Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Firing rate distributions in a feedforward network of neural oscillators with intrinsic and network heterogeneity

Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, Richmond, VA 23284, USA

The level of firing rate heterogeneity in a population of cortical neurons has consequences for how stimuli are processed. Recent studies have shown that the right amount of firing rate heterogeneity (not too much or too little) is a signature of efficient coding, thus quantifying the relative amount of firing rate heterogeneity is important. In a feedforward network of stochastic neural oscillators, we study the firing rate heterogeneity stemming from two sources: intrinsic (different individual cells) and network (different effects from presynaptic inputs). We find that the relationship between these two forms of heterogeneity can lead to significant changes in firing rate heterogeneity. We consider several networks, including noisy excitatory synaptic inputs, and noisy inputs with both excitatory and inhibitory inputs. To mathematically explain these results, we apply a phase reduction and derive asymptotic approximations of the firing rate statistics assuming weak noise and coupling. Our analytic calculations reveals how the interaction between intrinsic and network heterogeneity results in different firing rate distributions. Our work shows the importance of the phase-resetting curve (and various transformations of it revealed by our analytic calculations) in controlling firing rate statistics.
  Article Metrics


1. J. Gjorgjieva, R. A. Mease, W. J. Moody, et al., Intrinsic neuronal properties switch the mode of information transmission in networks, PLoS Comput. Biol., 10 (2014), e1003962.

2. A. Kohn, R. Coen-Cagli, I. Kanitscheider, et al., Correlations and neuronal population information, Annu. Rev. Neurosci., 39 (2016), 237–256.

3. C. Ly and G. Marsat, Variable synaptic strengths controls the firing rate distribution in feedforward neural networks, J. Comput. Neurosci., 44 (2018), 75–95.

4. G. Marsat and L. Maler, Neural heterogeneity and efficient population codes for communication signals, J. Neurophysiol.y, 104 (2010), 2543–2555.

5. R. A. Mease, M. Famulare, J. Gjorgjieva, et al., Emergence of adaptive computation by single neurons in the developing cortex, J. Neurosci., 33 (2013), 12154–12170.

6. K. Padmanabhan and N. N. Urban, Intrinsic biophysical diversity decorrelates neuronal firing while increasing information content, Nature Neurosci., 13 (2010), 1276–1282.

7. M. I. Chelaru and V. Dragoi, Efficient coding in heterogeneous neuronal populations, P. Nat. Acad. Sci., 105 (2008), 16344–16349.

8. J. Mejias and A. Longtin, Optimal heterogeneity for coding in spiking neural networks, Phys. Rev. Lett., 108 (2012), 228102.

9. A. S. Ecker, P. Berens, A. S. Tolias, et al., The effect of noise correlations in populations of diversely tuned neurons, J. Neurosci., 31 (2011), 14272–14283.

10. C. Ly, Firing rate dynamics in recurrent spiking neural networks with intrinsic and network heterogeneity, J. Comput. Neurosci., 39 (2015), 311–327.

11. E. Marder, Variability, compensation, and modulation in neurons and circuits, P. Nat. Acad. Sci., 108 (2011), 15542–15548.

12. E. Marder and J.-M. Goaillard, Variability, compensation and homeostasis in neuron and network function, Nature Rev. Neurosci., 7 (2006), 563.

13. A. Roxin, N. Brunel, D. Hansel, et al., On the distribution of firing rates in networks of cortical neurons, J. Neurosci., 31 (2011), 16217–16226.

14. S. Ratté, S. Hong, E. De Schutter, et al., Impact of neuronal properties on network coding: roles of spike initiation dynamics and robust synchrony transfer, Neuron, 78 (2013), 758–772.

15. N.W. Schultheiss, A. A. Prinz and R. J. Butera, Phase Response Curves in Neuroscience: Theory, Experiment, and Analysis, vol. 6 of Springer Series in Computational Neuroscience, Springer, 2012.

16. S. D. Burton, G. B. Ermentrout and N. N. Urban, Intrinsic heterogeneity in oscillatory dynamics limits correlation-induced neural synchronization, J. Neurophysiol., 108 (2012), 2115–2133.

17. J. A. Acebrón, L. Bonilla, C. J. P. Vincente,et al., The kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137–185.

18. J. Ahn, L. Kreeger, S. Lubejko, et al., Heterogeneity of intrinsic biophysical properties among cochlear nucleus neurons improves the population coding of temporal information, J. Neurophysiol., 111 (2014), 2320–2331.

19. I. Aihara, Modeling synchronized calling behavior of japanese tree frogs, Phys. Rev. E, 80 (2009), 011918.

20. K. Arai and H. Nakao, Phase coherence in an ensemble of uncoupled limit-cycle oscillators receiving common poisson impulses, Phys. Rev. E, 77 (2008), 036218.

21. P. Ashwin, S. Coombes and R. Nicks, Mathematical frameworks for oscillatory network dynamics in neuroscience, J. Math. Neurosci., 6 (2016), 2.

22. R. Azouz and C. M. Gray, Dynamic spike threshold reveals a mechanism for synaptic coincidence detection in cortical neurons in vivo, P. Nat. Acad. Sci., 97 (2000), 8110–8115.

23. A. Bremaud, D. West and A. Thomson, Binomial parameters differ across neocortical layers and with different classes of connections in adult rat and cat neocortex, P. Nat. Acad. Sci., 104 (2007), 14134–14139.

24. R. J. Butera Jr, J. Rinzel and J. C. Smith, Models of respiratory rhythm generation in the prebotzinger complex. ii. populations of coupled pacemaker neurons, J. Neurophysiol., 82 (1999), 398–415.

25. F. P. Chabrol, A. Arenz, M. T. Wiechert, et al., Synaptic diversity enables temporal coding of coincident multisensory inputs in single neurons, Nature Neurosci., 18 (2015), 718.

26. B. Ermentrout, Type i membranes, phase resetting curves, and synchrony, Neural. Comput., 8 (1996), 979–1001.

27. B. Ermentrout, Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students, vol. 14, Siam, 2002.

28. G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience, vol. 35 of Interdisciplinary Applied Mathematics, Springer, 2010.

29. A. Georgopoulos, A. Schwartz and R. Kettner, Neuronal population coding of movement direction, Science, 233 (1986), 1416–1419.

30. J. Gjorgjieva, G. Drion and E. Marder, Computational implications of biophysical diversity and multiple timescales in neurons and synapses for circuit performance, Curr. Opin. Neurobiol., 37 (2016), 44–52.

31. P. Goel and B. Ermentrout, Synchrony, stability, and firing patterns in pulse-coupled oscillators, Physica D: Nonlinear Phenomena, 163 (2002), 191–216.

32. R. Grashow, T. Brookings and E. Marder, Compensation for variable intrinsic neuronal excitability by circuit-synaptic interactions, J. Neurosci., 30 (2010), 9145–9156.

33. G. Hermann and J. Touboul, Heterogeneous connections induce oscillations in large-scale networks, Phys. Rev. Lett., 109 (2012), 018702.

34. X. Jiang, S. Shen, C. R. Cadwell, et al., Principles of connectivity among morphologically defined cell types in adult neocortex, Science, 350 (2015), aac9462.

35. S. Kay, Fundamentals of Statistical Signal Processing, Volume 1: Estimation Theory, Prentice Hall PTR, 1993.

36. Y. Kuramoto, Chemical oscillations, waves and turbulence, Berlin: Springer, 1984.

37. Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics (ed. H. Araki), Springer Berlin Heidelberg, Berlin, Heidelberg, 1975, 420–422.

38. R. B. Levy and A. D. Reyes, Spatial profile of excitatory and inhibitory synaptic connectivity in mouse primary auditory cortex, J Neurosci., 32 (2012), 5609–5619.

39. C. Ly and B. Doiron, Noise-enhanced coding in phasic neuron spike trains, PLoS ONE, 4 (2017), e0176963.

40. C. Ly and G. B. Ermentrout, Analysis of recurrent networks of pulse-coupled noisy neural oscillators, SIAM J. Appl. Dyn. Syst., 9 (2010), 113–137.

41. C. Ly, T. Melman, A. L. Barth, et al., Phase-resetting curve determines how bk currents affect neuronal firing, J. Comput. Neurosci., 30 (2011), 211–223.

42. T. McGeer et al., Passive dynamic walking, I. J. Robotic Res., 9 (1990), 62–82.

43. A. L. Meredith, S.W.Wiler, B. H. Miller, et al., Bk calcium-activated potassium channels regulate circadian behavioral rhythms and pacemaker output, Nature Neurosci., 9 (2006), 1041.

44. J. Middleton, A. Longtin, J. Benda, et al., Postsynaptic receptive field size and spike threshold determine encoding of high-frequency information via sensitivity to synchronous presynaptic activity, J. Neurophysiol., 101 (2009), 1160–1170.

45. C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophys. J., 35 (1981), 193–213.

46. J. M. Newby and M. A. Schwemmer, Effects of moderate noise on a limit cycle oscillator: Counterrotation and bistability, Phys. Rev. Lett., 112 (2014), 114101.

47. S. Oprisan, A. Prinz and C. Canavier, Phase resetting and phase locking in hybrid circuits of one model and one biological neuron, Biophys. J., 87 (2004), 2283–2298.

48. A. Oswald, B. Doiron, J. Rinzel, et al., Spatial profile and differential recruitment of gabab modulate oscillatory activity in auditory cortex, J. Neurosci., 29 (2009), 10321–10334.

49. D. Parker, Variable properties in a single class of excitatory spinal synapse, J. Nneurosci., 23 (2003), 3154–3163.

50. N. Priebe and D. Ferster, Inhibition, spike threshold, and stimulus selectivity in primary visual cortex, Neuron, 57 (2008), 482–497.

51. H. Risken, The fokker-planck equation. methods of solution and applications, vol. 18 of, Springer series in synergetics, 301.

52. J. T. Schwabedal and A. Pikovsky, Phase description of stochastic oscillations, Phys. Rev. Lett., 110 (2013), 204102.

53. M. Shamir and H. Sompolinsky, Implications of neuronal diversity on population coding, Neural computation, 18 (2006), 1951–1986.

54. A. A. Sharp, F. K. Skinner and E. Marder, Mechanisms of oscillation in dynamic clamp constructed two-cell half-center circuits, J. Neurophysiol., 76 (1996), 867–883.

55. K. M. Stiefel, B. S. Gutkin and T. J. Sejnowski, Cholinergic neuromodulation changes phase response curve shape and type in cortical pyramidal neurons, PloS one, 3 (2008), e3947.

56. J.-N. Teramae, H. Nakao and G. B. Ermentrout, Stochastic phase reduction for a general class of noisy limit cycle oscillators, Phys. Rev. Lett., 102 (2009), 194102.

57. P. J. Thomas and B. Lindner, Asymptotic phase for stochastic oscillators, Physical review letters, 113 (2014), 254101.

58. T. I. Tóth, M. Grabowska, N. Rosjat, et al., Investigating inter-segmental connections between thoracic ganglia in the stick insect by means of experimental and simulated phase response curves, Biol.l cybern., 109 (2015), 349–362.

59. S. J. Tripathy, K. Padmanabhan, R. C. Gerkin, et al., Intermediate intrinsic diversity enhances neural population coding, P. Natl. Acad. Sci. USA, 110 (2013), 8248–8253.

60. A. T.Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15–42.

61. A. T. Winfree, The geometry of biological time, vol. 12, Springer Science & Business Media, 2001.

62. L. Yassin, B. L. Benedetti, J.-S. Jouhanneau, et al., An embedded subnetwork of highly active neurons in the neocortex, Neuron, 68 (2010), 1043–1050.

63. R. A. York and T. Itoh, Injection-and phase-locking techniques for beam control [antenna arrays], IEEE T. Microw. Theory, 46 (1998), 1920–1929.

64. R. Yuste, J. N. MacLean, J. Smith, et al., The cortex as a central pattern generator, Nature Rev. Neurosci., 6 (2005), 477.

65. V. Zampini, J. K. Liu, M. A. Diana, et al., Mechanisms and functional roles of glutamatergic synapse diversity in a cerebellar circuit, eLife, 5 (2016), e15872.

66. P. Zhou, S. Burton, N. Urban, et al., Impact of neuronal heterogeneity on correlated colored noise-induced synchronization, Front. comput. neurosci., 7 (2013), 113.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved