Carmen C Canavier

CC Canavier

MJ Jutras

PJ Uhlaas

Robert J Butera

S Liebe

Sharon E Norman

SK Maran

T Sigurdsson

XJ Wang

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Fixed point topology and robustness to perturbations between pairs of coupled neurons

Norman Sharon E

Canavier Carmen C

Butera Robert J

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BMC Neuroscience

POSTER PRESENTATION Open Access
Fixed point topology and robustness to
perturbations between pairs of coupled neurons
Sharon E Norman1*, Carmen C Canavier2,3, Robert J Butera1,4
From Twenty First Annual Computational Neuroscience Meeting: CNS*2012
Decatur, GA, USA. 21-26 July 2012
Synchronization of neural firing plays an important role
in nervous system function. Synchronized and phase
locked relationships have been implicated in the crea-
tion and retrieval of memories [1,2], while the disrup-
tion of normal synchrony and phase locking appears to
be a factor in disorders like schizophrenia [3,4]. Here we
explore synchronization mechanisms in small two-cell
networks and the robustness of these networks to
perturbations.
Phase response curves (PRCs) describe how neurons
respond to perturbations applied at specific times during
the interspike interval. The PRC can be plotted in terms
of the time to next spike as a function of when the
spike arrived; this curve describes the dynamics of a sin-
gle neuron, and the intersection points of two of these
interaction curves represent the fixed points of the
coupled system [5,6].
Achuthan and colleagues have recently shown (manu-
script submitted) that interaction curves with 0, 1, or 2
fixed points can exist in hybrid circuits of one biological
and one model neuron. In this work, we use two recipro-
cally coupled Wang Buzsaki [7] model neurons to investi-
gate how different numbers of coupled system fixed points
affect the network robustness against perturbation. In
addition, we investigate the effect of interaction curve
shape on network robustness. The perturbation applied to
test robustness is a delay or advance of a single spike of
one model neuron.
Like Achuthan et al, we show that systems with 1 or 2
fixed points exhibit phase locking, while a system with 0
fixed points, but interaction curves in close proximity,
shows temporary locking with some phase walkthrough.
We are currently quantifying the effects of perturbation
strength and timing on transient disturbances to the phase
locking of the coupled system as well as convergence time
back to the pre-perturbation phase relationship. In addi-
tion, we are exploring how the proximity of interaction
curves affects robustness of the coupled system to pertur-
bation. Preliminary results show that interaction curves
that are closer together take a longer amount of time and
more cycles to recover from perturbation than interaction
curves that are further apart. Experimental validation of
these modeling results will be shown.
Author details
1School of Electrical and Computer Engineering, Georgia Institute of
Technology, Atlanta, GA 30332, USA. 2Department of Cell Biology and
Anatomy, Louisiana State University Health Science Center, New Orleans, LA
70112, USA. 3Neuroscience Center of Excellence, Louisiana State University
Health Science Center, New Orleans, LA 70112, USA. 4Wallace H. Coulter
Department of Biomedical Engineering, Georgia Institute of Technology,
Atlanta, GA 30332, USA.
Published: 16 July 2012
References
1. Jutras MJ, Buffalo EA: Synchronous neural activity and memory formation.
Curr Opin Neurobiol 2010, 20:150-155.
2. Liebe S, Hoerzer GM, Logothetis NK, Rainer G: Theta coupling between V4
and prefrontal cortex predicts visual short-term memory performance.
Nat Neurosci 2012, published online, doi:10.1038/nn.3038.
3. Sigurdsson T, Stark KL, Karayiorgou M, Gogos JA, Gordon JA: Impaired
hippocampal-prefrontal synchrony in a genetic mouse model of
schizophrenia. Nature 2010, 464:763-767.
4. Uhlaas PJ, Singer W: Neural synchrony in brain disorders: relevance for
cognitive dysfunctions and pathophysiology. Neuron 2006, 52:155-168.
5. Canavier CC, Achuthan S: Pulse coupled oscillators and the phase
resetting curve. Math Biosci 2010, 226(2):77-96.
6. Maran SK, Canavier CC: Using phase resetting to predict 1:1 and 2:2
locking in two neuron networks in which firing order is not always
preserved. J Comput Neurosci 2008, 24:37-55.
7. Wang XJ, Buzsaki G: Gamma oscillation by synaptic inhibition in a
hippocampal interneuronal network model. J Neurosci 1996,
16(20):6402-6413.
* Correspondence: sbyers3@gatech.edu
1School of Electrical and Computer Engineering, Georgia Institute of
Technology, Atlanta, GA 30332, USA
Full list of author information is available at the end of the article
Norman et al. BMC Neuroscience 2012, 13(Suppl 1):P174
http://www.biomedcentral.com/1471-2202/13/S1/P174
© 2012 Norman et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
doi:10.1186/1471-2202-13-S1-P174
Cite this article as: Norman et al.: Fixed point topology and robustness
to perturbations between pairs of coupled neurons. BMC Neuroscience
2012 13(Suppl 1):P174.
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Fixed point topology and robustness to perturbations between pairs of coupled neurons

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