University of Minnesota
University of Minnesota

IMA New Directions Program

Jonathan E. Rubin's References

for the New Directions Short Course

Mathematical Neuroscience

June 16-27, 2008

Photo Gallery | Lecture Videos

DAY 1: June 16, 2008 – Introduction to the Nervous System



3) M.C. Diamond, A.B. Scheibel, L.M. Elson, ``The Human Brain Coloring Book'', 1985, HarperCollinsPublishers, New York, NY

4) "Principles of Neural Science", edited by E. Kandel et al.

5) J. Lubke, V. Egger, B. Sakmann, and D. Feldmeyer, "Columnar organization of dendrites and axons of single and synaptically coupled excitatory spiny neurons in layer 4 of the rat barrel cortex", J. Neuorsci., 20:5300–5311, 2000.

6) A. Gupta, Y. Wang, and H. Markram, "Organizing principles for a diversity of GABAergic interneurons and synapses in the neocortex", Science, 287:273–278, 2000.

7) H. Markram, M. Toledo-Rodriguez, Y. Wang, A. Gupta, G. Silberberg, and C. Wu, "Interneurons of the neocortical inhibitory system", Nat. Rev. Neurosci., 5:793–807, 2004.

DAY 2: June 17, 2008 – Simple Models and Networks

1) A.V.M. Herz et al., Modeling Single-Neuron Dynamics and Computations: A Balance of Detail and Abstraction, Science, 314:80–85, 2006.

2) E.M. Izhikevich, Simple Model of Spiking Neurons, IEEE Trans. Neural Networks, 14:1569–1572, 2003.

3) E.M. Izhikevich, Which Model to Use for Cortical Spiking Neurons?, IEEE Trans. Neural Networks, 2004.

4) J. Keener, F. Hoppensteadt, and J. Rinzel, Integrate-and-fire models of nerve membrane response to oscillatory input, SIAM J. Appl. Math., 41:503–517, 1981.

5) S. Coombes and P. Bressloff, Mode-locking and Arnold tongues in integrate-and-fire neural oscillators, Phys. Rev. E 60:2086–2096, 1999.

6) P. Bressloff, Lectures in Mathematical Neuroscience, PCMI Lecture Series (AMS), 2008.

7) E. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (2006), MIT Press, Cambridge, MA.

DAY 3: June 18, 2008 – Hodgkin-Huxley Theory

Hodgkin, A., and Huxley, A. (1952): A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117:500–544.

Johnston, D., and Wu, S. (1997): Foundations of Cellular Neurophysiology, MIT Press, Cambridge, MA.

Dayan, P., and Abbott, L. (2001): Theoretical Neuroscience, MIT Press, Cambridge, MA.

Kepler, T.B., Abbott, L.F. and Marder, E. (1992) Reduction of Conductance-Based Neuron Models. Biol. Cybern. 66: 381–387.

Troy, William C. The bifurcation of periodic solutions in the Hodgkin-Huxley equations. Quart. Appl. Math. 36 (1978/79), no. 1, 73–83.

DAY 4: June 19, 2008 – Membrane Dynamics, Singular Perturbation, Bursting, Synapses

J. Drover, J. Rubin, J. Su, and B. Ermentrout, Analysis of a canard mechanism by which excitatory synaptic coupling can synchronize neurons at low firing frequencies, SIAM J. Appl. Math., 65:69–92, 2004.

J. Su, J. Rubin, and D. Terman, Effects of noise on elliptic bursters, Nonlinearity, 17:133–157, 2004.

J. Rubin, Surprising effects of synaptic excitation, J. Comp. Neurosci., 18:333–342, 2005.

J.Rubin and M. Wechselberger, Giant squid-hidden canard: the 3D geometry of the Hodgkin-Huxley model, Biol. Cybern. 97:5–32, 2007.

J.Rubin and M. Wechselberger, The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales, Chaos, 18:015105, 2008.

E. Lee and D. Terman, Uniqueness and stability of periodic bursting solutions, J. Diff. Equations, 158:48–78, 1999.

E.M. Izhikevich, Neural excitability, spiking and bursting, Int. J. Bif. Chaos, 10:1171–1266, 2000.

Rinzel, J. Bursting oscillations in an excitable membrane model. Ordinary and partial differential equations (Dundee, 1984), 304–316, Lecture Notes in Math., 1151, Springer, Berlin, 1985.

Rinzel, John A formal classification of bursting mechanisms in excitable systems. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 1578–1593, Amer. Math. Soc., Providence, RI, 1987.

Rinzel, J. A formal classification of bursting mechanisms in excitable systems. Mathematical topics in population biology, morphogenesis and neurosciences (Kyoto, 1985), 267–281, Lecture Notes in Biomath., 71, Springer, Berlin, 1987.

M. Pedersen and M. Sorensen, The effect of noise on beta-cell burst period, SIAM J. Appl. Math, 67: 530–542, 2007.

Terman, D. The transition from bursting to continuous spiking in excitable membrane models. J. Nonlinear Sci. 2 (1992), no. 2, 135–182.

Terman, David Chaotic spikes arising from a model of bursting in excitable membranes. SIAM J. Appl. Math. 51 (1991), no. 5, 1418–1450.

