In neuroscience, specifically electrophysiology, it is common to replace a measured sequence of action potentials or spike trains with delta functions prior to analysis. We apply a method called circular statistics to a time series of delta functions and show that the method is equivalent to the power spectrum. This technique allows us to easily visualize the idea of the power spectrum of spike trains and easily reveals oscillatory and stochastic behavior. We provide several illustrations of the method and an example suitable for students, and suggest that the method might be useful for courses in introductory biophysics and neuroscience.
REFERENCES
1.
L.
Glass
and M. C.
Mackey
, From Clocks to Chaos: The Rhythms of Life
(Princeton U. P.
, Princeton, NJ
, 1988
).2.
L.
Glass
, “Synchronization and rhythmic processes in physiology
,” Nature (London)
410
, 277
–284
(2001
).3.
L.
Schimansky-Geier
and C.
Zulicke
, “Harmonic noise: Effect on bistable systems
,” Z. Phys. B: Condens. Matter
79
, 451
–460
(1990
).4.
H. A.
Braun
, H.
Wissing
, K.
Schäfer
, and M. C.
Hirsch
, “Oscillation and noise determine signal transduction in shark multimodal sensory cells
,” Nature (London)
367
, 270
–273
(1994
).5.
K.
Schäfer
, H. A.
Braun
, R. C.
Peters
, and F.
Bretschneider
, “Periodic firing pattern in afferent discharges from electroreceptor organs of catfish
,” Pfluegers Arch. Gesamte Physiol. Menschen Tiere
429
, 378
–385
(1995
).6.
A.
Neiman
, X.
Pei
, D. F.
Russell
, W.
Wojtenek
, L. A.
Wilkens
, F.
Moss
, H. A.
Braun
, M.
Huber
, and K.
Voigt
, “Synchronization of the electrosensitive noisy cells in the paddlefish
,” Phys. Rev. Lett.
82
, 660
–663
(1999
).7.
A.
Neiman
and D. F.
Russell
, “Stochastic biperiodic oscillations in the electroreceptors of paddlefish
,” Phys. Rev. Lett.
86
, 3443
–3446
(2001
).8.
A.
Neiman
and D. F.
Russell
, “Synchronization of noise-induced bursts in noncoupled sensory neurons
,” Phys. Rev. Lett.
88
, 138103
–1
(2002
).9.
A. B.
Neiman
and D. F.
Russell
, “Two distinct types of noisy oscillators in electroreceptors of paddlefish
,” J. Neurophysiol.
92
, 492
–509
(2004
).10.
M.
Heinz
, K.
Schäfer
, and H. A.
Braun
, “Analysis of facial cold receptor activity in the rat
,” Brain Res.
521
, 289
–295
(1990
).11.
P.
Martin
, D.
Bozovic
, Y.
Choe
, and A. J.
Hudspeth
, “Spontaneous oscillation by hair bundles of the bullfrog’s sacculus
,” J. Neurosci.
23
, 4533
–4548
(2003
).12.
S.
Camalet
, T.
Duke
, F.
Jülicher
, and J.
Prost
, “Auditory sensitivity provided by self-tuned critical oscillations of hair cells
,” Proc. Natl. Acad. Sci. U.S.A.
97
, 3183
–3188
(2000
).13.
S. M.
Bezrukov
and I.
Vodyanoy
, “Noise-induced enhancement of signal transduction across voltage-dependent ion channels
,” Nature (London)
378
, 362
–364
(1995
).14.
15.
16.
N. A.
Hill
and D.-P.
Hader
, “A biased random walk model for the trajectories of swimming micro-organisms
,” J. Theor. Biol.
186
, 503
–526
(1997
).17.
J.
Parra
, S. N.
Kalitzin
, J.
Iriarte
, W.
Blanes
, D. N.
Velis
, and F. H.
Lopes da Silva
, “Gamma-band phase clustering and photosensitivity: Is there an underlying mechanism common to photosensitive epilepsy and visual perception?
,” Brain
126
, 1164
–1172
(2003
).18.
P. A.
Tass
, “Transmission of stimulus-locked responses in two oscillators with bistable coupling
,” Biol. Cybern.
91
, 203
–211
(2004
).19.
P. A.
Tass
, “Transmission of stimulus-locked responses in two coupled phase oscillators
,” Phys. Rev. E
69
, 051909
(2004
).20.
21.
J. S.
Bentad
and A. G.
Piersol
, Random Data: Analysis and Measurement Procedures
(Wiley
, New York
, 2000
).22.
D. B.
Percival
and A. T.
Walden
, Spectral Analysis for Physical Applications
(Cambridge U. P.
, New York
, 1993
).23.
24.
G. B.
Arfken
and H. J.
Weber
, Mathematical Methods for Physicists
(Academic
, San Diego
, 1995
).25.
X.
Pei
, L. A.
Wilkens
, and F.
Moss
, “Light enhances hydrodynamic signaling in the multimodal caudal photoreceptor interneurons of the crayfish
,” J. Neurophysiol.
76
, 3002
–3011
(1996
).26.
S.
Bahar
, “Effect of light on stochastic phase synchronization in the crayfish caudal photoreceptor
,” Biol. Cybern.
89
, 200
–213
(2003
).27.
Z.
Brzezniak
and T.
Zastawniak
, Basic Stochastic Processes
(Springer
, New York
, 2000
).28.
P. G.
Hoel
, S. C.
Port
, and C. J.
Stone
, Introduction to Stochastic Processes
(Houghton Mifflin
, Boston
, 1972
).29.
T. I.
Netoff
, R.
Clewley
, S.
Arno
, T.
Keck
, and J. A.
White
, “Epilepsy in small-world networks
,” J. Neurosci.
24
, 8075
–8083
(2004
).30.
R. L.
Stratonovich
, Topics in the Theory of Random Noise
(Gordon and Breach
, New York
, 1967
), Vol. 2
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2009
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