Diffusion approximation of neuronal models revisited

Previous Article
A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model
MBE Home
This Issue
Next Article
Cross nearest-spike interval based method to measure synchrony dynamics

2014, 11(1): 11-25. doi: 10.3934/mbe.2014.11.11

Diffusion approximation of neuronal models revisited

1.	Institute of Physiology, Academy of Sciences of the Czech Republic, Videnska 1083, 142 20 Prague 4, Czech Republic

Received December 2012 Revised May 2013 Published September 2013

Abstract
Related Papers

Leaky integrate-and-fire neuronal models with reversal potentials have a number of different diffusion approximations, each depending on the form of the amplitudes of the postsynaptic potentials. Probability distributions of the first-passage times of the membrane potential in the original model and its diffusion approximations are numerically compared in order to find which of the approximations is the most suitable one. The properties of the random amplitudes of postsynaptic potentials are discussed. It is shown on a simple example that the quality of the approximation depends directly on them.

Keywords: reversal potentials, Integrate-and-fire model, Stein's model, diffusion approximation..

Mathematics Subject Classification: 60J60, 60J70, 92C2.

Citation: Jakub Cupera. Diffusion approximation of neuronal models revisited. Mathematical Biosciences & Engineering, 2014, 11 (1) : 11-25. doi: 10.3934/mbe.2014.11.11

References:

[1]	J. M. Bower and D. Beeman, "The Book of GENESIS: Exploring Realistic Neural Models with the GEneral NEural Simulation System,", Springer-Verlag, (1998). Google Scholar
[2]	S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Feller neuronal model,, Phys. Rev. E (3), 73 (2006). doi: 10.1103/PhysRevE.73.061910. Google Scholar
[3]	L. C. Giancarlo, M. Giugliano, W. Senn and S. Fusi, The response of cortical neurons to in vivo-like input current: Theory and experiment,, Biol. Cybern., 99 (2008), 279. Google Scholar
[4]	F. B. Hanson and H. C. Tuckwell, Diffusion approximations for neuronal activity including synaptic reversal potentials,, J. Theor. Neurobiol., 2 (1983), 127. Google Scholar
[5]	A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve,, J. Physiol., 117 (1952), 500. Google Scholar
[6]	C. Koch and I. Segev, "Methods in Neuronal Modeling: From Synapses to Networks,", Mass. MIT Press, (1989). Google Scholar
[7]	L. Kostal, Approximate information capacity of the perfect integrate-and-fire neuron using the temporal code,, Brain Res., 1434 (2012), 136. doi: 10.1016/j.brainres.2011.07.007. Google Scholar
[8]	V. Lanska and P. Lansky and C. E. Smith, Synaptic transmission in a diffusion model for neural activity,, J. Theor. Biol., 166 (1994), 393. Google Scholar
[9]	P. Lansky, On approximations of Stein's neuronal model,, J. Theor. Biol., 107 (1984), 631. Google Scholar
[10]	P. Lánský and V. Lánská, Diffusion approximation of the neuronal model with synaptic reversal potentials,, Biol. Cybern., 56 (1987), 19. doi: 10.1007/BF00333064. Google Scholar
[11]	P. Lánský, L. Sacerdote and F. Tomassetti, On the comparison of Feller and Ornstein-Uhlenbeck models for neural activity,, Biol. Cybern., 73 (1995), 457. Google Scholar
[12]	M. Musila and P. Lánský, Generalized Stein's model for anatomically complex neurons,, Biosystems, 25 (1991), 179. doi: 10.1016/0303-2647(91)90004-5. Google Scholar
[13]	M. Musila and P. Lánský, On the interspike intervals calculated from diffusion approximations of Stein's neuronal model with reversal potentials,, J. Theor. Biol., 171 (1994), 225. doi: 10.1006/jtbi.1994.1226. Google Scholar
[14]	L. M. Ricciardi, "Diffusion Processes and Related Topics in Biology,", Notes taken by Charles E. Smith, (1977). Google Scholar
[15]	L. M. Ricciardi and L. Sacerdote, Ornstein-Uhlenbeck process as a model for neuronal activity,, Biol. Cybern., 35 (1979), 1. doi: 10.1007/BF01845839. Google Scholar
[16]	M. J. E. Richardson, Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive,, Phys. Rev. E, 76 (2007). doi: 10.1103/PhysRevE.76.021919. Google Scholar
[17]	H. Risken, "The Fokker-Planck Equation: Methods of Solution and Applications,", Springer Series in Synergetics, 18 (1989). doi: 10.1007/978-3-642-61544-3. Google Scholar
[18]	M. Rudolph and A. Destexhe, An extended analytic expression for the membrane potential distribution of conductance-based synaptic noise,, Neural Comput., 17 (2005), 2301. doi: 10.1162/0899766054796932. Google Scholar
[19]	R. F. Schmidt, "Fundamentals of Neurophysiology,", Springer-Verlag, (1978). Google Scholar
[20]	C. E. Smith and M. W. Smith, Moments of voltage trajectories for Stein's model with synaptic reversal potentials,, J. Theor. Neurobiol., 3 (1984), 67. Google Scholar
[21]	R. B. Stein, A theoretical analysis of neuronal variability,, Biophys. J., 5 (1965), 173. doi: 10.1016/S0006-3495(65)86709-1. Google Scholar
[22]	H. C. Tuckwell, Synaptic transmission in a model for stochastic neural activity,, J. Theor. Biol., 77 (1979), 65. doi: 10.1016/0022-5193(79)90138-3. Google Scholar
[23]	H. C. Tuckwell and D. K. Cope, Accuracy of neuronal interspike times calculated from a diffusion approximation,, J. Theor. Biol., 83 (1980), 377. doi: 10.1016/0022-5193(80)90045-4. Google Scholar
[24]	H. C. Tuckwell and P. Lánský, On the simulation of biological diffusion processes,, Comput. Biol. Med., 27 (1997), 1. doi: 10.1016/S0010-4825(96)00033-9. Google Scholar
[25]	W. J. Wilbur and J. Rinzel, A theoretical basis for large coefficient of variation and bimodality in neuronal interspike interval distributions,, J. Theor. Biol., 105 (1983), 345. doi: 10.1016/S0022-5193(83)80013-7. Google Scholar

