prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How many people attend the 233rd ACS national meeting?

Read the following text and answer briefly.

Context 1: The major actions taken by the board of directors and council during the national meeting in Chicago were reported in C&EN, April 30 (page 32) 
The Society Committee on Budget & Finance met on Saturday, March 24, to review the society's 2006 financial performance  The society ended 2006 with a net contribution from operations of $12 2 million, on revenues of $424 0 million and expenses of $411 8 million  This was $7 8 million favorable to the approved budget 
After including the results of the Member Insurance Program and new ventures, the society's overall net contribution for 2006 was $11 5 million, which was $7 4 million favorable to the approved budget  The favorable variance was primarily attributable to higher than budgeted electronic services revenue and investment income, as well as expense savings from lower than budgeted health care costs and reduced IT spending  In addition, the society ended the year in compliance with the board-established financial guidelines 
The Society Committee on Education (SOCED) received an update from President Catherine Hunt on the thematic programming featured in Chicago focusing on the sustainability of energy, food, and water  President-Elect Bruce Bursten solicited input from the committee pertaining to the central role of education in his agenda  SOCED received a presentation from the Membership Affairs Committee on its white paper on membership requirements 
Committee members strongly support the proposal to include undergraduates as members of the society, but they requested that financial arrangements be clearly spelled out in the petition to ensure that the highly successful Student Affiliates program remains intact  The committee discussed the Education Division programs that were reviewed in 2006 and those that will be reviewed in 2007, under the auspices of the Program Review Advisory Group  SOCED received an update from the Committee on Professional Training regarding the draft ACS guidelines for approval of bachelor's degree programs in chemistry 
Committee members discussed the report prepared by the Globalization Task Force, focusing on those sections relevant to education  The committee suggested initiatives related to the new ACS strategic plan, including a potential program that would engage retired chemists in the K-12 classroom  SOCED created a task force to consider the role of online, or "virtual," simulations in the chemistry laboratory, recognizing the value of online/virtual experiences as a supplement to, but not a replacement for, hands-on laboratory experiments 
The committee ratified two interim actions taken since the Dec  3, 2006, meeting: to remove the financial restriction of the ACS Petroleum Research Fund (ACS PRF) Supplement for Underrepresented Minority Research Programs (SUMR) and to contact nominators whose nominations for the Volunteer Service Award had expired and to invite them to reactivate their nomination packet for the 2008 Volunteer Service Award 
Acting under delegated authority, the committee voted to accept the recommendations of the ACS Petroleum Research Fund Advisory Board (February 2007 meeting) for funding grants totaling $5 2 million; voted to recommend to the board a screened list of six nominees (due to a two-way tie for fifth place) for the 2008 Priestley Medal; voted to recommend to the board a screened list of five nominees for the 2008 Award for Volunteer Service to ACS; on the recommendation of the ACS Committee on Frasch Foundation Grants, voted to recommend to the board that it recommend to the trustee (US Trust) of the Frasch Foundation 12 grants for research in agricultural chemistry for the period of 2007–12; voted to recommend to the ACS Board of Directors that a new national award be established, the "ACS Award for Affordable Green Chemistry," sponsored by Rohm and Haas; and voted to recommend to the ACS Board of Directors that a new endowment be established, the "Affordable Green Chemistry Endowment Fund," to support the award 
The committee also reviewed the final report from the Special Board Task Force on the Review of the ACS National Awards Program, chaired by Ronald Breslow; established a Canvassing & Selection Subcommittee; and reviewed a list of external awards for which ACS may want to nominate candidates  The committee agreed to include the list of significant external awards in the awards locator database that is being developed 
The committee was updated on efforts to reconcile ACS's technical divisions' desires to leverage national meeting content using the Internet with our journal editors' concerns about prior publication issues  A conference call on this issue was scheduled for April 21, 2007 
The committee received a presentation on the recent actions of the ACS Board of Directors International Strategy Group (ISG)  The group's charge is to develop recommendations for a short- and long-term international strategy for the society 
The committee was updated on the status of the activities of the Board Oversight Group on Leadership Development (BOG)  Potential solutions for the unexpectedly high cost of facilitator training and transitioning from the current Leaders Conference format to the newly designed curriculum were presented to the committee 
The committee reviewed plans for conducting the 2007 Membership Satisfaction Survey  Preliminary results are expected in May or June with a final report to be delivered to the board at the 2007 Boston national meeting 
The committee received a briefing on the status of the MORE Project: Multidisciplinary Opportunities though Resource Enhancement  Twenty-eight proposals were received, and a decision on which proposals to support will be made in early May 
The chair led a discussion on draft 2007 committee goals, and committee members offered several suggestions related to successfully meeting them  One suggestion was to modify a communications goal to make it more completely reflect the duties of the committee outlined in the board regulations  The chair and committee members will examine the suggestion and revisit the question after the board retreat where board committee duties will be examined 
ACS President Hunt discussed her 2007-08 Presidential Task Force on Enhancing Science & Technology, which is charged with developing advocacy best practices that can enhance ACS's attainment of its public policy priorities  The task force is composed of a diverse set of ACS members as well as former U S  Representative and chairman of the House Science Committee, Sherwood Boehlert, who will cochair the task force 
• Results of the 2007 Public Policy Priorities Survey, which resulted in a four-tiered ranking of ACS's 2007 public policies  The ranking will help focus staff resources in conducting outreach and advocacy on behalf of ACS members 
• The hiring of a communications consulting firm for 2007 to assist ACS in implementing the initial phase of the ACS Strategic Communications Plan 
• Creation of a pilot ACS state government affairs advocacy program  Committee members agreed to the creation of a pilot, and staff will propose an initial list of states, policy focus, and a budget to carry out the program 
The committee met in executive session on March 23 and in open session jointly with the Joint Board-Council Committee on Publications and the Division of Chemical Information on March 26 
The committee heard from Chemical Abstracts Service (CAS) management on a range of issues including a report on continuing database building efforts, product enhancements, and CAS's centennial celebration plans 
The Committee on Chemical Safety (CCS) provides advice on the handling of chemicals and seeks to ensure safe facilities, designs, and operations by calling attention to potential hazards and stimulating education in safe practices 
CCS has several publications (many downloadable), including the flagship publication, "Safety in Academic Chemistry Labs" (SACL)  Work has recently started on the translation of SACL into Arabic  This is in addition to the online Spanish version of SACL  Also online are the "Student Lab Code of Conduct for Secondary Science Programs" and a security vulnerability analysis checklist  A K-12 restricted hazardous substances list is under development  The third edition of the "Chemical Safety Manual for Small Businesses" will be ready soon 
The committee's Task Force on Laboratory Environment, Health & Safety is working on a new edition of "Laboratory Waste Management " Task force members also commented on the recent Environmental Protection Agency Proposed Rule for Hazardous Waste in Academic Laboratories  Our Video Safety Resources Task Force is developing video resources to be distributed over the Web 
CCS has been involved in collaborations for the updating of publications like "Prudent Practices in the Laboratory" and "ACS Guidelines for the Teaching of High School Chemistry " Along with other ACS units, CCS is exploring participating in the EPA's School Chemicals Cleanout Campaign 
The Committee on Chemists with Disabilities (CWD) met at the 233rd ACS national meeting, Chicago, on Monday, March 26  Judy Summers-Gates reported on the Joint Subcommittee on Diversity meeting  This subcommittee is made up of representatives of the five committees that support people in chemistry (as opposed to a category of the profession): CWD, Committee on Minority Affairs, Committee on Technician Affairs, Women Chemists Committee, and Younger Chemists Committee, and its goal is to develop ways to coordinate the efforts of the five groups 
The CWD Ambassador Program that was announced at CWD's 25th anniversary celebration at the Washington, D C , meeting was discussed  Zelda Wasserman reported on the status of the letter from CWD to the ACS Board regarding captioning of ACS video materials  Janelle Kasper-Wolf, of ACS staff, discussed adding new questions to the ACS annual employment salary survey to obtain information for the committee 
At the Chicago national meeting, the Committee on Community Activities (CCA) partnered with the ACS Education Division and the Office of the President to host "Chemistry In Action—It's Easy Being Green" at the Peggy Notebaert Nature Museum on Saturday, March 24  More than 250 children participated in the hands-on activities focused on recycling  ACS President Hunt presented a Salutes to Excellence plaque to the museum for its dedication to community outreach 
The Chemists Celebrate Earth Day celebration occurred in 120 local sections with 138 coordinators leading the efforts within their communities  This represents an increase of more than 30% in local section and coordinator participation from 2006 
CCA was featured in C&EN's April 16th issue on page 53  A shortcut to CCA's homepage was created: chemistry org/committees/cca html 
During the Boston national meeting, CCA and the Office of Community Activities will celebrate National Chemistry Week's 20th Anniversary and its theme, "The Many Faces of Chemistry " A special outreach event is being planned for Sunday, Aug  19  Hands-on activities will focus on health and wellness 
The Committee on Corporation Associates (CCA) advises and influences ACS to ensure that its products and services are of value to industrial members and their companies  CCA vice chair, Roslyn White (SC Johnson), provided an overview of recent interactions between Corporation Associates and the U K -based Society of Chemical Industry (SCI) 
CCA gave feedback to a recommendations report from the ACS Board Committee on Professional & Member Relations Task Force on Globalization  Presentations were also received from the ACS Green Chemistry Institute and SCI 
Staff reported on the Department of Industry Member Programs' activities since the San Francisco meeting  The report covered the Regional Industrial Innovation Awards, the World Congress on Industrial Biotechnology, the Analytical Pavilion sponsored by C&EN, and the ACS/Pharma Leaders Meeting 
The Awards/Finance & Grants Subcommittee reported that CCA received two funding proposals that total $7,500  Funding was provided to the following: The Committee on Economic & Professional Affairs at $3,000 for the Chicago symposium on "Benefits Trends for the Chemical Workforce" and the Office of Graduate Education and the Department of Career Development & Management at $4,500 for a workshop on "Preparing for Life after Graduate School," to be held in conjunction with the 39th Central Regional Meeting 
The subcommittee also requested that ACS staff provide CCA with an official annual statement of Corporation Associates' financial reserves as of Jan  1 of each year 
The Programs Subcommittee reported on planned programming activities in 2007 and beyond between CCA and SCI  The subcommittee gave an update on a Boston symposium cosponsored by Corporation Associates and the Medicinal Chemistry Division featuring past ACS Heroes of Chemistry from the pharmaceutical industry 
By request of the subcommittee, representatives from Chemical Abstracts Service gave an overview on AnaVist—a tool with potential applications for CCA's efforts to provide member companies with industry-relevant information and reports  The subcommittee also requested that CCA earmark approximately $20,000 of Corporation Associates funds in 2008 for its activities 
The Educational Outreach Subcommittee reported on its decision to collaborate with the Graduate Student Symposium Programming Committee of the Chemical Education Division on a graduate student industry roundtable program in Boston 
The subcommittee requested $5,000 in support of this effort  The subcommittee also discussed a request for corporate executive support of an American Association of Physics Teachers initiative to promote undergraduate research 
The Committee on Environmental Improvement (CEI) continues to be focused on the sustainability of the chemical enterprise  In Chicago, the committee introduced a multiyear effort to make ACS meetings more sustainable  This effort is designed to take advantage of the large size and diverse programs of the society to lead in the sustainability arena by "walking the talk "
