2016, 13(3): 551-568. doi: 10.3934/mbe.2016007

Effect of spontaneous activity on stimulus detection in a simple neuronal model

1. 

Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlarska 2a, 611 37 Brno, Czech Republic

Received  March 2015 Revised  October 2015 Published  January 2016

It is studied what level of a continuous-valued signal is optimally estimable on the basis of first-spike latency neuronal data. When a spontaneous neuronal activity is present, the first spike after the stimulus onset may be caused either by the stimulus itself, or it may be a result of the prevailing spontaneous activity. Under certain regularity conditions, Fisher information is the inverse of the variance of the best estimator. It can be considered as a function of the signal intensity and then indicates accuracy of the estimation for each signal level. The Fisher information is normalized with respect to the time needed to obtain an observation. The accuracy of signal level estimation is investigated in basic discharge patterns modelled by a Poisson and a renewal process and the impact of the complex interaction between spontaneous activity and a delay of the response is shown.
Citation: Marie Levakova. Effect of spontaneous activity on stimulus detection in a simple neuronal model. Mathematical Biosciences & Engineering, 2016, 13 (3) : 551-568. doi: 10.3934/mbe.2016007
References:
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[2]

D. G. Albrecht and D. B. Hamilton, Striate cortex of monkey and cat: Contrast response function,, J. Neurosci., 48 (1982), 217.

[3]

S. Amari and H. Nakahara, Difficulty of singularity in population coding,, Neural Comput., 17 (2005), 839. doi: 10.1162/0899766053429426.

[4]

S. N. Baker and G. L. Gerstein, Determination of response latency and its application to normalization of cross-correlation measures,, Neural Comput., 13 (2001), 1351. doi: 10.1162/08997660152002889.

[5]

M. J. Berry, D. K. Warland and M. Meister, The structure and precision of retinal spike trains,, Proc. Natl. Acad. Sci. USA, 94 (1997), 5411. doi: 10.1073/pnas.94.10.5411.

[6]

M. Bethge, D. Rottermund and K. Pawelzik, Optimal short-term population coding: When Fisher information fails,, Neural Comput., 14 (2002), 2317. doi: 10.1162/08997660260293247.

[7]

R. Brasselet, S. Panzeri, N. K. Logothetis and C. Kayser, Neurons with stereotyped and rapid responses provide a reference frame for relative temporal coding in primate auditory cortex,, J. Neurosci., 32 (2012), 2998. doi: 10.1523/JNEUROSCI.5435-11.2012.

[8]

N. Brunel and J. P. Nadal, Mutual information, Fisher information, and population coding,, Neural Comput., 10 (1998), 1731. doi: 10.1162/089976698300017115.

[9]

M. Chastrette, T. Thomas-Danguin and E. Rallet, Modelling the human olfactory stimulus-response function,, Chem. Senses, 23 (1998), 181. doi: 10.1093/chemse/23.2.181.

[10]

C. C. Chow and J. A. White, Spontaneous action potentials due to channel fluctuations,, Biophys. J., 71 (1996), 3013. doi: 10.1016/S0006-3495(96)79494-8.

[11]

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events,, Methuen, (1966).

[12]

Y. Dan, J. M. Alonso, W. Usrey and R. Reid, Coding of visual information by the precisely correlated spikes in the lateral geniculate nucleus,, Nature Neurosci., 1 (1998), 501.

[13]

I. Dean, N. Harper and D. McAlpine, Neural population coding of sound level adapts to stimulus statistics,, Nature Neurosci., 8 (2005), 1684. doi: 10.1038/nn1541.

[14]

R. deCharms and M. Merzenich, Primary cortical representation of sounds by the coordination of action-potential timing,, Nature, 381 (1996), 610. doi: 10.1038/381610a0.

[15]

S. Durant, C. W. G. Clifford, N. A. Crowder, N. S. C. Price and M. R. Ibbotson, Characterizing contrast adaptation in a population of cat primary visual cortical neurons using Fisher information,, J. Opt. Soc. Am. A, 24 (2007), 1529. doi: 10.1364/JOSAA.24.001529.

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J. J. Eggermont, Azimuth coding in primary auditory cortex of the cat. II. Relative latency and interspike interval representation,, J. Neurophysiol., 80 (1998), 2151.

