Journal of Mathematical Biology

, Volume 68, Issue 1–2, pp 303–340 | Cite as

The robustness of phase-locking in neurons with dendro-dendritic electrical coupling

Article

Abstract

We examine the effects of dendritic filtering on the existence, stability, and robustness of phase-locked states to heterogeneity and noise in a pair of electrically coupled ball-and-stick neurons with passive dendrites. We use the theory of weakly coupled oscillators and analytically derived filtering properties of the dendritic coupling to systematically explore how the electrotonic length and diameter of dendrites can alter phase-locking. In the case of a fixed value of the coupling conductance (\(g_c\)) taken from the literature, we find that repeated exchanges in stability between the synchronous and anti-phase states can occur as the electrical coupling becomes more distally located on the dendrites. However, the robustness of the phase-locked states in this case decreases rapidly towards zero as the distance between the electrical coupling and the somata increases. Published estimates of \(g_c\) are calculated from the experimentally measured coupling coefficient (\(CC\)) based on a single-compartment description of a neuron, and therefore may be severe underestimates of \(g_c\). With this in mind, we re-examine the stability and robustness of phase-locking using a fixed value of \(CC\), which imposes a limit on the maximum distance the electrical coupling can be located away from the somata. In this case, although the phase-locked states remain robust over the entire range of possible coupling locations, no exchanges in stability with changing coupling position are observed except for a single exchange that occurs in the case of a high somatic firing frequency and a large dendritic radius. Thus, our analysis suggests that multiple exchanges in stability with changing coupling location are unlikely to be observed in real neural systems.

Keywords

Mathematical Neuroscience Ball-and-stick neuronal model Phase response curves Electrical Coupling Dendrites Synchronization 

Mathematics Subject Classification (2000)

