# American Institute of Mathematical Sciences

2014, 11(1): 81-104. doi: 10.3934/mbe.2014.11.81

## Non-Markovian spiking statistics of a neuron with delayed feedback in presence of refractoriness

 1 Bogolyubov Institute for Theoretical Physics, Metrologichna str., 14-B, 03680 Kyiv, Ukraine, Ukraine

Received  December 2012 Revised  June 2013 Published  September 2013

Spiking statistics of a self-inhibitory neuron is considered. The neuron receives excitatory input from a Poisson stream and inhibitory impulses through a feedback line with a delay. After triggering, the neuron is in the refractory state for a positive period of time.
Recently, [35,6], it was proven for a neuron with delayed feedback and without the refractory state, that the output stream of interspike intervals (ISI) cannot be represented as a Markov process. The refractory state presence, in a sense limits the memory range in the spiking process, which might restore Markov property to the ISI stream.
Here we check such a possibility. For this purpose, we calculate the conditional probability density $P(t_{n+1}\mid t_{n},\ldots,t_1,t_{0})$, and prove exactly that it does not reduce to $P(t_{n+1}\mid t_{n},\ldots,t_1)$ for any $n\ge0$. That means, that activity of the system with refractory state as well cannot be represented as a Markov process of any order.
We conclude that it is namely the delayed feedback presence which results in non-Markovian statistics of neuronal firing. As delayed feedback lines are common for any realistic neural network, the non-Markovian statistics of the network activity should be taken into account in processing of experimental data.
Citation: Kseniia Kravchuk, Alexander Vidybida. Non-Markovian spiking statistics of a neuron with delayed feedback in presence of refractoriness. Mathematical Biosciences & Engineering, 2014, 11 (1) : 81-104. doi: 10.3934/mbe.2014.11.81
##### References:
 [1] A. Antonov and T. Misirpashaev, Markovian projection onto a displaced diffusion: Generic formulas with applications, working paper series, (2006). doi: 10.2139/ssrn.937860. [2] V. Aroniadou-Anderjaska, M. Ennis and M. T. Shipley, Dendrodendritic recurrent excitation in mitral cells of the rat olfactory bulb, J. Neurophysiol. 82 (1999), 489-494. [3] A. Bacci, J. R. Huguenard and D. A. Prince, Functional autaptic neurotransmission in fast-spiking interneurons: A novel form of feedback inhibition in the neocortex, J. Neurosci., 23 (2003), 859-866. [4] E. Benedetto and L. Sacerdote, On dependency properties of the ISIs generated by a two-compartmental neuronal model, Biological Cybernetics, 107 (2013), 95-106. doi: 10.1007/s00422-012-0536-0. [5] J. M. Bekkers and C. F. Stevens, Excitatory and inhibitory autaptic currents in isolated hippocampal neurons maintained in cell culture, PNAS, 88 (1991), 7834-7838. [6] G. G. Borst, J. C. Lodder and K. S. Kits, Large amplitude variability of GABAergic IPSC in melanotrophs from Xenopus laevis: Evidence that quantal size differs between synapses, J. Neurophysiol., 71 (1994), 639-655. [7] T. Britvina and J. J. Eggermont, A Markov model for interspike interval distributions of auditory cortical neurons that do not show periodic firings, Formal Aspects of Computing, 96 (2007), 245-264. [8] V. Chan-Palay, The recurrent collaterals of Purkinje cell axons: A correlated study of rat's cerebellar cortex with electron microscopy and the Golgi-method, Z. Anat. Entwicklungsgesch, 134 (1971), 210-234. [9] J. L. Doob, "Stochastic Processes," John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. [10] F. Farkhooi, M. F. Strube-Bloss and M. P. Nawrot, Serial correlation in neural spike trains: Experimental evidence, stochastic modelling, and single neuron variability, Phys. Rev. E, 79 (2009), 021905. [11] S. Ghosh-Dastidar and H. Adeli, Spiking neural networks, International Journal of Neural Systems, 19 (2009), 295-308. [12] A. I. Gulyas, R. Miles, A. Sík, K. Tóth, N. Tamamaki and T. F. Freund, Hippocampal pyramidal cells excite inhibitory