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H. S. Friedman and C. E. Priebe, Estimating stimulus response latency,, J. Neurosci. Meth., 83 (1998), 185. doi: 10.1016/S0165-0270(98)00075-2.

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S. Furukawa, L. Xu and J. C. Middlebrooks, Coding of sound-source location by ensembles of cortical neurons,, J. Neurosci., 20 (2000), 1216.

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S. Furukawa and J. C. Middlebrooks, Cortical representation of auditory space: Information-bearing features of spike patterns,, J. Neurophysiol., 87 (2002), 1749.

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T. J. Gawne, T. W. Kjaer and B. J. Richmond, Latency: Another potential code for feature binding in striate cortex,, J. Neurophysiol., 76 (1996), 1356.

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G. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron,, Biophys. J., 4 (1964), 41. doi: 10.1016/S0006-3495(64)86768-0.

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G. Gerstein, P. Bedenbaugh and A. Aertsen, Neural assemblies,, IEEE Trans. Biomed. Engineering, 36 (): 4.

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W. Gilles, T. Michèle and P. Khashayar, Intrinsic variability of latency to first-spike,, Biol. Cybern., 103 (2010), 43. doi: 10.1007/s00422-010-0384-8.

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P. E. Greenwood and P. Lansky, Optimum signal in a simple neuronal model with signal-dependent noise,, Biol. Cybern., 92 (2005), 199. doi: 10.1007/s00422-005-0545-3.

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P. E. Greenwood, L. M. Ward and W. Wefelmeyer, Statistical analysis of stochastic resonance in a simple setting,, Phys. Rev. E, 60 (1999), 4687. doi: 10.1103/PhysRevE.60.4687.

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P. E. Greenwood, L. M. Ward, D. F. Russel, A. Neiman and F. Moss, Stochastic resonance enhances the electrosensory information available to paddlefish for prey capture,, Phys. Rev. Lett., 84 (2000). doi: 10.1103/PhysRevLett.84.4773.

[27]

S. Grün and S. Rotter (ed.), Analysis of Parallel Spike Trains,, Springer, (2010).

[28]

B. S. Hansson, Olfaction in lepidoptera,, Experientia, 51 (1995), 1003. doi: 10.1007/BF01946910.

[29]

P. Heil, Auditory cortical onset responses revisited: First-spike timing,, J. Neurophysiol., 77 (1997), 2616.

[30]

M. A. Hietanen, N. A. Crowder and M. R. Ibbotson, Contrast gain control is drift-rate dependent: An informational analysis,, J. Neurophysiol., 97 (2007), 1078. doi: 10.1152/jn.00991.2006.

[31]

R. L. Jenison, Decoding first-spike latency: A likelihood approach,, Neurocomputing, 38 (2001), 239. doi: 10.1016/S0925-2312(01)00355-1.

[32]

D. H. Johnson and W. Ray, Optimal stimulus coding by neural populations using rate codes,, J. Comput. Neurosci., 16 (2004), 129. doi: 10.1023/B:JCNS.0000014106.09948.83.

[33]

L. Kostal, P. Lansky and J. P. Rospars, Efficient olfactory coding in the pheromone receptor neuron of a moth,, PLoS Comput. Biol., 4 (2008). doi: 10.1371/journal.pcbi.1000053.

[34]

L. Kostal and P. Lansky, Coding accuracy is not fully determined by the neuronal model,, Neural Comput., 27 (2015), 1051. doi: 10.1162/NECO_a_00724.

[35]

S. Koyama and L. Kostal, The effect of interspike interval statistics on the information gain under the rate coding hypothesis,, Math. Biosci. Eng., 11 (2014), 63.

[36]

P. Lansky and P. E. Greenwood, Optimal signal estimation in neuronal models,, Neural Comput., 17 (2005), 2240. doi: 10.1162/0899766054615653.

[37]

P. Lansky and P. E. Greenwood, Optimal signal in sensory neurons under an extended rate coding concept,, BioSystems, 89 (2007), 10. doi: 10.1016/j.biosystems.2006.04.010.

[38]

P. Lansky, L. Sacerdote and C. Zucca, Optimum signal in a diffusion leaky integrate-and-fire neuronal model,, Math. Biosci., 207 (2007), 261. doi: 10.1016/j.mbs.2006.08.027.

[39]

P. Lansky and S. Sato, The stochastic diffusion models of nerve membrane depolarization and interspike interval generation,, J. Peripher. Nerv. Syst., 4 (1998), 27.

