American Institute of Mathematical Sciences

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2014, 11(2): 217-231. doi: 10.3934/mbe.2014.11.217

On a spike train probability model with interacting neural units

Antonio Di Crescenzo 1, , Maria Longobardi 2, and  Barbara Martinucci 1,

1. 

Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, I-84084 Fisciano (SA), Italy, Italy
2. 

Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Cintia, I-80126 Napoli, Italy

Received  October 2012 Revised  April 2013 Published  October 2013

We investigate an extension of the spike train stochastic model based on the conditional intensity, in which the recovery function includes an interaction between several excitatory neural units. Such function is proposed as depending both on the time elapsed since the last spike and on the last spiking unit. Our approach, being somewhat related to the competing risks model, allows to obtain the general form of the interspike distribution and of the probability of consecutive spikes from the same unit. Various results are finally presented in the two cases when the free firing rate function (i) is constant, and (ii) has a sinusoidal form.
Keywords: recovery function., firing rate, cumulative firing rate, refractory period, conditional intensity, Poisson process, Neural model.
Mathematics Subject Classification: Primary: 60J28, 92B20; Secondary: 60K2.
Citation: Antonio Di Crescenzo, Maria Longobardi, Barbara Martinucci. On a spike train probability model with interacting neural units. Mathematical Biosciences & Engineering, 2014, 11 (2) : 217-231. doi: 10.3934/mbe.2014.11.217
References:
[1]

D. J. Amit and N. Brunel, Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex, Cerebral Cortex, 7 (1997), 237-252. doi: 10.1093/cercor/7.3.237.  Google Scholar

[2]

M. Barbi, S. Chillemi, A. Di Garbo and L. Reale, Stochastic resonance in a sinusoidally forced LIF model with noisy threshold, Biosystems, 71 (2003), 23-28. doi: 10.1016/S0303-2647(03)00106-0.  Google Scholar

[3]

O. Bernander, C. Koch and M. Usher, The effect of synchronized inputs at the single neuron level, Neural Computation, 6 (1994), 622-641. Google Scholar

[4]

A. Buonocore, A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, A Markov chain-based model for actomyosin dynamics, Sci. Math. Japon., 70 (2009), 159-174.  Google Scholar

[5]

A. Buonocore, A. Di Crescenzo, B. Martinucci and L. M. Ricciardi, A stochastic model for the stepwise motion in actomyosin dynamics, Sci. Math. Japon., 58 (2003), 245-254.  Google Scholar

[6]

M. J. Berry and M. Meister, Refractoriness and neural precision, J. Neurosci., 18 (1998), 2200-2211. Google Scholar

[7]

A. N. Burkitt, A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input, Biol. Cybern., 95 (2006), 1-19. doi: 10.1007/s00422-006-0068-6.  Google Scholar

[8]

A. N. Burkitt, A review of the integrate-and-fire neuron model. II. Inhomogeneous synaptic input and network properties, Biol. Cybern., 95 (2006), 97-112. doi: 10.1007/s00422-006-0082-8.  Google Scholar

[9]

H. P. Chan and W.-L. Loh, Some theoretical results on neural spike train probability models, Ann. Stat., 35 (2007), 2691-2722. doi: 10.1214/009053607000000280.  Google Scholar

[10]

M. Crowder, Classical Competing Risks, Chapman & Hall/CRC, Boca Raton, 2001. doi: 10.1201/9781420035902.  Google Scholar

[11]

M. Deger, S. Cardanobile, M. Helias and S. Rotter, The Poisson process with dead time captures important statistical features of neural activity, BMC Neuroscience, 10 (2009), P110. doi: 10.1186/1471-2202-10-S1-P110.  Google Scholar

[12]

A. Di Crescenzo, E. Di Nardo, A. G. Nobile, E. Pirozzi and L. M. Ricciardi, On some computational results for single neurons' activity modeling, BioSystems, 58 (2000), 19-26. Google Scholar

[13]

A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, Stochastic population models with interacting species, J. Math. Biol., 42 (2001), 1-25. doi: 10.1007/PL00000070.  Google Scholar

[14]

