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The effect of interspike interval statistics on the information gain under the rate coding hypothesis
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NonMarkovian spiking statistics of a neuron with delayed feedback in presence of refractoriness
1.  Bogolyubov Institute for Theoretical Physics, Metrologichna str., 14B, 03680 Kyiv, Ukraine, Ukraine 
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 nonMarkovian statistics of neuronal firing. As delayed feedback lines are common for any realistic neural network, the nonMarkovian statistics of the network activity should be taken into account in processing of experimental data.
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. AroniadouAnderjaska, M. Ennis and M. T. Shipley, Dendrodendritic recurrent excitation in mitral cells of the rat olfactory bulb,, J. Neurophysiol. 82 (1999), 82 (1999), 489. 
[3] 
A. Bacci, J. R. Huguenard and D. A. Prince, Functional autaptic neurotransmission in fastspiking interneurons: A novel form of feedback inhibition in the neocortex,, J. Neurosci., 23 (2003), 859. 
[4] 
E. Benedetto and L. Sacerdote, On dependency properties of the ISIs generated by a twocompartmental neuronal model,, Biological Cybernetics, 107 (2013), 95. doi: 10.1007/s0042201205360. 
[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. 
[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. 
[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. 
[8] 
V. ChanPalay, The recurrent collaterals of Purkinje cell axons: A correlated study of rat's cerebellar cortex with electron microscopy and the Golgimethod,, Z. Anat. Entwicklungsgesch, 134 (1971), 210. 
[9] 
J. L. Doob, "Stochastic Processes,", John Wiley & Sons, (1953). 
[10] 
F. Farkhooi, M. F. StrubeBloss and M. P. Nawrot, Serial correlation in neural spike trains: Experimental evidence, stochastic modelling, and single neuron variability,, Phys. Rev. E, 79 (2009). 
[11] 
S. GhoshDastidar and H. Adeli, Spiking neural networks,, International Journal of Neural Systems, 19 (2009), 295. 
[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. 
[13] 
A. L. Hodgkin, "The Conduction of the Nervous Impulse,", Liverpool University Press, (1971). 
[14] 
A. V. Holden, "Models of the Stochastic Activity of Neurones,", Lecture Notes in Biomathematics, (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. 
[16] 
K. G. Kravchuk and A. K. Vidybida, Firing statistics of inhibitory neuron with delayed feedback. II: NonMarkovian behavior,, BioSystems, 112 (2013), 233. 
[17] 
M. W. Levine, Firing rates of a retinal neuron are not predictable from interspike interval statistics,, Biophys. J., 30 (1980), 9. 
[18] 
S. B. Lowen and M. C. Teich, Auditorynerve action potentials form a nonrenewal point process over short as well as long time scales,, J. Acoust. Am., 92 (1992), 803. 
[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. 
[20] 
D. M. MacKay, Selforganization in the time domain,, in, (1962), 37. 
[21] 
R. Miles, Synaptic excitation of inhibitory cells by single CA3 hippocampal pyramidal cells of the guineapig in vitro,, J. Physiol., 428 (1990), 61. 
[22] 
J. W. Moore, N. Stockbridge and M. Westerfield, On the site of impulse initiation in a neurone,, J. Physiol., 336 (1983), 301. 
[23] 
M. P. Nawrot, C. Boucsein, V. RodriguezMolina, 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. 
[24] 
J. G. Nicholls, A. R. Martin, B. G. Wallace and P. A. Fuchs, "From Neuron to Brain,", Sinauer Associates, (2001). 
[25] 
R. A. Nicoll and C. E. Jahr, Selfexcitation of olfactory bulb neurones,, Nature, 296 (1982), 441. 
[26] 
M. R. Park, J. W. Lighthall and S. T. Kitai, Recurrent inhibition in the rat neostriatum,, Brain Res., 194 (1980), 359. 
[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. 
[28] 
R. F. Schmidt, "Fundamentals of Neurophysiology,", Springer Study Edition, (1981). 
[29] 
G. Tamás, E. H. Buhl and P. Somogyi, Massive autaptic selfinnervation of GABAergic neurons in cat visual cortex,, J. Neurosci., 17 (1997), 6352. 
[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. 
[31] 
A. K. Vidybida, Inhibition as binding controller at the single neuron level,, BioSystems, 48 (1998), 263. 
[32] 
A. K. Vidybida, Output stream of a binding neuron,, Ukrainian Mathematical Journal, 59 (2007), 1819. doi: 10.1007/s1125300800285. 
[33] 
A. K. Vidybida, Inputoutput relations in binding neuron,, BioSystems, 89 (2007), 160. 
[34] 
A. K. Vidybida, Output stream of binding neuron with instantaneous feedback,, Eur. Phys. J. B, 65 (2008), 577. 
[35] 
A. K. Vidybida and K. G. Kravchuk, Delayed feedback causes nonMarkovian behavior of neuronal firing statistics,, Ukrainian Mathematical Journal, 64 (2012), 1587. 
[36] 
A. K. Vidybida and K. G. Kravchuk, Firing statistics of inhibitory neuron with delayed feedback. I. Output ISI probability density,, BioSystems, 112 (2013), 224. 
