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Ermentrout GB, Terman DH (2010) Mathematical Foundations of Neuroscience Interdisciplinary Applied Mathematics, Antman SS:Marsden JE:Sirovich L:Wiggins, ed. pp.1
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Agmon-Snir H, Carr CE, Rinzel J (1998) The role of dendrites in auditory coincidence detection. Nature 393:268-72 [Journal] [PubMed] Allen T (1983) On the arithmetic of phase locking: coupled neurons as a lattice on R2 Phys D 6:305-320 Amari S (1977) Dynamics of pattern formation in lateral-inhibition type neural fields. Biol Cybern 27:77-87 [PubMed] Amitai Y, Gibson JR, Beierlein M, Patrick SL, Ho AM, Connors BW, Golomb D (2002) The spatial dimensions of electrically coupled networks of interneurons in the neocortex. J Neurosci 22:4142-52 [Journal] [PubMed] Anderson J, Rosenfeld E (1998) Talking Nets: An Oral History of Neural Networks Aronson DG, Weinberger HF (1975) Nonlinear diffusion in population genetics, combustion and nerve pulse propagation Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, Goldstein J, ed. pp.5 Baer SM, Erneux T, Rinzel J (1989) The slow passage through Hopf bifurcation: delay, memory ecects, and resonance. J Appl Math 49:55-71 Bai L, Huang X, Yang Q, Wu JY (2006) Spatiotemporal patterns of an evoked network oscillation in neocortical slices: coupled local oscillators. J Neurophysiol 96:2528-38 [Journal] [PubMed] Bao W, Wu JY (2003) Propagating wave and irregular dynamics: spatiotemporal patterns of cholinergic theta oscillations in neocortex in vitro. J Neurophysiol 90:333-41 [Journal] [PubMed] BARLOW HB, HILL RM, LEVICK WR (1964) RETINAL GANGLION CELLS RESPONDING SELECTIVELY TO DIRECTION AND SPEED OF IMAGE MOTION IN THE RABBIT. J Physiol 173:377-407 [PubMed] Beer RD (1995) On the dynamics of small continuous-time recurrent neural networks Adaptive Behavior 3:469-509 Beierlein M, Gibson JR, Connors BW (2003) Two dynamically distinct inhibitory networks in layer 4 of the neocortex. J Neurophysiol 90:2987-3000 [Journal] [PubMed] Bellman R, Cooke KL (1963) Differential-Difference Equations Bertram R (2001) Differential filtering of two presynaptic depression mechanisms. Neural Comput 13:69-85 [PubMed] Bertram R, Butte MJ, Kiemel T, Sherman A (1995) Topological and phenomenological classification of bursting oscillations. Bull Math Biol 57:413-39 [PubMed] Bevan MD, Magill PJ, Terman D, Bolam JP, Wilson CJ (2002) Move to the rhythm: oscillations in the subthalamic nucleus-external globus pallidus network. Trends Neurosci 25:525-31 [PubMed] Borg-Graham LJ (1991) Modeling the nonlinear conductances of excitable membranes Cellular and Molecular Neurobiology: A Practical Approach, Wheal H:Chad J, ed. pp.247 Borisyuk RM, Kirillov AB (1992) Bifurcation analysis of a neural network model. Biol Cybern 66:319-25 [PubMed] Bose A, Kopell N, Terman D (2000) Almost-synchronous solutions for mutually coupled excitatory neurons. Physica D 140:69-94 Bressloff PC (2000) Traveling waves and pulses in a one-dimensional network of excitable integrate-and-fire neurons. J Math Biol 40:169-98 [PubMed] Brette R, Gerstner W (2005) Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. J Neurophysiol 94:3637-42 [Journal] [PubMed]
Brodfuehrer PD, Debski EA, O'Gara BA, Friesen WO (1995) Neuronal control of leech swimming. J Neurobiol 27:403-18 [Journal] [PubMed] Brown E, Moehlis J, Holmes P (2004) On the phase reduction and response dynamics of neural oscillator populations. Neural Comput 16:673-715 [Journal] [PubMed] Brown TG (1911) The intrinsic factors in the act of progression in the mammal Proc R Soc London Ser B 84:308-319 Brunel N, Chance FS, Fourcaud N, Abbott LF (2001) Effects of synaptic noise and filtering on the frequency response of spiking neurons. Phys Rev Lett 86:2186-9 [Journal] [PubMed] Brunel N, Latham PE (2003) Firing rate of the noisy quadratic integrate-and-fire neuron. Neural Comput 15:2281-306 [Journal] [PubMed] Brunel N, Sergi S (1998) Firing frequency of leaky intergrate-and-fire neurons with synaptic current dynamics. J Theor Biol 195:87-95 [Journal] [PubMed] Buice MA, Cowan JD (2009) Statistical mechanics of the neocortex. Prog Biophys Mol Biol 99:53-86 [Journal] [PubMed] Butera RJ, Rinzel J, Smith JC (1999) Models of respiratory rhythm generation in the pre-Bötzinger complex. I. Bursting pacemaker neurons. J Neurophysiol 82:382-97 [Journal] [PubMed]
Butera RJ, Rinzel J, Smith JC (1999) Models of respiratory rhythm generation in the pre-Bötzinger complex. II. Populations Of coupled pacemaker neurons. J Neurophysiol 82:398-415 [Journal] [PubMed] Cai D, Tao L, Rangan AV, Mclaughlin DW (2006) Kinetic theory for neuronal network dynamics Commun Math Sci 4(1):97-127 Carnevale NT, Hines ML (2006) The NEURON Book [Journal] Carpenter GA (1977) A geometric approach to singular perturbation problems with applications to nerve impulse equations J Diff Eqn 23:335-367 Carpio A, Chapman SJ, Hastings S, Mcleod JB (2000) Wave solutions for a discrete reaction-diffusion equation Eur J Appl Math 11(4):399-412 Castro-Alamancos MA (2002) Properties of primary sensory (lemniscal) synapses in the ventrobasal thalamus and the relay of high-frequency sensory inputs. J Neurophysiol 87:946-53 [Journal] [PubMed] Chen X (1997) Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations Adv Differ Equat 2(1):125-160 Chen Z, Ermentrout B, Wang XJ (1998) Wave propagation mediated by GABAB synapse and rebound excitation in an inhibitory network: a reduced model approach. J Comput Neurosci 5:53-69 [PubMed] Chow CC, White JA (1996) Spontaneous action potentials due to channel fluctuations. Biophys J 71:3013-21 [Journal] [PubMed]