The committee held a dialogue with representatives of the U S  Environmental Protection Agency who are trying to "green" federal conferences and work with the travel and tourism industry to change practices and shrink the environmental footprint of meetings  The committee also used advertising and student volunteers to engage individual meeting participants in a campaign to increase recycling by asking "Are you sustainable?" Moving forward, CEI looks forward to working closely with the Committee on Meetings & Expositions to advance this agenda 
CEI was also pleased to participate in the meeting theme on sustainability through the ACS presidential programming  CEI cohosted the Monday presidential luncheon to discuss sustainability issues with the Committee on Science and is leading the follow-up to that luncheon, which will include recommendations on advancing sustainability in the three focal areas of the meeting—energy, food, and water 
The committee also continued its dialogue with the Committee on Corporation Associates about a collaborative workshop  This activity, tentatively slated for the New Orleans meeting, will seek additional insights from chemical and allied products companies about public policy barriers that limit adoption of more sustainable products and practices as well as policy incentives that would lead to increased sustainability in the chemical enterprise 
At its Chicago meeting, the committee welcomed the president of the Jordanian Chemical Society and the past-president of the Arab Union of Chemists 
The committee was briefed on Pittcon 2007, where, with financial support from the Society of Analytical Chemists of Pittsburgh, ACS coordinated participation of a scientific delegation from Adriatic nations 
The committee heard reports on the 2007 Frontiers of Chemical Science III: Research & Education in the Middle East meeting; the 2007 Transatlantic Frontiers of Chemistry meeting, which was jointly sponsored by ACS, the German Chemical Society, and the Royal Society of Chemistry; planned workshops to engage U S  and Chinese early-career scientists in chemical biology, supramolecular, and new materials chemistry; and ACS Discovery Corps U S /Brazil Research Collaboration Project in Biomass Conversion to Biofuels, Biomaterials & Chemicals 
The committee discussed Latin American engagement opportunities created through Puerto Rico's involvement in three key chemical science events there: the 2009 ACS Southeast Regional Meeting, the 2008 Federation of Latin American Chemical Associations (FLAQ) meeting, and the proposed IUPAC 2011 Congress & General Assembly 
The committee heard reports on letter-writing efforts by the ACS president to government officials in Libya and Mexico expressing concerns about challenges to the scientific freedom and human rights of scientists there 
The Committee on Minority Affairs (CMA) approved new vision, mission, and values statements at the Chicago national meeting  The mission of CMA is to increase the participation of minority chemical scientists and influence policy on behalf of minorities in ACS and the chemical enterprise 
An aggressive new strategic plan was approved by CMA to guide its activities over the next three years  By the end of 2009, CMA will increase the number of ACS Scholars that graduate to 100 per year, add 100 new minorities to leadership positions in ACS, engage in several collaborations, and increase the number of minority members of ACS by 5,000  CMA will focus initially on increasing minorities in ACS leadership  In working toward this goal, CMA began work on two new leadership-development programs for minority chemists 
CMA continues to support the work of the Joint Subcommittee on Diversity (JSD) in developing programs, products, and services to ensure full participation of all members in ACS  In Chicago, JSD premiered a diversity booth at the meeting exposition hall and cosponsored symposia 
The Committee on Patents & Related Matters (CPRM) discussed proposed legislative and regulatory changes to the U S  patent system as well as open-access legislation and the potential effects such matters might have on industry and academia as well as on ACS 
CPRM also continued its work on several new educational tools to assist and inform members on patent issues and other intellectual property matters important to a successful career in the chemical enterprise  Many of these tools are now available on the committee's expanded website, membership acs org/C/CPRM/ 
At the March 2007 meeting, the Committee on Professional Training (CPT) reviewed 42 new and additional information reports from ACS-approved chemistry programs  CPT held conferences with four schools seeking approval, discussed three updates and five site visit reports, and approved three new schools  The total number of ACS-approved chemistry programs is now 642 
The committee released the full draft of the ACS guidelines for review and comment  Copies of the draft were distributed to the department chairs at all ACS-approved schools, the chairs of all ACS committees, and the chairs of all ACS technical divisions 
Several CPT members met with the ACS technical divisions during the Chicago meeting to present an overview of the draft and obtain feedback  The draft guidelines document is available on the CPT website, and the committee invites any comments to be sent to cpt@acs org 
In other business, the committee continued development of the two workshops with minority-serving institutions that will be held in 2007  The committee reviewed the editorial policies for the 2007 edition of the ACS Directory of Graduate Research, which is using a new protocol for retrieving research publication titles in an effort to improve the accuracy of the directory 
C&EN finished 2006 with an exceptionally strong editorial package  The first months of 2007 are proving to be equally successful in fulfilling the magazine's mission of keeping its readers informed  On the advertising side, revenues in 2006 increased for the second year in a row, and the early months of 2007 show continuing positive signs  The most significant editorial achievement was the successful launch of the redesign of the print edition of C&EN with the Oct  16, 2006, issue 
The Subcommittee on Copyright has successfully updated the Copyright Module on the ACS Publications website  The subcommittee is looking into the possibility of conducting copyright programs at future ACS national and regional meetings 
The final monitoring reports for Chemistry of Materials, Journal of Agricultural & Food Chemistry, and Molecular Pharmaceutics were presented and accepted by the committee  Journal of Chemical Information & Modeling, Organic Letters, Accounts of Chemical Research, and the Journal of Chemical Theory & Computation will be monitored next 
3  Examining the scientific basis of public policies related to the chemical sciences and making recommendations to the appropriate ACS units 
In the first of these areas, ComSci partnered with President Hunt and the Committee on Environmental Improvement in planning and hosting a sustainability luncheon that featured roundtable discussions centering on a key sustainability question  At the Boston national meeting, ComSci will deliver a full-day program on the subject of "Partnerships in Innovation & Competitiveness "
Regarding the second thrust, ComSci will present two programs in Boston: a box lunch that will feature two speakers taking opposing sides on the subject of "Genetic Screening & Diagnostic Testing: Do You Really Want to Know?" and a symposium titled "Creating & Sustaining International Research Collaborations "
In support of the last thrust, ComSci is planning two events for 2008: "Balancing Security & Openness" will gather data to determine if the recent emphasis on security is hindering scientific progress and "Transitioning Chemical Science to Commercially Successful Products "
The Women Chemists Committee (WCC) hosted more than 70 attendees at its open meeting recently in Chicago, where representatives from Iota Sigma Pi, Women in Science & Engineering, the Association of Women in Science, and the Chicago local section helped WCC celebrate the committee's 80th anniversary 
The Women in Industry Breakfast was also highly successful with a new format of speed networking  More than 100 participants had the opportunity to practice their elevator speeches and make several professional connections  A related workshop will be offered by WCC in Boston 
In Chicago, WCC sponsored two symposia, "Women Achieving Success: The ACS as a Platform in Leadership Development" in honor of Madeleine Joullié's 80th birthday and the ACS Award for Encouraging Women into Careers in the Chemical Sciences: Symposium in Honor of Bojan H  Jennings 
More than 225 ACS meeting attendees were present for the biannual WCC Luncheon and heard the keynote speaker Laura Kiessling, 2007 Francis P  Garvan-John Olin Medal Recipient  Twelve women presented their research at this meeting with funding by the WCC/Eli Lilly Travel Grant Award  WCC members also spent time educating expo attendees on programs offered by the ACS Office of Diversity Programs at its new booth 
In Chicago, the Younger Chemists Committee (YCC) welcomed its new committee members with an information session centered on YCC's charter as well as on its strategic plan: to make ACS relevant to younger chemists, to involve younger chemists in all levels of the society, and to integrate younger chemists into the profession 
In January, YCC again hosted a Leadership Development Workshop during the ACS Leaders Conference  There were more than 80 applications for the 15 awards, which covered travel and registration for the conference  YCC plans to again fund the travel awards and provide leadership training for young chemists in 2008  YCC also solicited applications and selected a new graduate student representative on the Graduate Education Advisory Board 
During the Chicago meeting, YCC programs included "Starting a Successful Research Program at a Predominantly Undergraduate Institution," "Career Experiences at the Interface of Chemistry & Biology," and "Chemistry Pedagogy 101 "
In addition to these programs, YCC cosponsored five programs with various committees and divisions  YCC continues to reach out to ACS committees and divisions and has initiated liaisonships with 11 technical divisions to encourage technical programming that highlights the contributions of younger chemists  Looking forward to Boston, YCC is planning symposia including "The Many Faces of Chemistry: International Opportunities for Chemists"; "Being a Responsible Chemist: Ethics, Politics & Policy"; and "Changing Landscapes of the Bio-Pharma Industry "
The Committee on Committees (ConC) conducted its annual training session for new national committee chairs at the ACS Leaders Conference in January 2007  ConC's interactive session for committee chairs in Chicago served as an opportune follow-on and a forum for informative interchange among seasoned and new chairs 
ConC began developing its recommendations for the 2008 committee chair appointments for consideration by the president-elect and chair of the board  ConC continues to focus efforts to identify members with the skills and expertise specified by the committee chairs using the councilor preference form 
The form will be sent to councilors in May  ConC also seeks the names of noncouncilor members for consideration for service on council-related committees, especially those with no prior appointment 
As part of ongoing activities with the joint CPC-Board Governance Review Task Force, ConC has collected data on committee liaisons to other committees  This information will be distributed to committee chairs  The number of liaisons indicates that unofficial but strong communication channels exist within the ACS committee structure 
On Sunday evening, the Committee on Nominations & Elections (N&E) sponsored its fifth successful Town Hall Meeting for President-Elect Nominees  An estimated 200 people attended this session  This forum facilitated communication among the 2008 president-elect nominees, councilors, and other members  N&E will hold another Town Hall Meeting featuring the candidates for director-at-large at the fall meeting in Boston 
Now that voting over the Internet has become an accepted procedure for ACS national elections, the ACS technical divisions and local sections have expressed strong interest in using this method for their elections  N&E has developed protocols for elections for local sections and divisions  This document will be forwarded to the appropriate committees for their review and distribution 
N&E is responsible for reviewing annually the distribution of member populations within the six electoral districts to ensure that the districts have equitable representation  According to bylaw V, section 4(a), the member population of each electoral district must be within 10% of the result of dividing by six the number of members whose addresses lie within these districts  The committee is happy to report that the six electoral districts are in compliance 
The committee has developed a petition on election procedures for president-elect and district director  The proposed election mechanism provides for a preferential (ranked) ballot and an "instant runoff " N&E continues to address the areas of campaigning and the timing of our national election process  Between the Chicago and Boston meetings, the committee plans to sponsor online forums for input from councilors and other interested members on these issues 
In response to member concerns regarding the collection of signatures for petition candidates, N&E reviewed the society's bylaws  The bylaws state that an endorsement is required, but does not stipulate the method of endorsement  N&E has determined that original or electronic signatures are acceptable and will establish appropriate procedures for receipt of electronic signatures 
