Quality end-of-life care depends on understanding patients’ end-of-life choices. Individuals and cultures may hold end-of-life priorities at different hierarchy. Forced ranking rather than independent rating, and by-person factor analysis rather than averaging may reveal otherwise masked typologies.n = 120) rank-ordered 47 opinion statements on end-of-life care following a 9-category symmetrical distribution. Statements’ scores were analyzed by averaging analysis and factor analysis (Q-methodology).We explored Saudi males’ forced-ranked, end-of-life priorities and dis-priorities. Respondents ; 52 % reported average religiosity, 88 and 83 % ≥ very good health and life-quality, respectively, and 100 % ≥ high school education. Averaging analysis revealed that the extreme five end-of-life priorities were to, be at peace with God, be able to say the statement of faith, maintain dignity, resolve conflicts, and have religious death rituals respected, respectively. The extreme five dis-priorities were to, die in the hospital, not receive intensive care if in coma, die at peak of life, be informed about impending death by family/friends rather than doctor, and keep medical status confidential from family/friends, respectively. Q-methodology classified 67 % of respondents into five highly transcendent opinion types. Type-I  and Type-V  reported the lowest and highest religiosity, respectively. Type-II  and Type-III  reported the best and worst life-quality, respectively. Type-I respondents were the oldest with the lowest general health, in contrast to Type-IV . Of the extreme 14 priorities/dis-priorities for the five types, 29, 14, 14, 50, and 36 %, respectively, were not among the extreme 20 priorities/dis-priorities identified by averaging analysis for the entire cohort.1) Transcendence was the extreme end-of-life priority, and dying in the hospital was the extreme dis-priority. 2) Quality of life was conceptualized differently with less emphasize on its physiological aspects. 3) Disclosure of terminal illness to family/close friends was preferred as long it is through the patient. 4) Q-methodology identified five types of constellations of end-of-life priorities and dis-priorities that may be related to respondents’ demographics and are partially masked by averaging analysis.The online version of this article (doi:10.1186/s12904-015-0064-5) contains supplementary material, which is available to authorized users. Quality end-of-life care is increasingly viewed as a global public health problem as well as health systems problem in need of better information , 2. ThisSeveral North American and European studies explored end-of-life priorities from the point of view of patients with life-limiting illness, –21 outpaThree instruments have been widely used to evaluate the dying experience and explore end-of-life priorities, namely, Quality of Dying and Death (QODD), , 39, 40 The aims of this study were to explore Saudi males’ opinions regarding end-of-life priorities and the usefulness of Q-methodology in this setting.This exploratory cross-sectional study was conducted in accordance with the ethical principles contained in the Declaration of Helsinki after approval of the Research Ethics Committee (REC) of the King Faisal Specialist Hospital and Research Center (KFSH&RC). All respondents provided verbal informed consent before participating in the study.The study instrument (Q-set) was developed, validated, and piloted by the authors. The initial Q-concourse (collection of opinion statements to represent possible end-of-life priorities) was constructed in Arabic language based on published conceptual frameworks , 13 and n = 10). Qualitative evaluation of Q-sorts after assuming extreme point of view as well as focused probing indicated that the Q-sorts reflected sorters’ point of view.Criterion validity does not apply to Q-methodology because of its subjective nature. Content validity was examined thorough literature review and by eliciting domain experts’ advice. Reliability was studied through test-retest assessment, re-sorting after assuming certain extreme points of view , and focused probing in an interview session following completion of Q-sorts. Participants were asked to Q-sort the same Q-set three times: first according to their point of view, then after assuming a certain extreme point of view, and finally according to their point of view for a second time (without prior knowledge that the Q-sorting will be repeated). The aim was to ensure that the results are reproducible and reflect the point of view of the sorter rather than being a function of the particular Q-set used. Intra-individual correlation coefficient was consistently more than 0.8 (n = 7), treatment preferences (n = 5), whole-person-concerns (n = 8), moment of death (n = 5), family/friends (n = 6), achieving sense of completion/spirituality/religiosity (n = 5), preparation for death (n = 5), and relationship with healthcare professionals (n = 6). The first three domains are most related to life quality vs quantity concerns, the fourth and fifth to connectedness, the sixth to transcendence, the seventh to coping, and the eighth to information-disclosure and decision-making. An English translation (accuracy confirmed by back translation) of the final Q-set is available in Additional file The developed Q-set and sorting instructions were piloted on 20 Saudis of different demographics. The primary emphasis was to identify statements that are ambiguous, unclear, or leading, as well as to identify potential areas that were not covered by the initial Q-set. The Q-sorts used for validation or pilot testing were not included in this report. The final Q-set consisted of 47 statements in 8 thematic domains: symptoms and personal control  was checked and respondents were asked to correct any identified mistake.Sample size was based on convenience and practicality, consistent with Q-methodology exploratory nature. KFSH&RC Saudi employees as well as patients and patients’ companions attending outpatient clinics were invited to participate through direct contact and advertisement. Saudi adults (≥18 year old) who had completed at least a high school education, who were able to understand the purpose and procedures of the study, and who provided informed consent, were eligible to participate. The study recruited both males and females, however, due to limitation of the statistical program used for Q-methodology analysis (maximum 120 Q-sorts), and the fact that analyzing mixed male and female Q-sorts obscured important gender differences; male and female Q-sorts were analyzed separately; only the results of males are reported here.The following data were also collected: age, self-declared religiosity , general health , life quality , employment status , living arrangement , and death experience in family/close friends . Respondents also rated six statements related to attitude toward death and one statement related to life satisfaction on a 5-point scale .p < 0.01) loading ≥ 0.38 on one, and only one, factor were considered definer Q-sorts for the factor. A model Q-sort for each factor was composed from statements’ scores calculated as weighted (based on factor loading) average across definer Q-sorts. This idealized Q-sort represented how a hypothesized respondent with 100 % loading on the factor would have ordered all the Q-set statements. Interpretation of factors involved comparing composite statement scores across factors and reviewing respondents’ post-sorting comments. Respondents who loaded significantly on one of the identified factors were compared as groups, in regards to their age, sorting time, religiosity, general health, life quality, employment status, living arrangement, death experience in family/close friends, attitude to death, and life satisfaction. Wilcoxon Signed Ranks, student t, Mann–Whitney, Kruskal-Wallis, and Fisher exact test were used to compare opinion types based on the variable examined (IBM SPSS Statistics 20). Two-tailed p values are reported.Data were verified by double entry and validity checks. As pre-specified in the study protocol, due to complexity of sorting the Q-set into nine categories, Q-sorts that took <20 min were not considered valid and were excluded from analysis. Q-sorts were analyzed by by-person centroid factor analysis , using PCQ for Windows . Data analysis in Q-methodology involves sequential application of correlation, factor analysis, and computation of factor scores . Mean (SD) sorting time was 38.2 (16.6) minutes.Main demographic data of respondents are summarized in Table Respondents ranked 47 end-of-life opinion statements on a 9-point scale  following a systematic forced distribution , 8.7 (0.9), 6.5 (1.8), and 6.4 (1.9), respectively). Three statements in the whole-person-concerns domain: “I want to die maintaining my dignity.” (6.8 (1.7)), “I want to die without having my body exposed.” (6.3 (1.8)), and “I want to die clean.” (5.9 (1.7)); and one statement in the symptoms and personal control domain: “I want to die free of pain.” (5.7 (2.0)) also received priority scores.Four statements related to life quality/quantity, “If I go into coma, I do not want to be placed in an intensive care unit.”, “I want to die at the peak of my life.”, “I want to have no tubes inserted into my body.”, and “I want to live longer regardless of my medical condition.” received dis-priority scores (3.5 (1.8), 3.5 (2.1), 3.9 (1.9), and 3.9 (2.0), respectively). Two statements in the family/friends domain, “I want my family/friends, rather than my doctor to inform me about my impending death.” and “I want my medical status to be kept confidential from my family/friends.” also received dis-priority scores (3.5 (1.6) and 3.5 (1.7), respectively).In the moment of death domain, one statement “I want to have my family/friends with me at my last moments.” received a priority score (5.9 (1.9)) and one statement “I want to die in hospital.” received the most extreme dis-priority score (3.0 (1.7)).Finally, three statements in the preparation for death domain, “If I have a fatal illness, I don’t want to know.”, “I want to discuss my fears about dying with my physician.”, and “I want to discuss my fears about dying with my family/friends.” received dis-priority scores (3.7 (2.1), 3.9 (1.6), and 4.08 (1.7), respectively). However, statement, “I want to have my financial affairs in order before I die.” received a priority score (6.0 (2.2)).Using the 120 Q-sorts as variables, we were able to extract 9 factors with eigenvalues higher than 1. However, it was not practical to apply all these factors for factor rotation or to meaningfully interpret them. By inspecting the scree plot  for areas of sudden decrease in eigenvalues and by logically analyzing the results after extracting 3 to 9 factors, we determined that the appropriate number of factors to extract was five. This five-factor solution accounted for 44 % of the total variance and 67 % of the 120 Q-sorts. Of the remaining Q-sorts, ten did not have significant loading on any of the factors, and 30 were confounded (loaded significantly on more than one factor).There were three consensus statements and three differentiating statements. All the five factors (opinion types) gave the most extreme priority (rounded score 9) to “I want to die at peace with God.” and to “I want to die being able to say the statement of faith.” Respondents’ justification included “Without peace with God my life is worth nothing.”, “We are created to worship Him”, “If God is pleased, people will be pleased, so I will die with peace of mind.”, “It is the dream of every Muslim.”, “Because of my sins.”, and “To go to Paradise.” Some respondents cited the following verses from The Quran “I have only created Jinns and humans, that they may serve Me. No Sustenance do I require of them, nor do I require that they should feed Me. For Allah is He Who gives  Sustenance, − Lord of Power,- Steadfast (for ever).”  