# American Institute of Mathematical Sciences

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

 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.
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. 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. 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. doi: 10.2307/1427102. Google Scholar [4] D. R. Cox and P. A. W. Lewis, "The Statistical Analysis of Series of Events,", Methuen & Co., (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). 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). 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. 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. 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. doi: 10.1007/s004220050390. Google Scholar [10] J. Keilson and H. F. Ross, Passage time distributions for Gaussian Markov (Ornstein-Uhlenbeck) statistical processes,, in, (1975), 233. Google Scholar [11] H. Kim and S. Shinomoto, Estimating nonstationary input signals from a single neuronal spike train,, Phys. Rev. E, 86 (2012). 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. 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. doi: 10.1007/s00422-008-0237-x. Google Scholar [14] N. N. Lebedev, "Special Functions and Their Applications,", Revised edition, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (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. 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. 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. 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. 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. 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. Google Scholar [31] Y. Shu, A. Hasenstaub and D. A. McCormick, Turning on and off recurrent balanced cortical activity,, Nature, 423 (2003), 288. 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. 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. doi: 10.1162/neco.1997.9.5.971. Google Scholar [34] H. C. Tuckwell, "Introduction to Theoretical Neurobiology,", Cambridge Studies in Mathematical Biology, (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. 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. 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. doi: 10.1162/neco.2009.06-08-807. Google Scholar

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##### 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. 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. 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. doi: 10.2307/1427102. Google Scholar [4] D. R. Cox and P. A. W. Lewis, "The Statistical Analysis of Series of Events,", Methuen & Co., (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). 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). 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. 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. 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. doi: 10.1007/s004220050390. Google Scholar [10] J. Keilson and H. F. Ross, Passage time distributions for Gaussian Markov (Ornstein-Uhlenbeck) statistical processes,, in, (1975), 233. Google Scholar [11] H. Kim and S. Shinomoto, Estimating nonstationary input signals from a single neuronal spike train,, Phys. Rev. E, 86 (2012). 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. 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. doi: 10.1007/s00422-008-0237-x. Google Scholar [14] N. N. Lebedev, "Special Functions and Their Applications,", Revised edition, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (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. 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. 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. 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. 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. 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. Google Scholar [31] Y. Shu, A. Hasenstaub and D. A. McCormick, Turning on and off recurrent balanced cortical activity,, Nature, 423 (2003), 288. 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. 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. doi: 10.1162/neco.1997.9.5.971. Google Scholar [34] H. C. Tuckwell, "Introduction to Theoretical Neurobiology,", Cambridge Studies in Mathematical Biology, (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. 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. 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. doi: 10.1162/neco.2009.06-08-807. Google Scholar
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