Cohen AH, Ermentrout GB, Kiemel T, Kopell N, Sigvardt KA, Williams TL (1992) Modelling of intersegmental coordination in the lamprey central pattern generator for locomotion. Trends Neurosci 15:434-8 [PubMed] Cohen M, Grossberg S (1983) Absolute stability of global pattern formation and parallel memory storage by competitive neural network. Ieee Trans Smc 13:815-826 Cole KS, Guttman R, Bezanilla F (1970) Nerve membrane excitation without threshold. Proc Natl Acad Sci U S A 65:884-91 [PubMed] Compte A, Brunel N, Goldman-Rakic PS, Wang XJ (2000) Synaptic mechanisms and network dynamics underlying spatial working memory in a cortical network model. Cereb Cortex 10:910-23 [PubMed] Conley C (1978) Isolated invariant Sets and the Morse Index CBMS Lecture Notes in Math Connor JA, Stevens CF (1971) Prediction of repetitive firing behaviour from voltage clamp data on an isolated neurone soma. J Physiol 213:31-53 [PubMed] Coombes S (2005) Waves, bumps, and patterns in neural field theories. Biol Cybern 93:91-108 [Journal] [PubMed] Coombes S, Bressloff PC (2005) Bursting: The genesis of rhythm in the nervous system, Coombes S:Bressloff PC, ed. Coombes S, Owen MR (2004) Evans functions for integral neural field equations with heaviside firing rate function Siam J Appl Dyn Syst 3:574-600 Cowan J, Sharp D (1988) Neural networks and artificial intelligence Daedalus 117:85-121 Cowan JD, Ermentrout GB (1978) Some aspects of the eigenbehavior of neural nets Studies in mathematical biology 1: Cellular behavior and the development of pattern, Levin SA, ed. pp.67 Crook SM, Ermentrout GB, Bower JM (1998) Dendritic and synaptic effects in systems of coupled cortical oscillators. J Comput Neurosci 5:315-29 [PubMed] Curtu R, Ermentrout B (2001) Oscillations in a refractory neural net. J Math Biol 43:81-100 [PubMed] Dayan P, Abbott LF (2001) Theoretical Neuroscience. Computational and Mathematical Modeling of Neural Systems Del Negro CA, Koshiya N, Butera RJ, Smith JC (2002) Persistent sodium current, membrane properties and bursting behavior of pre-bötzinger complex inspiratory neurons in vitro. J Neurophysiol 88:2242-50 [Journal] [PubMed] Destexhe A, Babloyantz A (1991) Pacemaker-induced coherence in cortical networks Neural Comput 3:145-154 Destexhe A, Babloyantz A, Sejnowski TJ (1993) Ionic mechanisms for intrinsic slow oscillations in thalamic relay neurons. Biophys J 65:1538-52 [Journal] [PubMed]
Destexhe A, Contreras D, Sejnowski TJ, Steriade M (1994) A model of spindle rhythmicity in the isolated thalamic reticular nucleus. J Neurophysiol 72:803-18 [Journal] [PubMed]
Destexhe A, Gaspard P (1993) Bursting oscillations from a homoclinic tangency in a time delay system Phys Lett A 173:386-391 Destexhe A, Mainen ZF, Sejnowski TJ (1994) Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formalism. J Comput Neurosci 1:195-230 [PubMed]
Destexhe A, McCormick DA, Sejnowski TJ (1993) A model for 8-10 Hz spindling in interconnected thalamic relay and reticularis neurons. Biophys J 65:2473-7 [Journal] [PubMed] Destexhe A, Paré D (1999) Impact of network activity on the integrative properties of neocortical pyramidal neurons in vivo. J Neurophysiol 81:1531-47 [Journal] [PubMed] Diekmann O (1977) On a nonlinear integral equation arising in mathematical epidemiology Differential Equations And Applications(Proceedings of Third Scheveningen Conference, Scheveningen, 1977, Eckhaus W:de_Jager EM, ed. pp.133 Egorov AV, Hamam BN, Fransén E, Hasselmo ME, Alonso AA (2002) Graded persistent activity in entorhinal cortex neurons. Nature 420:173-8 [Journal] [PubMed] Ermentrout B (1996) Type I membranes, phase resetting curves, and synchrony. Neural Comput 8:979-1001 [PubMed] Ermentrout B (1998) Linearization of F-I curves by adaptation. Neural Comput 10:1721-9 [PubMed] Ermentrout B, Flores J, Gelperin A (1998) Minimal model of oscillations and waves in the Limax olfactory lobe with tests of the model's predictive power. J Neurophysiol 79:2677-89 [Journal] [PubMed] Ermentrout B, Kopell N (1984) Frequency plateaus in a chain of weakly coupled oscillators. Siam J Math Anal 15:215-237 Ermentrout GB (1992) Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators Siam J Appl Math 52:1665-1687 Ermentrout GB (1995) A heuristic description of spiral wave instability in discrete media Phys D 82:54-164 Ermentrout GB, Cowan JD (1980) Large scale spatially organized activity in neural nets. Siam J Appl Math 38:1-21 Ermentrout GB, Dutta-moscato J, Pinto D (2005) Elliptic bursters, depolarization block, and waves Bursting, Bressloff P:Coombes S, ed. pp.385 Ermentrout GB, Galán RF, Urban NN (2008) Reliability, synchrony and noise. Trends Neurosci 31:428-34 [Journal] [PubMed] Ermentrout GB, Kleinfeld D (2001) Traveling electrical waves in cortex: insights from phase dynamics and speculation on a computational role. Neuron 29:33-44 [PubMed] Ermentrout GB, Kopell N (1986) Parabolic bursting in an excitable system coupled with a slow oscillation. Siam J Appl Math 46:233-253 Ermentrout GB, Kopell N (1998) Fine structure of neural spiking and synchronization in the presence of conduction delays. Proc Natl Acad Sci U S A 95:1259-64 [PubMed] Ermentrout GB, Mcleod JB (1993) Existence and uniqueness of travelling waves for a neural network Proc Roy Soc Edinburgh A 123:461-478 Ermentrout GB, Rinzel J (1981) Waves in a simple, excitable or oscillatory, reaction-diffusion model Reaction-diffusion Model J Math Biol 11:269-294 Ermentrout GB, Rinzel J (1984) Beyond a pacemaker's entrainment limit: phase walk-through. Am J Physiol 246:R102-6 [Journal] [PubMed] Evans JW (1975) Nerve axon equations iv Indiana Univ Math J 24:1169-1190 Evans JW, Fenichel N, Feroe JA (1982) Double impulse solutions in nerve axon equations Siam J Appl Math 42:219-234 Faisal AA, Selen LP, Wolpert DM (2008) Noise in the nervous system. Nat Rev Neurosci 9:292-303 [Journal] [PubMed] Fall CP, Marland ES, Wagner JM, Tyson JJ (2002) Computational Cell Biology Feinerman O, Segal M, Moses E (2005) Signal propagation along unidimensional neuronal networks. J Neurophysiol 94:3406-16 [Journal] [PubMed] Fenichel N (1979) Geometric singular perturbation theory for ordinary differential equations J Diff Eqn 31:53-98 Feroe JA (1982) Existence and stability of multiple impulse solutions of a nerve axon equation Siam J Appl Math 42:235-246 Fife PC, Mcleod JB (1977) The approach of solutions of nonlinear diffusion equations to traveling wave solutions Arch Ration Mech Anal 65:355-361 Fitzhugh R (1961) Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophys J 1:445-66 [PubMed] Fourcaud N, Brunel N (2002) Dynamics of the firing probability of noisy integrate-and-fire neurons. Neural Comput 14:2057-110 [Journal] [PubMed] Fourcaud-Trocmé N, Hansel D, van Vreeswijk C, Brunel N (2003) How spike generation mechanisms determine the neuronal response to fluctuating inputs. J Neurosci 23:11628-40 [PubMed] Fox RF, Lu Yn (1994) Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 49:3421-3431 [PubMed] Fransén E, Tahvildari B, Egorov AV, Hasselmo ME, Alonso AA (2006) Mechanism of graded persistent cellular activity of entorhinal cortex layer v neurons. Neuron 49:735-46 [Journal] [PubMed] Fuster JM (1973) Unit activity in prefrontal cortex during delayed-response performance: neuronal correlates of transient memory. J Neurophysiol 36:61-78 [Journal] [PubMed] Fuster JM, Alexander GE (1971) Neuron activity related to short-term memory. Science 173:652-4 [PubMed] Gardiner CW (2004) Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences Springer Series in Synergetics, 3rd ed. Gavrikov KE, Nilson JE, Dmitriev AV, Zucker CL, Mangel SC (2006) Dendritic compartmentalization of chloride cotransporters underlies directional responses of starburst amacrine cells in retina. Proc Natl Acad Sci U S A 103:18793-8 [Journal] [PubMed] Gerstner W, Kistler WM (1992) Spiking NeuronModels: Single Neurons, Populations, Plasticity Gerstner W, van Hemmen JL, Cowan JD (1996) What matters in neuronal locking? Neural Comput 8:1653-76 [PubMed] Gibson JR, Beierlein M, Connors BW (1999) Two networks of electrically coupled inhibitory neurons in neocortex. Nature 402:75-9 [Journal] [PubMed] Gibson JR, Beierlein M, Connors BW (2005) Functional properties of electrical synapses between inhibitory interneurons of neocortical layer 4. J Neurophysiol 93:467-80 [Journal] [PubMed] Glass L, Mackey MC (1988) From Clocks To Chaos: The Rhythms Of Life Golomb D, Amitai Y (1997) Propagating neuronal discharges in neocortical slices: computational and experimental study. J Neurophysiol 78:1199-211 [Journal] [PubMed] Golomb D, Wang XJ, Rinzel J (1994) Synchronization properties of spindle oscillations in a thalamic reticular nucleus model. J Neurophysiol 72:1109-26 [Journal] [PubMed] Golubitsky M, Stewart I, Buono PL, Collins JJ (1999) Symmetry in locomotor central pattern generators and animal gaits. Nature 401:693-5 [Journal] [PubMed] Gray CM, Engel AK, König P, Singer W (1992) Synchronization of oscillatory neuronal responses in cat striate cortex: temporal properties. Vis Neurosci 8:337-47 [PubMed] Gray CM, König P, Engel AK, Singer W (1989) Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature 338:334-7 [Journal] [PubMed] Grillner S, Wallén P (2002) Cellular bases of a vertebrate locomotor system-steering, intersegmental and segmental co-ordination and sensory control. Brain Res Brain Res Rev 40:92-106 [PubMed] Grossberg S, Carpenter GA (1987) A massively parallel architecture for a self-organizing neural pattern recognition machine Computer Vision, Graphics, and Image Processing :54-115 Guckenheimer J, Holmes P (1990) Nonlinear oscillations, dynamical systems and bifurcations of vector fields Appl Math Sci Guckenheimer J, Oliva RA (1992) Chaos in the hodgkin-huxley model Siam J Apl Dyn Syst 1:105-114 Gulledge AT, Stuart GJ (2003) Excitatory actions of GABA in the cortex. Neuron 37:299-309 [PubMed] Gutkin BS, Ermentrout GB, Reyes AD (2005) Phase-response curves give the responses of neurons to transient inputs. J Neurophysiol 94:1623-35 [Journal] [PubMed] Hale J, Kocak H (1991) Dynamics and Bifurcations Hall BJ, Delaney KR (2002) Contribution of a calcium-activated non-specific conductance to NMDA receptor-mediated synaptic potentials in granule cells of the frog olfactory bulb. J Physiol 543:819-34 [PubMed] Han F, Caporale N, Dan Y (2008) Reverberation of recent visual experience in spontaneous cortical waves. Neuron 60:321-7 [Journal] [PubMed] Hastings SP (1976) On the existence of homoclinic and paeriodic orbits for the fitzhugh-naguo equations Quart J Math Oxford 27:123-124 Hastings SP (1982) Single and multiple pulse waves for the fitzhugh-nagumo equations Siam J Appl Math 42:247-260 Helmchen F, Imoto K, Sakmann B (1996) Ca2+ buffering and action potential-evoked Ca2+ signaling in dendrites of pyramidal neurons. Biophys J 70:1069-81 [Journal] [PubMed] Hertz J, Krogh A, Palmer RG (2001) Introduction to the theory of neural computation Hille B (2001) Ionic Channels of Excitable Membranes Hindmarsh JL, Rose RM (1984) A model of neuronal bursting using three coupled first order differential equations. Proc R Soc Lond B Biol Sci 221:87-102 [Journal] [PubMed]
HODGKIN AL, HUXLEY AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500-44 [Journal] [PubMed] Holcman D, Tsodyks M (2006) The emergence of Up and Down states in cortical networks. PLoS Comput Biol 2:e23 [Journal] [PubMed] Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci U S A 79:2554-8 [PubMed] Hopfield JJ (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci U S A 81:3088-92 [PubMed] Hoppensteadt FC, Izhikevich EM (1997) Weakly Connected Neural Networks :90 Hoyle RB (2006) Pattern Formation: An Introduction to Methods Huang X, Troy WC, Yang Q, Ma H, Laing CR, Schiff SJ, Wu JY (2004) Spiral waves in disinhibited mammalian neocortex. J Neurosci 24:9897-902 [Journal] [PubMed] Huguenard JP, Mccormick DA (1994) Electrophysiology of the Neuron: An Interactive Tutorial
Izhikevich EM (2000) Neural excitability, spiking and bursting Int J Bifurcat Chaos Appl Sci Eng 10:1171-1266 Izhikevich EM (2000) Phase equations for relaxation oscillations Siam J Appl Mat 60:1789-1804 Izhikevich EM (2003) Simple model of spiking neurons. IEEE Trans Neural Netw 14:1569-72 [Journal] [PubMed]
Izhikevich EM (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting [Journal]
Jack JJB, Noble D, Tsien RW (1975) Electric Current Flow in Excitable Cells Jahr CE, Stevens CF (1990) A quantitative description of NMDA receptor-channel kinetic behavior. J Neurosci 10:1830-7 [PubMed] Johnston D, Hoffman DA, Colbert CM, Magee JC (1999) Regulation of back-propagating action potentials in hippocampal neurons. Curr Opin Neurobiol 9:288-92 [PubMed] Johnston D, Magee JC, Colbert CM, Cristie BR (1996) Active properties of neuronal dendrites. Annu Rev Neurosci 19:165-86 [Journal] [PubMed] Johnston D, Sms WU (1994) Foundations of cellular neurophysiology Johnston D, Wu S (1997) Cellular Neurophysiology Jones CK (1984) Stability of the traveling wave solution of the fitzhugh-nagumo equations Trans A M S 286:431-469 Kandel E, Schwartz J, Jessell T (1991) The Principles of Neural Science Karbowski J, Kopell N (2000) Multispikes and synchronization in a large neural network with temporal delays. Neural Comput 12:1573-606 [PubMed] Keener J, Sneyd J (1998) Mathematical Physiology Interdisciplinary Applied Mathematics Keener JP (1987) Proagation and its failure in coupled systems of discrete excitable cells Siam J Appl Math 47:556-572 Keener JP (2000) Principles of Applied Mathematics: Transformation and Approximation Advanced Book Program, revised edition Kelso J (1995) Dynamic patterns: The self-organization of brain and behavior Kepler TB, Abbott LF, Marder E (1992) Reduction of conductance-based neuron models. Biol Cybern 66:381-7 [PubMed] Kilpatrick ZP, Bressloff PC (2010) Spatially structured oscillations in a two-dimensional excitatory neuronal network with synaptic depression. J Comput Neurosci 28:193-209 [Journal] [PubMed] Kim U, Bal T, McCormick DA (1995) Spindle waves are propagating synchronized oscillations in the ferret LGNd in vitro. J Neurophysiol 74:1301-23 [Journal] [PubMed] Kishimoto K, Amari S (1979) Existence and stability of local excitations in homogeneous neural fields. J Math Biol 7:303-18 [PubMed] Kloeden PE, Platen E (1992) Numerical Solution Of Stochastic Differential Equations Kluver H (1969) Mescal and the Mechanisms of Hallucination Koch C (1999) Biophysics Of Computation: Information Processing in Single Neurons Koch C, Segev I (1998) Methods In Neuronal Modeling Kocsis B, Li S (2004) In vivo contribution of h-channels in the septal pacemaker to theta rhythm generation. Eur J Neurosci 20:2149-58 [Journal] [PubMed] Kopell N (1988) Toward a theory of modelling central pattern generators Neural Control Of Rhythmic Movements In Vertebrates, Cohen AH:Rossignol S:Grillner S, ed. pp.369 Kopell N, Ermentrout GB (1986) Symmetry and phaselocking in chains of weakly coupled oscillators. Comm Pure Appl Mat 39:623-660 Kopell N, Ermentrout GB (2002) Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators Handbook of dynamical systems, Fiedler B, ed, ed. pp.5 Kopell N, Howard LN (1973) Plane wave solutions to reaction-diffusion equations Studies Appl Mat 52:291-328 Kopell N, Somers D (1995) Anti-phase solutions in relaxation oscillators coupled through excitatory interactions. J Math Biol 33:261-80 [PubMed] Krinskii VI, Kokoz IuM (1973) [Analysis of the equations of excitable membranes. I. Reduction of the Hodgkins-Huxley equations to a 2d order