The Committee on Constitution & Bylaws (C&B), acting for the council, issued new certified bylaws to the Corning Section, the Portland Section, the Division of Colloid & Surface Chemistry, and the Division of Chemical Education  The committee reviewed new proposed amendments for the Division of Medicinal Chemistry, the Columbus Section, the Detroit Section, and the Southern Arizona Section 
Three petitions were presented to council for action at this meeting  Regarding the "Petition on Election Procedures 2006," a motion to separate the petition was approved, and the petition was divided  Provisions affecting bylaw V, sec  2d, bylaw V, sec  3c, and bylaw V, sec  4f, which deal with election procedures and the timing of run-off elections, were approved by council and will become effective following confirmation by the board of directors 
The second part of the petition regarding bylaw V, sec  2c, and bylaw V, sec  3b, which deal with signature requirements for petition candidates for president-elect and director-at-large respectively, was recommitted to the Committee on Nominations & Elections, which has primary substantive responsibility for the petition 
The Committee on Nominations & Elections was asked to reconsider the signature requirements, procedures for acceptance of electronic signatures, and recommendations from the Governance Review Task Force on election procedures 
The second petition presented to council for action was the "Petition on Rules for Nominating Members of N&E for National Offices " This petition was not approved by council  The third petition, the "Petition on Multiyear Dues," was amended by incidental motion on the council floor, calling for the petition to become effective when technical components are instituted to track payments, but no later than Jan  1, 2010  Council approved the incidental motion and then approved the petition 
The committee reviewed one petition for consideration, the "Petition on Local Section Affiliations," which will be submitted to council for action at the fall 2007 meeting in Boston 
The committee met with representatives of the Committee on Membership Affairs and the Governance Review Task Force to continue discussions on proposals currently being formulated on membership requirements and student membership  In addition, the committee discussed election issues of concern to the Southern California Section 
We hope you enjoyed the presidential and division thematic program, "Sustainability of Energy, Food & Water" in Chicago  A small, dedicated group of volunteers and staff labored tirelessly to create and coordinate this programming; to them the Committee on Divisional Activities (DAC) offers sincere thanks 
DAC has committed to transfer the process of choosing and organizing future national meeting themes to a body that represents all divisions  We made substantial progress in Chicago, where division, secretariat, and committee representatives convened to discuss national meeting program concepts  They proposed themes for the 2008 Philadelphia national meeting as well as a framework for a future national programming group 
Divisions have successfully served their members fortunate enough to attend national meetings  To maximize benefits to division members, DAC encourages divisions to consider extending the reach of the content they deliver at national meetings through Internet-based distribution channels and will support worthy efforts in this direction via Innovative Program Grants 
The committee voted in Chicago to propose modifications to the division funding formula that will more greatly reward interdisciplinary programming  The new formula will also be simpler and more transparent to divisions  DAC will present the revised plan to council for action in Boston 
The Committee on Economic & Professional Affairs (CEPA), working with ACS staff in the Departments of Career Management & Development and Member Research & Technology, continues to update and implement its strategic plan to address the career needs of society members 
Specifically, the committee reviewed and revised existing workshops and materials to help ACS members get jobs  CEPA is developing new programs to address the needs of mid- and late-career chemists to ensure their continued competitiveness in the workplace and to ease their career transitions  New initiatives in these areas include the development of workshops, online training, surveys to assess member needs, suggested changes to public policies, and updates to professional and ethical workplace guidelines  As a result of discussions at the Public Policy Roundtable, which was held in San Francisco, a background paper is being developed on trends in health care issues 
The newly revised "Chemical Professional's Code of Conduct" was presented to council, which approved it  The Standards & Ethics Subcommittee is preparing a revision of the "Academic Professional Guidelines" to be presented to council for consideration in Boston 
CEPA reviewed the Globalization Task Force Report  As our science diffuses around the globe, we want to make sure that our members are aware of the economic and professional challenges they will face and that they have the tools they need to succeed  Therefore, CEPA made a commitment to work with other committees, divisions, and ACS staff to develop programs and policies that position our membership to compete in the global workforce 
CEPA heard and discussed a presentation on the proposal from the Membership Affairs Committee on broadening the requirements of membership  CEPA supports the spirit of this proposal and encourages further detailed studies to assess financial impacts on local sections and student affiliates chapters 
The Local Section Activities Committee (LSAC) recognized local sections celebrating significant anniversaries in 2007, including Savannah River (50 years), Northeast Tennessee (75 years), and the St  Louis and Syracuse local sections (both celebrating 100 years) 
LSAC hosted the local section leaders track in conjunction with the ACS Leaders Conference in Baltimore on Jan  26–28  A total of 135 delegates from 124 local sections participated in the weekend leadership conference 
LSAC also hosted a Local Section Summit on March 2–4 in Arlington, Va  The summit focused on practical operational issues that will support local sections' long-term success  Specific areas that were discussed include the development of a multiyear plan to expand or develop programming for local sections, opportunities to encourage innovation and experimentation within and among local sections, and capitalizing on existing opportunities to facilitate partnerships between local sections and other ACS groups 
Following the San Francisco national meeting, LSAC launched a local section Science Café minigrant program  Fifty-five local sections accepted LSAC invitation to host Science Cafés in 2007 
A DVD entitled "ACS Close to Home: Local Sections Connecting Chemistry & the Community" was released earlier this year  The video provides a seven-minute overview of the many outreach and educational programs sponsored by local sections and the critical role they play in positively influencing the public's perception of chemistry and its practitioners  Copies of the DVD were sent to all local section officers 
The Committee on Meetings & Expositions (M&E) reported that the 233rd ACS national meeting hosted 14,520 attendees  This included 7,152 chemical scientists, 5,059 students, 1,283 exhibitors, 119 precollege teachers, 573 exposition visitors, and 453 guests  The exposition had 424 booths with 268 companies 
The 10 2006 regional meetings set a new standard for excellence with attendance exceeding 8,000, a 30% increase in average meeting attendance compared to the 2005 meetings  A total of 4,717 abstracts were submitted  A region summit was held in February at which the final report of the ReACT study group was reviewed 
The practice of tracking the number of presenter no-shows continues  M&E will collaborate with the Committee on Divisional Activities to study options for addressing this problem  Suggestions will be presented at the Boston meeting for implementation in 2008 
It is the intent of M&E to pursue the goal of making our meetings "greener " We will communicate with staff and governance units to identify actions for both the short and long term 
The American Institute of Chemical Engineers (AIChE) and ACS will hold their 2008 spring meetings simultaneously in New Orleans  An ad hoc working group consisting of members from M&E, DAC, and AIChE are actively exploring joint programming opportunities for this meeting 
The Committee on Membership Affairs (MAC) met in executive session on Saturday and Sunday in Chicago and reported that the ACS closed 2006 with 160,491 members, our highest year-end membership count since 2002  Of the 17,857 applications processed in 2006, more than 1,000 came from the Member-Get-a-Member campaign in which many councilors participated  The society's retention rate in 2006 remained strong at 92%  The committee also reported that recruitment for the first two months of 2007 netted 2,844 new applications—729 more than for the same time period last year 
MAC continues to work with deliberate speed on the proposed new bylaw language for members, student members, and society affiliates-the three ways to connect to the society  The committee received input from the Governance Review Task Force and its action teams, the Council Policy Committee, the board of directors, the Committee on Constitution & Bylaws, and several other committees between the San Francisco and Chicago meetings  These interactions have resulted in the current bylaw change recommendations 
In Chicago, representatives from MAC attended several committee meetings and all seven councilor caucuses to summarize the current proposal for membership changes, answer questions, and seek input  In addition, all committee chairs were invited to have their respective committees review these bylaw changes and respond to MAC—if possible—before council met on Wednesday  MAC received 11 responses: eight supported the proposed changes as is, and three supported the proposed language with specified changes or considerations 
The comprehensive petition will likely represent the most significant and voluminous change in the ACS bylaws that has occurred in decades, and MAC is proud to be among the leaders in its development and in efforts to get it right the first time  Hundreds of individuals have contributed to this major effort, since MAC began such discussions at the spring 2004 national meeting 
The Committee on Ethics met in Chicago and discussed the possibility of organizing and scheduling a committee retreat in the near future to enable the committee to move from the current stage of exploring the needs and interests of ACS members to setting priorities for the next few years 
The Project SEED program offers summer research opportunities for high school students from economically disadvantaged families  Since its inception in 1968, the program has had a significant impact on the lives of more than 8,400 students  At the selection meeting in March, the committee approved research projects for 340 SEED I students and 98 SEED II students for this summer in more than 100 institutions 
The 2006 annual assessment surveys from 300 students indicate that 78% of the Project SEED participants are planning to major in a chemistry-related science, and 66% aspire to continue to graduate education  This program is made possible by contributions from industry, academia, local sections, ACS friends and members, the ACS Petroleum Research Fund, and the Project SEED Endowment 
The committee formally submitted a request to ConC to amend the Project SEED acronym and the committee duties described in the Supplementary Information of the "ACS Charter, Constitution, Bylaws & Regulations "
In Chicago, the committee's agenda focused on the ACS Strategic Plan and how Project SEED fits into it, the Program Review Advisory Group (PRAG) review of the Project SEED program, the committee's review of an online application form, and planning of the 40th anniversary celebration to be held at the Philadelphia meeting in the fall of 2008  The committee selected a task force to review the criteria for selection of the Project SEED ChemLuminary Award 
3  Making ACS relevant to technicians 
Last year, CTA, along with the Division of Chemical Technicians, the Committee on Economic & Professional Affairs, and ChemTechLinks, started the Equipping the 2015 Chemical Technology Workforce initiative  This year, the initiative awarded six $500 minigrants to activities and programs that support the educational and professional development of chemical technicians 
We are pleased to announce that the winners of the minigrants are the ACS Division of Environmental Chemistry; the Chemical Technician Program Chair for the 39th ACS Central Regional Meeting in Covington, Ky ; Delta College, University Center, Mich ; Grand Rapids Community College, in Michigan; Mount San Antonio College, Walnut, Calif ; and Southwestern College in Chula Vista, Calif 
The winners are collaborating with industry, academia, and ACS local sections on such activities as chemical technology career fairs for high school students, discussion panels on employability skills for technicians, and technical programming at regional and national meetings on the vital role technicians have in the chemical enterprise 
Because of the enthusiastic response to the minigrants, Equipping the 2015 Chemical Technology Workforce will be supporting another round of minigrants to be distributed in the fall  Details will be available on the website  For more information, go to www ChemTechLinks org and click on "Equipping the 2015 Chemical Technology Workforce "
CTA has also joined with the Joint Subcommittee on Diversity, formerly known as the Collaboration of Committees Working Group  Because this group is focused on increasing diversity in ACS and the chemical enterprise, we believe that this is an opportunity to raise awareness of the value of technicians  CTA looks forward to collaborating on the promotion of traditionally underrepresented chemical professionals 
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Context 2: Paper Info

Title: Generalized Pole-Residue Method for Dynamic Analysis of Nonlinear Systems based on Volterra Series
Publish Date: March 7, 2023
Author List: Qianying Cao (from State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology), Anteng Chang (from College of Engineering, Ocean University of China), Junfeng Du (from College of Engineering, Ocean University of China), Lin Lu (from State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology)

Figure

Fig  1: Procedure to compute the response by a combination of Volterra series and Laguerre polynomials
Fig  2: Linear frequency response function: (a) Modulus of H 1 (ω), (b) phase angle of H 1 (ω)
Fig  6: Comparison of h 1 (t) based on the analytical and reconstructed by Laguerre polynomials
Fig  11: Response for Case 1: (a) comparison between the proposed method and Runge-Kutta method, (b) contribution of the three components
Fig  18: Comparison of original excitations and reconstructed results: (a) Case 1, (b) Case 2 (c) Case 3
Fig  19: Response to irregular excitation for Case 1: (a) comparison between the proposed method and Runge-Kutta method, (b) contribution of the three components
Fig  23: Input-output dataset used to identify Volterra series: (a) input excitation, (b) output response
Fig  26: Comparison of responses between the predicted and numerical results: (a) response to regular excitation, (b) response to irregular excitation
Parameter values of the irregular excitation

abstract

Dynamic systems characterized by second-order nonlinear ordinary differential equations appear in many fields of physics and engineering  To solve these kinds of problems, time-consuming stepby-step numerical integration methods and convolution methods based on Volterra series in the time domain have been widely used 
In contrast, this work develops an efficient generalized pole-residue method based on the Volterra series performed in the Laplace domain  The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method 
Compared to the traditional pole-residue method for a linear system, one of the novelties of the pole-residue method in this paper is how to deal with the higher-order poles and their corresponding coefficients  Because the proposed method derives an explicit, continuous response function of time, it is much more efficient than traditional numerical methods 
Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations  Because the natural response, forced response and cross response are naturally obtained in the solution procedure, meaningful mathematical and physical insights are gained  In numerical studies, systems with a known equation of motion and an unknown equation of motion are investigated 
For each system, regular excitations and complex irregular excitations with different parameters are studied  Numerical studies validate the good accuracy and high efficiency of the proposed method by comparing it with the fourth-order Runge-Kutta method 

Introduction

Most real dynamic systems, as encountered in mechanical and civil engineering, are inherently nonlinear and include geometric nonlinearities, nonlinear constitutive relations in material or nonlinear resistances, etc    Nonlinear problems are attracting increasing attention from engineers and scientists 
This work focuses on solving nonlinear system vibration problems, i e , computing transient responses of nonlinear oscillators under arbitrary irregular excitations based on a combination of a pole-residue operation and Volterra series  Because Volterra series are single-valued, the scope of the present study is restricted to nonlinear behaviours without bifurcations  
To analyse nonlinear vibration problems, researchers have performed extensive studies and developed various mathematical methods  Popular methods include step-by-step numerical integration methods in the time domain, such as the Runge-Kutta method  This kind of method not only requires a small time-step resolution for obtaining high-precision solutions but also is prone to numerical instability  
For a long response with small time steps, the time domain methods are very costly in computational time  Volterra series is another widely used method, which is the extension of the Duhamel integral for linear systems   Volterra series can reproduce many nonlinear phenomena, but they are very complex due to higher-dimensional convolution integrals  
Since 1980's, significant progress has been made in the general area of the Volterra series  The reader is referred to Ref  for a quite thorough literature review on the relevant topics  After 2017, most papers focus on Volterra series identification  De Paula and Marques proposed a method for the identification of Volterra kernels, which was based on time-delay neural networks 
Son and Kim presented a method for a direct estimation of the Volterra kernel coefficients  Dalla Libera et al  introduced two new kernels for Volterra series identification  Peng et al  used the measured response to identify the kernel function and performed the nonlinear structural damage detection  Only a few papers concentrated on simplifying the computation of convolution integrals 
Traditional methods for computing convolution integrals involved in the Volterra series have been performed in three distinct domains: time, frequency and Laplace  The time domain method based on Volterra series refers to discrete time convolution methods, which also suffer computational cost problems  
Both the frequency domain method and the Laplace domain method based on the Volterra series consist of three steps: (1) Volterra series are transformed into an algebraic equation in the frequency domain or Laplace domain; the algebraic equation is solved by purely algebraic manipulations; and (3) the solution in Step ( ) is transformed back to the time domain 
Many researchers have used the frequency domain method to compute the responses of nonlinear systems  Billings et al  developed a new method for identifying the generalized frequency response function (GFRF) of nonlinear systems and then predicted the nonlinear response based on these GFRFs  Carassale et al  introduced a frequency domain approach for nonlinear bridge aerodynamics and aeroelasticity 
Ho et al  computed an output frequency domain function of a nonlinear damped duffing system modelled by a Volterra series under a sinusoidal input  Kim et al  identified the higher order frequency response functions by using the nonlinear autoregressive with exogenous input technique and the harmonic probing method 
This type of frequency domain method is much more efficient than the time domain method due to the fast Fourier transform algorithm  However, the frequency domain method not only is limited by frequency resolutions but also suffers from leakage problems due to the use of discrete Fourier transforms  In addition, the frequency domain method calculates only a steady-state response 
A natural response generated by initial conditions and a cross response caused by interactions between a system and an excitation are ignored  In contrast, the Laplace domain method can calculate all response components because initial conditions are considered in the computational procedure  However, it has been restricted to analytical operations for simple excitations, such as sinusoidal excitations and exponential excitations  
The proposed method falls into the category of the Volterra series method computed in the Laplace domain  Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations  Because the proposed method follows a similar path as a pole-residue method for linear systems , the proposed method to solve nonlinear system vibration problems is called the generalized pole-residue method 
The main concept of the pole-residue method developed by Hu et al  was that the poles and residues of the response could be easily obtained from those of the input and system transfer function to obtain the closed-form response solution of linear systems  This method included three steps: (1) writing the system transfer function into pole-residue form; (2) writing the excitation into pole-residue form by the Prony-SS method; (3) computing the poles and residues of the response by an algebraic operation based on those from system and excitation 
Compared to Hu et al  , which was regarded as an efficient tool to compute responses of linear systems, the generalized pole-residue method in this paper is introduced to compute responses of nonlinear systems  The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method 
Compared to the traditional pole-residue method for a linear system, one of the novelties of the generalized pole-residue method is how to deal with the higher-order poles and their corresponding coefficients  Similar to the Taylor series, the Volterra series representation is an infinite series, and convergence conditions are needed to assure that the representation is meaningful 
Because the proposed method is based on the Volterra series, only the system with convergent Volterra series representation can be treated by the proposed method  The paper is organized as follows  In Section 2, the nonlinear response is modelled by a Volterra series, and Volterra kernel functions are decoupled by Laguerre polynomials 
Then, the pole-residue method for computing explicit responses is developed in Section 3  Numerical studies and discussions are given in Section 4  Finally, the conclusions are drawn in Section 5 

Response calculation based on Volterra series

A nonlinear oscillator, whose governing equation of motion is given by where z(t, y, ẏ) represents an arbitrary nonlinear term; m, c, and k are the mass, damping and linear stiffness, respectively; y(t), ẏ(t) and ÿ(t) are the displacement, velocity and acceleration, respectively; and f (t) is the time-dependent excitation 
If the energy of excitation f (t) is limited, the nonlinear response under zero initial conditions (i e , zero displacement and zero velocity) can be represented by the Volterra series : where N is the order of Volterra series and In Eq  3, h 1 (τ ) is called the first-order Volterra kernel function, which represents the linear behaviour of the system; h n (τ 1 ,      
, τ n ) for n > 1 are the higher-order Volterra kernel functions, which describe the nonlinear behaviour of the system  The complete formulation of y(t) includes infinite series where the labour of calculating the n th term increases quickly with the growth of n  Fortunately, the response accuracy may be ensured by the first several order Volterra series 
This is proved here in numerical studies  The commonly known Laguerre polynomials are represented as : where p i is the order of the Laguerre polynomials and a i is the damping rate  The Laguerre polynomials satisfy the orthogonal relationship expressed as: By using Laguerre polynomials, the Volterra kernel function h n (t 1 ,      
, t n ) in Eq  3 can be decoupled as follows : where the coefficient is computed resorting to the orthogonal relationship in Eq  5: Substituting Eq  6 into Eq  3 yields       The above operation that uses the Laguerre polynomials to decouple Volterra higher order kernel functions has been well-developed 
The reader is referred to Refs  for details about the adopted technique  After decoupling Volterra higher order kernel functions in time, one can regroup Eq  8 into:       By denoting Eq  9 becomes The above procedure to compute the nonlinear response by a combination of Volterra series and Laguerre polynomials is schematically shown in Fig   
Volterra kernel functions h n (t 1 ,       , t n ) can be obtained by either an equation of motion or measured input-output signals  To derive a closedform solution of the response, we must obtain a closed-form solution of x i (t) first  In the following presentation, a closed-form solution of the aforementioned x i (t) and y n (t) is derived by using the pole-residue method 
3  Pole-residue method for calculating x i (t) and y n (t) Performing the Laplace transform of x i (t) in Eq  10 yields where in which Eq  13 includes a single pole and several higher-order poles  For k = 0, −a i is a single pole, and b p i (0) is a corresponding coefficient, namely, the residue  For k > 0, −a i are higher-order poles, and b p i (k) are corresponding coefficients 
For an irregular excitation signal f (t) of a finite duration of T , it can always be approximated into a pole-residue form by using the complex exponential signal decomposition method-Prony-SS : where N ℓ is the number of components; α ℓ and λ ℓ are constant coefficients, which either are real numbers or occur in complex conjugate pairs 
We define λ ℓ = −δ ℓ + iΩ ℓ , where Ω ℓ is the excitation frequency and δ ℓ is the damping factor of the ℓ th component  We denote α ℓ = A ℓ e iθ ℓ , where A ℓ is the amplitude and θ ℓ is the sinusoidal initial phase in radians  Taking the Laplace transform of Eq  15 yields Note that the concept of the Prony-SS method is similar to that of a principal component method 
A smooth excitation usually requires just several terms to achieve a good approximation  For high irregular loadings, including more terms would achieve a better approximation  Substituting Eqs  13 and 16 into Eq  12 yields Expressing xi (s) in its pole-residue form yields where λ ℓ are simple poles, and the corresponding residues are easily obtained by
and −a i are higher-order poles, and the corresponding coefficients are firstly derived as: By taking the inverse Laplace transform of Eq  18, a closed-form solution is obtained: Substituting Eqs  11 and 21 into Eq  2 yields Theoretically speaking, the proposed method for deriving the closed-form solution of the nonlinear response is applicable to any order of the Volterra series 
For practical engineering, usually only the first several order responses dominate  By setting up N = 2, Eq  22 can be simplified into three components: where the natural response, which is only related to system poles, is given by and the cross response, which is related to both system poles and excitation poles, is given by
and the forced response, which is related only to excitation poles, is given by The first term in Eq  26 is the first-order forced response governed by the excitation frequency, i e , the imaginary part of the pole λ ℓ   The second term corresponds to the second-order nonlinear forced response, which includes the sum frequency and difference frequency responses governed by λ ℓ + λ j  
Eq  26 straightforwardly offers visible information about the possible nonlinear vibrations by the cooperation of excitation frequencies  Particularly, consider a sinusoidal excitation f (t) = sin ω r t, which can be expressed as f (t) = γe λt + γ * e λ * t , where γ = −0 5i and λ = iω r   Substituting these values into Eq 
26, the second term of Eq  26 is simplified as where the first term is the difference frequency response, and the second term is the sum frequency response 

