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2016, 13(3): 483-493. doi: 10.3934/mbe.2016002

A leaky integrate-and-fire model with adaptation for the generation of a spike train

Aniello Buonocore 1, , Luigia Caputo 2, , Enrica Pirozzi 2, and  Maria Francesca Carfora 3,

1. 

Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, 80126 Napoli
2. 

Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Cintia, Napoli
3. 

Istituto per le Appplicazioni del Calcolo "Mauro Picone", Consiglio Nazionale delle Ricerche, Via Pietro Castellino, Napoli

Received  April 2015 Revised  November 2015 Published  January 2016

A model is proposed to describe the spike-frequency adaptation observed in many neuronal systems. We assume that adaptation is mainly due to a calcium-activated potassium current, and we consider two coupled stochastic differential equations for which an analytical approach combined with simulation techniques and numerical methods allow to obtain both qualitative and quantitative results about asymptotic mean firing rate, mean calcium concentration and the firing probability density. A related algorithm, based on the Hazard Rate Method, is also devised and described.
Keywords: hazard rate method., Calcium-activated potassium current, fast-slow analysis.
Mathematics Subject Classification: Primary: 60J60; Secondary: 60H3.
Citation: Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. A leaky integrate-and-fire model with adaptation for the generation of a spike train. Mathematical Biosciences & Engineering, 2016, 13 (3) : 483-493. doi: 10.3934/mbe.2016002
References:
[1]

J. Benda and A. W. M. Herz, A universal model for spike-frequency adaptation, Neural Computation, 15 (2003), 2523-2564. doi: 10.1162/089976603322385063.

[2]

R. Brette and W. Gerstner, Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, Journal of Neurophysiology, 94 (2005), 3637-3642. doi: 10.1152/jn.00686.2005.

[3]

D. A. Brown and P. R. Adams, Muscarinic supression of a novel voltage-sensitive K+ current in a vertebrate neuron, Nature, 183 (1980), 673-676.

[4]

A. Buonocore, L. Caputo, E. Pirozzi and M. F. Carfora, A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model, Mathematical Biosciences and Engineering, 11 (2014), 1-10.

[5]

A. Buonocore, L. Caputo, E. Pirozzi and L. M. Ricciardi, On a generalized leaky integrate-and-fire model for single neuron activity, in Computer Aided Systems Theory - EUROCAST 2005, Lecture Notes in Computer Science, 5717, Springer, Berlin-Heidelberg, 2009, 152-158. doi: 10.1007/978-3-642-04772-5_21.

[6]

A. Buonocore, L. Caputo, E. Pirozzi and L. M. Ricciardi, On a stochastic leaky integrate-and-fire neuronal model, Neural Computation, 22 (2010), 2558-2585. doi: 10.1162/NECO_a_00023.

[7]

A. Buonocore , A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities, Advances in Applied Probability, 19 (1987), 784-800. doi: 10.2307/1427102.

[8]

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

[9]

M. J. Chacron, K. Pakdaman and A. Longtin, Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire model with threshold fatigue, Neural Computation, 15 (2003), 253-278. doi: 10.1162/089976603762552915.

[10]

S. M. Crook, G. B. Ermentrout and J. M. Bower, Spike frequency adaptation affects the synchronization properties of networks of cortical oscillators, Neural Computation, 10 (1998), 837-854. doi: 10.1162/089976698300017511.

[11]

Y. Dong, F. Mihalas and E. Niebur, Improved integral equation solution for the first passage time of leaky integrate-andfire neurons, Neural Computation, 23 (2011), 421-434. doi: 10.1162/NECO_a_00078.

[12]

B. Ermentrout, M. Pascal and B. Gutkin, The effects of spike frequency adaptation and negative feedback on the synchronization of neural oscillators, Neural Computation, 13 (2001), 1285-1310. doi: 10.1162/08997660152002861.

[13]

I. A. Fleidervish, A. Friedman and M. J. Gutnick, Slow inactivation of Na+ current and slow cumulative spike adaptation in mouse and guinea-pig neocortical neurones in slices, The Journal of Physiology, 493 (1996), 83-97. doi: 10.1113/jphysiol.1996.sp021366.

[14]

G. Fuhrmann, H. Markram and M. Tsodyks, Spike frequency adaptation and neocortical rhythms, Journal of Neurophysiology, 88 (2002), 761-770.

[15]

R. Granit, D. Kernell and G. K. Shortess, Quantitative aspects of repetitive firing of mammalian motoneurones, caused by injected currents, The Journal of Physiology, 168 (1963), 911-931. doi: 10.1113/jphysiol.1963.sp007230.

[16]

B. Hille, Ion Channels of Excitable Membranes, Sinauer Associates, Sunderland, MA, 2001.

[17]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology, 117 (1952), 500-544.

[18]

A. V. Holden, Models of the Stochastic Activity of Neurones, Springer-Verlag, New York, 1976.

[19]

R. Jolivet, R. Kobayashi, A. Rauch, R. Naud, S. Shinomoto and W. Gerstner, A benchmark test for a quantitative assessment of simple neuron models, Journal of Neuroscience Methods, 169 (2008), 417-424. doi: 10.1016/j.jneumeth.2007.11.006.

[20]

R. Kobayashi, Y. Tsubo and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Frontiers in Computational Neuroscience, 3 (2009), p9. doi: 10.3389/neuro.10.009.2009.

[21]

G. La Camera, A. Rauch, H. R. Luescher, W. Senn and S. Fusi, Minimal models of adapted neuronal response to in vivo-like input currents, Neural Computation, 16 (2004), 2101-2124. doi: 10.1162/0899766041732468.

[22]

Y. H. Liu and X. J. Wang, Spike-frequency adaptation of a generalized leaky integrate-and-fire model neuron, Journal of Computational Neuroscience, 10 (2001), 24-45.

[23]

D. V. Madison and R. A. Nicoll, Control of the repetitive discharge of rat CA1 pyramidal neurones in vitro, The Journal of Physiology, 354 (1984), 319-331.

[24]

A. G. Nobile, L. M. Ricciardi and L. Sacerdote, Exponential trends of Ornstein-Uhlenbeck first-passage-time densities, Journal of Applied Probability, 22 (1985), 360-369. doi: 10.2307/3213779.

[25]

R. K. Powers, A. Sawczuk, J. R. Musick and M. D. Binder, Multiple mechanisms of spike-frequency adaptation in motoneurones, Journal of Physiology, 93 (1999), 101-114. doi: 10.1016/S0928-4257(99)80141-7.

[26]

A. Rauch, G. La Camera, H. R. Luescher, W. Senn and S. Fusi, Neocortical cells respond as integrate-and-fire neurons to in vivo-like input currents, Journal of Neurophysiology, 90 (2003), 1598-1612. doi: 10.1152/jn.00293.2003.

[27]

L. Sacerdote and M. Giraudo, Stochastic integrate and fire models: A review on mathematical methods and their applications, in Stochastic Biomathematical Models with Applications to Neuronal Modeling (eds. Bachar, Batzel and Ditlevsen), Lecture Notes in Mathematics 2058, Springer, Berlin-Heidelberg (2013), 99-148. doi: 10.1007/978-3-642-32157-3_5.

[28]

P. Sah, Ca2+-activated K+ currents in neurones: Types, physiological roles and modulation, Trends in Neurosciences, 19 (1996), 150-154. doi: 10.1016/S0166-2236(96)80026-9.

[29]

S. Shinomoto, Y. Sakai and S. Funahashi, The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex, Neural Computation, 11 (1999), 935-951. doi: 10.1162/089976699300016511.

[30]

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

[31]

H. C. Tuckwell, Introduction to Theoretical Neurobiology, Vol. 2, Cambridge University Press, Cambridge, England, 1988.

show all references

References:
[1]

J. Benda and A. W. M. Herz, A universal model for spike-frequency adaptation, Neural Computation, 15 (2003), 2523-2564. doi: 10.1162/089976603322385063.

[2]

R. Brette and W. Gerstner, Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, Journal of Neurophysiology, 94 (2005), 3637-3642. doi: 10.1152/jn.00686.2005.

[3]

D. A. Brown and P. R. Adams, Muscarinic supression of a novel voltage-sensitive K+ current in a vertebrate neuron, Nature, 183 (1980), 673-676.

[4]

A. Buonocore, L. Caputo, E. Pirozzi and M. F. Carfora, A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model, Mathematical Biosciences and Engineering, 11 (2014), 1-10.

[5]

A. Buonocore, L. Caputo, E. Pirozzi and L. M. Ricciardi, On a generalized leaky integrate-and-fire model for single neuron activity, in Computer Aided Systems Theory - EUROCAST 2005, Lecture Notes in Computer Science, 5717, Springer, Berlin-Heidelberg, 2009, 152-158. doi: 10.1007/978-3-642-04772-5_21.

[6]

A. Buonocore, L. Caputo, E. Pirozzi and L. M. Ricciardi, On a stochastic leaky integrate-and-fire neuronal model, Neural Computation, 22 (2010), 2558-2585. doi: 10.1162/NECO_a_00023.

[7]

A. Buonocore , A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities, Advances in Applied Probability, 19 (1987), 784-800. doi: 10.2307/1427102.

[8]

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

[9]

M. J. Chacron, K. Pakdaman and A. Longtin, Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire model with threshold fatigue, Neural Computation, 15 (2003), 253-278. doi: 10.1162/089976603762552915.

[10]

S. M. Crook, G. B. Ermentrout and J. M. Bower, Spike frequency adaptation affects the synchronization properties of networks of cortical oscillators, Neural Computation, 10 (1998), 837-854. doi: 10.1162/089976698300017511.

[11]

Y. Dong, F. Mihalas and E. Niebur, Improved integral equation solution for the first passage time of leaky integrate-andfire neurons, Neural Computation, 23 (2011), 421-434. doi: 10.1162/NECO_a_00078.

[12]

B. Ermentrout, M. Pascal and B. Gutkin, The effects of spike frequency adaptation and negative feedback on the synchronization of neural oscillators, Neural Computation, 13 (2001), 1285-1310. doi: 10.1162/08997660152002861.

[13]

I. A. Fleidervish, A. Friedman and M. J. Gutnick, Slow inactivation of Na+ current and slow cumulative spike adaptation in mouse and guinea-pig neocortical neurones in slices, The Journal of Physiology, 493 (1996), 83-97. doi: 10.1113/jphysiol.1996.sp021366.

[14]

G. Fuhrmann, H. Markram and M. Tsodyks, Spike frequency adaptation and neocortical rhythms, Journal of Neurophysiology, 88 (2002), 761-770.

[15]

R. Granit, D. Kernell and G. K. Shortess, Quantitative aspects of repetitive firing of mammalian motoneurones, caused by injected currents, The Journal of Physiology, 168 (1963), 911-931. doi: 10.1113/jphysiol.1963.sp007230.

[16]

B. Hille, Ion Channels of Excitable Membranes, Sinauer Associates, Sunderland, MA, 2001.

[17]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology, 117 (1952), 500-544.

[18]

A. V. Holden, Models of the Stochastic Activity of Neurones, Springer-Verlag, New York, 1976.

[19]

R. Jolivet, R. Kobayashi, A. Rauch, R. Naud, S. Shinomoto and W. Gerstner, A benchmark test for a quantitative assessment of simple neuron models, Journal of Neuroscience Methods, 169 (2008), 417-424. doi: 10.1016/j.jneumeth.2007.11.006.

[20]

R. Kobayashi, Y. Tsubo and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Frontiers in Computational Neuroscience, 3 (2009), p9. doi: 10.3389/neuro.10.009.2009.

[21]

G. La Camera, A. Rauch, H. R. Luescher, W. Senn and S. Fusi, Minimal models of adapted neuronal response to in vivo-like input currents, Neural Computation, 16 (2004), 2101-2124. doi: 10.1162/0899766041732468.

[22]

Y. H. Liu and X. J. Wang, Spike-frequency adaptation of a generalized leaky integrate-and-fire model neuron, Journal of Computational Neuroscience, 10 (2001), 24-45.

[23]

D. V. Madison and R. A. Nicoll, Control of the repetitive discharge of rat CA1 pyramidal neurones in vitro, The Journal of Physiology, 354 (1984), 319-331.

[24]

A. G. Nobile, L. M. Ricciardi and L. Sacerdote, Exponential trends of Ornstein-Uhlenbeck first-passage-time densities, Journal of Applied Probability, 22 (1985), 360-369. doi: 10.2307/3213779.

[25]

R. K. Powers, A. Sawczuk, J. R. Musick and M. D. Binder, Multiple mechanisms of spike-frequency adaptation in motoneurones, Journal of Physiology, 93 (1999), 101-114. doi: 10.1016/S0928-4257(99)80141-7.

[26]

A. Rauch, G. La Camera, H. R. Luescher, W. Senn and S. Fusi, Neocortical cells respond as integrate-and-fire neurons to in vivo-like input currents, Journal of Neurophysiology, 90 (2003), 1598-1612. doi: 10.1152/jn.00293.2003.

[27]

L. Sacerdote and M. Giraudo, Stochastic integrate and fire models: A review on mathematical methods and their applications, in Stochastic Biomathematical Models with Applications to Neuronal Modeling (eds. Bachar, Batzel and Ditlevsen), Lecture Notes in Mathematics 2058, Springer, Berlin-Heidelberg (2013), 99-148. doi: 10.1007/978-3-642-32157-3_5.

[28]

P. Sah, Ca2+-activated K+ currents in neurones: Types, physiological roles and modulation, Trends in Neurosciences, 19 (1996), 150-154. doi: 10.1016/S0166-2236(96)80026-9.

[29]

S. Shinomoto, Y. Sakai and S. Funahashi, The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex, Neural Computation, 11 (1999), 935-951. doi: 10.1162/089976699300016511.

[30]

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

[31]

H. C. Tuckwell, Introduction to Theoretical Neurobiology, Vol. 2, Cambridge University Press, Cambridge, England, 1988.

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