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Sloan Foundation","doi-asserted-by":"publisher","award":["FG-2017-9554"],"award-info":[{"award-number":["FG-2017-9554"]}],"id":[{"id":"10.13039\/100000879","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2113213"],"award-info":[{"award-number":["DMS-2113213"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.ploscompbiol.org"],"crossmark-restriction":false},"short-container-title":["PLoS Comput Biol"],"abstract":"<jats:p>Characterizing metastable neural dynamics in finite-size spiking networks remains a daunting challenge. We propose to address this challenge in the recently introduced replica-mean-field (RMF) limit. In this limit, networks are made of infinitely many replicas of the finite network of interest, but with randomized interactions across replicas. Such randomization renders certain excitatory networks fully tractable at the cost of neglecting activity correlations, but with explicit dependence on the finite size of the neural constituents. However, metastable dynamics typically unfold in networks with mixed inhibition and excitation. Here, we extend the RMF computational framework to point-process-based neural network models with exponential stochastic intensities, allowing for mixed excitation and inhibition. Within this setting, we show that metastable finite-size networks admit multistable RMF limits, which are fully characterized by stationary firing rates. Technically, these stationary rates are determined as the solutions of a set of delayed differential equations under certain regularity conditions that any physical solutions shall satisfy. We solve this original problem by combining the resolvent formalism and singular-perturbation theory. Importantly, we find that these rates specify probabilistic pseudo-equilibria which accurately capture the neural variability observed in the original finite-size network. We also discuss the emergence of metastability as a stochastic bifurcation, which can be interpreted as a static phase transition in the RMF limits. In turn, we expect to leverage the static picture of RMF limits to infer purely dynamical features of metastable finite-size networks, such as the transition rates between pseudo-equilibria.<\/jats:p>","DOI":"10.1371\/journal.pcbi.1010215","type":"journal-article","created":{"date-parts":[[2022,6,17]],"date-time":"2022-06-17T17:38:07Z","timestamp":1655487487000},"page":"e1010215","update-policy":"https:\/\/doi.org\/10.1371\/journal.pcbi.corrections_policy","source":"Crossref","is-referenced-by-count":10,"title":["Metastable spiking networks in the replica-mean-field limit"],"prefix":"10.1371","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0953-6939","authenticated-orcid":true,"given":"Luyan","family":"Yu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3538-6882","authenticated-orcid":true,"given":"Thibaud 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Sch\u00f6lvinck","year":"2010","journal-title":"Proceedings of the National Academy of Sciences"},{"issue":"7459","key":"pcbi.1010215.ref004","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1038\/nature12276","article-title":"Attention enhances synaptic efficacy and the signal-to-noise ratio in neural circuits","volume":"499","author":"F Briggs","year":"2013","journal-title":"Nature"},{"issue":"3764","key":"pcbi.1010215.ref005","doi-asserted-by":"crossref","first-page":"842","DOI":"10.1126\/science.155.3764.842","article-title":"Synaptic noise as a source of variability in the interval between action potentials","volume":"155","author":"WH Calvin","year":"1967","journal-title":"Science"},{"issue":"1","key":"pcbi.1010215.ref006","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1146\/annurev.bb.16.060187.002323","article-title":"Physical limits to sensation and perception","volume":"16","author":"W Bialek","year":"1987","journal-title":"Annual review of biophysics and biophysical