Math 299: Mathematics of the Brain,   Spring quarter 2007/08

  TuTh 2:15:3:30pm, Building 380 -- 380F,
Instructor: Victor Eliashberg   (victore@stanford.edu)

• About the course
• Announcements
• Lecture slides
• Lecture slides
• Suggested readings
• Homeworks

The course is an overview of several levels of mathematics of the brain with an emphasis on the problem of a universal learning computer (ULC) with the basic cognitive characteristics similar to those of the human brain. We present evidence that a brainlike ULC capable of learning to simulate a broad range of human cognitive functions can have a relatively short initial (untrained) representation. We examine different hypotheses about the structure of this representation.
See Course syllabus.

There will be no final exam. The grade for the class will be based on the attendance, class activity, and return of the homeworks.

No announcements.

• lecture 1 (.ppt)
• lectures 2, 3, and 4 (.ppt)
• lectures 5, 6, and 7 (.ppt)
• the last lecture (.ppt)

• Matlab 1 (.zip)

• Lecture 1:
  • W.J.H. Nauta and M. Feirtag. Fundamental neuroanatomy. W.H. Freeman and Co. 1986.
    It is one of the best books on functional neuroanatomy. You can buy it on Amazon.com for less than $10.00.
• Lecture 2:
  • V.Eliashberg. A relationship between neural networks and programmable logic arrays. International Conf. on Neural Networks, San Francisco, March 28 -Apr.1 1993, vol III. Download .pdf file
• Lectures 3-4:
  • G. W. Zopf jr. Attitude and context. Trans. of the Illinois Symposium on Principles of Self-organization. .    Download .pdf file.
    Note. Rotate pages 90 degree clockwize to read the scanned text.
  • V.Eliashberg. The concept of E-machine: On Brain Hardware and the Algorithms of Thinking. Proceedngs of the Third Annual Conference of the Cognitive Science Society, U.C. Berkeley, 1981, pp. 289-292.    Download .pdf file.
• Lectures 5-9:
  • E.R. Kandel. In search of memory. Norton and Co. 2006, Chapter 16. Molecules and short-term memory. Chapter 17. Long-term memory. Chapter 18. Memory genes. Chapter 19. A dialoge between genes and synapses. pp. 221-276. (You need to clearly understand that a bilogical synapse is more than just a variable weight.)
  • B. Hille. Ion channels of excitable membranes. Third edition. Sinaur Associates, Inc. 2001, pp. 575-602. Chapter 18. Gating mechanisms: kinetic thinking. (Just to get some basic idea about the current situtaion in this important area.)
  • J.G. Nichols, A.R. Martin, B.G. Wallace. From neuron to brain. Third edition. Sinaur Associates, Inc. 1992. Chapter 3. Ionic basis of the resting potential. pp. 66-89.
  • J. Szentagothai. Structuro-Functional Considerations of the Cerebellar Neiron Network, Proc. of the IEEE, Vol. 56, No. 6, June 1968, pp. 960-966. Download .pdf file
  • V. Eliashberg. Ensembles of membrane proteins as statistical mixed-signal computers. Proc. of the Internationl Joint Conference on Neural Networks. Montreal, Canada. July 31 - August 4, 2005. Download .pdf file

• Lecture 1:
  • Questions:
    1. What is the difference between biologically-inspired engineering (bionics) and scientific/engineering (reverse engineering)? What is the falsification strategy? Why, in science, negative facts have more power than positive facts?
    2. How big is the human genome? How long would be the human DNA molecule stretched in a single line? How long would be all DNA molecules from the human body connected in a single string?
    3. Why a formal representation of the untrained human brain cannot be too long?
    4. What are the main anatomically distinct homogeneous areas of the human neocortex?
    5. What is the purpose of association fibers? Why the neocortex cannot have a crossbar connectivity?
    6. The neural networks of the neocortex have rather simple topology and are quite slow. Where does the functional complexity of the neocortex come from?
• Lecture 2:
  • Questions:
    1. What is the difference between an analog system and a symbolic system?
    2. What is a deterministic combinatorial machine, a probabilistic combinatorial machine, and a machine universal with respect to the class of combinatorial machines = universal combinatorial machine?
    3. What is a programmable logic array (PLA)? How does a three-layer artificial neural network (3 layer ANN) solve the arbitration problem in PLA in the case of ambiguous associations?
    4. What is a deterministic finite-state machine, a probabilistic finite-state machine, and a machine universal with respect to the class of finite-state machines = universal finite-state machine ?
    5. How can one build any deterministic finite-state machine given a universal combinatorial machine, e.g. a PLA?
• Lectures 3 and 4:
  • Questions:
    1. What is a conventional RAM? What is a GRAM? What is the difference? Why does the external system (W,D) from slide 17 behave as a GRAM?
    2. Let a GRAM have address set A={1,2,3} and data set D={a,b,c,ε}. What will be the output sequence dout(1),... dout(6) if the input sequence (addr(1),din(1)),... (addr(6),din(6)) is (1,a),(2,b),(3,c),(1,ε),(2,ε),(3,ε)?
    3. A complete memory machine (CMM) is a system that stores its complete input/output sequence in its long-term memory (LTM). Let M be a CMM with the structure shown in slide 20 that stores its input sequence in gx, and its output sequence (signals after switches N) in gy. Let the work of M be described by expressions in slide 26. Let m=2 and p=1. Explain how M can be taught to simulate the GRAM from item 2.
    4. If you can answer question 3, can you also explain how the primitive E-machine from slides 28 and 29 can be taught to simulate the above GRAM? Assume that this E-machine uses complete memory learning algorithm.
• Lectures 5-9:
  • Problems:
    1. Consider the differential equation:    τ du/dt + u =s , where τ is a time constant that depends on s-u as follows:
       if( s - u > 0) τ = τ1 ; else τ = τ2. Name at least one system that is described by this equation with τ1 < τ2. Where does this nonlinearity come from?
    2. Explain how a differential equation containing time derivatives of any order can be simulated by a circuitry build from integrating operational amplifiers, constant coefficients, and nonlinear elements.
    3. Explain how differential equations with different time constants can result from the statistical dynamics of ensembles of probabilistic molecular machines (EPMM).
    4. What is Monte Carlo simulation?
    5. Think of some possible sources of differential equations with big time constant (seconds, minutes, hours) in the brain.