92B25 37N25 

References

  1. Amitai Y, Gibson JR, Beierlein M, Patrick PL, Ho AM, Connors BW, Golomb D (2004) The spatial dimensions of electrically coupled networks of interneurons in the neocortex. J Neurosci 22(10): 4142–4152Google Scholar
  2. Averbeck BB, Lee D (2004) Coding and transmission of information by neural ensembles. Trends Neurosci 27(4):225–230CrossRefGoogle Scholar
  3. Beierlein M, Gibson JR, Connors BW (2000) A network of electronically coupled interneurons drives synchronized activity in neocortex. Nat Neurosci 3:904–910CrossRefGoogle Scholar
  4. Bennett MVL (1977) The nervous system, part I. In: Brookhart JM, Mountcastle VB (eds) Handbook of physiology, section I. American Physiological Society, Bethesda, pp 357–416Google Scholar
  5. Bressloff PC, Coombes S (1997) Physics of the extended neuron. J Mod Phys B 11:2343–2392CrossRefGoogle Scholar
  6. Bressloff PC, Coombes S (1997) Synchrony in an array of integrate-and-fire neurons with dendritic structure. Phys Rev Lett 78:4665–4668CrossRefGoogle Scholar
  7. Buzsáki G (1997) Functions for hippocampal interneurons. Can J Physiol Pharm 75:508–515CrossRefGoogle Scholar
  8. Buzsáki G, Draguhn A (2004) Neuronal oscillations in cortical networks. Science 304(5679):1926–1929CrossRefGoogle Scholar
  9. Cardin JA, Carleń M, Meletis K, Knoblich U, Zhang F, Deisseroth K, Tsai L, Moore CI (2009) Driving fast-spiking cells induces gamma rhythm and controls sensory responses. Nature 459:663–667CrossRefGoogle Scholar
  10. Connors BW, Long MA (2004) Electrical synapses in the mammalian brain. Annu Rev Neurosci 27:393–418CrossRefGoogle Scholar
  11. Crook SM, Ermentrout GB, Bower JM (1998) Dendritic and synaptic effects in systems of coupled cortical oscillators. J Comput Neurosci 5:315–329CrossRefMATHGoogle Scholar
  12. Erisir A, Lau D, Rudy B, Leonard CS (1999) Function of specific K+ channels in sustained high-frequency firing of fast-spiking neocortical interneurons. J Neurophysiol 82:2476–2489Google Scholar
  13. Ermentrout B, Pascal M, Gutkin B (2001) The effects of spike frequency adaptation and negative feedback on the synchronization of neural oscillators. Neural Comput 13(6):1285–1310CrossRefMATHGoogle Scholar
  14. Ermentrout GB (1996) Type 1 membranes, phase resetting curves, and synchrony. Neural Comput 8: 979–1001CrossRefGoogle Scholar
  15. Ermentrout GB, Kopell N (1984) Frequency plateaus in a chain of weakly coupled oscillators, I. SIAM J Math Anal 15(2):215–237CrossRefMATHMathSciNetGoogle Scholar
  16. Fink C, Booth V, Zochowski M (2011) Cellularly-driven differences in network synchronization capacity are differentially modulated by firing frequency. PLOS Comput Biol 7(5):e1002062Google Scholar
  17. Fukuda T, Kosaka T (2000) Gap junctions linking the dendritic network of gabaergic interneurons in the hippocampus. J Neurosci 20:1519–1528Google Scholar
  18. Fukuda T, Kosaka T (2003) Ultrastructural study of gap junctions between dendrites of parvalbumin-containing gabaergic neurons in various neocortical areas of the adult rat. Neuroscience 120:5–20CrossRefGoogle Scholar
  19. Fukuda T, Kosaka T, Singer W, Galuske RAW (2006) Gap junctions among dendrites of cortical gabaergic neurons establish a dense and widespread intercolumnar network. J Neurosci 26:3434–3443CrossRefGoogle Scholar
  20. Galaretta M, Hestrin S (1999) A network of fast spiking cells in the neocortex connected by electrical synapses. Nature 402:72–75CrossRefGoogle Scholar
  21. Gibson JR, Beierlein M, Connors BW (1999) Two networks of electrically coupled inhibitory neurons in neocortex. Nature 402:75–79CrossRefGoogle Scholar
  22. Gibson JR, Beierlein M, Connors BW (2005) Functional properties of electrical synapses between inhibitory interneurons of neocortical layer 4. J Neurosci 93:467–480Google Scholar
  23. Gonzalez-Burgos G, Lewis DA (2008) Gaba neurons and the mechanisms of network oscillations: Implications for understanding cortical dysfunction in schizophrenia. Schizophrenia Bull 34(5):944–961CrossRefGoogle Scholar
  24. Hestrin S, Galarreta M (2005) Electrical synapses define networks of neocortical gabaergic neurons. Trends Neurosci 28:304–309CrossRefGoogle Scholar
  25. Hoppensteadt FC, Izikevich EM (1997) Weakly connected neural networks. Springer, New YorkCrossRefGoogle Scholar
  26. Hu H, Martina M, Jonas P (2010) Dendritic mechanisms underlying rapid synaptic activation of fast-spiking hippocampal interneurons. Science 327:52–58CrossRefGoogle Scholar
  27. Johnston D, Narayanan R (2008) Active dendrites: colorful wings of the mysterious butterfly. Trends Neurosci 31(6):309–316CrossRefGoogle Scholar
  28. Koch C (1999) Biophysics of computation: information processing in single neurons. Oxford University Press, New YorkGoogle Scholar
  29. Kopell N, Ermentrout GB (2003) Chemical and electrical synapses perform complementary roles in the synchronization of interneuronal networks. Proc Natl Acad Sci 101:15482–15487CrossRefGoogle Scholar
  30. Kuramoto Y (1984) Chemical oscillations, waves, and turbulence. Springer, BerlinCrossRefMATHGoogle Scholar
  31. Lewis TJ, Rinzel J (2003) Dynamics of spiking neurons connected by both inhibitory and electrical coupling. J Comput Neurosci 14:283–309CrossRefGoogle Scholar
  32. Lewis TJ, Rinzel J (2004) Dendritic effects in networks of electrically coupled fast-spiking interneurons. Neurocomputing 58–60:145–150CrossRefMathSciNetGoogle Scholar
  33. Lewis TJ, Skinner FK (2012) Understanding activity in electrically coupled networks using PRCs and the theory of weakly coupled oscillators. In: Schultheiss NW, Prinz A, Butera RJ (eds) Phase response curves in neuroscience: theory, experiment, and analysis. Springer, New York, pp 329–359Google Scholar
