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A minimum-error, energy-constrained neural code is an instantaneous-rate code

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Abstract

Sensory neurons code information about stimuli in their sequence of action potentials (spikes). Intuitively, the spikes should represent stimuli with high fidelity. However, generating and propagating spikes is a metabolically expensive process. It is therefore likely that neural codes have been selected to balance energy expenditure against encoding error. Our recently proposed optimal, energy-constrained neural coder (Jones et al. Frontiers in Computational Neuroscience, 9, 61 2015) postulates that neurons time spikes to minimize the trade-off between stimulus reconstruction error and expended energy by adjusting the spike threshold using a simple dynamic threshold. Here, we show that this proposed coding scheme is related to existing coding schemes, such as rate and temporal codes. We derive an instantaneous rate coder and show that the spike-rate depends on the signal and its derivative. In the limit of high spike rates the spike train maximizes fidelity given an energy constraint (average spike-rate), and the predicted interspike intervals are identical to those generated by our existing optimal coding neuron. The instantaneous rate coder is shown to closely match the spike-rates recorded from P-type primary afferents in weakly electric fish. In particular, the coder is a predictor of the peristimulus time histogram (PSTH). When tested against in vitro cortical pyramidal neuron recordings, the instantaneous spike-rate approximates DC step inputs, matching both the average spike-rate and the time-to-first-spike (a simple temporal code). Overall, the instantaneous rate coder relates optimal, energy-constrained encoding to the concepts of rate-coding and temporal-coding, suggesting a possible unifying principle of neural encoding of sensory signals.

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References

  • Adrian, ED (1926). The impulses produced by sensory nerve endings. The Journal of Physiology, 61(1), 49–72.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Aldworth, ZN, Dimitrov, AG, Cummins, GI, Gedeon, T, & Miller, JP (2011). Temporal encoding in a nervous system. PLoS Computational Biology, 7(5), e1002041.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Attwell, D, & Laughlin, SB (2001). An energy budget for signaling in the grey matter of the brain. Journal of Cerebral Blood Flow & Metabolism, 21(10), 1133–1145.

    Article  CAS  Google Scholar 

  • Baddeley, R, Abbott, LF, Booth, MC, Sengpiel, F, Freeman, T, Wakeman, EA, & Rolls, ET (1997). Responses of neurons in primary and inferior temporal visual cortices to natural scenes. Proceedings of the Royal Society of London Series B: Biological Sciences, 264(1389), 1775–1783.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Benda, J, & Herz, AV (2003). A universal model for spike-frequency adaptation. Neural computation, 15(11), 2523–2564.

    Article  PubMed  Google Scholar 

  • Berger, T, & Levy, WB (2010). A mathematical theory of energy efficient neural computation and communication. IEEE Transactions on Information Theory, 56(2), 852–874.

    Article  Google Scholar 

  • Bialek, W, Rieke, F, de Ruyter van Steveninck, RR, & Warland, D (1991). Reading a neural code. Science, 252(5014), 1854– 1857.

    Article  CAS  PubMed  Google Scholar 

  • Boerlin, M, Machens, C, Deneve, S, & Sporns, O (2013). Predictive coding of dynamical variables in balanced spiking networks. PLoS Computational Biology, 9(11), e1003258.

    Article  PubMed  PubMed Central  Google Scholar 

  • Brandman, R, & Nelson, ME (2002). A simple model of long-term spike train regularization. Neural Computation, 14(7), 1575–1597.

    Article  PubMed  Google Scholar 

  • Brown, D, & Adams, P (1980). Muscarinic suppression of a novel voltage-sensitive k + ; current in a vertebrate neurone. Nature, 283(5748), 673–676.

    Article  CAS  PubMed  Google Scholar 

  • Chacron, MJ, Longtin, A, & Maler, L (2001). Negative interspike interval correlations increase the neuronal capacity for encoding time-dependent stimuli. The Journal of Neuroscience, 21(14), 5328–5343.

    CAS  PubMed  Google Scholar 

  • Chacron, MJ, Pakdaman, K, & Longtin, A (2003). Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire model with threshold fatigue. Neural Computation, 15(2), 253–278.

    Article  PubMed  Google Scholar 

  • Dayan, P, & Abbott, LF (2001). Theoretical neuroscience, Vol. 806. Cambridge: MIT Press.

  • Delmas, P, & Brown, DA (2005). Pathways modulating neural KCNQ/M (Kv7) potassium channels. Nature Reviews Neuroscience, 6(11), 850–862.