Numerical studies

In practical engineering, some systems have an accurate equation of motion  Additionally, some systems have difficulty constructing their equations of motion because of complex nonlinear dynamic behaviours and uncertain system parameters  In this article, a system with a known equation of motion is called a known system, and a system with an unknown equation of motion is called an unknown system for simplicity 
In this section, two numerical studies are presented  The first study verifies the proposed method using a known nonlinear oscillator, and the second study demonstrates the applicability of the proposed method to an unknown system  Throughout the numerical studies, the unit system is the metre-kilogramme-second (MKS) system; for conciseness, explicit units for quantities are omitted 

A known nonlinear system

This study chooses a nonlinear oscillator written as: where mass m = 1, damping c = 1, linear stiffness k 1 = 10, quadratic stiffness k 2 = 20 and cubic stiffness k 3 = 20  It is a case that has been studied in a previously published article   The linear natural frequency of the system ω 0 = k 1 /m = 3 16 and the damping ratio ζ = c/(2mω 0 ) = 15 8% 
This kind of oscillator occurs in many engineering problems, such as a model of fluid resonance in a narrow gap between large vessels   In the model, k 1 y represents the linear restoring force of the fluid, and k 2 y 2 and k 3 y 3 are respectively the quadratic and cubic nonlinear restoring forces of the fluid 

Volterra kernel functions

Generally, the first several order responses dominate the total response of a system  Hence, the order of the Volterra series in Eq  22 is chosen to be 3, namely, N = 3  For computing the first three order responses from Eq  22, the first three order Volterra kernel functions need to be known  Since Volterra kernel functions and corresponding frequency response functions are related by a specific Fourier transform pair, we can first write the first three orders of frequency response functions directly from Eq  28 
Then, Volterra kernel functions are obtained by the inverse Fourier transform  Based on the harmonic probing algorithm , the linear frequency response function (LFRF) H 1 (ω), the quadratic frequency response function (QFRF) H 2 (ω 1 , ω 2 ) and the cubic frequency response function (CFRF) H 3 (ω 1 , ω 2 , ω 3 ) are analytically given by:
and Figures show H 1 (ω), H 2 (ω 1 , ω 2 ) and H 3 (ω 1 , ω 2 , ω 3 ), respectively, which agree well with those reported in Ref    As expected, the modulus of H 1 (ω) in Fig  peaks near the linear natural frequency ω 0 , and the phase angle decreases monotonically from 0 to -π with increasing frequency 
Figure shows the sum frequency QFRF, where the energy converges along the line of ω 1 +ω 2 ≈ ω 0   Therefore, when the sum frequency of a two-tone excitation equals the linear resonant frequency, the second-order response may reach its maximum  Additionally, those pairs of excitations in line ω 1 + ω 2 ≈ ω 0 may produce non-negligible vibration magnitudes due to second-order nonlinear effects 
For the difference frequency QFRF in Fig  (b), the energy converges along two main lines, i e , ω 1 ≈ ω 0 and ω 2 ≈ ω 0   Figures show moduli of H 3 (ω, ω, ω) and H 3 (ω, ω, −ω), which are diagonal terms of the sum frequency CFRF and the difference frequency CFRF, respectively  While the modulus of H 3 (ω, ω, ω) peaks near ω ≈ ω 0 /3 and ω 0 , that of H 3 (ω, ω, −ω) peaks near ω ≈ ω 0 with a small hump around ω ≈ ω 0 /2 
Values at ω ≈ ω 0 /3 and ω 0 /2 may be magnified by higher-order stiffness terms in Eq  28  By performing the inverse fast Fourier transform to Eqs  29-31, the corresponding linear impulse response function h 1 (t), quadratic impulse response function h 2 (t 1 , t 2 ) and cubic impulse response function h 3 (t 1 , t 2 , t 3 ) are obtained 
Here, h 1 (t) and h 2 (t 1 , t 2 ) are plotted in Figs  , respectively, and h 3 (t, t, t) is shown in Fig    In the numerical implementation, Eqs  29-31 have been utilized with the frequency interval ∆ω = 0 1, number of frequency components N n = 1025, and cut-off frequencies 102 4 and −102 4  For decoupling Volterra kernel functions by using Laguerre polynomials, the damping rate and number of Laguerre polynomials for each order Volterra kernel function need to be determined (see Eqs  4 and 6) 
In this example, we set a 1 = a 2 = a 3 = 2 and R 1 = R 2 = R 3 = 24 because coefficients c p 1    pn become very small when R n > 24, n = 1, 2, 3  According to Eq  7, the coefficients of the first three order Volterra kernel functions are calculated, which are shown in Figs  9 and 10  For convenience, Fig  plots only c p 1 p 2 p 3 for p 3 = 0 
With the increase of the order of Laguerre polynomials, coefficients in Figs  9 and 10 gradually decrease, which illustrates how the first several orders of Laguerre polynomials dominate all orders of the Volterra kernel function  With the known Laguerre polynomials and corresponding coefficients, Volterra kernel functions are reconstructed by Eq  6 
For comparison, reconstructed Volterra kernel functions are also plotted in Figs    The reconstructed results agree well with the analytical values, which verifies the accuracy of the decomposition 

Sinusoidal excitation

From Eq  28, we consider a sinusoidal excitation where A and Ω are the amplitude and the frequency, respectively  Five cases of A and Ω are shown in Table   Excitation frequencies in Cases 1 and 2 are larger than the linear natural frequency (ω 0 ≈ 3 16), those in Case 3 are very close to ω 0 , and those in Cases 4 and 5 are smaller than ω 0  
All cases have same amplitudes  The poles of a sinusoidal excitation are λ 1,2 = ±iΩ, and the residues are α 1,2 = ∓iA/2  Numerical values of excitation poles and residues for different cases are listed in Table   Table : Parameter values, poles and residues of the sinusoidal excitation Substituting poles and residues of the excitation, as well as those of the system into Eqs 
20 and 19, response coefficients β p i ,k corresponding to system poles −a i and response coefficients γ p i ,ℓ corresponding to excitation poles λ ℓ are calculated, respectively  According to Eq  22, the first three orders of responses for each case in Table are calculated  Figures )-15(a) show the comparison of responses obtained by the proposed method and the fourth-order Runge-Kutta method with ∆t = 10 −4  
For Cases 1 and 2, the first-order responses agree well with the total responses obtained by the Runge-Kutta method, and the higher-order responses only slightly improve the transient parts  For Cases 3-5, the sum of the first three orders of responses is in good agreement with the Runge-Kutta solution 
When the response nonlinearity increases, higher-order responses need to be considered  In other words, the proposed method can accurately compute the nonlinear responses by choosing a small number N of Volterra series terms  Figures )-15(b) show the contributions of the three response components for the five cases 
In each case, the first-order response is the most dominant component, and the contributions of secondand third-order responses are much less than those of the first-order response  Especially for Cases 1 and 2, whose excitation frequencies are far from the linear natural frequency, second-and thirdorder responses are close to zero 
This may be because the QFRF and CFRF approach zero when the frequency is larger than 4 rad/s (see Figs  )  Furthermore, the mean values of the first-order responses are approximately zero, and those of the second-order responses are always smaller than zero, which are the difference frequency components in Eq  27 
Moreover, it is clearly observed that second-order responses for Cases 3-5 exhibit a periodic oscillation with a period near half of that for the first-order response, which is excited by the sum frequency component of the excitation (see second part of Eq  27)  Compared with steady-state solutions of first-and second-order responses, those of third-order responses in Cases 3-5 are no longer single regular motions 
By performing the FFT, frequency spectra of these three third-order responses are shown in Fig    We find that these three third-order responses are all dominated by their own fundamental harmonic component and the third harmonic (triple frequency) component  Figure shows the computational time to calculate the response of the oscillator for Case 1 by the proposed method, the fourth-order Runge-Kutta method and the convolution method 
The proposed method, which has an explicit solution, is much more efficient in computational time than the latter two methods, which need small time steps to obtain high-precision solutions  In particular, the efficiency of the proposed method increases with the length of the response time  Computation time (sec )
t=0 02s Convolution, t=0 02s Convolution, t=0 001s Runge-Kutta, t=0 001s Fig  : Comparison of computation efficiency of the proposed method, the fourth-fifth order Runge-Kutta method and the convolution method regular loading in Case 1

Irregular excitation

In Eq  28, considering an irregular excitation consisting of several cosine functions where N f is the number of cosine components; A n , Ω n and θ n are the amplitude, frequency and phase angle of the n th component, respectively  Table lists three cases of these parameters  In each case, the amplitudes of all components are the same, and phase angles θ n uniformly distributed between 0 and 2π are randomly generated 
To decompose the excitation into a pole-residue form, the Prony-SS method is used, whose concept is similar to that of a principal component method  The readers are referred to Ref  for details  The chosen rank of each case is also shown in Table   Figure shows the comparison of original excitations and reconstructed results of these three cases, which all have excellent agreement 
show the results computed by the fourth-order Runge-Kutta method  In all cases, the sums of the first three orders of responses agree well with those obtained by the Runge-Kutta method  The contributions of the first three orders of responses for each case are plotted in Figs  )-21(b)  Similarly, the system vibration is dominated by the first-order response 
However, the contributions of second-and third-order significantly grow with increasing excitation magnitude and frequency number  Furthermore, when the magnitude of the nonlinear response becomes large, sharp troughs are present  This phenomenon may be induced by the nonlinear stiffness  While the first-order response fails to capture these troughs, the higher-order responses successfully capture these troughs 
Figure plots the computational time to calculate the response of the oscillator for the irregular loading in Case 1 by the proposed method and the fourth-fifth order Runge-Kutta method, respectively  While the fourth-fifth order Runge-Kutta method is more efficient under a small response length, the proposed method becomes much more efficient when the response length is larger than about 130 s 
In addition, the proposed method obtains the explicit response solution, so one can directly obtain the response value at a specific time t p instead of integrating from 0 to t p for traditional numerical methods  Computation time (sec ) Proposed, t=0 02s Runge-Kutta, t=0 001s Fig  : Comparison of computation efficiency of the method and the fourth-fifth order Runge-Kutta method for irregular loading in Case 1