  34. Mainen ZF, Sejnowski TJ (1996) Influence of dendritic structure on firing pattern in model neocortical neurons. Nature 382:363–366CrossRefGoogle Scholar
  35. Mancilla JG, Lewis TJ, Pinto DJ, Rinzel J, Connors BW (2007) Synchronization of electrically coupled pairs of inhibitory interneurons in neocortex. J Neurosci 27(8):2058–2073CrossRefGoogle Scholar
  36. Mann EO, Paulsen O (2005) Mechanisms underlying gamma (40 Hz) network oscillations in the hippocampusa mini-review. Prog Biophys Mol Biol 87(1):67–76CrossRefGoogle Scholar
  37. McBain CJ, Fisahn A (2001) Interneurons unbound. Nat Rev 2:11–23CrossRefGoogle Scholar
  38. Netoff T, Schwemmer MA, Lewis TJ (2012) Experimentally estimating phase response curves of neurons: theoretical and practical issues. In: Schultheiss NW, Prinz A, Butera RJ (eds) Phase response curves in neuroscience: theory, experiment, and analysis. Springer, New York, pp 95–129CrossRefGoogle Scholar
  39. Neu JC (1979) Coupled chemical oscillators. SIAM J Appl Math 37(2):307–315CrossRefMATHMathSciNetGoogle Scholar
  40. Ono T, Sasaki K, Nishino H, Fukuda M, Shibata R (1986) Feeding and diurnal related activity of lateral hypothalamic neurons in freely behaving rats. Brain Res 373(1–2):92–102CrossRefGoogle Scholar
  41. Pfeuty B, Mato G, Golomb D, Hansel D (2003) Electrical synapses and synchrony: the role of intrinsic currents. J Neurosci 23(15):6280–6294Google Scholar
  42. Pfeuty B, Mato G, Golomb D, Hansel D (2005) The combined effects of inhibitory and electrical synapses in synchrony. Neural Comput 17:633–670CrossRefMATHMathSciNetGoogle Scholar
  43. Prinz AA, Fromherz P (2002) Effect of neuritic cables on conductance estimates for remote electrical synapses. J Neurophysiol 89:2215–2224CrossRefGoogle Scholar
  44. Rall W (1957) Membrane time constants of motoneurons. Science 126:454CrossRefGoogle Scholar
  45. Rall W (1959) Branching dendritic trees and motoneuron membrane resistivity. Exp Neurol 2:491–527CrossRefGoogle Scholar
  46. Rall W (1960) Membrane potential transients and membrane time constant of motoneurons. Exp Neurol 2:503–532CrossRefGoogle Scholar
  47. Rall W (1977) Time constants and electrotonic length of membrane cylinders and neurons. In: Brookhart JM, Mountcastle VB (eds) Handbook of physiology. The nervous system I. American Physiological Society, BethesdaGoogle Scholar
  48. Rekling JC, Feldman JL (1998) Pre-bötzinger complex and pacemaker neurons: hypothesized site and kernel for respiratory rhythm generation. Annu Rev Physiol 60:385–405CrossRefGoogle Scholar
  49. Remme M, Lengyel M, Gutkin BS (2009) The role of ongoing dendritic oscillations in single-neuron dynamics. PLoS Comput Biol 5(9):e1000493CrossRefMathSciNetGoogle Scholar
  50. Risken H (1989) The Fokker–Planck equation: methods of solution and applications. Springer, New YorkMATHGoogle Scholar
  51. Salinas E, Sejnowski TJ (2001) Correlated neuronal activity and the flow of neural information. Nat Rev Neurosci 2:539–550CrossRefGoogle Scholar
  52. Saraga F, Ng L, Skinner FK (2006) Distal gap junctions and active dendrites can tune network dynamics. J Neurophysiol 95:1669–1682CrossRefGoogle Scholar
  53. Saraga F, Skinner FK (2004) Location, location, location (and density) of gap junctions in multi-compartment models. Neurocomputing 58–60:713–719CrossRefMathSciNetGoogle Scholar
  54. Schwemmer MA, Lewis TJ (2011) Effects of dendritic load on the firing frequency of oscillating neurons. Phys Rev E 83:013906CrossRefMathSciNetGoogle Scholar
  55. Schwemmer MA, Lewis TJ (2012) Bistability in a leaky-integrate-and-fire neuron with a passive dendrite. SIAM J Appl Dyn Syst 11:507–539CrossRefMATHMathSciNetGoogle Scholar
  56. Schwemmer MA, Lewis TJ (2012) The theory of weakly coupled oscillators. In: Schultheiss NW, Prinz A, Butera RJ (eds) Phase response curves in neuroscience: theory, experiment, and analysis. Springer, New York, pp 3–32CrossRefGoogle Scholar
  57. Sohal VS, Zhang F, Yizhar O, Deisseroth K (2009) Parvalbumin neurons and gamma rhythms enhance cortical circuit performance. Nature 459:698–702CrossRefGoogle Scholar
  58. Stratonovich RL (1967) Topics in the theory of random noise. Gordon and Breach, New YorkMATHGoogle Scholar
  59. Teramae J, Nakao H, Ermentrout GB (2009) Stochastic phase reduction for a general class of noisy limit cycle oscillators. Phys Rev Lett 102:194102CrossRefGoogle Scholar
  60. Teramae J, Tanaka D (2004) Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators. Phys Rev Lett 93(20):204103CrossRefGoogle Scholar
  61. Traub RD, Kopell N, Bibbig A, Buhl E, LeBeau FEN, Whittington MA (2001) Gap junctions between interneuron dendrites can enhance synchrony of gamma oscillations in distributed networks. J Neurosci 21(23):9478–9486Google Scholar
  62. Tresch MC, Kiehn O (2002) Synchronization of motor neurons during locomotion in the neonatal rat: predictors and mechanisms. J Neurosci 22(22):9997–10008Google Scholar
  63. van Kampen NG (1981) Stochastic processes in physics and chemistry. Elsevier Science, AmsterdamMATHGoogle Scholar
  64. Wang X-J, Buzsáki G (1996) Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. J Neurosci 16(20):6402–6413Google Scholar
  65. Ward LM (2003) Synchronous neural oscillations and cognitive processes. Trends Cogn Sci 7(12):553–559CrossRefGoogle Scholar
  66. Zahid T, Skinner FK (2009) Predicting synchronous and asynchronous network groupings of hippocampal interneurons coupled with dendritic gap junctions. Brain Res 1262:115–129CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Program in Applied and Computational Mathematics and Princeton Neuroscience InstitutePrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsUniversity of California, DavisDavisUSA
  3. 3.Mathematical Biosciences InstituteThe Ohio State UniversityColumbusUSA

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