# Linearization of excitatory synaptic integration at no extra cost

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## Abstract

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.

## Keywords

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

### Acknowledgements

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.

## References

- 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 - 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 - 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 - 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 - 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 - Barber, M.J., Clark, J.W., & Anderson, C.H. (2003). Neural representation of probabilistic information.
*Neural Computation*,*15*(8), 1843–1864.CrossRefPubMedGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - Carnevale, N.T., & Hines, M.L. (2006).
*The NEURON book*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Cash, S., & Yuste, R. (1999). Linear summation of excitatory inputs by ca1 pyramidal neurons.
*Neuron*,*22*(2), 383–394.CrossRefPubMedGoogle Scholar - 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 - 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 - Deneve, S., & Pouget, A. (2004). Bayesian multisensory integration and cross-modal spatial links.
*Journal of Physiology-Paris*,*98*(1), 249–258.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - Levy, W.B., & Baxter, R.A. (1996). Energy efficient neural codes.
*Neural Computation*,*8*(3), 531–543.CrossRefPubMedGoogle Scholar - 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 - 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 - 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 - 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 - Magee, J.C. (1999). Dendritic ih normalizes temporal summation in hippocampal ca1 neurons.
*Nature Neuroscience*,*2*(6), 508– 514.CrossRefPubMedGoogle Scholar - 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 - Magee, J.C. et al. (2000). Dendritic integration of excitatory synaptic input.
*Nature Reviews Neuroscience*,*1*(3), 181–190.CrossRefPubMedGoogle Scholar - 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 - 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 - Migliore, M., & Shepherd, G.M. (2002). Emerging rules for the distributions of active dendritic conductances.
*Nature Reviews Neuroscience*,*3*(5), 362–370.CrossRefPubMedGoogle Scholar - 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 - Morel, D., & Levy, W.B. (2009). The cost of linearization.
*Journal of Computational Neuroscience*,*27*(2), 259–275.CrossRefPubMedGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - Stuart, G., & Sakmann, B. (1995). Amplification of epsps by axosomatic sodium channels in neocortical pyramidal neurons.
*Neuron*,*15*(5), 1065–1076.CrossRefPubMedGoogle Scholar - Vincent, B.T., & Baddeley, R.J. (2003). Synaptic energy efficiency in retinal processing.
*Vision Research*,*43*(11), 1285–1292.CrossRefGoogle Scholar - 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