[37] 
Y. Wu, F. Kawasaki and R. W. Ordway, Properties of shortterm synaptic depression at larval neuromuscular synapses in wildtype and temperaturesensitive paralytic mutants of drosophila,, J. Neurophysiol., 93 (2005), 2396. 
show all references
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. AroniadouAnderjaska, M. Ennis and M. T. Shipley, Dendrodendritic recurrent excitation in mitral cells of the rat olfactory bulb,, J. Neurophysiol. 82 (1999), 82 (1999), 489. 
[3] 
A. Bacci, J. R. Huguenard and D. A. Prince, Functional autaptic neurotransmission in fastspiking interneurons: A novel form of feedback inhibition in the neocortex,, J. Neurosci., 23 (2003), 859. 
[4] 
E. Benedetto and L. Sacerdote, On dependency properties of the ISIs generated by a twocompartmental neuronal model,, Biological Cybernetics, 107 (2013), 95. doi: 10.1007/s0042201205360. 
[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. 
[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. 
[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. 
[8] 
V. ChanPalay, The recurrent collaterals of Purkinje cell axons: A correlated study of rat's cerebellar cortex with electron microscopy and the Golgimethod,, Z. Anat. Entwicklungsgesch, 134 (1971), 210. 
[9] 
J. L. Doob, "Stochastic Processes,", John Wiley & Sons, (1953). 
[10] 
F. Farkhooi, M. F. StrubeBloss and M. P. Nawrot, Serial correlation in neural spike trains: Experimental evidence, stochastic modelling, and single neuron variability,, Phys. Rev. E, 79 (2009). 
[11] 
S. GhoshDastidar and H. Adeli, Spiking neural networks,, International Journal of Neural Systems, 19 (2009), 295. 
[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. 
[13] 
A. L. Hodgkin, "The Conduction of the Nervous Impulse,", Liverpool University Press, (1971). 
[14] 
A. V. Holden, "Models of the Stochastic Activity of Neurones,", Lecture Notes in Biomathematics, (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. 
[16] 
K. G. Kravchuk and A. K. Vidybida, Firing statistics of inhibitory neuron with delayed feedback. II: NonMarkovian behavior,, BioSystems, 112 (2013), 233. 
[17] 
M. W. Levine, Firing rates of a retinal neuron are not predictable from interspike interval statistics,, Biophys. J., 30 (1980), 9. 
[18] 
S. B. Lowen and M. C. Teich, Auditorynerve action potentials form a nonrenewal point process over short as well as long time scales,, J. Acoust. Am., 92 (1992), 803. 
[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. 
[20] 
D. M. MacKay, Selforganization in the time domain,, in, (1962), 37. 
[21] 
R. Miles, Synaptic excitation of inhibitory cells by single CA3 hippocampal pyramidal cells of the guineapig in vitro,, J. Physiol., 428 (1990), 61. 
[22] 
J. W. Moore, N. Stockbridge and M. Westerfield, On the site of impulse initiation in a neurone,, J. Physiol., 336 (1983), 301. 
[23] 
M. P. Nawrot, C. Boucsein, V. RodriguezMolina, 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. 
[24] 
J. G. Nicholls, A. R. Martin, B. G. Wallace and P. A. Fuchs, "From Neuron to Brain,", Sinauer Associates, (2001). 
[25] 
R. A. Nicoll and C. E. Jahr, Selfexcitation of olfactory bulb neurones,, Nature, 296 (1982), 441. 
[26] 
M. R. Park, J. W. Lighthall and S. T. Kitai, Recurrent inhibition in the rat neostriatum,, Brain Res., 194 (1980), 359. 
[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. 
[28] 
R. F. Schmidt, "Fundamentals of Neurophysiology,", Springer Study Edition, (1981). 
[29] 
G. Tamás, E. H. Buhl and P. Somogyi, Massive autaptic selfinnervation of GABAergic neurons in cat visual cortex,, J. Neurosci., 17 (1997), 6352. 
[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. 
[31] 
A. K. Vidybida, Inhibition as binding controller at the single neuron level,, BioSystems, 48 (1998), 263. 
[32] 
A. K. Vidybida, Output stream of a binding neuron,, Ukrainian Mathematical Journal, 59 (2007), 1819. doi: 10.1007/s1125300800285. 
[33] 
A. K. Vidybida, Inputoutput relations in binding neuron,, BioSystems, 89 (2007), 160. 
[34] 
A. K. Vidybida, Output stream of binding neuron with instantaneous feedback,, Eur. Phys. J. B, 65 (2008), 577. 
[35] 
A. K. Vidybida and K. G. Kravchuk, Delayed feedback causes nonMarkovian behavior of neuronal firing statistics,, Ukrainian Mathematical Journal, 64 (2012), 1587. 
[36] 
A. K. Vidybida and K. G. Kravchuk, Firing statistics of inhibitory neuron with delayed feedback. I. Output ISI probability density,, BioSystems, 112 (2013), 224. 
[37] 
Y. Wu, F. Kawasaki and R. W. Ordway, Properties of shortterm synaptic depression at larval neuromuscular synapses in wildtype and temperaturesensitive paralytic mutants of drosophila,, J. Neurophysiol., 93 (2005), 2396. 
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