Professor Engineering Sciences and Applied Mathematics McCormick School of Engineering Northwestern University 2145 Sheridan Road Evanston, Illinois 60208-3125 Co-Director Northwestern Institute on Complex Systems Other Affiliations: Neurobiology and Physiology Center for Photonic Computing & Communication Fellow of the Optical Society of America phone: 847-491-8784          fax: 847-491-2178 main office: 847-491-3345

Fiber Optics The goal of this research area is the mathematical modeling of high bit-rate fiber-optic communications. We study linear and nonlinear pulse propagation in optical fibers, including solitons. One emphasis has been the application of parametric amplification to optical fiber systems and devices; this work has involved the modelling of experiments performed by Prem Kumar's group at Northwestern. Most recently, we have been using importance sampling and related methods to investigate the significance of rare events in lightwave transmission systems. Studies have considered polarization mode dispersion (PMD), amplitude and timing jitter in soliton transmission systems caused by amplified spontaneous emission (ASE) noise, and phase jitter in differential phase-shift keyed (DPSK) systems.

Simulated eye-diagram for a 10 Gb/s DPSK system indicating the probability for the received voltage to have a particular value at any given time. The colorbar on right gives the logarithm (base 10) of each color.

Computational Neuroscience With Nelson Spruston (Neurobiology and Physiology) and David Chopp (Applied Math) this NIH-funded project is a experimental and computational study of microcircuits composed of principal neurons and interneurons in the CA1 region of the hippocampus. Computational models are being developed based on the latest data available from ongoing experimental studies, which include patch-clamp experiments measuring ionic conductances, calcium imaging, and full-cell morphological reconstructions. More results are available on the dichotomy of backpropagating action potentials and the gating of distal dendritic spikes.

Group Members

Current graduate students: M. S. McCallum, R. E. Trana, V. Menon, G. Donovan, Y. Katz

Current postdoctoral associates: J. Li

Former graduate students: A. K. Hobbs, D. J. Muraki, T. Ueda, C. V. Hile, J. N. Kutz, A. Niculae, M. J. Mills, B. S. Marks, R. O. Moore, S. K. Bhatt, E. T. Spiller

Former undergraduate students: A. D. Kim, S. L. Fogal, A. C. Narayan

Former postdoctoral associates: C. G. Goedde, T.-S. Yang, G. Biondini, T. Horikis

Sample neuroscience publications:

1. Y. Katz, W. L. Kath, N. Spruston and M. Hasselmo, Coincidence Detection of Place and Temporal Context in a Network Model of Spiking Hippocampal Neurons, PLoS Computational Biology, 3(12): e234 (2007).
2. D. A. Nicholson, R. Trana, Y. Katz, W. L. Kath, N. Spruston and Y. Geinisman, Distance-dependent differences in synapse number and AMPA receptor expression in hippocampal CA1 pyramidal neurons, Neuron 50 (2006), pp. 431-442.
3. T. Jarsky, A. Roxin, W. L. Kath and N. Spruston, Conditional dendritic spike propagation following distal synaptic activation of hippocampal CA1 pyramidal neurons, Nature Neuroscience 8 (2005), pp. 1667-1676.
4. N. L. Golding, T. J. Mickus, Y. Katz, W. L. Kath, and N. Spruston, Factors mediating powerful voltage attenuation along CA1 pyramidal neuron dendrites, J. Physiology 568 (2005), pp. 69-82.
5. W. L. Kath, Computational modeling of dendrites, J. Neurobiology 64 (2005), pp. 91-99.
6. N. Spruston and W. L. Kath, Dendritic arithmetic, Nature Neurosci. 7 (2004), pp. 567-569.
7. N. Golding, W. L. Kath and N. Spruston, Dichotomy of action potential backpropagation in CA1 pyramidal neuron dendrites, J. Neurophysiology, 86 (2001), pp. 2998-3010; correction, 87 (2002), pp. U1-U2.

Sample optics publications:

1. R. O. Moore, G. Biondini and W. L. Kath, A Method to Compute Statistics of Large, Noise-Induced Perturbations of Nonlinear Schrödinger Solitons, SIAM J. Applied Math 67 (2007), pp. 1418-1439.
2. E. T. Spiller, W. L. Kath, R. O. Moore and C. J. McKinstrie, Computing large signal distortions and bit-error ratios in DPSK transmission systems, Photonics Tech. Letts. 17 (2005), pp. 1022-1024.
3. G. Biondini, W. L. Kath and C. R. Menyuk, Importance sampling for polarization-mode dispersion: Techniques and applications, J. Lightwave Tech. 22 (2004), pp. 1201-1215.
4. G. Biondini and W. L. Kath, Polarization mode dispersion emulation with Maxwellian lengths and importance sampling, Photonics Tech. Letts. 16 (2004), pp. 789-791.
5. R. O. Moore, B. Biondini and W. L. Kath, Importance sampling for noise-induced amplitude and timing jitter in soliton transmission systems, Opt. Lett., 28 (2003), pp. 105-107.
6. S. L. Fogal, G. Biondini and W. L. Kath, Multiple importance sampling for first- and second-order polarization-mode dispersion, Phot. Tech. Lett., 14 (2002), pp. 1273-1275; correction, pp. 1487.
7. G. Biondini, W. L. Kath and C. R. Menyuk, Importance sampling for polarization-mode dispersion, Phot. Tech. Lett., 14 (2002), pp. 310-312.
8. R. Holzloehner, C. R. Menyuk, V. S. Grigoryan and W. L. Kath, Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization, J. Lightwave Technology, 20 (2002), pp. 389-400.
9. Bill & Bonnie Kath, Making waves: Solitons and Their Applications, SIAM News, 31 #2, March 1998.

Winter 2008 courses:

Numerical Methods for Random Processes

Fall 2007 courses:

Math 234 Multivariable Calculus

Winter 2006 courses:

ESAM 495, Special Topics in Applied Mathematics -- Computational Neuroscience