American Institute of Mathematical Sciences

Advanced
2014, 11(1): 49-62. doi: 10.3934/mbe.2014.11.49

Estimating nonstationary inputs from a single spike train based on a neuron model with adaptation

Hideaki Kim 1, and  Shigeru Shinomoto 2,

1. 

NTT Service Evolution Laboratories, NTT Corporation, Yokosuka-shi, Kanagawa, 239-0847, Japan
2. 

Department of Physics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan

Received  December 2012 Revised  June 2013 Published  September 2013

Because every spike of a neuron is determined by input signals, a train of spikes may contain information about the dynamics of unobserved neurons. A state-space method based on the leaky integrate-and-fire model, describing neuronal transformation from input signals to a spike train has been proposed for tracking input parameters represented by their mean and fluctuation [11]. In the present paper, we propose to make the estimation more realistic by adopting an LIF model augmented with an adaptive moving threshold. Moreover, because the direct state-space method is computationally infeasible for a data set comprising thousands of spikes, we further develop a practical method for transforming instantaneous firing characteristics back to input parameters. The instantaneous firing characteristics, represented by the firing rate and non-Poisson irregularity, can be estimated using a computationally feasible algorithm. We applied our proposed methods to synthetic data to clarify that they perform well.
Keywords: method of moments, spike frequency adaptation, Input estimation, neuronal model, spike data, state-space method, Bayesian analysis, first-passage time..
Mathematics Subject Classification: Primary: 60H30, 62P10; Secondary: 65C3.
Citation: Hideaki Kim, Shigeru Shinomoto. Estimating nonstationary inputs from a single spike train based on a neuron model with adaptation. Mathematical Biosciences & Engineering, 2014, 11 (1) : 49-62. doi: 10.3934/mbe.2014.11.49
References:
[1]

O. Avila-Akerberg and M. J. Chacron, Nonrenewal spike train statistics: Causes and functional consequences on neural coding, Exp. Brain Res., 210 (2011), 353-371. Google Scholar

[2]

J. Benda, L. Maler and A. Longtin, Linear versus nonlinear signal transmission in neuron models with adaptation currents or dynamic thresholds, J. Neurophysiol., 104 (2010), 2806-2820. doi: 10.1152/jn.00240.2010.  Google Scholar

[3]

A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities, Adv. Appl. Probab., 19 (1987), 784-800. doi: 10.2307/1427102.  Google Scholar

[4]

D. R. Cox and P. A. W. Lewis, "The Statistical Analysis of Series of Events," Methuen & Co., Ltd., London; John Wiley & Sons, Inc., New York, 1966.  Google Scholar

[5]

S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model, Phys. Rev. E, 71 (2005), 011907, 9 pp. doi: 10.1103/PhysRevE.71.011907.  Google Scholar

[6]

F. Farkhooi, M. F. Strube-Bloss and M. P. Nawrot, Serial correlation in neural spike trains: Experimental evidence, stochastic modeling, and single neuron variability, Phys. Rev. E, 79 (2009), 021905. doi: 10.1103/PhysRevE.79.021905.  Google Scholar

[7]

M. J. Higley and D. Contreras, Balanced excitation and inhibition determine spike timing during frequency adaptation, J. Neurosci., 26 (2006), 448-457. doi: 10.1523/JNEUROSCI.3506-05.2006.  Google Scholar

[8]

J. Inoue, S. Sato and L. M. Ricciardi, On the parameter estimation for diffusion models of single neuron's activities, Biol. Cybern., 73 (1995), 209-221. doi: 10.1007/BF00201423.  Google Scholar

[9]

S. Iyengar and Q. Liao, Modeling neural activity using the generalized inverse Gaussian distribution, Biol. Cybern., 77 (1997), 289-295. doi: 10.1007/s004220050390.  Google Scholar

[10]

J. Keilson and H. F. Ross, Passage time distributions for Gaussian Markov (Ornstein-Uhlenbeck) statistical processes, in "Selected tables in mathematical statistics, Vol. III," Amer. Math. Soc., Providence, RI, (1975), 233-327.  Google Scholar

[11]

H. Kim and S. Shinomoto, Estimating nonstationary input signals from a single neuronal spike train, Phys. Rev. E, 86 (2012), 051903. doi: 10.1103/PhysRevE.86.051903.  Google Scholar

[12]

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

[13]

P. Lánský and S. Ditlevsen, A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models, Biol. Cybern., 99 (2008), 253-262. doi: 10.1007/s00422-008-0237-x.  Google Scholar

[14]