and some cited Prophet Muhammad’s saying “He, whose last words are: there is no God but Allah will enter Paradise.” All opinion types gave the same dis-priority (rounded score 2) to “I want my medical status to be kept confidential from my family/friends.” Statement, “I want to have an Islamic clergy with me at my last moments.” differentiated opinion type I from the rest , statement, “I want to die being able to bathe and feed myself.” differentiated opinion type III from the rest , and statement, “I want to die being able to control my bladder.” differentiated opinion type V from the rest .To facilitate interpretation of results, we considered that an issue was important to respondents belonging to a particular opinion type if its related statement received one of the seven highest (priority) or seven lowest (dis-priority) scores  in the model Q-sort of the opinion type. Issues related to the rest of the statements were considered non-priority. As shown in Table The eigenvalue and explained variance for opinion type I were 8.9 and 7 %, respectively. Thirteen respondents belonged to this opinion type only; 9 respondents belonged both to this and other opinion types.In addition to the two common transcendence priorities, resolving conflicts before dying was a priority for opinion type I (in common with 2 out of the 4 other types). Respondents’ justifications included “So that I would forgive/be forgiven by those whom I aggrieved/who aggrieved me.”, “I don’t want any conflict after death.”, and “Good deeds/worships would not be accepted without forgiveness by others.” However, in variance with the other four types, having an Islamic clergy at time of death and receiving care from religiously trustworthy professionals were the most extreme dis-priorities. Justifications included, “What is the benefit?”, “It will make me feel depressed.”, “I think healthcare professionals have their own code of ethics regardless of their religions.”, and “There is no relation between religion and health issues.” Opinion type I could be described as family-caring, since it assigned priority scores to avoiding emotional and financial burdens to family/friends and dis-priority scores to “I want my family/friends rather than my doctor to inform me about my impending death.”, “I want my doctor to discuss any concerns related to my illness and care in the presence of my family/friends.”, and “I want my medical status to be kept confidential from family/friends”; suggesting that they would rather inform family/friends themselves in a way that would ease their emotional burden. Justifications included “They are believers, but I want to tell them in my own way.” This together with assigning a dis-priority score to “If I have a fatal illness I don’t want to know.” indicated a monitoring style of coping. Finally, opinion type I was life-quality-concerned, assigning priority scores to “I want to die maintaining dignity.” and “I want to die without having my body exposed.”, and dis-priority score to “I want to live longer regardless of my medical condition.” Justifications included, “God gave me dignity, I should maintain it.”, “If I die with dignity everything else is easy, everything follows dignity and respect.”, and “Long life with good deeds is good thing, but life just for staying alive is not important at all.”; suggesting that for this opinion type, transcendence is part of life quality.It is of note that avoidance of emotional and financial burdens to family/friends, two priorities for this opinion type, were not among the ten priorities identified by averaging analysis for the entire cohort. Further, its two most extreme dis-priorities, having an Islamic clergy at the last moments and receiving care from religiously trustworthy professionals, were not among the ten dis-priorities for the entire cohort.The eigenvalue and explained variance for opinion type II were 12.4 and 10 %, respectively. Seventeen respondents belonged to this type only; 16 respondents belonged both to this and other opinion types.This opinion type was also highly transcendent. However, in variance with opinion type I, it was described as rituals-apt as it gave priority scores to “I want my religious death rituals to be respected” and “I want to have an Islamic clergy with me at my last moments.” Justifications for the later included, “To remind me and presage me.”, “His presence will remind me of the statement of faith and to ask for forgiveness.”, and “Good thing to have, maybe he will help me (after God), to ease death and prepare to meet God.” Opinion type II could be described as family-dependent, as it gave priority score to “I want to have my family/friends with me at my last moments.” and dis-priority scores to “I want to die in the hospital.” and “I want my medical status to be kept confidential from my family/friends.” Justifications included, “I just want my family with me. It is not important where I die.” and “The opposite is true, they should know everything.” This opinion type strongly opposed “If I have a fatal illness I don’t want to know.” and strongly embraced “I want to have my financial affairs in order before I die.”; indicating a monitoring coping style. Justifications included, “I want to be prepared.”, “To the contrary, I should know.” “I need to know to seek treatment.”, “Illness is a test just as health is, knowing will rejuvenate faith.”, and “Fatal illness is a test and there would be big rewards for people who are patients.” for the first and “It affects my next life.”, “I will be questioned about it.”, “I don’t want conflict between my children.”, “It is a debt in my neck.”, “Financial matters may be a burden on me in my next life and in my grave.”, and “Debt is the most important thing one leaves after death, we will be stopped in front of God till the debt is given back.” for the second. Finally, it reacted strongly against “I don’t want to be kept on life support when there is little hope for a meaningful recovery”, “I want to have no tubes inserted into my body”, and “If I go into coma, I don’t want to be placed in an intensive care unit.” Justifications included, “I don’t care, I want cure at any price.”; “It is not a shame.”, “It is not important what is inserted in the body.”; “It is the doctor decision not mine.”, “It provides better care.”, and “When in coma, the decision is to the first of kin.” This opinion type even gave dis-priority score to “I want to die free of pain” (justified as “Pain would be expected when the soul departs from the flesh.”), indicating a strong life-quantity concern.Having an Islamic clergy present at the last moments, a priority for this opinion type, and no life support with little hope, the most extreme dis-priority for this opinion type, were not among the 20 priorities and dis-priorities identified by averaging analysis for the entire cohort. Further, this opinion type gave dis-priority score to “I want to die free of pain”, which received a priority score by averaging analysis.The eigenvalue and explained variance for opinion type III were 17.2 and 14 %, respectively. Thirty one respondents belonged to this type only; 24 respondents belonged both to this and other opinion types.Despite being also highly transcendent, opinion type III was silent (gave non-priority scores) regarding “I want my religious death rituals to be respected.” and “I want to have an Islamic clergy with me at my last moments.” It could be described as self/family-neutral, as it was also silent regarding “I want to have my family/friends with me at my last moments.”, “I want to avoid being an emotional burden to my family/friends.”, and “I want to avoid being a financial burden to my family/friends.”; and gave weak dis-priority scores to “I want my family/friends, rather than my doctor to inform me about my impending death.” and “I want my medical status to be kept confidential from my family/friends.” It reacted strongly against “I want to discuss my fears about dying with my family/friends.” and “I want to discuss my fears about dying with my physician.” and was silent regarding “if I have a fatal illness, I don’t want to know.”; suggesting an avoidance coping style. In clear contrast to the other opinion types, the third highest priority for opinion type III was to be able to self bathe and feed. In addition, it gave priority scores to “I want to die maintaining my dignity.”, “I want to die free of pain.”, “I want to die being able to control my bowels.”, and “I want to die without having my body exposed.” and dis-priority score to “I want to die in hospital.”; suggesting a strong life-quality concern. Justifications included, “I don’t want to appear weak in the eyes of others.”, “I don’t want to be a burden on others.”, “Exposing the body is forbidden for Muslims.”, “I want to die at home.”, “Rarely people get out alive, more suffering in hospital.”, and “I want to die at home but near a hospital.” However, it also gave dis-priority scores to “If I go into coma, I do not want to be placed in an intensive care unit.” and “I want to die at the peak of my life.” and were silent regarding “I want to live longer regardless of my medical condition.”; suggesting an equally strong life-quantity concern as well. Justifications included, “It is the doctor who knows and should decide.”, “I want to use my life to worship God and obey my parents.”, “It is my goal to see my grandchildren and provide them with good life.”, and “I want to live longer.”This opinion type contributed the largest number of respondents to the entire cohort. Nevertheless, its third and sixth highest priorities (being able to self bathe and feed and being able to control bowels) were not among the ten priorities identified by averaging analysis.The eigenvalue and explained variance for opinion type IV were 8.6 and 7 %, respectively. Nine respondents belonged to this opinion type only; ten respondents belonged both to this and other opinion types.In communality with the other four opinion types, this opinion type was highly transcendent. In agreement with opinion type II, it gave priority score to “I want my religious death rituals to be respected.” It also gave priority score to “I want to die clean.”, and thus was classified as rituals-apt. Justifications included, “Because cleanliness originates from/is part of belief.” Nevertheless, they were silent in regard to “I want to have an Islamic clergy with me at my last moments.” and gave dis-priority score to “I want to die well dressed.” Justifications included, “I wish I will die pure not well-dressed because I will go into soil.”, and “In the last moments, what is important is the inside (essence) not the outside (appearance).”; indicating different conceptualization of death rituals. Opinion type IV can be described as self-centered; it gave dis-priority scores to “I want to die at home.”, “I don’t want to die alone.”, and “I want to avoid being an emotional burden to my family/friends.”, and “I want to die in hospital.” and was silent in regards to “I want to resolve any conflict before I die.” and “I want to avoid being a financial burden to my family/friends.” Justifications included, “Place is not important, what is important is how to die.”, “Death in any place is the same death.”, and “Death is unavoidable; I wish I will die before my father, sibs, and friends.” Interestingly, in agreement with the other four opinion types, it gave dis-priority score to “I want my medical status to be kept confidential from my family/friends.”, probably here in order for the family/friends to be available to help. It gave priority score to “I want the doctor to inform me about my impending death before informing my family.” and dis-priority score to “I want my family/friends rather than my doctor to inform me about my impending death.”; suggesting a strong monitoring coping style. Finally, although it gave