Abstract
Neurons transmit information as action potentials or spikes. Due to the inherent randomness of the inter-spike intervals (ISIs), probabilistic models are often used for their description. Cumulative damage (CD) distributions are a family of probabilistic models that has been widely considered for describing time-related cumulative processes. This family allows us to consider certain deterministic principles for modeling ISIs from a probabilistic viewpoint and to link its parameters to values with biological interpretation. The CD family includes the Birnbaum–Saunders and inverse Gaussian distributions, which possess distinctive properties and theoretical arguments useful for ISI description. We expand the use of CD distributions to the modeling of neural spiking behavior, mainly by testing the suitability of the Birnbaum–Saunders distribution, which has not been studied in the setting of neural activity. We validate this expansion with original experimental and simulated electrophysiological data.
References
Balakrishnan N, Leiva V, Sanhueza A, Cabrera E (2009) Mixture inverse Gaussian distribution and its transformations, moments and applications. Statistics 43:91–104
Baranauskas G, Mukovskiy A, Wolf F, Volgushev M (2010) The determinants of the onset dynamics of action potentials in a computational model. Neuroscience 167:1070–1090
Bazaes A, Olivares J, Schmachtenberg O (2013) Properties, projections and tuning of teleost olfactory receptor neurons. J Chem Ecol 39:451–464
Birnbaum ZW, Saunders SC (1969) A new family of life distributions. J Appl Probab 6:319–327
Brette R (2013) Sharpness of spike initiation in neurons explained by compartmentalization. PLoS Comput Biol 9:e1003338
Brown EN, Barbieri R, Ventura V, Kass RE, Frank LM (2002) The time-rescaling theorem and its application to neural spike train data analysis. Neural Comput 14:325–346
Brunel N, Chance FS, Fourcaud N, Abbott LF (2001) Effects of synaptic noise and filtering on the frequency response of spiking neurons. Phys Rev Lett 86:2186–2189
Burkitt A (2006) A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input. Biol Cybern 95:1–19
Butts DA, Weng C, Jin J, Yeh CI, Lesica NA, Alonso JM, Stanley GB (2007) Temporal precision in the neural code and the timescales of natural vision. Nature 449:92–95
Castro-Kuriss C, Leiva V, Athayde E (2014) Graphical tools to assess goodness-of-fit in non-location-scale distributions. Colomb J Stat 37:341–365 (special issue on “Current Topics in Statistical Graphics”)
Chen Y, Nitz DA (2011) A unified description of cerebellar inter-spike interval distributions and variabilities using summation of Gaussians. Netw Comput Neural Syst 22:7496
Chow CC, White JA (1996) Spontaneous action potentials due to channel fluctuations. Biophys J 71:3013–3021
Citi L, Ba D, Brown EN, Barbieri R (2013) Likelihood methods for point processes with refractoriness. Neural Comput 26:237–263
Dayan P, Abbott LF (2005) Theoretical neuroscience: computational and mathematical modeling of neural systems. The MIT Press, Cambridge
Desmond A (1986) On the relationship between two fatigue life models. IEEE Trans Reliab 35:167–169
Figueira JC, Andrade JM (2011) A neural network approach to fatigue life prediction. Int J Fatigue 33:313322
Fierro R, Leiva V, Ruggeri F, Sanhueza A (2013) On a Birnbaum–Saunders distribution arising from a non-homogeneous Poisson process. Stat Probab Lett 83:1233–1239
Geman S (1979) Some averaging and stability results for random differential equations. SIAM J Appl Math 36:86–105
Guiraud P, Leiva V, Fierro R (2009) A non-central version of the Birnbaum–Saunders distribution for reliability analysis. IEEE Trans Reliab 58:152–160
Hille B (2001) Ion channels of excitable membranes. Sinauer Associates Inc, Sunderland
Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544
Ilin V, Malyshev A, Wolf F, Volgushev M (2013) Fast computations in cortical ensembles require rapid initiation of action potentials. J Neurosci 33:2281–2292
Inoue J, Sato S, Ricciardi L (1995) On the parameter estimation for diffusion models of single neurons. Biol Cybern 73:209–221
Iyengar S, Liao Q (1997) Modeling neural activity using by the generalized inverse Gaussian distribution. Biol Cybern 77:289–295
Izhikevich EM (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. The MIT press, Cambridge
Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1–2. Wiley, New York
Jorgensen B (1982) Statistical properties of the generalized inverse Gaussian distribution. Springer, New York
Kass RE, Raftery AE (1995) Bayes factors. J Am Stat Soc 90:773–795
Karatzas I, Shreve E (1991) Brownian motion and stochastic calculus. Springer, New York
Kotz S, Leiva V, Sanhueza A (2010) Two new mixture models related to the inverse Gaussian distribution. Methodol Comput Appl Probab 12:199–212