[40]

M. Levakova, S. Ditlevsen and P. Lansky, Estimating latency from inhibitory input,, Biol. Cybern., 108 (2014), 475. doi: 10.1007/s00422-014-0614-6.

[41]

M. Levakova, M. Tamborrino, S. Ditlevsen and P. Lansky, A review of the methods for neuronal response latency estimation,, BioSystems, 136 (2015), 23. doi: 10.1016/j.biosystems.2015.04.008.

[42]

I. Nelken, G. Chechik, T. D. Mrsic-Flogel, A. J. King and J. W. H. Schnupp, Encoding stimulus information by spike numbers and mean response time in primary auditory cortex,, J. Comput. Neurosci., 19 (2005), 199. doi: 10.1007/s10827-005-1739-3.

[43]

S. Nirenberg, S. Carcieri, A. Jacobs and P. Latham, Retinal ganglion cells act largely as independent encoders,, Nature, 411 (2001), 698.

[44]

L. Nizami, Estimating auditory neuronal dynamic range using a fitted function,, Hearing Res., 167 (2002), 13. doi: 10.1016/S0378-5955(02)00293-9.

[45]

H. Nover, C. H. Anderson and G. C. DeAngelis, A logarithmic, scale-invariant representation of speed in macaque middle temporal area accounts for speed discrimination performance,, J. Neurosci., 25 (2005), 10049. doi: 10.1523/JNEUROSCI.1661-05.2005.

[46]

Z. Pawlas, L. B. Klebanov, V. Beneš, M. Prokešová, J. Popelář and P. Lansky, First-spike latency in the presence of spontaneous activity,, Neural Comput., 22 (2010), 1675. doi: 10.1162/neco.2010.11-09-1118.

[47]

S. Panzeri, R. S. Petersen, S. R. Schultz, M. Lebedev, Michael and M. E. Diamond, The role of spike timing in the coding of stimulus location in rat somatosensory cortex,, Neuron, 29 (2001), 769. doi: 10.1016/S0896-6273(01)00251-3.

[48]

S. Panzeri, R. A. A. Ince, M. E. Diamond and C. Kayser, Reading spike timing without a clock: Intrinsic decoding of spike trains,, Phil. Trans. R. Soc. B, 369 (2014). doi: 10.1098/rstb.2012.0467.

[49]

D. Perkel and G. Bullock, Neuronal coding,, Neurosci. Res. Prog. Bull., 6 (1968), 221.

[50]

R. S. Petersen, S. Panzeri and M. E. Diamond, Population coding of stimulus location in rat somatosensory cortex,, Neuron, 32 (2001), 503. doi: 10.1016/S0896-6273(01)00481-0.

[51]

R. S. Petersen, S. Panzeri and M. E. Diamond, The role of individual spikes and spike patterns in population coding of stimulus location in rat somatosensory cortex,, BioSystems, 67 (2002), 187. doi: 10.1016/S0303-2647(02)00076-X.

[52]

D. S. Reich, F. Mechler and J. D. Victor, Temporal coding of contrast in primary visual cortex: When, what, and why,, J. Neurophysiol., 85 (2001), 1039.

[53]

J. P. Rospars, P. Lansky, A. Duchamp and P. Duchamp-Viret, Relation between stimulus and response in frog olfactory receptor neurons in vivo,, Eur. J. Neurosci., 18 (2003), 1135. doi: 10.1046/j.1460-9568.2003.02766.x.

[54]

M. Stemmler, A single spike suffices: The simplest form of stochastic resonance in model neurons,, Network, 7 (1996), 687. doi: 10.1088/0954-898X_7_4_005.

[55]

M. Tamborrino, S. Ditlevsen and P. Lansky, Identification of noisy response latency,, Phys. Rev. E, 86 (2012). doi: 10.1103/PhysRevE.86.021128.

[56]

M. Tamborrino, S. Ditlevsen and P. Lansky, Parametric inference of neuronal response latency in presence of a background signal,, BioSystems, 112 (2013), 249. doi: 10.1016/j.biosystems.2013.01.009.

[57]

M. C. K. Tweedie, Statistical properties of inverse Gaussian distributions. I,, Ann. Math. Stat., 28 (1957), 362. doi: 10.1214/aoms/1177706964.