neurons through a single release site, Nature, 366 (1993), 683-687. [13] A. L. Hodgkin, "The Conduction of the Nervous Impulse," Liverpool University Press, Liverpool, 1971. [14] A. V. Holden, "Models of the Stochastic Activity of Neurones," Lecture Notes in Biomathematics, Vol. 12, Springer-Verlag, Berlin-New York, 1976. [15] P. Jonas, J. Bischofberger, D. Fricker and R. Miles, Fast in, fast out temporal and spatial signal processing in hippocampal interneurons, Trends in Neurosciences, 27 (2004), 30-40. [16] K. G. Kravchuk and A. K. Vidybida, Firing statistics of inhibitory neuron with delayed feedback. II: Non-Markovian behavior, BioSystems, 112 (2013), 233-248. [17] M. W. Levine, Firing rates of a retinal neuron are not predictable from interspike interval statistics, Biophys. J., 30 (1980), 9-26. [18] S. B. Lowen and M. C. Teich, Auditory-nerve action potentials form a nonrenewal point process over short as well as long time scales, J. Acoust. Am., 92 (1992), 803-806. [19] J. Lübke, H. Markram, M. Frotscher and B. Sakmann, Frequency and dendritic distribution of autapses established by layer 5 pyramidal neurons in the developing rat neocortex: Comparison with synaptic innervation of adjacent neurons of the same class, J. Neurosci., 16 (1996), 3209-3218. [20] D. M. MacKay, Self-organization in the time domain, in "Self-Organizing Systems" (eds. M. C. Yovitts and G. T. Jacobi, et al.), Spartan Books, Washington, (1962), 37-48. [21] R. Miles, Synaptic excitation of inhibitory cells by single CA3 hippocampal pyramidal cells of the guinea-pig in vitro, J. Physiol., 428 (1990), 61-77. [22] J. W. Moore, N. Stockbridge and M. Westerfield, On the site of impulse initiation in a neurone, J. Physiol., 336 (1983), 301-311. [23] M. P. Nawrot, C. Boucsein, V. Rodriguez-Molina, A. Aertsen, S. Grün and S. Rotter, Serial interval statistics of spontaneous activity in cortical neurons in vivo and in vitro, Neurocomputing, 70 (2007), 1717-1722. [24] J. G. Nicholls, A. R. Martin, B. G. Wallace and P. A. Fuchs, "From Neuron to Brain," Sinauer Associates, Sunderland, 2001. [25] R. A. Nicoll and C. E. Jahr, Self-excitation of olfactory bulb neurones, Nature, 296 (1982), 441-444. [26] M. R. Park, J. W. Lighthall and S. T. Kitai, Recurrent inhibition in the rat neostriatum, Brain Res., 194 (1980), 359-369. [27] R. Ratnam and M. E. Nelson, Nonrenewal statistics of electrosensory afferent spike trains: Implications for the detection of weak sensory signals, J. Neurosci., 20 (2000), 6672-6683. [28] R. F. Schmidt, "Fundamentals of Neurophysiology," Springer Study Edition, Springer, 1981. [29] G. Tamás, E. H. Buhl and P. Somogyi, Massive autaptic self-innervation of GABAergic neurons in cat visual cortex, J. Neurosci., 17 (1997), 6352-6364. [30] H. Van der Loos and E. M. Glaser, Autapses in neocortex cerebri: Synapses between a pyramidal cell's axon and its own dendrites, Brain Res., 48 (1972), 355-360. [31] A. K. Vidybida, Inhibition as binding controller at the single neuron level, BioSystems, 48 (1998), 263-267. [32] A. K. Vidybida, Output stream of a binding neuron, Ukrainian Mathematical Journal, 59 (2007), 1819-1839. doi: 10.1007/s11253-008-0028-5. [33] A. K. Vidybida, Input-output relations in binding neuron, BioSystems, 89 (2007), 160-165. [34] A. K. Vidybida, Output stream of binding neuron with instantaneous feedback, Eur. Phys. J. B, 65 (2008), 577-584; Erratum: Eur. Phys. J. B, 69 (2009), 313. [35] A. K. Vidybida and K. G. Kravchuk, Delayed feedback causes non-Markovian behavior of neuronal firing statistics, Ukrainian Mathematical Journal, 64 (2012), 1587-1609. [36] A. K. Vidybida and K. G. Kravchuk, Firing statistics of inhibitory neuron with delayed feedback. I. Output ISI probability density, BioSystems, 112, 3 (2013), 224-232. [37] Y. Wu, F. Kawasaki and R. W. Ordway, Properties of short-term synaptic depression at larval neuromuscular synapses in wild-type and temperature-sensitive paralytic mutants of drosophila, J. Neurophysiol., 93 (2005), 2396-2405.

