Networks of neurons as dynamical systems: from geometry to biophysics
Author:
Nancy Kopell
Journal:
Quart. Appl. Math. 56 (1998), 707-718
MSC:
Primary 92C05; Secondary 34C35
DOI:
https://doi.org/10.1090/qam/1668734
MathSciNet review:
MR1668734
Full-text PDF Free Access
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Additional Information
V. Murthy and E. Fetz, Coherent 25- to 35-Hz oscillations in the sensorymotor cortex of awake behaving monkeys, Proc. Nat. Acad. Sci. USA 5690-5674
C. M. Gray, P. Konig, 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)
G. Laurent and H. Davidowitz, Encoding of olfactory information with oscillating neural assemblies, Science 265, 1872–1875 (1994)
E. Basar, C. Basar-Eroglu, R. Parnefjord, E. Rahn, and M. Schurmann, Evoked potentials: ensembles of brain induced rhythmicities in the alpha, theta and gamma ranges, in Induced Rhythms in the Brain, E. Basar and T. H. Bullock, eds., Birkhäuser, Boston, 1992
J. O’Keefe, Hippocampus, theta and spatial memory, Current Opinion in Neurobiology 3, 917–924 (1993)
F. A. Lenz, J. Vitek, and M. DeLong, Role of the thalamus in parkinsonian tremor: evidence from studies in patients and primate models, Stereotactic and Functional Neurosurgery 60, 94–103 (1993)
M. Steriade, D. Contreras, and F. Amzica, Synchronized sleep oscillations and their paroxysmal developments, Trends in the Neurosciences 17, 199–208 (1994)
- N. Kopell and G. B. Ermentrout, Symmetry and phaselocking in chains of weakly coupled oscillators, Comm. Pure Appl. Math. 39 (1986), no. 5, 623–660. MR 849426, DOI https://doi.org/10.1002/cpa.3160390504
- N. Kopell and G. B. Ermentrout, Phase transitions and other phenomena in chains of coupled oscillators, SIAM J. Appl. Math. 50 (1990), no. 4, 1014–1052. MR 1053917, DOI https://doi.org/10.1137/0150062
Kandel, Schwartz, and Jessel, Principles of Neuroscience, Simon and Schuster, N.Y., 1991
J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and oscillations, in Methods of Neural Modeling, C. Koch and I. Segev, eds., MIT Press, Cambridge, MA, 1989, pp. 135–170
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W. Yamada, C. Koch, and P. Adams, Multiple channels and calcium dynamics in Methods of Neuronal Modeling, C. Koch and I. Segev, eds., MIT Press, 1989
W. Rall, Cable theory for dendritic neurons, in Methods of Neuronal Modeling, C. Koch and I. Segev, eds., MIT Press, 1989
D. A. McCormick, Neurotransmitter actions in the thalamus and cerebral cortex and their role in the neuromodulation of thalamocortical activity, Prog. Neurobiol. 39, 337–388 (1992)
R. Harris-Warrick and E. Marder, Modulation of neural networks for behavior, Ann. Rev. Neurosci. 14, 39–57 (1991)
G. LeMasson, E. Marder, and L. F. Abbott, Activity-dependent regulation of conductances in model neurons, Science 259, 1915–1917 (1993)
C. Morris and H. Lecar, Voltage oscillations in the barnicle giant muscle fiber, Biophysical J. 35, 193–213 (1981)
J. Rinzel, Excitation dynamics: insights from simplified network models, Federation Proc. 44, 2944–2946 (1985)
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D. Somers and N. Kopell, Rapid synchronization through fast threshold modulation, Biol. Cybern. 68, 393–407 (1993)
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C. M. Gray, Synchronous oscillations in neuronal systems: mechanisms and functions, J. Comput. Neuro. 1, 11–38 (1994)
D. L. Wang and D. Terman, Image segmentation based on oscillatory correlation, Neural Comp. 9, 805–836 (1997)
R. Harris-Warrick, E. Marder, A. Selverston, and M. Moulins, eds., Dynamic Biological Networks: The Stomatogastric Nervous System, MIT Press, 1992
L. Abbott, E. Marder, and S. Hooper, Oscillating networks: control of burst duration by electrically coupled neurons, Neural Comp. 3, 487–497 (1991)
N. Kopell, L. Abbott, and C. Soto-Trevino, On the behavior of a neural oscillator electrically coupled to a bistable cell, Physica D, to appear
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V. Booth and J. Rinzel, A minimal compartmental model for a dendritic origin of bistability of motoneuron firing patterns, J. Comput. Neurosci. 2, 1–14 (1995)
I. Segev, J. Fleshman, and R. Burke, Compartmental models of complex neurons, in Methods of Neuronal Modeling, C. Koch and I. Segev, eds., MIT Press, 1989
- Cristina Soto-Trevino, Geometric methods for periodic orbits in singularly perturbed systems, ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–Boston University. MR 2696661
W. Friesen, Reciprocal inhibition, a mechanism underlying oscillatory animal movements, Neurosci. Behavior 18, 547–553 (1994); P. Rowat and A. Selverston, Oscillatory mechanisms in pairs of neurons connected with fast inhibitory synapses, J. Comput. Neurosci. 4, 103–127 (1997); F. Skinner, N. Kopell, and E. Marder, Mechanisms for oscillations and frequency control in networks of mutually inhibitory relaxation oscillators, J. Comput. Neurosci. 1, 69–87 (1994)
X. J. Wang and J. Rinzel, Alternating and synchronous rhythms in reciprocally inhibitory model neurons, Neural Comp. 4, 84–97 (1992)
X. J. Wang and J. Rinzel, Spindle rhythmicity in the reticularis thalami nucleus: synchronization among mutually inhibitory neurons, Neuroscience 53, 899–904 (1993)
M. Steriade, E. G. Jones, and R. R. Llinas, Thalamic oscillations and signaling, Wiley, New York, 1990
C. van Vreeswijk, L. Abbott, and G. B. Ermentrout, When inhibition, not excitation synchronized neural firing, J. Comput. Neurosci. 1, 313–321 (1994)
W. Gerstner, J. L. van Hemmen, and J. Cowen, What matters in neuronal locking?, Neural Comp. 8, 1653–1676 (1996)
- Carson C. Chow, Phase-locking in weakly heterogeneous neuronal networks, Phys. D 118 (1998), no. 3-4, 343–370. MR 1633022, DOI https://doi.org/10.1016/S0167-2789%2898%2900082-7
- D. Terman, N. Kopell, and A. Bose, Dynamics of two mutually coupled slow inhibitory neurons, Phys. D 117 (1998), no. 1-4, 241–275. MR 1630243, DOI https://doi.org/10.1016/S0167-2789%2897%2900312-6
D. Golomb and J. Rinzel, Clustering in globally coupled inhibitory neurons, Physica D 72, 259–282 (1994); X.-J. Wang, D. Golomb, and J. Rinzel, Emergent spindle oscillations and intermittent burst firing in a thalamic model: specific neuronal mechanisms, Proc. Natl. Acad. Sci. (USA) 92, 5577–5581 (1995)
N. Kopell and G. LeMasson, Rhythmogenesis, amplitude modulation and multiplexing in a cortical architecture, Proc. Nat. Acad. Sci (USA) 91, 10586–10590 (1994)
D. Terman, A. Bose, and N. Kopell, Functional reorganization in thalamocortical networks: Transition between spindling and delta sleep rhythms, Proc. Nat. Acad. Sci. USA 93, 15417–15422 (1996)
A. Destexhe, D. A. McCormick, and T. J. Sejnowski, A model for 8–10 spindling in interconnected thalamic relay and reticularis neurons, Biophysical J. 65, 2473–2477 (1993)
D. Golomb, X.-J. Wang, and J. Rinzel, Synchronization properties of spindle oscillations in a thalamic reticular nucleus model, J. Neurophys. 72 (1994)
Y. Manor, F. Nadim, E. Marder, S. Epstein, J. Ritt, and N. Kopell, Frequency regulation of a slow rhythm by a fast periodic input: a geometric analysis, submitted
J. Collins, C. Chow, and T. Imhoff, Stochastic resonance without tuning, Nature 376, 236–238 (1995)
L. F. Abbott, J. A. Varela, K. Sen, and S. B. Nelson, Synaptic depression and cortical gain control, Science 275, 220–224 (1997)
V. Murthy and E. Fetz, Coherent 25- to 35-Hz oscillations in the sensorymotor cortex of awake behaving monkeys, Proc. Nat. Acad. Sci. USA 5690-5674
C. M. Gray, P. Konig, 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)
G. Laurent and H. Davidowitz, Encoding of olfactory information with oscillating neural assemblies, Science 265, 1872–1875 (1994)
E. Basar, C. Basar-Eroglu, R. Parnefjord, E. Rahn, and M. Schurmann, Evoked potentials: ensembles of brain induced rhythmicities in the alpha, theta and gamma ranges, in Induced Rhythms in the Brain, E. Basar and T. H. Bullock, eds., Birkhäuser, Boston, 1992
J. O’Keefe, Hippocampus, theta and spatial memory, Current Opinion in Neurobiology 3, 917–924 (1993)
F. A. Lenz, J. Vitek, and M. DeLong, Role of the thalamus in parkinsonian tremor: evidence from studies in patients and primate models, Stereotactic and Functional Neurosurgery 60, 94–103 (1993)
M. Steriade, D. Contreras, and F. Amzica, Synchronized sleep oscillations and their paroxysmal developments, Trends in the Neurosciences 17, 199–208 (1994)
N. Kopell, N. Ermentrout, and G. B. Ermentrout, Symmetry and phaselocking in chains of weakly coupled oscillators, Comm. Pure Appl. Math. 39, 623–660 (1986)
N. Kopell and G. B. Ermentrout, Phase transitions and other phenomena in chains of oscillators, SIAM J. Appl. Math. 50, 1014–1052 (1990)
Kandel, Schwartz, and Jessel, Principles of Neuroscience, Simon and Schuster, N.Y., 1991
J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and oscillations, in Methods of Neural Modeling, C. Koch and I. Segev, eds., MIT Press, Cambridge, MA, 1989, pp. 135–170
H. Tuckwell, Stochastic Processes in the Neurosciences, SIAM, Philadelphia, 1989
W. Yamada, C. Koch, and P. Adams, Multiple channels and calcium dynamics in Methods of Neuronal Modeling, C. Koch and I. Segev, eds., MIT Press, 1989
W. Rall, Cable theory for dendritic neurons, in Methods of Neuronal Modeling, C. Koch and I. Segev, eds., MIT Press, 1989
D. A. McCormick, Neurotransmitter actions in the thalamus and cerebral cortex and their role in the neuromodulation of thalamocortical activity, Prog. Neurobiol. 39, 337–388 (1992)
R. Harris-Warrick and E. Marder, Modulation of neural networks for behavior, Ann. Rev. Neurosci. 14, 39–57 (1991)
G. LeMasson, E. Marder, and L. F. Abbott, Activity-dependent regulation of conductances in model neurons, Science 259, 1915–1917 (1993)
C. Morris and H. Lecar, Voltage oscillations in the barnicle giant muscle fiber, Biophysical J. 35, 193–213 (1981)
J. Rinzel, Excitation dynamics: insights from simplified network models, Federation Proc. 44, 2944–2946 (1985)
F. Hoppensteadt and E. Izhikevich, Weakly Connected Neural Networks, Springer-Verlag, N.Y., 1997
G. B. Ermentrout and N. Kopell, Multiple pulse interactions and averaging in coupled neural oscillators, Journal of Math. Biol. 29, 195–217 (1991)
D. Somers and N. Kopell, Rapid synchronization through fast threshold modulation, Biol. Cybern. 68, 393–407 (1993)
D. Somers and N. Kopell, Waves and synchrony in arrays of oscillators of relaxation and non-relaxation type, Physica D 89, 169–183 (1995)
D. Terman and D. L. Wang, Global competition and local cooperation in a network of neural oscillators, Physica D 81, 148–176 (1995)
C. M. Gray, Synchronous oscillations in neuronal systems: mechanisms and functions, J. Comput. Neuro. 1, 11–38 (1994)
D. L. Wang and D. Terman, Image segmentation based on oscillatory correlation, Neural Comp. 9, 805–836 (1997)
R. Harris-Warrick, E. Marder, A. Selverston, and M. Moulins, eds., Dynamic Biological Networks: The Stomatogastric Nervous System, MIT Press, 1992
L. Abbott, E. Marder, and S. Hooper, Oscillating networks: control of burst duration by electrically coupled neurons, Neural Comp. 3, 487–497 (1991)
N. Kopell, L. Abbott, and C. Soto-Trevino, On the behavior of a neural oscillator electrically coupled to a bistable cell, Physica D, to appear
E. F. Mischenko and N. Kh. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations, Plenum Press, N.Y., 1980
J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci. USA 79, 2554–2558 (1982)
V. Booth and J. Rinzel, A minimal compartmental model for a dendritic origin of bistability of motoneuron firing patterns, J. Comput. Neurosci. 2, 1–14 (1995)
I. Segev, J. Fleshman, and R. Burke, Compartmental models of complex neurons, in Methods of Neuronal Modeling, C. Koch and I. Segev, eds., MIT Press, 1989
C. Soto-Trevino, Geometric methods for periodic orbits in singularly perturbed systems, Thesis, Boston University, 1997
W. Friesen, Reciprocal inhibition, a mechanism underlying oscillatory animal movements, Neurosci. Behavior 18, 547–553 (1994); P. Rowat and A. Selverston, Oscillatory mechanisms in pairs of neurons connected with fast inhibitory synapses, J. Comput. Neurosci. 4, 103–127 (1997); F. Skinner, N. Kopell, and E. Marder, Mechanisms for oscillations and frequency control in networks of mutually inhibitory relaxation oscillators, J. Comput. Neurosci. 1, 69–87 (1994)
X. J. Wang and J. Rinzel, Alternating and synchronous rhythms in reciprocally inhibitory model neurons, Neural Comp. 4, 84–97 (1992)
X. J. Wang and J. Rinzel, Spindle rhythmicity in the reticularis thalami nucleus: synchronization among mutually inhibitory neurons, Neuroscience 53, 899–904 (1993)
M. Steriade, E. G. Jones, and R. R. Llinas, Thalamic oscillations and signaling, Wiley, New York, 1990
C. van Vreeswijk, L. Abbott, and G. B. Ermentrout, When inhibition, not excitation synchronized neural firing, J. Comput. Neurosci. 1, 313–321 (1994)
W. Gerstner, J. L. van Hemmen, and J. Cowen, What matters in neuronal locking?, Neural Comp. 8, 1653–1676 (1996)
C. Chow, Phase-locking in weakly heterogeneous neuronal networks, Physica D 118, 343–370 (1998)
D. Terman, N. Kopell, and A. Bose, Dynamics of two mutually coupled slow inhibitory neurons, Physica D, to appear
D. Golomb and J. Rinzel, Clustering in globally coupled inhibitory neurons, Physica D 72, 259–282 (1994); X.-J. Wang, D. Golomb, and J. Rinzel, Emergent spindle oscillations and intermittent burst firing in a thalamic model: specific neuronal mechanisms, Proc. Natl. Acad. Sci. (USA) 92, 5577–5581 (1995)
N. Kopell and G. LeMasson, Rhythmogenesis, amplitude modulation and multiplexing in a cortical architecture, Proc. Nat. Acad. Sci (USA) 91, 10586–10590 (1994)
D. Terman, A. Bose, and N. Kopell, Functional reorganization in thalamocortical networks: Transition between spindling and delta sleep rhythms, Proc. Nat. Acad. Sci. USA 93, 15417–15422 (1996)
A. Destexhe, D. A. McCormick, and T. J. Sejnowski, A model for 8–10 spindling in interconnected thalamic relay and reticularis neurons, Biophysical J. 65, 2473–2477 (1993)
D. Golomb, X.-J. Wang, and J. Rinzel, Synchronization properties of spindle oscillations in a thalamic reticular nucleus model, J. Neurophys. 72 (1994)
Y. Manor, F. Nadim, E. Marder, S. Epstein, J. Ritt, and N. Kopell, Frequency regulation of a slow rhythm by a fast periodic input: a geometric analysis, submitted
J. Collins, C. Chow, and T. Imhoff, Stochastic resonance without tuning, Nature 376, 236–238 (1995)
L. F. Abbott, J. A. Varela, K. Sen, and S. B. Nelson, Synaptic depression and cortical gain control, Science 275, 220–224 (1997)
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© Copyright 1998
American Mathematical Society