chemistry"},{"issue":"5","key":"pcbi.1010215.ref007","doi-asserted-by":"crossref","first-page":"e1002059","DOI":"10.1371\/journal.pcbi.1002059","article-title":"How structure determines correlations in neuronal networks","volume":"7","author":"V Pernice","year":"2011","journal-title":"PLoS computational biology"},{"key":"pcbi.1010215.ref008","volume-title":"Cortex: statistics and geometry of neuronal connectivity","author":"V Braitenberg","year":"2013"},{"issue":"7539","key":"pcbi.1010215.ref009","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1038\/nature14182","article-title":"Functional organization of excitatory synaptic strength in primary visual cortex","volume":"518","author":"L Cossell","year":"2015","journal-title":"Nature"},{"issue":"3","key":"pcbi.1010215.ref010","doi-asserted-by":"crossref","first-page":"e68","DOI":"10.1371\/journal.pbio.0030068","article-title":"Highly nonrandom features of synaptic connectivity in local cortical circuits","volume":"3","author":"S Song","year":"2005","journal-title":"PLoS biology"},{"issue":"4","key":"pcbi.1010215.ref011","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1038\/nrn3687","article-title":"The log-dynamic brain: how skewed distributions affect network operations","volume":"15","author":"G Buzs\u00e1ki","year":"2014","journal-title":"Nature Reviews Neuroscience"},{"issue":"2","key":"pcbi.1010215.ref012","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1093\/cercor\/bhs006","article-title":"Interpyramid spike transmission stabilizes the sparseness of recurrent network activity","volume":"23","author":"Y Ikegaya","year":"2013","journal-title":"Cerebral Cortex"},{"issue":"6","key":"pcbi.1010215.ref013","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1038\/nrn2148","article-title":"The action potential in mammalian central neurons","volume":"8","author":"BP Bean","year":"2007","journal-title":"Nature Reviews Neuroscience"},{"issue":"5","key":"pcbi.1010215.ref014","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1038\/nrn1668","article-title":"Neuronal variability: noise or part of the signal?","volume":"6","author":"RB Stein","year":"2005","journal-title":"Nature Reviews Neuroscience"},{"issue":"4","key":"pcbi.1010215.ref015","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1038\/nrn2258","article-title":"Noise in the nervous system","volume":"9","author":"AA Faisal","year":"2008","journal-title":"Nature reviews neuroscience"},{"key":"pcbi.1010215.ref016","article-title":"Bring the noise: Reconceptualizing spontaneous neural activity","author":"LQ Uddin","year":"2020","journal-title":"Trends in Cognitive Sciences"},{"issue":"19","key":"pcbi.1010215.ref017","doi-asserted-by":"crossref","first-page":"8616","DOI":"10.1073\/pnas.92.19.8616","article-title":"Cortical activity flips among quasi-stationary states","volume":"92","author":"M Abeles","year":"1995","journal-title":"Proceedings of the National Academy of Sciences"},{"issue":"1","key":"pcbi.1010215.ref018","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.neuron.2013.12.022","article-title":"The metastable brain","volume":"81","author":"E Tognoli","year":"2014","journal-title":"Neuron"},{"issue":"1","key":"pcbi.1010215.ref019","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00422-006-0068-6","article-title":"A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input","volume":"95","author":"AN Burkitt","year":"2006","journal-title":"Biological cybernetics"},{"issue":"4","key":"pcbi.1010215.ref020","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/BF00339367","article-title":"Homogeneous nets of neuron-like elements","volume":"17","author":"SI Amari","year":"1975","journal-title":"Biological cybernetics"},{"key":"pcbi.1010215.ref021","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3389\/neuro.10.001.2009","article-title":"A constructive mean-field analysis of multi population neural networks with random synaptic weights and stochastic inputs","volume":"3","author":"OD Faugeras","year":"2009","journal-title":"Frontiers in computational neuroscience"},{"issue":"1","key":"pcbi.1010215.ref022","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1137\/110832392","article-title":"Noise-induced behaviors in neural mean field dynamics","volume":"11","author":"J Touboul","year":"2012","journal-title":"SIAM Journal on Applied Dynamical Systems"},{"issue":"4","key":"pcbi.1010215.ref023","doi-asserted-by":"crossref","first-page":"1756","DOI":"10.1137\/19M1243877","article-title":"Replica-mean-field limits for intensity-based neural networks","volume":"18","author":"F Baccelli","year":"2019","journal-title":"SIAM Journal on Applied Dynamical Systems"},{"issue":"11-12","key":"pcbi.1010215.ref024","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1016\/j.peva.2008.03.005","article-title":"A class of mean field interaction models for computer and communication systems","volume":"65","author":"M Benaim","year":"2008","journal-title":"Performance evaluation"},{"issue":"10","key":"pcbi.1010215.ref025","first-page":"4950","article-title":"An analytical framework for coverage in cellular networks leveraging vehicles","volume":"66","author":"CS Choi","year":"2018","journal-title":"IEEE Transactions on Communications"},{"issue":"17","key":"pcbi.1010215.ref026","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1051\/jphyslet:019850046017077100","article-title":"Replicas and optimization","volume":"46","author":"M M\u00e9zard","year":"1985","journal-title":"Journal de Physique Lettres"},{"issue":"29","key":"pcbi.1010215.ref027","first-page":"6655","article-title":"Statistical physics of structural glasses","volume":"12","author":"M M\u00e9zard","year":"2000","journal-title":"Journal of Physics: Condensed Matter"},{"key":"pcbi.1010215.ref028","unstructured":"Baccelli F, Davydov M, Taillefumier T. Replica-Mean-Field Limits of Fragmentation-Interaction-Aggregation Processes. arXiv preprint arXiv:200507962. 2020."},{"issue":"2","key":"pcbi.1010215.ref029","doi-asserted-by":"crossref","first-page":"e1005390","DOI":"10.1371\/journal.pcbi.1005390","article-title":"On the stability and dynamics of stochastic spiking neuron models: Nonlinear Hawkes process and point process GLMs","volume":"13","author":"F Gerhard","year":"2017","journal-title":"PLoS computational biology"},{"key":"pcbi.1010215.ref030","doi-asserted-by":"crossref","unstructured":"Labarre D, Meissner W, Boraud T. Measure of the regularity of events in stochastic point processes, application to neuron activity analysis. In: 2008 IEEE International Conference on Acoustics, Speech and Signal Processing. IEEE; 2008. p. 489\u2013492.","DOI":"10.1109\/ICASSP.2008.4517653"},{"issue":"2","key":"pcbi.1010215.ref031","doi-asserted-by":"crossref","first-page":"1074","DOI":"10.1152\/jn.00697.2004","article-title":"A point process framework for relating neural spiking activity to spiking history, neural ensemble, and extrinsic covariate effects","volume":"93","author":"W Truccolo","year":"2005","journal-title":"Journal of neurophysiology"},{"key":"pcbi.1010215.ref032","volume-title":"Monte carlo methods","author":"J Hammersley","year":"2013"},{"key":"pcbi.1010215.ref033","unstructured":"Matthes K. Zur Theorie der Bedienungsprozesse. In: Trans. Third Prague Conf. Information Theory, Statist. Decision Functions, Random Processes (Liblice, 1962). Publ. House Czech. Acad. Sci., Prague; 1964. p. 513\u2013528."