Academics

Doctoral Program


Overview


The requirements for the Ph.D. and Eng.Sc.D. degrees are identical. Both require a dissertation based on the candidate’s original research, conducted under the supervision of a faculty member. The work may be theoretical or experimental or both. Students who wish to become candidates for the doctoral degree in electrical engineering have the option of applying for admission to the Eng.Sc.D. program or the Ph.D. program. Students who elect the Eng.Sc.D. degree register in the School of Engineering and Applied Science; those who elect the Ph.D. degree register in the Graduate School of Arts and Sciences. Doctoral candidates must obtain a minimum of 60 points of formal course credit beyond the bachelor’s degree. A master’s degree from an accredited institution may be accepted as equivalent to 30 points. A minimum of 30 points beyond the master’s degree must be earned while in residence in the doctoral program.

For application information click here.

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Admission Requirements


1. General Remarks

Admission as a graduate student generally requires a bachelor of science degree. Most graduate students in the Department of Electrical Engineering have, as undergraduates, majored in electrical engineering, physics, or mathematics. Outstanding students with a bachelor's degree in other fields of science and engineering may also be admitted as graduate students. Usually, however, they will be required to remove certain undergraduate deficiencies without gaining credit toward the advanced degree. All applicants for the M.S., Ph.D., and Eng.Sc.D. degrees, including Columbia undergraduates, are required to take the graduate record examination (GRE) general test. The application deadline is December 15 for students who wish to be considered for financial aid. Only completed applications will be considered. All students must have a satisfactory command of the English language. Students whose native language is not English are required to submit results of the Test of English as a Foreign Language (TOEFL) with their application. Students who score below 550 on the TOEFL will not be admitted. Students whose native language is not English must pass the English Language Test given by the American Language Program (ALP) at Columbia before they graduate. An arrangement to take the test must be made with the ALP in 505 Lewisohn Hall (tel: 212-854-3584) to make an appointment. Usually results are available three days following the test. Candidates for the M.S. degree must reach level 8 and candidates for the doctoral degree must reach level 10 before they defend their thesis. Students failing to reach those levels on the placement examination will usually be required to register for an English course.

2. Classification of Graduate Students

Students are admitted under the following classifications:

  • M.S. Candidate
  • M.S. Doctoral-Track Candidate
  • Ph.D. Candidate
  • Eng.Sc.D. Candidate
  • Professional-Degree Candidate
  • Matriculated Special Student
  • One-Time Special Student


3. M.S. Doctoral-Track Candidates

Exceptionally qualified students may be admitted to M.S. doctoral-track candidacy directly after receiving the B.S. degree. These students may gain admission to the doctoral program upon satisfactory completion of the requirements for the M.S. degree, although a formal application (without new reference letters) must be approved by their adviser. The usual practice for potential doctoral degree candidates is, however, to apply for doctoral candidacy after completing the requirements for the M.S. degree.

4. Doctoral Degree (Ph.D. and Eng.Sc.D.)

The basic requirement is a master's degree in engineering, science, or mathematics with a superb record. It is very difficult to meet all doctoral-degree requirements by evening attendance only.

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Rules Applicable to all Graduate Students


1. Academic Regulations and Requirements
Rules and requirements, applicable to all graduate students, are contained in the current bulletin of the School of Engineering and Applied Science (SEAS) and the bulletin of the Graduate School of Arts and Sciences (GSAS).

2. Scholastic Standing
All degree candidates must maintain a minimum scholastic standing, measured by a grade-point average, as indicated under the various degree requirements.

A special student must maintain the following schedule of averages from the time of first registration in order to be allowed to continue:

  • 1-4 points. No minimum requirement on average
  • 5 points 1.00
  • 6 points 1.25
  • 7 points 1.50
  • 8 points 1.75
  • 9 points + 2.00
Averages are calculated on the basis of A=4, B=3, C=2, D=1, and F=0. Courses in which the candidate receives the grade of R (registration only) or F (failure) are not credited toward a degree. However, F grades are included in calculating the point average. A grade of P (pass) gives course credit but does not affect the grade-point average.

3. Dropping a Course
Students in the Electrical Engineering Department may drop a course after the change-of-program period, but before the first examination or the midterm date (whichever occurs earlier), provided they receive the permission of their adviser and the dean. Withdrawal from a course at a later time requires the permission of the chair of the Department of Electrical Engineering and the dean. Such permission is not generally granted to students with poor performance in the course.

