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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Peter Thomas at 2013-10-08 00:09:12 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{WainribThieullenPakdaman2010BiolCyb,
Abstract = {The assessment of the variability of neuronal spike timing is fundamental to gain understanding of latency coding. Based on recent mathematical results, we investigate theoretically the impact of channel noise on latency variability. For large numbers of ion channels, we derive the asymptotic distribution of latency, together with an explicit expression for its variance. Consequences in terms of information processing are studied with Fisher information in the Morris-Lecar model. A competition between sensitivity to input and precision is responsible for favoring two distinct regimes of latencies.},
Author = {Wainrib, Gilles and Thieullen, Mich\`{e}le and Pakdaman, Khashayar},
Date-Added = {2013-10-07 22:08:03 +0000},
Date-Modified = {2013-10-07 22:09:10 +0000},
Doi = {10.1007/s00422-010-0384-8},
Journal = {Biol Cybern},
Journal-Full = {Biological cybernetics},
Mesh = {Action Potentials; Animals; Cell Membrane; Central Nervous System; Humans; Ion Channel Gating; Neurons; Reaction Time; Synaptic Transmission},
Month = {Jul},
Number = {1},
Pages = {43-56},
Pmid = {20372920},
Pst = {ppublish},
Title = {Intrinsic variability of latency to first-spike},
Volume = {103},
Year = {2010},
Bdsk-Url-1 = {http://dx.doi.org/10.1007/s00422-010-0384-8}}
@article{PakdamanThieullenWainrib2010AdvAppProb,
Author = {Pakdaman, K and Thieullen, M and Wainrib, G},
Journal = {Advances in Applied Probability},
Number = {3},
Pages = {761--794},
Publisher = {Applied Probability Trust},
Title = {Fluid limit theorems for stochastic hybrid systems with application to neuron models},
Volume = {42},
Year = {2010}}
@article{WainribThieullenPakdaman2012JCNS,
Abstract = {We introduce a method for systematically reducing the dimension of biophysically realistic neuron models with stochastic ion channels exploiting time-scales separation. Based on a combination of singular perturbation methods for kinetic Markov schemes with some recent mathematical developments of the averaging method, the techniques are general and applicable to a large class of models. As an example, we derive and analyze reductions of different stochastic versions of the Hodgkin Huxley (HH) model, leading to distinct reduced models. The bifurcation analysis of one of the reduced models with the number of channels as a parameter provides new insights into some features of noisy discharge patterns, such as the bimodality of interspike intervals distribution. Our analysis of the stochastic HH model shows that, besides being a method to reduce the number of variables of neuronal models, our reduction scheme is a powerful method for gaining understanding on the impact of fluctuations due to finite size effects on the dynamics of slow fast systems. Our analysis of the reduced model reveals that decreasing the number of sodium channels in the HH model leads to a transition in the dynamics reminiscent of the Hopf bifurcation and that this transition accounts for changes in characteristics of the spike train generated by the model. Finally, we also examine the impact of these results on neuronal coding, notably, reliability of discharge times and spike latency, showing that reducing the number of channels can enhance discharge time reliability in response to weak inputs and that this phenomenon can be accounted for through the analysis of the reduced model.},
Author = {Wainrib, Gilles and Thieullen, Mich\`{e}le and Pakdaman, Khashayar},
Date-Added = {2013-10-07 22:05:25 +0000},
Date-Modified = {2013-10-07 22:06:03 +0000},
Doi = {10.1007/s10827-011-0355-7},
Journal = {J Comput Neurosci},
Journal-Full = {Journal of computational neuroscience},
Mesh = {Action Potentials; Animals; Biophysical Processes; Computer Simulation; Humans; Ion Channels; Models, Neurological; Neurons; Reproducibility of Results; Stochastic Processes; Time Factors},
Month = {Apr},
Number = {2},
Pages = {327-46},
Pmid = {21842259},
Pst = {ppublish},
Title = {Reduction of stochastic conductance-based neuron models with time-scales separation},
Volume = {32},
Year = {2012},
Bdsk-Url-1 = {http://dx.doi.org/10.1007/s10827-011-0355-7}}
@book{Haas2002,
Address = {New York},
Author = {Peter J. Haas},
Edition = {first},
Editor = {Peter Glynn and Stephen Robinson},
Publisher = {Springer},
Title = {Stochastic Petri Nets: Modelling Stability, Simulation},
Year = {2002}}
@article{Glynn89,
Author = {Peter W. Glynn},
Journal = {Proc. of the IEEE},
Number = {1},
Pages = {14--23},
Title = {A {GSMP} Formalism for Discrete Event Systems},
Volume = {77},
Year = {1989}}
@article{BuonocoreNobileRicciardi1987AdvApplProb,
Abstract = {The first-passage-time p.d.f. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions. Use of this equation is made to prove that for the Wiener and the Ornstein-Uhlenbeck processes the singularity of the kernel can be removed by a suitable choice of these functions. A simple and efficient numerical procedure for the solution of the integral equation is provided and its convergence is briefly discussed. Use of this equation is finally made to obtain closed-form expressions for first-passage-time p.d.f.'s in the case of various time-dependent boundaries.},
Author = {Buonocore, A. and Nobile, A. G. and Ricciardi, L. M.},
Copyright = {Copyright � 1987 Applied Probability Trust},
Issn = {00018678},
Journal = {Advances in Applied Probability},
Language = {English},
Number = {4},
Pages = {pp. 784-800},
Publisher = {Applied Probability Trust},
Title = {A New Integral Equation for the Evaluation of First-Passage-Time Probability Densities},
Url = {http://www.jstor.org/stable/1427102},
Volume = {19},
Year = {1987},
Bdsk-Url-1 = {http://www.jstor.org/stable/1427102}}
@article{Fortet1943JMathPuresAppl,
Author = {R. Fortet},
Date-Added = {2013-03-26 11:15:15 +0000},
Date-Modified = {2013-03-26 11:17:36 +0000},
Journal = {J. Math. Pures Appl.},
Pages = {177-243},
Title = {Les fonctions al\'{e}atoires due type de {Markoff} associ\'{e}es \`{a} certaines \'{e}quations line\'{a}ires aux d\'{e}riv\'{e}es partialles due type parabolique},
Volume = {22},
Year = {1943}}
@article{RubinTerman2002SIADS,
Author = {Rubin, J. and Terman, D.},
Date-Added = {2012-11-13 20:41:26 +0000},
Date-Modified = {2012-11-13 20:44:43 +0000},
Doi = {10.1137/S111111110240323X},
Eprint = {http://epubs.siam.org/doi/pdf/10.1137/S111111110240323X},
Journal = {SIAM Journal on Applied Dynamical Systems},
Number = {1},
Pages = {146-174},
Title = {Synchronized Activity and Loss of Synchrony Among Heterogeneous Conditional Oscillators},
Volume = {1},
Year = {2002},
Bdsk-Url-1 = {http://epubs.siam.org/doi/abs/10.1137/S111111110240323X},
Bdsk-Url-2 = {http://dx.doi.org/10.1137/S111111110240323X}}
@article{LaingKevrekidis2008PhysicaD,
Abstract = {We study a network of 500 globally-coupled modified van der Pol oscillators. The value of a parameter associated with each oscillator is drawn from a normal distribution, giving a heterogeneous network. For strong enough coupling the oscillators all have the same period, and we consider periodic forcing of the network when it is in this state. By exploiting the correlations that quickly develop between the state of an oscillator and the value of its parameter we obtain an approximate low-dimensional description of the system in terms of the first few coefficients in a polynomial chaos expansion. Standard bifurcation analysis can then be performed on the low-dimensional system which results from this computational coarse-graining, and the results obtained from this predict very well the behaviour of the high-dimensional system for any set of realisations of the random parameter. Situations in which the method begins to fail are also discussed.},
Author = {Carlo R. Laing and Ioannis G. Kevrekidis},
Date-Modified = {2012-11-13 20:51:06 +0000},
Doi = {10.1016/j.physd.2007.08.013},
Issn = {0167-2789},
Journal = {Physica D: Nonlinear Phenomena},
Keywords = {Polynomial chaos},
Number = {2},
Pages = {207 - 215},
Title = {Periodically-forced finite networks of heterogeneous globally-coupled oscillators: A low-dimensional approach},
Volume = {237},
Year = {2008},
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0167278907003168},
Bdsk-Url-2 = {http://dx.doi.org/10.1016/j.physd.2007.08.013}}
@article{TaillefumierMagnasco2012arXiv,
Author = {Taillefumier, T. and Magnasco, M.O.},
Journal = {arXiv preprint arXiv:1206.6129},
Title = {A phase transition in the first passage of a Brownian process through a fluctuating boundary: implications for neural coding},
Year = {2012}}
@article{ClewleyRotsteinKopell2005MMS,
Author = {Robert Clewley and Horacio G. Rotstein and Nancy Kopell},
Date-Added = {2012-10-22 04:20:45 +0000},
Date-Modified = {2012-10-22 04:22:12 +0000},
Journal = {Multiscale Modeling \& Simulation},
Title = {A computational tool for the reduction of nonlinear {ODE} systems posessing multiple scales},
Year = {2005}}
@article{MatveevBoseNadim2007JCNS,
Abstract = {Out-of-phase bursting is a functionally important behavior displayed by central pattern generators and other neural circuits. Understanding this complex activity requires the knowledge of the interplay between the intrinsic cell properties and the properties of synaptic coupling between the cells. Here we describe a simple method that allows us to investigate the existence and stability of anti-phase bursting solutions in a network of two spiking neurons, each possessing a T-type calcium current and coupled by reciprocal inhibition. We derive a one-dimensional map which fully characterizes the genesis and regulation of anti-phase bursting arising from the interaction of the T-current properties with the properties of synaptic inhibition. This map is the burst length return map formed as the composition of two distinct one-dimensional maps that are each regulated by a different set of model parameters. Although each map is constructed using the properties of a single isolated model neuron, the composition of the two maps accurately captures the behavior of the full network. We analyze the parameter sensitivity of these maps to determine the influence of both the intrinsic cell properties and the synaptic properties on the burst length, and to find the conditions under which multistability of several bursting solutions is achieved. Although the derivation of the map relies on a number of simplifying assumptions, we discuss how the principle features of this dimensional reduction method could be extended to more realistic model networks.},
Author = {Matveev, Victor and Bose, Amitabha and Nadim, Farzan},
Date-Added = {2012-10-22 04:17:50 +0000},
Date-Modified = {2012-10-22 04:18:10 +0000},
Doi = {10.1007/s10827-007-0026-x},
Journal = {J Comput Neurosci},
Journal-Full = {Journal of computational neuroscience},
Mesh = {Action Potentials; Animals; Cell Communication; Models, Neurological; Nerve Net; Neural Inhibition; Neural Networks (Computer); Neurons; Nonlinear Dynamics},
Month = {Oct},
Number = {2},
Pages = {169-87},
Pmc = {PMC2606977},
Pmid = {17440801},
Pst = {ppublish},
Title = {Capturing the bursting dynamics of a two-cell inhibitory network using a one-dimensional map},
Volume = {23},
Year = {2007},
Bdsk-Url-1 = {http://dx.doi.org/10.1007/s10827-007-0026-x}}
@article{RubinTerman2012JMathNsci,
Author = {Jonathan E. Rubin and David H. Terman},
Date-Added = {2012-10-22 04:15:19 +0000},
Date-Modified = {2012-10-22 04:16:11 +0000},
Journal = {Journal of Mathematical Neuroscience},
Number = {1},
Pages = {4},
Title = {Explicit maps to predict activation order in multi-phase rhythms of a coupled cell network},
Volume = {2},
Year = {2012}}
@article{LajoieSheaBrown2011SIADS,
Author = {Guillaume Lajoie and Eric Shea-Brown},
Date-Added = {2012-10-22 04:11:02 +0000},
Date-Modified = {2012-10-22 04:12:58 +0000},
Journal = {Siam J. Applied Dynamical Systems},
Number = {4},
Pages = {1232-1271},
Title = {Shared inputs, entrainment, and desynchrony in elliptic bursters: from slow passage to discontinuous circle maps},
Volume = {10},
Year = {2011}}
@unpublished{SherwoodGuckenheimer2009arXiv,
Author = {William Erik Sherwood and John Guckenheimer},
Date-Added = {2012-10-22 02:27:50 +0000},
Date-Modified = {2012-10-22 02:29:57 +0000},
Month = {11 Oct.},
Note = {arxiv.org:0910.1970},
Title = {Dissecting the Phase Response of a Model Bursting Neuron},
Year = {2009}}
@article{SherwoodHarris-WarrickGuckenheimer2011JCNS,
Abstract = {Establishing, maintaining, and modifying the phase relationships between extensor and flexor muscle groups is essential for central pattern generators in the spinal cord to coordinate the hindlimbs well enough to produce the basic walking rhythm. This paper investigates a simplified computational model for the spinal hindlimb central pattern generator (CPG) that is abstracted from experimental data from the rodent spinal cord. This model produces locomotor-like activity with appropriate phase relationships in which right and left muscle groups alternate while extensor and flexor muscle groups alternate. Convergence to this locomotor pattern is slow, however, and the range of parameter values for which the model produces appropriate output is relatively narrow. We examine these aspects of the model's coordination of left-right activity through investigation of successively more complicated subnetworks, focusing on the role of the synaptic architecture in shaping motoneuron phasing. We find unexpected sensitivity in the phase response properties of individual neurons in response to stimulation and a need for high levels of both inhibition and excitation to achieve the walking rhythm. In the absence of cross-cord excitation, equal levels of ipsilateral and contralateral inhibition result in a strong preference for hopping over walking. Inhibition alone can produce the walking rhythm, but contralateral inhibition must be much stronger than ipsilateral inhibition. Cross-cord excitatory connections significantly enhance convergence to the walking rhythm, which is achieved most rapidly with strong crossed excitation and greater contralateral than ipsilateral inhibition. We discuss the implications of these results for CPG architectures based on unit burst generators.},
Author = {Sherwood, William Erik and Harris-Warrick, Ronald and Guckenheimer, John},
Date-Added = {2012-10-22 02:23:42 +0000},
Date-Modified = {2012-10-22 02:24:05 +0000},
Doi = {10.1007/s10827-010-0259-y},
Journal = {J Comput Neurosci},
Journal-Full = {Journal of computational neuroscience},
Mesh = {Animals; Computer Simulation; Electric Stimulation; Functional Laterality; Hindlimb; Locomotion; Models, Biological; Motor Neurons; Neural Inhibition; Neural Pathways; Periodicity; Rodentia; Spinal Cord; Synapses},
Month = {Apr},
Number = {2},
Pages = {323-60},
Pmid = {20644988},
Pst = {ppublish},
Title = {Synaptic patterning of left-right alternation in a computational model of the rodent hindlimb central pattern generator},
Volume = {30},
Year = {2011},
Bdsk-Url-1 = {http://dx.doi.org/10.1007/s10827-010-0259-y}}
@book{Winfree2000,
Address = {New York},
Author = {A.T. Winfree},
Date-Added = {2012-10-22 02:19:56 +0000},
Date-Modified = {2012-10-22 02:21:05 +0000},
Edition = {Second},
Publisher = {Springer-Verlag},
Title = {The Geometry of Biological Time},
Year = {2000}}
@article{ErmentroutGlassOldeman2012NeCo,
Abstract = {We introduce a simple two-dimensional model that extends the Poincar{\'e} oscillator so that the attracting limit cycle undergoes a saddle node bifurcation on an invariant circle (SNIC) for certain parameter values. Arbitrarily close to this bifurcation, the phase-resetting curve (PRC) continuously depends on parameters, where its shape can be not only primarily positive or primarily negative but also nearly sinusoidal. This example system shows that one must be careful inferring anything about the bifurcation structure of the oscillator from the shape of its PRC.},
Author = {Ermentrout, G Bard and Glass, Leon and Oldeman, Bart E},
Date-Added = {2012-10-22 02:17:21 +0000},
Date-Modified = {2012-10-22 02:17:39 +0000},
Doi = {10.1162/NECO_a_00370},
Journal = {Neural Comput},
Journal-Full = {Neural computation},
Month = {Sep},
Pmid = {22970869},
Pst = {aheadofprint},
Title = {The Shape of Phase-Resetting Curves in Oscillators with a Saddle Node on an Invariant Circle Bifurcation},
Year = {2012},
Bdsk-Url-1 = {http://dx.doi.org/10.1162/NECO_a_00370}}
@article{KopellErmentrout1988MatBiol,
Abstract = {Much can be deduced about the behavior of chains of oscillators under
minimal assumptions about the nature of the oscillators or the coupling. This paper
reviews work on such chains, and provides a framework within which implications
may be drawn about the neural networks that govern undulatory locomotion in lower
vertebrates.},
Author = {N. Kopell and G.B. Ermentrout},
Doi = {10.1016/0025-5564(88)90059-4},
Issn = {0025-5564},
Journal = {Mathematical Biosciences},
Number = {1-2},
Pages = {87 - 109},
Title = {Coupled oscillators and the design of central pattern generators},
Url = {http://www.sciencedirect.com/science/article/pii/0025556488900594},
Volume = {90},
Year = 1988,
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/0025556488900594},
Bdsk-Url-2 = {http://dx.doi.org/10.1016/0025-5564(88)90059-4}}
@book{HirschSmaleDevaney2004,
Author = {Morris W. Hirsch and Stephen Smale and Robert L. Devaney},
Date-Added = {2012-10-07 20:43:47 +0000},
Date-Modified = {2012-10-07 20:45:16 +0000},
Edition = {2nd},
Publisher = {Elsevier Academic Press},
Title = {{Differential Equations, Dynamical Systems \& An Introduction to Chaos}},
Year = {2004}}
@unpublished{BorisyukRassoul-Agha2012-submitted,
Author = {Alla Borisyuk and Firas Rassoul-Agha},
Date-Added = {2012-10-05 02:28:02 +0000},
Date-Modified = {2012-10-05 02:29:30 +0000},
Month = {October},
Note = {submitted},
Title = {Quasiperiodicity and Phase Locking in Stochastic Circle Maps: a Spectral Approach},
Year = {2012}}
@article{LinWedgwoodCoombesYoung2012JMathBiol,
Abstract = {{Perturbation theory is an important tool in the analysis of oscillators and their response to external stimuli. It is predicated on the assumption that the perturbations in question are "sufficiently weak", an assumption that is not always valid when perturbative methods are applied. In this paper, we identify a number of concrete dynamical scenarios in which a standard perturbative technique, based on the infinitesimal phase response curve (PRC), is shown to give different predictions than the full model. Shear-induced chaos, i.e., chaotic behavior that results from the amplification of small perturbations by underlying shear, is missed entirely by the PRC. We show also that the presence of "sticky" phase-space structures tend to cause perturbative techniques to overestimate the frequencies and regularity of the oscillations. The phenomena we describe can all be observed in a simple 2D neuron model, which we choose for illustration as the PRC is widely used in mathematical neuroscience.}},
Author = {Lin, Kevin K and Wedgwood, Kyle C A and Coombes, Stephen and Young, Lai-Sang},
Date-Added = {2012-10-05 02:23:49 +0000},
Date-Modified = {2012-10-05 02:24:26 +0000},
Doi = {10.1007/s00285-012-0506-0},
Journal = {J Math Biol},
Journal-Full = {Journal of mathematical biology},
Month = {Jan},
Pmid = {22290314},
Pst = {aheadofprint},
Title = {Limitations of perturbative techniques in the analysis of rhythms and oscillations},
Year = {2012},
Bdsk-Url-1 = {http://dx.doi.org/10.1007/s00285-012-0506-0}}
@article{GogOpreaProctorRucklidge1999ProcRSocLondA,
Author = {J.R. Gog and I. Oprea and M.R.E. Proctor and A.M. Rucklidge},
Date-Added = {2012-10-05 01:30:17 +0000},
Date-Modified = {2012-10-05 01:32:20 +0000},
Journal = {Proceedings of the Royal Society A},
Keywords = {doi: 10.1098/rspa.1999.0498},
Pages = {4205-4222},
Title = {Destabilization by noise of transverse perturbations to heteroclinic cycles: a simple model and an example from dynamo theory},
Volume = {455},
Year = {1999}}
@book{Oksendal2007,
Author = {Bernt {\O}ksendal},
Date-Added = {2012-08-01 17:42:28 +0000},
Date-Modified = {2012-08-01 17:45:22 +0000},
Edition = {6th},
Publisher = {Springer},
Title = {Stochastic Differential Equations: An Introduction with Applications},
Year = {2007}}
@unpublished{LajoieSheaBrown2011arXiv,
Author = {Guillaume Lajoie and Eric Shea-Brown},
Date-Added = {2012-08-01 16:53:35 +0000},
Date-Modified = {2012-08-01 16:55:01 +0000},
Keywords = {elliptic bursting, circle maps, perturbed oscillators, synchrony, mathematical neuroscience},
Month = {May 25},
Note = {http://arxiv.org/pdf/1010.2809v2.pdf},
Title = {Shared inputs, entrainment, and desynchrony in elliptic bursters: from slow passage to discontinuous circle maps},
Year = {2011}}
@article{Rubin:2000uq,
Abstract = {We develop geometric dynamical systems methods to determine how various components contribute to a neuronal network's emergent population behaviors. The results clarify the multiple roles inhibition can play in producing different rhythms. Which rhythms arise depends on how inhibition interacts with intrinsic properties of the neurons; the nature of these interactions depends on the underlying architecture of the network. Our analysis demonstrates that fast inhibitory coupling may lead to synchronized rhythms if either the cells within the network or the architecture of the network is sufficiently complicated. This cannot occur in mutually coupled networks with basic cells; the geometric approach helps explain how additional network complexity allows for synchronized rhythms in the presence of fast inhibitory coupling. The networks and issues considered are motivated by recent models for thalamic oscillations. The analysis helps clarify the roles of various biophysical features, such as fast and slow inhibition, cortical inputs, and ionic conductances, in producing network behavior associated with the spindle sleep rhythm and with paroxysmal discharge rhythms. Transitions between these rhythms are also discussed.},
Author = {Rubin, J and Terman, D},
Date-Added = {2012-08-01 16:47:29 +0000},
Date-Modified = {2012-08-01 16:47:29 +0000},
Journal = {Neural Comput},
Journal-Full = {Neural computation},
Mesh = {Cerebral Cortex; Mathematics; Models, Neurological; Neural Inhibition; Neural Networks (Computer); Neurons; Periodicity; Synapses; Thalamus},
Month = {Mar},
Number = {3},
Pages = {597-645},
Pmid = {10769324},
Pst = {ppublish},
Title = {Geometric analysis of population rhythms in synaptically coupled neuronal networks},
Volume = {12},
Year = {2000}}
@article{BovierEckhoffGayrardKlein2002CommMathPhys,
Abstract = {We study a large class of reversible Markov chains with discrete state space and transition matrix P N . We define the notion of a set of metastable points as a subset of the state space Γ N such that (i) this set is reached from any point x ∈Γ N without return to x with probability at least b N , while (ii) for any two points x , y in the metastable set, the probability T − 1 x , y to reach y from x without return to x is smaller than a N − 1 < b N . Under some additional non-degeneracy assumption, we show that in such a situation: (i) To each metastable point corresponds a metastable state, whose mean exit time can be computed precisely. (ii) To each metastable point corresponds one simple eigenvalue of 1 − P N which is essentially equal to the inverse mean exit time from this state. Moreover, these results imply very sharp uniform control of the deviation of the probability distribution of metastable exit times from the exponential distribution.},
Affiliation = {Weierstrass-Institut f{\"u}r Angewandte Analysis und Stochastik, Mohrenstrasse 39, 10117 Berlin, Germany. E-mail: [email protected] DE},
Author = {Bovier, Anton and Eckhoff, Michael and Gayrard, V{\'e}ronique and Klein, Markus},
Date-Added = {2012-08-01 00:33:09 +0000},
Date-Modified = {2012-08-01 00:33:44 +0000},
Issn = {0010-3616},
Issue = {2},
Journal = {Communications in Mathematical Physics},
Keyword = {Physics and Astronomy},
Note = {10.1007/s002200200609},
Pages = {219-255},
Publisher = {Springer Berlin / Heidelberg},
Title = {Metastability and Low Lying Spectra¶in Reversible Markov Chains},
Url = {http://dx.doi.org/10.1007/s002200200609},
Volume = {228},
Year = {2002},
Bdsk-Url-1 = {http://dx.doi.org/10.1007/s002200200609}}
@book{PinskyKarlin2011,
Author = {Mark A. Pinsky and Samuel Karlin},
Date-Added = {2012-07-30 00:07:59 -0400},
Date-Modified = {2012-07-30 00:07:59 -0400},
Edition = {4th},
Publisher = {Academic Press},
Title = {An Introduction to Stochastic Modeling},
Year = {2011}}
@incollection{Rubin+Terman:2002,
Author = {J. Rubin, D. Terman},
Booktitle = {Handbook of Dynamical Systems, vol. 2: Towards Applications},
Date-Added = {2012-07-22 10:15:00 -0400},
Date-Modified = {2012-07-22 10:18:18 -0400},
Editor = {B. Fiedler},
Pages = {93-146},
Publisher = {Elsevier},
Title = {Geometric singular pertubation analysis of neuronal dynamics},
Year = {2002}}
@article{Winfree1974JMB,
Author = {A. T. Winfree},
Date-Added = {2012-07-27 01:51:15 +0000},
Date-Modified = {2012-07-27 01:52:11 +0000},
Journal = {Journal of Mathematical Biology},
Pages = {73--95},
Title = {Patterns of Phase Compromise in Biological Cycles},
Volume = {1},
Year = {1974}}
@article{ChiconeLiu2004JDiffEqs,
Author = {Carmen Chicone and Weishi Liu},
Date-Added = {2012-07-27 01:48:10 +0000},
Date-Modified = {2012-07-27 01:49:55 +0000},
Journal = {Journal of Differential Equations},
Keywords = {doi:10.1016/j.jde.2004.03.011},
Pages = {227--246},
Title = {Asymptotic phase revisited},
Volume = {204},
Year = {2004}}
@book{CoddingtonLevinson,
Author = {Earl A. Coddington and Norman Levinson},
Date-Added = {2012-07-23 04:29:45 +0000},
Date-Modified = {2012-07-23 04:32:47 +0000},
Publisher = {McGraw-Hill Book Company, Inc.},
Title = {Theory of Ordinary Differential Equations},
Year = {1955}}
@article{KeplerElston2001BPJ,
Author = {Thomas B. Kepler and Timothy C. Elston},
Doi = {10.1016/S0006-3495(01)75949-8},
Issn = {0006-3495},
Journal = {Biophysical Journal},
Number = {6},
Pages = {3116 - 3136},
Title = {Stochasticity in Transcriptional Regulation: Origins, Consequences, and Mathematical Representations},
Url = {http://www.sciencedirect.com/science/article/pii/S0006349501759498},
Volume = {81},
Year = {2001},
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0006349501759498},
Bdsk-Url-2 = {http://dx.doi.org/10.1016/S0006-3495(01)75949-8}}
@inbook{Ermentrout2001inBowerBook,
Author = {Bard Ermentrout},
Chapter = {11. Simplifying and reducing complex models},
Date-Added = {2012-07-22 11:55:48 +0000},
Date-Modified = {2012-07-22 11:59:18 +0000},
Pages = {307-323},
Publisher = {MIT Press},
Title = {Computational Modeling of Genetic and Biochemical Networks},
Year = {2001}}
@article{Ermentrout1996NeuralComput,
Abstract = {Type I membrane oscillators such as the Connor model (Connor et al. 1977) and the Morris-Lecar model (Morris and Lecar 1981) admit very low frequency oscillations near the critical applied current. Hansel et al. (1995) have numerically shown that synchrony is difficult to achieve with these models and that the phase resetting curve is strictly positive. We use singular perturbation methods and averaging to show that this is a general property of Type I membrane models. We show in a limited sense that so called Type II resetting occurs with models that obtain rhythmicity via a Hopf bifurcation. We also show the differences between synapses that act rapidly and those that act slowly and derive a canonical form for the phase interactions.},
Author = {Ermentrout, B},
Date-Added = {2012-07-21 03:50:00 +0000},
Date-Modified = {2012-07-23 04:33:54 +0000},
Journal = {Neural Comput},
Journal-Full = {Neural computation},
Mesh = {Cell Membrane; Neural Networks (Computer); Neurons; Synapses},
Month = {Jul},
Number = {5},
Pages = {979-1001},
Pmid = {8697231},
Pst = {ppublish},
Title = {Type {{I}} membranes, phase resetting curves, and synchrony},
Volume = {8},
Year = {1996}}
@article{ErmentroutKopell1986SIAMJAM,
Author = {G.B. Ermentrout and N. Kopell},
Date-Added = {2012-07-21 03:46:35 +0000},
Date-Modified = {2012-07-21 03:48:19 +0000},
Journal = {SIAM J. Appl. Math},
Pages = {233-253},
Title = {Parabolic bursting in an excitable system coupled with a slow oscillation.},
Volume = {46},
Year = {1986}}
@article{BrunelLatham2003NeuralComp,
Abstract = {We calculate the firing rate of the quadratic integrate-and-fire neuron in response to a colored noise input current. Such an input current is a good approximation to the noise due to the random bombardment of spikes, with the correlation time of the noise corresponding to the decay time of the synapses. The key parameter that determines the firing rate is the ratio of the correlation time of the colored noise, tau(s), to the neuronal time constant, tau(m). We calculate the firing rate exactly in two limits: when the ratio, tau(s)/tau(m), goes to zero (white noise) and when it goes to infinity. The correction to the short correlation time limit is O(tau(s)/tau(m)), which is qualita tively different from that of the leaky integrate-and-fire neuron, where the correction is O( radical tau(s)/tau(m)). The difference is due to the different boundary conditions of the probability density function of the membrane potential of the neuron at firing threshold. The correction to the long correlation time limit is O(tau(m)/tau(s)). By combining the short and long correlation time limits, we derive an expression that provides a good approximation to the firing rate over the whole range of tau(s)/tau(m) in the suprathreshold regime-that is, in a regime in which the average current is sufficient to make the cell fire. In the subthreshold regime, the expression breaks down somewhat when tau(s) becomes large compared to tau(m).},
Author = {Brunel, Nicolas and Latham, Peter E},
Date-Added = {2012-07-20 19:08:20 +0000},
Date-Modified = {2012-07-20 19:08:42 +0000},
Doi = {10.1162/089976603322362365},
Journal = {Neural Comput},
Journal-Full = {Neural computation},
Mesh = {Action Potentials; Afferent Pathways; Animals; Artifacts; Cell Membrane; Central Nervous System; Humans; Models, Neurological; Models, Statistical; Nerve Net; Neurons; Reaction Time; Synapses; Synaptic Transmission; Time Factors},
Month = {Oct},
Number = {10},
Pages = {2281-306},
Pmid = {14511522},
Pst = {ppublish},
Title = {Firing rate of the noisy quadratic integrate-and-fire neuron},
Volume = {15},
Year = {2003},
Bdsk-Url-1 = {http://dx.doi.org/10.1162/089976603322362365}}
@book{LaingLord2010,
Date-Added = {2012-07-19 14:47:52 +0000},
Date-Modified = {2012-07-19 14:49:29 +0000},
Editor = {Carlo Laing and Gabriel J. Lord},
Publisher = {Oxford University Press},
Title = {Stochastic Methods in Neuroscience},
Year = {2010}}
@unpublished{DenkerRodrigues2011arXiv,
Author = {Manfred Denker and Ana Rodrigues},
Date-Added = {2012-07-19 11:06:58 -0400},
Date-Modified = {2012-07-19 11:08:10 -0400},
Month = {22 Nov},
Note = {http://arxiv.org/pdf/1111.5071v1.pdf},
Title = {The Combinatorics of Avalanche Dynamics},
Year = {2011}}
@article{BenayounCowanVanDrongelenWallace2010PLoSCompBiol,
Abstract = {{Neuronal avalanches are a form of spontaneous activity widely observed in cortical slices and other types of nervous tissue, both in vivo and in vitro. They are characterized by irregular, isolated population bursts when many neurons fire together, where the number of spikes per burst obeys a power law distribution. We simulate, using the Gillespie algorithm, a model of neuronal avalanches based on stochastic single neurons. The network consists of excitatory and inhibitory neurons, first with all-to-all connectivity and later with random sparse connectivity. Analyzing our model using the system size expansion, we show that the model obeys the standard Wilson-Cowan equations for large network sizes ( neurons). When excitation and inhibition are closely balanced, networks of thousands of neurons exhibit irregular synchronous activity, including the characteristic power law distribution of avalanche size. We show that these avalanches are due to the balanced network having weakly stable functionally feedforward dynamics, which amplifies some small fluctuations into the large population bursts. Balanced networks are thought to underlie a variety of observed network behaviours and have useful computational properties, such as responding quickly to changes in input. Thus, the appearance of avalanches in such functionally feedforward networks indicates that avalanches may be a simple consequence of a widely present network structure, when neuron dynamics are noisy. An important implication is that a network need not be "critical" for the production of avalanches, so experimentally observed power laws in burst size may be a signature of noisy functionally feedforward structure rather than of, for example, self-organized criticality.}},
Author = {Benayoun, Marc and Cowan, Jack D and van Drongelen, Wim and Wallace, Edward},
Date-Added = {2012-07-19 10:58:37 -0400},
Date-Modified = {2012-07-19 10:59:25 -0400},
Doi = {10.1371/journal.pcbi.1000846},
Journal = {PLoS Comput Biol},
Journal-Full = {PLoS computational biology},