priority scores to “I want to receive medical care with compassion.” and to “I don’t want to be kept on life support when there is little hope for a meaningful recovery.” it was otherwise silent to issues related to life quantity and quality.Three priorities  and four dis-priorities  for this opinion type were not among the 20 priorities and dis-priorities identified by averaging analysis for the entire cohort.The eigenvalue and explained variance for type V were 6.9 and 6 %, respectively. Ten respondents belonged to this opinion type only; five respondents belonged both to this and other opinion types.In agreement with opinion types II and IV, opinion type V was described as rituals-apt. Respondents justifications included, “Every human wants to be shrouded and buried according to his religion.” and “Because Islam respects the dead body.” Opinion type V was described as family-centered (both family-dependent and family-caring), as it embraced both “I want to have my family/friends with me at my last moments” and “I want to die knowing that my family /friends are prepared to accept my death”. Justifications included, “To pray for me and to forgive me.”, “To be comfortable.”, “I want to end with my family as I started with them.”, and “I expect to be happier with them.” It was silent in regards to coping style and information disclosure. However, it reacted strongly against “I want to die being able to control my bladder”, “I want to die well dressed.”, “I want to have no tubes inserted into my body.”, “If I go into coma, I don’t want to be placed in an intensive care unit.”, “I want to die being able to control my bowels.”, “I want to die at the peak of my life.”, and “I want to die maintaining my sense of humor.”; indicating a strong life-quantity-concern. Justifications included, “Controlling bladder does not mean anything to me.”, “When on death bed appearance is not important.”, “I want tubes inserted in my body to continue my treatment.”, and “Controlling bowels is not important at time of death.” Nevertheless, it gave priority score to “I want to die free of pain.”One priority (having family/friends prepared to accept death) and four dis-priorities  for this opinion type were not among the 20 priorities and dis-priorities identified by averaging analysis for the entire cohort.Contrasting normalized scores for each of the 47 statement among the five opinion types supported our interpretation of the factors. For example, normalized score differences between opinion type I  and opinion type II (life-quantity-concerned) in relation to “I do not want to be kept on life support when there is little hope for a meaningful recovery.”, “I want to die free of pain.”, “I want to die maintaining my dignity.”, and “I want to die free of depression.” were 1.79, 1.61, 1.07, 1.01, respectively. Similarly, score differences between opinion type I and opinion type V (life-quantity-concerned) in relation to “I want to die maintaining my dignity.” and “I want to live longer regardless of my medical condition.” were 1.52 and −1.57, respectively; and between opinion type II and type III  in relation to “I want to die free of pain.”, “I want to die having no difficulty breathing.”, “I want to die free of anxiety.”,, and “I want to die being able to bathe and feed myself.” were −1.84, −1.18, −1.15, and −1.08, respectively.In the same vein, differences in normalized scores between opinion type I  and opinion types II, IV, and V  in relation to “I want an Islamic clergy with me at my last moments.”, and “I want to receive care from healthcare professionals whom I religiously trust” were −2.76 and −1.79, −2.01 and −1.75, and −1.91 and −1.90, respectively. Similarly, differences in normalized scores between opinion type III (avoidance-coping) and opinion types I, II, and IV (monitoring-coping) in relation to “I want to discuss my fears about dying with my family/friends” were −0.44, −1.03, and −1.04, respectively.Although all five opinion types were highly transcendent there were quantitative differences. Opinion types I, II, and III were more transcendent than opinion types IV and V .Table p > 0.11). There was no significant difference in sorting time (ANOVA p = 0.42) and borderline significant difference in age , which was due to a significant difference between opinion types I and IV (p = 0.04). Using Kruskal-Wallis test, there were no significant differences (p > 0.27) among the five opinion types in relation to religiosity, general health, life quality, life satisfaction, or death attitude, except in regard to statement “I often think about dying.” (p = 0.08). For this statement, there were significant differences (Mann–Whitney U test between opinion types II and I (p = 0.007), II and III (p = 0.04), and II and V (p = 0.02), with opinion type II having stronger agreement with the statement than the other 3 types. Further, there were significant differences between opinion types I and IV in life satisfaction and general health .There were no significant differences among the five opinion types in relation to employment status, living arrangement, or death experience in family/close friends  by averaging analysis, the extreme ten priorities were to, be at peace with God, be able to say the statement of faith, maintain dignity, resolve conflicts, have religious death rituals respected, not have the body exposed, have financial affairs in order, die clean, have family/ friends at last moments, and die free of pain, respectively. The extreme ten dis-priorities were to, die in the hospital, not receive intensive care if in coma, die at peak of life, be informed about impending death by family/friends rather than the doctor, keep medical status confidential from family/friends, not know if one has a fatal illness, not have tubes inserted in one’s body, live longer regardless of medical condition, discuss fears about dying with the physician, and discuss fear about dying with family/friends, respectively. 2) Q-methodology analysis classified 67 % of the respondents into five opinion types: rituals-averse transcendent, family-caring, monitoring-coping, life-quality-concerned; rituals-apt transcendent, family-dependent, monitoring-coping, life-quantity-concerned; rituals-silent transcendent, self/family-neutral, avoidance-coping, life-quality & quantity-concerned; rituals-apt transcendent, self-centered, monitoring-coping, life-quality/quantity-neutral; and rituals-apt transcendent, family-centered, neutral-coping, life-quantity-concerned. 3) Out of the extreme 14 priorities/dis-priorities for the five types, 29, 14, 14, 50, and 36 %, respectively, were not among the extreme 20 priorities/dis-priorities identified by averaging analysis for the entire cohort. 4) Eleven issues were identified as non-priority both on averaging analysis and Q-methodology analysis: to die having no breathing difficulty, anxiety, or depression, to receive all available treatments no matter what the chances of success are, to be able to communicate with others and referred to as a person, to avoid being a financial burden to society, to die instantaneously, to make own medical decisions, to have doctors available to answer questions, and to receive medical information regularly.Religious traditions often provide a framework for understanding death and dying as well as norms for end-of-life care. Islamic traditions, in common with Judeo-Christian traditions, view life as sacred and a mean to prepare the soul, and death as inevitable and a transition to another life. Our respondents were highly transcendent, eight of the top ten end-of-life priorities were clearly  or arguably  in the transcendence domain. In contrast, in a North American study, coming to peace with God was ranked the second or third (after freedom from pain and presence of family), depending on the subgroup studied,  and in aAccording to Islamic traditions, when a person is dying, close relatives/friends kindly remind him/her of the vast Mercy of Allah and to say the statement of faith (there is no God but Allah). Dying rituals include having the dying person to lie or sit facing toward Mecca. After death, the body is ritually washed (modesty preserved), anointed, shrouded , prayed over, and buried with a simple rite. In this study, the importance of rituals to respondents was variable. Although three opinion types were rituals-apt, one was rituals-silent and one was rituals-averse, suggesting that to some, resolving issues of faith within oneself may be more important than social or interpersonal expressions of spirituality. Further, having religious death rituals respected, but not having an Islamic clergy at the last moments, was one of the overall priorities. Furthermore, respondents who belonged to the rituals-averse and rituals-silent opinion types had lower self-declared religiosity, which is likely related to ritual practices rather than strength of beliefs or inner piety, as these two opinion types had higher normalized factor scores for “I want to die at peace with God.” and “I want to die being able to say the statement of faith.” than two of the three rituals-apt opinion types. Interestingly, in a North American study, having funeral arrangements planned received stronger importance rating from patients compared to physicians, , 13 and Arguably, sanctity of life is related not only to its intrinsic value but also to some of its qualities, such as self-consciousness, ability to establish relationship, and ability to derive pleasure . At the In our study, three out of the extreme ten dis-priorities indicated preference for life-quantity ; and one dis-priority  and four out of the extreme ten priorities  indicated preference for life-quality. Our results may be a reflection of the Islamic views of life sanctity, life as a mean to develop the soul, mandating relief of suffering whenever possible, and enduring and accepting irrelievable suffering as a spiritual reward. Nevertheless, Q-methodology showed that our respondents were not homogeneous in this regard. We were able to stratify respondents into life-quality-concerned, life-quantity-concerned (2 opinion types), life-quality and quantity-concerned, and life-quality/quantity-neutral. Preference for life quality was qualitatively associated with lower self-declared life quality, life satisfaction, and religiosity. A North American study on patients with advanced heart failure identified two distinct groups independent of functional or symptoms status, one preferring treatments that prolonged survival time and another that favored strategies that improved quality of life but reduced survival time . HoweverConceptualizing what constitute life quality may differ among individuals, individuals in different roles, and cultures. Interestingly, to die free of pain was the tenth priority for our respondents, while other physiological  and psychological  indicators of life-quality were non-priorities. Further, to die free of pain was a priority for only two of the five opinion types, and in fact a weak dis-priority for one. On the other hand, to be able to self bathe and feed, to be able to control bowels, to die clean, and to die without having body exposed were variably among the top seven priorities of the five opinion types. The importance of having pain free death has been documented in previous studies. Freedom from pain was ranked as the most important by seriously ill patients, recently bereaved family, physicians, and non-physician care providers,  higher oOne of the extreme ten priorities (to have family/close friends at last minutes) and two of the extreme ten dis-priorities  indicated the importance of connectedness to our respondents overall. Q-methodology analysis was able to group respondents into, family-caring, family-dependent, family-centered (both caring and dependent), self-centered, and self/family-neutral. A study on terminally ill men revealed heterogeneity in