La Camera G, Giugliano M, Senn W, Fusi S (2008) The response of cortical neurons to in vivo-like input current: theory and experiment. Biol Cybern 99:279–301
Lánský P, Smith CE (1989) The effect of a random initial value in neural first-passage time models. Math Biosci 93:191–215
Leiva V, Saulo H, Leao J, Marchant C (2014) A family of autoregressive conditional duration models applied to financial data. Comput Stat Data Anal 79:175–191
Leiva V, Sanhueza A, Saunders S (2015) New developments and applications on life distributions under cumulative damage. Under 2nd review in Applied Stochastic Models in Business and Industry
Levine MW (1991) The distribution of intervals between neural impulses in the maintained discharges of retinal ganglion cells. Biol Cybern 65:459–467
Magloczky Z, Freundemail TF (2005) Impaired and repaired inhibitory circuits in the epileptic human hippocampus. Trends Neurosci 28:334–340
Marchant C, Leiva V, Cysneiros FJA (2015) Multivariate Birnbaum-Saunders regression models for metal fatigue. Under 2nd review in IEEE Transactions on Reliability
McCormick D, Shu Y, Yu Y (2007) Neurophysiology: Hodgkin and Huxley model-still standing? Nature 445:E1–E2
Mensi S, Naud R, Pozzorini C, Avermann M, Petersen CCH, Gerstner W (2012) Parameter extraction and classification of three cortical neuron types reveals two distinct adaptation mechanisms. J Neurophysiol 107:1756–1775
Mountcastle VB, Talbot WH, Kornhuber HH (1966) The neural transformation of mechanical stimuli delivered to the monkey’s hand. In: de Reuck AVS, Knight J (eds) Ciba Foundation Symposium: touch, heat and pain. Churchill, London, pp 325–351
Naundorf B, Wolf F, Volgushev M (2006) Unique features of action potential initiation in cortical neurons. Nature 440:1060–1063
Nikulin MS, Limnios N, Balakrishnan N, Kahle W, Huber-Carol C (2010) Advances in degradation modeling: applications to reliability, survival analysis, and finance. Birkhauser, Berlin
Paninski L, Pillow JW, Simoncelli EP (2004) Maximum likelihood estimation of a stochastic integrate-and-fire neural encoding model. Neural Comput 16:2533–2561
Peng YR, Zeng SY, Song HL, Li MY, Yamada MK, Yu X (2010) Postsynaptic spiking homeostatically induces cell-autonomous regulation of inhibitory inputs via retrograde signaling. J Neurosci 30:16220–16231
Resnick S (1992) Adventures in stochastic processes. Birkhauser, New York
Saunders SC (2007) Reliability, life testing and prediction of service lives. Springer, New York
Schmachtenberg O (2006) Histological and electrophysiological properties of crypt cells from the olfactory epithelium of the marine teleost Trachurus symmetricus. J Comp Neurol 495:113–121
Steimer A, Douglas R (2013) Spike-based probabilistic inference in analog graphical models using interspike-interval coding. Neural Comput 25:2303–2354
Truccolo W, Eden UT, Fellows MR, Donoghue JP, Brown EN (2005) A point process framework for relating neural spiking activity to spiking history, neural ensemble, and extrinsic covariate effects. J Neurophysiol 93:1074–1089
Tuckwell HC (1989) Stochastic processes in the neurosciences. Society for Industrial and Applied Mathematics, Pennsylvania
Volgushev M, Malyshev A, Balaban P, Volgushev S, Wolf F (2008) Onset dynamics of action potentials in rat neocortical neurons and identified snail neurons: quantification of the difference. PLoS One 3:e1962
Wald A (1947) Sequential analysis. Wiley, New York
Yu Y, Shu Y, McCormick D (2008) Cortical action potential backpropagation explains spike threshold variability and rapid-onset kinetics. J Neurosci 28:7260–7272
Acknowledgments
The authors thank the Editor-in-Chief, Prof. Dr. J. Leo van Hemmen, and two anonymous referees for their valuable comments on an earlier version of this manuscript, which resulted in this improved version. The authors thank Rosie Gronthos from Australia for proofreading the first revised version of this manuscript and Patricia Van Roon from Canada for proofreading its second version. The research of V. Leiva was supported by FONDECYT 1120879 grant from the Chilean government; of M. Tejo by FONDECYT 3140613 postdoctorate grant; of P. Guiraud by PIA-Anillo ACT1112 grant from the Chilean government; of P. Orio by FONDECYT 1130862 and PIA-Anillo ACT-1113; and of P. Orio and O. Schmachtenberg by The Centro Interdisciplinario de Neurociencia de Valparaíso, which is a Millennium Institute supported by the Millennium Scientific Initiative of the Ministerio de Economía, Fomento y Turismo of the Chilean government.
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Leiva, V., Tejo, M., Guiraud, P. et al. Modeling neural activity with cumulative damage distributions. Biol Cybern 109, 421–433 (2015). https://doi.org/10.1007/s00422-015-0651-9
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DOI: https://doi.org/10.1007/s00422-015-0651-9