Mesh = {Action Potentials; Algorithms; Animals; Computer Simulation; Markov Chains; Models, Neurological; Nerve Net; Rats; Stochastic Processes},
Number = {7},
Pages = {e1000846},
Pmc = {PMC2900286},
Pmid = {20628615},
Pst = {epublish},
Title = {Avalanches in a stochastic model of spiking neurons},
Volume = {6},
Year = {2010},
Bdsk-Url-1 = {http://dx.doi.org/10.1371/journal.pcbi.1000846}}
@article{ButlerBenayounWallaceVanDrongelenGoldenfeldCowan2012PNAS,
Abstract = {In the cat or primate primary visual cortex (V1), normal vision corresponds to a state where neural excitation patterns are driven by external visual stimuli. A spectacular failure mode of V1 occurs when such patterns are overwhelmed by spontaneously generated spatially self-organized patterns of neural excitation. These are experienced as geometric visual hallucinations. The problem of identifying the mechanisms by which V1 avoids this failure is made acute by recent advances in the statistical mechanics of pattern formation, which suggest that the hallucinatory state should be very robust. Here, we report how incorporating physiologically realistic long-range connections between inhibitory neurons changes the behavior of a model of V1. We find that the sparsity of long-range inhibition in V1 plays a previously unrecognized but key functional role in preserving the normal vision state. Surprisingly, it also contributes to the observed regularity of geometric visual hallucinations. Our results provide an explanation for the observed sparsity of long-range inhibition in V1--this generic architectural feature is an evolutionary adaptation that tunes V1 to the normal vision state. In addition, it has been shown that exactly the same long-range connections play a key role in the development of orientation preference maps. Thus V1's most striking long-range features--patchy excitatory connections and sparse inhibitory connections--are strongly constrained by two requirements: the need for the visual state to be robust and the developmental requirements of the orientational preference map.},
Author = {Butler, Thomas Charles and Benayoun, Marc and Wallace, Edward and van Drongelen, Wim and Goldenfeld, Nigel and Cowan, Jack},
Date-Added = {2012-07-19 10:58:29 -0400},
Date-Modified = {2012-07-19 11:00:30 -0400},
Doi = {10.1073/pnas.1118672109},
Journal = {Proc Natl Acad Sci U S A},
Journal-Full = {Proceedings of the National Academy of Sciences of the United States of America},
Mesh = {Adaptation, Biological; Biological Evolution; Hallucinations; Humans; Models, Neurological; Neurons; Orientation; Pattern Recognition, Visual; Visual Cortex},
Month = {Jan},
Number = {2},
Pages = {606-9},
Pmc = {PMC3258647},
Pmid = {22203969},
Pst = {ppublish},
Title = {Evolutionary constraints on visual cortex architecture from the dynamics of hallucinations},
Volume = {109},
Year = {2012},
Bdsk-Url-1 = {http://dx.doi.org/10.1073/pnas.1118672109}}
@article{WallaceBenayounVanDrongelenCowan2011PLoSOne,
Abstract = {Networks of neurons produce diverse patterns of oscillations, arising from the network's global properties, the propensity of individual neurons to oscillate, or a mixture of the two. Here we describe noisy limit cycles and quasi-cycles, two related mechanisms underlying emergent oscillations in neuronal networks whose individual components, stochastic spiking neurons, do not themselves oscillate. Both mechanisms are shown to produce gamma band oscillations at the population level while individual neurons fire at a rate much lower than the population frequency. Spike trains in a network undergoing noisy limit cycles display a preferred period which is not found in the case of quasi-cycles, due to the even faster decay of phase information in quasi-cycles. These oscillations persist in sparsely connected networks, and variation of the network's connectivity results in variation of the oscillation frequency. A network of such neurons behaves as a stochastic perturbation of the deterministic Wilson-Cowan equations, and the network undergoes noisy limit cycles or quasi-cycles depending on whether these have limit cycles or a weakly stable focus. These mechanisms provide a new perspective on the emergence of rhythmic firing in neural networks, showing the coexistence of population-level oscillations with very irregular individual spike trains in a simple and general framework.},
Author = {Wallace, Edward and Benayoun, Marc and van Drongelen, Wim and Cowan, Jack D},
Date-Added = {2012-07-19 10:58:27 -0400},
Date-Modified = {2012-07-19 10:59:55 -0400},
Doi = {10.1371/journal.pone.0014804},
Journal = {PLoS One},
Journal-Full = {PloS one},
Mesh = {Action Potentials; Animals; Humans; Models, Theoretical; Neurons; Periodicity},
Number = {5},
Pages = {e14804},
Pmc = {PMC3089610},
Pmid = {21573105},
Pst = {epublish},
Title = {Emergent oscillations in networks of stochastic spiking neurons},
Volume = {6},
Year = {2011},
Bdsk-Url-1 = {http://dx.doi.org/10.1371/journal.pone.0014804}}
@article{BuiceCowanChow2010NeuralComput,
Abstract = {Population rate or activity equations are the foundation of a common approach to modeling for neural networks. These equations provide mean field dynamics for the firing rate or activity of neurons within a network given some connectivity. The shortcoming of these equations is that they take into account only the average firing rate, while leaving out higher-order statistics like correlations between firing. A stochastic theory of neural networks that includes statistics at all orders was recently formulated. We describe how this theory yields a systematic extension to population rate equations by introducing equations for correlations and appropriate coupling terms. Each level of the approximation yields closed equations; they depend only on the mean and specific correlations of interest, without an ad hoc criterion for doing so. We show in an example of an all-to-all connected network how our system of generalized activity equations captures phenomena missed by the mean field rate equations alone.},
Author = {Buice, Michael A and Cowan, Jack D and Chow, Carson C},
Date-Added = {2012-07-19 10:53:45 -0400},
Date-Modified = {2012-07-19 10:55:12 -0400},
Doi = {10.1162/neco.2009.02-09-960},
Journal = {Neural Comput},
Journal-Full = {Neural computation},
Mesh = {Action Potentials; Algorithms; Animals; Artificial Intelligence; Brain; Humans; Mathematical Computing; Mathematical Concepts; Nerve Net; Neural Networks (Computer); Neurons},
Month = {Feb},
Number = {2},
Pages = {377-426},
Pmc = {PMC2805768},
Pmid = {19852585},
Pst = {ppublish},
Title = {Systematic fluctuation expansion for neural network activity equations},
Volume = {22},
Year = {2010},
Bdsk-Url-1 = {http://dx.doi.org/10.1162/neco.2009.02-09-960}}
@article{BuiceCowan2009ProgBiophysMolBiol,
Abstract = {{We analyze neocortical dynamics using field theoretic methods for non-equilibrium statistical processes. Assuming the dynamics is Markovian, we introduce a model that describes both neural fluctuations and responses to stimuli. We show that at low spiking rates, neocortical activity exhibits a dynamical phase transition which is in the universality class of directed percolation (DP). Because of the high density and large spatial extent of neural interactions, there is a "mean field" region in which the effects of fluctuations are negligible. However as the generation and decay of spiking activity becomes balanced, there is a crossover into the critical fluctuation driven DP region, consistent with measurements in neocortical slice preparations. From the perspective of theoretical neuroscience, the principal contribution of this work is the formulation of a theory of neural activity that goes beyond the mean-field approximation and incorporates the effects of fluctuations and correlations in the critical region. This theory shows that the scaling laws found in many measurements of neocortical activity, in anesthetized, normal and epileptic neocortex, are consistent with the existence of DP and related phase transitions at a critical point. It also shows how such properties lead to a model of the origins of both random and rhythmic brain activity.}},
Author = {Buice, Michael A and Cowan, Jack D},
Date = {2009 Feb-Apr},
Date-Added = {2012-07-19 10:53:43 -0400},
Date-Modified = {2012-07-19 10:54:30 -0400},
Doi = {10.1016/j.pbiomolbio.2009.07.003},
Journal = {Prog Biophys Mol Biol},