4. "R" Credit
  1. Students must enroll for R credit at the time of registration. They cannot change to R credit at a later time without special permission from the dean, which is not easily granted.
  2. A course which has been taken for R credit may not be repeated for examination credit.
5. Auditing
  1. The privilege of auditing courses is granted to two kinds of regular degree candidates: Those who in any term are registered for 15 points or more or those who have completed their course requirements for the doctorate and are registered for full-time research.
  2. Application is made at the registrar's office during the change of program period (Monday through Friday following registration).
6. Teaching Assistants
Teaching Assistants may not register for more than three courses without the written permission of the chairman of the Financial Aid Committee.
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Requirements for the Doctoral Degree


1. The Ph.D. and Eng.Sc.D. Degrees
Students who wish to become candidates for the doctoral degree in electrical engineering have the option of applying for admission to either the Eng.Sc.D. or the Ph.D. program. Students who elect the Eng.Sc.D. degree register in the School of Engineering and Applied Science, and those who choose the Ph.D., register in the Graduate School of Arts and Sciences. Candidates for either degree must earn a minimum of 60 points of credit beyond the bachelor's degree. A master's degree from an accredited institution may be accepted as the equivalent of 30 points. A minimum of 30 points beyond the master's degree must be earned while in residence at Columbia , and at least 15 of these points must be obtained while registered in the doctoral program. A student for the Eng.Sc.D. must fulfill the additional point requirements stated in Part 2 below. For Ph.D. candidates, the 60-point minimum will represent a partial fulfillment of the residence-unit requirements stated in part 3 below. One residence unit is essentially equivalent to 15 points, but the student should read carefully the detailed specifications as given in the appropriate university bulletin and in Part 3 below.

2. Special Point Requirements for the Eng.Sc.D. Degree
  1. Beginning with the semester immediately following the one in which the minimum point requirements of Part 1 are completed, the candidate must register continuously for EE E9800 (3, 6, 9, or 12 points) until 12 points have been accumulated.
  2. Registration for EE E9800 at times other than as prescribed in (a) is not allowed except by written permission of the dean.
  3. No part of the credit for EE E9800 may be counted toward the minimum requirements stated in Part 1. These credits are in addition to the required 30 credits of course and research work.
  4. Continuous registration is required to maintain enrollment in any degree program. Continuous registration refers to the autumn and spring semesters, but not the summer session. Failure to register will result in withdrawal from the program. If a student completes the 9800 requirement and has not yet completed the thesis requirements he or she may opt to take ELEN E9900 for 0 points until the dissertation is complete. ELEN E9800 and E9900 courses are for DES students only. Ph.D. students should not register for these courses at any time.
3. Residence-Unit Requirements for the Ph.D. Degree
A residence unit is awarded for one term of full-time study, as determined by the student's adviser. In some cases a student may be allowed to register for either one-quarter or one-half residence unit for one semester of part-time study. Six residence units beyond the bachelor's degree are required for the Ph.D. Continuous registration until completion of all requirements is obligatory. Students whose degree programs are not yet complete at the end of the minimum residence requirements must register for extended residence. A maximum of two residence units may be credited for academic work completed at another university or in another division of Columbia University. Students are directed to the Graduate School of Arts and Sciences (GSAS) bulletin for further information
.
4. Scholastic Average
A candidate is required to maintain a grade-point average of at least 3.5 at any time in the program. Students with an unsatisfactory average may be asked to withdraw.

5. Program of Study
A prospective doctoral candidate follows a program of study formulated in consultation with his/her faculty adviser. The program must include the following:

  1. Courses used to satisfy the 30-point minimum beyond the M.S. degree will usually be at the 6000 level or higher.
  2. Not more than 6 points of seminar.
  3. At most 12 points of doctoral research (ELEN E9001, E9002, E9011, E9012, E6001, and E6002) may be credited toward the degree. However, in exceptional cases, in recognition of a student's advanced standing and professional proficiency, this 12-point limitation may be waived at the discretion of the chair of the doctoral committee.
6. Doctoral Qualifying Examination (D.Q.E.)
Students enrolled in the Ph.D., Eng.Sc.D., or doctoral-track M.S. programs must pass the Ph.D. qualifying examination, as part of the requirements for the doctoral degree. The exam is given each January, during the week before classes start. Students in the above programs must take this exam in their first year at Columbia. It is emphasized that research and academic advisers, or the student coordinator, are not empowered to allow a student to postpone taking this examination. A student who does not take the examination during his/her first year following admission, without an extraordinary legitimate excuse approved by the doctoral committee, will be assumed to have failed it. Students studying for their M.S. degree, not currently in the M.S.-Ph.D. track, but who plan to apply for a position in the Ph.D. program, can also take the exam with the permission of the chair of the doctoral committee. Passing the exam is not connected to Ph.D. admissions (see Admission Requirements). However, if a student is admitted to the Ph.D. program and has already passed the exam, he/she will not have to take it again.
The qualifying exam consists of two parts: a written part and an oral part.

The written part is a breadth exam, and is the same for all students. Its duration is four hours. Its purpose is to test:

  1. Breadth of knowledge
  2. Understanding of fundamentals
  3. Ability to do research

The exam is based only on topics typically covered in undergraduate classes, but the types of questions asked in it assume graduate level maturity in terms of thinking ability and initiative. The examination consists of 18 problems, three from each of the following six areas:

  1. Circuits and Electronics
  2. Signals, Systems, and Communications
  3. Solid-State Devices and Electromagnetics
  4. Digital Computing Systems
  5. Networking
  6. Systems Biology
The exam is not tied to particular courses. A syllabus for the material is given in Section D below. Students must work out six and only six out of the 18 problems. If more than six problems are handed in, six of those problems will be picked at random for grading.

If a student comes from a department other than electrical engineering and computer science, and brings significant expertise in an area distinct from those covered in the EE written D.Q.E. examination, and that knowledge is expected to be usable in a fundamental manner in the student's Ph.D. thesis, then the student may request from the department to only be responsible for four problems in the written D.Q.E. exam. If the student research adviser concurs with that request, then the student will have to make an oral presentation prior to the D.Q.E., to a committee of three faculty members explaining the topic of his/her proposed research, and demonstrating the use of that knowledge in combination with a sufficiently broad area within electrical engineering. The student's performance in that oral examination will then be considered by the department in lieu of the additional problems in the written D.Q.E. exam.

No calculators will be allowed into the written examination.

The oral part is a depth exam in one of the above areas, selected by the student. This examination is not limited to the material of the written examination. The oral exam is given within three working days of the written exam. Each student is examined by several designated faculty members on a one-to-one basis. These exams take place in faculty offices, and each typically lasts no more than 15 minutes. The scheduling of the slots for the oral exam is done by the graduate student coordinator.

The evaluation of the results of the oral and written exams is made by the entire faculty, and students are judged on a department-wide basis. Faculty members are not empowered to discuss any aspect of the exam or its evaluation with the students. Questions concerning procedural matters are to be addressed to the graduate student coordinator in the Electrial Engineering Department office.

The department notifies the students of the results of the exam in writing, within two weeks of the date of the written part. Students who fail the exam the first time may take it once more in the following year. Under no circumstances will a student be allowed to take the exam more than twice.

Students who must take the qualifying exam in January are expected to have taken by that time at least 6 credits of graduate (6000-level) or senior/graduate (4000-level) courses at Columbia, and are expected to obtain an average score of A- or better in those courses. However, failure to meet this expectation is not a valid reason for postponing to take the examination.

Students who want to take the qualifying exam must file the required form by November 30. If they already have identified an adviser who is tentatively willing to supervise their research, they should indicate his/her name on the form. However, having a research adviser is not a requirement for taking the D.Q.E.; in many instances a research adviser will be identified after a student has passed the examination.

