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

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.
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]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[2]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[3]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[4]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[5]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464

[6]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[7]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

[8]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344

[9]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349

[10]

Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020459

[11]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[12]

Yolanda Guerrero–Sánchez, Muhammad Umar, Zulqurnain Sabir, Juan L. G. Guirao, Muhammad Asif Zahoor Raja. Solving a class of biological HIV infection model of latently infected cells using heuristic approach. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020431

[13]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275

[14]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

[15]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[16]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[17]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[18]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[19]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[20]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]