},{"issue":"12","key":"pcbi.1010215.ref034","doi-asserted-by":"crossref","first-page":"3145","DOI":"10.1162\/NECO_a_00346","article-title":"Exact Event-Driven Implementation for Recurrent Networks of Stochastic Perfect Integrate-and-Fire Neurons","volume":"24","author":"T Taillefumier","year":"2012","journal-title":"Neural Computation"},{"key":"pcbi.1010215.ref035","doi-asserted-by":"crossref","first-page":"051903","DOI":"10.1103\/PhysRevE.82.051903","article-title":"Metastable states and quasicycles in a stochastic Wilson-Cowan model of neuronal population dynamics","volume":"82","author":"PC Bressloff","year":"2010","journal-title":"Phys Rev E"},{"issue":"3","key":"pcbi.1010215.ref036","doi-asserted-by":"crossref","first-page":"1125","DOI":"10.1152\/jn.00116.2007","article-title":"Noise-induced alternations in an attractor network model of perceptual bistability","volume":"98","author":"R Moreno-Bote","year":"2007","journal-title":"Journal of neurophysiology"},{"issue":"7599","key":"pcbi.1010215.ref037","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1038\/nature17192","article-title":"Anatomy and function of an excitatory network in the visual cortex","volume":"532","author":"WCA Lee","year":"2016","journal-title":"Nature"},{"key":"pcbi.1010215.ref038","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BFb0095237","volume-title":"Dynamical systems","author":"L Arnold","year":"1995"},{"key":"pcbi.1010215.ref039","volume-title":"Delay differential equations","author":"Y Kuang","year":"2012"},{"issue":"1","key":"pcbi.1010215.ref040","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/0377-0427(89)90071-X","article-title":"Stability of numerical methods for delay differential equations","volume":"25","author":"L Torelli","year":"1989","journal-title":"Journal of Computational and Applied Mathematics"},{"key":"pcbi.1010215.ref041","volume-title":"Ordinary and delay differential equations","author":"RD Driver","year":"2012"},{"issue":"2","key":"pcbi.1010215.ref042","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1115\/1.1568121","article-title":"Analysis of a system of linear delay differential equations","volume":"125","author":"FM Asl","year":"2003","journal-title":"J Dyn Sys, Meas, Control"},{"issue":"5","key":"pcbi.1010215.ref043","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1063\/1.1703667","article-title":"Liouville Equation and the Resolvent Formalism","volume":"1","author":"S Teitler","year":"1960","journal-title":"Journal of Mathematical Physics"},{"issue":"5","key":"pcbi.1010215.ref044","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1070\/RM1975v030n05ABEH001522","article-title":"Asymptotic behaviour of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equations","volume":"30","author":"IM Gel\u2019fand","year":"1975","journal-title":"Russian Mathematical Surveys"},{"issue":"9","key":"pcbi.1010215.ref045","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1007\/BF02781012","article-title":"Physical applications of the resolvent operators on the mathematical formalism of Feynman\u2019s theory of the positron","volume":"8","author":"M Sch\u00f6nberg","year":"1951","journal-title":"Il Nuovo Cimento (1943-1954)"},{"key":"pcbi.1010215.ref046","unstructured":"Van Assche W. Pad\u00e9 and Hermite-Pad\u00e9 approximation and orthogonality. arXiv preprint math\/0609094. 2006."},{"key":"pcbi.1010215.ref047","volume-title":"Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory","author":"CM Bender","year":"2013"},{"issue":"3","key":"pcbi.1010215.ref048","doi-asserted-by":"crossref","first-page":"679","DOI":"10.17323\/1609-4514-2005-5-3-679-704","article-title":"Poisson hypothesis for information networks. I","volume":"5","author":"AN Rybko","year":"2005","journal-title":"Moscow mathematical journal"},{"issue":"4","key":"pcbi.1010215.ref049","doi-asserted-by":"crossref","first-page":"927","DOI":"10.17323\/1609-4514-2005-5-4-927-959","article-title":"Poisson hypothesis for information networks. 