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##### References:
 [1] A. Antonov and T. Misirpashaev, Markovian projection onto a displaced diffusion: Generic formulas with applications, working paper series, (2006). doi: 10.2139/ssrn.937860. [2] V. Aroniadou-Anderjaska, M. Ennis and M. T. Shipley, Dendrodendritic recurrent excitation in mitral cells of the rat olfactory bulb, J. Neurophysiol. 82 (1999), 489-494. [3] A. Bacci, J. R. Huguenard and D. A. Prince, Functional autaptic neurotransmission in fast-spiking interneurons: A novel form of feedback inhibition in the neocortex, J. Neurosci., 23 (2003), 859-866. [4] E. Benedetto and L. Sacerdote, On dependency properties of the ISIs generated by a two-compartmental neuronal model, Biological Cybernetics, 107 (2013), 95-106. doi: 10.1007/s00422-012-0536-0. [5] J. M. Bekkers and C. F. Stevens, Excitatory and inhibitory autaptic currents in isolated hippocampal neurons maintained in cell culture, PNAS, 88 (1991), 7834-7838. [6] G. G. Borst, J. C. Lodder and K. S. Kits, Large amplitude variability of GABAergic IPSC in melanotrophs from Xenopus laevis: Evidence that quantal size differs between synapses, J. Neurophysiol., 71 (1994), 639-655. [7] T. Britvina and J. J. Eggermont, A Markov model for interspike interval distributions of auditory cortical neurons that do not show periodic firings, Formal Aspects of Computing, 96 (2007), 245-264. [8] V. Chan-Palay, The recurrent collaterals of Purkinje cell axons: A correlated study of rat's cerebellar cortex with electron microscopy and the Golgi-method, Z. Anat. Entwicklungsgesch, 134 (1971), 210-234. [9] J. L. Doob, "Stochastic Processes," John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. [10] F. Farkhooi, M. F. Strube-Bloss and M. P. Nawrot, Serial correlation in neural spike trains: Experimental evidence, stochastic modelling, and single neuron variability, Phys. Rev. E, 79 (2009), 021905. [11] S. Ghosh-Dastidar and H. Adeli, Spiking neural networks, International Journal of Neural Systems, 19 (2009), 295-308. [12] A. I. Gulyas, R. Miles, A. Sík, K. Tóth, N. Tamamaki and T. F. Freund, Hippocampal pyramidal cells excite inhibitory neurons through a single release site, Nature, 366 (1993), 683-687. [13] A. L. Hodgkin, "The Conduction of the Nervous Impulse," Liverpool University Press, Liverpool, 1971. [14] A. V. Holden, "Models of the Stochastic Activity of Neurones," Lecture Notes in Biomathematics, Vol. 12, Springer-Verlag, Berlin-New York, 1976. [15] P. Jonas, J. Bischofberger, D. Fricker and R. Miles, Fast in, fast out temporal and spatial signal processing in hippocampal interneurons, Trends in Neurosciences, 27 (2004), 30-40. [16] K. G. Kravchuk and A. K. Vidybida, Firing statistics of inhibitory neuron with delayed feedback. II: Non-Markovian behavior, BioSystems, 112 (2013), 233-248. [17] M. W. Levine, Firing rates of a retinal neuron are not predictable from interspike interval statistics, Biophys. J., 30 (1980), 9-26. [18] S. B. Lowen and M. C. Teich, Auditory-nerve action potentials form a nonrenewal point process over short as well as long time