Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
The Hodgkin–Huxley Equations Read chapter Read first chapter

2010 | OriginalPaper | Chapter

8. Neural Oscillators: Weak Coupling

Authors : G. Bard Ermentrout, David H. Terman

Published in: Mathematical Foundations of Neuroscience

Publisher: Springer New York

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

This chapter begins the second part of the book. By now, we hope that the reader has a thorough knowledge of single cell dynamics and is ready to move onto networks. There are two main approaches to the analysis and modeling of networks of neurons. In one approach, the details of the action potentials (spikes) matter a great deal. In the second approach, we do not care about the timing of individual neurons; rather, we are concerned only with the firing rates of populations. This division is reflected in the sometimes acrimonious battles between those who believe that actual spike times matter and those who believe that the rates are all that the brain cares about. On these issues, we have our own opinions, but for the sake of the reader, we will remain agnostic and try to present both sorts of models.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Synaptic Channels
next chapter Neuronal Networks: Fast/Slow Analysis
Literature
1.
H. Agmon-Snir, C. E. Carr, and J. Rinzel. A case study for dendritic function: Improving the performance of auditory coincidence detectors. Nature, 393:268–272, 1998.CrossRef
2.
8.
L. Bai, X. Huang, Q. Yang, and J. Y. Wu. Spatiotemporal patterns of an evoked network oscillation in neocortical slices: Coupled local oscillators. J. Neurophysiol., 96:2528–2538, 2006.CrossRef
9.
W. Bao and J. Y. Wu. Propagating wave and irregular dynamics: spatiotemporal patterns of cholinergic theta oscillations in neocortex in vitro. J. Neurophysiol., 90:333–341, 2003.CrossRef
20.
P. C. Bressloff. Traveling waves and pulses in a one-dimensional network of excitable integrate-and-fire neurons. J. Math. Biol., 40:169–198, 2000.MathSciNetMATHCrossRef
23.
P. D. Brodfuehrer, E. A. Debski, B. A. O’Gara, and W. O. Friesen. Neuronal control of leech swimming. J. Neurobiol., 27:403–418, 1995.CrossRef
25.
E. Brown, J. Moehlis, and P. Holmes. On the phase reduction and response dynamics of neural oscillator populations. Neural Comput., 16:673–715, 2004.MATHCrossRef
41.
A. H. Cohen, G. B. Ermentrout, T. Kiemel, N. Kopell, K. A. Sigvardt, and T. L. Williams. Modelling of intersegmental coordination in the lamprey central pattern generator for locomotion. Trends Neurosci., 15:434–438, 1992.CrossRef
51.
S. M. Crook, G. B. Ermentrout, J. M. Bower Dendritic and synaptic effects in systems of coupled cortical oscillators. J. Comput. Neurosci., 5:315–29, 1998.MATHCrossRef
64.
A. V. Egorov, B. N. Hamam, E. Fransén, M. E. Hasselmo, and A. A. Alonso. Graded persistent activity in entorhinal cortex neurons. Nature, 420:173–178, 2002.CrossRef
69.
G. B. Ermentrout and J. D. Cowan. Large scale spatially organized activity in neural nets. SIAM J. Appl. Math., 38(1):1–21, 1980.MathSciNetMATHCrossRef
70.
G. B. Ermentrout and D. Kleinfeld. Traveling electrical waves in cortex: insights from phase dynamics and speculation on a computational role. Neuron, 29:33–44, 2001.CrossRef
72.
G. B. Ermentrout and N. Kopell. Parabolic bursting in an excitable system coupled with a slow oscillation. SIAM J. Appl. Math., 46:223–253, 1986.MathSciNet
76.
G. B. Ermentrout and J. Rinzel. Beyond a pacemaker’s entrainment limit: phase walk-through. Am. J. Physiol. Regul. Integr. Comp. Physiol., 246:102–106, 1984.
84.
