Wavelets of Excitability in Sensory Neurons
JEFF HASTY,1 J. J. COLLINS,1 KURT WIESENFELD,3 AND PETER GRIGG2
Center for BioDynamics and Department of Biomedical Engineering, Boston University, Boston 02215; 2Department of
Physiology, University of Massachusetts Medical School, Worcester, Massachusetts 01655; and 3School of Physics, Georgia
Institute of Technology, Atlanta, Georgia 30332
Received 27 March 2001; accepted in final form 9 May 2001
Hasty, Jeff, J. J. Collins, Kurt Wiesenfeld, and Peter Grigg.
Wavelets of excitability in sensory neurons. J Neurophysiol 86:
2097–2101, 2001. We have investigated variations in the excitability
of mammalian cutaneous mechanoreceptor neurons. We focused on
the phase dynamics of an action potential relative to a periodic
stimulus, showing that the excitability of these sensory neurons has
interesting nonstationary oscillations. Using a wavelet analysis, these
oscillations were characterized through the depiction of their period as
a function of time. It was determined that the induced oscillations are
weakly dependent on the stimulus frequency, and that lower temperatures significantly reduce the frequency of the phase response. Our
results reveal novel excitability properties in sensory neurons, and,
more generally, could prove significant in the deduction of mechanistic attributes underlying the nonstationary excitability in neuronal
systems. Since peripheral neurons feed information to the CNS,
variable responses observed in higher regions may be generated in
part at the site of sensory detection.
When certain neurons are subjected to repeated presentations of
an identical stimulus, the action potentials encoding the stimulus
information have variable responses between presentations (Arieli
et al. 1996; Britten et al. 1993; Dean 1981; de Ruyter van
Steveninck et al. 1997; Hunter et al. 1998; Mainen and Sejnowski
1995; Rieke et al. 1997; Rose et al. 1969; Schiller et al. 1976;
Shadlen and Newsome 1998; Snowden et al. 1992). While such
variability is thought to be important in the processing of information by neurons in the CNS (de Ruyter van Steveninck et al.
1997; Mainen and Sejnowski 1995; Rieke et al. 1997; Shadlen
and Newsome 1998), its potential role in peripheral neurons has
not been appreciated (Koltzenburg et al. 1997; Merzenich and
Harrington 1969). One central issue is whether such variability is
utilized in the transfer of information (de Ruyter van Steveninck
et al. 1997; Gerstner et al. 1996; Pei et al. 1996; Rieke et al. 1997)
or whether it is merely a stochastic effect attributable to some
underlying process. In other words, does a neuronal system reliably encode due to or in spite of such variability? A key to the
resolution of this question is the determination of the source of the
variability, as well as the deduction of an underlying mechanism
for its generation. Here we report that when mechanoreceptors,
recorded in isolated skin, are stimulated with mechanical sinusoids, the timing of responses in relation to the stimulus exhibits
nonstationary, wavelike variability.
Address for reprint requests: J. Hasty, Dept. of Biomedical Engineering,
Boston University, 44 Cummington St., Boston, MA 02215 (E-mail:
Rapidly adapting (RA) mechanoreceptor neurons were recorded in
a preparation of skin and nerve that was isolated from the hindlimb of
adult rats, and studied in vitro. Single guard hair afferents were
activated by moving hairs using periodic (sinusoidal) stimuli. We
were interested in the phase relationship between the stimuli and the
responses of individual neurons.
Adult rats were anesthetized with pentobarbital sodium (Nembutal),
administered intraperitoneally. The experimental preparation was an
isolated sample of skin and nerve, taken from the inner thigh, and
studied in vitro. The fur along the thigh was clipped to a length of
approximately 2 mm. Then the skin sample, approximately 14 mm
square, was excised along with its sensory innervation, a branch of the
saphenous nerve. The sample was removed to an apparatus where it
was maintained in a bath of artificial interstitial fluid kept at room
temperature (20°C). The skin was supported from underneath by a
platinum mesh. Thus the under side of the skin was maintained in the
bath while the upper surface was dry. The cutaneous nerve was pulled
into a small oil-filled plastic chamber for recording. The nerve was
dissected into small fibers that were placed on a fine gold wire
electrode for recording. The indifferent electrode was placed in the
bath. Signals were amplified with a PARC 118 amplifier and filtered
with a Riverbend Electronics Learning Filter. Guard hair afferents
were sought by gently stroking the clipped hairs while recording from
fibers. Recordings were often made from filaments containing several
active neurons whose active hairs in the skin were far enough apart to
allow them to be stimulated independently of each other.