J. Best, A. Borisyuk, J. Rubin, D. Terman and M. Wechselberger, "The dynamic range of bursting in a model respiratory pacemaker network," SIAM J. Appl. Dyn. Syst., 4: 1107–1139, 2005.

G.S. Medvedev, Reduction of a model of an excitable cell to a one-dimensional map, Physica D, 202(1–2), 37–59, 2005.

DAY 5: June 20, 2008 – Small Networks and Synchrony

J. Rubin and D. Terman, Geometric analysis of population rhythms in synaptically coupled neuronal networks, Neural Comp., 12:597–645, 2000.

J. Rubin and D. Terman, Analysis of clustered firing patterns in synaptically coupled networks of oscillators, J. Math. Biol., 41:513–545, 2000.

J. Rubin, Bursting indcued by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters, Phys. Rev. E, 74:021917, 2006.

Jonathan Rubin and David Terman, "Geometric Singular Perturbation Analysis of Neuronal Dynamics," in B. Fiedler, editor, Handbook of Dynamical Systems, Vol. 2, Elsevier, 2002.

D. Somers and N. Kopell, "Rapid synchronization through fast threshold modulation", Biol. Cybern. 68 (1993), 393–407.

D. Somers and N. Kopell, "Waves and synchrony in arrays of oscillators of relaxation and non-relaxation type", Physica D 89 (1995) 169–183.

D. Terman, N. Kopell and A. Bose, ``Dynamics of two mutually inhibitory neurons'' Physica D 117:241–275 (1998).

J. Rubin, "Bursting induced by excitatory synaptic coupling in non-identical conditional relaxation oscillators or square-wave bursters", Phys. Rev. E, 74, 021917, 2006.

DAY 8: June 25, 2008 – Synaptic Plasticity

L.F. Abbott et al., Synaptic depression and cortical gain control, Science 275:221–224, 1997.

M. Tsodyks and H. Markram, The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability, Proc. Nat. Acad. Sci. USA, 94:719–723, 1997.

J. Trommershauser, J. Marienhagen, and A. Zippelius, Stochastic model of central synapses: slow diffusion of transmitter interacting with spatial distributed receptors and transporters, J. Theor. Biol., 198:101–120, 1999.

M.C.W. van Rossum, G.Q. Bi, and G.G. Turrigiano, Stable Hebbian learning from spike-timing-dependent plasticity, J. Neurosci., 20:8812–8821, 2000.

A. Bose, Y. Manor, and F. Nadim, Bistable oscillations arising from synaptic depression, SIAM J. Appl. Math., 62:706–727, 2001.

J. Rubin, D. Lee, and H. Sompolinsky, Equilibrium properties of temporally asymmetric Hebbian plasticity, Phys. Rev. Lett., 86:364–367, 2001.

J. Rubin, R. Gerkin, G. Bi and C. Chow, Calcium time course as a signal for spike-timing-dependent plasticity, J. Neurophysiol., 93:2600–2613, 2004.

Q. Zou and A. Destexhe, Kinetic models of spike-timing dependent plasticity and their functional consequences in detecting correlations, Biol. Cybern., 2008.

B. Earnshaw and P. Bressloff, Modeling the role of lateral membrane diffusion in AMPA receptor trafficking along a spiny dendrite, J. Comput. Neurosci., DOI10.1007/s10827-008-0084-8, 2008; see also Bressloff and Earnshaw, Phys. Rev. E, 2007 and Bressloff, Earnshaw and Ward, SIAM J. Appl. Math, 2007, referenced in the Earnshaw and Bressloff paper

A. Destexhe, Z. Mainen and T. Sejnowski, Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formalism, J. Comput. Neurosci., 1: 195–231, 1994.

S. Song, K. Miller and L. Abbott, Competitive Hebbian learning through spike-timing-dependent synaptic plasticity, Nat. Neurosci. 3:919–926, 2000.

Song, S. and Abbott, L.F. (2001) Column and Map Development and Cortical Re-Mapping Through Spike-Timing Dependent Plasticity. Neuron 32:339–350.

L. Abbott and W. Regehr, Synaptic computation, Nature 431:796–803, 2004.

J. Karbowski and G.B. Ermentrout, Synchrony arising from a balanced synaptic plasticity in a network of heterogeneous neural oscillators, Phys. Rev. E, 65:031902, 2002.

DAY 9: June 26, 2008 – Waves, Evans Functions, Spatial Models

References (pdf)

DAY 10: June 27, 2008 – Development, Pattern Formation

Y. Guo, J. Rubin, C. McIntyre, J. Vitek, and D. Terman, "Thalamocortical relay fidelity varies across subthalamic nucleus deep brain stimulation protocols in a data-driven computational model", J. Neurophysiol., 99: 1477–1492, 2008.

J. Rubin and K. Josic, "The firing of an excitable neuron in the presence of stochastic trains of strong inputs", Neural Comp., 19: 1251–1294, 2007.

Jonathan Rubin and David Terman, "High frequency stimulation of the subthalamic nucleus eliminates pathological rhythmicity in a computational model," J. Comp. Neurosci., 16: 211–235, 2004.

D. Terman, J.E. Rubin, A.C. Yew and C.J. Wilson, "Activity patterns in a model for the Subthalamopallidal Network of the Basal Ganglia," J. Neurosci., 22: 2963–2976, 2002.