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.

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.
E.
Batschelet
,
Circular Statistics in Biology
(
Academic
,
New York
,
1981
).
15.
K.
Mardia
,
Statistics of Directional Data
(
Academic
,
London
,
1972
).
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.
P.
Dayan
and
L. F.
Abbot
,
Theoretical Neuroscience
(
MIT
,
Cambridge, MA
,
2001
).
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.
P. P.
Mitra
and
H.
Bokil
,
Observed Brain Dynamics
(
Oxford U. P.
,
New York
,
2008
).
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
.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.