Journal of Computational Neuroscience

, Volume 44, Issue 2, pp 173–188 | Cite as

Linearization of excitatory synaptic integration at no extra cost

  • Danielle Morel
  • Chandan Singh
  • William B LevyEmail author


In many theories of neural computation, linearly summed synaptic activation is a pervasive assumption for the computations performed by individual neurons. Indeed, for certain nominally optimal models, linear summation is required. However, the biophysical mechanisms needed to produce linear summation may add to the energy-cost of neural processing. Thus, the benefits provided by linear summation may be outweighed by the energy-costs. Using voltage-gated conductances in a relatively simple neuron model, this paper quantifies the cost of linearizing dendritically localized synaptic activation. Different combinations of voltage-gated conductances were examined, and many are found to produce linearization; here, four of these models are presented. Comparing the energy-costs to a purely passive model, reveals minimal or even no additional costs in some cases.


Voltage-gated conductances Metabolic cost Sodium channel Mixed-cation channel Biophysical model 



The authors thank the University of Virginia Department of Neurosurgery for their support.

Compliance with Ethical Standards

Conflict of interests

The authors declare that they have no conflict of interest.


  1. Agrawal, N., Hamam, B., Magistretti, J., Alonso, A., & Ragsdale, D. (2001). Persistent sodium channel activity mediates subthreshold membrane potential oscillations and low-threshold spikes in rat entorhinal cortex layer v neurons. Neuroscience, 102(1), 53–64.CrossRefPubMedGoogle Scholar
  2. Araya, R., Eisenthal, K.B., & Yuste, R. (2006). Dendritic spines linearize the summation of excitatory potentials. Proceedings of the National Academy of Sciences, 103(49), 18,799–18,804.CrossRefGoogle Scholar
  3. Attwell, D., & Laughlin, S.B. (2001). An energy budget for signaling in the grey matter of the brain. Journal of Cerebral Blood Flow & Metabolism, 21(10), 1133–1145.CrossRefGoogle Scholar
  4. Balasubramanian, V., & Berry, M.J. (2002). A test of metabolically efficient coding in the retina. Network: Computation in Neural Systems, 13(4), 531–552.CrossRefGoogle Scholar
  5. Baranauskas, G., David, Y., & Fleidervish, I.A. (2013). Spatial mismatch between the na+ flux and spike initiation in axon initial segment. Proceedings of the National Academy of Sciences, 110(10), 4051–4056.CrossRefGoogle Scholar
  6. Barber, M.J., Clark, J.W., & Anderson, C.H. (2003). Neural representation of probabilistic information. Neural Computation, 15(8), 1843–1864.CrossRefPubMedGoogle Scholar
  7. Bekkers, J.M. (2000). Distribution and activation of voltage-gated potassium channels in cell-attached and outside-out patches from large layer 5 cortical pyramidal neurons of the rat. The Journal of Physiology, 525(3), 611–620.CrossRefPubMedPubMedCentralGoogle Scholar
  8. Berger, T., & Levy, W.B. (2010). A mathematical theory of energy efficient neural computation and communication. IEEE Transactions on Information Theory, 56(2), 852–874.CrossRefGoogle Scholar
  9. Bernander, O., Koch, C., & Douglas, R.J. (1994). Amplification and linearization of distal synaptic input to cortical pyramidal cells. Journal of Neurophysiology, 72(6), 2743–2753.CrossRefPubMedGoogle Scholar
  10. Carandini, M., & Ferster, D. (2000). Membrane potential and firing rate in cat primary visual cortex. The Journal of Neuroscience, 20(1), 470–484.PubMedGoogle Scholar
  11. Carandini, M., Heeger, D.J., & Movshon, J.A. (1997). Linearity and normalization in simple cells of the macaque primary visual cortex. The Journal of Neuroscience, 17(21), 8621–8644.PubMedGoogle Scholar
  12. Carandini, M., Heeger, D.J., & Anthony Movshon, J. (1999). Linearity and gain control in V1 simple cells. In P.S. Ulinski, E.G. Jones, & A. Peters (Eds.) Models of cortical circuits. Cerebral cortex, Vol. 13. Boston: Springer.Google Scholar
  13. Carnevale, N.T., & Hines, M.L. (2006). The NEURON book. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  14. Cash, S., & Yuste, R. (1999). Linear summation of excitatory inputs by ca1 pyramidal neurons. Neuron, 22 (2), 383–394.CrossRefPubMedGoogle Scholar