views about presence of others at the very end of life . WantingOverall, to die in the hospital was the most extreme dis-priority for our respondents, whereas to die at home was a non-priority. However, there was clear heterogeneity, to die in the hospital was a dis-priority for three opinion types, to die at home was a dis-priority for one opinion type, and two opinion types were silent as to the preferred place of death. Rather constant with our results, 51 %  to 84 % (Netherland) of European general public,  51 % of Five of the extreme 10 dis-priorities were related to information disclosure and coping style , indicating that overall respondents wanted to be informed about their fatal illness (but not to discuss it), to inform their family/friends about their medical condition, and to be informed before their family/friends. Nevertheless, the following were non-priorities, being able to communicate with others, having the doctor available to answer questions, and receiving medical information regularly. Our results are consistent with results in other cultures. In a North American study, preparation for the end of life was consistently rated as important by seriously ill patients, recently bereaved family, physicians, and non-physician care providers,  and abouDenying one’s death fits with a medical culture that defines death as a negative outcome in the fight against disease (rather than a transition to another life); therefore, one would fear that disclosing information about a terminal illness would remove hope and may not be favored over hiding it . On the Although not to know if one has a fatal illness was a dis-priority, making own medical decisions was a non-priority for our respondents. People value the right to not have their physical, emotional, and dispositional privacy invaded (negative liberty); however, the value of decisional privacy (positive liberty) is less certain . FurtherDevelopment of an optimal instrument for studying end-of-life care continues to be a work in progress , 54. SevWe hypothesized that there are latent constellations of end-of-life choices that may be masked by averaging analysis. Indeed, we found that 29, 14, 14, 50, and 36 % of the extreme 14 priorities/dis-priorities for the five opinion types, respectively, were not among the extreme 20 priorities/dis-priorities identified by averaging analysis for the entire cohort. Using a smaller Q-set of 34 statements, a study on 37 South Korean university students’ perception of good death identified four types, a resolute acceptance type, a reasonable, natural life span type, a relational, sentimental type, and an altruistic, satisfied type .This study had a number of limitations. First, it was based on convenience sampling and performed at a single tertiary healthcare institution in a major metropolitan city. Further, because forced-ranking is mentally demanding, only educated and committed individuals were suitable for the study. Thus our results could be generalized only with caution even though the institution is a governmental referral center for the entire country.Second, the study confronted rather healthy individuals with a hypothetical scenario. Responses to hypothetical scenarios may not accurately indicate what people would choose in real-life situations. However, such responses have enriched our understanding of end-of-life choices, , 32–37 aThird, Q-methodology is exploratory and not exhaustive in nature. Opinion types are defined as prototypical exemplars rather than as discrete categories; Q-methodology assumes neither discontinuous data nor clear cut-off points between categories. Thus the five opinion types extracted in this study should be seen as impressionistic conclusions. Further, it is likely that there are opinion types other than those identified in the study. Moreover, the prevalence of the identified opinion types among the larger population requires large sample surveys and standard analytic methods.Despite these limitations, our data support four main conclusions. 1) In young adult Saudi males transcendence was the extreme end-of-life priority, and dying in the hospital was the extreme dis-priority. 2) Quality of life was conceptualized differently with less emphasize on its physiological aspects. 3) Disclosure of terminal illness to family/close friends was preferred as long it is through the patient. 4) Q-methodology identified five types of constellations of end-of-life priorities and dis-priorities that may be related to respondents’ demographics and are partially masked by averaging analysis. The results emphasize the need to consider broader quality of life meanings as well as patients’ faith and religion to achieve the goal of quality death and dying. Patient-centered approaches to end-of-life care needs to take into account the identified priorities, dis-priorities, and typologies to sensitize health workers caring for Muslims/ Middle Eastern patients. This is the first study, to our knowledge to assess end-of-life priorities in such cultures; our findings need to be expanded and confirmed. The results also emphasize diversity in prioritizing end-of-life issues; one average opinion profile may not be useful as concerns can appear in varying combinations and hierarchy.


Acta Cryst. (2016), E72, 976–979.Erratum to  et al. , the name and affiliation of one of the authors is incorrect.In the paper by Çelik


Moreover, we propose a D-delay transmission strategy to ensure that every packet can be delivered to its destination nodes with limited delay. Different from most existing network schemes, our network model has a novel two-tier architecture. The existence of helping nodes greatly improves the network capacity. Four types of mobile networks are studied in this paper: i.i.d. fast mobility model and slow mobility model in two-dimensional space, i.i.d. fast mobility model and slow mobility model in three-dimensional space. Using the virtual channel model, we present an intuitive analysis of the capacity of two-dimensional mobile networks and three-dimensional mobile networks, respectively. Given a delay constraint D, we derive the asymptotic expressions for the capacity of the four types of mobile networks. Furthermore, the impact of D and m to the capacity of the whole network is analyzed. Our findings provide great guidance for the future design of the next generation of networks.This paper investigates the capacity problem of heterogeneous wireless networks in mobility scenarios. A heterogeneous network model which consists of  Their finding implies that as the size of the network goes to infinity, the per-node throughput will decrease to zero. Following this work, extensive studies have been conducted to achieve a tighter capacity bound . We have the following theorem:Theorem 2. In two-dimensional i.i.d fast mobility model, without considering relaying progress, we can bound the number of bits directly delivered to destination nodes as:Proof. Consider inequality Equation (3), according to the Cauchy-Schwarz inequality, we have:Now we investigate the throughput capacity contributed by iX and jX, let dist(t) denote their distance in time slot t. Based on the assumption that the mobility model are i.i.d, we have:The inequality above gives a bound on the expected distance which the bits take. For two arbitrary nodes T time slots, we have:In W1 bits in single time slot, hence we can obtain:The network can carry at most According to the inequalities above, we have:By Jensen’s inequality, combining inequality Equation (4) and inequality Equation (5), we can obtain the following inequality:Let E[Λd[T]]. □Simplify the inequality above, we can obtain Theorem 3. In two-dimensional i.i.d fast mobility model, if packet relaying exists, the bound on the number of bits transmitted to the destination number through relay nodes is:D is delay constant.Proof. T time slots, let Λr[T] denote the number of bits which are transmitted from relay nodes to destination nodes in T time slots, we have Λ[T] = Λr[T] + Λd[T]. According to Cauchy-Schwarz inequality and inequality Equation (3), we obtain:In Theorem 2 has given number of bits directly delivered to destination nodes. Now we consider the scenario where packets can be relayed in the networks.B, let Bd denote its destination node and Be denote the node storing bit B. We assume that from time slot Bt to time slot Bt + D − 1, the minimum distance between Bd and Be is BL. Then we have:Consider bit B which is in the set R[T], we have:For an arbitrary bit According to Taylor’s formula, equality Equation (14) can be expressed as:Therefore, we obtain:Obviously, we have:Combining Equations (7) and (8), we have:Hence, we can obtain the following inequalities:According to Jensen's inequality, combining inequality Equation (6) and inequality Equation (9), we have:E[Λr[T]]. □Theorem 4. In two-dimensional i.i.d. fast mobility networks, the ad hoc transmission mode capacity is:Proof. Combining Theorems 2 and 3, we have:Let In the meaning of order, the throughput capacity contributed by relaying nodes plays a dominating role compared with the throughput without relaying node. Thus, we have Theorem 5. In two-dimensional networks, the per-node throughput helping transmission mode contributes is:Proof. p, q) as the source and the helping node on  as the destination. In our network model, two neighbor helping nodes can communicate directly, hence the minimum number of hops is h = |p − i| + |j − q|. Summing up all the possible h, we have:E[|p − i|] can be obtained by the following equation:Since the node distribution is i.i.d., each helping node is chosen as a source or a destination with the same probability. We randomly choose the helping node on  and (12) into Equation (10), we have:W2, for each helping node, it can carry at most W2 bits in each time slot. Thus, we have:Hλ is the per-node throughput capacity of backbone helping node.Since the bandwidth of uplink sub-channel is iY, its interference region is a circle with radius (1 + ΔH)Hr. Each helping node occupies a region of iY is:For an arbitrary helping node iY is bounded by c8 = c7 − 1. It implies that there exists a temporal and spatial scheduling strategy that every helping node can obtain at least one slot to communicate in every (1 + c8) slots. Therefore, we have:Hence the number of nodes interfered by Combining Equations (14) and (15), we can obtain the per-node throughput capacity of backbone:λ = Aλ(n) + Hλ(m). According to Theorems 4 and 5, we can obtain Theorem 6 as follows:Theorem 6. In two-dimensional i.i.d. fast mobility networks, under protocol model, the network capacity is:Proof. Combining Theorems 4 and 5, we can prove the theorem above. □Since the total throughput capacity is contributed by multi-hop mode and helping mode, we have Theorem 7. In two-dimensional i.i.d. slow mobility networks, we have the following inequalities:Proof. Similarly with the proof of Theorem 1, we can prove the theorem above. □Similar to the fast mobility model, we can calculate the throughput capacity of slow mobility model. First we present some important inequalities, as shown in Theorem 7:Theorem 8. In two-dimensional i.i.d. slow mobility networks, if relaying progress is not allowed, the number of bits directly delivered to destination nodes is bounded as follows:Proof. According to Theorem 1 and the Cauchy-Schwarz inequality, we have:First we consider the case that all the packets are directly transmitted to the destination nodes without relaying.Since Similar to the analysis in Theorem 2, the following inequality can be obtained:Combining inequality Equation (18) and inequality Equation (19), we have:Let Theorem 9. In two-dimensional i.i.d. slow mobility network model, if packet relaying is allowed, the bound on the number of bits which are transmitted successfully from relay nodes to destination nodes is as follows:Proof. Similar to the steps in capacity analysis