    Article  CAS  PubMed  Google Scholar 

  • Eggermont, J, Johannesma, P, & Aertsen, A (1983). Reverse-correlation methods in auditory research. Quarterly Reviews of Biophysics, 16(03), 341–414.

    Article  CAS  PubMed  Google Scholar 

  • Eggermont, JJ (1998). Is there a neural code? Neuroscience & Biobehavioral Reviews, 22(2), 355–370.

    Article  CAS  Google Scholar 

  • Farkhooi, F, Strube-Bloss, MF, & Nawrot, MP (2009). Serial correlation in neural spike trains: experimental evidence, stochastic modeling, and single neuron variability. Physical Review E, 79(2), 021905.

    Article  Google Scholar 

  • Gabbiani, F (1996). Coding of time-varying signals in spike trains of linear and half-wave rectifying neurons. Network: Computation in Neural Systems, 7(1), 61–85.

    Article  Google Scholar 

  • Gautrais, J, & Thorpe, S (1998). Rate coding versus temporal order coding: a theoretical approach. Biosystems, 48(1), 57–65.

    Article  CAS  PubMed  Google Scholar 

  • Gerstner, W, & Naud, R (2009). How good are neuron models? Science, 326(5951), 379–380.

    Article  CAS  PubMed  Google Scholar 

  • Goense, J, & Ratnam, R (2003). Continuous detection of weak sensory signals in afferent spike trains: the role of anti-correlated interspike intervals in detection performance. Journal of Comparative Physiology A, 189(10), 741–759.

    Article  CAS  Google Scholar 

  • Gollisch, T, & Meister, M (2008). Rapid neural coding in the retina with relative spike latencies. Science, 319(5866), 1108– 1111.

    Article  CAS  PubMed  Google Scholar 

  • Hille, B, & et al. (2001). Ion channels of excitable membranes. Sinauer Sunderland, MA.

  • Hodgkin, AL, & Huxley, AF (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117(4), 500.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Johnson, EC, Jones, DL, & Ratnam, R (2015). Minimum squared-error, energy-constrained encoding by adaptive threshold models of neurons. In 2015 IEEE international symposium on information theory proceedings (ISIT), IEEE (pp. 1337–1341).

  • Jones, DL, Johnson, EC, & Ratnam, R (2015). A stimulus-dependent spike threshold is an optimal neural coder. Frontiers in Computational Neuroscience, 9, 61.

    Article  PubMed  PubMed Central  Google Scholar 

  • Kayser, C, Logothetis, NK, & Panzeri, S (2010). Millisecond encoding precision of auditory cortex neurons. Proceedings of the National Academy of Sciences, 107(39), 16,976–16,981.

    Article  CAS  Google Scholar 

  • Kiang, NYS, Wantanabe, T, Thomas, EC, & Clark, LF. (1965). Discharge patterns of single fibers in the cat’s auditory nerve. Cambridge: MIT Press.

    Google Scholar 

  • Kistler, W, Gerstner, W, & Hemmen, J (1997). Reduction of the Hodgkin-Huxley equations to a single-variable threshold model. Neural Computation, 9(5), 1015–1045.

    Article  Google Scholar 

  • Kobayashi, R, Tsubo, Y, & Shinomoto, S (2009). Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold. Frontiers in Computational Neuroscience, 3, 9.

    Article  PubMed  PubMed Central  Google Scholar 

  • Laughlin, SB (2001). Energy as a constraint on the coding and processing of sensory information. Current Opinion in Neurobiology, 11(4), 475–480.

    Article  CAS  PubMed  Google Scholar 

  • Laughlin, SB, De Ruyter van Steveninck, RR, & Anderson, JC (1998). The metabolic cost of neural information. Nature Neuroscience, 1(1), 36–41.

    Article  CAS  PubMed  Google Scholar 

  • Levy, WB, & Baxter, RA (1996). Energy efficient neural codes. Neural Computation, 8(3), 531–543.

    Article  CAS  PubMed  Google Scholar 

  • London, M, Roth, A, Beeren, L, Häusser, M, & Latham, PE (2010). Sensitivity to perturbations in vivo implies high noise and suggests rate coding in cortex. Nature, 466, 123–127.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Lüdtke, N, & Nelson, ME (2006). Short-term synaptic plasticity can enhance weak signal detectability in nonrenewal spike trains. Neural Computation, 18(12), 2879–2916.

    Article  PubMed  Google Scholar 

  • MacKay, DM, & McCulloch, WS (1952). The limiting information capacity of a neuronal link. The Bulletin of Mathematical Biophysics, 14(2), 127–135.

    Article  Google Scholar 

  • Masuda, N, & Aihara, K (2003). Duality of rate coding and temporal coding in multilayered feedforward networks. Neural Computation, 15(1), 103–125.

    Article  PubMed  Google Scholar 

  • Nabatiyan, A, Poulet, J, De Polavieja, G, & Hedwig, B (2003). Temporal pattern recognition based on instantaneous spike rate coding in a simple auditory system. Journal of Neurophysiology, 90(4), 2484–2493.

    Article  CAS  PubMed  Google Scholar 

  • Nelson, M, Xu, Z, & Payne, J (1997). Characterization and modeling of p-type electrosensory afferent responses to amplitude modulations in a wave-type electric fish. Journal of Comparative Physiology A, 181(5), 532–544.

    Article  CAS  Google Scholar 

  • Nesse, WH, Maler, L, & Longtin, A (2010). Biophysical information representation in temporally correlated spike trains. Proceedings of the National Academy of Sciences, 107(51), 21,973–21,978.

    Article  CAS  Google Scholar 

  • Niven, JE, & Laughlin, SB (2008). Energy limitation as a selective pressure on the evolution of sensory systems. Journal of Experimental Biology, 211(11), 1792–1804.

    Article  CAS  PubMed  Google Scholar 

  • Oswald, AMM, Doiron, B, & Maler, L (2007). Interval coding. I. Burst interspike intervals as indicators of stimulus intensity. Journal of Neurophysiology, 97(4), 2731–2743.

    Article  PubMed  Google Scholar 

  • Panzeri, S, & Schultz, SR (2001). A unified approach to the study of temporal, correlational, and rate coding. Neural Computation, 13(6), 1311–1349.

    Article  CAS  PubMed  Google Scholar 

  • Prescott, SA, & Sejnowski, TJ (2008). Spike-rate coding and spike-time coding are affected oppositely by different adaptation mechanisms. The Journal of Neuroscience, 28(50), 13,649–13,661.

    Article  CAS  Google Scholar 

  • Ratnam, R, & Nelson, ME (2000). Nonrenewal statistics of electrosensory afferent spike trains: implications for the detection of weak sensory signals. The Journal of Neuroscience, 20(17), 6672–6683.

    CAS  PubMed  Google Scholar 

  • Rudolph, M, & Destexhe, A (2003). Tuning neocortical pyramidal neurons between integrators and coincidence detectors. Journal of Computational Neuroscience, 14(3), 239– 251.

    Article  PubMed  Google Scholar 

  • Sengupta, B, Stemmler, M, Laughlin, SB, & Niven, JE (2010). Action potential energy efficiency varies among neuron types in vertebrates and invertebrates. PLoS Computational Biology, 6(7), e1000840.

    Article  PubMed  PubMed Central  Google Scholar 

  • Strong, SP, Koberle, R, De Ruyter van Steveninck, RR, & Bialek, W (1998). Entropy and information in neural spike trains. Physical Review Letters, 80(1), 197.

    Article  CAS  Google Scholar 

  • Van Rullen, R, & Thorpe, SJ (2001). Rate coding versus temporal order coding: what the retinal ganglion cells tell the visual cortex. Neural Computation, 13(6), 1255–1283.

    Article  CAS  PubMed  Google Scholar 

  • Yager, D, & Hopkins, C (1993). Directional characteristics of tuberous electroreceptors in the weakly electric fish, hypopomus (gymnotiformes). Journal of Comparative Physiology A, 173(4), 401–414.

    Article  CAS  Google Scholar 

Download references

Acknowledgments

This research was supported by National Science Foundation grants EFRI-BSBA-0938007 and IGERT 0903622, research funds from the College of Engineering, UIUC, Coordinated Science Laboratory, UIUC and the Advanced Digital Sciences Center, Illinois at Singapore. Electric fish data were collected in the laboratory of Mark E. Nelson, UIUC, through the National Institute of Health grant R01MH49242 and National Science Foundation grant IBN-0078206. We gratefully acknowledge the availability of rat cortical pyramidal neuron data in the public domain through the International Neuroinformatics Coordinating Facility.

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Correspondence to Erik C. Johnson.

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Johnson, E.C., Jones, D.L. & Ratnam, R. A minimum-error, energy-constrained neural code is an instantaneous-rate code. J Comput Neurosci 40, 193–206 (2016). https://doi.org/10.1007/s10827-016-0592-x

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  • DOI: https://doi.org/10.1007/s10827-016-0592-x

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