7. Declaration of a Research Adviser
Students who are admitted to the Ph.D. program and pass the D.Q.E. must still find an adviser willing to supervise their work in order to continue in the Ph.D. program. Those who pass the D.Q.E. exam and do not yet have a adviser should speak to potential research advisers and identify one willing to supervise them as soon as possible. Once the name of the adviser is known, it should be reported to the Graduate Student Coordinator in the EE office. Students are expected to find and declare a research adviser within three semesters (not including summer) after passing their qualifying exam.
Students who pass the qualifying exam but cannot find an adviser in the area of their choice willing to supervise them and/or support them, are allowed to change to a different area without having to retake any portion of the qualifying exam. Students who are receiving financial support but fail the qualifying exam in January will generally continue to be supported through the spring; however, support beyond that point is not automatic.
8. Thesis Proposal Examination
Doctoral students must pass the Ph.D. thesis-proposal exam within two years after they have passed the D.Q.E. Students who want to take this exam will be expected to have a grade point average of at least 3.5 for courses taken at Columbia. The thesis proposal exam is oral, and its topic is chosen in coordination with the student's research adviser. The exam is administered by a committee of at least three faculty members, including the student's research adviser, and can last up to two hours. The student makes an oral presentation of the state of the art and his/her proposed research, possibly with new results. This oral presentation typically lasts no more than twenty minutes, and is followed by a period of questions by the committee. These can be any questions the committee feels are necessary to judge the candidate's knowledge and readiness for doctoral research in his/her chosen field, and are not limited to the topic of the presentation. The student is notified of the outcome shortly after the exam.

9. Breadth Course Requirement
Doctoral students must complete a breadth requirement through course work in a technical area clearly outside their main area of interest. This course work should normally be outside the EE department, or even outside the FFSEAS. In special cases, some of the courses can be inside the EE department if it is clear that, together with the courses outside the department, they form a cohesive whole in an area clearly distinct from the main area of interest. In such cases, the student may use up to two courses inside the EE department to fulfill the Breadth Course Requirement, subject to the approval of the student’s adviser. This course work should be planned in consultation with the student's research or academic adviser in such a way as to expose the student to concepts in a different discipline, and especially to different approaches and ways of thinking than those prevalent in the student's main area of research. This program should consist of at least three graduate courses (4000- or 6000-level) in one area. If the area chosen is truly far removed from the student's main area, the student may be allowed to replace one of the graduate courses with an undergraduate course although such a course will not count toward the 60-point graduate course requirement. The courses should be selected so that they form a cohesive whole.
10. Language Proficiency Requirement
Every candidate in the the Eng.Sc.D. or the Ph.D. program who did not get his/her bachelor's degree in a country where English is the native language must take the American Language Examination and reach Level 10.
11. Progress Review
The performance of each student in the doctoral program will be reviewed regularly, at least once a semester, to ensure satisfactory progress towards completion of study. When necessary, students failing the requirements may be placed on probation or asked to withdraw from the doctoral program.

12. Master of Philosophy Degree
The degree of master of philosophy is conferred upon a student who has successfully fulfilled all of the following requirements of GSAS and our department:
  1. A minimum of 30 points beyond the M.S. must have been taken
  2. Six residence units (R.U.s) must have been completed; two of these are awarded for the M.S. degree and four are accumulated while the student is a Ph.D. degree candidate
  3. The research proposal examination must have been passed
  4. If the student did not get his/her machelor degree in a country where English is the native language he/she must take the American Language Examination and reach Level 10.
When the above requirements are met, students should see the Graduate Student Coordinator to apply for this degree.

13. Location of Research
The research should be conducted on the Columbia University campus. In unusual cases the thesis work may be performed at the student's place of employment subject to the following conditions:
  1. The research must be unclassified.
  2. The dissertation must represent the student's own efforts.
  3. The employer must have full cognizance that the student is using company facilities and possibly company time to carry out doctoral research.
  4. The employer must have no objections to the publication of results.
  5. Supervisory committee members must be free to visit the location where the research is being conducted whenever they wish to do so.
  6. The student must be prepared to consult with his/her thesis adviser on the campus as often as two half-days per week, if the adviser so requires.
  7. At the time the research program is submitted for approval, the student must obtain a letter from a responsive member of his/her firm, addressed to the chair of the doctoral committee, stating that the company will abide by the conditions set forth in this section.
14. Application for Final Thesis Defense Examination
Upon completion of a thesis acceptable to the candidate's thesis adviser, the candidate should file application for the final oral thesis defense examination in the office of the appropriate dean. Examinations will not be scheduled during the summer except in unusual cases. Seven copies of the dissertation are required for use by the members of the final examination committee. The examination committee is approved by the Dean and must consist of at least two faculty members from departments other than electrical engineering as well as at least three regular members of the electrical engineering department. It may also include scientists or engineers from outside the University who are interested in the research problem. The candidate must distribute thesis copies to each member of the final examination committee at least three weeks before the date of the examination. Eng.Sc.D. candidates should consult the academic calendar of the School of Engineering and Applied Science for pertinent deadlines.

15. Thesis Defense
The candidate stands for examination before the final examination committee on the date agreed upon by all members of the committee. It is customary for the candidate to present initially a brief and concise summary of his/her research with emphasis on original contributions to the particular area. The oral presentation part is typically not more than forty minutes. The examination is either passed, passed with revision required, or failed. (NOTE: Candidates must be registered at the time of the thesis defense.)

16. Thesis Deposit and Publication
Two copies of the accepted dissertation must be submitted to the dissertation secretary in the dean's office of the Graduate School of Arts of Sciences after the defense. Rules governing the form and deposit of the dissertation can be found in a number of booklets available from the Graduate School of Arts and Sciences, as well as in mimeographed information available from the dean's office of the School of Engineering and Applied Science, or the dissertation secretary of the Graduate School of Arts and Sciences. It is absolutely essential to secure these rules and to prepare the dissertation in accordance with them. Acknowledgement should be made to indicate that the work was carried out in the Department of Electrical Engineering.

17. Time Limit
All degree requirements must be completed within seven full-time-equivalent years from the beginning of the first course credited toward the doctoral degree. Students who do not meet this time limit will no longer be considered degree candidates.

18. Leave of Absence
A student who must interrupt studies for a compelling reason, for example, sustained ill health, may be granted a leave of absence for a stated period, usually not to exceed one year. If a leave is granted, the fact is entered on the student's permanent academic record. The period of leave of absence is not counted as part of the time allowed for the completion of degree requirements. A student who leaves the University without securing a leave of absence must apply for readmission, which is not easily granted.
19. Transfer to Professional Degree Candidacy
Doctoral candidates may transfer to professional degree candidacy at any time and may apply all accumulated credits toward meeting the point requirements specified in section C1.
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Syllabus for the Written Part of the Ph.D. Qualifying Examination


NOTE: In the topics listed below, the emphasis is meant to be on FUNDAMENTAL PRINCIPLES, and not on dry facts. Proper understanding of these principles, in the familiar context outlined below, should enable one to apply them to less familiar situations.

Area 1: Circuits and Electronics
  • Resistors, capacitors, inductors, ideal transformers, independent sources, dependent sources and operational amplifiers
  • Charge, current, voltage, and power
  • Kirchhoff's voltage and current laws
  • Node and mesh analysis
  • Analysis of RL, RC and RLC circuits (up to second order) using differential equations
  • Natural and forced response
  • Linearity and superposition; Thevenin and Norton equivalents
  • Phasor analysis
  • Transfer functions, the complex plane, poles and zeros
  • Frequency response and Bode plots
  • Resonant circuits
  • Diodes and diode circuits
  • MOS and bipolar transistor characteristics
  • Basic logic gate circuits (CMOS, TTL, ECL)
  • Noise margins, logic delay, rise and fall times, fanin and fanout
  • Circuits for latches and flip-flops
  • Transient response of logic gate circuits
  • Small-signal equivalent circuits for diodes and transistors
  • Single-transistor amplifiers and differential pairs, and their dc bias analysis, large-signal analysis, small-signal analysis and frequency response
  • Circuits using operational amplifiers

Area 2: Signals, Systems, and Communications
  • Continuous-time and discrete-time systems
  • Linear time-invariant systems
  • Convolution
  • Fourier series
  • Continuous-time Fourier transform
  • Discrete-time Fourier transform
  • Filters and difference and differential equations
  • One-sided and two-sided Laplace transforms (including the handling of initial conditions)
  • One-sided and two-sided z-transforms (including the handling of initial conditions)
  • Time and frequency characteristics of linear time-invariant systems
  • Frequency domain analysis
  • Transfer function
  • Sampling theorem
  • Frequency response, Bode plots
  • Filtering, allpass, minimum-phase, linear phase systems
  • Linear feedback systems
  • Stability
  • Gain and phase margin
  • State-space analysis of continuous-time and discrete-time systems
  • Analysis of basic communication systems
  • Amplitude modulation
  • Pulse amplitude modulations (PAM), pulse code modulation (PCM)
  • Demodulation/detection
  • Multiplexing signals: time division multiplexing (TDM), frequency division multiplexing (FDM)
  • Elementary probability, sample spaces
  • Discrete and continuous random variables
  • Distribution functions (Bernoulli, Poisson, geometric, binomial, normal, exponential)
  • Conditional probability and Bayes's formula
  • Independent events and random variables
  • Expectation, conditional expectation
  • Covariance, variance, correlation

Area 3: Solid-State Devices and Electromagnetics
  • Electronic classification of solids: metals, semiconductors, insulators
  • Density-of-states, Fermi-Dirac distribution function, quasi-Fermi levels
  • Donors and acceptors, electrons and holes, majority and minority carrier
  • Drift and diffusion, fundamental transport equations
  • Recombination and generation
  • P-N junctions: I-V characteristics and small-signal models, breakdown
  • Bipolar junction transistors: Basic structure and operation, I-V characteristics and small-signal models
  • MOS devices: C-V characteristics of MOS capacitors, accumulation, depletion, inversion, threshold voltage, I-V characteristics, and small-signal models
  • Fundamental fabrication processes: photolithography, oxidation, epitaxy, diffusion, ion implantation
  • Electric and magnetic fields
  • Gauss, Coulomb, Ampere, Biot-Savart laws
  • Electromotive force; magnetomotive force
  • Current, current density, conservation of charge
  • Current element; dipole
  • Ohm's law, conductivity, electric power density
  • Faraday's law, Maxwell's equations
  • Transmission lines; wave propagation
  • Standing waves; impedance matching
  • Poynting theorem: real, complex
  • Plane waves; polarization: linear, circular
  • Snell's laws


Area 4: Digital Computing Systems
  • Elementary logic design
  • Digital coding, parity, error correction
  • Switching functions, truth tables, boolean algebra.
  • Logic gates, canonical forms, K-maps and 2-level logic minimization
  • Digital building blocks
  • Decoders, multiplexers, encoders, logic arrays, read-only memory
  • Latches, flip-flops, setup and hold times
  • Registers, shift registers, counters, register files, read-write memories
  • State machine design
  • Synchronous and asynchronous systems, sequential systems
  • State transition diagrams, finite-state machines, state minimization
  • Computer architecture
  • Data paths and control, busses, bit slices, register transfer language
  • Microcode, microprogrammed controllers, hardwired controllers
  • Instruction set architecture, assembly language, CPU, main memory, cache, I/O, interrupts
  • Fundamental data structures and algorithms
  • Arrays, linked lists, stacks, hash tables, queues, trees
  • Sorting, searching, storage management
  • Software engineering and operating systems concepts
  • Basic programming and use of compilers/interpreters; operator precedence, naming, control structures, nesting and scope, definition of new data structures; subroutines, iteration, recursion, exception handling; virtual memory, processor scheduling, process management, interprocess communication, device management, file systems

Area 5: Networking
  • Basic probability and combinatorics
  • Bernoulli and Poisson processes
  • Binomial, geometric, exponential distributions
  • Error detecting and correcting codes
  • 1D and 2D parity check codes
  • CRC check codes
  • Hamming codes
  • Multiple access resolution protocols
  • ALOHA
  • Slotted ALOHA / round-based methods
  • Carrier sensing MAC protocols
  • CDMA
  • Routing and spanning tree algorithms
  • Minimum spanning tree
  • Dijkstra's algorithm
  • Bellman-Ford / distance-vector algorithms
  • Reliable transfer protocols
  • Alternating-bit protocol
  • Selective repeat
  • Go-back-n
  • Finite state methods
  • Congestion and flow control
  • Fluid models of congestion collapse
  • Flow control (application of Little's law)
  • AIMD adaptive congestion control
  • Queueing and fairness
  • FIFO queuing (M/M/k/n queueing systems)
  • Round-robin
  • Virtual clock
  • (Weighted) fair queueing
  • Markov model representations of queueing systems
  • Max-min fairness
  • Network analysis
  • Little's law
  • Kleinrock's independence approximation

Area 6: Systems Biology
  • Biomolecular sequence alignment
  • Building of phylogenetic trees
  • Gene prediction
  • Biomolecular structure prediction
  • Analysis of microarray data
  • Modeling of biomolecular networks
  • Hodgkin-Huxley neuron
  • Reduced Hodgkin-Huxley models
  • Integrate-and-fire neuron
  • Stimulus representation and the neural code
  • Algorithms for stimulus recovery
  • Synaptic plasticity and learning algorithms
  • Computation by excitatory and inhibitory networks
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