of network capacity of two-dimensional i.i.d. fast mobility networks, we have:Simplify the inequality above, we can obtain Theorem 8. □Therefore, we can obtain the following inequalities:Combining inequality Equation (18) and inequality Equation (20), we have:Theorem 10. In two-dimensional i.i.d. slow mobility networks, the throughput capacity contributed by ad hoc transmission mode is:Proof. Combining Theorems 9 and 10, we have:Let Inequality Equation (21) then follows the proof of Theorem 4. □Theorem 11. In two-dimensional i.i.d. slow mobility networks, under protocol model, the capacity is:Proof. Combining Theorems 5 and 10, we can prove the theorem above. □Theorem 5 has given the throughput capacity contributed by backbone networks. Combining the results of Theorems 5 and 10, we can obtain the network capacity of two-dimensional i.i.d. slow mobility networks:Theorem 12. In three-dimensional i.i.d fast mobility model, we have:Proof. The proof progress is similar with Theorem 1. □Theorem 13. In three-dimensional i.i.d fast mobility model, we can obtain that the bounds on the number of bits directly delivered to destination nodes by ad hoc transmission mode:Proof. According to the Cauchy-Schwarz inequality, we can obtain the following inequality:Inequality Equation (1) and inequality Equation (2) which hold in two-dimensional networks are also suitable for three-dimensional networks. However, in three-dimensional i.i.d fast mobility model, we should modify inequality Equation (3) to analyze the network capacity of the networks.Since the node distribution is i.i.d in a three-dimensional space, according to the knowledge of Probability Theory, we have that:According to the inequality above, we conclude that:W1 bits. Hence we have:In single time slot, each node can carry at most Taking expectation on the inequality above, we have:Combining inequality Equation (23) and inequality Equation (24), we have:Let Then Theorem 14 can be proved by simplifying the inequality above. □ad hoc mode considering relaying progress.Theorem 14. In three-dimensional i.i.d fast mobility networks, if relaying exists, we can obtain the bound on the number of bits successfully delivered to the destination nodes through relay nodes as follows:D is the delay constant.Proof. According to the Cauchy-Schwarz inequality and inequality Equation (48), we have the following inequalities:Next we study the throughput contributed by B, in time , let Be and destination node Bd, i.e.:Consider bit Since the node distribution is i.i.d in each time slot, we have:Using the Taylor formula, we can obtain the following inequality:Then we can get the bound on the expectation of the number of bits which satisfy Since Furthermore, we have:Combining inequality Equation (25) and inequality Equation (26), we can obtain the following inequalities:Let ad hoc transmission mode:Theorem 15. In three-dimensional i.i.d. fast mobility networks, the ad hoc transmission mode capacity is:Proof. ad hoc transmission mode as follows:Based on Theorems 13 and 14, we have the total number of bits transmitted by Combining Theorems 13 and 14, we can obtain the capacity contributed by ad hoc transmission mode:According to the definition of order, we can obtain the capacity contributed by Theorem 16. In three-dimensional networks, the per-node throughput capacity contributed by the backbone network is:Proof. p, q, u) and the helping node on  as source and destination respectively. Hence we have that the expectation of the hops is E[h] = 3E[E[|p − i|]].In three-dimensional backbone network, we randomly choose the helping node on  in Theorem 7 is suitable for the three-dimensional i.i.d. slow mobility network. Besides, we have the following inequality which will be used in the following analysis:Now we consider the three-dimensional i.i.d. slow mobility networks. Similar to the method used in network capacity analysis of two-dimensional i.i.d. slow mobility networks, we first calculate the bounds on the number of bits which are directly transmitted from source nodes to destination nodes without packet relaying. Furthermore, we analyze the number of bits which are delivered successfully from relay nodes to destination nodes.Hence, we have:As Similar to the steps in Theorem 8, we have:Therefore, we can conclude that:Theorem 19. In three-dimensional i.i.d. slow mobility networks, if packet relaying is allowed, we have the following inequality:Proof. Similar to the proof process, we have:Let Hence, we have the following inequalities:Based on the argument of Theorem 1, we have:Hence, we have:Theorem 20. In three-dimensional i.i.d. slow mobility networks, the capacity contributed by ad hoc transmission mode is:Proof. Similar to the proof of Theorem 4, we have:Let Hence we have Theorem 21. In three-dimensional i.i.d slow mobility networks, under the protocol model, the capacity is:Proof. Combining Theorems 16 and 20, we can prove the theorem above. □Theorem 16 implies that in three-dimensional networks, the capacity contributed by helping transmission mode is We have obtained the capacities of the four types of heterogeneous mobile networks in this section. By analyzing these capacity expressions, we will get some interesting conclusions. The analysis is presented in the following section.n, D and m to the network capacity. Firstly we list the capacity contributed by ad hoc transmission mode as follows: in two-dimensional i.i.d. fast mobility model, we have Aλ(n) of the four types of networks. In the Section above we have obtained the throughput capacity of the four types of mobile networks. In this section we analyze the impact of some important parameters such as Aλ(n) and n is shown in n goes to infinity, the number of mobile nodes offers many opportunities for source nodes to meet destination nodes, even in slow mobility networks. Hence, if the number of mobile nodes is large enough, the capacities of the four types of networks are similar.The relationship of D = Ω(n2), we have Aλ = Θ(1). In this case, the four types of mobile networks have the same capacity, which is Θ(1). It means that in delay tolerant networks, the network capacity can achieve constant order, and the price is that the delay will go to infinity. Of course, the delay constraint D is approximated in the analysis above. Concretely, in two-dimensional mobile network model, D = Ω(n); in three-dimensional mobile network model, D = Ω(n2). However, since n is assumed to be large enough, D is not limited in both the two cases.Let D = O(n), we have Aλ = 0. In this case, the capacity will decrease to zero as n goes to infinity. Concretely, in two-dimensional mobile network model, D = O(n); in three-dimensional network model, D = O(n2). It implies that if the delay is assumed to be very small, the number of bits transmitted by ad hoc transmission mode is very little. Our analysis is shown in Let D falls in between O(n) and Ω(n), there is a positive correlation between Aλ and the delay constraint. Theorems 5 and 16 offer the throughput capacity of backbone networks. When (1)In two-dimensional networks:Combining the analysis above, we reach the following conclusions:D = Ω(n), the capacity is m = Ω(n), most of packets can be delivered to the destination nodes with the help of backbone network. It implies that if the number of helping nodes is large enough, network capacity can stay constant order, and the delay can be limited, too. Similarly, if m = o(n), most of the packets will be transmitted by ad hoc transmission mode. It means that if the number of helping nodes is small, though the capacity is also constant order, the delay can not be ensured.If the delay is allowed to be larger than some fixed constant D = Ω(n)), the network capacity is (2)In three-dimensional mobile networks:If the delay is limited , which means that the network capacity of the whole network can achieve a constant order. If D is constrained, the capacity contributed by ad hoc mode is almost zero and most of packets have to be delivered by helping transmission mode.We can reach similar conclusions to two-dimensional mobile networks. According the analysis above, we can conclude that the number of helping nodes in the network is a key factor which affects the throughput capacity. If the number of helping nodes is large enough, the capacity can stay constant with the help of the backbone network. In this case, the time delay can be ensured. If the number of helping node is small, most of the packets have to be delivered in multi-hop fashion. In this case, though the throughput capacity can achieve constant order too, the delay will go to infinity. The delay constraint D is not limited and D is limited, where D is the delay constraint. We find that if D is not limited, the capacity contributed by ad hoc transmission mode can achieve constant order, while the price is that the delay will go to infinity. If D is limited, the capacity contributed by ad hoc transmission mode will decrease to zero. Hence there is a positive correlation between the throughput capacity and D. We also find that the number of helping nodes in the network is an important factor on the network capacity greatly. If the number of helping nodes is large enough, most of packets can be transmitted by helping transmission mode. In this condition the network capacity can achieve constant order without a long time delay. If the number of helping nodes is small, most of packets have to be transmitted by helping transmission mode, and the time delay can not be ensured.In this paper we propose a new heterogeneous mobile network architecture, which is composed of normal nodes and more powerful helping nodes. Four types of mobile network model are investigated: two-dimensional i.i.d. fast mobility model, two-dimensional i.i.d. slow mobility model, three-dimensional i.i.d. fast mobility model and three-dimensional i.i.d. slow mobility model. First we provide a brief description of our network model. We analyze the throughput capacity of the four types of network model intuitively by virtual channel systems. Then we obtain the network capacity by strict mathematical proof respectively. We find that the throughput capacity is