When a suitable afferent was identified, the appropriate hair was
actuated with a mechanical stimulator that consisted of a Cambridge
Technology 300B lever system. This is a DC servomotor that rotates
a shaft through controlled angular displacements. The motor actuated
a 60-mm-long cantilever whose tip was brought into contact with the
hair. The displacements that we used were small (⬍0.5 mm) so that
the motion of the tip was essentially linear. The stimulator had a
mechanical bandwidth of 0 to approximately 120 Hz. Stimuli were
displacement-controlled sinusoids. Stimulus waveforms, along with a
synchronization pulse, were generated with a Wavetek waveform
generator. The stimulus amplitude was adjusted so as to elicit one
spike per cycle and was kept constant throughout experimental runs.
The phase of each action potential was measured in relation to the
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J. HASTY, J. J. COLLINS, K. WIESENFELD, AND P. GRIGG
position and force signals was 500 Hz, and the acquisition rate for the
synchronization pulse and neuronal spikes was 500 kHz.
The phase of the ith spike relative to the synchronization pulse is
i ⫽ 共si ⫺ zi兲2f
where si and zi are the spike and syncronization times (si ⬎ zi), and f
is the frequency of indentation. Defined in this way, the phase has
units of radians and is a measure of the threshold or excitability of the
mechanoreceptor system, i.e., it represents the magnitude of indentation required for action potential generation.
The continuous wavelet transform (CWT) of a time series (t) of
length T is given by
It is a measure of the degree to which the function (t) is periodic,
with period proportional to the scale parameter s, and at a particular
time denoted by . The function (t) is typically of Gaussian form
(with compact support), and we utilized the function
FIG. 1. Typical phase data from a mechanoreceptor. A trigger pulse is
generated at the beginning of each actuation cycle, so that data collection
periods contain the times of the trigger pulse and any subsequent action
potentials. The phase, which represents the excitability of the mechanoreceptor
response, is defined as i ⫽ (si ⫺ zi)2f, where si and zi are the spike and
trigger times (si ⬎ zi), and f is the frequency of actuation. A: data for a 15-Hz
stimulus frequency. During the entire run, the phase is bounded by /2 as the
mechanoreceptor is responding to the inward motion of the indenter. The phase
slowly increases, indicating a slow decrease in excitability, which is likely due
to adaptation. Inset: a section of the data is displayed at higher resolution,
elucidating oscillations in the phase data. B: the power spectrum of the phase
data in A. There are 2 distinct maxima at 0.18 and 0.50 Hz, along with a strong
peak at very low frequency.
stimulus waveform (Del Prete and Grigg 1998; Hunter et al. 1998;
Neiman et al. 1999; Read and Siegal 1996).
Both mechanical and neuronal data were acquired using a Cambridge Electronic Design micro 1401 data acquisition system. We
collected four analog signals. Two signals represented actuator position and the force applied by the actuator. In addition, we acquired the
synchronization pulse that denoted the start of each stimulus cycle and
the amplified neural recording. The data acquisition rate for the
The scale parameter s in Eq. 2 is proportional to the width of the
Gaussian and is, by analogy with the fast Fourier transform, a measure
of the period of the signal. Operationally, the meaning of the wavelet
transform can be illustrated as follows. Since the CWT is a function
of both time and scale, consider fixing the width of the Gaussian in
Eq. 2 by letting s ⫽ 1 and superimposing it along with the time series
at time t ⫽ 0. Then the CWT for ⫽ 0 and s ⫽ 1 is obtained by
integrating Eq. 2 over time, and physically, this corresponds to the
degree to which the Gaussian function and the time series overlay; i.e.,
the value of their convolution (the prefactor 1/公s is necessary for
normalization purposes so that the transformed signal will have the
same “energy” for each s). The Gaussian (at scale s ⫽ 1) is then
translated in time to the location t ⫽ , and the integral is recomputed
to obtain the CWT at t ⫽ and s ⫽ 1 in the time-frequency plane. This
procedure is repeated until the Gaussian has been translated to the end
of the time series, resulting in the calculation of the CWT for fixed s,
i.e., ⌽(, s ⫽ 1). Then, s is increased by a small value and the entire
procedure is repeated. Note that this is a continuous transform, and
therefore both and s must be incremented continuously. However,
since the CWT is obtained numerically, both parameters are increased
FIG. 2. Time-period diagrams obtained from a continuous wavelet analysis of the phase data in Fig. 1A. A:
there are oscillations of several different periods, and, at
times, there is significant frequency drift. The frequency
drift is quite noticeable from 40 to 100 s and again from
150 to 200 s. The regions from 100 to 140 s and 200 to
250 s indicate coexisting waves of several different
periods. B: a blowup of the time scale reveals more
precisely dominant periods of approximately 2 and 6 – 8
s, respectively. These correspond to the peaks in the
power spectrum of Fig. 1B.
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WAVELETS OF EXCITABILITY IN SENSORY NEURONS
FIG. 3. Representative time-period diagrams
for actuation frequencies of 5 and 10 Hz. The
data sets were obtained under identical experimental conditions as the 15-Hz data used in Fig.
2, enabling comparison between Fig. 3 and Fig.
2B. In all 3 data sets, we observe periodicities of
approximately 2 and 6 s, with the only major
difference being the intermittency at 5 Hz. Compared with the 10- and 15-Hz data, the 5-Hz
response shows significantly fewer periodic regions at low frequency.
by a sufficiently small step size, which corresponds to sampling the
Data were collected from 14 neurons, and the results presented below are representative of the qualitative features
observed in all of the neurons. Figure 1 shows the phase
behavior of a typical neuron. The response shows a gradual,
time-dependent increase in phase, a signature of the adaptation
process. In addition, the variability of the response increases
with time. We find that there is significant structure to the
variability, which is seen to exhibit waves (Fig. 1, inset). To
quantify the properties of the waves, Fourier analysis was
utilized. The power spectrum of an entire data run of 300-s
exhibits peaks at 0.18 and 0.50 Hz (Fig. 1B). Since the time
series appears to be nonstationary, we investigated the temporal properties associated with the oscillations by partitioning
the data into four sequential segments and analyzing each
independently. We found significant qualitative differences in
the power spectra generated from quarter segments in the data
sets. For example, the power spectrum obtained using the first
quarter of the time series in Fig. 1A does not clearly exhibit the
0.18- and 0.50-Hz maxima seen in Fig. 1B. However, analysis
of the second and third quarter segments reveals significant
power at these frequencies and, in addition, the birth of substantial peaks at low frequencies. Analysis of the fourth quarter
segment exhibits maxima at 0.18 and 0.50 Hz, but relatively
little low-frequency content.
Wavelet analysis is well-suited for describing a nonstationary time series such as the phase data in Fig. 1A. While the
classical Fourier technique yields information only in the frequency domain, the wavelet approach quantifies periodicity in
both the time and frequency domains and is thus capable of
elucidating the nature of the time-dependent oscillatory regions
noted above (see the caption of Fig. 2 for the definition of the
continuous wavelet transform). Figure 2 shows the fluctuations
in frequency and power during the run depicted in Fig. 1. We
observe phase oscillations of several different frequencies and
find that, at times, these waves exhibit a significant frequency
drift. Such a variable response is of interest for two primary
reasons. First, since the current opinion on population coding is
that the function of a population of neurons is to produce
responses in phase with a common stimulus, variability in the
responses of single afferents should adversely affect this funcJ Neurophysiol • VOL
tion. Further work is thus needed to determine what effect the
waves have on the degree of synchronous behavior observed in
groups of cells with a common stimulus. Second, the waves
may provide insight into peripheral mechanisms involved in
It is perhaps natural to conjecture that the adaptation process
is somehow responsible for the oscillatory behavior. For example, in vertebrate auditory hair cells, the firing of an action
potential induces a transport of calcium cations into the neuronal cell (Lumpkin and Hudspeth 1998; Lumpkin et al. 1997).
These cations then act to increase the firing threshold through
their interaction with the ion channel gating process. Thus for
a periodic stimulus, each drive period could lead to a buildup
of intracellular calcium, provided passive mechanisms responsible for export are unable to keep pace with the drive. Oscillations could then occur if an active export process, such as the
transport of calcium out of the cell by protein pumps (Lumpkin
and Hudspeth 1998), was triggered after a number of actuation
cycles. If such an adaptation mechanism is indeed related to the
phase periodicity, then the wavelet frequency response should
depend on the actuation frequency. In Fig. 3, we plot representative time-period diagrams for actuation frequencies of 5
and 10 Hz. We observe variable periodicities of approximately
2 and 6 s for multiple stimulus frequencies. Since the stimulus
represents the only periodic signal available to the neuron, it is
of interest that the frequency of the phase response does not
appear to be influenced by the drive frequency.
FIG. 4. The power spectrum of 2 sets of representative data obtained at 2
temperatures for an actuation frequency of 5 Hz. There is a temperatureinduced shift from 0.81 Hz at 20.4°C to 0.62 Hz at 14.3°C. Additionally, at low
frequency, there appears to be a 2nd shift downward, although the power
spectrum in this regime renders it difficult to characterize this change
86 • OCTOBER 2001 •
J. HASTY, J. J. COLLINS, K. WIESENFELD, AND P. GRIGG
FIG. 5. Time-period diagram obtained from a wavelet
analysis of the data used for Fig. 4. In both sets, there are
significant oscillations at many different frequencies. Both
sets have similar structure, and this has been highlighted
with rectangles drawn around what appear to be complementary regions. By focusing on these regions, note that,
relative to the data at 20.4° (top), the 14.3° (bottom) data
have increased oscillatory periods at nearly all frequencies.
The time delay in the occurrence of the regions in the
bottom diagram is attributable to variations between data
runs and is not a property of the temperature difference.
If the phase oscillations are linked to some metabolic process, then one might anticipate a decrease in the frequency of
the waves as the sensory system is cooled. In Fig. 4, we plot the
power spectrum of two sets of representative phase data obtained at two temperatures for an actuation frequency of 5 Hz.
As the temperature is decreased, we observe a significant
decrease of approximately 20% in the frequency of the phase
response. In Fig. 5, the time-period diagram demonstrates that
the oscillatory response is slowed throughout an entire experimental run.
tics (Lowen et al. 1999), and such models might provide
insight into the underlying mechanism for the excitability
waves. Along these lines, further studies are needed to
determine whether the phenomenon is a necessary component of the encoding apparatus or whether the system simply
tolerates its existence.
We thank N. Kopell, A. Neiman, and J. White.
This work was supported by The Fetzer Institute (J. Hasty) and National
Institute of Neurological Disorders and Stroke Grant NS-10783.
The characterization of the waves of excitability reported
here raises several significant issues. Of central importance
is our finding that structured variations occur at the periphery of the sensory system. While variations in neuronal
responses within the CNS have been the subject of much
discussion (Arieli et al. 1996; Britten et al. 1993; Dean
1981; de Ruyter van Steveninck et al. 1997; Gerstner et al.
1996; Mainen and Sejnowski 1995; Pei et al. 1996; Rieke et
al. 1997; Rose et al. 1969; Schiller et al. 1976; Shadlen and
Newsome 1998; Snowden et al. 1992), it has been assumed
that periphery responses that feed the central system are
regular. Additionally, since the experiment entailed the
stimulation of individual hairs and therefore single neurons,
the oscillatory response cannot be attributed to a network of
neurons. This suggests that the transduction mechanism in
these sensory neurons is more complex than is typically
appreciated. The fact that the waves appear to be independent of the stimulus frequency is remarkable, given that the
only obvious time scales present are those set by the periodic drive and adaptation process. Perhaps the cells possess
an underlying internal clock that operates independently of
the drive. From an information-transfer point of view, the
observed temperature effects could imply that the excitability waves are conveying useful information regarding the
environment. It is also conceivable that the oscillations
could lead to fractal statistics similar to those observed in
the neuronal spike trains of other preparations (Ivanov et al.
1999; Teich 1989; Teich et al. 1997). Importantly, models
incorporating correlations in the underlying ion channel
kinetics have been shown to generate nonstationary statisJ Neurophysiol • VOL
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