American Institute of Mathematical Sciences

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

Diffusion approximation of neuronal models revisited

Jakub Cupera 1,

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.
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, New York, 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), 061910, 9 pp. 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), 279-301. Google Scholar

[4]

F. B. Hanson and H. C. Tuckwell, Diffusion approximations for neuronal activity including synaptic reversal potentials, J. Theor. Neurobiol., 2 (1983), 127-153. 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), 500-544. Google Scholar

[6]

C. Koch and I. Segev, "Methods in Neuronal Modeling: From Synapses to Networks," Mass. MIT Press, Cambridge, 1989. Google Scholar

[7]

L. Kostal, Approximate information capacity of the perfect integrate-and-fire neuron using the temporal code, Brain Res., 1434 (2012), 136-141. 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), 393-406. Google Scholar

[9]

P. Lansky, On approximations of Stein's neuronal model, J. Theor. Biol., 107 (1984), 631-647. Google Scholar

[10]

P. Lánský and V. Lánská, Diffusion approximation of the neuronal model with synaptic reversal potentials, Biol. Cybern., 56 (1987), 19-26. 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), 457-465. Google Scholar

[12]

M. Musila and P. Lánský, Generalized Stein's model for anatomically complex neurons, Biosystems, 25 (1991), 179-191. 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), 225-232. 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, Lecture Notes in Biomathematics, Vol. 14, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[15]

L. M. Ricciardi and L. Sacerdote, Ornstein-Uhlenbeck process as a model for neuronal activity, Biol. Cybern., 35 (1979), 1-9. 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), 021919. 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, Springer-Verlag, Berlin, 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), 2301-2315. doi: 10.1162/0899766054796932.  Google Scholar

[19]

R. F. Schmidt, "Fundamentals of Neurophysiology," Springer-Verlag, Berlin, 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), 67-77. Google Scholar

[21]

R. B. Stein, A theoretical analysis of neuronal variability, Biophys. J., 5 (1965), 173-194. 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), 65-81. 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), 377-387. 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), 1-7. 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), 345-368. 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, New York, 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), 061910, 9 pp. 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), 279-301. Google Scholar

[4]

F. B. Hanson and H. C. Tuckwell, Diffusion approximations for neuronal activity including synaptic reversal potentials, J. Theor. Neurobiol., 2 (1983), 127-153. 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), 500-544. Google Scholar

[6]

C. Koch and I. Segev, "Methods in Neuronal Modeling: From Synapses to Networks," Mass. MIT Press, Cambridge, 1989. Google Scholar

[7]

L. Kostal, Approximate information capacity of the perfect integrate-and-fire neuron using the temporal code, Brain Res., 1434 (2012), 136-141. 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), 393-406. Google Scholar

[9]

P. Lansky, On approximations of Stein's neuronal model, J. Theor. Biol., 107 (1984), 631-647. Google Scholar

[10]

P. Lánský and V. Lánská, Diffusion approximation of the neuronal model with synaptic reversal potentials, Biol. Cybern., 56 (1987), 19-26. 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), 457-465. Google Scholar

[12]

M. Musila and P. Lánský, Generalized Stein's model for anatomically complex neurons, Biosystems, 25 (1991), 179-191. 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), 225-232. 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, Lecture Notes in Biomathematics, Vol. 14, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[15]

L. M. Ricciardi and L. Sacerdote, Ornstein-Uhlenbeck process as a model for neuronal activity, Biol. Cybern., 35 (1979), 1-9. 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), 021919. 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, Springer-Verlag, Berlin, 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), 2301-2315. doi: 10.1162/0899766054796932.  Google Scholar

[19]

R. F. Schmidt, "Fundamentals of Neurophysiology," Springer-Verlag, Berlin, 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), 67-77. Google Scholar

[21]

R. B. Stein, A theoretical analysis of neuronal variability, Biophys. J., 5 (1965), 173-194. 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), 65-81. 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), 377-387. 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), 1-7. 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), 345-368. doi: 10.1016/S0022-5193(83)80013-7.  Google Scholar

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