Stimulus dimensionality-reduction methods in neuroscience seek to identify a low-dimensional space of stimulus features that affect a neuron’s probability of spiking. One popular method, known as maximally informative dimensions (MID), uses an information-theoretic quantity known as “single-spike information” to identify this space. Here we examine MID from a model-based perspective. We show that MID is a maximum-likelihood estimator for the parameters of a linear-nonlinear-Poisson (LNP) model, and that the empirical single-spike information corresponds to the normalized log-likelihood under a Poisson model. This equivalence implies that MID does not necessarily find maximally informative stimulus dimensions when spiking is not well described as Poisson. We provide several examples to illustrate this shortcoming, and derive a lower bound on the information lost when spiking is Bernoulli in discrete time bins. To overcome this limitation, we introduce model-based dimensionality reduction methods for neurons with non-Poisson firing statistics, and show that they can be framed equivalently in likelihood-based or information-theoretic terms. Finally, we show how to overcome practical limitations on the number of stimulus dimensions that MID can estimate by constraining the form of the non-parametric nonlinearity in an LNP model. We illustrate these methods with simulations and data from primate visual cortex. A popular approach to the neural coding problem is to identify a low-dimensional linear projection of the stimulus space that preserves the aspects of the stimulus that affect a neuron’s probability of spiking. Previous work has focused on both information-theoretic and likelihood-based estimators for finding such projections. Here, we show that these two approaches are in fact equivalent. We show that maximally informative dimensions (MID), a popular information-theoretic method for dimensionality reduction, is identical to the maximum-likelihood estimator for a particular linear-nonlinear encoding model with Poisson spiking. One implication of this equivalence is that MID may not find the information-theoretically optimal stimulus projection when spiking is non-Poisson, which we illustrate with a few simple examples. Using these insights, we propose novel dimensionality-reduction methods that incorporate non-Poisson spiking, and suggest new parametrizations that allow for tractable estimation of high-dimensional subspaces. The neural coding problem, an important topic in systems and computational neuroscience, concerns the probabilistic relationship between environmental stimuli and neural spike responses. Characterizing this relationship is difficult in general because of the high dimensionality of natural signals. A substantial literature therefore has focused on dimensionality reduction methods for identifying which stimuli affect a neuron’s probability of firing. The basic idea is that many neurons compute their responses in a low dimensional subspace, spanned by a small number of stimulus features. By identifying this subspace, we can more easily characterize the nonlinear mapping from stimulus features to spike responses –5.K, such that the probability of response r given a stimulus s depends only on the linear projection of s onto these filters, i.e., p(r|s) ≈ p(r|K⊤s). Existing methods differ in computational complexity, modeling assumptions, and stimulus requirements. Typically, moment-based estimators have low computational cost but succeed only for restricted classes of stimulus distributions, whereas information-theoretic and likelihood-based estimators allow for arbitrary stimuli but have high computational cost. Previous work has established theoretical connections between moment-based and likelihood-based estimators [Neural dimensionality-reduction methods can be coarsely divided into three classes: 1) moment-based estimators, such as spike-triggered average (STA) and covariance (STC)  moment-b–8; (2) mtimators , 19, 23,timators , 21, 24.Here we focus on maximally informative dimensions (MID), a well-known information-theoretic estimator introduced by Sharpee, Rust & Bialek . We showThe connection between the MID estimator and the LNP model makes clear that MID does not incorporate information carried by non-Poisson statistics of the response. We illustrate this shortcoming by showing that MID can fail to find information-maximizing filters for simulated neurons with binary or other non-Poisson spike count distributions. To overcome this limitation, we introduce new dimensionality-reduction estimators based on non-Poisson noise models, and show that they can be framed equivalently in information-theoretic or likelihood-based terms.Finally, we show that a model-based perspective leads to strategies for overcoming a limitation of traditional MID, that it cannot tractably estimate more than two or three filters. The difficulty arises from the intractability of using histograms to estimate densities in high-dimensional subspaces. However, the single-spike information depends only on the ratio of densities, which is proportional to the nonlinearity in the LNP model. We show that by restricting the parametrization of this nonlinearity so that the number of parameters does not grow exponentially with the number of dimensions, we can obtain flexible yet computationally tractable estimators for models with many filters or dimensions.Linear-nonlinear cascade models provide a useful framework for describing neural responses to high-dimensional stimuli. These models define the response in terms of a cascade of linear, nonlinear, and probabilistic spiking stages see . The linθ = {K,α} denote the parameters of the LNP model, where K is a  matrix whose columns contain the stimulus filters , and α are parameters governing the nonlinear function f from feature space to instantaneous spike rate. Under this model, the probability of a spike response r given stimulus s is governed by a Poisson distribution:λ denotes the stimulus-driven spike rate  and Δ denotes a time bin size. The defining feature of a Poisson process is that responses in non-overlapping time bins are conditionally independent given the spike rate. In a discrete-time LNP model, the conditional probability of a dataset D = {}, consisting of stimulus-response pairs indexed by t ∈ {1,…,N}, is the product of independent terms. The log-likelihood is therefore a sum over time bins:rt!) is a constant that does not depend on θ. The ML estimate for θ is simply the maximizer of the log-likelihood: Let Iss, is the the average information that the time of a single spike (considered independently of other spikes) carries about the stimulusThe maximally informative dimensions (MID) estimator seeks to find an informative low-dimensional projection of the stimulus by maximizing an information-theoretic quantity known as the single-spike information . This qup(s) denotes the marginal or “raw” distribution over stimuli, and p(s|spike) is the distribution over stimuli conditioned on observing a spike, also known as the “spike-triggered” stimulus distribution. Note that p(s|spike) is not the same as p(s|r = 1), the distribution of stimuli conditioned on a spike count of r = 1, since a stimulus that elicits two spikes will contribute twice as much to the spike-triggered distribution as a stimulus that elicits only one spike.Although first introduced as a quantity that can be computed from the peri-stimulus time histogram (PSTH) measured in response to a repeated stimulus, the single-spike information can also be expressed as the Kullback-Leibler (KL) divergence between two distributions over the stimulus  and p(K⊤s|spike) are the raw and spike-triggered stimulus distributions projected onto the subspace defined by the columns of K, respectively. In practice, the MID estimator maximizes an estimate of the projected single-spike information:Îss(K) denotes an empirical estimate of Iss(K). The columns of The MID estimator  seeks top(s) and p(s|spike), is exactly the ML estimator for the parameters of an LNP model, regardless of spike rate, the time bins used to count spikes, or the amount of data.Previous work has shown that MID converges asymptotically to the maximum-likelihood (ML) estimator for an LNP model in the limit of small time bins , 24. Herp(K⊤s) and p(K⊤s|spike). These density estimates are then used to compute Îss(K), the plug-in estimate of single-spike information in a subspace defined by K  that partition the range of the projected stimuli K⊤s. In the one-dimensional case, we typically choose these sets to be intervals Bi = [bi−1,bi), defined by bin edges {b0, …, bm}, where b0 = −∞ and bm = +∞. Then let p(K⊤s) and p(K⊤s|spike), respectively, given by:xt = K⊤st denotes the linear projection of the stimulus st, 1Bi(⋅) is the indicator function for the set Bi, defined as:Let {ĝ(x) denotes the ratio of density estimates:The estimates ĝ(x) is a piece-wise constant function that takes the value q̂i/p̂i over the ith histogram bin Bi.Note that f is parametrized as a piece-wise constant function, taking the value fi over histogram bin Bi. Given a projection matrix K, the ML estimate for the parameter vector α =  is the average number of spikes per stimulus in each histogram bin, divided by time bin width Δ, that is:lnp denotes the Poisson log-likelihood under a “null” model in which spike rate does not depend on the stimulus, but takes constant rate lnp can be considered an estimate for the marginal entropy of the response distribution, Iss can be equally regarded as “LNP information”.Now, consider an LNP model in which the nonlinearity kelihood :ℒlnp, meaning that the MID estimate is the same as an ML estimate for the filters in an LNP model:Empirical single-spike information is therefore equal to LNP model log-likelihood per spike, plus a constant that does not depend on model parameters. This equality holds independent of time bin size Δ, the number of samples Îss, correspond to a particular parametrization of the LNP model nonlinearity f as a piece-wise constant function over histogram bins. The ratio of these plug-in estimates gives rise to the ML estimate for f. MID is thus formally equivalent to an ML estimator for both the linear filters and the nonlinearity of an LNP model.Moreover, the histogram-based estimates of the raw and spike-triggered stimulus densities Iss and yields large errors in the filter estimate. Even with 1000 stimuli and 200 spikes, a 20-bin histogram gives substantial upward bias in the estimate of single-spike information } isK has a single filter and the nonlinearity is restricted to be a logistic function, f(x) = 1/(1+exp(−x)), this reduces to the logistic regression model. Note that the spike probability λ is analogous to the single-bin Poisson rate λΔ from the LNP model (p(r = 1) goes to zero [We can define the linear-nonlinear-Bernoulli (LNB) model by analogy to the LNP model, but with Bernoulli instead of Poisson spiking. The parameters NP model , and the to zero , 26.x = K⊤s and a Bernoulli spike response r ∈ {0,1} is given by:Iss is the single-spike information  and p(x), weighted by (N−nsp)/nsp, the ratio of the number of silences to the number of spikes. It is straightforward to show that empirical Bernoulli information equals the LNB model log-likelihood per spike plus a constant:θ, the maximum of the empirical Bernoulli information is identical to the maximum of the LNB model likelihood, meaning that once again, we have an exact equivalence between likelihood-based and information-based estimators.Let ormation , and Î0Î0 > 0 and these spike absences are neglected by the single-spike information measure. Only in the limit of infinitesimal time bins, where p(r = 1) → 0, does ÎBer converge to Îss[Îss, and the LNB filter estimate The empirical Bernoulli information is strictly greater than the estimated single-spike (or “Poisson”) information for a binary spike train that is not all zeros or ones, since e to Îss, 26. As λ increases linearly as a function of stimulus angle: λ = (s−π/2)/π, for s ∈  and Î0 (silences) as a function of projection angle, which illustrates the relative biases of the two quantities.The first example  uses stis1 and suppressive in stimulus axis s2 . For this neuron, both stimulus axes are clearly informative, but the  axis s2 carries 13% more information than the  axis s1. However, the standard MID estimator identifies s1 as the most informative axis  carried by silences, as a function of marginal probability of a spike, p(r = 1).In the limit of infinitesimal time bins, the information carried by silences goes to zero, and the plug-in estimates for Bernoulli and single-spike information converge: p(r = 1) → 0, we find that:p(r = 1) (see p(r = 1). The bound is tight in the Poisson limit, p(r = 1) → 0, but is substantially loose in the limit where spiking is common p(r = 1) → 1, in which all information is carried by silences. In the limit of rare spiking, For neural responses binned at the stimulus refresh rate , it is not uncommon to observe multiple spikes in a single bin. For the general case, then, we must consider an arbitrary distribution over counts conditioned on a stimulus. As we will see, maximizing the mutual information based on histogram estimators is once again equivalent to maximizing the likelihood of an LN model with piece-wise constant mappings from the linear stimulus projection to count probabilities.s with a spike count r ∈ {0,…,rmax}, where rmax is the maximum possible number of spikes within the time bin (constrained by the refractory period or firing rate saturation). The linear-nonlinear-count (LNC) model, which includes LNB as a