Diffusion approximation of neuronal models revisited

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A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model

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

Abstract
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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.
[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.
[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.
[4]	F. B. Hanson and H. C. Tuckwell, Diffusion approximations for neuronal activity including synaptic reversal potentials, J. Theor. Neurobiol., 2 (1983), 127-153.
[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.
[6]	C. Koch and I. Segev, "Methods in Neuronal Modeling: From Synapses to Networks," Mass. MIT Press, Cambridge, 1989.
[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.
[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.
[9]	P. Lansky, On approximations of Stein's neuronal model, J. Theor. Biol., 107 (1984), 631-647.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	R. F. Schmidt, "Fundamentals of Neurophysiology," Springer-Verlag, Berlin, 1978.
[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.
[21]	R. B. Stein, A theoretical analysis of neuronal variability, Biophys. J., 5 (1965), 173-194. doi: 10.1016/S0006-3495(65)86709-1.
[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.
[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.
[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.
[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.

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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.
[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.
[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.
[4]	F. B. Hanson and H. C. Tuckwell, Diffusion approximations for neuronal activity including synaptic reversal potentials, J. Theor. Neurobiol., 2 (1983), 127-153.
[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.
[6]	C. Koch and I. Segev, "Methods in Neuronal Modeling: From Synapses to Networks," Mass. MIT Press, Cambridge, 1989.
[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.
[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.
[9]	P. Lansky, On approximations of Stein's neuronal model, J. Theor. Biol., 107 (1984), 631-647.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	R. F. Schmidt, "Fundamentals of Neurophysiology," Springer-Verlag, Berlin, 1978.
[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.
[21]	R. B. Stein, A theoretical analysis of neuronal variability, Biophys. J., 5 (1965), 173-194. doi: 10.1016/S0006-3495(65)86709-1.
[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.
[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.
[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.
[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.

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