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Approximate analytic solution of Hodgkin-Huxley equations

Thesis Info

Author

Khairunnisa

Program

MS

Institute

Institute of Business Administration

Institute Type

Private

City

Karachi

Province

Sindh

Country

Pakistan

Thesis Completing Year

2016

Page

46

Subject

Mathematics

Language

English

Other

CallNo: 519

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676720939439

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In the last century, there has been extensive research on our brain and many mathematical models and theories have been developed which describe the dynamical behavior of neurons. One of them is the widely known, Hodgkin-Huxley model. The Hodgkin-Huxley model for space clamp situation (uniform voltage over a patch of nerve membrane) is a mathematical model consisting of 4 nonlinear ordinary differential equations that describe membrane action potentials in neurons. Before this work, these equations could only be solved by numerical techniques and analytical solutions were not found. In this work, efforts are put to find the analytic solution of the Hodgkin-Huxley model by using Homotopy Perturbation Method. Homotopy Perturbation Method was developed by Ji-Huan He (1998) by merging two techniques, the standard homotopy and the perturbation technique for solving linear, nonlinear, initial and boundary value problems. Further, the solution is compared with the experimental results found by Hodgkin and Huxley. In this work, the first-order approximate analytic solution of the space-clamped Hodgkin Huxley model has been computed and algorithm for a higher-order solution is given. For plotting the solution, MATHEMATICA is used. It is found that the first-order solution can describe many key properties of the Hodgkin-Huxley model. Further, besides some differences, the general agreement of the first-order solution of space-clamped Hodgkin-Huxley Equations by Homotopy Perturbation Method with experimental results is good. Homotopy Perturbation The method is proved to be a convenient and efficient method to find an approximate or exact analytic solution of nonlinear differential equations
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مولانا شوکت علی

مولانا شوکت علی مرحوم
اسی مہینہ کادوسرا المناک سانحہ مولانا شوکت علی خادم کعبہ کی وفات ہے، شوکت علی مرحوم ہندوستان کے اُن پرچند مسلمانوں میں ایک تھے جن کی شہرت نہ صر ف ہندوستان تک محدود ہے بلکہ دنیائے اسلام کے دوردراز گوشوں تک ان کانام عزت واحترام کے ساتھ لیاجاتاہے اوریہ واقعہ ہے کہ مرحوم بجا طور پر اس شہرت و احترام کے مستحق تھے، پچھلے چند برسوں کوچھوڑ کر بلاخوف تردید کہا جا سکتاہے کہ مرحوم کی زندگی قربانی، ایثار، ولولہ اورجوش عمل کے اعتبار سے مسلمانوں کے لیے قابل تقلید نمونہ تھی،جنگ طرابلس اورجنگ بلقان سے لے کراب تک ہندوستانی مسلمانوں کی اجتماعی اورسیاسی زندگی کے جتنے دور گذرے ہیں مرحوم کی خدمات اُن تمام دوروں میں اس قدر نمایاں اوراس قدر روشن ہیں جنہیں کسی طرح فراموش نہیں کیا جاسکتا’’علی برادران‘‘ہندوستان کی دوشخصیتوں سے مرکب ایک ایسی حقیقت کانام ہے جس کے زبان پرآتے ہی کرداروعمل اورشجاعت وبسالت کاایک سبق آموز نقشہ آنکھوں کے سامنے آجاتاہے۔
صد حسرت وافسوس کہ ہندوستان اپنے ایک جانباز،بہادر سپاہی اورپرانے خادم سے ہمیشہ کے لیے محروم ہوگیا۔انا ﷲ واناالیہ راجعون۔ حق تعالیٰ مرحوم کی خدمات کوقبول فرمائے اوردامانِ رحمت میں جگہ دے۔
[دسمبر۱۹۳۸ء]

مغرب کا نظریہ ملکیت اور اسلام تقابلی جائزہ

Islam recognizes the right of individual ownership of material things in this world. A person can hold all kinds of Halal material things in his individual possession. However, Islamic Shari'a doesn't allow such a concept of individual ownership which is given in Capitalism and as adopted by Western world. The Western world's concept about individual ownership is very liberal and without any restrictions. While, Islam doesn't give full liberty to any individual but rather instructs them to own and possess Halal material things via legitimate sources, and also instructs the right usage of these material things in the light of Qur'an and Sunnah. The benefit of this Islamic law is that disqualified individuals, such as an insane person or children, have no right on disposing his or her individual property. Similarly, in the eyes of Islam, an individual person is not allowed to dispose his or her property in such a way which causes trouble and inconvenience to others, for example a person cannot dig a well on his own land which causes trouble and inconvenience to others. Islam prohibits such disposing of an individual's property.

Combining Ability and Gene Action Studies of Maize Zea Mays L. under Water Deficit Conditions

Sixty inbred lines of maize (Zea mays L) were screened in the green house at seedling stage both under normal and water deficit conditions. The layout of design was complete randomized desidn (CRD) with two replications. Six water deficit tolerant inbred lines were selected on the basis of dry root weight, root/shoot ratio and relative leaf water content. These inbred lines were sown in the field conditions to attempt all possible crosses in a diallel fashion. The layout of the design was randomized complete block design (RCBD). F1 crosses, their reciprocal crosses along with their parental lines were planted under both normal and water deficit conditions. Normal irrigations(8) were applied to normal set of experiment whereas 50% of yhe normal irrigations(4) were applied to water deficit condition. The data pertaining to various morpho-physiological parameters was recorded at physiological maturity of the crop. The data was subjected to statistical analysis. The coefficient correlation among seedling traits like fresh root weight, dry root weight, root/shoot ratio and relative leaf water contents showed strong association with each other in both normal and water deficit conditions. Mean squares for all the parameters were significant under both conditions, indicating the presence of sufficient genetic variability. Relative general combining ability, specific combining ability and reciprocal effects were calculated for all the characters (Griffing, 1956, Method I, Model I). Estimates of variance components for general combining ability (GCA), specific combining ability (SCA) and reciprocal effects for all the parameters i.e., plant height, leaf temperature, days taken to tasseling, days taken to silking, anthesis-silking interval, no kernels per ear,100 grain weight, grain yield per plant, no of kernel per row, ear height, relative leaf water content, no of ears per plant under normal and water deficit regimes. These components were computed in order to obtain an estimate of relative importance of additive and non-additive or dominance type of gene action. Estimation of components of variation indicate larger values of GCA variance as compared to SCA variance depicting the preponderance of additive genetic effects for all the traits except number of kernels per ear having greater SCA variance than GCA variance 15 showing non additive genetic effects under normal conditions. Heritability estimates for grain yield and yield related parameters shoed maximum ability to transfer the genes to the next generation. Inbred lines W-10× W64SP proved to be best grain yielders on mean basis under normal and water deficit conditions. Hence, these inbred lines might be exploited in future breeding programs.Under water deficit condition, best performing cross was N-18×W-64SP. Hence this cross can be utilized for future breeding program under water deficit conditions.