RT Journal Article
T1 A framework for macroscopic phase-resetting curves for generalised spiking neural networks
A1 Dumont, Grégory
A1 Pérez Cervera, Alberto
A1 Gutkin, Boris
AB Brain rhythms emerge from synchronization among interconnected spiking neurons. Key properties of such rhythms can be gleaned from the phase-resetting curve (PRC). Inferring the PRC and developing a systematic phase reduction theory for large-scale brain rhythms remains an outstanding challenge. Here we present a theoretical framework and methodology to compute the PRC of generic spiking networks with emergent collective oscillations. We adopt a renewal approach where neurons are described by the time since their last action potential, a description that can reproduce the dynamical feature of many cell types. For a sufficiently large number of neurons, the network dynamics are well captured by a continuity equation known as the refractory density equation. We develop an adjoint method for this equation giving a semi-analytical expression of the infinitesimal PRC. We confirm the validity of our framework for specific examples of neural networks. Our theoretical framework can link key biological properties at the individual neuron scale and the macroscopic oscillatory network properties. Beyond spiking networks, the approach is applicable to a broad class of systems that can be described by renewal processes.
PB Public Library of Science
SN 1553-734X
YR 2022
FD 2022-08-01
LK https://hdl.handle.net/20.500.14352/73039
UL https://hdl.handle.net/20.500.14352/73039
LA eng
NO 1.Winfree A. The Geometry of Biological Time. Springer-Verlag, London, 1980.2.Ashwin P., Coombes S., and Nicks R. Mathematical frameworks for oscillatory network dynamics in neuroscience. The Journal of Mathematical Neuroscience, 6(1):2, 2016. pmid:267391333.Nakao H. Phase reduction approach to synchronisation of nonlinear oscillators. Contemporary Physics, 57(2):188–214, 2016.4.Achuthan S. and Canavier C. C. Phase-resetting curves determine synchronization, phase locking, and clustering in networks of neural oscillators. Journal of Neuroscience, 29(16):5218–5233, 2009. pmid:193869185.Dumont G. and Gutkin B. Macroscopic phase resetting-curves determine oscillatory coherence and signal transfer in inter-coupled neural circuits. PLOS Computational Biology, 15(5):1–34, 05 2019. pmid:310710856.Kirst C., Timme M., and Battaglia D. Dynamic information routing in complex networks. Nature Communications, 7:11061 EP–, 04 2016. pmid:270672577.Stiefel K. M. and Ermentrout G. B. Neurons as oscillators. Journal of Neurophysiology, 2016. pmid:276838878.Brown S. A., Kunz D., Dumas A., Westermark P. O., Vanselow K., Tilmann-Wahnschaffe A., Herzel H., and Kramer A. Molecular insights into human daily behavior. Proceedings of the National Academy of Sciences, 105(5):1602–1607, 2008. pmid:182275139.Brown E., Moehlis J., and Holmes P. On the phase reduction and response dynamics of neural oscillator populations. Neural Computation, 16(4):673–715, 2016/10/31 2004. pmid:1502582610.Buzsaki G. Rhythms of the Brain. Oxford University Press, 2006.11.Kopell N., Kramer M., Malerba P., and Whittington M. Are different rhythms good for different functions? Frontiers in Human Neuroscience, 4:187, 2010. pmid:2110301912.Akao A., Ogawa Y., Jimbo Y., Ermentrout G. B., and Kotani K. Relationship between the mechanisms of gamma rhythm generation and the magnitude of the macroscopic phase response function in a population of excitatory and inhibitory modified quadratic integrate-and-fire neurons. Phys. Rev. E, 97:012209, Jan 2018. pmid:2944839113.Dumont G., Ermentrout G. B., and Gutkin B. Macroscopic phase-resetting curves for spiking neural networks. Phys. Rev. E, 96:042311, Oct 2017. pmid:2934756614.Canavier C. C. Phase-resetting as a tool of information transmission. Current Opinion in Neurobiology, 31:206–213., 2015. SI: Brain rhythms and dynamic coordination. pmid:2552900315.Sharpe F. and Lotka A. A problem in age-distribution. Philosophical Magazine, 6:435–438, 1911.16.McKendrick A. Applications of mathematics to medical problems. Proc. Edinburgh Math. Soc, 44(98-130), 1926.17.Castillo-Chavez C., Hethcote H. W., Andreasen V., Levin S. A., and Liu W. M. Epidemiological models with age structure, proportionate mixing, and cross-immunity. Journal of Mathematical Biology, 27(3):233–258, 1989. pmid:274614018.Franceschetti A. and Pugliese A. Threshold behaviour of a sir epidemic model with age structure and immigration. Journal of Mathematical Biology, 57(1):1–27, 2008. pmid:1798513119.Keyfitz B. and Keyfitz N. The mckendrick partial differential equation and its uses in epidemiology and population study. Mathematical and Computer Modelling, 26(6):1–9, 1997.20.Billy F., Clairambault J., Delaunay F., Feillet C., and Robert N. Age-structured cell population model to study the influence of growth factors on cell cycle dynamics. Math Biosci Eng., 10(1):1–17, 2013. pmid:2331135921.Billy F., Clairambaultt J., Fercoq O., Gaubertt S., Lepoutre T., and Ouillon T. Synchronisation and control of proliferation in cycling cell population models with age structure. Mathematics and Computers in Simulation, 96:66–94., 2014. Differential and Integral Equations with Applications in Biology and Medicine.22.Gabriel P., Garbett S. P., Quaranta V., Tyson D. R., and Webb G. F. The contribution of age structure to cell population responses to targeted therapeutics. Journal of Theoretical Biology, 311:19–27, 2012. pmid:2279633023.Betti M. I., Wahl L. M., and Zamir M. Age structure is critical to the population dynamics and survival of honeybee colonies. Royal Society Open Science, 3(11), 2016. pmid:2801862724.Betti M. I., Wahl L. M., and Zamir M. Reproduction number and asymptotic stability for the dynamics of a honey bee colony with continuous age structure. Bulletin of Mathematical Biology, Jun 2017. pmid:2863110825.Foerster H. V. Some remarks on changing populations. Kinetics of Cellular Proliferation, pages 382–399, 1959.26.Britton N. Essential Mathematical Biology. Springer-Verlag, London, 2003.27.Murray J. D. Mathematical Biology: an introduction. Interdisciplinary Applied Mathematics. Mathematical Biology, 2002.28.Perthame B. Transport equation in biology. Birkhauser Verlag, Basel, 2007.29.Gerstner W. and van Hemmen J. L. Associative memory in a network of spiking neurons. Network: Computation in Neural Systems, 3(2):139–164, 1992.30.Chevallier J., Caceres M. J., Doumic M., and Reynaud-Bouret P. Microscopic approach of a time elapsed neural model. Mathematical Models and Methods in Applied Sciences, 25(14):2669–2719, 2015.31.Pakdaman K., Perthame B., and Salort D. Dynamics of a structured neuron population. Nonlinearity, 23(1):55, 2010.32.Gerstner W. Population dynamics of spiking neurons: Fast transients, asynchronous states, and locking. Neural Computation, 12(1):43–89, 2000. pmid:1063693333.Deger M., Helias M., Cardanobile S., Atay F. M., and Rotter S. Nonequilibrium dynamics of stochastic point processes with refractoriness. Phys. Rev. E, 82:021129, Aug 2010. pmid:2086679734.Pietras B., Gallice N., and Schwalger T. Low-dimensional firing-rate dynamics for populations of renewal-type spiking neurons. Phys. Rev. E, 102:022407, Aug 2020. pmid:3294245035.Deger M., Schwalger T., Naud R., and Gerstner W. Fluctuations and information filtering in coupled populations of spiking neurons with adaptation. Phys. Rev. E, 90:062704, Dec 2014. pmid:2561512636.Dumont G., Payeur A., and Longtin A. A stochastic-field description of finite-size spiking neural networks. PLOS Computational Biology, 13(8):1–34, 08 2017. pmid:2878744737.Meyer C. and Vreeswijk C. v. Temporal correlations in stochastic networks of spiking neurons. Neural Computation, 14(2):369–404, 2002. pmid:1180291738.Schwalger T., Deger M., and Gerstner W. Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size. PLOS Computational Biology, 13(4):1–63, 04 2017. pmid:2842295739.Gerstner W., Kistler W. M., Naud R., and Paninski L. Neuronal dynamics: From single neurons to networks and models of cognition. Cambridge University Press, Cambridge, 2014.40.Dumont G., Henry J., and Tarniceriu C. O. Noisy threshold in neuronal models: connections with the noisy leaky integrate-and-fire model. Journal of mathematical biology, 73(6-7):1413–1436, 2016. pmid:2704097041.Dumont G., Henry J., and Tarniceriu C. O. Theoretical connections between mathematical neuronal models corresponding to different expressions of noise. Journal of Theoretical Biology, 406:31–41, 2016. pmid:2733454742.Dumont G., Henry J., and Tarniceriu C. O. A theoretical connection between the noisy leaky integrate-and-fire and the escape rate models: The non-autonomous case. Math. Model. Nat. Phenom., 15:59, 2020.43.Chizhov A. V. Conductance-based refractory density model of primary visual cortex. Journal of Computational Neuroscience, 36(2):297–319, Apr 2014. pmid:2388831344.Chizhov A. V. and Graham L. J. Population model of hippocampal pyramidal neurons, linking a refractory density approach to conductance-based neurons. Phys. Rev. E, 75:011924, Jan 2007. pmid:1735820145.Chizhov A. V. and Graham L. J. Efficient evaluation of neuron populations receiving colored-noise current based on a refractory density method. Phys. Rev. E, 77:011910, Jan 2008. pmid:1835187946.Weber A. I. and Pillow J. W. Capturing the Dynamical Repertoire of Single Neurons with Generalized Linear Models. Neural Computation, 29(12):3260–3289, 12 2017. pmid:2895702047.Naud R. and Gerstner W. Coding and decoding with adapting neurons: A population approach to the peri-stimulus time histogram. PLOS Computational Biology, 8(10):1–14, 10 2012.48.Schwalger T. and Chizhov A. V. Mind the last spike — firing rate models for mesoscopic populations of spiking neurons. Current Opinion in Neurobiology, 58:155–166., 2019. Computational Neuroscience. pmid:3159000349.Dumont G. and Henry J. Synchronization of an Excitatory Integrate-and-Fire Neural Network. Bulletin of Mathematical Biology, 75: 629–648, 2013. pmid:2343564550.Gutkin B., Ermentrout G. B., and Reyes A. D. Phase response curves give the responses of neurons to transient inputs Journal of Neurophysiology, 94: 1623–1635, 2005. pmid:1582959551.Vreeswijk C. v., Abbott L and Ermentrout G. B. When inhibition not excitation synchronizes neural firing Journal of Computational Neuroscience, 1:313–321, 1994. pmid:879223752.Chizhov A. V. Conductance-based refractory density approach: comparison with experimental data and generalization to lognormal distribution of input current. JBiological Cybernetics, 111:353364, Aug 2017. pmid:2881969053.Akam T., Oren I., Mantoan L., Ferenczi E., and Kullmann D. M. Oscillatory dynamics in the hippocampus support dentate gyrus-ca3 coupling. Nat Neurosci, 15(5):763–768, 05 2012. pmid:2246650554.Gerhard F., Deger M., and Truccolo W. On the stability and dynamics of stochastic spiking neuron models: Nonlinear hawkes process and point process glms. PLOS Computational Biology, 13(2):1–31, 02 2017. pmid:2823489955.Ermentrout G. B., Galan R. F., and Urban N. N. Relating neural dynamics to neural coding. Phys. Rev. Lett., 99:248103, Dec 2007. pmid:1823349456.Ermentrout G. B. and Terman D. Mathematical foundations of neuroscience. Springer, 2010.57.Pérez-Cervera A., Ashwin P., Huguet G., Seara T. M., and Rankin J. The uncoupling limit of identical hopf bifurcations with an application to perceptual bistability. The Journal of Mathematical Neuroscience, 9(1):1–33, 2019. pmid:3138515058.Pérez-Cervera A., Seara T. M., and Huguet G. Phase-locked states in oscillating neural networks and their role in neural communication. Communications in Nonlinear Science and Numerical Simulation, 80:104992, 2020.59.Pérez-Cervera A., and Hlinka J. Perturbations both trigger and delay seizures due to generic properties of slow-fast relaxation oscillators. PLoS Computational Biology, 17.3 (2021): e1008521. pmid:3378043760.Pariz A., Fischer I., Valizadeh A. and Mirasso C. Transmission delays and frequency detuning can regulate information flow between brain regions. PLOS Computational Biology, 17.4 (2021): e1008129. pmid:3385713561.Reyner-Parra D. and Huguet G. Phase-locking patterns underlying effective communication in exact firing rate models of neural networks. PLOS Computational Biology, 18.5 (2022): e1009342. pmid:3558414762.Torben-Nielsen B., Uusisaari M. and Stiefel K. M. NA comparison of methods to determine neuronal phase-response curves. Frontiers in Neuroinformatics, 2010. pmid:20431724
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RD 3 may 2024