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Stochastic processes. SIAM; 1999.","DOI":"10.1137\/1.9781611971125"},{"key":"pcbi.1010215.ref055","volume-title":"Probability and its Applications","author":"DJ Daley","year":"2003"},{"key":"pcbi.1010215.ref056","volume-title":"An introduction to the theory of point processes: volume II: general theory and structure","author":"DJ Daley","year":"2007"},{"issue":"1","key":"pcbi.1010215.ref057","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1093\/biomet\/58.1.83","article-title":"Spectra of some self-exciting and mutually exciting point processes","volume":"58","author":"AG Hawkes","year":"1971","journal-title":"Biometrika"},{"issue":"3","key":"pcbi.1010215.ref058","doi-asserted-by":"crossref","first-page":"493","DOI":"10.2307\/3212693","article-title":"A cluster process representation of a self-exciting process","volume":"11","author":"AG Hawkes","year":"1974","journal-title":"Journal of Applied Probability"},{"key":"pcbi.1010215.ref059","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511815706","volume-title":"Spiking neuron models: Single neurons, populations, plasticity","author":"W Gerstner","year":"2002"},{"issue":"4","key":"pcbi.1010215.ref060","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1167\/6.4.13","article-title":"Spike-triggered neural characterization","volume":"6","author":"O Schwartz","year":"2006","journal-title":"Journal of vision"},{"issue":"2","key":"pcbi.1010215.ref061","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1080\/713663221","article-title":"A simple white noise analysis of neuronal light responses","volume":"12","author":"E Chichilnisky","year":"2001","journal-title":"Network: computation in neural systems"},{"issue":"4","key":"pcbi.1010215.ref062","doi-asserted-by":"crossref","first-page":"866","DOI":"10.1007\/s10955-014-1145-1","article-title":"Hydrodynamic limit for interacting neurons","volume":"158","author":"A De Masi","year":"2015","journal-title":"Journal of Statistical Physics"},{"issue":"4-6","key":"pcbi.1010215.ref063","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/S0375-9601(96)00878-X","article-title":"Stochastic resonance in a model neuron with reset","volume":"225","author":"HE Plesser","year":"1997","journal-title":"Physics Letters A"},{"issue":"5","key":"pcbi.1010215.ref064","doi-asserted-by":"crossref","first-page":"896","DOI":"10.1007\/s10955-013-0733-9","article-title":"Infinite systems of interacting chains with memory of variable length\u2014a stochastic model for biological neural nets","volume":"151","author":"A Galves","year":"2013","journal-title":"Journal of Statistical Physics"},{"issue":"3","key":"pcbi.1010215.ref065","doi-asserted-by":"crossref","first-page":"518","DOI":"10.2307\/1427522","article-title":"Stability of Markovian processes III: Foster\u2013Lyapunov criteria for continuous-time processes","volume":"25","author":"SP Meyn","year":"1993","journal-title":"Advances in Applied Probability"},{"key":"pcbi.1010215.ref066","doi-asserted-by":"crossref","first-page":"408","DOI":"10.2307\/3213783","article-title":"The intensity conservation law for queues with randomly changed service rate","author":"M Miyazawa","year":"1985","journal-title":"Journal of applied probability"},{"key":"pcbi.1010215.ref067","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/978-3-662-11657-9","volume-title":"Elements of Queueing Theory","author":"F Baccelli","year":"2003"},{"key":"pcbi.1010215.ref068","volume-title":"Perturbation theory for linear operators","author":"T Kato","year":"2013"},{"issue":"2","key":"pcbi.1010215.ref069","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1016\/j.jmaa.2005.05.009","article-title":"An iterative method for solving nonlinear functional equations","volume":"316","author":"V Daftardar-Gejji","year":"2006","journal-title":"Journal of Mathematical Analysis and Applications"},{"key":"pcbi.1010215.ref070","volume-title":"Principles of multiscale modeling","author":"E Weinan","year":"2011"},{"key":"pcbi.1010215.ref071","volume-title":"Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory","author":"CM Bender","year":"2013"},{"issue":"5293","key":"pcbi.1010215.ref072","doi-asserted-by":"crossref","first-page":"1724","DOI":"10.1126\/science.274.5293.1724","article-title":"Chaos in neuronal networks with balanced excitatory and inhibitory activity","volume":"274","author":"C Van