Abstract
This paper studies a nonlinear dynamical phenomenon called the multiple firing event (MFE) in a spatially heterogeneous stochastic neural field model, which is extended from that in our previous paper (Li et al. in J Math Biol 78:83–115, 2018). MFEs are a partially synchronized spiking barrages that are believed to be responsible for the Gamma oscillation. Rigorous results about the stochastic stability and the law of large numbers are proved, which further imply the well-definedness and computability of many quantities related to MFEs. Then we devote to study spatial and temporal properties of MFEs. Our key finding is that MFEs are spatially correlated but the spatial correlation decays quickly. Detailed mathematical justifications are made based on our qualitative models that aim to demonstrate the mechanism of MFEs.
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Athreya Krishna B, Ney Peter E, Ney PE (2004) Branching processes. Courier Corporation
Beggs JM, Plenz D (2003) Neuronal avalanches in neocortical circuits. J Neurosci 23(35):11167–11177
Börgers C, Epstein S, Kopell NJ (2005) Background gamma rhythmicity and attention in cortical local circuits: a computational study. Proc Natl Acad Sci USA 102(19):7002–7007
Börgers C, Kopell N (2003) Synchronization in networks of excitatory and inhibitory neurons with sparse, random connectivity. Neural Comput 15(3):509–538
Börgers C, Kopell N (2005) Effects of noisy drive on rhythms in networks of excitatory and inhibitory neurons. Neural Comput 17(3):557–608
Cai D, Tao L, Shelley M, McLaughlin DW (2004) An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex. Proc Natl Acad Sci USA 101(20):7757–7762
Cai D, Tao L, Rangan AV, McLaughlin DW et al (2006) Kinetic theory for neuronal network dynamics. Commun Math Sci 4(1):97–127
Chariker L, Young L-S (2015) Emergent spike patterns in neuronal populations. J Comput Neurosci 38(1):203–220
Churchland MM, Yu Byron M, Cunningham JP, Sugrue LP, Cohen MR, Corrado GS, Newsome WT, Clark AM, Hosseini P, Scott BB et al (2010) Stimulus onset quenches neural variability: a widespread cortical phenomenon. Nature Neurosci 13(3):369
Goddard C, Sridharan D, Huguenard JR, Knudsen EI (2012) Gamma oscillations are generated locally in an attention-related midbrain network. Neuron 73(3):567–580
Hairer M (2010) Convergence of Markov processes, Lecture notes. http://www.hairer.org/notes/Convergence.pdf
Hairer M, Mattingly JC (2011) Yet another look at harris’ ergodic theorem for markov chains. In: Seminar on stochastic analysis, random fields and applications VI. Springer, pp. 109–117
Haskell E, Nykamp DQ, Tranchina D (2001) A population density method for large-scale modeling of neuronal networks with realistic synaptic kinetics. Neurocomputing 38:627–632
Henrie JA, Shapley R (2005) Lfp power spectra in v1 cortex: the graded effect of stimulus contrast. J Neurophysiol 94(1):479–490
Hubel DH (1995) Eye, brain, and vision. Scientific American Library/Scientific American Books
Kaschube M, Schnabel M, Löwel S, Coppola DM, White LE, Wolf F (2010) Universality in the evolution of orientation columns in the visual cortex. Science 330(6007):1113–1116
Lee K-H, Williams LM, Breakspear M, Gordon E (2003) Synchronous gamma activity: a review and contribution to an integrative neuroscience model of schizophrenia. Brain Res Rev 41(1):57–78
Li Y, Chariker L, Young L-S (2018) How well do reduced models capture the dynamics in models of interacting neurons? J Math Biol 78:83–115
Mazzoni A, Broccard FD, Garcia-Perez E, Bonifazi P, Ruaro ME, Torre V (2007) On the dynamics of the spontaneous activity in neuronal networks. PloS ONE 2(5):e439
Menon V, Freeman WJ, Cutillo BA, Desmond JE, Ward MF, Bressler SL, Laxer KD, Barbaro N, Gevins AS (1996) Spatio-temporal correlations in human gamma band electrocorticograms. Electroencephalogr Clin Neurophysiol 98(2):89–102
Meyn SP, Tweedie RL (2009) Markov chains and stochastic stability. Cambridge University Press, Cambridge
Newhall KA, Kovačič G, Kramer PR, Cai D (2010) Cascade-induced synchrony in stochastically driven neuronal networks. Phys Rev E 82(4):041903
Petermann T, Thiagarajan TC, Lebedev MA, Nicolelis MAL, Chialvo DR, Plenz D (2009) Spontaneous cortical activity in awake monkeys composed of neuronal avalanches. Proc Natl Acad Sci 106(37):15921–15926
Plenz D, Stewart CV, Shew W, Yang H, Klaus A, Bellay T (2011) Multi-electrode array recordings of neuronal avalanches in organotypic cultures. J Vis Exp 54:e2949
Rangan AV, Young L-S (2013) Dynamics of spiking neurons: between homogeneity and synchrony. J Comput Neurosci 34(3):433–460
Rangan AV, Young L-S (2013) Emergent dynamics in a model of visual cortex. J Comput Neurosci 35(2):155–167
Samonds JM, Zhou Z, Bernard MR, Bonds AB (2006) Synchronous activity in cat visual cortex encodes collinear and cocircular contours. J Neurophysiol 95(4):2602–2616
Shew WL, Yang H, Yu S, Roy R, Plenz D (2011) Information capacity and transmission are maximized in balanced cortical networks with neuronal avalanches. J Neurosci 31(1):55–63
Stout W (1974) Almost sure convergence, vol 95. Academic Press, New York
Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12(1):1–24
Wilson HR, Cowan JD (1973) A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Biol Cybern 13(2):55–80
Yu J, Ferster D (2010) Membrane potential synchrony in primary visual cortex during sensory stimulation. Neuron 68(6):1187–1201
Yu S, Yang H, Nakahara H, Santos GS, Nikolić D, Plenz D (2011) Higher-order interactions characterized in cortical activity. J Neurosci 31(48):17514–17526
Zhang J, Newhall K, Zhou D, Rangan A (2014) Distribution of correlated spiking events in a population-based approach for integrate-and-fire networks. J Comput Neurosci 36(2):279–295
Zhang J, Zhou D, Cai D, Rangan AV (2014) A coarse-grained framework for spiking neuronal networks: between homogeneity and synchrony. J Comput Neurosci 37(1):81–104
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Yao Li and Hui Xu were partially supported by the University of Massachusetts Amherst FRG/HEG grant. Yao Li is partially supported by NSF DMS-1813246.
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Li, Y., Xu, H. Stochastic neural field model: multiple firing events and correlations. J. Math. Biol. 79, 1169–1204 (2019). https://doi.org/10.1007/s00285-019-01389-6
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DOI: https://doi.org/10.1007/s00285-019-01389-6