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2016, 13(3): 509-520. doi: 10.3934/mbe.2016004

Efficient information transfer by Poisson neurons

1. 

Institute of Physiology of the Czech Academy of Sciences, Videnska 1083, 14220 Prague 4, Czech Republic

2. 

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

Received  March 2015 Revised  August 2015 Published  January 2016

Recently, it has been suggested that certain neurons with Poissonian spiking statistics may communicate by discontinuously switching between two levels of firing intensity. Such a situation resembles in many ways the optimal information transmission protocol for the continuous-time Poisson channel known from information theory. In this contribution we employ the classical information-theoretic results to analyze the efficiency of such a transmission from different perspectives, emphasising the neurobiological viewpoint. We address both the ultimate limits, in terms of the information capacity under metabolic cost constraints, and the achievable bounds on performance at rates below capacity with fixed decoding error probability. In doing so we discuss optimal values of experimentally measurable quantities that can be compared with the actual neuronal recordings in a future effort.
Citation: Lubomir Kostal, Shigeru Shinomoto. Efficient information transfer by Poisson neurons. Mathematical Biosciences & Engineering, 2016, 13 (3) : 509-520. doi: 10.3934/mbe.2016004
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables, Dover, New York, 1992.

[2]

E. D. Adrian, Basis of Sensation, W. W. Norton and Co., New York, 1928.

[3]

D. Attwell and S. B. Laughlin, An energy budget for signaling in the grey matter of the brain, J. Cereb. Blood Flow Metab., 21 (2001), 1133-1145. doi: 10.1097/00004647-200110000-00001.

[4]

V. Balasubramanian and M. J. Berry, A test of metabolically efficient coding in the retina, Netw. Comput. Neural Syst., 13 (2002), 531-552. doi: 10.1088/0954-898X_13_4_306.

[5]

W. Bialek, Biophysics: Searching for Principles, Princeton University Press, Princeton, 2012.

[6]

P. Brémaud, Point Processes and Queues, Martingale Dynamics, Springer, New York, USA, 1981.

[7]

G. Caire, S. Shamai and S. Verdu, Noiseless data compression with low-density parity-check codes, in Advances in Network Information Theory (eds. P. Gupta, G. Kramer and A. J. van Wijngaarden), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 66, American Mathematical Society, 2004, 263-284.

[8]

K. Chakraborty and P. Naryan, The poisson fading channel, IEEE Trans. Inf. Theory, 53 (2007), 2349-2364. doi: 10.1109/TIT.2007.899559.

[9]

M. Davis, Capacity and cutoff rate for Poisson-type channels, IEEE Trans. Inf. Theory, 26 (1980), 710-715. doi: 10.1109/TIT.1980.1056262.

[10]

P. Dayan and L. F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, MIT Press, 2001.

[11]

R. R. de Ruyter van Steveninck and S. B. Laughlin, The rate of information transfer at graded-potential synapses, Nature, 379 (1996), 642-644.

[12]

R. C. Dorf, The Electrical Engineering Handbook, CRC Press, Boca Raton, USA, 1997.

[13]

P. Duchamp-Viret, L. Kostal, M. Chaput, P. Lansky and J.-P. Rospars, Patterns of spontaneous activity in single rat olfactory receptor neurons are different in normally breathing and tracheotomized animals, J. Neurobiol., 65 (2005), 97-114. doi: 10.1002/neu.20177.

[14]

M. R. Frey, Information capacity of the poisson channel, IEEE Trans. Inf. Theory, 37 (1991), 244-256. doi: 10.1109/18.75239.

[15]

R. G. Gallager, Information Theory and Reliable Communication, John Wiley and Sons, Inc., New York, USA, 1972. doi: 10.1007/978-3-7091-2945-6.

[16]

W. Gerstner and W. M. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511815706.

[17]

A. Gremiaux, T. Nowotny, D. Martinez, P. Lucas and J.-P. Rospars, Modelling the signal delivered by a population of first-order neurons in a moth olfactory system, Brain Res., 1434 (2012), 123-135. doi: 10.1016/j.brainres.2011.09.035.

[18]

A. Hasenstaub, S. Otte, E. Callaway and T. J. Sejnowski, Metabolic cost as a unifying principle governing neuronal biophysics, Proc. Natl. Acad. Sci. U.S.A., 107 (2010), 12329-12334. doi: 10.1073/pnas.0914886107.