A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, A note on birth-death processes with catastrophes, Stat. Prob. Lett., 78 (2008), 2248-2257. doi: 10.1016/j.spl.2008.01.093.  Google Scholar

[15]

A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, On time non-homogeneous stochastic processes with catastrophes, in Cybernetics and Systems 2010 (ed. R. Trappl), Austrian Society for Cybernetic Studies, Vienna, 2010, 169-174. Google Scholar

[16]

A. Di Crescenzo and M. Longobardi, On the NBU ageing notion within the competing risks model, J. Stat. Plann. Infer., 136 (2006), 1638-1654. doi: 10.1016/j.jspi.2004.08.022.  Google Scholar

[17]

A. Di Crescenzo and M. Longobardi, Competing risks within shock models, Sci. Math. Japon., 67 (2008), 125-135.  Google Scholar

[18]

A. Di Crescenzo, B. Martinucci, E. Pirozzi and L. M. Ricciardi, On the interaction between two Stein's neuronal units, in Cybernetics and Systems 2004 (ed. R. Trappl), Austrian Society for Cybernetic Studies, Vienna, 2004, 205-210. Google Scholar

[19]

G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophy. J., 4 (1964), 41-68. doi: 10.1016/S0006-3495(64)86768-0.  Google Scholar

[20]

D. Hampel and P. Lansky, On the estimation of refractory period, J. Neurosci. Meth., 171 (2008), 288-295. doi: 10.1016/j.jneumeth.2008.03.003.  Google Scholar

[21]

D. H. Johnson, Point process models of single-neuron discharges, J. Comput. Neurosci., 3 (1996), 275-299. doi: 10.1007/BF00161089.  Google Scholar

[22]

D. H. Johnson and A. Swami, The transmission of signals by auditory-nerve fiber discharge patterns, J. Acoust. Soc. Am., 74 (1983), 493-501. doi: 10.1121/1.389815.  Google Scholar

[23]

R. E. Kass and V. Ventura, A spike-train probability model, Neural Comput., 13 (2001), 1713-1720. doi: 10.1162/08997660152469314.  Google Scholar

[24]

A. Mazzoni, F. D. Broccard, E. Garcia-Perez, P. Bonifazi, M. E. Ruaro and V. Torre, On the dynamics of the spontaneous activity in neuronal networks, PLoS ONE, 2 (2007), e439. doi: 10.1371/journal.pone.0000439.  Google Scholar

[25]

M. I. Miller, Algorithms for removing recovery-related distortion from auditory nerve discharge patterns, J. Acoust. Soc. Am., 77 (1985), 1452-1464. doi: 10.1121/1.392040.  Google Scholar

[26]

U. Picchini, S. Ditlevsen, A. De Gaetano and P. Lansky, Parameters of the diffusion leaky integrate-and-fire neuronal model for a slowly fluctuating signal, Neural Comput., 20 (2008), 2696-2714. doi: 10.1162/neco.2008.11-07-653.  Google Scholar

[27]

L. M. Ricciardi, Diffusion Processes and Related Topics in Biology, Notes taken by Charles E. Smith, Lecture Notes in Biomathematics, Vol. 14, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[28]

L. M. Ricciardi, Modeling single neuron activity in the presence of refractoriness: New contributions to an old problem, in Imagination and Rigor. Essays on Eduardo R. Caianiello's Scientific Heritage (ed. S. Termini), Springer-Verlag Italia, 2006, 133-145. doi: 10.1007/88-470-0472-1_11.  Google Scholar

[29]

L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. G. Nobile, On the instantaneous return process for neuronal diffusion models, in Structure: from Physics to General Systems - Festschrift Volume in Honour of E.R. Caianiello on his Seventieth Birthday (eds. M. Marinaro and G. Scarpetta), World Scientific, Singapore, 1992, 78-94. Google Scholar

[30]

L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. G. Nobile, An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling, Math. Japon., 50 (1999), 247-322.  Google Scholar

[31]

W. R. Softky and C. Koch, The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs, The Journal of Neuroscience, 13 (1993), 334-350. Google Scholar

[32]