[58]

S. D. Wilke and C. W. Eurich, Representational accuracy of stochastic neural populations,, Neural Comp., 14 (2002), 155. doi: 10.1162/089976602753284482.

[59]

R. L. Winslow and M. B. Sachs, Single-tone intensity discrimination based on auditory-nerve rate responses in background of quiet, noise, and with stimulation of the crossed olivocochlear bundle,, Hearing Res., 35 (1988), 165. doi: 10.1016/0378-5955(88)90116-5.

show all references

References:
[1]

L. F. Abbott and P. Dayan, The effect of correlated variability on the accuracy of a population code,, Neural Comput., 11 (1999), 91. doi: 10.1162/089976699300016827.

[2]

D. G. Albrecht and D. B. Hamilton, Striate cortex of monkey and cat: Contrast response function,, J. Neurosci., 48 (1982), 217.

[3]

S. Amari and H. Nakahara, Difficulty of singularity in population coding,, Neural Comput., 17 (2005), 839. doi: 10.1162/0899766053429426.

[4]

S. N. Baker and G. L. Gerstein, Determination of response latency and its application to normalization of cross-correlation measures,, Neural Comput., 13 (2001), 1351. doi: 10.1162/08997660152002889.

[5]

M. J. Berry, D. K. Warland and M. Meister, The structure and precision of retinal spike trains,, Proc. Natl. Acad. Sci. USA, 94 (1997), 5411. doi: 10.1073/pnas.94.10.5411.

[6]

M. Bethge, D. Rottermund and K. Pawelzik, Optimal short-term population coding: When Fisher information fails,, Neural Comput., 14 (2002), 2317. doi: 10.1162/08997660260293247.

[7]

R. Brasselet, S. Panzeri, N. K. Logothetis and C. Kayser, Neurons with stereotyped and rapid responses provide a reference frame for relative temporal coding in primate auditory cortex,, J. Neurosci., 32 (2012), 2998. doi: 10.1523/JNEUROSCI.5435-11.2012.

[8]

N. Brunel and J. P. Nadal, Mutual information, Fisher information, and population coding,, Neural Comput., 10 (1998), 1731. doi: 10.1162/089976698300017115.

[9]

M. Chastrette, T. Thomas-Danguin and E. Rallet, Modelling the human olfactory stimulus-response function,, Chem. Senses, 23 (1998), 181. doi: 10.1093/chemse/23.2.181.

[10]

C. C. Chow and J. A. White, Spontaneous action potentials due to channel fluctuations,, Biophys. J., 71 (1996), 3013. doi: 10.1016/S0006-3495(96)79494-8.

[11]

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events,, Methuen, (1966).

[12]

Y. Dan, J. M. Alonso, W. Usrey and R. Reid, Coding of visual information by the precisely correlated spikes in the lateral geniculate nucleus,, Nature Neurosci., 1 (1998), 501.

[13]

I. Dean, N. Harper and D. McAlpine, Neural population coding of sound level adapts to stimulus statistics,, Nature Neurosci., 8 (2005), 1684. doi: 10.1038/nn1541.

[14]

R. deCharms and M. Merzenich, Primary cortical representation of sounds by the coordination of action-potential timing,, Nature, 381 (1996), 610. doi: 10.1038/381610a0.

[15]

S. Durant, C. W. G. Clifford, N. A. Crowder, N. S. C. Price and M. R. Ibbotson, Characterizing contrast adaptation in a population of cat primary visual cortical neurons using Fisher information,, J. Opt. Soc. Am. A, 24 (2007), 1529. doi: 10.1364/JOSAA.24.001529.

[16]

J. J. Eggermont, Azimuth coding in primary auditory cortex of the cat. II. Relative latency and interspike interval representation,, J. Neurophysiol., 80 (1998), 2151.

[17]

H. S. Friedman and C. E. Priebe, Estimating stimulus response latency,, J. Neurosci. Meth., 83 (1998), 185. doi: 10.1016/S0165-0270(98)00075-2.

[18]

S. Furukawa, L. Xu and J. C. Middlebrooks, Coding of sound-source location by ensembles of cortical neurons,, J. Neurosci., 20 (2000), 1216.

[19]

S. Furukawa and J. C. Middlebrooks, Cortical representation of auditory space: Information-bearing features of spike patterns,, J. Neurophysiol., 87 (2002), 1749.