[19]

J. Huang and S. P. Meyn, Characterization and computation of optimal distributions for channel coding, IEEE Trans. Inf. Theory, 51 (2005), 2336-2351. doi: 10.1109/TIT.2005.850108.

[20]

S. Ihara, Information Theory for Continuous Systems, World Scientific Publishing, Singapore, 1993. doi: 10.1142/9789814355827.

[21]

E. Javel and N. F. Viemeister, Stochastic properties of cat auditory nerve responses to electric and acoustic stimuli and application to intensity discrimination, J. Acoust. Soc. Am., 107 (2000), 908-921. doi: 10.1121/1.428269.

[22]

D. H. Johnson, Point process models of single-neuron discharges, J. Comput. Neurosci., 3 (1996), 275-299. doi: 10.1007/BF00161089.

[23]

D. H. Johnson and I. N. Goodman, Inferring the capacity of the vector Poisson channel with a Bernoulli model, Netw. Comput. Neural Syst., 19 (2008), 13-33. doi: 10.1080/09548980701656798.

[24]

Y. M. Kabanov, The capacity of a channel of the poisson type, Theor. Prob. App., 23 (1978), 143-147. doi: 10.1137/1123013.

[25]

R. Kobayashi, S. Shinomoto and P. Lansky, Estimation of time-dependent input from neuronal membrane potential, Neural Comput., 23 (2011), 3070-3093. doi: 10.1162/NECO_a_00205.

[26]

R. Kobayashi, Y. Tsubo and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Front. Comput. Neurosci., 3 (2009), p9. doi: 10.3389/neuro.10.009.2009.

[27]

L. Kostal, Information capacity in the weak-signal approximation, Phys. Rev. E, 82 (2010), 026115. doi: 10.1103/PhysRevE.82.026115.

[28]

L. Kostal, Approximate information capacity of the perfect integrate-and-fire neuron using the temporal code, Brain Res., 1434 (2012), 136-141. doi: 10.1016/j.brainres.2011.07.007.

[29]

L. Kostal and R. Kobayashi, Optimal decoding and information transmission in Hodgkin-Huxley neurons under metabolic cost constraints, Biosystems, 136 (2015), 3-10. doi: 10.1016/j.biosystems.2015.06.008.

[30]

L. Kostal, P. Lansky and M. D. McDonnell, Metabolic cost of neuronal information in an empirical stimulus-response model, Biol. Cybern., 107 (2013), 355-365. doi: 10.1007/s00422-013-0554-6.

[31]

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), 63-80.

[32]

A. Lapidoth, On the reliability function of the ideal poisson channel with noiseless feedback, IEEE Trans. Inf. Theory, 39 (1993), 491-503. doi: 10.1109/18.212279.

[33]

S. B. Laughlin, A simple coding procedure enhances a neuron's information capacity, Z. Naturforsch., 36 (1981), 910-912.

[34]

S. B. Laughlin, R. R. de Ruyter van Steveninck and J. C. Anderson, The metabolic cost of neural information, Nat. Neurosci., 1 (1998), 36-41.

[35]

J. L. Massey, Capacity, cutoff rate, and coding for a direct-detection optical channel, IEEE Trans. Commun., 29 (1981), 1615-1621. doi: 10.1109/TCOM.1981.1094916.

[36]

M. D. McDonnell, S. Ikeda and J. H. Manton, An introductory review of information theory in the context of computational neuroscience, Biol. Cybern., 105 (2011), 55-70. doi: 10.1007/s00422-011-0451-9.

[37]

R. J. McEliece, Practical codes for photon communication, IEEE Trans. Inf. Theory, 27 (1981), 393-398. doi: 10.1109/TIT.1981.1056380.

[38]

Y. Mochizuki and S. Shinomoto, Analog and digital codes in the brain, Phys. Rev. E, 89 (2014), 022705. doi: 10.1103/PhysRevE.89.022705.

[39]

R. Moreno-Bote, Poisson-like spiking in circuits with probabilistic synapses, PLoS Comput. Biol., 10 (2014), e1003522. doi: 10.1371/journal.pcbi.1003522.

[40]

D. H. Perkel and T. H. Bullock, Neural coding, Neurosci. Res. Prog. Sum., 3 (1968), 405-527.

[41]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, Inc., San Francisco, 1964.

[42]

F. Rieke, R. de Ruyter van Steveninck, D. Warland and W. Bialek, Spikes: Exploring the Neural Code, MIT Press, Cambridge, 1999.

[43]

B. Rimoldi, Beyond the separation principle: A broader approach to source-channel coding, in 4th Int. ITG Conf., VDE Verlag, Berlin, 2002, 233-238.

[44]

P. Sadeghi, P. O. Vontobel and R. Shams, Optimization of information rate upper and lower bounds for channels with memory, IEEE Trans. Inf. Theory, 55 (2009), 663-688. doi: 10.1109/TIT.2008.2009581.

[45]

B. Sengupta, S. B. Laughlin and J. E. Niven, Balanced excitatory and inhibitory synaptic currents promote efficient coding and metabolic efficiency, PLoS Comput. Biol., 9 (2013), e1003263. doi: 10.1371/journal.pcbi.1003263.

[46]

B. S. Sengupta and M. B. Stemmler, Power consumption during neuronal computation, Proc. IEEE, 102 (2014), 738-750. doi: 10.1109/JPROC.2014.2307755.

[47]

S. Shamai and A. Lapidoth, Bounds on the capacity of a spectrally constrained poisson channel, IEEE Trans. Inf. Theory, 39 (1993), 19-29. doi: 10.1109/18.179338.

[48]

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.

[49]

P. Suksompong and T. Berger, Capacity analysis for integrate-and-fire neurons with descending action potential thresholds, IEEE Trans. Inf. Theory, 56 (2010), 838-851. doi: 10.1109/TIT.2009.2037042.

[50]

P. J. Thomas and A. W. Eckford, Capacity of a simple intercellular signal transduction channel, preprint, arXiv:1411.1650.

[51]

H. C. Tuckwell, Introduction to Theoretical Neurobiology, Vol. 2, Cambridge University Press, New York, 1988.

[52]

S. Verdu, On channel capacity per unit cost, IEEE Trans. Inf. Theory, 36 (1990), 1019-1030. doi: 10.1109/18.57201.

[53]

S. Verdu and T. S. Han, A general formula for channel capacity, IEEE Trans. Inf. Theory, 40 (1994), 1147-1157. doi: 10.1109/18.335960.

[54]

A. D. Wyner, Capacity and error exponent for the direct detection photon channel - part I, IEEE Trans. Inf. Theory, 34 (1988), 1449-1461. doi: 10.1109/18.21284.

[55]

A. D. Wyner, Capacity and error exponent for the direct detection photon channel - part II, IEEE Trans. Inf. Theory, 34 (1988), 1462-1471. doi: 10.1109/18.21284.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables, Dover, New York, 1992.

[2]

E. D. Adrian, Basis of Sensation, W. W. Norton and Co., New York, 1928.

[3]

D. Attwell and S. B. Laughlin, An energy budget for signaling in the grey matter of the brain, J. Cereb. Blood Flow Metab., 21 (2001), 1133-1145. doi: 10.1097/00004647-200110000-00001.

[4]

V. Balasubramanian and M. J. Berry, A test of metabolically efficient coding in the retina, Netw. Comput. Neural Syst., 13 (2002), 531-552. doi: 10.1088/0954-898X_13_4_306.

[5]

W. Bialek, Biophysics: Searching for Principles, Princeton University Press, Princeton, 2012.

[6]

P. Brémaud, Point Processes and Queues, Martingale Dynamics, Springer, New York, USA, 1981.

[7]

G. Caire, S. Shamai and S. Verdu, Noiseless data compression with low-density parity-check codes, in Advances in Network Information Theory (eds. P. Gupta, G. Kramer and A. J. van Wijngaarden), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 66, American Mathematical Society, 2004, 263-284.

[8]

K. Chakraborty and P. Naryan, The poisson fading channel, IEEE Trans. Inf. Theory, 53 (2007), 2349-2364. doi: 10.1109/TIT.2007.899559.

[9]

M. Davis, Capacity and cutoff rate for Poisson-type channels, IEEE Trans. Inf. Theory, 26 (1980), 710-715. doi: 10.1109/TIT.1980.1056262.

[10]

P. Dayan and L. F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, MIT Press, 2001.

[11]

R. R. de Ruyter van Steveninck and S. B. Laughlin, The rate of information transfer at graded-potential synapses, Nature, 379 (1996), 642-644.

[12]

R. C. Dorf, The Electrical Engineering Handbook, CRC Press, Boca Raton, USA, 1997.

[13]

P. Duchamp-Viret, L. Kostal, M. Chaput, P. Lansky and J.-P. Rospars, Patterns of spontaneous activity in single rat olfactory receptor neurons are different in normally breathing and tracheotomized animals, J. Neurobiol., 65 (2005), 97-114. doi: 10.1002/neu.20177.

[14]

M. R. Frey, Information capacity of the poisson channel, IEEE Trans. Inf. Theory, 37 (1991), 244-256. doi: 10.1109/18.75239.

[15]

R. G. Gallager, Information Theory and Reliable Communication, John Wiley and Sons, Inc., New York, USA, 1972. doi: 10.1007/978-3-7091-2945-6.

[16]

W. Gerstner and W. M. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511815706.

[17]

A. Gremiaux, T. Nowotny, D. Martinez, P. Lucas and J.-P. Rospars, Modelling the signal delivered by a population of first-order neurons in a moth olfactory system, Brain Res., 1434 (2012), 123-135. doi: 10.1016/j.brainres.2011.09.035.

[18]

A. Hasenstaub, S. Otte, E. Callaway and T. J. Sejnowski, Metabolic cost as a unifying principle governing neuronal biophysics, Proc. Natl. Acad. Sci. U.S.A., 107 (2010), 12329-12334. doi: 10.1073/pnas.0914886107.

[19]

J. Huang and S. P. Meyn, Characterization and computation of optimal distributions for channel coding, IEEE Trans. Inf. Theory, 51 (2005), 2336-2351. doi: 10.1109/TIT.2005.850108.

[20]

S. Ihara, Information Theory for Continuous Systems, World Scientific Publishing, Singapore, 1993. doi: 10.1142/9789814355827.

[21]

E. Javel and N. F. Viemeister, Stochastic properties of cat auditory nerve responses to electric and acoustic stimuli and application to intensity discrimination, J. Acoust. Soc. Am., 107 (2000), 908-921. doi: 10.1121/1.428269.

[22]

D. H. Johnson, Point process models of single-neuron discharges, J. Comput. Neurosci., 3 (1996), 275-299. doi: 10.1007/BF00161089.

[23]

D. H. Johnson and I. N. Goodman, Inferring the capacity of the vector Poisson channel with a Bernoulli model, Netw. Comput. Neural Syst., 19 (2008), 13-33. doi: 10.1080/09548980701656798.

[24]

Y. M. Kabanov, The capacity of a channel of the poisson type, Theor. Prob. App., 23 (1978), 143-147. doi: 10.1137/1123013.

[25]

R. Kobayashi, S. Shinomoto and P. Lansky, Estimation of time-dependent input from neuronal membrane potential, Neural Comput., 23 (2011), 3070-3093. doi: 10.1162/NECO_a_00205.

[26]

R. Kobayashi, Y. Tsubo and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Front. Comput. Neurosci., 3 (2009), p9. doi: 10.3389/neuro.10.009.2009.

[27]

L. Kostal, Information capacity in the weak-signal approximation, Phys. Rev. E, 82 (2010), 026115. doi: 10.1103/PhysRevE.82.026115.

[28]

L. Kostal, Approximate information capacity of the perfect integrate-and-fire neuron using the temporal code, Brain Res., 1434 (2012), 136-141. doi: 10.1016/j.brainres.2011.07.007.

[29]

L. Kostal and R. Kobayashi, Optimal decoding and information transmission in Hodgkin-Huxley neurons under metabolic cost constraints, Biosystems, 136 (2015), 3-10. doi: 10.1016/j.biosystems.2015.06.008.

[30]

L. Kostal, P. Lansky and M. D. McDonnell, Metabolic cost of neuronal information in an empirical stimulus-response model, Biol. Cybern., 107 (2013), 355-365. doi: 10.1007/s00422-013-0554-6.

[31]

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), 63-80.

[32]

A. Lapidoth, On the reliability function of the ideal poisson channel with noiseless feedback, IEEE Trans. Inf. Theory, 39 (1993), 491-503. doi: 10.1109/18.212279.

