Outline for Orals Examination
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General:
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1) Real Variables

2) Complex Variables

3) Ordinary Differential Equations

4) Partial Differential Equations

5) Asympotics


Special:  Mathematical Neurobiology
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1) Hodgkin-Huxley Equations
	i) Derivation of the Equations
		a. Cable equation
		b. H-H channel gating model and equations
	ii) Qualitative behavior of space-clamped (ODE)
		a. Fundamentals of phase plane and 2-D stability analysis
		b. Fast m/v phase plane
		c. Slow manifold in n/v plane
	iii) Singular perturbation treatment
	iv) Numerical methods for HH PDE

2) Synaptic Dynamics
	i) Ca++ channel dynamics
	ii) Ca++ diffusion within pre-synaptic terminal
	iii) Synaptic noise: statistics of vesicle release
	iv) Post-synaptic channel dynamics

3) Simplified models for interacting neurons and networks of interacting 
populations
	i) Fitzhugh-Nagumo 
	ii) Integrate-and-fire
	iii) Wilson-Cowan population model
	iv) Population density approach (Knight, Manin and Sirovich)

4) Probabilistic Modelling/Programming
	i) Random variable generation
	   	a. Basic pseudo-random number generators
	   	b. Generating r.v.s with arbitrary distribution
	   	c. Gaussion r.v.s (Box-Muller)
	ii) Monte Carlo techniques 
	   	a. Methods of controlling variance (e.g. control variates)
	ii) Poisson point processes
		a. Generating inhomogeneous r.v.s
		a. Application: optimal filter for the photon image --
			causal and non-causal	
		 
 
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