special case, is defined by a linear dimensionality reduction matrix K and a set of nonlinear functions {f(0) …, frmax)((x) = 1,∀x, since the probabilities over possible counts must add to 1 for each stimulus.Suppose that a neuron responds to a stimulus θ = , … αrmax) given data D = {} can be written:1j(rt) is an indicator function selecting time bins t in which the spike count is j. As before, we consider the case where fj). The Icount estimator, by contrast, is sensitive to the non-Poisson statistics of the response, and combines information from all spike counts  , 7, infowhy MID is impractical for high-dimensional feature spaces. The problem isn’t the number of filter parameters: these scale linearly with dimensionality, since a p-filter model with D-dimensional stimuli requires only Dp parameters, or indeed only p(x) and p(x|spike). For histogram-based density estimators, the number of parameters grows exponentially with dimension: a histogram with m bins along each of p filter axes requires mp parameters, a phenomenon sometimes called the “curse of dimensionality”.Before we continue, it is helpful to consider p(x|spike) and p(x) to a single nonlinear function f. This change in focus makes it easier to scale the likelihood approach to high dimensions for a few different reasons. First, direct estimation of a single nonlinearity in place of two densities immediately halves the number of parameters required to achieve a similarly detailed picture of the neuron’s response to the filtered stimulus. Second, the dependence of the MID cost function on the logarithm of the ratio p(x|spike)/p(x) makes it very sensitive to noise in the estimated value of the denominator p(x) when that value is near 0. Unfortunately, as p(x) is also the probability with which samples are generated, these low-value regions are precisely where the fewest samples are available. This is a common difficulty in the empirical estimation of information-theoretic quantities, and others working in more general machine-learning settings have suggested direct estimation of the ratio rather than its parts [A key benefit of the LNP model likelihood framework is that it shifts the focus of estimation away from the separate densities ts parts –33. In Lf  instead of regularizing the estimates of stimulus densities. For example, consider an experiment using natural visual images. While natural images presumably form a smooth manifold within the space of all possible pixel patterns, the structure of this manifold is neither simple nor known. The natural distribution of images does not factor over disjoint sets of pixels, nor over linear projections of pixel values. A small random perturbation in all pixels makes a natural image appear unnaturally noisy, violating the underlying presumption of kernel density estimators that local perturbations do not alter the density much. Indeed the question of how best to model the distribution of natural stimuli is a matter of active research. By contrast, we might expect to be able to develop better parametric forms to describe the non-linearities employed by neural systems. For instance, we might expect the neural nonlinearity to vary smoothly in the space of photoreceptor activation, and thus of filter outputs. Thus, locally kernel-smoothed estimates of the non-linear mapping—or even parametric choices of function class, such as low-order polynomials—might be valid, even if the stimulus density changes abruptly. Alternatively, subunits within the neural receptive field might lead to additively or multiplicatively separable components of the nonlinearity that act on the outputs of different filters. In this case, it would be possible to factor f between two subsets of filter outputs, say to give f(x) = f1(x1)f2(x2), even though there is no reason for the stimulus distribution to factor p(x) ≠ p(x1)p(x2). This reduction of f to two (or more) lower-dimensional functions would avoid the exponential parameter explosion implied by the curse of dimensionality.Third, … as the nonlinearity is a property of the neuron rather than the stimulus, it may be more straightforward to construct a smoothed or structured parametrization for f is parametrized by a quadratic form embedded in a 1D nonlinearity [f as a sum of one-dimensional nonlinearities, in which case the number of parameters grows only linearly with the number of filters.Indeed, such strategies for parametrization of the nonlinear mapping are already implicit in likelihood-based estimators inspired by the spike-triggered average and covariance. In many such cases, inearity , so thatinearity , 22, 24.inearity , parametf is parametrized using basis functions {φi(⋅)}, i = 1,…,nφ, which are linearly combined with weights αi and then passed through a scalar nonlinearity g. We refer to g as the output nonlinearity; its primary role is to ensure the spike rate λ is positive regardless of weights αi. This can also be considered a special case of an LNLN model [Here we provide a general formulation that encompasses both standard MID and constrained methods that scale to high-dimensional subspaces. We can rewrite the LNP model  as folloLN model , 34, 35.g and the basis functions {φi} in advance, fitting the nonlinearity simply involves estimating the parameters αi from the projected stimuli and associated spike counts. If g is convex and log-concave, then the log-likelihood is concave in {αi} given K, meaning the parameters governing f can be fit without getting stuck in non-global maxima [If we fix l maxima .g to the identity function and the basis functions φi to histogram-bin indicator functions (denoted 1Bi(⋅) in i’th histogram bin , cubic splines, radial basis functions, or any mixture of these bases. Alternatively, we could use non-parametric models such as Gaussian processes, which have been used to model low-dimensional tuning curves and firing rate maps , 37. TheWe propose to parametrize the nonlinearity for many-filter LNP models using cylindrical basis functions (CBFs), which we introduce by analogy to radial basis functions (RBFs). These functions are restricted in some directions of the feature space (like RBFs), but are constant along other dimensions. They are therefore the function-domain analogues to the probability “experts” used in product-of-experts models  in that μ and a characteristic width σ:xi and is invariant along xj ≠ i. Parametrizing f with first-order CBFs is tantamount to assuming f can be parametrized as the sum of 1D functions along each filter axis, that is f(x) = g(f1(x1) + … fm(xm)), where each function fi is parametrized with a linear combination of “bump” functions. This setup resembles the parametrization used in the extended projection-pursuit regression (ePPR) model [g confers some added flexibility. For example, we can have multiplicative combination when g(⋅) = exp(⋅), resulting in a separable f, or rectified additive combination when g(⋅) = max, which is closer to ePPR. If we use d basis functions along each filter axis, the resulting nonlinearity requires kd parameters for a k-filter LNP model.We define a “first-order” CBF as a Gaussian bump in one direction of the feature space, parametrized by center location R) model , althougμi and μj determine the center of the basis function in the  plane. A second-order basis represents f as a (transformed) sum of these bivariate functions, giving d2 basis functions for each of the k filter outputs, or merely k filters into disjoint pairs. Higher-order CBFs can be defined analogously: k′th order CBFs are Gaussian RBFs in a k′-dimensional subspace while remaining constant in the remaining k−k′ dimensions. Of course, there is no need to represent the entire nonlinearity using CBFs of the same order. It might make sense, for example, to represent the nonlinear combination of the first two filter responses with second order CBFs (which is comparable to standard MID with a 2D histogram representation of the nonlinearity), and then use first order CBFs to represent the contributions of additional (less-informative) filter outputs.We can define second-order CBFs as functions that have Gaussian dependence on two dimensions of the input space and that are insensitive to all others:To illustrate the feasibility of this approach, we applied dimensionality reduction methods to a previously published dataset from macaque V1 . This dap parameters for the nonlinearity), and a grid with 3 RBFs per dimension for the rbf-LNP model (3p parameters). By contrast, the exponentiated-quadratic nonlinearity underlying the iSTAC estimator requires O(p2) parameters.To compare performance, we analyzed the growth in empirical single-spike information (computed on a “test” dataset) as a function of the number of filters. Note that this is equivalent to computing test log-likelihood under the LNP model. For a subset of neurons determined to have 8 or more informative filters (16/59 cells), the cbf-LNP filters captured more information than the iSTAC filters . This inn filters identified .Rowekampmp et al ), and deet al[f(k⊤s+s⊤Cs). This method is one-dimensional in a quadratic stimulus space  and therefore avoids the curse of dimensionality. Other work has used independent component analysis to find directions in stimulus space in which the spike-triggered distribution has maximal deviations from Gaussianity [Other proposals can be seen as assuming specific quadratic-based parametrizations for the nonlinearity, that are more restrictive than the CBF form. The iSTAC estimator, introduced by Pillow & Simoncelli , is basen et al.  considerl. [et al, 46 usedssianity .We have studied the estimator known as maximally informative dimensions (MID),  a populaEstimators defined by the optima of information-theoretic functionals have attractive theoretical properties, including that they provide well-defined and  distribution-agnostic characterizations of data. In practice, however, such agnosticism can be difficult to achieve, as the need to estimate information-theoretic quantities from data requires the choice of a particular estimator. MID has the virtue of using a non-parametric estimator for raw and spike-triggered stimulus densities, meaning that the number of parameters  can grow flexibly with the amount of data. This allows it to converge for arbitrary densities, in the limit of infinite data. However, for a finite dataset, the choice of number of bins is critical for obtaining an accurate estimate. As we show in A second kind of distributional assumption arises from MID’s reliance on single-spike information, which is tantamount to an assumption of Poisson spiking. To be clear, the single-spike information represents a valid information-theoretic quantity that does not explicitly assume any model. As noted in , it is sThe distributional-dependence of MID should therefore be considered when interpreting its estimates of filters and nonlinearities. MID makes different, but not necessarily fewer, assumptions when compared to other LN estimators. For instance, although the maximum-likelihood estimator for a generalized linear model assumes a less-flexible model for the neural nonlinearity than does MID, it readily permits estimation of certain forms of spike-interdependence that MID neglects. In particular, MID-derived estimates are subject to concerns regarding model mismatch that arise whenever the true generative family is unknown .In light of the danger that these distributional assumptions be obscured by the information-theoretic framing of MID, our belief is that the safer approach is to specify a model explicitly and adopt a likelihood-based estimation framework. Where the information theoretic and likelihood-based estimators are identical, nothing is lost by this approach. However, besides making assumptions explicit, the likelihood-based framework also readily facilitates the introduction of suitable priors for regularization of suitable priors, or hierarchical