R. B. Stein, A theoretical analysis of neuronal variability, Biophys. J., 5 (1965), 173-194. doi: 10.1016/S0006-3495(65)86709-1.  Google Scholar

[33]

T. Tateno, S. Doi, S. Sato and L. M. Ricciardi, Stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise: A first-passage-time approach, J. Stat. Phys., 78 (1995), 917-935. doi: 10.1007/BF02183694.  Google Scholar

[34]

K. Yoshino, T. Nomura, K. Pakdaman and S. Sato, Synthetic analysis of periodically stimulated excitable and oscillatory membrane models, Phys. Rev. E (3), 59 (1999), 956-969. doi: 10.1103/PhysRevE.59.956.  Google Scholar

show all references

References:
[1]

D. J. Amit and N. Brunel, Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex, Cerebral Cortex, 7 (1997), 237-252. doi: 10.1093/cercor/7.3.237.  Google Scholar

[2]

M. Barbi, S. Chillemi, A. Di Garbo and L. Reale, Stochastic resonance in a sinusoidally forced LIF model with noisy threshold, Biosystems, 71 (2003), 23-28. doi: 10.1016/S0303-2647(03)00106-0.  Google Scholar

[3]

O. Bernander, C. Koch and M. Usher, The effect of synchronized inputs at the single neuron level, Neural Computation, 6 (1994), 622-641. Google Scholar

[4]

A. Buonocore, A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, A Markov chain-based model for actomyosin dynamics, Sci. Math. Japon., 70 (2009), 159-174.  Google Scholar

[5]

A. Buonocore, A. Di Crescenzo, B. Martinucci and L. M. Ricciardi, A stochastic model for the stepwise motion in actomyosin dynamics, Sci. Math. Japon., 58 (2003), 245-254.  Google Scholar

[6]

M. J. Berry and M. Meister, Refractoriness and neural precision, J. Neurosci., 18 (1998), 2200-2211. Google Scholar

[7]

A. N. Burkitt, A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input, Biol. Cybern., 95 (2006), 1-19. doi: 10.1007/s00422-006-0068-6.  Google Scholar

[8]

A. N. Burkitt, A review of the integrate-and-fire neuron model. II. Inhomogeneous synaptic input and network properties, Biol. Cybern., 95 (2006), 97-112. doi: 10.1007/s00422-006-0082-8.  Google Scholar

[9]

H. P. Chan and W.-L. Loh, Some theoretical results on neural spike train probability models, Ann. Stat., 35 (2007), 2691-2722. doi: 10.1214/009053607000000280.  Google Scholar

[10]

M. Crowder, Classical Competing Risks, Chapman & Hall/CRC, Boca Raton, 2001. doi: 10.1201/9781420035902.  Google Scholar

[11]

M. Deger, S. Cardanobile, M. Helias and S. Rotter, The Poisson process with dead time captures important statistical features of neural activity, BMC Neuroscience, 10 (2009), P110. doi: 10.1186/1471-2202-10-S1-P110.  Google Scholar

[12]

A. Di Crescenzo, E. Di Nardo, A. G. Nobile, E. Pirozzi and L. M. Ricciardi, On some computational results for single neurons' activity modeling, BioSystems, 58 (2000), 19-26. Google Scholar

[13]

A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, Stochastic population models with interacting species, J. Math. Biol., 42 (2001), 1-25. doi: 10.1007/PL00000070.  Google Scholar

[14]

A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, A note on birth-death processes with catastrophes, Stat. Prob. Lett., 78 (2008), 2248-2257. doi: 10.1016/j.spl.2008.01.093.  Google Scholar

[15]

A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, On time non-homogeneous stochastic processes with catastrophes, in Cybernetics and Systems 2010 (ed. R. Trappl), Austrian Society for Cybernetic Studies, Vienna, 2010, 169-174. Google Scholar

[16]

A. Di Crescenzo and M. Longobardi, On the NBU ageing notion within the competing risks model, J. Stat. Plann. Infer., 136 (2006), 1638-1654. doi: 10.1016/j.jspi.2004.08.022.  Google Scholar

[17]