show all references

References:

[1]	J. M. Bower and D. Beeman, "The Book of GENESIS: Exploring Realistic Neural Models with the GEneral NEural Simulation System,", Springer-Verlag, (1998). Google Scholar
[2]	S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Feller neuronal model,, Phys. Rev. E (3), 73 (2006). doi: 10.1103/PhysRevE.73.061910. Google Scholar
[3]	L. C. Giancarlo, M. Giugliano, W. Senn and S. Fusi, The response of cortical neurons to in vivo-like input current: Theory and experiment,, Biol. Cybern., 99 (2008), 279. Google Scholar
[4]	F. B. Hanson and H. C. Tuckwell, Diffusion approximations for neuronal activity including synaptic reversal potentials,, J. Theor. Neurobiol., 2 (1983), 127. Google Scholar
[5]	A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve,, J. Physiol., 117 (1952), 500. Google Scholar
[6]	C. Koch and I. Segev, "Methods in Neuronal Modeling: From Synapses to Networks,", Mass. MIT Press, (1989). Google Scholar
[7]	L. Kostal, Approximate information capacity of the perfect integrate-and-fire neuron using the temporal code,, Brain Res., 1434 (2012), 136. doi: 10.1016/j.brainres.2011.07.007. Google Scholar
[8]	V. Lanska and P. Lansky and C. E. Smith, Synaptic transmission in a diffusion model for neural activity,, J. Theor. Biol., 166 (1994), 393. Google Scholar
[9]	P. Lansky, On approximations of Stein's neuronal model,, J. Theor. Biol., 107 (1984), 631. Google Scholar
[10]	P. Lánský and V. Lánská, Diffusion approximation of the neuronal model with synaptic reversal potentials,, Biol. Cybern., 56 (1987), 19. doi: 10.1007/BF00333064. Google Scholar
[11]	P. Lánský, L. Sacerdote and F. Tomassetti, On the comparison of Feller and Ornstein-Uhlenbeck models for neural activity,, Biol. Cybern., 73 (1995), 457. Google Scholar
[12]	M. Musila and P. Lánský, Generalized Stein's model for anatomically complex neurons,, Biosystems, 25 (1991), 179. doi: 10.1016/0303-2647(91)90004-5. Google Scholar
[13]	M. Musila and P. Lánský, On the interspike intervals calculated from diffusion approximations of Stein's neuronal model with reversal potentials,, J. Theor. Biol., 171 (1994), 225. doi: 10.1006/jtbi.1994.1226. Google Scholar
[14]	L. M. Ricciardi, "Diffusion Processes and Related Topics in Biology,", Notes taken by Charles E. Smith, (1977). Google Scholar
[15]	L. M. Ricciardi and L. Sacerdote, Ornstein-Uhlenbeck process as a model for neuronal activity,, Biol. Cybern., 35 (1979), 1. doi: 10.1007/BF01845839. Google Scholar
[16]	M. J. E. Richardson, Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive,, Phys. Rev. E, 76 (2007). doi: 10.1103/PhysRevE.76.021919. Google Scholar
[17]	H. Risken, "The Fokker-Planck Equation: Methods of Solution and Applications,", Springer Series in Synergetics, 18 (1989). doi: 10.1007/978-3-642-61544-3. Google Scholar
[18]	M. Rudolph and A. Destexhe, An extended analytic expression for the membrane potential distribution of conductance-based synaptic noise,, Neural Comput., 17 (2005), 2301. doi: 10.1162/0899766054796932. Google Scholar
[19]	R. F. Schmidt, "Fundamentals of Neurophysiology,", Springer-Verlag, (1978). Google Scholar
[20]	C. E. Smith and M. W. Smith, Moments of voltage trajectories for Stein's model with synaptic reversal potentials,, J. Theor. Neurobiol., 3 (1984), 67. Google Scholar
[21]	R. B. Stein, A theoretical analysis of neuronal variability,, Biophys. J., 5 (1965), 173. doi: 10.1016/S0006-3495(65)86709-1. Google Scholar
[22]	H. C. Tuckwell, Synaptic transmission in a model for stochastic neural activity,, J. Theor. Biol., 77 (1979), 65. doi: 10.1016/0022-5193(79)90138-3. Google Scholar
[23]	H. C. Tuckwell and D. K. Cope, Accuracy of neuronal interspike times calculated from a diffusion approximation,, J. Theor. Biol., 83 (1980), 377. doi: 10.1016/0022-5193(80)90045-4. Google Scholar
[24]	H. C. Tuckwell and P. Lánský, On the simulation of biological diffusion processes,, Comput. Biol. Med., 27 (1997), 1. doi: 10.1016/S0010-4825(96)00033-9. Google Scholar
[25]	W. J. Wilbur and J. Rinzel, A theoretical basis for large coefficient of variation and bimodality in neuronal interspike interval distributions,, J. Theor. Biol., 105 (1983), 345. doi: 10.1016/S0022-5193(83)80013-7. Google Scholar