C. P. Fall, E. S. Marland, J. M. Wagner, and J. J. Tyson, editors. Computational Cell Biology, volume 20 of Interdisciplinary Applied Mathematics. Springer, New York, 2002.MATH
102.
J. R. Gibson, M. Beierlein, and B. W. Connors. Functional properties of electrical synapses between inhibitory interneurons of neocortical layer 4. J. Neurophysiol., 93:467–480, 2005.CrossRef
104.
D. Golomb and Y. Amitai. Propagating neuronal discharges in neocortical slices: computational and experimental study. J. Neurophysiol., 78:1199–1211, Sep 1997.
106.
M. Golubitsky, I. Stewart, P. L. Buono, and J. J. Collins. Symmetry in locomotor central pattern generators and animal gaits. Nature, 401:693–695, 1999.CrossRef
107.
C. M. Gray, P. König, A. K. Engel, and W. Singer. Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature, 338:334–337, 1989.CrossRef
108.
C. M. Gray, A. K. Engel, P. König, and W. Singer. Synchronization of oscillatory neuronal responses in cat striate cortex: temporal properties. Vis. Neurosci., 8:337–347, 1992.CrossRef
112.
113.
A. T. Gulledge and G. J. Stuart. Excitatory actions of GABA in the cortex. Neuron, 37:299–309, 2003.CrossRef
115.
J. K. Hale and H. Koçak. Dynamics and Bifurcations, volume 3 of Texts in Applied Mathematics. Springer, New York, 1991.MATHCrossRef
128.
F. C. Hoppensteadt and E. M. Izhikevich. Weakly Connected Neural Networks, volume 126 of Applied Mathematical Sciences. Springer, New York, 1997.CrossRef
131.
J. Huguenard and D. McCormick. Electrophysiology of the Neuron: An Interactive Tutorial. Oxford University Press, Oxford, 1994.
134.
E. M. Izhikevich. Simple model of spiking neurons. IEEE Trans Neural Netw., 14:1569–1572, 2003.CrossRef
147.
J. P. Keener. Principles of Applied Mathematics: Transformation and Approximation. Advanced Book Program, Perseus Books, Cambridge, MA, revised edition, 2000.MATH
157.
C. Koch and e. I. Segev. Methods in Neuronal Modeling: From Synapses to Networks. MIT, Cambridge, MA, 1998.
159.
N. Kopell. Toward a theory of modelling central pattern generators. In A. H. Cohen, S. Rossignol, and S. Grillner, editors, Neural Control of Rhythmic Movements in Vertebrates, pages 265–284. Wiley, New York, 1988.
160.
N. Kopell and B. Ermentrout. Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators. In B. Fiedler, G. Iooss, and N. Kopell, editors, Handbook of Dynamical Systems II: Towards Applications. Elsevier, Amsterdam, 2002.
164.
V. I. Krinski and I. u. M. Kokoz. Analysis of the equations of excitable membranes. I. Reduction of the Hodgkins–Huxley equations to a 2d order system. Biofizika, 18:506–511, 1973.
165.
E. P. Krisner. Homoclinic orbit solutions of a one dimensional Wilson-Cowan type model. Electron. J. Differ. Equat., 107:30, 2008.MathSciNet
190.
Y. Manor, A. Bose, V. Booth, and F. Nadim. Contribution of synaptic depression to phase maintenance in a model rhythmic network. J. Neurophysiol., 90:3513–3528, Nov 2003.CrossRef
191.
H. Markram, Y. Wang, and M. Tsodyks. Differential signaling via the same axon of neocortical pyramidal neurons. Proc. Natl. Acad. Sci. U.S.A., 95:5323–5328, 1998.CrossRef
195.
J. M. Mayville, S. L. Bressler, A. Fuchs, and J. A. Kelso. Spatiotemporal reorganization of electrical activity in the human brain associated with a timing transition in rhythmic auditory-motor coordination. Exp. Brain Res., 127:371–381, 1999.CrossRef
200.
B. D. Mensh, E. Aksay, D. D. Lee, H. S. Seung, and D. W. Tank. Spontaneous eye movements in goldfish: oculomotor integrator performance, plasticity, and dependence on visual feedback. Vis. Res., 44:711–726, 2004.CrossRef
203.
M. Migliore, L. Messineo, and M. Ferrante. Dendritic Ih selectively blocks temporal summation of unsynchronized distal inputs in CA1 pyramidal neurons. J. Comput. Neurosci., 16:5–13, 2004.CrossRef
204.
R. E. Mirollo and S. H. Strogatz. Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math., 50(6):1645–1662, 1990.MathSciNetMATHCrossRef
205.
J. D. Murray. Mathematical Biology. II, volume 18 of Interdisciplinary Applied Mathematics. Spatial models and biomedical applications. Springer, New York, third edition, 2003.
206.
M. Muller and R. Wehner. Path integration in desert ants, Cataglyphis fortis. Proc. Natl. Acad. Sci. U.S.A., 85:5287–5290, 1988.CrossRef
209.
R. Osan, R. Curtu, J. Rubin, and B. Ermentrout. Multiple-spike waves in a one-dimensional integrate-and-fire neural network. J. Math. Biol., 48:243–274, 2004.MathSciNetMATHCrossRef
210.
J. E. Paullet and G. B. Ermentrout. Stable rotating waves in two-dimensional discrete active media. SIAM J. Appl. Math., 54(6):1720–1744, 1994.MathSciNetMATHCrossRef
214.
B. Pfeuty, G. Mato, D. Golomb, and D. Hansel. Electrical synapses and synchrony: the role of intrinsic currents. J. Neurosci., 23:6280–6294, 2003.
215.
P. Pinsky and J. Rinzel. Intrinsic and network rhythmogenesis in a reduced traub model of ca3 neurons. J. Comput. Neurosci., 1:39–60, 1994.CrossRef
218.
D. J. Pinto, J. A. Hartings, J. C. Brumberg, and D. J. Simons. Cortical damping: analysis of thalamocortical response transformations in rodent barrel cortex. Cereb. Cortex, 13:33–44, Jan 2003.CrossRef
220.
J. C. Prechtl, L. B. Cohen, B. Pesaran, P. P. Mitra, and D. Kleinfeld. Visual stimuli induce waves of electrical activity in turtle cortex. Proc. Natl. Acad. Sci. U.S.A., 94:7621–7626, 1997.CrossRef
235.
R. Romo, C. D. Brody, A. Hernandez, and L. Lemus. Neuronal correlates of parametric working memory in the prefrontal cortex. Nature, 399:470–473, 1999.CrossRef
246.
A. Shpiro, R. Curtu, J. Rinzel, and N. Rubin. Dynamical characteristics common to neuronal competition models. J. Neurophysiol., 97:462–473, 2007.CrossRef
249.
F. Skinner, N. Kopell, and E. Marder. Mechanisms for oscillation and frequency control in networks of mutually inhibitory relaxation oscillators. J. Comput. Neurosci., 1:69–87, 1994.MATHCrossRef
270.
R. D. Traub, R. K. Wong, R. Miles, and H. Michelson. A model of a ca3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances. J. Neurophysiol., 66:635–650, 1991.
273.
M. Tsodyks, A. Uziel, and H. Markram. Synchrony generation in recurrent networks with frequency-dependent synapses. J. Neurosci., 20:RC50, 2000.
278.
C. Van Vreeswijk, L. F. Abbott, and G. B. Ermentrout. When inhibition not excitation synchronizes neural firing. J. Comput. Neurosci., 1:313–321, 1994.CrossRef
282.
X. J. Wang. Calcium coding and adaptive temporal computation in cortical pyramidal neurons. J. Neurophysiol., 79:1549–1566, 1998.
283.
X. J. Wang. Synaptic basis of cortical persistent activity: the importance of NMDA receptors to working memory. J. Neurosci., 19:9587–9603, 1999.
284.
X.-J. Wang and J. Rinzel. Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Comput., 4:84–97, 1992.CrossRef
286.
M. A. Whittington, R. D. Traub, and J. G. Jefferys. Synchronized oscillations in interneuron networks driven by metabotropic glutamate receptor activation. Nature, 373:612–615, 1995.CrossRef
Metadata
Title
Neural Oscillators: Weak Coupling
Authors
G. Bard Ermentrout
David H. Terman
Copyright Year
2010
Publisher
Springer New York
Book
Mathematical Foundations of Neuroscience

Print ISBN: 978-0-387-87707-5

Electronic ISBN: 978-0-387-87708-2

Copyright Year: 2010

DOI
https://doi.org/10.1007/978-0-387-87708-2_8

Premium Partner

    Image Credits