×

Estimating parameters of generalized integrate-and-fire neurons from the maximum likelihood of spike trains. (English) Zbl 1302.92022

Summary: When a neuronal spike train is observed, what can we deduce from it about the properties of the neuron that generated it? A natural way to answer this question is to make an assumption about the type of neuron, select an appropriate model for this type, and then choose the model parameters as those that are most likely to generate the observed spike train. This is the maximum likelihood method. If the neuron obeys simple integrate-and-fire dynamics, L. Paninski et al. [Neural Comput. 16, No. 12, 2533–2561 (2004; Zbl 1180.62179)] showed that its negative log-likelihood function is convex and that, at least in principle, its unique global minimum can thus be found by gradient descent techniques. Many biological neurons are, however, known to generate a richer repertoire of spiking behaviors than can be explained in a simple integrate-and-fire model. For instance, such a model retains only an implicit (through spike-induced currents), not an explicit, memory of its input; an example of a physiological situation that cannot be explained is the absence of firing if the input current is increased very slowly. Therefore, we use an expanded model, which is capable of generating a large number of complex firing patterns while still being linear. Linearity is important because it maintains the distribution of the random variables and still allows maximum likelihood methods to be used. In this study, we show that although convexity of the negative log-likelihood function is not guaranteed for this model, the minimum of this function yields a good estimate for the model parameters, in particular if the noise level is treated as a free parameter. Furthermore, we show that a nonlinear function minimization method (\(r\)-algorithm with space dilation) usually reaches the global minimum.

MSC:

92C20 Neural biology

Citations:

Zbl 1180.62179
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1152/jn.00686.2005 · doi:10.1152/jn.00686.2005
[2] DOI: 10.1201/9780203494462.ch9 · doi:10.1201/9780203494462.ch9
[3] DOI: 10.2307/1427102 · Zbl 0632.60079 · doi:10.2307/1427102
[4] DOI: 10.1093/imanum/drn008 · Zbl 1185.90166 · doi:10.1093/imanum/drn008
[5] DOI: 10.1007/s00422-006-0068-6 · Zbl 1161.92315 · doi:10.1007/s00422-006-0068-6
[6] DOI: 10.1162/NECO_a_00078 · Zbl 1208.92011 · doi:10.1162/NECO_a_00078
[7] Efstratiadis A., Proceedings of the Fifth International Conference on Hydroinformatis pp 1423– (2002)
[8] Figueiredo J., Rev. Brasil Cienc. Mec./J. Braz. Soc. Mesh. Science 19 pp 371– (1997)
[9] DOI: 10.1126/science.1181936 · doi:10.1126/science.1181936
[10] Hille B., Ionic channels of excitable membranes (1992)
[11] DOI: 10.1109/TNN.2003.820440 · doi:10.1109/TNN.2003.820440
[12] DOI: 10.1016/j.jneumeth.2007.11.006 · doi:10.1016/j.jneumeth.2007.11.006
[13] DOI: 10.1109/TBCAS.2009.2032396 · doi:10.1109/TBCAS.2009.2032396
[14] DOI: 10.1162/neco.1997.9.5.1015 · doi:10.1162/neco.1997.9.5.1015
[15] Lapicque L., J. Physiol. Pathol. Gen. 9 pp 620– (1907)
[16] DOI: 10.1007/BF02478259 · Zbl 0063.03860 · doi:10.1007/BF02478259
[17] DOI: 10.1109/CISS.2011.5766209 · doi:10.1109/CISS.2011.5766209
[18] DOI: 10.1162/neco.2008.12-07-680 · Zbl 1156.92008 · doi:10.1162/neco.2008.12-07-680
[19] DOI: 10.1162/089976603321192068 · Zbl 1084.68733 · doi:10.1162/089976603321192068
[20] DOI: 10.1109/TNN.2004.832708 · doi:10.1109/TNN.2004.832708
[21] DOI: 10.1162/neco.2008.07-07-570 · Zbl 1148.92010 · doi:10.1162/neco.2008.07-07-570
[22] DOI: 10.1016/S1352-2310(96)00354-8 · doi:10.1016/S1352-2310(96)00354-8
[23] DOI: 10.1088/0954-898X/15/4/002 · doi:10.1088/0954-898X/15/4/002
[24] DOI: 10.1007/s10827-007-0042-x · Zbl 05537031 · doi:10.1007/s10827-007-0042-x
[25] DOI: 10.1162/0899766042321797 · Zbl 1180.62179 · doi:10.1162/0899766042321797
[26] DOI: 10.1016/S0375-9601(96)00878-X · doi:10.1016/S0375-9601(96)00878-X
[27] DOI: 10.1038/nn1253 · doi:10.1038/nn1253
[28] DOI: 10.4249/scholarpedia.1903 · doi:10.4249/scholarpedia.1903
[29] DOI: 10.3389/neuro.11.002.2010 · doi:10.3389/neuro.11.002.2010
[30] DOI: 10.1109/TNN.2010.2083685 · doi:10.1109/TNN.2010.2083685
[31] Schrödinger E., Physikalische Zeitschrift 16 pp 289– (1915)
[32] DOI: 10.1007/978-3-642-82118-9 · doi:10.1007/978-3-642-82118-9
[33] DOI: 10.1111/j.1469-7793.1997.617ba.x · doi:10.1111/j.1469-7793.1997.617ba.x
[34] DOI: 10.1162/neco.2007.19.12.3226 · Zbl 1144.68049 · doi:10.1162/neco.2007.19.12.3226
[35] DOI: 10.1162/089976601300014321 · Zbl 0979.92011 · doi:10.1162/089976601300014321
[36] Victor J. D., J. Neurophysiol. 76 (2) pp 1310– (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.