Abstract
Five parameters of one of the most common neuronal models, the diffusion leaky integrate-and-fire model, also known as the Ornstein-Uhlenbeck neuronal model, were estimated on the basis of intracellular recording. These parameters can be classified into two categories. Three of them (the membrane time constant, the resting potential and the firing threshold) characterize the neuron itself. The remaining two characterize the neuronal input. The intracellular data were collected during spontaneous firing, which in this case is characterized by a Poisson process of interspike intervals. Two methods for the estimation were applied, the regression method and the maximum-likelihood method. Both methods permit to estimate the input parameters and the membrane time constant in a short time window (a single interspike interval). We found that, at least in our example, the regression method gave more consistent results than the maximum-likelihood method. The estimates of the input parameters show the asymptotical normality, which can be further used for statistical testing, under the condition that the data are collected in different experimental situations. The model neuron, as deduced from the determined parameters, works in a subthreshold regimen. This result was confirmed by both applied methods. The subthreshold regimen for this model is characterized by the Poissonian firing. This is in a complete agreement with the observed interspike interval data.
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References
Brillinger DR, Segundo JP (1979) Empirical examination of the threshold model of neuron firing. Biol. Cybern. 35: 213–220.
Christodoulou C, Bugmann G (2000) Near poisson-type firing produced by concurrent excitation and inhibition. BioSystems 58: 41–48.
Dayan P, Abbot LF (2001) Theoretical Neuroscience. MIT Press, Cambridge, MA.
Ditlevsen S, Lansky P (2005) Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model. Phys. Rev. E 71: Art. No. 011907.
Duchamp-Viret P, Kostal L, Chaput M, Lansky P, Rospars J-P (2005) Patterns of spontaneous activity in single rat olfactory receptor neurons are different in normally breathing and tracheotomized animals. J Neurobiol.
Eggermont JJ, Smith GM, Bowman D (1993) Spontaneous burst firing in cat primary auditory-cortex—Age and depth dependency and its effect on neural interaction measures. J. Neurophysiol. 69: 1292–1313.
Feigin P (1976) Maximum likelihood estimation for stochastic processes—A martingale approach. Adv. Appl. Probab. 8: 712–736.
Gardiner CW (1983) Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, Berlin.
Gerstner W, Kistler W (2002) Spiking Neuron Models. Cambridge University Press, Cambridge.
He J (2003) Slow oscillation in non-lemniscal auditory thalamus. J Neurosci. 23: 8281–8290.
Inoue J, Sato S, Ricciardi LM (1995) On the parameter estimation for diffusion models of single neurons' activities. Biol. Cybern. 73: 209–221.
Johnson DH (1996) Point process models of single-neuron discharges. J. Comput. Neurosci. 3: 275–300.
Jolivet R, Rauch A, Lüscher H-R, Gerstner W (2006) Integrate-and-fire models with adaptation are good enough: Predicting spike times under random current injection. In: Y Weiss, B Schölkopf, J Platt, eds. Advances in Neural Information Processing Systems 18, MIT Press, Cambridge MA.
Jones TA, Jones SM (2000) Spontaneous activity in the statoacoustic ganglion of the chicken embryo. J. Neurphysiol. 83: 1452–1468.
Keat J, Reinagel P, Reid RC, Meister M (2001) Predicting every spike: A model for the responses of visual neurons. Neuron 30: 803–817.
Kistler WM, Gerstner W, van Hemmen JL (1997) Reduction of the Hodgkin-Huxley equations to a single-variable threshold model. Neural Comput. 9: 1015–1045.
Kloeden PE, Platen E (1992) Numerical Solution of Stochastic Differential Equations. Springer, Berlin.
Koch C (1999) Biophysics of Computation: Information Processing in Single Neurons. Oxford University Press, Oxford.
Koyama S, Shinomnoto S (2005) Empirical Bayes interpretation of random point events. J. Phys. A: Math. Gen. 38: L531–L537.
La Camera G, Rauch A, Luscher HR, Senn W, Fusi S (2004) Minimal models of adapted neuronal response to in vivo-like input currents. Neural Comput. 16: 2101–2124.
Lansky P (1983) Inference for the diffusion models of neuronal activity. Math. Biosci. 67: 247–260.
Lansky P (1997) Sources of periodical force in noisy integrate-and-fire models of neuronal dynamics. Phys. Rev. E 55: 2040–2043.
Lansky P, Lanska V (1987) Diffusion approximation of the neuronal model with synaptic reversal potentials. Biol. Cybern. 56:19–26.
Lansky P, Giorno V, Nobile AG, Ricciardi LM (1998) A diffusion neuronal model and its parameters. In: LM Ricciardi, ed. Proceedings of International Workshop Biomathematics and related Computational Problems. Kluwer, Dordrecht.
Lansky P, Lanska V (1994) First-passage-time problem for simulated stochastic diffusion processes. Comp. Biol. Med. 24: 91–101.
Lansky P, Smith CE (1989) The effect of a random initial value in neural 1st-passage-time models. Math. Biosci. 93: 191–215.
Laughlin SB (2001) Energy as a constraint on the coding and processing of sensory information. Curr. Opin. Neurobiol. 11: 475–480.
Lin X, Chen SP (2000) Endogenously generated spontaneous spiking activities recorded from postnatal spiral ganglion neurons in vitro. Developmental Brain Res. 119: 297–305.
Nobile AG, Ricciardi LM, Sacerdote L (1985) Exponential trends of Ornstein-Uhlenbeck 1st-passage-time densities. J. Appl. Prob. 22: 360–369.
Paninski L, Pillow J, Simoncelli E (2005) Comparing integrate-and-fire models estimated using intracellular and extracellular data. Neurocomputing 65: 379–385.
Paninski L, Pillow JW, Simoncelli EP (2004) Maximum likelihood estimation of a stochastic integrate-and-fire neural encoding model. Neural Comput. 16: 2533–2561.
Pinsky PF, Rinzel J (1994) Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons 1: 39–60.
Prakasa Rao BLS (1999) Statistical inference for diffusion type processes. Arnold, London.
Rauch A, La Camera G, Luscher HR, Senn W, Fusi S (2003) Neocortical pyramidal cells respond as integrate-and-fire neurons to in vivo-like input currents. J. Neurophysiol. 90: 1598–1612.
Ricciardi LM, Lansky P (2003) Diffusion models of neuronal activity. In: MA Arbib, ed. The Handbook of the Brain Theory and Neural Networks, (2nd edn.) MIT Press, Cambridge, MA.
Richardson MJE, Gerstner W (2005) Synaptic shot noise and conductance fluctuations affect the membrane voltage with equal significance Neural Comput. 17: 923–947.
Rodriguez R, Lansky P (2000) Effect of spatial extension on noise enhanced phase-locking in a leaky integrate-and-fire model of a neuron. Phys. Rev. E 62: 8427–8437.
Rospars J-P, Lansky P, Vaillant J, Duchamp-Viret P, Duchamp A (1994) Spontaneous activity of first- and second-order neurons in the olfactory system. Brain Res. 662: 31–44.
Segev I (1992) Single neurone models: Oversimple, complex and reduced. TINS 15:414–421.
Shinomoto S, Sakai Y, Funahashi S (1999) The ornstein-uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex. Neural Comp. 11: 935–951.
Stein RB (1965) A theoretical analysis of neuronal variability. Biophys. J. 5: 173–195.
Stevens CF, Zador AM (1998) Novel Integrate-and-fire-like Model of repetitive firing in cortical neurons. Proceedings of the 5th Joint Symposium on Neural Computation, UCSD, La Jolla CA.
Stevens CF (1964) Letter to the editor. Biophys. J. 4: 417–419.
Tateno T, Kawana A, Jimbo Y (2002) Analytical characterization of spontaneous firing in networks of developing rat cultured cortical neurons. Phys. Rev. E 65: Art. No. 051924.
Tuckwell HC (1988) Introduction to Theoretical Neurobiology. Cambridge Univ. Press, Cambridge.
Tuckwell HC, Lansky P (1997) On the simulation of biological diffusion processes. Comput. Biol. Med. 27: 1–7.
Tuckwell HC, Richter W (1978) Neuronal interspike time distribution and the estimation of neurophysiological and neuroanatomical parameters. J. theor. Biol. 71: 167–183.
Xiong Y, Yu YQ, Chan YS He J (2003) An in-vivo intracellular study of the auditory thalamic neurons. Thalamus Related Sys. 2: 253–260.
Yu YQ, Xiong Y, Chan YS He JF (2004) Corticofugal gating of auditory information in the thalamus: An in vivo intracellular recording study. J. Neurosci. 24: 3060–3069.
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Lansky, P., Sanda, P. & He, J. The parameters of the stochastic leaky integrate-and-fire neuronal model. J Comput Neurosci 21, 211–223 (2006). https://doi.org/10.1007/s10827-006-8527-6
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DOI: https://doi.org/10.1007/s10827-006-8527-6