scales, J. Acoust. Am., 92 (1992), 803-806. [19] J. Lübke, H. Markram, M. Frotscher and B. Sakmann, Frequency and dendritic distribution of autapses established by layer 5 pyramidal neurons in the developing rat neocortex: Comparison with synaptic innervation of adjacent neurons of the same class, J. Neurosci., 16 (1996), 3209-3218. [20] D. M. MacKay, Self-organization in the time domain, in "Self-Organizing Systems" (eds. M. C. Yovitts and G. T. Jacobi, et al.), Spartan Books, Washington, (1962), 37-48. [21] R. Miles, Synaptic excitation of inhibitory cells by single CA3 hippocampal pyramidal cells of the guinea-pig in vitro, J. Physiol., 428 (1990), 61-77. [22] J. W. Moore, N. Stockbridge and M. Westerfield, On the site of impulse initiation in a neurone, J. Physiol., 336 (1983), 301-311. [23] M. P. Nawrot, C. Boucsein, V. Rodriguez-Molina, A. Aertsen, S. Grün and S. Rotter, Serial interval statistics of spontaneous activity in cortical neurons in vivo and in vitro, Neurocomputing, 70 (2007), 1717-1722. [24] J. G. Nicholls, A. R. Martin, B. G. Wallace and P. A. Fuchs, "From Neuron to Brain," Sinauer Associates, Sunderland, 2001. [25] R. A. Nicoll and C. E. Jahr, Self-excitation of olfactory bulb neurones, Nature, 296 (1982), 441-444. [26] M. R. Park, J. W. Lighthall and S. T. Kitai, Recurrent inhibition in the rat neostriatum, Brain Res., 194 (1980), 359-369. [27] R. Ratnam and M. E. Nelson, Nonrenewal statistics of electrosensory afferent spike trains: Implications for the detection of weak sensory signals, J. Neurosci., 20 (2000), 6672-6683. [28] R. F. Schmidt, "Fundamentals of Neurophysiology," Springer Study Edition, Springer, 1981. [29] G. Tamás, E. H. Buhl and P. Somogyi, Massive autaptic self-innervation of GABAergic neurons in cat visual cortex, J. Neurosci., 17 (1997), 6352-6364. [30] H. Van der Loos and E. M. Glaser, Autapses in neocortex cerebri: Synapses between a pyramidal cell's axon and its own dendrites, Brain Res., 48 (1972), 355-360. [31] A. K. Vidybida, Inhibition as binding controller at the single neuron level, BioSystems, 48 (1998), 263-267. [32] A. K. Vidybida, Output stream of a binding neuron, Ukrainian Mathematical Journal, 59 (2007), 1819-1839. doi: 10.1007/s11253-008-0028-5. [33] A. K. Vidybida, Input-output relations in binding neuron, BioSystems, 89 (2007), 160-165. [34] A. K. Vidybida, Output stream of binding neuron with instantaneous feedback, Eur. Phys. J. B, 65 (2008), 577-584; Erratum: Eur. Phys. J. B, 69 (2009), 313. [35] A. K. Vidybida and K. G. Kravchuk, Delayed feedback causes non-Markovian behavior of neuronal firing statistics, Ukrainian Mathematical Journal, 64 (2012), 1587-1609. [36] A. K. Vidybida and K. G. Kravchuk, Firing statistics of inhibitory neuron with delayed feedback. I. Output ISI probability density, BioSystems, 112, 3 (2013), 224-232. [37] Y. Wu, F. Kawasaki and R. W. Ordway, Properties of short-term synaptic depression at larval neuromuscular synapses in wild-type and temperature-sensitive paralytic mutants of drosophila, J. Neurophysiol., 93 (2005), 2396-2405.
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