[1]	Michele Barbi, Angelo Di Garbo, Rita Balocchi. Improved integrate-and-fire model for RSA. Mathematical Biosciences & Engineering, 2007, 4 (4) : 609-615. doi: 10.3934/mbe.2007.4.609
[2]	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
[3]	Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (1) : 1-10. doi: 10.3934/mbe.2014.11.1
[4]	Katarzyna PichÓr, Ryszard Rudnicki. Stability of stochastic semigroups and applications to Stein's neuronal model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 377-385. doi: 10.3934/dcdsb.2018026
[5]	Timothy J. Lewis. Phase-locking in electrically coupled non-leaky integrate-and-fire neurons. Conference Publications, 2003, 2003 (Special) : 554-562. doi: 10.3934/proc.2003.2003.554
[6]	Qing-Qing Yang, Wai-Ki Ching, Wanhua He, Tak-Kuen Siu. Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales. Journal of Industrial & Management Optimization, 2019, 15 (1) : 293-318. doi: 10.3934/jimo.2018044
[7]	Stephen Thompson, Thomas I. Seidman. Approximation of a semigroup model of anomalous diffusion in a bounded set. Evolution Equations & Control Theory, 2013, 2 (1) : 173-192. doi: 10.3934/eect.2013.2.173
[8]	Benoît Perthame, Delphine Salort. On a voltage-conductance kinetic system for integrate & fire neural networks. Kinetic & Related Models, 2013, 6 (4) : 841-864. doi: 10.3934/krm.2013.6.841
[9]	Roberta Sirovich, Luisa Testa. A new firing paradigm for integrate and fire stochastic neuronal models. Mathematical Biosciences & Engineering, 2016, 13 (3) : 597-611. doi: 10.3934/mbe.2016010
[10]	Frank Jochmann. Power-law approximation of Bean's critical-state model with displacement current. Conference Publications, 2011, 2011 (Special) : 747-753. doi: 10.3934/proc.2011.2011.747
[11]	Corinna Burkard, Aurelia Minut, Karim Ramdani. Far field model for time reversal and application to selective focusing on small dielectric inhomogeneities. Inverse Problems & Imaging, 2013, 7 (2) : 445-470. doi: 10.3934/ipi.2013.7.445
[12]	Pierre Guiraud, Etienne Tanré. Stability of synchronization under stochastic perturbations in leaky integrate and fire neural networks of finite size. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5183-5201. doi: 10.3934/dcdsb.2019056
[13]	Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015
[14]	Peng Jiang. Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2045-2063. doi: 10.3934/dcds.2017087
[15]	Beata Jackowska-Zduniak, Urszula Foryś. Mathematical model of the atrioventricular nodal double response tachycardia and double-fire pathology. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1143-1158. doi: 10.3934/mbe.2016035
[16]	Wei-Ming Ni, Yaping Wu, Qian Xu. The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5271-5298. doi: 10.3934/dcds.2014.34.5271
[17]	Monica De Angelis, Gaetano Fiore. Diffusion effects in a superconductive model. Communications on Pure & Applied Analysis, 2014, 13 (1) : 217-223. doi: 10.3934/cpaa.2014.13.217
[18]	Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model. Mathematical Biosciences & Engineering, 2013, 10 (3) : 637-647. doi: 10.3934/mbe.2013.10.637
[19]	Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Fast algorithms for the approximation of a traffic flow model on networks. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 427-448. doi: 10.3934/dcdsb.2006.6.427
[20]	Jishan Fan, Tohru Ozawa. An approximation model for the density-dependent magnetohydrodynamic equations. Conference Publications, 2013, 2013 (special) : 207-216. doi: 10.3934/proc.2013.2013.207

2018 Impact Factor: 1.313

American Institute of Mathematical Sciences