  15. Cook, E.P., Guest, J.A., Liang, Y., Masse, N.Y., & Colbert, C.M. (2007). Dendrite-to-soma input/output function of continuous time-varying signals in hippocampal ca1 pyramidal neurons. Journal of Neurophysiology, 98 (5), 2943–2955.CrossRefPubMedGoogle Scholar
  16. DeAngelis, G.C., Ohzawa, I., & Freeman, R. (1993). Spatiotemporal organization of simple-cell receptive fields in the cat’s striate cortex. i. general characteristics and postnatal development. Journal of Neurophysiology, 69 (4), 1091–1117.CrossRefPubMedGoogle Scholar
  17. Deneve, S., & Pouget, A. (2004). Bayesian multisensory integration and cross-modal spatial links. Journal of Physiology-Paris, 98(1), 249–258.CrossRefGoogle Scholar
  18. Destexhe, A., Rudolph, M., & Paré, D. (2003). The high-conductance state of neocortical neurons in vivo. Nature Reviews Neuroscience, 4(9), 739–751.CrossRefPubMedGoogle Scholar
  19. Ferster, D. (1994). Linearity of synaptic interactions in the assembly of receptive fields in cat visual cortex. Current Opinion in Neurobiology, 4(4), 563–568.CrossRefPubMedGoogle Scholar
  20. Gasparini, S., & Magee, J.C. (2006). State-dependent dendritic computation in hippocampal ca1 pyramidal neurons. The Journal of Neuroscience, 26(7), 2088–2100.CrossRefPubMedGoogle Scholar
  21. Gillespie, D.C., Lampl, I., Anderson, J.S., & Ferster, D. (2001). Dynamics of the orientation-tuned membrane potential response in cat primary visual cortex. Nature Neuroscience, 4(10), 1014–1019.CrossRefPubMedGoogle Scholar
  22. Goldberg, D.H., Sripati, A.P., & Andreou, A.G. (2003). Energy efficiency in a channel model for the spiking axon. Neurocomputing, 52, 39–44.CrossRefGoogle Scholar
  23. Hoffman, D.A., Magee, J.C., Colbert, C.M., & Johnston, D. (1997). K+ channel regulation of signal propagation in dendrites of hippocampal pyramidal neurons. Nature, 387(6636), 869–875.CrossRefPubMedGoogle Scholar
  24. Howarth, C., Peppiatt-Wildman, C.M., & Attwell, D. (2010). The energy use associated with neural computation in the cerebellum. Journal of Cerebral Blood Flow & Metabolism, 30(2), 403–414.CrossRefGoogle Scholar
  25. Hu, W., Tian, C., Li, T., Yang, M., Hou, H., & Shu, Y. (2009). Distinct contributions of nav1. 6 and nav1. 2 in action potential initiation and backpropagation. Nature Neuroscience, 12(8), 996–1002.CrossRefPubMedGoogle Scholar
  26. Jagadeesh, B., Wheat, H.S., & Ferster, D. (1993). Linearity of summation of synaptic potentials underlying direction selectivity in simple cells of the cat visual cortex. Science-AAAS-Weekly Paper Edition-including Guide to Scientific Information, 262(5141), 1901–1905.Google Scholar
  27. Kerti, K., Lorincz, A., & Nusser, Z. (2012). Unique somato-dendritic distribution pattern of kv4. 2 channels on hippocampal ca1 pyramidal cells. European Journal of Neuroscience, 35(1), 66–75.CrossRefPubMedPubMedCentralGoogle Scholar
  28. Kole, M.H., Ilschner, S.U., Kampa, B.M., Williams, S.R., Ruben, P.C., & Stuart, G.J. (2008). Action potential generation requires a high sodium channel density in the axon initial segment. Nature Neuroscience, 11(2), 178–186.CrossRefPubMedGoogle Scholar
  29. Levy, W.B., & Baxter, R.A. (1996). Energy efficient neural codes. Neural Computation, 8(3), 531–543.CrossRefPubMedGoogle Scholar
  30. Levy, W.B., & Baxter, R.A. (2002). Energy-efficient neuronal computation via quantal synaptic failures. The Journal of Neuroscience, 22(11), 4746–4755.PubMedGoogle Scholar
  31. Levy, W.B., Colbert, C.M., & Desmond, N.L. (1990). Elemental adaptive processes of neurons and synapses: a statistical/computational perspective Vol. 187. Hillsdale: Erlbaum.Google Scholar
  32. Levy, W.B., Berger, T., & Sunkgar, M. (2016). Neural computation from first principles: Using the maximum entropy method to obtain an optimal bits-per-joule neuron. IEEE Transactions on Molecular, Biological and Multi-Scale Communications, 2(2), 154–165.CrossRefGoogle Scholar
  33. Magee, J.C. (1998). Dendritic hyperpolarization-activated currents modify the integrative properties of hippocampal ca1 pyramidal neurons. The Journal of Neuroscience, 18(19), 7613–7624.PubMedGoogle Scholar
  34. Magee, J.C. (1999). Dendritic ih normalizes temporal summation in hippocampal ca1 neurons. Nature Neuroscience, 2(6), 508– 514.CrossRefPubMedGoogle Scholar
  35. Magee, J.C., & Cook, E.P. (2000). Somatic epsp amplitude is independent of synapse location in hippocampal pyramidal neurons. Nature Neuroscience, 3(9), 895–903.CrossRefPubMedGoogle Scholar
  36. Magee, J.C. et al. (2000). Dendritic integration of excitatory synaptic input. Nature Reviews Neuroscience, 1 (3), 181–190.CrossRefPubMedGoogle Scholar
  37. Magistretti, J., & Alonso, A. (1999). Biophysical properties and slow voltage-dependent inactivation of a sustained sodium current in entorhinal cortex layer-ii principal neurons a whole-cell and single-channel study. The Journal of General Physiology, 114(4), 491–509.CrossRefPubMedPubMedCentralGoogle Scholar
  38. McCulloch, W.S., & Pitts, W. (1943). A logical calculus of the ideas immanent in nervous activity. The Bulletin of Mathematical Biophysics, 5(4), 115–133.CrossRefGoogle Scholar
  39. Migliore, M., & Shepherd, G.M. (2002). Emerging rules for the distributions of active dendritic conductances. Nature Reviews Neuroscience, 3(5), 362–370.CrossRefPubMedGoogle Scholar
  40. Morel, D., & Levy, W.B. (2007). Persistent sodium is a better linearizing mechanism than the hyperpolarization-activated current. Neurocomputing, 70(10), 1635–1639.CrossRefGoogle Scholar
  41. Morel, D., & Levy, W.B. (2009). The cost of linearization. Journal of Computational Neuroscience, 27(2), 259–275.CrossRefPubMedGoogle Scholar
  42. Paré, D., Shink, E., Gaudreau, H., Destexhe, A., & Lang, E.J. (1998). Impact of spontaneous synaptic activity on the resting properties of cat neocortical pyramidal neurons in vivo. Journal of Neurophysiology, 79(3), 1450–1460.CrossRefPubMedGoogle Scholar
  43. Poirazi, P., Brannon, T., & Mel, B.W. (2003). Arithmetic of subthreshold synaptic summation in a model ca1 pyramidal cell. Neuron, 37, 977–987.CrossRefPubMedGoogle Scholar
  44. Powers, R.K., & Binder, M.D. (2000). Summation of effective synaptic currents and firing rate modulation in cat spinal motoneurons. Journal of Neurophysiology, 83(1), 483–500.CrossRefPubMedGoogle Scholar
  45. Priebe, N.J., & Ferster, D. (2005). Direction selectivity of excitation and inhibition in simple cells of the cat primary visual cortex. Neuron, 45(1), 133–145.CrossRefPubMedGoogle Scholar
  46. Revah, O., Libman, L., Fleidervish, I.A., & Gutnick, M.J. (2015). The outwardly rectifying current of layer 5 neocortical neurons that was originally identified as non-specific cationic is essentially a potassium current. PloS One, 10(7), e0132,108.CrossRefGoogle Scholar
  47. Sengupta, B., Stemmler, M.B., & Friston, K.J. (2013). Information and efficiency in the nervous system—a synthesis. PLoS Computational Biology, 9(7), e1003,157.CrossRefGoogle Scholar
  48. Singh, C., & Levy, W.B. (2017). A consensus layer v pyramidal neuron can sustain interpulse-interval coding. PloS One, 12(7), e0180,839.CrossRefGoogle Scholar
  49. Smith, M.A., Ellis-Davies, G.C., & Magee, J.C. (2003). Mechanism of the distance-dependent scaling of schaffer collateral synapses in rat ca1 pyramidal neurons. The Journal of Physiology, 548(1), 245–258.CrossRefPubMedPubMedCentralGoogle Scholar
  50. Stuart, G., & Sakmann, B. (1995). Amplification of epsps by axosomatic sodium channels in neocortical pyramidal neurons. Neuron, 15(5), 1065–1076.CrossRefPubMedGoogle Scholar
  51. Vincent, B.T., & Baddeley, R.J. (2003). Synaptic energy efficiency in retinal processing. Vision Research, 43 (11), 1285–1292.CrossRefGoogle Scholar
  52. Vincent, B.T., Baddeley, R.J., Troscianko, T., & Gilchrist, I.D. (2005). Is the early visual system optimised to be energy efficient? Network: Computation in Neural Systems, 16(2-3), 175–190.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Danielle Morel
    • 1
  • Chandan Singh
    • 2
  • William B Levy
    • 2
    Email author
  1. 1.Physics DepartmentEmory & Henry CollegeEmoryUSA
  2. 2.Departments of Neurosurgery and of PsychologyUniversity of VirginiaCharlottesvilleUSA

Personalised recommendations