American Institute of Mathematical Sciences

Advanced
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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[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.

[21]

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

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[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.

[21]

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

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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.

[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.

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