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

  • Michael A. Schwemmer
  • Timothy J. Lewis


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.


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

Mathematics Subject Classification (2000)

92B25 37N25 



This work was supported by the National Science Foundation under grant DMS-0921039. MS is supported by NIH grant T32-MH065214-1 through the Princeton Neuroscience Institute. The authors would also like to thank Jiawei Zhang for comments on the manuscript.


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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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