An unknown nonlinear

To check the applicability of the proposed method to an unknown nonlinear system, a known input excitation and its corresponding response are used to identify its Volterra kernel functions  When the Volterra kernel functions are known, we can follow the procedure in Section 4 1 to predict system responses 
In this study, the input excitation is white noise with a constant power spectrum S 0 = 0 001, and the corresponding response is obtained by solving Eq  28 by the fourth-order Runge-Kutta method, which is shown in Fig    From Section 4 1, we determine that the sum of the first two orders of responses agrees well with the total response 
In this study, the order of Volterra series N is chosen to be 2, damping rates of Laguerre polynomials are a 1 = a 2 = 2, and numbers of Laguerre polynomials are R 1 = R 2 = 24  To estimate the first two orders of Volterra kernel functions, a matrix equation is constructed using excitation data and response data 
By using the least square method to solve this matrix equation, coefficients c p 1 and c p 1 p 2 in Eq  8 are identified  Figure plots c p 1 and c p 1 p 2 , respectively, which have good agreement with the exact results in Fig    Then, the first two order Volterra kernel functions are constructed by Eq  6 
Compared with the exact results in Figs  , the identified Volterra kernel functions in Fig  completely agree well with the exact solutions  Note that the white noise excitation, which can excite more frequency components of the response, is chosen to obtain good Volterra kernel functions  A regular excitation f (t) = sin(πt) and an irregular excitation f (t) = N f n=1 A n cos(Ω n t + θ n ) with A n = 0 3 and Ω n varying from 0 to 40 with equal interval 1 are chosen as input excitations 
The predicted responses, along with results obtained by the fourth-order Runge-Kutta method, are shown in Fig    In both cases, the proposed method accurately predicts system responses  As presented in Eq  23, a nonlinear response is the sum of three terms: natural response y s (t), forced response y f (t) and cross response y c (t) 
These individual terms, as well as their sum to two excitations, are shown in Figs  27 and 28, respectively  As shown in Figs  and 28, both first-and second-order responses include the natural response y s (t) and the forced response y f (t), but the cross response y c (t) only exists in second-order responses 
When t becomes larger, both y s (t) and y c (t) diminish due to the presence of system damping, and the total response is entirely governed by y f (t)  Moreover, we notice some features at t = 0 for these components, including y s (0) = −y f (0) for the first-order response and y s (0) + y f (0) = −y c (0) for the second-order response, which are due to imposed zero initial conditions 

Conclusions

Considering arbitrary irregular excitations, an efficient generalized pole-residue method to compute the nonlinear dynamic response modelled by the Volterra series was developed  A core of the proposed method was obtaining poles and corresponding coefficients of Volterra kernel functions, then those of each order response modelled by each order Volterra series 
Once the poles and corresponding coefficients of Volterra kernel functions and excitations were both available, the remaining derivation could follow a similar pole-residue method that had been developed for ordinary linear oscillators  To obtain the poles and corresponding coefficients of Volterra kernel functions, two steps were included: (1) using Laguerre polynomials to decouple higher-order Volterra kernel functions with respect to time and (2) obtaining poles and corresponding coefficients of Laguerre polynomials in the Laplace domain 
Because the proposed method gave an explicit, continuous response function of time, it was much more efficient than traditional numerical methods  Moreover, many meaningful physical and mathematical insights were gained because not only each order response but also the natural response, the forced response and the cross response of each order were obtained in the solution procedure 
To demonstrate that the proposed method was not only suitable for a system with a known equation of motion but also applicable to a system with an unknown equation of motion, two numerical studies were conducted  For each study, regular excitations and complex irregular excitations with different parameters were investigated 
The efficiency of the proposed method was verified by the fourth-order Runge-Kutta method  This paper only computes the response under zero initial conditions  The response under non-zero initial conditions will be investigated in our future work 



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Context 3: Paper Info

Title: Deep Spiking Neural Networks with High Representation Similarity Model Visual Pathways of Macaque and Mouse
Publish Date: 22 May 2023
Author List: Zhengyu Ma (from Department of Networked Intelligence, Peng Cheng Laboratory), Yu Liutao (from Department of Networked Intelligence, Peng Cheng Laboratory), Huihui Zhou (from Department of Networked Intelligence, Peng Cheng Laboratory), Allen Brain
Author Affiliation: CORNet-S ConvNeXt-Tiny ConvNeXt-Small EfficientNet, AlexNet RegNetY, ResNet34 ConvNeXt-Base CORNetSEW, ResNet8 ResNet101 SEW-ResNet18 ViT-L, GoogLeNet SEW-ResNet34 SEW-ResNet8 Wide

Figure

Figure 1: To conduct neural representation similarity experiments, we apply three similarity metrics to a layer-by-layer comparison between the responses of models and the neural activities of visual cortex 
Figure 2: For three datasets and three similarity metrics, each point indicates the final representation similarity score of a model Each pair of SEW ResNet and ResNet with the same depth are linked by a gray solid line In almost all conditions, SEW ResNet outperforms ResNet by a large margin 
Figure3: For three datasets and three similarity metrics, we plot the trajectories of similarity score with model layer depth The models are divided into two groups: ResNet and SEW ResNet The normalized layer depth ranges from 0 (the first layer) to 1 (the last layer) Because the depths of models are not the same, we first discretize the normalized depth into 50 bins, and then apply the cubic spline interpolation to the scores of each model, yielding the smooth trajectories shown in the plot The fine, semitransparent lines are the trajectories of each model The thick lines are the average trajectories among each group 
Figure 5: For Macaque-Synthetic dataset, trajectories of similarity score with model layer depth are plotted The models are divided into two groups: ViT and CNN&SNN The normalized layer depth ranges from 0 (the first layer) to 1 (the last layer) The calculation and plotting of the trajectories are the same as Figure 3 
Figure6: The basic block of SpikingMobileNet "PW CONV" is the pointwise convolution and "DW CONV" is the depthwise convolution "SN" is the spiking neuron 
Figure 7: Overall model rankings of the similarity scores on Allen Brain mouse dataset The similarity scores of CNNs, SNNs and vision transformers are shown by blue, green and orange bars, respectively 
Figure 9: Overall model rankings of the similarity scores on Macaque-Synthetic dataset 
Figure 10: The Spearman's rank correlation between the overall model rankings of different metrics There is a strong correlation between SVCCA and TSVD-Reg, but RSA has weaker correlations with them 
The correlation between the similarity scores and the model depth r is Spearman's rank correlation coefficient "-" indicates that there is no significant correlation 
Architectures of SNNs "sn" denotes the spiking neuron "g = 32" denotes the grouped convolutions with 32 groups The hyper-parameters of the spike-element-wise block are shown in the brackets with the number of stacked blocks outside 

