.At t =2000,the NO concentration is raised from 1to 2.5at a single site to mimic the uncaging experiment.The LFP of sites 0and 20are shown before and after uncaging.The middle panel shows that apical uncaging has an effect which propagates to the basal end as in Fig.7in the middle panels.However Fig.8also shows that an in-crement in the basal NO,while transiently increasing the basal LFP,does not have any affect on the apical LFP.While apical application of NO has an effect on the basal LFP,there is no entrainment.That is,the api-cal and basal ends do not show transient phaselocking.(The gap junctions are not completely blocked so that the LFP remains in Figs.8and 9.Figure 11shows com-plete blockage of the gap junctions.)This Fig.8also shows that NO has different effects on the amplitude of the LFP depending on whether it is applied basally or apically.This can be explained by looking at Fig.3D which shows a nonmonotone dependence of the voltage amplitude on the T-current activation threshold.The value of V th at which the amplitude peaks depends on the leak potential.For higher values of the leak potential this peak is shifted to the right.Cells at the apical end of the lobe are in the regime where decreasing V th through application of NO decreases the amplitude while those at the basal end (more hyperpolarized)increase their amplitude when V th is lowered.These differential ef-fects also provide an explanation for the asymmetry in propagation with low gap junctions.NO speeds up。