American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Successive spike times predicted by a stochastic neuronal model with a variable input signal
  • MBE Home
  • This Issue
  • Next Article
    The effect of positive interspike interval correlations on neuronal information transmission
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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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. Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[31]

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

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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. Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[31]

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

[1]

Alexandre Caboussat, Allison Leonard. Numerical solution and fast-slow decomposition of a population of weakly coupled systems. Conference Publications, 2009, 2009 (Special) : 123-132. doi: 10.3934/proc.2009.2009.123
[2]

Daniele Andreucci, Dario Bellaveglia, Emilio N.M. Cirillo, Silvia Marconi. Effect of intracellular diffusion on current--voltage curves in potassium channels. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1837-1853. doi: 10.3934/dcdsb.2014.19.1837
[3]

Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112
[4]

C. Xiong, J.P. Miller, F. Gao, Y. Yan, J.C. Morris. Testing increasing hazard rate for the progression time of dementia. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 813-821. doi: 10.3934/dcdsb.2004.4.813
[5]

C. Connell Mccluskey. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences & Engineering, 2006, 3 (4) : 603-614. doi: 10.3934/mbe.2006.3.603
[6]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184
[7]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[8]

Yong Zhang, Francis Y. L. Chin, Francis C. M. Lau, Haisheng Tan, Hing-Fung Ting. Constant competitive algorithms for unbounded one-Way trading under monotone hazard rate. Mathematical Foundations of Computing, 2018, 1 (4) : 383-392. doi: 10.3934/mfc.2018019
[9]

Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2233-2256. doi: 10.3934/dcdsb.2015.20.2233
[10]

Seung-Yeal Ha, Dohyun Kim, Jinyeong Park. Fast and slow velocity alignments in a Cucker-Smale ensemble with adaptive couplings. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4621-4654. doi: 10.3934/cpaa.2020209
[11]

Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021
[12]

Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257
[13]

Renato Huzak. Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 171-215. doi: 10.3934/dcds.2016.36.171
[14]

Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069
[15]

Chunlai Mu, Jun Zhou, Yuhuan Li. Fast rate of dead core for fast diffusion equation with strong absorption. Communications on Pure & Applied Analysis, 2010, 9 (2) : 397-411. doi: 10.3934/cpaa.2010.9.397
[16]

Yaonan Ma, Li-Zhi Liao. The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1681-1711. doi: 10.3934/jimo.2020040
[17]

Marek Fila, Michael Winkler. Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation. Communications on Pure & Applied Analysis, 2015, 14 (1) : 107-119. doi: 10.3934/cpaa.2015.14.107
[18]

Ilya Schurov. Duck farming on the two-torus: Multiple canard cycles in generic slow-fast systems. Conference Publications, 2011, 2011 (Special) : 1289-1298. doi: 10.3934/proc.2011.2011.1289
[19]

Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure & Applied Analysis, 2016, 15 (1) : 219-241. doi: 10.3934/cpaa.2016.15.219
[20]

Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences