Journal of Computational Neuroscience

, Volume 27, Issue 2, pp 277–290 | Cite as

Sensitivity of firing rate to input fluctuations depends on time scale separation between fast and slow variables in single neurons

  • Brian Nils Lundstrom
  • Michael Famulare
  • Larry B. Sorensen
  • William J. Spain
  • Adrienne L. Fairhall


Neuronal responses are often characterized by the firing rate as a function of the stimulus mean, or the fI curve. We introduce a novel classification of neurons into Types A, B−, and B+ according to how fI curves are modulated by input fluctuations. In Type A neurons, the fI curves display little sensitivity to input fluctuations when the mean current is large. In contrast, Type B neurons display sensitivity to fluctuations throughout the entire range of input means. Type B− neurons do not fire repetitively for any constant input, whereas Type B+ neurons do. We show that Type B+ behavior results from a separation of time scales between a slow and fast variable. A voltage-dependent time constant for the recovery variable can facilitate sensitivity to input fluctuations. Type B+ firing rates can be approximated using a simple “energy barrier” model.


Noise Gain fI curve Stimulus fluctuations Single neuron Time scales Dynamical systems Phase portrait Hodgkin-Huxley Slow adaptation Slow AHP 



We thank Bard Ermentrout for helpful conversations during initial stages of the project at the Marine Biological Laboratory’s Methods in Computational Neuroscience 2007 course, Matthew Higgs and Michele Giugliano for helpful discussions and providing data for a figure, and Randy Powers and Sungho Hong for comments on a draft of this manuscript.


This work was supported by a Burroughs-Wellcome Careers at the Scientific Interface grant and a McKnight Scholar Award; BNL was supported by grant number F30NS055650 from the National Institute of Neurological Disorders and Stroke, the Medical Scientist Training Program at UW supported by the National Institute of General Medical Sciences, and an ARCS fellowship; WJS was supported by a VA Merit Review.

Author contributions

Conceived of, designed, and performed the simulations: BL. Analyzed the data: BL MF LS AF. Wrote the paper: BL MF WS AF. Developed the conceptual framework: BL MF LS WS AF.


  1. Arfken, G. B., & Weber, H. -J. (1995). Mathematical methods for physicists (4th ed.). San Diego: Academic.Google Scholar
  2. Arsiero, M., Luscher, H. R., Lundstrom, B. N., & Giugliano, M. (2007). The impact of input fluctuations on the frequency-current relationships of layer 5 pyramidal neurons in the rat medial prefrontal cortex. The Journal of Neuroscience, 27, 3274–3284. doi: 10.1523/JNEUROSCI.4937-06.2007.PubMedCrossRefGoogle Scholar
  3. Benda, J., Longtin, A., & Maler, L. (2005). Spike-frequency adaptation separates transient communication signals from background oscillations. The Journal of Neuroscience, 25, 2312–2321. doi: 10.1523/JNEUROSCI.4795-04.2005.PubMedCrossRefGoogle Scholar
  4. Chance, F. S., Abbott, L. F., & Reyes, A. D. (2002). Gain modulation from background synaptic input. Neuron, 35, 773–782. doi: 10.1016/S0896-6273(02)00820-6.PubMedCrossRefGoogle Scholar
  5. Connor, J. A., & Stevens, C. F. (1971). Prediction of repetitive firing behaviour from voltage clamp data on an isolated neurone soma. The Journal of Physiology, 213, 31–53.PubMedGoogle Scholar
  6. Dayan, P., & Abbott, L. F. (2001). Theoretical neuroscience : Computational and mathematical modeling of neural systems. Cambridge, MA: Massachusetts Institute of Technology Press.Google Scholar
  7. Destexhe, A., Rudolph, M., Fellous, J. M., & Sejnowski, T. J. (2001). Fluctuating synaptic conductances recreate in vivo-like activity in neocortical neurons. Neuroscience, 107, 13–24. doi: 10.1016/S0306-4522(01)00344-X.PubMedCrossRefGoogle Scholar
  8. Destexhe, A., Rudolph, M., & Pare, D. (2003). The high-conductance state of neocortical neurons in vivo. Nature Reviews. Neuroscience, 4, 739–751. doi: 10.1038/nrn1198.PubMedCrossRefGoogle Scholar
  9. DeVille, R. E., Vanden-Eijnden, E., & Muratov, C. B. (2005). Two distinct mechanisms of coherence in randomly perturbed dynamical systems. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 72, 031105. doi: 10.1103/PhysRevE.72.031105.Google Scholar
  10. Ermentrout, B. (1998). Linearization of F–I curves by adaptation. Neural Computation, 10, 1721–1729. doi: 10.1162/089976698300017106.PubMedCrossRefGoogle Scholar
  11. Fairhall, A. L., Lewen, G. D., Bialek, W., & de Ruyter Van Steveninck, R. R. (2001). Efficiency and ambiguity in an adaptive neural code. Nature, 412, 787–792. doi: 10.1038/35090500.PubMedCrossRefGoogle Scholar
  12. Fellous, J. M., Rudolph, M., Destexhe, A., & Sejnowski, T. J. (2003). Synaptic background noise controls the input/output characteristics of single cells in an in vitro model of in vivo activity. Neuroscience, 122, 811–829. doi: 10.1016/j.neuroscience.2003.08.027.PubMedCrossRefGoogle Scholar
  13. Fleidervish, I. A., Friedman, A., & Gutnick, M. J. (1996). Slow inactivation of Na+ current and slow cumulative spike adaptation in mouse and guinea-pig neocortical neurones in slices. The Journal of Physiology, 493(Pt 1), 83–97.PubMedGoogle Scholar
  14. Gerstner, W., & Kistler, W. M. (2002). Spiking neuron models : Single neurons, populations, plasticity. Cambridge, UK: Cambridge University Press.Google Scholar
  15. Gutkin, B. S., & Ermentrout, G. B. (1998). Dynamics of membrane excitability determine interspike interval variability: a link between spike generation mechanisms and cortical spike train statistics. Neural Computation, 10, 1047–1065. doi: 10.1162/089976698300017331.PubMedCrossRefGoogle Scholar
  16. Higgs, M. H., Slee, S. J., & Spain, W. J. (2006). Diversity of gain modulation by noise in neocortical neurons: regulation by the slow afterhyperpolarization conductance. The Journal of Neuroscience, 26, 8787–8799. doi: 10.1523/JNEUROSCI.1792-06.2006.PubMedCrossRefGoogle Scholar
  17. Hodgkin, A. L. (1948). The local electric changes associated with repetitive action in a non-medullated axon. The Journal of Physiology, 107, 165–181.PubMedGoogle Scholar
  18. Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117, 500–544.PubMedGoogle Scholar
  19. Hong, S., Aguera y Arcas, B., & Fairhall, A. L. (2007). Single neuron computation: from dynamical system to feature detector. Neural Computation, 19, 3133–3172. doi: 10.1162/neco.2007.19.12.3133.PubMedCrossRefGoogle Scholar
  20. Hong, S., Lundstrom, B. N., & Fairhall, A. L. (2008). Intrinsic gain modulation and adaptive neural coding. PLoS Computational Biology, 4, e1000119. doi: 10.1371/journal.pcbi.1000119.PubMedCrossRefGoogle Scholar
  21. Izhikevich, E. M. (2007). Dynamical systems in neuroscience : The geometry of excitability and bursting. Cambridge, MA: MIT Press.Google Scholar
  22. Koch, C. (1999). Biophysics of computation: information processing in single neurons. New York: Oxford University Press.Google Scholar
  23. Konig, P., Engel, A. K., & Singer, W. (1996). Integrator or coincidence detector? The role of the cortical neuron revisited. Trends in Neurosciences, 19, 130–137. doi: 10.1016/S0166-2236(96)80019-1.PubMedCrossRefGoogle Scholar
  24. Lundstrom, B. N., & Fairhall, A. L. (2006). Decoding stimulus variance from a distributional neural code of interspike intervals. The Journal of Neuroscience, 26, 9030–9037. doi: 10.1523/JNEUROSCI.0225-06.2006.PubMedCrossRefGoogle Scholar
  25. Lundstrom, B. N., Hong, S., Higgs, M. H., & Fairhall, A. L. (2008). Two computational regimes of a single-compartment neuron separated by a planar boundary in conductance space. Neural Comput, 20, 1239–1260.PubMedCrossRefGoogle Scholar
  26. Moreno, R., de la Rocha, J., Renart, A., & Parga, N. (2002). Response of spiking neurons to correlated inputs. Physical Review Letters, 89, 288101. doi: 10.1103/PhysRevLett.89.288101.PubMedCrossRefGoogle Scholar
  27. Morris, C., & Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal, 35, 193–213. doi: 10.1016/S0006-3495(81)84782-0.PubMedCrossRefGoogle Scholar
  28. Prescott, S. A., Ratte, S., De Koninck, Y., & Sejnowski, T. J. (2006). Nonlinear interaction between shunting and adaptation controls a switch between integration and coincidence detection in pyramidal neurons. The Journal of Neuroscience, 26, 9084–9097. doi: 10.1523/JNEUROSCI.1388-06.2006.PubMedCrossRefGoogle Scholar
  29. Rauch, A., La Camera, G., Luscher, H. R., Senn, W., & Fusi, S. (2003). Neocortical pyramidal cells respond as integrate-and-fire neurons to in vivo-like input currents. Journal of Neurophysiology, 90, 1598–1612. doi: 10.1152/jn.00293.2003.PubMedCrossRefGoogle Scholar
  30. Richardson, M. J. (2004). Effects of synaptic conductance on the voltage distribution and firing rate of spiking neurons. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 69, 051918. doi: 10.1103/PhysRevE.69.051918.Google Scholar
  31. Rinzel, J., & Ermentrout, B. (1998). Analysis of neural excitability and oscillations. In C. Koch, I. Segev, & (Eds.), (pp. 251–291, 2nd ed.). Cambridge, Massachusetts: MIT Press.Google Scholar
  32. Robinson, H. P., & Harsch, A. (2002). Stages of spike time variability during neuronal responses to transient inputs. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 66, 061902. doi: 10.1103/PhysRevE.66.061902.Google Scholar
  33. Rudolph, M., & Destexhe, A. (2005). An extended analytic expression for the membrane potential distribution of conductance-based synaptic noise. Neural Computation, 17, 2301–2315. doi: 10.1162/0899766054796932.PubMedCrossRefGoogle Scholar
  34. Rudolph, M., & Destexhe, A. (2006). On the use of analytical expressions for the voltage distribution to analyze intracellular recordings. Neural Computation, 18, 2917–2922. doi: 10.1162/neco.2006.18.12.2917.PubMedCrossRefGoogle Scholar
  35. Rush, M. E., & Rinzel, J. (1995). The potassium A-current, low firing rates and rebound excitation in Hodgkin–Huxley models. Bulletin of Mathematical Biology, 57, 899–929.PubMedGoogle Scholar
  36. Shadlen, M. N., & Newsome, W. T. (1994). Noise, neural codes and cortical organization. Current Opinion in Neurobiology, 4, 569–579. doi: 10.1016/0959-4388(94)90059-0.PubMedCrossRefGoogle Scholar
  37. Slee, S. J., Higgs, M. H., Fairhall, A. L., & Spain, W. J. (2005). Two-dimensional time coding in the auditory brainstem. The Journal of Neuroscience, 25, 9978–9988. doi: 10.1523/JNEUROSCI.2666-05.2005.PubMedCrossRefGoogle Scholar
  38. Strogatz, S. H. (1994). Nonlinear dynamics and Chaos: with applications to physics, biology, chemistry, and engineering. Reading, Mass.: Addison-Wesley.Google Scholar
  39. Tateno, T., & Pakdaman, K. (2004). Random dynamics of the Morris–Lecar neural model. Chaos (Woodbury, N.Y.), 14, 511–530. doi: doi:10.1063/1.1756118.CrossRefGoogle Scholar
  40. VanRullen, R., Guyonneau, R., & Thorpe, S. J. (2005). Spike times make sense. Trends in Neurosciences, 28, 1–4. doi: 10.1016/j.tins.2004.10.010.PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Brian Nils Lundstrom
    • 1
  • Michael Famulare
    • 2
  • Larry B. Sorensen
    • 2
  • William J. Spain
    • 1
    • 3
  • Adrienne L. Fairhall
    • 1
  1. 1.Department of Physiology and BiophysicsUniversity of WashingtonSeattleUSA
  2. 2.Department of PhysicsUniversity of WashingtonSeattleUSA
  3. 3.Neurology SectionVeterans Affairs Puget Sound Health Care SystemSeattleUSA

Personalised recommendations