system]. Biofizika 18:506-11 [PubMed] Krisner EP (2008) Homoclinic orbit solutions of a one dimensional Wilson-Cowan type model Electron J Differ Equat 107:30 Kuramoto Y (1984) Chemical oscillations, waves, and turbulence Kuznetsov YA (2004) Elements of applied bifurcation theory, 3rd Edition Kyriazi HT, Simons DJ (1993) Thalamocortical response transformations in simulated whisker barrels. J Neurosci 13:1601-15 [PubMed] Laing C, Lord G (2009) Stochastic Methods in Neuroscience Laing CR, Troy WC (2003) PDE methods for nonlocal models Siam J Appl Dyn Syst 2:487-516 Laing CR, Troy WC (2003) Two-bump solutions of Amari-type models of neuronal pattern formation Physica D 187:190-218 Laing CR, Troy WC, Gutkin B, Ermentrout GB (2002) Multiple bumps in a neuronal model of working memory SIAM J Appl Math 63:62-97 Lambe EK, Aghajanian GK (2006) Hallucinogen-induced UP states in the brain slice of rat prefrontal cortex: role of glutamate spillover and NR2B-NMDA receptors. Neuropsychopharmacology 31:1682-9 [Journal] [PubMed] Lapicque L (1907) Recherches quantitatives sur lexcitation electrique des nerfs traitee comme une polarisation J Physiol Pathol Gen 9:620-635 [Journal] Latham PE, Richmond BJ, Nelson PG, Nirenberg S (2000) Intrinsic dynamics in neuronal networks. I. Theory. J Neurophysiol 83:808-27 [Journal] [PubMed] Laurent G (2002) Olfactory network dynamics and the coding of multidimensional signals. Nat Rev Neurosci 3:884-95 [Journal] [PubMed] Lee E, Terman D (1999) Uniqueness and stability of periodic bursting solutions Journal Of Difference Equations 158:48-78 Lehky SR (1988) An astable multivibrator model of binocular rivalry. Perception 17:215-28 [Journal] [PubMed] Lindner B (2004) Interspike interval statistics of neurons driven by colored noise. Phys Rev E Stat Nonlin Soft Matter Phys 69:022901 [Journal] [PubMed] Lindner B, Longtin A, Bulsara A (2003) Analytic expressions for rate and CV of a type I neuron driven by white gaussian noise. Neural Comput 15:1760-87 [Journal] [PubMed] Lindner B, Schimansky-Geier L (2001) Transmission of noise coded versus additive signals through a neuronal ensemble. Phys Rev Lett 86:2934-7 [Journal] [PubMed] Lisman JE, Fellous JM, Wang XJ (1998) A role for NMDA-receptor channels in working memory. Nat Neurosci 1:273-5 [Journal] [PubMed] Loebel A, Tsodyks M (2002) Computation by ensemble synchronization in recurrent networks with synaptic depression. J Comput Neurosci 13:111-24 [PubMed] Loewenstein Y, Sompolinsky H (2003) Temporal integration by calcium dynamics in a model neuron. Nat Neurosci 6:961-7 [Journal] [PubMed] Maccaferri G, McBain CJ (1996) The hyperpolarization-activated current (Ih) and its contribution to pacemaker activity in rat CA1 hippocampal stratum oriens-alveus interneurones. J Physiol 497 ( Pt 1):119-30 [PubMed] Mackie GO, Meech RW (1985) Separate sodium and calcium spikes in the same axon. Nature 313:791-3 [PubMed] Magee J, Hoffman D, Colbert C, Johnston D (1998) Electrical and calcium signaling in dendrites of hippocampal pyramidal neurons. Annu Rev Physiol 60:327-46 [Journal] [PubMed] Maginu K (1985) Geometrical characteristics associated with stability and bifurcations of periodic traveling waves in reaction-diffusion systems Siam J Appl Math 45:750-774 Manor Y, Bose A, Booth V, Nadim F (2003) Contribution of synaptic depression to phase maintenance in a model rhythmic network. J Neurophysiol 90:3513-28 [Journal] [PubMed] Markram H, Wang Y, Tsodyks M (1998) Differential signaling via the same axon of neocortical pyramidal neurons. Proc Natl Acad Sci U S A 95:5323-8 [PubMed] Martin AR, Wallace BG, Fuchs PA, Nicholls JG (2001) From Neuron to Brain: A Cellular and Molecular Approach to the Function of the Nervous System Masland RH (2004) Directional selectivity in retinal ganglion cells The Visual Neurosciences, Chalupa LM:Werner JS, ed. pp.451 Matthews PC, Mirollo RE, Strogatz SH (1991) Physica D 52:293-331 Mayville JM, Bressler SL, Fuchs A, Kelso JA (1999) Spatiotemporal reorganization of electrical activity in the human brain associated with a timing transition in rhythmic auditory-motor coordination. Exp Brain Res 127:371-81 [PubMed] Mcclelland J, Rumelhart D (1987) Parallel Distributed Processes Mcclelland J, Rumelhart D (1987) Parallel Distributed Processes Mcclelland J, Rumelhart D (1987) Parallel Distributed Processes Mcclelland J, Rumelhart D (1987) Parallel Distributed Processes McCULLOCH WS, PITTS W (1948) The statistical organization of nervous activity. Biometrics 4:91-9 [PubMed] Medvedev G (2005) Reduction of a model of an excitable cell to a one-dimensional map Physica D 202:37-59 Meech RW, Mackie GO (1993) Ionic currents in giant motor axons of the jellyfish, Aglantha digitale. J Neurophysiol 69:884-93 [Journal] [PubMed] Mensh BD, Aksay E, Lee DD, Seung HS, Tank DW (2004) Spontaneous eye movements in goldfish: oculomotor integrator performance, plasticity, and dependence on visual feedback. Vision Res 44:711-26 [PubMed] Migliore M, Messineo L, Ferrante M (2004) Dendritic Ih selectively blocks temporal summation of unsynchronized distal inputs in CA1 pyramidal neurons. J Comput Neurosci 16:5-13 [PubMed]
Migliore M, Shepherd GM (2002) Emerging rules for the distributions of active dendritic conductances. Nat Rev Neurosci 3:362-70 [Journal] [PubMed]