Vreeswijk","year":"1996","journal-title":"Science"},{"issue":"5","key":"pcbi.1010215.ref073","doi-asserted-by":"crossref","first-page":"e1000072","DOI":"10.1371\/journal.pcbi.1000072","article-title":"Transient cognitive dynamics, metastability, and decision making","volume":"4","author":"MI Rabinovich","year":"2008","journal-title":"PLoS Comput Biol"},{"key":"pcbi.1010215.ref074","doi-asserted-by":"crossref","unstructured":"Kelso JS, Tognoli E. Metastability in the brain. In: The 2006 IEEE International Joint Conference on Neural Network Proceedings. IEEE; 2006. p. 363\u2013368.","DOI":"10.1109\/IJCNN.2006.246704"},{"issue":"2","key":"pcbi.1010215.ref075","doi-asserted-by":"crossref","first-page":"496","DOI":"10.1016\/j.biosystems.2006.12.001","article-title":"Metastability, criticality and phase transitions in brain and its models","volume":"90","author":"G Werner","year":"2007","journal-title":"Biosystems"},{"issue":"3","key":"pcbi.1010215.ref076","doi-asserted-by":"crossref","first-page":"1394","DOI":"10.1137\/120898978","article-title":"Metastability in a Stochastic Neural Network Modeled as a Velocity Jump Markov Process","volume":"12","author":"P Bressloff","year":"2013","journal-title":"SIAM Journal on Applied Dynamical Systems"},{"issue":"4","key":"pcbi.1010215.ref077","doi-asserted-by":"crossref","first-page":"2441","DOI":"10.1152\/jn.00924.2007","article-title":"Detecting neural-state transitions using hidden Markov models for motor cortical prostheses","volume":"100","author":"C Kemere","year":"2008","journal-title":"Journal of neurophysiology"},{"issue":"1","key":"pcbi.1010215.ref078","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1109\/MASSP.1986.1165342","article-title":"An introduction to hidden Markov models","volume":"3","author":"L Rabiner","year":"1986","journal-title":"ieee assp magazine"},{"key":"pcbi.1010215.ref079","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781107447615","volume-title":"Neuronal dynamics: From single neurons to networks and models of cognition","author":"W Gerstner","year":"2014"},{"key":"pcbi.1010215.ref080","first-page":"1563","article-title":"Stability of nonlinear Hawkes processes","author":"P Br\u00e9maud","year":"1996","journal-title":"The Annals of Probability"},{"issue":"1","key":"pcbi.1010215.ref081","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.neuron.2008.01.036","article-title":"Decoupling through synchrony in neuronal circuits with propagation delays","volume":"58","author":"EV Lubenov","year":"2008","journal-title":"Neuron"},{"issue":"1","key":"pcbi.1010215.ref082","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/0378-5955(85)90110-8","article-title":"Peripheral auditory adaptation and fatigue: a model oriented review","volume":"18","author":"J Eggermont","year":"1985","journal-title":"Hearing research"},{"issue":"1","key":"pcbi.1010215.ref083","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1749-8104-2-7","article-title":"Polarity and intracellular compartmentalization of Drosophila neurons","volume":"2","author":"MM Rolls","year":"2007","journal-title":"Neural development"},{"key":"pcbi.1010215.ref084","volume-title":"An introduction to the mathematics of neurons: modeling in the frequency domain","author":"FC Hoppensteadt","year":"1997"},{"issue":"2","key":"pcbi.1010215.ref085","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1002\/sapm1976552107","article-title":"A theorem on the convergence of Pad\u00e9 approximants","volume":"55","author":"GA Baker","year":"1976","journal-title":"Studies in Applied Mathematics"},{"key":"pcbi.1010215.ref086","volume-title":"Nonlinear oscillations, dynamical systems, and bifurcations of vector fields","author":"J Guckenheimer","year":"2013"},{"key":"pcbi.1010215.ref087","volume-title":"Handbook of stochastic methods","author":"CW Gardiner","year":"1985"},{"key":"pcbi.1010215.ref088","doi-asserted-by":"crossref","unstructured":"Varadhan SS. Large deviations and applications. 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