[20]

T. J. Gawne, T. W. Kjaer and B. J. Richmond, Latency: Another potential code for feature binding in striate cortex,, J. Neurophysiol., 76 (1996), 1356.

[21]

G. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron,, Biophys. J., 4 (1964), 41. doi: 10.1016/S0006-3495(64)86768-0.

[22]

G. Gerstein, P. Bedenbaugh and A. Aertsen, Neural assemblies,, IEEE Trans. Biomed. Engineering, 36 (): 4.

[23]

W. Gilles, T. Michèle and P. Khashayar, Intrinsic variability of latency to first-spike,, Biol. Cybern., 103 (2010), 43. doi: 10.1007/s00422-010-0384-8.

[24]

P. E. Greenwood and P. Lansky, Optimum signal in a simple neuronal model with signal-dependent noise,, Biol. Cybern., 92 (2005), 199. doi: 10.1007/s00422-005-0545-3.

[25]

P. E. Greenwood, L. M. Ward and W. Wefelmeyer, Statistical analysis of stochastic resonance in a simple setting,, Phys. Rev. E, 60 (1999), 4687. doi: 10.1103/PhysRevE.60.4687.

[26]

P. E. Greenwood, L. M. Ward, D. F. Russel, A. Neiman and F. Moss, Stochastic resonance enhances the electrosensory information available to paddlefish for prey capture,, Phys. Rev. Lett., 84 (2000). doi: 10.1103/PhysRevLett.84.4773.

[27]

S. Grün and S. Rotter (ed.), Analysis of Parallel Spike Trains,, Springer, (2010).

[28]

B. S. Hansson, Olfaction in lepidoptera,, Experientia, 51 (1995), 1003. doi: 10.1007/BF01946910.

[29]

P. Heil, Auditory cortical onset responses revisited: First-spike timing,, J. Neurophysiol., 77 (1997), 2616.

[30]

M. A. Hietanen, N. A. Crowder and M. R. Ibbotson, Contrast gain control is drift-rate dependent: An informational analysis,, J. Neurophysiol., 97 (2007), 1078. doi: 10.1152/jn.00991.2006.

[31]

R. L. Jenison, Decoding first-spike latency: A likelihood approach,, Neurocomputing, 38 (2001), 239. doi: 10.1016/S0925-2312(01)00355-1.

[32]

D. H. Johnson and W. Ray, Optimal stimulus coding by neural populations using rate codes,, J. Comput. Neurosci., 16 (2004), 129. doi: 10.1023/B:JCNS.0000014106.09948.83.

[33]

L. Kostal, P. Lansky and J. P. Rospars, Efficient olfactory coding in the pheromone receptor neuron of a moth,, PLoS Comput. Biol., 4 (2008). doi: 10.1371/journal.pcbi.1000053.

[34]

L. Kostal and P. Lansky, Coding accuracy is not fully determined by the neuronal model,, Neural Comput., 27 (2015), 1051. doi: 10.1162/NECO_a_00724.

[35]

S. Koyama and L. Kostal, The effect of interspike interval statistics on the information gain under the rate coding hypothesis,, Math. Biosci. Eng., 11 (2014), 63.

[36]

P. Lansky and P. E. Greenwood, Optimal signal estimation in neuronal models,, Neural Comput., 17 (2005), 2240. doi: 10.1162/0899766054615653.

[37]

P. Lansky and P. E. Greenwood, Optimal signal in sensory neurons under an extended rate coding concept,, BioSystems, 89 (2007), 10. doi: 10.1016/j.biosystems.2006.04.010.

[38]

P. Lansky, L. Sacerdote and C. Zucca, Optimum signal in a diffusion leaky integrate-and-fire neuronal model,, Math. Biosci., 207 (2007), 261. doi: 10.1016/j.mbs.2006.08.027.

[39]

P. Lansky and S. Sato, The stochastic diffusion models of nerve membrane depolarization and interspike interval generation,, J. Peripher. Nerv. Syst., 4 (1998), 27.

[40]

M. Levakova, S. Ditlevsen and P. Lansky, Estimating latency from inhibitory input,, Biol. Cybern., 108 (2014), 475. doi: 10.1007/s00422-014-0614-6.

[41]

M. Levakova, M. Tamborrino, S. Ditlevsen and P. Lansky, A review of the methods for neuronal response latency estimation,, BioSystems, 136 (2015), 23. doi: 10.1016/j.biosystems.2015.04.008.

