In this paper we investigate the rate coding capabilities of neurons whose input signal are alterations of the base state of balanced inhibitory and excitatory synaptic currents. We consider different regimes of excitation-inhibition relationship and an established conductance-based leaky integrator model with adaptive threshold and parameter sets recreating biologically relevant spiking regimes. We find that given mean post-synaptic firing rate, counter-intuitively, increased ratio of inhibition to excitation generally leads to higher signal to noise ratio (SNR). On the other hand, the inhibitory input significantly reduces the dynamic coding range of the neuron. We quantify the joint effect of SNR and dynamic coding range by computing the metabolic efficiency-the maximal amount of information per one ATP molecule expended (in bits/ATP). Moreover, by calculating the metabolic efficiency we are able to predict the shapes of the post-synaptic firing rate histograms that may be tested on experimental data. Likewise, optimal stimulus input distributions are predicted, however, we show that the optimum can essentially be reached with a broad range of input distributions. Finally, we examine which parameters of the used neuronal model are the most important for the metabolically efficient information transfer.

Charles University 1st Medical Faculty Prague Czech Republic

Institute of Ecology and Environmental Sciences INRA Versailles France

Institute of Physiology of the Czech Academy of Sciences Prague Czech Republic

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Dayan P, Abbott LF. Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems The MIT Press; 2005.

Barlow HB. Possible principles underlying the transformation of sensory messages In: Rosenblith W, editor. Sensory Communication. Cambridge: MIT Press; 1961. p. 217–234.

Laughlin S. A simple coding procedure enhances a neuron’s information capacity. Z Naturforsch [C]. 1981;36:910–912. 10.1515/znc-1981-9-1040 PubMed DOI

Atick JJ. Could information theory provide an ecological theory of sensory processing? Netw Comput Neural Syst. 1992;3(2):213–251. 10.1088/0954-898X_3_2_009 PubMed DOI

Lewicki MS. Efficient coding of natural sounds. Nat Neurosci. 2002;5(4):356–363. 10.1038/nn831 PubMed DOI

Machens CK, Gollisch T, Kolesnikova O, Herz AVM. Testing the efficiency of sensory coding with optimal stimulus ensembles. Neuron. 2005;47(3):447–456. 10.1016/j.neuron.2005.06.015 PubMed DOI

Smith EC, Lewicki MS. Efficient auditory coding. Nature. 2006;439(7079):978–982. 10.1038/nature04485 PubMed DOI

Hermundstad AM, Briguglio JJ, Conte MM, Victor JD, Balasubramanian V, Tkačik G. Variance predicts salience in central sensory processing. eLife. 2015;3:e03722 10.7554/eLife.03722 PubMed DOI PMC

Levakova M, Kostal L, Monsempès C, Jacob V, Lucas P. Moth olfactory receptor neurons adjust their encoding efficiency to temporal statistics of pheromone fluctuations. PLoS Comput Biol. 2018;14(11):e1006586 10.1371/journal.pcbi.1006586 PubMed DOI PMC

Shannon C. A mathematical theory of communication. Bell system technical journal. 1948;27 10.1002/j.1538-7305.1948.tb01338.x DOI

Attwell D, Laughlin SB. An Energy Budget for Signaling in the Grey Matter of the Brain. J Cereb Blood Flow Metab. 2001;21(10):1133–1145. 10.1097/00004647-200110000-00001 PubMed DOI

Harris JJ, Jolivet R, Attwell D. Synaptic energy use and supply. Neuron. 2012;75(5):762–777. 10.1016/j.neuron.2012.08.019 PubMed DOI

Harris JJ, Jolivet R, Engl E, Attwell D. Energy-Efficient Information Transfer by Visual Pathway Synapses. Curr Biol. 2015;25(24):3151–3160. 10.1016/j.cub.2015.10.063 PubMed DOI PMC

Levy WB, Baxter RA. Energy Efficient Neural Codes. Neural Comput. 1996;8(3):531–543. 10.1162/neco.1996.8.3.531 PubMed DOI

Stein RB, Gossen ER, Jones KE. Neuronal variability: noise or part of the signal? Nat Rev Neurosci. 2005;6(5):389–397. 10.1038/nrn1668 PubMed DOI

de Ruyter van Steveninck RR. Reproducibility and Variability in Neural Spike Trains. Science. 1997;275(5307):1805–1808. 10.1126/science.275.5307.1805 PubMed DOI

Strong SP, Koberle R, de Ruyter van Steveninck RR, Bialek W. Entropy and Information in Neural Spike Trains. Phys Rev Lett. 1998;80(1):197–200. 10.1103/PhysRevLett.80.197 DOI

Borst A, Theunissen FE. Information theory and neural coding. Nat Neurosci. 1999;2(11):947–957. 10.1038/14731 PubMed DOI

Stein RB. The Information Capacity of Nerve Cells Using a Frequency Code. Biophys J. 1967;7(6):797–826. 10.1016/S0006-3495(67)86623-2 PubMed DOI PMC

de Ruyter van Steveninck RR, Laughlin SB. The rate of information transfer at graded-potential synapses. Nature. 1996;379(6566):642–645. 10.1038/379642a0 DOI

Ikeda S, Manton JH. Capacity of a single spiking neuron channel. Neural Comput. 2009;21(6):1714–1748. 10.1162/neco.2009.05-08-792 PubMed DOI

Gallager RG. Information Theory and Reliable Communication. New York, NY, USA: John Wiley & Sons, Inc; 1968.

Dimitrov AG, Miller JP. Neural coding and decoding: communication channels and quantization. Netw Comput Neural Syst. 2001;12(4):441–472. 10.1080/net.12.4.441.472 PubMed DOI

Dimitrov AG, Lazar AL, Victor JD. Information theory in neuroscience. J Comput Neurosci. 2011;30:1–5. 10.1007/s10827-011-0314-3 PubMed DOI PMC

McDonnell MD, Ikeda S, Manton JH. An introductory review of information theory in the context of computational neuroscience. Biol Cybern. 2011;105:55–70. 10.1007/s00422-011-0451-9 PubMed DOI

Wibral M, Lizier JT, Priesemann V. Bits from brains for biologically inspired computing. Front Robot AI. 2015;2:5 10.3389/frobt.2015.00005 DOI

Laughlin SB, de Ruyter van Steveninck RR, Anderson JC. The metabolic cost of neural information. Nat Neurosci. 1998;1(1):36–41. 10.1038/236 PubMed DOI

Balasubramanian V, Kimber D, Berry MJ II. Metabolically Efficient Information Processing. Neural Comput. 2001;13(4):799–815. 10.1162/089976601300014358 PubMed DOI

McEliece RJ. The Theory of Information and Coding. Cambdridge, UK: Cambridge University Press; 2002.

de Polavieja GG. Errors Drive the Evolution of Biological Signalling to Costly Codes. J Theor Biol. 2002;214(4):657–664. 10.1006/jtbi.2001.2498 PubMed DOI

dePolavieja GG. Reliable biological communication with realistic constraints. Phys Rev E. 2004;70(6). PubMed

Adrian ED. The basis of sensation. W W Norton and Co, New York: 1928.

Treves A, Panzeri S, Rolls ET, Booth M, Wakeman EA. Firing rate distributions and efficiency of information transmission of inferior temporal cortex neurons to natural visual stimuli. Neural Comput. 1999;11:601–632. 10.1162/089976699300016593 PubMed DOI

Kostal L, Kobayashi R. Optimal decoding and information transmission in Hodgkin–Huxley neurons under metabolic cost constraints. Biosystems. 2015;136:3–10. 10.1016/j.biosystems.2015.06.008 PubMed DOI

Kostal L, Lansky P, McDonnell MD. Metabolic cost of neuronal information in an empirical stimulus-response model. Biol Cybern. 2013;107(3):355–365. 10.1007/s00422-013-0554-6 PubMed DOI

Suksompong P, Berger T. Capacity Analysis for Integrate-and-Fire Neurons With Descending Action Potential Thresholds. IEEE Trans Inf Theory. 2010;56(2):838–851. 10.1109/TIT.2009.2037042 DOI

Xing J, Berger T, Sungkar M, Levy WB. Energy Efficient Neurons With Generalized Inverse Gaussian Conditional and Marginal Hitting Times. IEEE Trans Inf Theory. 2015;61(8):4390–4398. 10.1109/TIT.2015.2444401 DOI

Sungkar M, Berger T, Levy WB. Mutual Information and Parameter Estimation in the Generalized Inverse Gaussian Diffusion Model of Cortical Neurons. IEEE Trans Mol Biol Multiscale Commun. 2016;2(2):166–182. 10.1109/TMBMC.2017.2656861 DOI

Sungkar M, Berger T, Levy WB. Capacity achieving input distribution to the generalized inverse Gaussian neuron model. In: 2017 55th Annual Allerton Conference on Communication, Control, and Computing (Allerton). IEEE; 2017.

Kobayashi R, Tsubo Y, Shinomoto S. Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold. Front Comput Neurosci. 2009;3:9 10.3389/neuro.10.009.2009 PubMed DOI PMC

Jolivet R, Kobayashi R, Rauch A, Naud R, Shinomoto S, Gerstner W. A benchmark test for a quantitative assessment of simple neuron models. J Neurosci Methods. 2008;169(2):417–424. 10.1016/j.jneumeth.2007.11.006 PubMed DOI

Jolivet R, Schürmann F, Berger TK, Naud R, Gerstner W, Roth A. The quantitative single-neuron modeling competition. Biol Cybern. 2008;99(4-5):417–426. 10.1007/s00422-008-0261-x PubMed DOI

Gerstner W, Naud R. How Good Are Neuron Models? Science. 2009;326(5951):379–380. 10.1126/science.1181936 PubMed DOI

Jahangiri AF, Gerling GJ. A multi-timescale adaptive threshold model for the SAI tactile afferent to predict response to mechanical vibration. Int IEEE EMBS Conf Neural Eng. 2011; p. 152–155. PubMed PMC

Kobayashi R, Kitano K. Impact of slow K+ currents on spike generation can be described by an adaptive threshold model. J Comput Neurosci. 2016;40(3):347–362. 10.1007/s10827-016-0601-0 PubMed DOI PMC

Gerstner W, Kistler WM, Naud R. Neuronal Dynamics. Cambridge University Press; 2019. Available from: https://www.ebook.de/de/product/22190732/wulfram_gerstner_werner_m_kistler_richard_naud_neuronal_dynamics.html.

Levakova M, Kostal L, Monsempès C, Lucas P, Kobayashi R. Adaptive integrate-and-fire model reproduces the dynamics of olfactory receptor neuron responses in a moth. J R Soc Interface. 2019;16(157):20190246 10.1098/rsif.2019.0246 PubMed DOI PMC

Witsenhausen H. Some aspects of convexity useful in information theory. IEEE Trans Inf Theory. 1980;26(3):265–271. 10.1109/TIT.1980.1056173 DOI

Destexhe A, Rudolph M, Fellous JM, Sejnowski TJ. Fluctuating synaptic conductances recreate in vivo-like activity in neocortical neurons. Neuroscience. 2001;107:13–24. 10.1016/s0306-4522(01)00344-x PubMed DOI PMC

Sengupta B, Stemmler M, Laughlin SB, Niven JE. Action Potential Energy Efficiency Varies Among Neuron Types in Vertebrates and Invertebrates. PLoS Comput Biol. 2010;6(7):e1000840 10.1371/journal.pcbi.1000840 PubMed DOI PMC

Harris JJ, Attwell D. The Energetics of CNS White Matter. J Neurosci. 2012;32(1):356–371. 10.1523/JNEUROSCI.3430-11.2012 PubMed DOI PMC

Butts DA, Goldman MS. Tuning Curves, Neuronal Variability, and Sensory Coding. PLoS Biol. 2006;4(4):e92 10.1371/journal.pbio.0040092 PubMed DOI PMC

Bezzi M. Quantifying the information transmitted in a single stimulus. Biosystems. 2007;89:4–9. 10.1016/j.biosystems.2006.04.009 PubMed DOI

Kostal L, D’Onofrio G. Coordinate invariance as a fundamental constraint on the form of stimulus-specific information measures. Biol Cybern. 2018;112(1–2):13–23. 10.1007/s00422-017-0729-7 PubMed DOI

Luenberger DG. Optimization by Vector Space Methods. 1st ed New York, NY, USA: John Wiley & Sons, Inc; 1997.

Smith JG. The Information Capacity of Amplitude- and Variance-Constrained Scalar Gaussian Channels. Information and Control. 1971;18(3):203–219. 10.1016/S0019-9958(71)90346-9 DOI

Verdu S. On channel capacity per unit cost. IEEE Trans Inf Theory. 1990;36(5):1019–1030. 10.1109/18.57201 DOI

Abou-Faycal IC, Trott MD, Shamai S. The capacity of discrete-time memoryless Rayleigh-fading channels. IEEE Trans Inf Theory. 2001;47(4):1290–1301. 10.1109/18.923716 DOI

Reinagel P, Reid RC. Temporal Coding of Visual Information in the Thalamus. J Neurosci. 2000;20(14):5392–5400. 10.1523/JNEUROSCI.20-14-05392.2000 PubMed DOI PMC

Richardson MJE. Effects of synaptic conductance on the voltage distribution and firing rate of spiking neurons. Phys Rev E. 2004;69(5). 10.1103/PhysRevE.69.051918 PubMed DOI

Monier C, Chavane F, Baudot P, Graham LJ, Frégnac Y. Orientation and Direction Selectivity of Synaptic Inputs in Visual Cortical Neurons. Neuron. 2003;37(4):663–680. 10.1016/s0896-6273(03)00064-3 PubMed DOI

Isomura Y, Harukuni R, Takekawa T, Aizawa H, Fukai T. Microcircuitry coordination of cortical motor information in self-initiation of voluntary movements. Nat Neurosci. 2009;12(12):1586–1593. 10.1038/nn.2431 PubMed DOI

Tsubo Y, Isomura Y, Fukai T. Power-Law Inter-Spike Interval Distributions Infer a Conditional Maximization of Entropy in Cortical Neurons. PLoS Comput Biol. 2012;8(4):e1002461 10.1371/journal.pcbi.1002461 PubMed DOI PMC

Kostal L. Information capacity in the weak-signal approximation. Phys Rev E. 2010;82(2). 10.1103/PhysRevE.82.026115 PubMed DOI

Shafi M, Zhou Y, Quintana J, Chow C, Fuster J, Bodner M. Variability in neuronal activity in primate cortex during working memory tasks. Neuroscience. 2007;146(3):1082–1108. 10.1016/j.neuroscience.2006.12.072 PubMed DOI

O’Connor DH, Peron SP, Huber D, Svoboda K. Neural Activity in Barrel Cortex Underlying Vibrissa-Based Object Localization in Mice. Neuron. 2010;67(6):1048–1061. 10.1016/j.neuron.2010.08.026 PubMed DOI

Buzsáki G, Mizuseki K. The log-dynamic brain: how skewed distributions affect network operations. Nat Rev Neurosci. 2014;15:264–278. 10.1038/nrn3687 PubMed DOI PMC

Yamauchi S, Kim H, Shinomoto S. Elemental Spiking Neuron Model for Reproducing Diverse Firing Patterns and Predicting Precise Firing Times. Front Comput Neurosci. 2011;5 10.3389/fncom.2011.00042 PubMed DOI PMC

Kostal L, Kobayashi R. Critical size of neural population for reliable information transmission. Phys Rev E (Rapid Commun). 2019;100(1):050401(R) 10.1103/PhysRevE.100.050401 PubMed DOI

Sengupta B, Laughlin SB, Niven JE. Balanced Excitatory and Inhibitory Synaptic Currents Promote Efficient Coding and Metabolic Efficiency. PLoS Comput Biol. 2013;9(10):e1003263 10.1371/journal.pcbi.1003263 PubMed DOI PMC

Richardson MJE, Gerstner W. Synaptic Shot Noise and Conductance Fluctuations Affect the Membrane Voltage with Equal Significance. Neural Comput. 2005;17(4):923–947. 10.1162/0899766053429444 PubMed DOI

Bernander O, Douglas RJ, Martin KA, Koch C. Synaptic background activity influences spatiotemporal integration in single pyramidal cells. Proc Natl Acad Sci USA. 1991;88(24):11569–11573. 10.1073/pnas.88.24.11569 PubMed DOI PMC

Paré D, Shink E, Gaudreau H, Destexhe A, Lang EJ. Impact of Spontaneous Synaptic Activity on the Resting Properties of Cat Neocortical Pyramidal Neurons In Vivo. J Neurophysiol. 1998;79(3):1450–1460. 10.1152/jn.1998.79.3.1450 PubMed DOI

Destexhe A, Rudolph M, Paré D. The high-conductance state of neocortical neurons in vivo. Nat Rev Neurosci. 2003;4(9):739–751. 10.1038/nrn1198 PubMed DOI

Mittmann W, Koch U, Häusser M. Feed-forward inhibition shapes the spike output of cerebellar Purkinje cells. J Physiol (Lond). 2005;563(2):369–378. 10.1113/jphysiol.2004.075028 PubMed DOI PMC

Wolfart J, Debay D, Masson GL, Destexhe A, Bal T. Synaptic background activity controls spike transfer from thalamus to cortex. Nat Neurosci. 2005;8(12):1760–1767. 10.1038/nn1591 PubMed DOI

Rudolph M, Pospischil M, Timofeev I, Destexhe A. Inhibition Determines Membrane Potential Dynamics and Controls Action Potential Generation in Awake and Sleeping Cat Cortex. J Neurosci. 2007;27(20):5280–5290. 10.1523/JNEUROSCI.4652-06.2007 PubMed DOI PMC

Harris JJ, Engl E, Attwell D, Jolivet RB. Energy-efficient information transfer at thalamocortical synapses. PLoS Comput Biol. 2019;15(8):e1007226 10.1371/journal.pcbi.1007226 PubMed DOI PMC

Kobayashi R, Kurita S, Kurth A, Kitano K, Mizuseki K, Diesmann M, et al. Reconstructing neuronal circuitry from parallel spike trains. Nat Commun. 2019;10(1). 10.1038/s41467-019-12225-2 PubMed DOI PMC

Kostal L, Lansky P. Information capacity and its approximations under metabolic cost in a simple homogeneous population of neurons. Biosystems. 2013;112(3):265–275. 10.1016/j.biosystems.2013.03.019 PubMed DOI

El Gamal A, Kim YH. Network Information Theory. New York: Cambridge University Press; 2011.

Shared input and recurrency in neural networks for metabolically efficient information transmission