abstract

Deep artificial neural networks (ANNs) play a major role in modeling the visual pathways of primate and rodent  However, they highly simplify the computational properties of neurons compared to their biological counterparts  Instead, Spiking Neural Networks (SNNs) are more biologically plausible models since spiking neurons encode information with time sequences of spikes, just like biological neurons do 
However, there is a lack of studies on visual pathways with deep SNNs models  In this study, we model the visual cortex with deep SNNs for the first time, and also with a wide range of state-of-the-art deep CNNs and ViTs for comparison  Using three similarity metrics, we conduct neural representation similarity experiments on three neural datasets collected from two species under three types of stimuli 
Based on extensive similarity analyses, we further investigate the functional hierarchy and mechanisms across species  Almost all similarity scores of SNNs are higher than their counterparts of CNNs with an average of 6 6%  Depths of the layers with the highest similarity scores exhibit little differences across mouse cortical regions, but vary significantly across macaque regions, suggesting that the visual processing structure of mice is more regionally homogeneous than that of macaques 
Besides, the multi-branch structures observed in some top mouse brain-like neural networks provide computational evidence of parallel processing streams in mice, and the different performance in fitting macaque neural representations under different stimuli exhibits the functional specialization of information processing in macaques 
Taken together, our study demonstrates that SNNs could serve as promising candidates to better model and explain the functional hierarchy and mechanisms of the visual system  Originally, the prototype of deep neural networks is inspired by the biological vision system   To date, deep neural networks not only occupy an unassailable position in the field of computer vision , but also become better models of the biological visual cortex compared to traditional models in the neuroscience community (Khaligh-Razavi and Kriegeskorte 2014;  
They have been successful at predicting the neural responses in primate visual cortex, matching the hierarchy of ventral visual stream (Güc ¸lü and van Gerven 2015; , and even controlling neural activity   Moreover, as training paradigms of mice and techniques for collecting neural activity (de Vries et al  2020) have been greatly improved, there is a strong interest in exploring mouse visual cortex 
Deep neural networks also play an important role in revealing the functional mechanisms and structures of mouse visual cortex   Compared to biological networks, Artificial Neural Networks discard the complexity of neurons   Spiking Neural Networks, incorporating the concept of time and spikes, are more biologically plausible models  
To be more specific, because of their capabilities of encoding information with spikes, capturing the dynamics of biological neurons, and extracting spatio-temporal features, deep SNNs are highly possible to yield brain-like representations )  However, deep SNNs have not been employed to model visual cortex due to the immaturity of training algorithms 
Recently, a state-ofthe-art directly trained deep SNN , makes it possible to use deep SNNs as visual cortex models  Contributions  In this work, we conduct large-scale neural representation similarity experiments on SNNs and other high-performing deep neural networks to study the brain's visual processing mechanisms, with three datasets and three similarity metrics (Figure ) 
Specifically, to the best of our knowledge, we are the first to use deep SNNs to fit complex biological neural representations and explore the biological visual cortex  We summarize our main contributions in four points as follows  • We find that SNNs outperform their counterparts of CNNs with the same depth and almost the same architectures in almost all experiments 
In addition, even with very different depths and architectures, SNNs can achieve top performance in most conditions  • By making a more direct comparison between macaques and mice for the first time, we reveal the differences in the visual pathways across the two species in terms of the homogeneity of visual regions and the increases of receptive field sizes across cortical visual pathways, which is consistent with previous physiological work 
• The multi-branch structures in neural networks benefit neural representation similarity to mouse visual cortex, providing computational evidence that parallel information processing streams are widespread between cortical regions in the mouse visual system  • Comparing the results of two macaque neural datasets under different stimuli, we reveal that the macaque vision system may have functional specialization for processing human faces and other natural scenes 
Altogether, as the first work to apply deep SNNs to fit neural representations, we shed light on visual processing mechanisms in both macaques and mice, demonstrating the potential of SNNs as a novel and powerful tool for research on the visual system  Our codes and appendix are available at https://github com/Grasshlw/SNN-Neural-Similarity 
There are plenty of computational models of macaque and mouse visual systems for exploring the visual processing mechanisms recently  We summarize some of the outstanding work in the following  The network models of macaque visual system  In the early days, studies basically used simple feedforward neural networks as the models of the macaque visual system (Khaligh-Razavi and Kriegeskorte 2014;  
Recently, some bio-inspired or more complex models achieved better performance in fitting the neural representations of macaque visual cortex   proposed a brainlike shallow CNN with recurrent connections to better match the macaque ventral visual stream  By mimicking the primary stage of the primate visual system, VOneNets ) performed more robustly in image recognition while better simulating macaque V1 
Moreover, the representations learned by unsupervised neural networks ) also effectively matched the neural activity of macaque ventral visual stream  Although the above work developed many bio-inspired structures, the networks are still traditional ANNs in nature  Our work introduces deep SNNs for the first time to explore the visual processing mechanisms of macaque visual system 
The network models of mouse visual system  Largescale mouse neural dataset provided an experimental basis for model studies of mouse visual system (de Vries et al  2020;   conducted comparisons between the representations of mouse visual cortex and the VGG16 trained on the Im-ageNet dataset  In , they developed a single neural network to model both the dorsal and ventral pathways with showing the functional specializations 
What's more, a large survey of advanced deep networks ) revealed some hierarchy and functional properties of mice  Similar to the studies of macaque visual system, deep SNNs have never been used to model the mouse visual system  In this work, we not only use SNNs as one of the candidates to fit the representations of mouse visual cortex, but also conduct direct comparisons between macaques and mice to further investigate the functional hierarchy and mechanisms of the two species 
Our work is conducted with three neural datasets  These datasets are recorded from two species under three types of stimuli  More specifically, there are neural responses of mouse visual cortex to natural scene stimuli, and responses of macaque visual cortex to face image and synthetic image stimuli  Allen Brain mouse dataset 
It is part of the Allen Brain Observatory Visual Coding dataset ) col-lected using Neuropixel probes from 6 regions simultaneously in mouse visual cortex  Compared to two-photon calcium imaging, Neuropixel probes simultaneously record the spikes across many cortical regions with high temporal resolution 
In these experiments, mice are presented with 118 250-ms natural scene stimuli in random orders for 50 times  Hundreds to thousands of neurons are recorded for each brain region  To get the stable neurons, we first concatenate the neural responses (average number of spikes in 10-ms bins across time) under 118 images for each neuron, and then preserve the neurons whose split-half reliability across 50 trials reaches at least 0 8 
Macaque-Face dataset  This dataset ) is composed of neural responses of 159 neurons in the macaque anterior medial (AM) face patch under 2,100 real face stimuli, recorded with Tungsten electrodes  For this dataset, we compute the average number of spikes in a time window of 50-350ms after stimulus onset and exclude eleven neurons with noisy responses by assessing the neurons' noise ceiling 
The details of the preprocessing procedure are the same as   Macaque-Synthetic dataset  This dataset is also about macaque neural responses which are recorded by electrodes under 3,200 synthetic image stimuli, and used for neural prediction in the initial version of Brain-Score   The image stimuli are generated by adding a 2D projection of a 3D object model to a natural background 
The objects consist of eight categories, each with eight subclasses  The position, pose, and size of each object are randomly selected  88 neurons of V4 and 168 neurons of IT are recorded  The neural responses are preprocessed to the form of average firing rate and can be downloaded from Brain-Score  Since the core visual function of macaque and mouse visual cortex is to recognize objects, the basic premise of model selection is that the model has good performance on object recognition tasks (e g 
classification on ImageNet)  Based on this premise, we employ 12 SNNs, 43 CNNs, and 26 vision transformers, all of which are pretrained on the Ima-geNet dataset and perform well in the classification task  As for SNNs, we use SEW ResNet as the base model, which is the deepest and SOTA directly trained SNN  
Furthermore, by combining the residual block used in SEW ResNet and the hierarchy of the visual cortex, we build several new SNNs and train them on the ImageNet using SpikingJelly ) (see Appendix A for model structures and the details of model training)  As for CNNs and vision transformers, we use 44 models from the Torchvision model zoo , 22 models from the Timm model zoo ) and 3 models from the brain-like CNNs, CORnet family ) 
In the feature extraction procedures of all models, we feed the same set of images used in biological experiments to the pretrained models and obtain features from all chosen layers  Different from CNNs and vision transformers, the features of SNNs are spikes in multiple time steps  To obtain the representation similarity between biological visual cortex and computational models, we apply three similarity metrics to computing similarity scores: representational similarity analysis (RSA) , regression-based encoding method and singular vector canonical correlation analysis (SVCCA)  
RSA has already been widely used to analyze neural representations of a model and a brain to different stimuli at the population level, while the regression-based encoding method directly fits the model features to neural activity data  SVCCA is originally proposed to compare features of deep neural networks, and then Buice 2019) used it to compare representation matrices from mouse visual cortex and DNNs, which demonstrated its effectiveness 
With the same model and same cortical region, we use these metrics for a layer-by-layer comparison to compute the similarity scores  The maximum similarity score across layers for a given cortical region is considered to be the level of representation similarity between the model and the cortical region 
Finally, in a given dataset, we take the average score of all cortical regions as the final similarity score for each model, which gives the overall model rankings  The implementation of each similarity metric is as follows  RSA  For two response matrices R ∈ R n×m from each layer of models and each cortical region, where n is the number of units/neurons and m is the number of stimuli, we calculate the representational similarity between the responses to each pair of image stimuli using the Pearson correlation coefficient r, yielding two representational dissimilarity matrices (RDM ∈ R m×m , where each element is the correlation distance 1 − r) 
Then, the Spearman rank correlation coefficient between the flattened upper triangles of these two matrices is the metric score  Regression-Based Encoding Method  Firstly, we run truncated singular value decomposition (TSVD) to reduce the feature dimension of model layers to 40  Secondly, the features after dimensionality reduction are fitted to the representations of each neuron by ridge regression 
Finally, we compute the Pearson correlation coefficient between the predicted and ground-truth representations of each neuron and take the mean of all correlation coefficients as the metric score  More specifically, we apply leave-one-out crossvalidation to obtain predicted representations of each neuron 
For simplicity, we name this method 'TSVD-Reg'  SVCCA  For both the responses of model layers and cortical regions, we use TSVD to reduce the dimension of unit/neuron to 40, yielding two reduced representation matrices  Then we apply canonical correlation analysis (CCA) to these two matrices to obtain a vector of correlation coefficients (the length of the vector is 40) 
The metric score is the mean of the vector  Because of the invariance of CCA to affine transformations , in this procedure, we only need to ensure that the stimulus dimension is consistent and aligned, even if the unit/neuron dimension is different  Dimensionality reduction plays an important role in this method to make the number of model features comparable to the number of neurons in cortical regions, since the former usually far exceeds the latter 
In addition, dimensionality reduction helps to determine which features are important to the original data, while CCA suffers in important feature detection  Using just CCA performs badly, which has been proven by   To check how similar the models are to the visual cortex's mechanisms in visual processing, we rank the final similarity scores of all models and conduct comparisons among three types of models (CNNs, SNNs, and vision transformers) 
Specially, we focus on comparing SNN (SEW ResNet) and CNN (ResNet) with the same depth and almost the same architectures (Figure )  The final similarity score of a model is the average similarity score across all cortical regions  (The overall rankings can be found in Appendix B and the comparisons among three types of models are shown in Appendix C )
Allen brain mouse dataset  No single model achieves the highest final similarity scores with all three metrics  For a fair comparison, we apply the paired t-test to SEW ResNet and ResNet with the same depth  For all three metrics, SEW ResNet performs better than ResNet by a large margin (t = 5 857, p = 0 004; t = 7 666, p = 0 002; t = 7 592, p = 0 002) 1   1 The results of the three similarity metrics are separated by semicolons, in the order of SVCCA, TSVD-Reg, and RSA 
Other Macaque-Face dataset



ground_truth: The paper aims to solve nonlinear system vibration problems efficiently.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the potential of SNNs in modeling the visual system?

Read the following text and answer briefly.

Context 3:  For both SVCCA and TSVD-Reg, Wide-SEW-ResNet14 and Wide-SEW-ResNet8 achieve the first and second highest final similarity scores respectively  But for RSA, TNT-S and Inception-ResNet-V2 take their place and outperform other models by a large margin  As for SEW ResNet and ResNet, the former performs significantly better than the latter for both SVCCA and TSVD-Reg (t = 8 195, p = 0 001; t = 7 528, p = 0 002) 
However, the difference is not significant for RSA (t = 1 117, p = 0 327)  Specifically, the similarity score of SEW ResNet152 is only slightly higher than that of ResNet152, and at the depth of 50 and 101, SEW ResNet's scores are lower than ResNet's  Macaque-Synthetic dataset  Similar to the results of Allen Brain dataset, no model performs best for all three metrics 
SEW ResNet performs moderately better than ResNet (t = 3 354, p = 0 028; t = 3 824, p = 0 019; t = 2 343, p = 0 079)  The only contrary is that SEW ResNet18 performs worse than ResNet18 for RSA  Further, to check the details of comparison between the SNNs and their CNN counterparts, we analyze the trajectories of similarity score across model layers (Figure ) 
As for ResNet and SEW ResNet with the same depth, the trends of their similarities across model layers are almost the same, but the former's trajectory is generally below the latter's  In other words, the similarity scores of SEW ResNet are higher than those of ResNet at almost all layers  Taken together, the results suggest that when the overall results that appear below also correspond to the three metrics in this order, unless the correspondence is stated in the text 
architectures and depth are the same, SNNs with spiking neurons perform consistently better than their counterparts of CNNs with an average increase of 6 6%  Besides, SEW ResNet14 also outperforms the brain-like recurrent CNN, CORnet-S, with the same number of layers (see more details in Appendix B)  Two properties of SNNs might contribute to the higher similarity scores 
On the one hand, IF neurons are the basic neurons of spiking neural networks  The IF neuron uses several differential equations to roughly approximate the membrane potential dynamics of biological neurons, which provides a more biologically plausible spike mechanism for the network  On the other hand, the spiking neural network is able to capture the temporal features by incorporating both time and binary signals, just like the biological visual system during information processing 
To figure out the distinctions in the functional hierarchy between macaques and mice, for each cortical region, we obtain the normalized depth of the layer that achieves the highest similarity score in each model  Then, we divide models (excluding vision transformers) into two groups based on their depths and conduct investigations on these two groups separately 
A nonparametric ANOVA is applied to each group for testing whether layer depths change significantly across cortical regions  For mouse visual cortex (Figure (a)), taking the deep model group as an example, ANOVA shows overall significant changes in depth across cortical regions for TSVD-Reg and RSA (Friedman's χ 2 = 49 169,
p = 2 0 × 10 −9 ; χ 2 = 19 455, p = 0 002)  But there is no significant change for SVCCA (χ 2 = 8 689, p = 0 122)  According to these results, the differences in depth across regions are indeterminacy and irregular  Meanwhile, the trends of layer depth between some regions contradict the hierarchy observed in physiological experiments of mice (those between VISp and VISrl for TSVD-Reg and between VISal and VISpm for RSA) 
However, for macaque visual cortex (Figure (b)), there are significant differences (t = −5 451, p = 6 5 × 10 −6 ; t = −8 312, p = 2 8 × 10 −9 ; t = −3 782, p = 6 9 × 10 −4 , also taking the deep model group as an example) between V4 and IT, and the trend is consistent with the information processing hierarchy in primate visual cortex 
The comparative analyses of the best layer depths of the shallow and deep model groups also exhibit the differences between macaques and mice  For mouse visual cortex, the best layer depths of shallow models are significantly higher than those of deep models  Compared to deep models, most shallow models achieve the top similarity scores in intermediate and even later layers 
Differently, for macaque visual cortex, the depth of models has little effect on the depth of the most similar layer  What's more, we find that the most similar layer of mouse visual cortex always occurs after the 28 × 28 feature map is downsampled to 14 × 14, which leads to the layer depths' difference between shallow and deep models 
Nevertheless, the best layer of macaque IT appears in the last part of networks, where the feature map has been downsampled more times  In summary, our results might reveal two distinctions in the functional hierarchy between macaques and mice  First, there is a distinct functional hierarchical structure of macaque ventral visual pathway, while there might be no clear sequential functional hierarchy in mouse visual cortex 
One explanation is that the mouse visual cortex is organized into a parallel structure and the function of mouse cortical regions are more generalized and homogeneous than those of macaques  Another possibility would be that even though the sequential relations exist among mouse cortical regions as proposed in anatomical and physiological work, they are too weak for the current deep neural networks to capture 
Additionally, mice perform more complex visual tasks than expected with a limited brain capacity   Consequently, the neural responses of mouse visual cortex may contain more information not related to object recognition that neural networks focus on  Secondly, it is well known that the units in the neural networks get larger receptive fields after downsampling, and through the analyses of differences between two groups of models based on depth, we find the feature map of the best layer for mouse is downsampled fewer times than that for macaque 
Based on these results, we provide computational evidence that the increased ratio of the receptive field size in cortical regions across the mouse visual pathway is smaller than those across the macaque visual pathways, which echoes some physio- Macaque-Face dataset --- Table : The correlation between the similarity scores and the number of parameters 
r is Spearman's rank correlation coefficient  "-" indicates that there is no significant correlation  To explore the processing mechanisms in the visual cortex of macaques and mice, we investigate the model properties from the whole to the details  As shown in Table and 2, we first measure the correlation between the similarity scores and the sizes (i e  the number of trainable parameters and the depth) of network models 
For Allen Brain mouse dataset, there are significant negative correlations between the similarity scores and the number of parameters for three metrics while there is no correlation with the depth  Conversely, for the two macaque neural datasets, the similarity scores are highly correlated with the depth of networks, but not with the number of parameters 
Specifically, there is a positive correlation for Macaque-Face dataset while a negative correlation for Macaque-Synthetic dataset  (We also apply the linear regression to analyze the correlation between the similarity scores and the model size  The results are consistent with Spearman's rank correlation and are shown in Appendix E) 
Based on these results, we further investigate more detailed properties of neural networks to explain the processing mechanisms in the visual cortex  For the mouse dataset, on the one hand, the best layer depths show non-significant changes across the mouse cortical regions as mentioned in the previous section 
On the other hand, the similarity scores of the mouse dataset are only correlated with the number of model parameters but not with the depth of models  It calls into the question whether any detailed structures in the neural networks help to reduce the number of parameters and improve its similarity to mouse visual cortex 
Therefore, we explore the commonalities between models that have the top 20% representation similarities (see Appendix D) for Allen Brain dataset  As expected, the top models contain similar structures, such as fire module, inception module, and depthwise separable convolution  All these structures essentially process information through multiple branches/channels and then integrate the features from each branch 
The models with this type of structure outperform other models (t = 2 411, p = 0 024; t = 3 030, p = 0 007; t = 1 174, p = 0 247)  Moreover, we apply the depthwise separable convolution to SNNs, which yields a positive effect  The representation similarity of Spiking-MobileNet is higher than SEW-ResNet50 with a similar depth (+0 8%; +3 9%; +12 1%) 
In fact, some studies using multiple pathways simulate the functions of mouse visual cortex to some extent   Our results further suggest that not only the mouse visual cortex might be an organization of parallel structures, but also there are extensive parallel information processing streams between each pair of cortical regions  
For the two macaque datasets with different stimuli, not only are the model rankings significantly different, but also the correlations between the similarity scores and the model depth are totally opposite  These results corroborate the following two processing mechanisms in macaques: the ventral visual stream of primate visual cortex possesses canonical coding principles at different stages; the brain exhibits a high degree of functional specialization, such as the visual recognition of faces and other objects, which is reflected in the different neural responses of the corresponding region (although the face patch AM is a sub-network of IT, they differ in the neural representations) 
Besides, as shown in Figure , The calculation and plotting of the trajectories are the same as Figure   the similarity scores of vision transformers reach the maximum in the early layers and then decrease  Differently, the scores of CNNs and SNNs keep trending upwards, reaching the maximum in almost the last layer 
On the other hand, Appendix C shows that vision transformers perform well in Macaque-Face dataset but poorly in Macaque-Synthetic dataset  Considering the features extraction mechanism of vision transformers, it divides the image into several patches and encodes each patch as well as their internal relation by self-attention 
This mechanism is effective for face images that are full of useful information  However, the synthetic image consists of a central target object and a naturalistic background  When vision transformers are fed with this type of stimuli, premature integration of global information can lead to model representations containing noise from the unrelated background 
What's more, when we take all models with the top 20% representation similarities as a whole for analyses, as described in the above paragraph, the properties that enable networks to achieve higher neural similarity are not yet clear  Taken together, the computational mechanism of the better models may reveal core processing divergence to different types of stimuli in the visual cortex 
In this work, we take large-scale neural representation similarity experiments as a basis, aided by analyses of the similarities across models and the visual cortical regions  Compared to other work, we introduce SNNs in the similarity analyses with biological neural responses for the first time, showing that SNNs achieve higher similarity scores than CNNs that have the same depth and almost the same architectures 
As analyzed in Section 3 1, two properties of SNNs might serve as the explanations for their high similarity scores  The subsequent analyses of the models' simulation performance and structures indicate significant differences in functional hierarchies between macaque and mouse visual cortex  As for macaques, we observed a clear sequential hi-erarchy 
However, as for mouse visual cortex, some work ) exhibits that the trend of the model feature complexity roughly matches the processing hierarchy, but other work suggests that the cortex ) is organized into a parallel structure  Our results are more supportive of the latter  Furthermore, we provide computational evidence not only that the increased ratio of the receptive field size in cortical regions across the mouse visual pathway is smaller than those across the macaque visual pathway, but also that there may be multiple pathways with parallel processing streams between mouse cortical regions 
Our results also clearly reveal that the processing mechanisms of macaque visual cortex differ to various stimuli  These findings provide us with new insights into the visual processing mechanisms of macaque and mouse, which are the two species that dominate the research of biological vision systems and differ considerably from each other 
Compared to CNNs, the study of task-driven deep SNNs is just in its initial state  Although we demonstrate that SNNs outperform their counterparts of CNNs, SNNs exhibit similar properties as CNNs in the further analyses  In this work, we only build several new SNNs by taking the hints from the biological visual hierarchy, while many well-established structures and learning algorithms in CNNs have not been applied to SNNs yet 
In addition, the neural datasets used in our experiments are all collected under static image stimuli, lacking rich dynamic information to some certain, which may not fully exploit the properties of SNNs  Given that SNNs perform well in the current experiments, we hope to explore more potential of SNNs in future work 
In conclusion, as more biologically plausible neural networks, SNNs may serve as a shortcut to explore the biological visual cortex  With studies on various aspects of SNNs, such as model architectures, learning algorithms, processing mechanisms, and neural coding methods, it's highly promising to better explain the sophisticated, complex, and diverse vision systems in the future 

Implementation Details of SNNs Spiking Neuron Model

For all SNNs, we use the Integrate-and-Fire (IF) model as the spiking neuron model, which acts as the activation layer in neural networks  As mentioned in , V t , X t and S t denote the state (membrane voltage), input (current) and output (spike) of the spiking neuron model respectively at time-step t, and the dynamics of the IF model can be described as follows:
(1) (2) (3) While V t is the membrane voltage after the trigger of a spike, H t is also the membrane voltage, but after charging and before a spike firing  Θ(x) is the unit step function, so S t equals 1 when H t is greater than or equal to the threshold voltage V thresh and 0 otherwise  Meanwhile, when a spike fires, V t is reset to V reset  
Here, we set V thresh = 1 and V reset = 0  In addition, because Θ(x) is non-differentiable at 0, the surrogate gradient method is applied to approximate the derivative function during back-propagation  Here, we use the inverse tangent function as the surrogate gradient function and the derivative function is
(5) In our experiments on SNNs, we not only use SEW ResNet proposed by ), but also build several new SNNs  On the one hand, we improve the spike-elementwise block in SEW ResNet with new architectures referring to studies on ResNet , as shown in Table   On the other hand, as the multi-branch structures in CNNs increase neural representation similarity to mouse visual cortex, we use depthwise separable convolutions and follow the overall architecture of MobileNetV2 to build the SpikingMobileNet, the basic block of which is shown in Figure  
Our implementation is based on SpikingJelly , an open-source framework of deep SNN  We use the ImageNet dataset to pre-train the new SNNs  Following the settings for training SEW ResNet , we train the models for 320 epochs on 8 GPUs (NVIDIA V100), using SGD with a mini-batch size of 32  The momentum is 0 9 and the weight decay is 0  The initial learning rate is 0 1 and we decay it with a cosine annealing, where the maximum number of iterations is the same as the number of epochs 
For all SNNs, we set the simulation duration T = 4 

Overall model rankings

The results of model rankings are shown in Figure , 8 and 9  We also apply the Spearman's rank correlation to the overall model rankings of different metrics, which is shown in Figure  

Score Comparisons among Model Groups

We conduct comparisons of similarity scores among CNNs, SNNs, and vision transformers  The results are shown in Figure  

Overall CNN rankings

The results of CNN rankings are shown in Figure , 13 and 14 

Correlations between the Model Sizes and the Similarity Scores

The results of linear regression to model sizes and the similarity scores are shown in Figure , 16 and 17 

The ImageNet Accuracy and the Similarity Scores

The results are shown in Figure  



ground_truth: SNNs have the potential to better model and explain the functional hierarchy and mechanisms of the visual system.