N. N. Lebedev, "Special Functions and Their Applications," Revised edition, Dover Publications, Inc., New York, 1972.  Google Scholar

[15]

B. Lindner and A. Longtin, Effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron, J. Theor. Biol., 232 (2005), 505-521. doi: 10.1016/j.jtbi.2004.08.030.  Google Scholar

[16]

Y.-H. Liu and X.-J. Wang, Spike-frequency adaptation of a generalized leaky integrate-and-fire model neuron, J. Comput. Neurosci., 10 (2001), 25-45. Google Scholar

[17]

A. Mason and A. Larkman, Correlations between morphology and electrophysiology of pyramidal neurons in slices of rat visual cortex. II. Electrophysiology, J. Neurosci., 10 (1990), 1415-1428. Google Scholar

[18]

A. Mason, A. Nicoll and K. Stratford, Synaptic transmission between individual pyramidal neurons of the rat visual cortex in vitro, J. Neurosci., 11 (1991), 72-84. Google Scholar

[19]

D. A. McCormick, B. W. Connors, J. W. Lighthall and D. A. Prince, Comparative electrophysiology of pyramidal and sparsely spiny stellate neurons of the neocortex, J. Neurophysiol., 54 (1985), 782-806. Google Scholar

[20]

P. Mullowney and S. Iyengar, Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data, J. Comput. Neurosci., 24 (2008), 179-194. doi: 10.1007/s10827-007-0047-5.  Google Scholar

[21]

M. P. Nawrot, C. Boucsein, V. Rodriguez-Molina, A. Aertsen, S. Grün and S. Rotter, Serial interval statistics of spontaneous activity in cortical neurons in vivo and in vitro, Neurocomput., 70 (2007), 1717-1722. doi: 10.1016/j.neucom.2006.10.101.  Google Scholar

[22]

L. Paninski, J. W. Pillow and E. P. Simoncelli, Maximum likelihood estimation of a stochastic integrate-and-fire neural encoding model, Neural Comput., 16 (2004), 2533-2561. doi: 10.1162/0899766042321797.  Google Scholar

[23]

L. Paninski, A. Haith and G. Szirtes, Integral equation methods for computing likelihoods and their derivatives in the stochastic integrate-and-fire model, J. Comput. Neurosci., 24 (2008), 69-79. doi: 10.1007/s10827-007-0042-x.  Google Scholar

[24]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes in C: The Art of Scientific Computing," $2^{nd}$ edition, Cambridge University Press, Cambridge, 1992.  Google Scholar

[25]

L. M. Ricciardi and S. Sato, First-passage-time density and moments of the Ornstein-Uhlenbeck process, J. Appl. Prob., 25 (1988), 43-57. doi: 10.2307/3214232.  Google Scholar

[26]

Y. Sakai, S. Funahashi and S. Shinomoto, Temporally correlated inputs to leaky integrate-and-fire models can reproduce spiking statistics of cortical neurons, Neural Netw., 12 (1999), 1181-1190. doi: 10.1016/S0893-6080(99)00053-2.  Google Scholar

[27]

T. Shimokawa and S. Shinomoto, Estimating instantaneous irregularity of neuronal firing, Neural Comput., 21 (2009), 1931-1951. doi: 10.1162/neco.2009.08-08-841.  Google Scholar

[28]

S. Shinomoto, Y. Sakai and S. Funahashi, The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex, Neural Comput., 11 (1999), 935-951. doi: 10.1162/089976699300016511.  Google Scholar

[29]

A. Smith and E. Brown, Estimating a state-space model from point process observations, Neural Comput., 15 (2003), 965-991. doi: 10.1162/089976603765202622.  Google Scholar

[30]

W. R. Softky and C. Koch, The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs, J. Neurosci., 13 (1993), 334-350. Google Scholar

[31]

Y. Shu, A. Hasenstaub and D. A. McCormick, Turning on and off recurrent balanced cortical activity, Nature, 423 (2003), 288-293. doi: 10.1038/nature01616.  Google Scholar

[32]

C. F. Stevens and A. M. Zador, Input synchrony and the irregular firing of cortical neurons, Nat. Neurosci., 1 (1998), 210-217. doi: 10.1038/659.  Google Scholar

[33]

T. W. Troyer and K. D. Miller, Physiological gain leads to high ISI variability in a simple model of a cortical regular spiking cell, Neural Comput., 9 (1997), 971-983. doi: 10.1162/neco.1997.9.5.971.  Google Scholar

[34]

H. C. Tuckwell, "Introduction to Theoretical Neurobiology," Cambridge Studies in Mathematical Biology, No. 8, Cambridge University Press, Cambridge, 1988. doi: 10.1017/CBO9780511623271.  Google Scholar

[35]

R. D. Vilela and B. Lindner, Are the input parameters of white noise driven integrate and fire neurons uniquely determined by rate and CV?, J. Theor. Biol., 257 (2009), 90-99. doi: 10.1016/j.jtbi.2008.11.004.  Google Scholar

[36]

M. Wehr and A. M. Zador, Balanced inhibition underlies tuning and sharpens spike timing in auditory cortex, Nature, 426 (2003), 442-446. doi: 10.1038/nature02116.  Google Scholar

[37]

X. Zhang, G. You, T. Chen and J. Feng, Maximum likelihood decoding of neuronal inputs from an interspike interval distribution, Neural Comput., 21 (2009), 3079-3105. doi: 10.1162/neco.2009.06-08-807.  Google Scholar

show all references

References:
[1]

O. Avila-Akerberg and M. J. Chacron, Nonrenewal spike train statistics: Causes and functional consequences on neural coding, Exp. Brain Res., 210 (2011), 353-371. Google Scholar

[2]

J. Benda, L. Maler and A. Longtin, Linear versus nonlinear signal transmission in neuron models with adaptation currents or dynamic thresholds, J. Neurophysiol., 104 (2010), 2806-2820. doi: 10.1152/jn.00240.2010.  Google Scholar

[3]

A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities, Adv. Appl. Probab., 19 (1987), 784-800. doi: 10.2307/1427102.  Google Scholar

[4]

D. R. Cox and P. A. W. Lewis, "The Statistical Analysis of Series of Events," Methuen & Co., Ltd., London; John Wiley & Sons, Inc., New York, 1966.  Google Scholar

[5]

S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model, Phys. Rev. E, 71 (2005), 011907, 9 pp. doi: 10.1103/PhysRevE.71.011907.  Google Scholar

[6]

F. Farkhooi, M. F. Strube-Bloss and M. P. Nawrot, Serial correlation in neural spike trains: Experimental evidence, stochastic modeling, and single neuron variability, Phys. Rev. E, 79 (2009), 021905. doi: 10.1103/PhysRevE.79.021905.  Google Scholar

[7]

M. J. Higley and D. Contreras, Balanced excitation and inhibition determine spike timing during frequency adaptation, J. Neurosci., 26 (2006), 448-457. doi: 10.1523/JNEUROSCI.3506-05.2006.  Google Scholar

[8]

J. Inoue, S. Sato and L. M. Ricciardi, On the parameter estimation for diffusion models of single neuron's activities, Biol. Cybern., 73 (1995), 209-221. doi: 10.1007/BF00201423.  Google Scholar

[9]

S. Iyengar and Q. Liao, Modeling neural activity using the generalized inverse Gaussian distribution, Biol. Cybern., 77 (1997), 289-295. doi: 10.1007/s004220050390.  Google Scholar

[10]

J. Keilson and H. F. Ross, Passage time distributions for Gaussian Markov (Ornstein-Uhlenbeck) statistical processes, in "Selected tables in mathematical statistics, Vol. III," Amer. Math. Soc., Providence, RI, (1975), 233-327.  Google Scholar

[11]

H. Kim and S. Shinomoto, Estimating nonstationary input signals from a single neuronal spike train, Phys. Rev. E, 86 (2012), 051903. doi: 10.1103/PhysRevE.86.051903.  Google Scholar

[12]

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

[13]

P. Lánský and S. Ditlevsen, A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models, Biol. Cybern., 99 (2008), 253-262. doi: 10.1007/s00422-008-0237-x.  Google Scholar

[14]

N. N. Lebedev, "Special Functions and Their Applications," Revised edition, Dover Publications, Inc., New York, 1972.  Google Scholar

[15]

B. Lindner and A. Longtin, Effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron, J. Theor. Biol., 232 (2005), 505-521. doi: 10.1016/j.jtbi.2004.08.030.  Google Scholar

[16]

Y.-H. Liu and X.-J. Wang, Spike-frequency adaptation of a generalized leaky integrate-and-fire model neuron, J. Comput. Neurosci., 10 (2001), 25-45. Google Scholar

[17]

A. Mason and A. Larkman, Correlations between morphology and electrophysiology of pyramidal neurons in slices of rat visual cortex. II. Electrophysiology, J. Neurosci., 10 (1990), 1415-1428. Google Scholar

[18]

A. Mason, A. Nicoll and K. Stratford, Synaptic transmission between individual pyramidal neurons of the rat visual cortex in vitro, J. Neurosci., 11 (1991), 72-84. Google Scholar

[19]

D. A. McCormick, B. W. Connors, J. W. Lighthall and D. A. Prince, Comparative electrophysiology of pyramidal and sparsely spiny stellate neurons of the neocortex, J. Neurophysiol., 54 (1985), 782-806. Google Scholar

[20]

P. Mullowney and S. Iyengar, Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data, J. Comput. Neurosci., 24 (2008), 179-194. doi: 10.1007/s10827-007-0047-5.  Google Scholar

[21]

M. P. Nawrot, C. Boucsein, V. Rodriguez-Molina, A. Aertsen, S. Grün and S. Rotter, Serial interval statistics of spontaneous activity in cortical neurons in vivo and in vitro, Neurocomput., 70 (2007), 1717-1722. doi: 10.1016/j.neucom.2006.10.101.  Google Scholar

[22]

L. Paninski, J. W. Pillow and E. P. Simoncelli, Maximum likelihood estimation of a stochastic integrate-and-fire neural encoding model, Neural Comput., 16 (2004), 2533-2561. doi: 10.1162/0899766042321797.  Google Scholar

[23]

L. Paninski, A. Haith and G. Szirtes, Integral equation methods for computing likelihoods and their derivatives in the stochastic integrate-and-fire model, J. Comput. Neurosci., 24 (2008), 69-79. doi: 10.1007/s10827-007-0042-x.  Google Scholar

[24]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes in C: The Art of Scientific Computing," $2^{nd}$ edition, Cambridge University Press, Cambridge, 1992.  Google Scholar

[25]

L. M. Ricciardi and S. Sato, First-passage-time density and moments of the Ornstein-Uhlenbeck process, J. Appl. Prob., 25 (1988), 43-57. doi: 10.2307/3214232.  Google Scholar

[26]

Y. Sakai, S. Funahashi and S. Shinomoto, Temporally correlated inputs to leaky integrate-and-fire models can reproduce spiking statistics of cortical neurons, Neural Netw., 12 (1999), 1181-1190. doi: 10.1016/S0893-6080(99)00053-2.  Google Scholar

[27]

T. Shimokawa and S. Shinomoto, Estimating instantaneous irregularity of neuronal firing, Neural Comput., 21 (2009), 1931-1951. doi: 10.1162/neco.2009.08-08-841.  Google Scholar

[28]

S. Shinomoto, Y. Sakai and S. Funahashi, The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex, Neural Comput., 11 (1999), 935-951. doi: 10.1162/089976699300016511.  Google Scholar

[29]

A. Smith and E. Brown, Estimating a state-space model from point process observations, Neural Comput., 15 (2003), 965-991. doi: 10.1162/089976603765202622.  Google Scholar

[30]

W. R. Softky and C. Koch, The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs, J. Neurosci., 13 (1993), 334-350. Google Scholar

[31]

Y. Shu, A. Hasenstaub and D. A. McCormick, Turning on and off recurrent balanced cortical activity, Nature, 423 (2003), 288-293. doi: 10.1038/nature01616.  Google Scholar

[32]

C. F. Stevens and A. M. Zador, Input synchrony and the irregular firing of cortical neurons, Nat. Neurosci., 1 (1998), 210-217. doi: 10.1038/659.  Google Scholar

[33]

T. W. Troyer and K. D. Miller, Physiological gain leads to high ISI variability in a simple model of a cortical regular spiking cell, Neural Comput., 9 (1997), 971-983. doi: 10.1162/neco.1997.9.5.971.  Google Scholar

[34]

H. C. Tuckwell, "Introduction to Theoretical Neurobiology," Cambridge Studies in Mathematical Biology, No. 8, Cambridge University Press, Cambridge, 1988. doi: 10.1017/CBO9780511623271.  Google Scholar

[35]

R. D. Vilela and B. Lindner, Are the input parameters of white noise driven integrate and fire neurons uniquely determined by rate and CV?, J. Theor. Biol., 257 (2009), 90-99. doi: 10.1016/j.jtbi.2008.11.004.  Google Scholar

[36]

M. Wehr and A. M. Zador, Balanced inhibition underlies tuning and sharpens spike timing in auditory cortex, Nature, 426 (2003), 442-446. doi: 10.1038/nature02116.  Google Scholar

[37]

X. Zhang, G. You, T. Chen and J. Feng, Maximum likelihood decoding of neuronal inputs from an interspike interval distribution, Neural Comput., 21 (2009), 3079-3105. doi: 10.1162/neco.2009.06-08-807.  Google Scholar

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