Journal-Full = {Progress in biophysics and molecular biology},
Mesh = {Animals; Biomechanics; Computer Simulation; Humans; Models, Biological; Neocortex; Nerve Net; Probability},
Number = {2-3},
Pages = {53-86},
Pmid = {19695282},
Pst = {ppublish},
Title = {Statistical mechanics of the neocortex},
Volume = {99},
Year = {2009},
Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.pbiomolbio.2009.07.003}}
@article{BuiceCowan2007PRE,
Abstract = {A well-defined stochastic theory for neural activity, which permits the calculation of arbitrary statistical moments and equations governing them, is a potentially valuable tool for theoretical neuroscience. We produce such a theory by analyzing the dynamics of neural activity using field theoretic methods for nonequilibrium statistical processes. Assuming that neural network activity is Markovian, we construct the effective spike model, which describes both neural fluctuations and response. This analysis leads to a systematic expansion of corrections to mean field theory, which for the effective spike model is a simple version of the Wilson-Cowan equation. We argue that neural activity governed by this model exhibits a dynamical phase transition which is in the universality class of directed percolation. More general models (which may incorporate refractoriness) can exhibit other universality classes, such as dynamic isotropic percolation. Because of the extremely high connectivity in typical networks, it is expected that higher-order terms in the systematic expansion are small for experimentally accessible measurements, and thus, consistent with measurements in neocortical slice preparations, we expect mean field exponents for the transition. We provide a quantitative criterion for the relative magnitude of each term in the systematic expansion, analogous to the Ginsburg criterion. Experimental identification of dynamic universality classes in vivo is an outstanding and important question for neuroscience.},
Author = {Buice, Michael A and Cowan, Jack D},
Date-Added = {2012-07-19 10:53:43 -0400},
Date-Modified = {2012-07-19 10:54:14 -0400},
Journal = {Phys Rev E Stat Nonlin Soft Matter Phys},
Journal-Full = {Physical review. E, Statistical, nonlinear, and soft matter physics},
Mesh = {Action Potentials; Biological Clocks; Computer Simulation; Models, Neurological; Models, Statistical; Neocortex; Nerve Net; Neurons; Stochastic Processes},
Month = {May},
Number = {5 Pt 1},
Pages = {051919},
Pmid = {17677110},
Pst = {ppublish},
Title = {Field-theoretic approach to fluctuation effects in neural networks},
Volume = {75},
Year = {2007}}
@article{NewbyKeener2011SIAMMM,
Author = {Jay M. Newby and James P. Keener},
Date-Added = {2012-04-28 18:30:34 -0400},
Date-Modified = {2012-04-28 18:31:42 -0400},
Journal = {SIAM J. Multiscale Modeling},
Number = {2},
Pages = {735-765},
Title = {An Asymptotic Analysis of the Spatially Inhomogeneous Velocity-Jump Process},
Volume = {9},
Year = {2011}}
@article{Bressloff2010PRE,
Abstract = {We analyze a stochastic model of neuronal population dynamics with intrinsic noise. In the thermodynamic limit N→∞ , where N determines the size of each population, the dynamics is described by deterministic Wilson-Cowan equations. On the other hand, for finite N the dynamics is described by a master equation that determines the probability of spiking activity within each population. We first consider a single excitatory population that exhibits bistability in the deterministic limit. The steady-state probability distribution of the stochastic network has maxima at points corresponding to the stable fixed points of the deterministic network; the relative weighting of the two maxima depends on the system size. For large but finite N , we calculate the exponentially small rate of noise-induced transitions between the resulting metastable states using a Wentzel-Kramers-Brillouin (WKB) approximation and matched asymptotic expansions. We then consider a two-population excitatory or inhibitory network that supports limit cycle oscillations. Using a diffusion approximation, we reduce the dynamics to a neural Langevin equation, and show how the intrinsic noise amplifies subthreshold oscillations (quasicycles).},
Author = {Bressloff, Paul C},
Date-Added = {2012-04-28 18:26:54 -0400},
Date-Modified = {2013-01-22 22:32:36 +0000},
Journal = {Phys Rev E Stat Nonlin Soft Matter Phys},
Journal-Full = {Physical review. E, Statistical, nonlinear, and soft matter physics},
Mesh = {Markov Chains; Models, Biological; Neurons},
Month = {Nov},
Number = {5 Pt 1},
Pages = {051903},
Pmid = {21230496},
Pst = {ppublish},
Title = {{Metastable states and quasicycles in a stochastic Wilson-Cowan model of neuronal population dynamics}},
Volume = {82},
Year = {2010}}
@inproceedings{RinzelBorisyuk2005LesHouches,
Author = {Alla Borisyuk and John Rinzel},
Booktitle = {Models and Methods in Neurophysics, Proc Les Houches Summer School 2003, (Session LXXX)},
Date-Added = {2012-03-14 18:09:14 -0400},
Date-Modified = {2012-03-14 18:10:45 -0400},
Pages = {19-72},
Publisher = {Elsevier},
Title = {Understanding neuronal dynamics by geometrical dissection of minimal models},
Year = {2005}}
@article{Papangelou2010JApplProb,
Author = {F. Papangelou},
Date-Added = {2012-02-14 15:24:39 -0500},
Date-Modified = {2012-02-14 15:26:28 -0500},
Journal = {J. Appl. Prob.},
Pages = {1164-1173},
Title = {{A Simple Model for Random Oscillations}},
Volume = {47},
Year = {2010}}
@book{Pinsky1991,
Address = {Singapore},
Author = {Mark A. Pinsky},
Date-Added = {2012-02-14 15:21:24 -0500},
Date-Modified = {2012-02-14 15:24:35 -0500},
Publisher = {World Scientific Publishing Co.},
Title = {{Lectures on Random Evolutions}},
Year = {1991}}
@book{KeenerSneyd2009,
Address = {New York},
Author = {James Keener and James Sneyd},
Date-Added = {2012-02-09 23:34:04 -0500},
Date-Modified = {2012-02-09 23:36:20 -0500},
Edition = {2nd},
Publisher = {Spring},
Title = {Mathematical Physiology},
Volume = {II. Systems Physiology},
Year = {2009}}
@article{KrivanVrkoc2007JMathBiol,
Author = {Vlastimil K\v{r}ivan and Ivo Vrko\v{c}},
Date-Added = {2011-10-20 10:56:15 -0400},
Date-Modified = {2011-10-20 11:08:02 -0400},
Journal = {J. Math. Biology},
Pages = {465-488},
Title = {A {Lyapunov} function for piecewise-independent differential equations: stability of the ideal free distribution in two patch environments},
Volume = {54},
Year = {2007}}
@article{BoukalKrivan1999JMathBiol,
Author = {David S. Boukal and Vlastimil K\v{r}ivan},
Date-Added = {2011-10-20 10:51:21 -0400},
Date-Modified = {2011-10-20 11:08:02 -0400},
Journal = {J. Math. Biology},
Pages = {493-517},
Title = {Lyapunov functions for {Lotka-Volterra} predator-prey models with optimal foraging behavior},
Volume = {39},
Year = {1999}}
@book{Filipov1988,
Address = {Dordrecht, The Netherlands},
Author = {Aleksei Fedorovich Filippov},
Date-Added = {2011-10-20 10:45:40 -0400},
Date-Modified = {2011-10-20 10:49:30 -0400},
Publisher = {Kluwer Academic Publishers},
Series = {Mathematics and Its Applications},
Title = {Differential Equations with Discontinuous Righthand Sides},
Year = {1988}}
@article{MorrisLecar1981BiophysJ,
Abstract = {Barnacle muscle fibers subjected to constant current stimulation produce a variety of types of oscillatory behavior when the internal medium contains the Ca++ chelator EGTA. Oscillations are abolished if Ca++ is removed from the external medium, or if the K+ conductance is blocked. Available voltage-clamp data indicate that the cell's active conductance systems are exceptionally simple. Given the complexity of barnacle fiber voltage behavior, this seems paradoxical. This paper presents an analysis of the possible modes of behavior available to a system of two noninactivating conductance mechanisms, and indicates a good correspondence to the types of behavior exhibited by barnacle fiber. The differential equations of a simple equivalent circuit for the fiber are dealt with by means of some of the mathematical techniques of nonlinear mechanics. General features of the system are (a) a propensity to produce damped or sustained oscillations over a rather broad parameter range, and (b) considerable latitude in the shape of the oscillatory potentials. It is concluded that for cells subject to changeable parameters (either from cell to cell or with time during cellular activity), a system dominated by two noninactivating conductances can exhibit varied oscillatory and bistable behavior.},
Author = {Morris, C and Lecar, H},
Date-Added = {2011-10-17 18:23:03 -0400},
Date-Modified = {2011-10-17 18:23:18 -0400},
Doi = {10.1016/S0006-3495(81)84782-0},
Journal = {Biophys J},
Journal-Full = {Biophysical journal},
Mesh = {Action Potentials; Animals; Calcium; Electric Conductivity; Models, Biological; Muscles; Potassium; Sarcolemma; Thoracica},
Month = {Jul},
Number = {1},
Pages = {193-213},
Pmc = {PMC1327511},
Pmid = {7260316},
Pst = {ppublish},
Title = {Voltage oscillations in the barnacle giant muscle fiber},
Volume = {35},
Year = {1981},
Bdsk-Url-1 = {http://dx.doi.org/10.1016/S0006-3495(81)84782-0}}
@article{NadimZhaoZhouBose2011JNE-inpress,
Author = {F. Nadim and S. Zhao and L. Zhou and A. Bose},
Date-Added = {2011-10-02 14:30:57 -0400},
Date-Modified = {2011-10-02 14:32:42 -0400},
Journal = {Journal of Neural Engineering},
Month = {Dec},
Number = {6},
Pages = {065001},
Title = {Inhibitory feedback promotes stability in an oscillatory network},
Volume = {8},
Year = {2011}}
@article{KeenerNewby2011PRE,
Abstract = {A stochastic interpretation of spontaneous action potential initiation is developed for the Morris-Lecar equations. Initiation of a spontaneous action potential can be interpreted as the escape from one of the wells of a double well potential, and we develop an asymptotic approximation of the mean exit time using a recently developed quasistationary perturbation method. Using the fact that the activating ionic channel's random openings and closings are fast relative to other processes, we derive an accurate estimate for the mean time to fire an action potential (MFT), which is valid for a below-threshold applied current. Previous studies have found that for above-threshold applied current, where there is only a single stable fixed point, a diffusion approximation can be used. We also explore why different diffusion approximation techniques fail to estimate the MFT.},
Author = {Keener, James P and Newby, Jay M},
Date-Added = {2011-09-29 00:15:59 -0400},
Date-Modified = {2011-10-17 18:19:50 -0400},
Journal = {Phys Rev E Stat Nonlin Soft Matter Phys},
Journal-Full = {Physical review. E, Statistical, nonlinear, and soft matter physics},
Month = {Jul},
Number = {1-1},
Pages = {011918},
Pmid = {21867224},
Pst = {ppublish},
Title = {Perturbation analysis of spontaneous action potential initiation by stochastic ion channels},
Volume = {84},
Year = {2011}}
@article{KoriKuramoto:2001:PRE,
Abstract = {The phenomenon of slow switching in populations of globally coupled oscillators is discussed. This characteristic collective dynamics, which was first discovered in a particular class of the phase oscillator model, is a result of the formation of a heteroclinic loop connecting a pair of clustered states of the population. We argue that the same behavior can arise in a wider class of oscillator models with the amplitude degree of freedom. We also argue how such heteroclinic loops arise inevitably and persist robustly in a homogeneous population of globally coupled oscillators. Although a heteroclinic loop might seem to arise only exceptionally, we find that it appears rather easily by introducing time delay into a population which would otherwise exhibit perfect phase synchrony. We argue that the appearance of the heteroclinic loop induced by the delayed coupling is then characterized by transcritical and saddle-node bifurcations. Slow switching arises when a system with a heteroclinic loop is weakly perturbed. This will be demonstrated with a vector model by applying weak noises. Other types of weak symmetry-breaking perturbations can also cause slow switching.},
Author = {Kori, H and Kuramoto, Y},
Date-Added = {2011-09-27 15:27:52 -0400},
Date-Modified = {2011-09-27 15:28:08 -0400},
Journal = {Phys Rev E Stat Nonlin Soft Matter Phys},
Journal-Full = {Physical review. E, Statistical, nonlinear, and soft matter physics},
Month = {Apr},
Number = {4 Pt 2},
Pages = {046214},
Pmid = {11308937},
Pst = {ppublish},
Title = {Slow switching in globally coupled oscillators: robustness and occurrence through delayed coupling},
Volume = {63},
Year = {2001}}
@article{UrbanErmentrout:2011:PRE,
Abstract = {Neuronal networks exhibit a variety of complex spatiotemporal patterns that include sequential activity, synchrony, and wavelike dynamics. Inhibition is the primary means through which such patterns are implemented. This behavior is dependent on both the intrinsic dynamics of the individual neurons as well as the connectivity patterns. Many neural circuits consist of networks of smaller subcircuits (motifs) that are coupled together to form the larger system. In this paper, we consider a particularly simple motif, comprising purely inhibitory interactions, which generates sequential periodic dynamics. We first describe the dynamics of the single motif both for general balanced coupling (all cells receive the same number and strength of inputs) and then for a specific class of balanced networks: circulant systems. We couple these motifs together to form larger networks. We use the theory of weak coupling to derive phase models which, themselves, have a certain structure and symmetry. We show that this structure endows the coupled system with the ability to produce arbitrary timing relationships between symmetrically coupled motifs and that the phase relationships are robust over a wide range of frequencies. The theory is applicable to many other systems in biology and physics.},
Author = {Urban, Alexander and Ermentrout, Bard},
Date-Added = {2011-09-27 15:06:09 -0400},
Date-Modified = {2011-09-27 15:06:35 -0400},
Journal = {Phys Rev E Stat Nonlin Soft Matter Phys},
Journal-Full = {Physical review. E, Statistical, nonlinear, and soft matter physics},
Month = {May},
Number = {5 Pt 1},
Pages = {051914},
Pmid = {21728578},
Pst = {ppublish},
Title = {Sequentially firing neurons confer flexible timing in neural pattern generators},
Volume = {83},
Year = {2011}}
@article{De-VriesShermanZhu:1998:BullMathBiol,
Abstract = {The interaction of a pair of weakly coupled biological bursters is examined.Bursting refers to oscillations in which an observable slowly alternates between phases of relative quiescence and rapid oscillatory behavior. The motivation for this work is to understand the role of electrical coupling in promoting the synchronization of bursting electrical activity (BEA) observed in the -cells of the islet of Langerhans, which secrete insulin in response to glucose. By studying the coupled fast subsystem of a model of BEA, we focus on the interaction that occurs during the rapid oscillatory phase. Coupling is weak, diffusive and non-scalar. In addition,non-identical oscillators are permitted. Using perturbation methods with the assumption that the uncoupled oscillators are near a Hopf bifurcation, a reduced system of equations is obtained. A detailed bifurcation study of this reduced system reveals a variety of patterns but suggests that asymmetrically phase-locked solutions are the most typical. Finally, the results are applied to the unreduced full bursting system and used to predict the burst pattern for a pair of cells with a given coupling strength and degree of heterogeneity.},
Author = {De Vries, G and Sherman, A and Zhu, H R},
Date-Added = {2011-09-15 11:02:27 -0400},
Date-Modified = {2011-09-15 11:02:45 -0400},
Journal = {Bull Math Biol},
Journal-Full = {Bulletin of mathematical biology},
Mesh = {Animals; Electromagnetic Phenomena; Humans; Insulin-Secreting Cells; Models, Biological},
Month = {Nov},
Number = {6},
Pages = {1167-200},
Pmid = {20957827},
Pst = {ppublish},
Title = {Diffusively coupled bursters: effects of cell heterogeneity},
Volume = {60},
Year = {1998}}
@article{ErmentroutKopell1984SIAMJMathAnal,
Author = {George Bard Ermentrout and Nancy Kopell},
Coden = {SJMAAH},
Date-Modified = {2011-09-15 10:53:37 -0400},
Doi = {DOI:10.1137/0515019},
Eissn = {10957154},
Issn = {00361410},
Journal = {SIAM J. on Mathematical Analysis},
Number = {2},
Pages = {215-237},
Publisher = {SIAM},
Title = {Frequency Plateaus in a Chain of Weakly Coupled Oscillators, I.},
Url = {http://dx.doi.org/doi/10.1137/0515019},
Volume = {15},
Year = {1984},
Bdsk-Url-1 = {http://dx.doi.org/doi/10.1137/0515019},
Bdsk-Url-2 = {http://dx.doi.org/10.1137/0515019}}