Migliore M, Shepherd GM (2008) Dendritic action potentials connect distributed dendrodendritic microcircuits. J Comput Neurosci 24:207-21 [Journal] [PubMed]
Mirollo RE, Strogatz SH (1990) Synchronization of pulse-coupled biological oscillators. Siam J Appl Math 6:1645-1657 Müller M, Wehner R (1988) Path integration in desert ants, Cataglyphis fortis. Proc Natl Acad Sci U S A 85:5287-90 [PubMed] Murray JD (2003) Spatial models and biomedical applications Mathematical Biology. II, volume 18 of Interdisciplinary Applied Mathematics, third edition Netoff TI, Banks MI, Dorval AD, Acker CD, Haas JS, Kopell N, White JA (2005) Synchronization in hybrid neuronal networks of the hippocampal formation. J Neurophysiol 93:1197-208 [Journal] [PubMed] Neu JC (1980) Large populations of coupled chemical oscillators Siam J Appl Math 38:305-316 Osan R, Curtu R, Rubin J, Ermentrout B (2004) Multiple-spike waves in a one-dimensional integrate-and-fire neural network. J Math Biol 48:243-74 [Journal] [PubMed] Paullet JE, Ermentrout GB (1994) Stable rotating waves in two-dimensional discrete active media Siam J Appl Math 54:1720-1744 Perkel DH, Mulloney B (1974) Motor pattern production in reciprocally inhibitory neurons exhibiting postinhibitory rebound. Science 185:181-3 [PubMed] Perko L (1996) Differential Equations and Dynamical Systems Petsche H, Prohaska O, Rappelsberger P, Vollmer R, Kaiser A (1974) Cortical seizure patterns in multidimensional view: the information content of equipotential maps. Epilepsia 15:439-63 [PubMed] Pfeuty B, Mato G, Golomb D, Hansel D (2003) Electrical synapses and synchrony: the role of intrinsic currents. J Neurosci 23:6280-94 [PubMed] Pinsky PF, Rinzel J (1994) Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons. J Comput Neurosci 1:39-60 [PubMed]
Pinto DJ, Brumberg JC, Simons DJ, Ermentrout GB (1996) A quantitative population model of whisker barrels: re-examining the Wilson-Cowan equations. J Comput Neurosci 3:247-64 [PubMed] Pinto DJ, Ermentrout GB (2001) Spatially structured activity in synaptically coupled neuronal networks. I. Traveling fronts and pulses Siam J Appl Math 62:206-225 Pinto DJ, Hartings JA, Brumberg JC, Simons DJ (2003) Cortical damping: analysis of thalamocortical response transformations in rodent barrel cortex. Cereb Cortex 13:33-44 [PubMed] Pinto DJ, Patrick SL, Huang WC, Connors BW (2005) Initiation, propagation, and termination of epileptiform activity in rodent neocortex in vitro involve distinct mechanisms. J Neurosci 25:8131-40 [Journal] [PubMed] Prechtl JC, Cohen LB, Pesaran B, Mitra PP, Kleinfeld D (1997) Visual stimuli induce waves of electrical activity in turtle cortex. Proc Natl Acad Sci U S A 94:7621-6 [PubMed] Rall W (1977) Core conductor theory and cable properties of neurons Handbook of Physiology. The Nervous System. Cellular Biology of Neurons, Kandel ER, ed. pp.39 Rall W, Rinzel J (1973) Branch input resistance and steady attenuation for input to one branch of a dendritic neuron model. Biophys J 13:648-87 [Journal] [PubMed] Ren L, Ermentrout GB (1998) Monotonicity of phaselocked solutions in chains and arrays of nearest-neighbor coupled oscillators Siam J Math Anal, electronic 29:208-234 Reyes AD (2003) Synchrony-dependent propagation of firing rate in iteratively constructed networks in vitro. Nat Neurosci 6:593-9 [Journal] [PubMed] Reyes AD, Fetz EE (1993) Effects of transient depolarizing potentials on the firing rate of cat neocortical neurons. J Neurophysiol 69:1673-83 [Journal] [PubMed] Rinzel J (1978) On repetitive activity in nerve. Fed Proc 37:2793-802 [PubMed] Rinzel J (1985) Bursting oscillations in an excitable membrane model, in ordinary and partial differential equations New Lecture Notes in Mathematics 1151:304-316 Rinzel J (1987) A formal classification of bursting mechanisms in excitable systems. Mathematical Topics in Population Biology, Morphogenesis, and Neurosciences., Teramoto E:Yamaguti M, ed. pp.267 Rinzel J (1990) Discussion: Electrical excitability of cells, theory and experiment: Review of the Hodgkin-Huxley foundation and update Bull Math Biol 52:5-23 Rinzel J, Ermentrout GB (1989) Analysis of neuronal excitability and oscillations Methods In Neuronal Modeling: From Synapses To Networks, Koch C:Segev I, ed. pp.135 Rinzel J, Maginu K (1984) Kinematic analysis of wave pattern formation in excitable media Nonequilibrium Dynamics In Chemical Systems, Springer Series Synergetics, Vidal C:Pacault A, ed. pp.107 Rinzel J, Terman D, Wang X, Ermentrout B (1998) Propagating activity patterns in large-scale inhibitory neuronal networks. Science 279:1351-5 [PubMed] Ritt J (2003) Evaluation of entrainment of a nonlinear neural oscillator to white noise. Phys Rev E Stat Nonlin Soft Matter Phys 68:041915 [Journal] [PubMed] Rodriguez R, Tuckwell HC (1996) Statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 54:5585-5590 [PubMed] Romo R, Brody CD, Hernández A, Lemus L (1999) Neuronal correlates of parametric working memory in the prefrontal cortex. Nature 399:470-3 [Journal] [PubMed] Rubin J, Terman D (2000) Analysis of clustered firing patterns in synaptically coupled networks of oscillators. J Math Biol 41:513-45 [PubMed] Rudolph M, Destexhe A (2003) Characterization of subthreshold voltage fluctuations in neuronal membranes. Neural Comput 15:2577-618 [Journal] [PubMed] Rudolph M, Destexhe A (2006) On the use of analytical expressions for the voltage distribution to analyze intracellular recordings. Neural Comput 18:2917-22 [Journal] [PubMed] Rush ME, Rinzel J (1995) The potassium A-current, low firing rates and rebound excitation in Hodgkin-Huxley models. Bull Math Biol 57:899-929 [PubMed] Sakaguchi H, Tobiishi S (2005) Spindle oscillation in noisy integrate-and-fire-or-burst neurons with inhibitory coupling Prog Theor Phys 79:1069-1079 Sakaguchi S, Shinimoto H, Kuramoto Y (1988) Mutual entrainment oscillator lattices with nonvariational coupling Prog. Theor. Phys. 79:1069-1079 Saper CB, Chou TC, Scammell TE (2001) The sleep switch: hypothalamic control of sleep and wakefulness. Trends Neurosci 24:726-31 [PubMed] Segev I, Rinzel J, Shepherd G (1994) The Theoretical Foundation of Dendritic Function: Selected Paper of Wilfrid Rall with Commentaries Seung HS, Lee DD, Reis BY, Tank DW (2000) Stability of the memory of eye position in a recurrent network of conductance-based model neurons. Neuron 26:259-71 [PubMed] Shilnikov A, Cymbalyuk G (2005) Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe. Phys Rev Lett 94:048101 [Journal] [PubMed] Shpiro A, Curtu R, Rinzel J, Rubin N (2007) Dynamical characteristics common to neuronal competition models. J Neurophysiol 97:462-73 [Journal] [PubMed] Shu Y, Hasenstaub A, McCormick DA (2003) Turning on and off recurrent balanced cortical activity. Nature 423:288-93 [Journal] [PubMed] Sigeti D, Horsthemke W (1989) Pseudo-regular oscillations induced by external noise J Stat Physics 54:1217-1222 Skinner FK, Kopell N, Marder E (1994) Mechanisms for oscillation and frequency control in reciprocally inhibitory model neural networks. J Comput Neurosci 1:69-87 [PubMed] Softky WR, Koch C (1993) The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. J Neurosci 13:334-50 [PubMed] Steriad M (2003) Neuronal Substrates of Sleep and Epilepsy Stoop R, Bunimovich LA, Steeb WH (2000) Generic origins of irregular spiking in neocortical networks. Biol Cybern 83:481-9 [Journal] [PubMed] Stoop R, Schindler K, Bunimovich LA (2000) When pyramidal neurons lock, when they respond chaotically, and when they like to synchronize. Neurosci Res 36:81-91 [PubMed] Strogatz S (2000) From Kuramoto to Crawford: Explaining the onset of synchronization in populations of globally coupled oscillators Physica D 143:1-20 Strogatz S (2003) How order emerges from chaos in the universe, nature, and daily life Sync. Hyperion Books, New York Strogatz SH (1984) Nonlinear Dynamics and Chaos Tabak J, Senn W, O'Donovan MJ, Rinzel J (2000) Modeling of spontaneous activity in developing spinal cord using activity-dependent depression in an excitatory network. J Neurosci 20:3041-56 [PubMed] Tanabe S, Pakdaman K (2001) Dynamics of moments of FitzHugh-Nagumo neuronal models and stochastic bifurcations. Phys Rev E Stat Nonlin Soft Matter Phys 63:031911 [Journal] [PubMed] Tateno T, Harsch A, Robinson HP (2004) Threshold firing frequency-current relationships of neurons in rat somatosensory cortex: type 1 and type 2 dynamics. J Neurophysiol 92:2283-94 [Journal] [PubMed] Teramae JN, Fukai T (2005) A Cellular Mechanism for Graded Persistent Activity in a Model Neuron and Its Implications in Working Memory J Comput Neurosci 18:105-121 [Journal] [PubMed] Teramae JN, Tanaka D (2004) Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators. Phys Rev Lett 93:204103 [Journal] [PubMed] Terman D (1991) Chaotic spikes arising from a model of bursting in excitable membranes. Siam J Appl Math 51:1418-1450 Terman D (1992) The transition from bursting to continuous spiking in excitable membrane models J Nonlinear Sci 2:135-182 Terman D, Ahn S, Wang X, Just W (2008) Reducing Neuronal Networks to Discrete Dynamics. Physica D 237:324-338 [Journal] [PubMed] Terman D, Kopell N, Bose A (1998) Dynamics of two mutually coupled slow inhibitory neurons. Physica D 117:241-275 Terman D, Wang DL (1995) Global competition and local cooperation in a network of neural oscillators Physica D 81:148-176 Terman DH, Ermentrout GB, Yew AC (2001) Propagating activity patterns in thalamic neuronal networks SIAM J Appl Math 61:1578-1604 Touboul J (2008) Bifurcation analysis of a general class of nonlinear integrate-and-fire neurons. Siam Appl Math 68:1045-1079 Traub RD, Miles R (1991) Neuronal Networks Of The Hippocampus Traub RD, Wong RK, Miles R, Michelson H (1991) A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances. J Neurophysiol 66:635-50 [Journal] [PubMed] Traynelis SF, Jaramillo F (1998) Getting the most out of noise in the central nervous system. Trends Neurosci 21:137-45 [PubMed] Troy WC (1978) The bifurcation of periodic solutions in the Hodgkin-Huxley equations. Quart Appl Math 36:73-83 Tsodyks M, Uziel A, Markram H (2000) Synchrony generation in recurrent networks with frequency-dependent synapses. J Neurosci 20:RC50 [PubMed] Tuckwell HC (1988) Introduction to Theoretical Neurobiology. Volume 2: Nonlinear and Stochastic Theories Tuckwell HC, Rodriguez R, Wan FY (2003) Determination of firing times for the stochastic Fitzhugh-Nagumo neuronal model. Neural Comput 15:143-59 [Journal] [PubMed] Turing AM (1990) The chemical basis of morphogenesis. 1953. Bull Math Biol 52:153-97; discussion 119-52 [PubMed] Ulrich D (2002) Dendritic resonance in rat neocortical pyramidal cells. J Neurophysiol 87:2753-9 [Journal] [PubMed] Van Vreeswijk C, Abbott LF, Ermentrout GB (1994) When inhibition not excitation synchronizes neural firing. J Comput Neurosci 1:313-21 [PubMed] Varela JA, Sen K, Gibson J, Fost J, Abbott LF, Nelson SB (1997) A quantitative description of short-term plasticity at excitatory synapses in layer 2/3 of rat primary visual cortex. J Neurosci 17:7926-40 [Journal] [PubMed]
Volgushev M, Chauvette S, Mukovski M, Timofeev I (2006) Precise long-range synchronization of activity and silence in neocortical neurons during slow-wave oscillations [corrected]. J Neurosci 26:5665-72 [Journal] [PubMed] von der Malsburg C, Schneider W (1986) A neural cocktail-party processor. Biol Cybern 54:29-40 [PubMed] Wang XJ (1998) Calcium coding and adaptive temporal computation in cortical pyramidal neurons. J Neurophysiol 79:1549-66 [Journal] [PubMed] Wang XJ (1999) Synaptic basis of cortical persistent activity: the importance of NMDA receptors to working memory. J Neurosci 19:9587-603 [PubMed] Wang XJ, Rinzel J (1992) Alternating and synchronous rhythms in reciprocally inhibitory model neurons Neural Comput 4:84-97 Wang XJ, Rinzel J (1995) Oscillatory and bursting properties of neurons The Handbook Of Brain Theory And Neural Networks, Arbib MA, ed. pp.686 Whittington MA, Traub RD, Jefferys JG (1995) Synchronized oscillations in interneuron networks driven by metabotropic glutamate receptor activation. Nature 373:612-5 [Journal] [PubMed] Wilson HR, Blake R, Lee SH (2001) Dynamics of travelling waves in visual perception. Nature 412:907-10 [Journal] [PubMed] Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:1-24 [Journal] [PubMed]
Wilson HR, Cowan JD (1973) A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13:55-80 [PubMed] | Ahn S, Rubchinsky LL (2017) Potential Mechanisms and Functions of Intermittent Neural Synchronization. Front Comput Neurosci 11:44 [Journal] [PubMed] Cattani A, Solinas S, Canuto C (2016) A Hybrid Model for the Computationally-Efficient Simulation of the Cerebellar Granular Layer. Front Comput Neurosci 10:30 [Journal] [PubMed] Conte C, Lee R, Sarkar M, Terman D (2018) A mathematical model of recurrent spreading depolarizations. J Comput Neurosci 44:203-217 [Journal] [PubMed]
DeWoskin D, Myung J, Belle MD, Piggins HD, Takumi T, Forger DB (2015) Distinct roles for GABA across multiple timescales in mammalian circadian timekeeping. Proc Natl Acad Sci U S A 112:E3911-9 [Journal] [PubMed]
Ferguson KA, Huh CY, Amilhon B, Manseau F, Williams S, Skinner FK (2015) Network models provide insights into how oriens-lacunosum-moleculare and bistratified cell interactions influence the power of local hippocampal CA1 theta oscillations. Front Syst Neurosci 9:110 [Journal] [PubMed]
Ferguson KA, Huh CY, Amilhon B, Williams S, Skinner FK (2013) Experimentally constrained CA1 fast-firing parvalbumin-positive interneuron network models exhibit sharp transitions into coherent high frequency rhythms. Front Comput Neurosci 7:144 [Journal] [PubMed]
Hübel N, Dahlem MA (2014) Dynamics from seconds to hours in Hodgkin-Huxley model with time-dependent ion concentrations and buffer reservoirs. PLoS Comput Biol 10:e1003941 [Journal] [PubMed]
Kömek K, Bard Ermentrout G, Walker CP, Cho RY (2012) Dopamine and gamma band synchrony in schizophrenia--insights from computational and empirical studies. Eur J Neurosci 36:2146-55 [Journal] [PubMed]
Krishnan GP, Filatov G, Shilnikov A, Bazhenov M (2015) Electrogenic properties of the Na?/K? ATPase control transitions between normal and pathological brain states. J Neurophysiol 113:3356-74 [Journal] [PubMed]
Maia PD, Kutz JN (2014) Compromised axonal functionality after neurodegeneration, concussion and/or traumatic brain injury. J Comput Neurosci 37:317-32 [Journal] [PubMed] Naze S, Bernard C, Jirsa V (2015) Computational modeling of seizure dynamics using coupled neuronal networks: factors shaping epileptiform activity. PLoS Comput Biol 11:e1004209 [Journal] [PubMed]
Nicola W, Campbell SA (2013) Bifurcations of large networks of two-dimensional integrate and fire neurons. J Comput Neurosci 35:87-108 [Journal] [PubMed]
Øyehaug L, Østby I, Lloyd CM, Omholt SW, Einevoll GT (2012) Dependence of spontaneous neuronal firing and depolarisation block on astroglial membrane transport mechanisms. J Comput Neurosci 32:147-65 [Journal] [PubMed]
Rakowski F, Karbowski J (2017) Optimal synaptic signaling connectome for locomotory behavior in Caenorhabditis elegans: Design minimizing energy cost. PLoS Comput Biol 13:e1005834 [Journal] [PubMed]
Singh C, Levy WB (2017) A consensus layer V pyramidal neuron can sustain interpulse-interval coding. PLoS One 12:e0180839 [Journal] [PubMed]
Soudry D, Meir R (2014) The neuronal response at extended timescales: a linearized spiking input-output relation. Front Comput Neurosci 8:29 [Journal] [PubMed]
Stiefel KM, Englitz B, Sejnowski TJ (2013) Origin of intrinsic irregular firing in cortical interneurons. Proc Natl Acad Sci U S A 110:7886-91 [Journal] [PubMed] Vo T, Tabak J, Bertram R, Wechselberger M (2014) A geometric understanding of how fast activating potassium channels promote bursting in pituitary cells. J Comput Neurosci 36:259-78 [Journal] [PubMed]
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