A. Di Crescenzo and M. Longobardi, Competing risks within shock models, Sci. Math. Japon., 67 (2008), 125-135.  Google Scholar

[18]

A. Di Crescenzo, B. Martinucci, E. Pirozzi and L. M. Ricciardi, On the interaction between two Stein's neuronal units, in Cybernetics and Systems 2004 (ed. R. Trappl), Austrian Society for Cybernetic Studies, Vienna, 2004, 205-210. Google Scholar

[19]

G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophy. J., 4 (1964), 41-68. doi: 10.1016/S0006-3495(64)86768-0.  Google Scholar

[20]

D. Hampel and P. Lansky, On the estimation of refractory period, J. Neurosci. Meth., 171 (2008), 288-295. doi: 10.1016/j.jneumeth.2008.03.003.  Google Scholar

[21]

D. H. Johnson, Point process models of single-neuron discharges, J. Comput. Neurosci., 3 (1996), 275-299. doi: 10.1007/BF00161089.  Google Scholar

[22]

D. H. Johnson and A. Swami, The transmission of signals by auditory-nerve fiber discharge patterns, J. Acoust. Soc. Am., 74 (1983), 493-501. doi: 10.1121/1.389815.  Google Scholar

[23]

R. E. Kass and V. Ventura, A spike-train probability model, Neural Comput., 13 (2001), 1713-1720. doi: 10.1162/08997660152469314.  Google Scholar

[24]

A. Mazzoni, F. D. Broccard, E. Garcia-Perez, P. Bonifazi, M. E. Ruaro and V. Torre, On the dynamics of the spontaneous activity in neuronal networks, PLoS ONE, 2 (2007), e439. doi: 10.1371/journal.pone.0000439.  Google Scholar

[25]

M. I. Miller, Algorithms for removing recovery-related distortion from auditory nerve discharge patterns, J. Acoust. Soc. Am., 77 (1985), 1452-1464. doi: 10.1121/1.392040.  Google Scholar

[26]

U. Picchini, S. Ditlevsen, A. De Gaetano and P. Lansky, Parameters of the diffusion leaky integrate-and-fire neuronal model for a slowly fluctuating signal, Neural Comput., 20 (2008), 2696-2714. doi: 10.1162/neco.2008.11-07-653.  Google Scholar

[27]

L. M. Ricciardi, Diffusion Processes and Related Topics in Biology, Notes taken by Charles E. Smith, Lecture Notes in Biomathematics, Vol. 14, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[28]

L. M. Ricciardi, Modeling single neuron activity in the presence of refractoriness: New contributions to an old problem, in Imagination and Rigor. Essays on Eduardo R. Caianiello's Scientific Heritage (ed. S. Termini), Springer-Verlag Italia, 2006, 133-145. doi: 10.1007/88-470-0472-1_11.  Google Scholar

[29]

L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. G. Nobile, On the instantaneous return process for neuronal diffusion models, in Structure: from Physics to General Systems - Festschrift Volume in Honour of E.R. Caianiello on his Seventieth Birthday (eds. M. Marinaro and G. Scarpetta), World Scientific, Singapore, 1992, 78-94. Google Scholar

[30]

L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. G. Nobile, An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling, Math. Japon., 50 (1999), 247-322.  Google Scholar

[31]

W. R. Softky and C. Koch, The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs, The Journal of Neuroscience, 13 (1993), 334-350. Google Scholar

[32]

R. B. Stein, A theoretical analysis of neuronal variability, Biophys. J., 5 (1965), 173-194. doi: 10.1016/S0006-3495(65)86709-1.  Google Scholar

[33]

T. Tateno, S. Doi, S. Sato and L. M. Ricciardi, Stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise: A first-passage-time approach, J. Stat. Phys., 78 (1995), 917-935. doi: 10.1007/BF02183694.  Google Scholar

[34]

K. Yoshino, T. Nomura, K. Pakdaman and S. Sato, Synthetic analysis of periodically stimulated excitable and oscillatory membrane models, Phys. Rev. E (3), 59 (1999), 956-969. doi: 10.1103/PhysRevE.59.956.  Google Scholar

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