@inbook{KopellErmentrout2002Handbook,
Author = {N. Kopell and G.B. Ermentrout},
Chapter = {1. Mechanisms of Phase-Locking and Frequency Control in Pairs of Coupled Neural Oscillators},
Date-Added = {2011-09-15 10:28:31 -0400},
Date-Modified = {2011-09-15 10:31:48 -0400},
Publisher = {North-Holland},
Title = {Handbook of Dynamical Systems},
Volume = {2},
Year = {2002}}
@article{SpardyEtAlRubin2011b,
Author = {Lucy E. Spardy and Sergey N. Markin and Natalia A. Shevtsova and Boris I. Prilutsky and Ilya A. Rybak and Jonathan E. Rubin},
Date-Added = {2011-09-05 06:12:59 -0400},
Date-Modified = {2011-09-05 06:14:02 -0400},
Journal = {Journal of Neural Engineering},
Month = {Dec},
Number = {6},
Pages = {065004},
Title = {A dynamical systems analysis of afferent control in a neuromechanical model of locomotion. {II. Phase asymmetry}},
Volume = {8},
Year = {2011}}
@article{SpardyEtAlRubin2011a,
Author = {Lucy E. Spardy and Sergey N. Markin and Natalia A. Shevtsova and Boris I. Prilutsky and Ilya A. Rybak and Jonathan E. Rubin},
Date-Added = {2011-09-05 06:10:35 -0400},
Date-Modified = {2011-09-05 06:12:54 -0400},
Journal = {Journal of Neural Engineering},
Month = {Dec},
Number = {6},
Pages = {065003},
Title = {A dynamical systems analysis of afferent control in a neuromechanical model of locomotion. {I. Rhythm generation}},
Volume = {8},
Year = {2011}}
@book{Axler:1997,
Address = {New York, NY},
Author = {Axler, Sheldon J.},
Title = {Linear Algebra Done Right},
Year = {1997}}
@book{Ermentrout2002XPP,
Author = {Bard Ermentrout},
Date-Added = {2011-07-23 00:53:37 -0400},
Date-Modified = {2011-07-23 00:55:58 -0400},
Publisher = {Society for Industrial and Applied Mathematics},
Title = {{Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students}},
Year = {2002}}
@book{CalvettiSomersalo2007,
Author = {Daniela Calvetti and Erkki Somersalo},
Date-Added = {2011-07-23 00:50:20 -0400},
Date-Modified = {2011-07-23 00:53:19 -0400},
Publisher = {Springer},
Title = {{Introduction to Bayesian Scientific Computing: Ten Lectures on Subjective Computing}},
Year = {2007}}
@article{Guckenheimer1975JMathBiol,
Affiliation = {Division of Natural Sciences University of California 95064 Santa Cruz CA USA},
Author = {Guckenheimer, J.},
Date-Added = {2011-07-22 21:55:13 -0400},
Date-Modified = {2011-07-22 21:55:29 -0400},
Issn = {0303-6812},
Issue = {3},
Journal = {Journal of Mathematical Biology},
Keyword = {Biomedical and Life Sciences},
Note = {10.1007/BF01273747},
Pages = {259-273},
Publisher = {Springer Berlin / Heidelberg},
Title = {Isochrons and phaseless sets},
Url = {http://dx.doi.org/10.1007/BF01273747},
Volume = {1},
Year = {1975},
Bdsk-Url-1 = {http://dx.doi.org/10.1007/BF01273747}}
@inproceedings{brent1973,
Author = {RICHARD P. BRENT},
Booktitle = {International Conference on Computational Linguistics},
Masid = {1375906},
Title = {ALGORITHMS FOR MINIMIZATION WITHOUT DERIVATIVES},
Year = {1973}}
@book{BerglundGentz2006,
Address = {London},
Author = {Nils Berglund and Barbara Gentz},
Date-Added = {2011-07-22 01:03:45 -0400},
Date-Modified = {2011-07-22 01:05:40 -0400},
Publisher = {Springer-Verlag},
Series = {Probability and Its Applications},
Title = {Noise-induced phenomena in slow-fast dynamical systems: a sample-paths approach},
Year = {2006}}
@article{GuckenheimerHolmes1988MathCambridge,
Abstract = { ABSTRACT This paper describes a previously undocumented phenomenon in dynamical systems theory; namely, the occurrence of heteroclinic cycles that are structurally stable within the space of Cr vector fields equivariant with respect to a symmetry group. In the space X(M) of Cr vector fields on a manifold M, there is a residual set of vector fields having no trajectories joining saddle points with stable manifolds of the same dimension. Such heteroclinic connections are a structurally unstable phenomenon [4]. However, in the space XG(M) ⊂ X(M) of vector fields equivariant with respect to a symmetry group G, the situation can be quite different. We give an example of an open set U of topologically equivalent vector fields in the space of vector fields on 3 equivariant with respect to a particular finite subgroup G ⊂ O(3) such that each X U has a heteroclinic cycle that is an attractor. The heteroclinic cycles consist of three equilibrium points and three trajectories joining them. },
Author = {Guckenheimer,John and Holmes,Philip},
Date-Added = {2011-07-16 07:30:07 -0400},
Date-Modified = {2011-07-16 07:30:37 -0400},
Doi = {10.1017/S0305004100064732},
Eprint = {http://journals.cambridge.org/article_S0305004100064732},
Journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
Number = {01},
Pages = {189-192},
Title = {Structurally stable heteroclinic cycles},
Url = {http://dx.doi.org/10.1017/S0305004100064732},
Volume = {103},
Year = {1988},
Bdsk-Url-1 = {http://dx.doi.org/10.1017/S0305004100064732}}
@article{Keener:2009:JMathBiol,
Abstract = {We show that many Markov models of ion channel kinetics have globally attracting stable invariant manifolds, even when the Markov process is time dependent. The primary implication of this is that, since the dimension of the invariant manifold is often substantially smaller than the full master equation system, simulations of ion channel kinetics can be substantially simplified, with no approximation. We show that this applies to certain models of potassium channels, sodium channels, ryanodine receptors and IP(3) receptors. We also use this to show that the original Hodgkin-Huxley formulations of potassium channel conductance and sodium channel conductance are the exact solutions of full Markov models for these channels.},
Author = {Keener, James P},
Date-Added = {2011-07-11 09:06:33 -0400},
Date-Modified = {2011-07-11 09:06:59 -0400},
Doi = {10.1007/s00285-008-0199-6},
Journal = {J Math Biol},
Journal-Full = {Journal of mathematical biology},
Mesh = {Computer Simulation; Ion Channels; Kinetics; Markov Chains; Models, Biological; Protein Conformation; Stochastic Processes},
Month = {Mar},
Number = {3},
Pages = {447-57},
Pmid = {18592240},
Pst = {ppublish},
Title = {Invariant manifold reductions for Markovian ion channel dynamics},
Volume = {58},
Year = {2009},
Bdsk-Url-1 = {http://dx.doi.org/10.1007/s00285-008-0199-6}}
@article{Keener:2010:JMathBiol,
Abstract = {We show that the cooperative model for the kinetics of a tetrameric potassium ion channel derived in Nekouzadeh et al. (Biophys J 95(7):3510-3520, 2008) is an invariant manifold reduction of the full master equation for the channel kinetics. We further establish the validity of this reduction for ion channel models consisting of multiple independent subunits with cooperative transitions from a single permissive state to a conducting state. Finally, we conclude that solutions of the reduced model are globally asymptotically stable solutions of the full master equation system.},
Author = {Keener, James P},
Date-Added = {2011-07-11 09:06:25 -0400},
Date-Modified = {2011-07-11 09:07:27 -0400},
Doi = {10.1007/s00285-009-0271-x},
Journal = {J Math Biol},
Journal-Full = {Journal of mathematical biology},
Mesh = {Kinetics; Markov Chains; Models, Biological; Potassium Channels},
Month = {Apr},
Number = {4},
Pages = {473-9},
Pmid = {19381633},
Pst = {ppublish},
Title = {Exact reductions of Markovian dynamics for ion channels with a single permissive state},
Volume = {60},
Year = {2010},
Bdsk-Url-1 = {http://dx.doi.org/10.1007/s00285-009-0271-x}}
@article{EarnshawKeener2010SIADS,
Author = {Berton A. Earnshaw and James P. Keener},
Date-Added = {2011-07-11 08:56:31 -0400},
Date-Modified = {2011-07-11 09:00:05 -0400},
Journal = {Siam J. Applied Dynamical Systems},
Month = {3 June},
Number = {2},
Pages = {568-588},
Title = {Invariant Manifolds of Binomial-Like Nonautonomous Master Equations},
Volume = {9},
Year = {2010}}
@book{Gard1988,
Address = {New York, NY},
Author = {Thomas C. Gard},
Date-Added = {2011-07-06 19:44:24 -0400},
Date-Modified = {2011-07-06 19:46:21 -0400},
Number = {114},
Publisher = {Marcel Dekker, Inc.},
Series = {Pure and Applied Mathematics},
Title = {Introduction to Stochastic Differential Equations},
Year = {1988}}
@book{Winfree1980,
Author = {A.T. Winfree},
Date-Added = {2011-07-04 22:54:16 -0400},
Date-Modified = {2011-07-04 22:56:37 -0400},
Publisher = {Springer-Verlag},
Title = {The Geometry of Biological Time},
Year = {1980}}