[33]

S. B. Laughlin, A simple coding procedure enhances a neuron's information capacity, Z. Naturforsch., 36 (1981), 910-912.

[34]

S. B. Laughlin, R. R. de Ruyter van Steveninck and J. C. Anderson, The metabolic cost of neural information, Nat. Neurosci., 1 (1998), 36-41.

[35]

J. L. Massey, Capacity, cutoff rate, and coding for a direct-detection optical channel, IEEE Trans. Commun., 29 (1981), 1615-1621. doi: 10.1109/TCOM.1981.1094916.

[36]

M. D. McDonnell, S. Ikeda and J. H. Manton, An introductory review of information theory in the context of computational neuroscience, Biol. Cybern., 105 (2011), 55-70. doi: 10.1007/s00422-011-0451-9.

[37]

R. J. McEliece, Practical codes for photon communication, IEEE Trans. Inf. Theory, 27 (1981), 393-398. doi: 10.1109/TIT.1981.1056380.

[38]

Y. Mochizuki and S. Shinomoto, Analog and digital codes in the brain, Phys. Rev. E, 89 (2014), 022705. doi: 10.1103/PhysRevE.89.022705.

[39]

R. Moreno-Bote, Poisson-like spiking in circuits with probabilistic synapses, PLoS Comput. Biol., 10 (2014), e1003522. doi: 10.1371/journal.pcbi.1003522.

[40]

D. H. Perkel and T. H. Bullock, Neural coding, Neurosci. Res. Prog. Sum., 3 (1968), 405-527.

[41]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, Inc., San Francisco, 1964.

[42]

F. Rieke, R. de Ruyter van Steveninck, D. Warland and W. Bialek, Spikes: Exploring the Neural Code, MIT Press, Cambridge, 1999.

[43]

B. Rimoldi, Beyond the separation principle: A broader approach to source-channel coding, in 4th Int. ITG Conf., VDE Verlag, Berlin, 2002, 233-238.

[44]

P. Sadeghi, P. O. Vontobel and R. Shams, Optimization of information rate upper and lower bounds for channels with memory, IEEE Trans. Inf. Theory, 55 (2009), 663-688. doi: 10.1109/TIT.2008.2009581.

[45]

B. Sengupta, S. B. Laughlin and J. E. Niven, Balanced excitatory and inhibitory synaptic currents promote efficient coding and metabolic efficiency, PLoS Comput. Biol., 9 (2013), e1003263. doi: 10.1371/journal.pcbi.1003263.

[46]

B. S. Sengupta and M. B. Stemmler, Power consumption during neuronal computation, Proc. IEEE, 102 (2014), 738-750. doi: 10.1109/JPROC.2014.2307755.

[47]

S. Shamai and A. Lapidoth, Bounds on the capacity of a spectrally constrained poisson channel, IEEE Trans. Inf. Theory, 39 (1993), 19-29. doi: 10.1109/18.179338.

[48]

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.

[49]

P. Suksompong and T. Berger, Capacity analysis for integrate-and-fire neurons with descending action potential thresholds, IEEE Trans. Inf. Theory, 56 (2010), 838-851. doi: 10.1109/TIT.2009.2037042.

[50]

P. J. Thomas and A. W. Eckford, Capacity of a simple intercellular signal transduction channel, preprint, arXiv:1411.1650.

[51]

H. C. Tuckwell, Introduction to Theoretical Neurobiology, Vol. 2, Cambridge University Press, New York, 1988.

[52]

S. Verdu, On channel capacity per unit cost, IEEE Trans. Inf. Theory, 36 (1990), 1019-1030. doi: 10.1109/18.57201.

[53]

S. Verdu and T. S. Han, A general formula for channel capacity, IEEE Trans. Inf. Theory, 40 (1994), 1147-1157. doi: 10.1109/18.335960.

[54]

A. D. Wyner, Capacity and error exponent for the direct detection photon channel - part I, IEEE Trans. Inf. Theory, 34 (1988), 1449-1461. doi: 10.1109/18.21284.

[55]

A. D. Wyner, Capacity and error exponent for the direct detection photon channel - part II, IEEE Trans. Inf. Theory, 34 (1988), 1462-1471. doi: 10.1109/18.21284.

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