models , 49, or Having clarified the relationship between MID and LNP model the, we introduced two generalizations designed to recover a maximally informative stimulus projection when neural response variability is not well described as Poisson. From a model-based perspective, the generalizations correspond to maximum likelihood estimators for a linear-nonlinear-Bernoulli (LNB) model (for binary spike counts), and the linear-nonlinear-Count (LNC) model (for arbitrary discrete spike counts). For both models, we obtained an equivalent relationship between log-likelihood and an estimate of mutual information between stimulus and response. This correspondence extends previous work that showed only approximate or asymptotic relationships between between information-theoretic and maximum-likelihood estimators , 24, 25.X stands in for any probability distribution over the neural response , and perform dimensionality reduction by maximizing likelihood for the filter parameters under this model. The log-likelihood under any such model can be associated with an information-theoretic quantity, analogous to single-spike, Bernoulli, and count information, using the difference of log-likelihoods  denotes the conditional response distribution associated with the LNX model with parameters θ, and px(r|θ0) describes the marginal distribution over r under the stimulus distribution p(s). The empirical or plug-in estimate of this information is equal to the LNX model log-likelihood plus the estimated marginal entropy:n denotes the number of samples and θ0 depends only on the marginal response distribution. The maximum likelihood estimate is therefore equally a maximal-information estimate.We could analogously define arbitrary “LNX” models, where see also ):Ilnx≜∑Note that all of the dimensionality-reduction methods we have discussed treat neural responses as conditionally independent given the stimulus, meaning that they do not capture dependencies between spike counts in different time bins . Spike-history dependencies can influence the single-bin spike count distribution—for example, a Bernoulli model is more accurate than a Poisson model when the bin size is smaller than or equal to the refractory period, since the Poisson model assigns positive probability to the event of having ≥ 2 two spikes in a single bin. The models we have considered can all be extended to capture spike history dependencies by augmenting the stimulus with a vector representation of spike history, as in both conditional renewal models and generalized linear models , 50–52.p(K⊤s) and p(K⊤s|spike). However, dimensionality and parameter count can be a crippling issue given limited data, and density estimation becomes intractable in dimensionalities > 3. Furthermore, the dependence of the information on the logarithm of the ratio of these densities amplifies sensitivity to errors in these estimates. The LNP-likelihood view suggests direct estimation of the nonlinearity f, rather than of the densities. Such estimates are naturally more robust, and are more sensibly regularized based on expectations about neuronal responses without reference to any regularities in the stimulus distribution. We have proposed a flexible yet tractable form for the nonlinearity in terms of linear combinations of basis functions cascaded with a second output nonlinearity. This approach yielded a flexible, computationally efficient, constrained version of MID that is able to estimate high-dimensional feature spaces. It is also general in the sense that it encompasses standard MID, generalized linear and quadratic models, and other constrained models that scale tractably to high-dimensional subspaces. Future work might seek to extend this flexible likelihood-based approach further, for example by including priors over the weights with which basis functions are combined to improve regularization, or perhaps by adjusting hyperparameters in a hierarchical model as has been successful with linear approaches [Lastly, we have shown that viewing MID from a model-based perspective provides insight into how to overcome practical limitations on the number of filters that can be estimated. Standard implementations of MID employ histogram-based density estimators for proaches , 49.In recent years, the ability to successfully characterize low-dimensional neural feature spaces using MID has proved useful to address questions relating to multidimensional feature selectivity –56. In aρ = p(r = 1) denote the marginal probability of a spike, so the probability of silence is p(r = 0) = 1−ρ. Let Q1 = p(s|r = 1) and Q0 = p(s|r = 0) denote the spike-triggered and silence-triggered stimulus distributions, respectively. Let Ps = p(s) denote the raw stimulus distribution. Note that we have the Ps = ρQ1+(1−ρ)Q0. The mutual information between the stimulus and one bin of the response , the mutual information is dominated by the first (Q1) term. Here we wish to show a bound on the fraction of information carried by the Q0 term. We can do this by computing a second-order Taylor expansion of I around ρ = 0, and show that their ratio is bounded below by ρ/2. Expanding in ρ, we havez−1) ≥ log(z). We therefore haveρ → 0.In the limit of small ρ. For the case of Q0 and Q1. We have been unable to find any counter-examples to this , but have so far been unable to find a general proof.We conjecture that the bound holds for all values of Îss based on a PSTH measured in response to repeated stimuli is also a Poisson log-likelihood per spike (plus a constant). Specifically, the empirical single-spike information is equal to the log-likelihood ratio between an inhomogeneous and homogeneous Poisson model of the repeat data :λ;r) denotes the log-likelihood of the repeat data r under a Poisson model with time-varying rate λ.An important general corollary to the equivalence between MID and an LNP maximum likelihood estimate is that the standard single-spike information estimate rjt} denote spike counts collected during a “frozen noise” experiment, with repeat index j ∈ {1,…,nrpt} and index t ∈ {1,…,nt} over time bins of width Δ. Then T = ntΔ is the duration of the stimulus and N = ntnrpt is the total number of time bins in the entire experiment. The single-spike information can be estimated with a discrete version of the formula for single-spike information provided in [t’th time bin in response to the stimulus sequence s, and T is long enough that an average over stimulus sequences is well approximated by the average across time.We can derive this equivalence as follows. Let {vided in  (see eq.s (as in ) that T nsp = ∑j,trjt is the total spike count. This allows us to rewrite single-spike information (The plug-in (ML) estimator for spike rate can be read off from the peri-stimulus time histogram (PSTH). It results from averaging the response across repeats for each time bin:ormation  as:I^ssr = {rjt} given rate vector Iss is equal to the difference between Poisson log-likelihoods under an inhomogeneous (rate-varying) and a homogeneous (constant rate) Poisson model, divided by spike count . These r from unseen test data. This results in the “cross-validated” single-spike information:Iss evaluated on training data tends to over-estimate information due to over-fitting.A similar quantity has been used to report the cross-validation performance of conditionally Poisson models, including the GLM , 58. To A’ and ‘B’, and two possible discrete stimulus sequences, s1 = AB and s2 = BA, each of which occurs with equal probability, so p(s1) = p(s2) = 0.5. Assume each sequence lasts T = 2s, so the natural time bin size for considering the spike response is Δ = 1s. Suppose that stimulus A always elicits 3 spikes, while B always elicits 1 spike. Thus, when sequence s1 is presented, we observe 3 spikes in the first time interval and 1 spike in the second interval; when s2 is presented, we observe 1 spike in the first time interval and 3 spikes in the second.To gain intuition for the different information measures we have considered , it is useful to consider how they differ for a simple idealized example. Consider a world with two stimuli, ‘λ1(t) and λ2(t), the spike rate in response to stimulus sequence s1 and s2, respectively. For this example, λ1(t), takes the value 3 during  takes values 1 and 3 during the corresponding intervals. The mean spike rate for both stimuli is λ̄ = 2 sp/s. Plugging these into Iss = 0.19 bits/spike. This result is slightly easier to grasp using an equivalent definition of single-spike information as the mutual information between the stimulus s and a single spike time τ (see [p(s|τ = 1) attaches 3/4 probability to s = s1 and 1/4 to s = s2. The posterior entropy is therefore −0.25 log 0.25−0.75 log 0.75 = 0.81 bits. We obtain the same entropy if the spike occurs in the second interval, so H(s|τ) = 0.81. The prior entropy is H(s) = 1 bit, so once again we have Iss = 1−0.81 = 0.19 bits/spike.Single-spike information can be computed exactly from e τ (see ). If oner takes values outside the set {0,1}, and therefore cannot have a Bernoulli distribution. To make Bernoulli information well defined, we would need to either truncate spike counts above 1  provides perfect information about the stimulus, since it occurs only in response to B.The Bernoulli information, by contrast, is undefined, since 1 , or elsr carries perfect information about the stimulus, so the mutual information between stimulus (A or B) and r is 1 bit. We defined Icount to be the mutual information normalized by the mean spike count per bin  nonlinearity. We performed joint optimization of filter parameters r = ⊤ and λ = (f(K⊤s1),…,f(K⊤snt))⊤, where nonlinearity f = g(∑αiφi) depends on basis function Φ = {φi} and weights α = {αi}  stimulus design matrix, Φ denotes the (nt×nφ) matrix whose ’th entry is φj(K⊤st), and Φi) is a (nt×1) vector composed of the point-wise derivatives of the inverse-link function g at its input, and ‘∘’ denotes Hadamard or component-wise vector product.Let λ′′ = g′′(Φα) and Φi,j). The spk-filter model, this resulted in 3k parameters for the nonlinearity, and (240+3)k parameters in total for a stimulus with 24 bars.The cbf-LNP model was implemented with a cylindrical basis function (CBF) nonlinearity using three first-order CBFs per filter. For a k-filter model, this resulted in 3k parameters for the nonlinearity, and 240k+3k parameters in total.The traditional MID estimator (rbf-LNP) was implemented using radial basis functions (RBFs) to represent the nonlinearity. Unlike the histogram-based parametrization discussed in the manuscript (which produces a piece-wise constant nonlinearity), this results in a smooth nonlinearity and, more importantly, a smooth log-likelihood with tractable analytic gradients. We defined a grid of RBFs with three grid points per dimension, so that CBF and RBF models were identical for a 1-filter model. For a g(⋅) = log(1+exp(⋅)), to ensure positive spike rates. We also evaluated the performance of an exponential function, g(⋅) = exp(⋅), which yielded slightly worse performance (reducing single-spike information by ∼ 0.02 bits/spike).For both models, the basis function responses were combined linearly and transformed by a “soft-rectification” function: θ = {K,α}. Both models were fit incrementally, with the N+1 dimensional model being initialized with the parameters of the N dimensional model, plus one additional filter . The joint likelihood in K and α was ascended using MATLAB’s fminunc optimization function, which exploits analytic gradients and Hessians. The models were fit to 80% of the data, with the remaining 20% used for validation.The cbf- and rbf-LNP models were both fit by maximizing the likelihood for the model parameters In order to calculate information contributed by excitatory filters under the cbf-LNP model , we remoMeasurements of computation time  were avehttp://pillowlab.princeton.edu/code.html.Code will be provided at