[42]

I. Nelken, G. Chechik, T. D. Mrsic-Flogel, A. J. King and J. W. H. Schnupp, Encoding stimulus information by spike numbers and mean response time in primary auditory cortex,, J. Comput. Neurosci., 19 (2005), 199. doi: 10.1007/s10827-005-1739-3.

[43]

S. Nirenberg, S. Carcieri, A. Jacobs and P. Latham, Retinal ganglion cells act largely as independent encoders,, Nature, 411 (2001), 698.

[44]

L. Nizami, Estimating auditory neuronal dynamic range using a fitted function,, Hearing Res., 167 (2002), 13. doi: 10.1016/S0378-5955(02)00293-9.

[45]

H. Nover, C. H. Anderson and G. C. DeAngelis, A logarithmic, scale-invariant representation of speed in macaque middle temporal area accounts for speed discrimination performance,, J. Neurosci., 25 (2005), 10049. doi: 10.1523/JNEUROSCI.1661-05.2005.

[46]

Z. Pawlas, L. B. Klebanov, V. Beneš, M. Prokešová, J. Popelář and P. Lansky, First-spike latency in the presence of spontaneous activity,, Neural Comput., 22 (2010), 1675. doi: 10.1162/neco.2010.11-09-1118.

[47]

S. Panzeri, R. S. Petersen, S. R. Schultz, M. Lebedev, Michael and M. E. Diamond, The role of spike timing in the coding of stimulus location in rat somatosensory cortex,, Neuron, 29 (2001), 769. doi: 10.1016/S0896-6273(01)00251-3.

[48]

S. Panzeri, R. A. A. Ince, M. E. Diamond and C. Kayser, Reading spike timing without a clock: Intrinsic decoding of spike trains,, Phil. Trans. R. Soc. B, 369 (2014). doi: 10.1098/rstb.2012.0467.

[49]

D. Perkel and G. Bullock, Neuronal coding,, Neurosci. Res. Prog. Bull., 6 (1968), 221.

[50]

R. S. Petersen, S. Panzeri and M. E. Diamond, Population coding of stimulus location in rat somatosensory cortex,, Neuron, 32 (2001), 503. doi: 10.1016/S0896-6273(01)00481-0.

[51]

R. S. Petersen, S. Panzeri and M. E. Diamond, The role of individual spikes and spike patterns in population coding of stimulus location in rat somatosensory cortex,, BioSystems, 67 (2002), 187. doi: 10.1016/S0303-2647(02)00076-X.

[52]

D. S. Reich, F. Mechler and J. D. Victor, Temporal coding of contrast in primary visual cortex: When, what, and why,, J. Neurophysiol., 85 (2001), 1039.

[53]

J. P. Rospars, P. Lansky, A. Duchamp and P. Duchamp-Viret, Relation between stimulus and response in frog olfactory receptor neurons in vivo,, Eur. J. Neurosci., 18 (2003), 1135. doi: 10.1046/j.1460-9568.2003.02766.x.

[54]

M. Stemmler, A single spike suffices: The simplest form of stochastic resonance in model neurons,, Network, 7 (1996), 687. doi: 10.1088/0954-898X_7_4_005.

[55]

M. Tamborrino, S. Ditlevsen and P. Lansky, Identification of noisy response latency,, Phys. Rev. E, 86 (2012). doi: 10.1103/PhysRevE.86.021128.

[56]

M. Tamborrino, S. Ditlevsen and P. Lansky, Parametric inference of neuronal response latency in presence of a background signal,, BioSystems, 112 (2013), 249. doi: 10.1016/j.biosystems.2013.01.009.

[57]

M. C. K. Tweedie, Statistical properties of inverse Gaussian distributions. I,, Ann. Math. Stat., 28 (1957), 362. doi: 10.1214/aoms/1177706964.

[58]

S. D. Wilke and C. W. Eurich, Representational accuracy of stochastic neural populations,, Neural Comp., 14 (2002), 155. doi: 10.1162/089976602753284482.

[59]

R. L. Winslow and M. B. Sachs, Single-tone intensity discrimination based on auditory-nerve rate responses in background of quiet, noise, and with stimulation of the crossed olivocochlear bundle,, Hearing Res., 35 (1988), 165. doi: 10.1016/0378-5955(88)90116-5.

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