Download Journal of Neuroscience Methods Estimation of neuronal firing rates

Journal of Neuroscience Methods 174 (2008) 281–291
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Journal of Neuroscience Methods
journal homepage: www.elsevier.com/locate/jneumeth
Estimation of neuronal firing rates with the three-state biological
point process model
Emanuel E. Zelniker a,∗ , Andrew P. Bradley b , Joanna E. Castner c ,
Helen J. Chenery c , David A. Copland c , Peter A. Silburn d
a
Department of Computer Science, Queen Mary, University of London, London E1 4NS, United Kingdom
School of Information Technology and Electrical Engineering, The University of Queensland, Queensland 4072, Australia
Division of Speech Pathology, The University of Queensland, Queensland 4072, Australia
d
School of Medicinie, The University of Queensland, Queensland 4072, Australia
b
c
a r t i c l e
i n f o
Article history:
Received 6 February 2007
Received in revised form 6 May 2008
Accepted 9 May 2008
Keywords:
Poisson process
Deep-brain stimulation
Micro-electrode recordings
Thresholding functional
Neuronal firing rates
Parameter estimation
a b s t r a c t
In the subcortex of the human brain, neuronal firing events are stochastic and the inter-arrival times
of action potentials (APs) are highly irregular. It has been shown that stimulation of the subthalamic
nucleus (STN), a small subcortical structure located within the basal ganglia, can help ameliorate the
motor symptoms associated with Parkinson’s disease (PD). However, success of image guided stereotactic
surgery is reliant upon the refinement of the anatomic target (in this case the STN) based on microelectrode recordings (MERs) of background activity and firing rate. In practice MERs must be analysed
on-line and in real-time. Currently, the most common method of performing on-line MER analysis is
a manual thresholding procedure. However, this is subjective in nature and often complicated by the
presence of variable amounts of background noise. Therefore, in this paper, we present an automated
adaptive thresholding technique, based on a modified ‘top-hat’ operator, which detects APs exceeding
the local background activity. We then go on to model these inter-arrival times using a coupled Poisson
process that provides improved estimates of both inter-burst and intra-burst neuronal firing activity in
the STN.
Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved.
1. Introduction
In the subcortex of the human brain, neuronal firing events
are stochastic in nature and so the inter-arrival times of action
potentials (APs) are highly irregular. While this phenomenon can
sometimes be adequately described using Poisson models Heeger
(2000), it is often confounded by the combination of the highly
irregular neuronal firing events and the fact that in vivo measurements are often contaminated by high levels of background noise.
This poses a great challenge to automated or semi-automated analysis of neural activity measured in vivo.
It has been demonstrated that stimulation of the subthalamic
nucleus (STN), a small subcortical structure located within the basal
ganglia, can help to improve the motor symptoms of Parkinson’s
Disease (Hutchison et al., 1998; Sterio et al., 2002; Chen et al., 2006).
However, in order to stimulate the STN one must first locate the target and then accurately position a stimulus electrode in this area.
∗ Corresponding author. Tel.: +44 207 882 5230.
E-mail addresses: zelniker@dcs.qmul.ac.uk (E.E. Zelniker),
bradley@itee.uq.edu.au (A.P. Bradley).
Currently, this is done by utilizing a stereotactic frame that guides
the micro-electrode to a target in the frontal subcortex (usually the
STN or globus pallidus interna, GPi). The precise location of the target is initially found using a combination of computed tomography
(CT) and magnetic resonance imaging (MRI). Data is then recorded
from the micro-electrode as it progresses towards the target. Features extracted from the micro-electrode recordings (MERs), such
as background activity and neuronal firing rate, are then used for
precise identification of the borders of the STN (Israel and Burchiel,
2004).
Micro-electrode recordings are commonly used in functionally
awake neurosurgical procedures, such as Deep-Brain Stimulation
(DBS), as they provide a method of verifying the location of target structures via their observed neurophysiological properties
(Hutchison et al., 1998; Sterio et al., 2002; Quiroga et al., 2004).
The high-impedance micrometer tip of the micro-electrode probe
allows for functional investigation of small populations of neurons
within the STN. Alternatively, Local Field Potentials (LFPs) have also
been used to confirm target location either during or post surgery
(Engel et al., 2005; Chen et al., 2006). Local Field Potentials are measured between the individual pairs of electrodes that are positioned
towards the terminus of the stimulus lead, adjacent electrodes
0165-0270/$ – see front matter. Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.jneumeth.2008.05.026
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E.E. Zelniker et al. / Journal of Neuroscience Methods 174 (2008) 281–291
being typically 2 mm apart (Chen et al., 2006). The primary advantage of MERs over LFPs are that they measure the localised activity
of small neuronal populations rather than the synchronous activity of larger assemblies (Engel et al., 2005). However, the limitation
of both MERs and LFPs are that the waveform analysis required to
confirm target location is largely based on a manual threshold of
the recorded signal (via pre-set trigger levels Cambridge Electronic
Designs, 2005) and an ad hoc analysis of resultant inter-arrival times
(Israel and Burchiel, 2004). For example, Hutchison et al. (1998) calculate a ‘burst index’ by dividing the time of the peak of the interval
time histogram by the mean firing rate of the spike train.
Whilst more complex methods of spike sorting and analysis have
been proposed, such as template matching and windowing, e.g., see
(Cambridge Electronic Designs, 2005; Castner et al., 2005), they
often require long sequences of data from which to extract reliable
spike templates. Therefore, they are only suitable for post hoc, offline, analysis as the selection of templates is currently an ad hoc
and subjective process. Undoubtedly, template matching can give
improved results, however, outcomes are highly dependent upon
the skills of the person performing the analysis (Israel and Burchiel,
2004). If the shape of the action potential (AP) was reliably known,
it would contain valuable information. However, the APs recorded
in vivo are often contaminated by large amounts of noise and so
shape is a much less reliable feature to analyse.
In this paper, we propose a general purpose analysis methodology that is robust both to the number of neurons and level
of background noise. Hence, the examples we will show will be
typical, not best-case recordings. We propose an automated ‘tophat’ thresholding technique for on-line, automated, spike detection
suitable for the analysis of short-time MER data. This method can be
seen as a possible replacement for manual thresholding currently
used for on-line spike analysis and complementary to the more complex and detailed off-line analysis methods (Wilson and Emerson,
2002). In addition, as the MERs recorded in this study relate to
multi-unit neuronal activity, we go on to model the resultant AP
inter-arrival times using the coupled Poisson process proposed by
Zhou et al. (1998). We then improve the robustness of this model
by describing two heuristics that can be used to estimate model
parameters in the presence of high levels of background noise. In
this way, we provide, for the first time, a principled methodology
for real-time analysis of both inter-burst and intra-burst neuronal
activity.
Zhou et al. (1998) model the egg-laying behaviour of the nematode C aenorhabditis elegans (or C. elegans) using a ‘coupled’ Poisson
process. This fundamental property of egg-laying events is characteristic of many other instances of clustered stochastic activity,
including neural activity (Hutchison et al., 1998; Heeger, 2000).
They observed that egg-laying events usually occur in clusters and
use one Poisson process to model egg-laying events over short periods of time and a second Poisson process to model egg-laying events
over long time periods that accounts for outlier egg-laying events.
Hutchison et al. (1998) specifically mention that they have observed
STN neurons that fire in irregular ‘bursts’ of activity and Zhou et al.
(1998) themselves also state that their model may be applicable
to cellular level phenomena. Therefore, in this paper, we investigate the application of this model to neuronal inter-arrival times
measured from in vivo MERs.
The paper is organised as follows. In Section 1.1, we review
the theory of Poisson neurons and the coupled Poisson process
described by Zhou et al. (1998). In Section 2, we briefly describe
how the MERs are acquired from the STN, the pre-processing and
adaptive thresholding of these signals and how the coupled Poisson process can be used to model neuronal firing rates. In Section
3, we provide results to demonstrate the efficacy of the proposed
adaptive thresholding technique and to show that the coupled Pois-
Fig. 1. Spike train signal and spike inter-arrival times generated from a Poisson
process.
son process is an appropriate model of neuronal inter-arrival times.
Finally, in Section 4, we offer some concluding remarks.
1.1. Background theory
1.1.1. Poisson neurons
A neuron is polarised in its steady state and then depolarizes
when it fires. Neuronal firings appear as spike trains. Action potentials are generated by a stochastic threshold process. The neuron
fires (i.e., depolarizes) if the action potential (AP), a voltage signal,
say u(t), reaches a certain threshold level. When this happens, a
firing time t̂ is recorded (Jolivet et al., 2004).
Given a signal s(t) with added noise n(t), i.e.,
y(t) = s(t) + n(t),
(1)
a spike train can be obtained by thresholding (1) to produce
z(t) = F[s(t) + n(t)],
(2)
where F[·] is some thresholding functional and z(t) is a list of firing times generated by the input. We will discuss our thresholding
procedure later on in Section 2.
1.1.2. Homogeneous Poisson process
The timing of successive action potentials in the cortex is highly
irregular. While there is value in single unit recordings in neuroscience research with clinical applications (Magnin et al., 2000),
it is also true that the irregular inter-spike intervals can be modelled as a random process (Heeger, 2000). If we use this modelling
approach, then looking at individual neuronal firing times is not a
meaningful exercise. However, it is possible to average the collective responses of many individual neurons, that is, the inter-spike
intervals in a short-time period, to get information on the average
spike rate over this interval.
If we assume that the generation of each spike depends only on
an underlying continuous/analogue driving signal, r(t), referred to
here as the instantaneous firing rate. It follows that the generation
of each spike is independent of all the other spikes. This is called
the independent spike hypothesis, which if assumed to be true, yields
a spike train described by a Poisson process (see Fig. 1). There is
evidence that these assumptions are largely correct (Heeger, 2000),
however, it is difficult at this moment in time to conclusively state
whether these assumptions are true or not.
If we assume that the average firing rate is constant, i.e.,
r(t) = , every sequence of n spikes over a fixed time interval has
an equal probability. Therefore, the spike train with a probability p(t̂1 , . . . , t̂n ) can be expressed by a probability function that
E.E. Zelniker et al. / Journal of Neuroscience Methods 174 (2008) 281–291
283
considers only the number of spikes, pT (n) during a time interval, T,
which is divided into M bins of size t = (T/M) and t is assumed
to be sufficiently small so as to avoid two spikes occurring within
any one bin.
The function pT (n) is the product of three factors:
• The probability of generating n spikes within M bins.
• The probability of not generating spikes in the remaining bins.
• A combinatorial factor equal to the number of ways of putting n
spikes into M bins.
and this results in the well known Poisson distribution LeonGarcia (1994)
pT (n) =
n
(T ) −T
.
e
n!
(3)
1.1.3. Spike inter-arrival distribution
Suppose that a spike occurs at time t̂i . The probability of a homogeneous Poisson process generating the next spike in the interval
ti + ≤ ti+1 < ti + + t for a small t is the probability that no
spike is fired for a time , times the probability t of generating
a spike within the following small interval t (Leon-Garcia, 1994),
i.e.,
0
p (0) =
() −
e
= e− ,
0!
(4)
p( ≤ ti+1 − ti < + t) = te− .
(5)
In the paper by Zhou et al. (1998), the authors propose a threestate biological point process to model the egg-laying behaviour of
a nematode. They do this as traditional point process models are
inappropriate to describe the observed temporal pattern of egglaying behavior. Egg-laying events were observed to be clustered
and neither the onset of egg-laying clusters nor individual egglaying events within those clusters appeared to be periodic. Thus,
a Poisson process was thought to be an accurate model for this
behaviour. However, as the egg-laying behaviour is clustered, the
simple inter-arrival distribution
x ≥ 0,
(6)
cannot be used to model both the short and long term intervals in
the data and so, a coupled Poisson process is proposed.
The linear log tail probabilities, log p(X ≥ x) = −x are observed
to have different rate constants, i.e., different ’s for short and long
intervals. As such, it is proposed to model the probabilistic structure
of the short and long intervals as two Poisson processes with different rate constants. This is why Zhou et al. (1998) refer to their model
as a ‘coupled’ Poisson process. The following model is proposed
fX (x) = a1 e−1 x + (1 − a)2 e−2 x ,
parameter 1 . Alternatively, none of the neurons may fire, thus they
enter the inactive, E1−p , state with probability 1 − p after some
delay represented by another exponential random variable with
rate parameter 2 . So, there are essentially only two states in the
model with time constants that relate to ‘bursts’ of neuronal firing
activity and ‘inter-bursts’ of inactivity (that relate to the neurons
not firing as they are either recovering or being inhibited from firing). It should also be noted that the A state acts as a ‘switch’ rather
than a state, there being no time delay or output associated with
this state.
The probability distribution function (pdf) of X is a weighted
sum of two exponential pdf’s Zhou et al. (1998) with parameters 1
and 2
fX (x) = k1 1 e−1 x + k2 (p2 )e−(p2 )x ,
x ≥ 0,
(7)
where 0 < a < 1 is the weight factor, and 1 and 2 are the time
constants for the short and long intervals. They let p be the probability of an event occurring after some delay which is represented
by a random variable.
For convenience, we reproduce the three-state diagram of Zhou
et al. (1998) in Fig. 2 (the relabelling of the states is justified in
Section 2.3). The probabilities of the state transitions are indicated
on the arrows. The model has three states, namely, the inactive
state (E1−p ), the active state (A) and the firing state (Ep ). Nominally, a group of neurons start at state A. A single neuron can then
activate and fire by entering the Ep state with probability p after
some delay represented by exponential random variable with rate
x ≥ 0,
(8)
where
k1 =
1.2. A three-state point process model
fX (x) = e−x ,
Fig. 2. Three-state model.
p(1 − 2 )
,
1 − p2
k2 =
1 (1 − p)
,
1 − p2
(9)
and the tail probability of X is
p(X ≥ x) = k1 e−1 x + k2 e−(p2 )x .
(10)
Now, since intervals are clustered over short periods of time and are
sparse at long intervals, the random variable Y = log(X) will have a
more uniform spread. The pdf of Y is related to that of X as follows
(Leon-Garcia, 1994)
dx .
dy
fY (y) = fX (x) (11)
Substituting x = ey and dx/dy = ey , we obtain the pdf of the log
intervals
y
y
fY (y) = [k1 1 e−1 e + k2 (p2 )e−(p2 )e ]ey .
(12)
From (12), it is not difficult to see that the pdf of Y is bimodal if 1
and p2 are sufficiently different, and consequently show that the
peaks of Y occur at y = − log 1 and y = − log(p2 ) with respective
peak magnitudes of k1 /e and k2 /e.
In order to estimate the parameters p, 1 , 2 from a noisy observation of intervals, a histogram of the intervals at uniformly spaced
bins, yi needs to be constructed. The histogram values are then normalised by the sample size so the scaled histogram serves as an
2 from the peak locaestimate of fY (y). Next, we calculate 1 and p
tions of the histogram and k̂1 /e and k̂2 /e from the corresponding
peak heights. Then, using (9), we have that
1 = k̂1 [ˆ 1 − p
2 ] + p
2 ,
p
p̂ =
1
p
,
ˆ1
ˆ 2 = p2 .
p̂
(13)
2. Materials and methods
Thus far, we have reviewed the theory of Poisson neurons, spike
trains, the notion of a thresholding functional and the coupled Poisson process proposed by Zhou et al. (1998). In this section, we will
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Fig. 3. An MER from the STN.
show the type of MERs that we have analysed, outline the thresholding of these signals and investigate whether the coupled Poisson
process is an appropriate model to describe neuronal inter-arrival
times. We note here that our analysis requires recordings from
many neurons. We do not require neurons to be separated on the
shape of their action potential (a step that is sensitive to noise).
Rather, we characterise multiple neurons by their inter- and intraburst activity. Also note that our method is statistical in nature and
is not dependent on the detection of all spikes, i.e., as we only estimate inter- and intra-burst activity, therefore, the method is robust
to some mis-detection of APs as a few missed APs will not adversely
affect the estimated rate constants.
Zhou et al. (1998) themselves suggest that their model may be
adapted to many other situations in which discrete events occur
in clusters. This is precisely what we have observed with our
MERs, but with our application, the intervals are not so sparse. This
‘bursty’ behaviour of MERs is also reported by others, for example,
(Hutchison et al., 1998). As we shall demonstrate in Section 3, we
have found that the coupled Poisson process works well in describing the probabilistic behaviour of neuronal firing rates from MERs
in the STN of patients who have Parkinson’s disease.
2.1. Acquisition of MERs from the STN
The data used in this study was recorded from the STN during functionally awake surgery known as deep-brain stimulation
(Sterio et al., 2002). In this study we acquired 247 micro-electrode
recordings (LeadPoint-Medtronic, Minneapolis), each approximately 1 s in length, from inside both the right and left STNs of
ten right-handed subjects (four female). It should be noted that
we specifically analyse relatively short lengths of MER data to
emphasize the suitability of our approach to real-time, on-line,
spike train analysis. The MERs were recorded as the subjects were
performing a specific motor task (pressing a button in response
to a linguistic task, as per Castner et al., 2005). Then, the MERs
were amplified, filtered to a bandwidth of 500–5000 Hz and then
sampled at 24 kHz. The purpose of the high-pass filter (corner frequency 500 Hz) is to minimise the effect of muscle artifacts, 50 Hz
mains interference and background electroencephalographic activity (Israel and Burchiel, 2004). Despite this, the data recorded is
frequently extremely noisy, the noise appearing as a spectrally
white baseline of activity (although it is difficult in this setting to
predict precisely the characteristics of the noise). Fig. 3 depicts one
such MER. From this figure, we can qualitatively observe how the
neuronal action potentials (firings) are instances of discrete events
which occur in clusters. We have recorded MER’s at every stage of
the linguistic task, i.e., before pressing the button (epoch 1), during
the pressing of the button (epoch 2) and after pressing of the button (epoch 3). However, in this paper we only present results for the
last epoch purely to evaluate the model’s ‘goodness-of-fit’ to MER
data. Further research, using this technique will investigate if there
are any significant trends in the STN activity related to the specific
language and motor tasks. Therefore, this paper concentrates on
how to best filter, threshold and model the MER data.
We believe that the background noise present in the recorded
MERs is primarily the result of many neurons firing in the vicinity
of the micro-electrode tip measuring activity in the STN. Therefore,
this noise can be considered as neural noise rather than instrument noise (Israel and Burchiel, 2004). There has been some success
in measuring firing rates of single unit recordings within the STN
(Hutchison et al., 1998). However, it is not normally known what
noise free multi-unit recordings look like. This is due to the way
that multi-unit recordings are measured and their additive nature
(discussed in more detail in the next section). In addition, the highpass noise reduction filter effectively acts as a differentiator, further
complicating the process of identifying different types of neurons
based on the morphology of their action potentials. Therefore, it is
common practice during the acquisition of MER during DBS surgery
to exclusively use gross features such as level of background activity
and neuronal firing rates to distinguish the different types of neurons present in different subcortical structures (Sterio et al., 2002;
Israel and Burchiel, 2004).
The band-pass filter in question is in the analogue bio-amplifier.
The filter settings are those recommended by Medtronic to reduce
the effect of EMG and hence we do not have any data without that
high-pass filter. The Leadpoint user manual says that “it is good
recording practice to set the filtering bandwidth (default settings
were used) to 500 Hz to 5 kHz”. In Israel and Burchiel (2004), it is
also stated that filters for MERs are usually set to attenuate all signals below 300 Hz and all signals above 2 kHz. Although these filter
settings will remove unwanted potentials and noise, there may also
be some distortion of the morphology of the spikes. It is important
to recognise these effects with respect to the electrical signal of
interest. It is stated that the ideal recording situation, therefore,
may be one in which concurrent recordings can be made from one
or two neurons of sufficient signal-to-noise-ratio and stability to
allow for electrophysiological properties of individual neurons to
be studied. Examples are given of separating individual neurons
based on “minimum versus maximum velocity and peak amplitude versus repolarisation velocity”. The velocity refers to the speed
with which the signal changes amplitude. However, any measure of
“velocity”, i.e., the derivative of the signal, will be sensitive to noise.
2.2. Thresholding functionals
Robust detection of spikes in the noisy MERs is a significant
problem on its own (Wilson and Emerson, 2002; Mtetwa and Smith,
2006). This is quite a separate problem to the one described thus
far. This process is complicated by the presence of the background
neuronal noise and so it is common practice to manually adjust an
amplitude threshold to distinguish maxima relating to neurons in
close proximity to the micro-electrode from the background activity in the region of the micro-electrode (Hutchison et al., 1997). In
this way, a list of inter-arrival times can be obtained that describe
the activity of the neurons of interest. This is the thresholding functional, F[·], briefly mentioned in Section 1.1.1.
Manual thresholding (Israel and Burchiel, 2004), which is often
considered to be the gold standard in a clinical setting such as
this, is unreliable because in many instances it is not clear how to
E.E. Zelniker et al. / Journal of Neuroscience Methods 174 (2008) 281–291
285
the positive-to-negative zero-crossings of the first derivative of the
absolute value of the MER. This is shown in Fig. 5(a).
Step two is the top-hat filter. For one-dimensional signals, we
use a structuring element with side brims and a middle section as
shown in Fig. 4. At every position, the structuring element windows
a section of the signal. The brims are positioned so that they sit on
the highest point (maxima) within the window. At the same time,
a bottom-hat is positioned with its brims lying on the lowest point
(minima) within the window (refer to Fig. 4). The brim positions of
the top- and bottom-hat are used to calculate adaptive thresholds,
T + and T − , respectively, i.e., if yw [n] is a windowed MER, then
Fig. 4. The basic principle of the top-hat filter.
select the ‘best’ threshold. Therefore, manual thresholds are often
selected in an arbitrary manner based on prior experience and the
observed firing rates. Furthermore, a manual threshold in a clinical
setting will typically only be positive, and therefore, would only be
applied to the positive valued components of an MER, i.e., negative
spikes whose absolute value also exceeds the manual threshold are
ignored. We will discuss this point further at the end of this section and show that in certain circumstances it can be argued that
both positive and negative thresholds should be taken. Another
disadvantage of manual thresholding is that it is subjective, and
therefore, potentially not repeatable. A manual threshold, or any
non-adaptive threshold setting technique, will require adjusting as
the micro-electrode recorder approaches the STN (i.e., the target)
as the background activity changes.
One potential way to automatically threshold the MERs is to
window the signal and at every position and to calculate the
variance of the windowed signal (Israel and Burchiel, 2004). This
variance can then be used to threshold the corresponding windowed signal, i.e., if the signal exceeds one or two (or more)
standard deviations, it is left unchanged, otherwise it is suppressed
to zero. We will refer to this thresholding method as the basic
thresholding functional. The problem with this approach is that
it may not be sufficiently robust, i.e., selecting one or two standard
deviations as the threshold may not always work well in practice. Therefore, in this section we describe an adaptive thresholding
functional that requires no manual intervention.
In this paper, we propose to use a ‘top-hat’ filter to threshold the
signal, which is an adaptive thresholding technique. The top-hat
filter is also known as the top-hat transform (Meyer and Beucher,
1990) and is a well known technique in image processing. Typically,
in order to apply a top-hat filter to a signal, we must subtract the
morphological opening of the signal from the original signal. However, our formulation of the top-hat filter differs to that in Meyer
and Beucher (1990), not only as we deal with one-dimensional signals, but also because we use a top-hat with side brims that sit on
the background noise. The basic idea, which we will elaborate on
in the next few paragraphs, is shown in Fig. 4. Additionally, we also
implement a bottom-hat filter with brims to threshold the negative
valued components of the MER. For the balance of this paper, unless
otherwise specified, when we use the term ‘top-hat filter’, we mean
(for convenience) the top- and bottom-hat filtering processes that
are explained next.
The first step is to find all index positions, n, of both the positive and negative peaks in the MER. For simplicity this is done by
identifying only the maximas in the absolute value of the MER, i.e.,
T + = max(max(ylb [n]), max(yrb [n])) + h,
(14)
T − = min(min(ylb [n]), min(yrb [n])) − h,
(15)
where ylb [n] is the section of the windowed MER under the left
brim, yrb [n] is the section of the windowed MER under the right
brim and h is the hat height, which can be set to zero or can be
selected according to a heuristic depending on the application at
hand. In this paper, we selected h to be one standard deviation of
the windowed MER, however, results appeared to be robust to the
choice of h. Note that a bottom-hat is simply a top-hat on −yw [n].
Then, if the signal at position n was marked as a positive or negative peak according to step one above and that peak exceeds T + or
T − , respectively, within the middle section of the windowed MER,
the spike train of inter-arrivals at n is set to one, otherwise, it is
suppressed to zero. Mathematically,
z[n] = ORall windows F[yw [n]],
where
F[yw [n]] =
(16)
⎧
1 if yms [n] > T + & yms [n] ≥ 0 & n marked as a peak,
⎪
⎪
⎨ 0 if yms [n] ≤ T + & yms [n] ≥ 0 & n marked as a peak,
1
if yms [n] < T − & yms [n] < 0 & n marked as a peak,
0
if nnot marked as a peak,
⎪
−
⎪
⎩ 0 if yms [n] ≥ T & yms [n] < 0 & n marked as a peak,
(17)
and yms [n] is the section of the windowed MER under the middle
section. Square brackets, [·], are used here to emphasize that an
MER is sampled (in this case at 24 kHz) and so exists at discrete
values of time, n, only.
In this paper, we implemented a top-hat filter with the middle
section set to 100 samples (≈ 4 ms). This value was selected based
on typical width observations of the action potentials. For simplicity, we also set both brim widths to 100 samples. To illustrate this
procedure, we applied a top-hat to the MER in Fig. 3 giving the
results in Fig. 5(b) (only a section of the signal is shown for clarity).
The third and final step is as follows: after the top-hat filtering step, we have observed that in certain instances, consecutive
positive–negative and negative–positive spikes remain in the spike
train (see Fig. 5(b)). This is due to the tri-phasic nature of neuronal activity measured external to the cell (Webster, 1998), and
the differential nature of the amplifier (Israel and Burchiel, 2004).
When a neuron fires, the profile of the action potential goes positive then negative and then positive again. If both of the positive
and negative sections of the action potential exceed the threshold of
the top-hat filter, a single action potential will produce two spikes
in the spike train. In order to avoid this, we remove one of these
spikes, i.e., if we encounter consecutive positive–negative spikes,
we suppress the one that is negative to zero, and similarly for consecutive negative–positive spikes. We do not assume an explicit
maximal time between positive–negative and negative–positive
spikes, however, the top-hat only marks a single maximum point
within the hat and so this effectively sets a minimum inter-spike
time (although, the hat width is chosen to be proportional to the
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Fig. 5. The MER in Fig. 3 is top-hat filtered. The result is overlayed on top. The spike train of inter-arrivals shows the temporal pattern of neuronal firings in the STN. Note,
positive or negative spikes which exceed the top-hat thresholds are considered to be neuronal firing events, i.e., a ‘1’, otherwise, the spike train is ‘0’. only a section of the
signal is shown for clarity. (a) Step one, peak detection, (b) step two, top-hat filtering, (c) step three, spike removal, (d) spike train.
spike width). This is shown in Fig. 5(c). From this, it is possible
to generate a spike train of inter-arrivals (see Fig. 5(d)). Compare this spike train to the spike train from the basic thresholded
signal in Fig. 6(b). We can immediately see that the top-hat procedure provides a far more robust thresholding, whereas the basic
thresholding procedure allows a lot more spikes from the baseband through, as expected. The basic threshold technique is very
simple and it involves setting the threshold to a multiple of the
standard deviation of the windowed MER signal. We arbitrarily set
this multiple to 1 in order to demonstrate this point.
Earlier in this subsection, we mentioned that manual thresholding ignores negative spikes whose absolute value also exceeds the
manual threshold. With our technique, we have shown that this
does not happen. We have chosen to threshold MERs in this way
for three reasons. Firstly, we believe it is important to threshold
both the positive and negative valued components of a signal and
use as much information as possible to obtain the spike train of
inter-arrivals. Secondly, the high-pass noise reduction filter acts as
a differentiator and so the MERs that we see (one such example is
shown in Fig. 3) are approximately the first derivative of the measured activity in the STN resulting in significant negative peaks that
we must deal with. Fig. 7(a) shows a simplified ideal profile of an
action potential and its first derivative. The action potential on the
left of Fig. 7(a) is an idealised representation of an intra-cellular
recording from a single neuron. On the right of Fig. 7(a) is the first
derivative of this, which is more like an extra-cellular action potential recorded by MER in vivo.1 Thirdly, we already know that an MER
is the result of many neurons (i.e., many action potentials like the
one depicted in Fig. 7(a)) firing in the vicinity of the STN. Therefore,
the measured MER is the additive sum of many action potentials
and this results in both constructive and destructive interference
(see Fig. 7(b)). Because of this, an MER can never be 100% accurate
and so should be considered only an estimate of the underlying firing rates. For these reasons, it is important to threshold both the
positive and negative valued components of the MER, as shown in
Fig. 7(c).
The paper in Smith and Mtetwa (2007) also talks about spike
detection and sorting techniques and how they cope with different levels of realistic interference (noise). However, the authors
concentrate on generating synthetic, but realistic, signals and interference taking into account the transfer characteristics between
neuron and electrode.
In this paper, we used a hat height set to one standard deviation
of a windowed MER to make sure that most of the ‘marginal’ spikes
1
In practice, the first derivative profile will be noisy but Fig. 7(a) is sufficient for
the purpose of our discussion here.
E.E. Zelniker et al. / Journal of Neuroscience Methods 174 (2008) 281–291
287
ment probe). The question is, what are short intervals and what are
long intervals? For the wild type C. elegans in Zhou et al. (1998),
short and long intervals were observed to be < 100 s and > 100 s,
respectively. What is this number for our application? To answer
this question, we need to estimate the log tail probability given a
finite number of intervals. This was also done in Zhou et al. (1998)
using the rank statistic.
An estimate of the log tail probability given the spike train (i.e.,
the entire spike train, not just the section shown in Fig. 5) is plotted
in Fig. 10. The plot suggests that short intervals are < 2.5 ms and
long intervals are > 2.5 ms (this matter, however, requires further
analysis).
The second cosmetic difference to the model presented in Zhou
et al. (1998) is a relabelling of the E and I states which the authors
propose in their paper (see Fig. 4 on page 2700 in Zhou et al.
(1998)). A neuron can either fire with probability p after some delay
represented by an exponential random variable or not fire with
probability 1 − p and be dormant for a period of time described by
the second exponential random variable. Since the coupled Poisson
model can be used for a larger class of applications, i.e., applications
where discrete events occur in clusters, and almost surely the only
thing that all these applications will have in common is the fact that
an event, E, happens with probability p or it does not happen with
probability 1 − p, we propose states Ep and E1−p in Fig. 2 instead of
E and I, respectively.
We have observed that with some MERs, the signal-to-noise
ratio (SNR) is around −20 dB. Traditionally, SNR is defined as
10 log
Fig. 6. The MER in Fig. 3 is thresholded using the basic approach. The result is overlayed on top. The spike train of inter-arrivals allows a lot more spikes from the
baseband through. (a) Basic threshold, (b) spike train.
which just stick out of the background noise band are also suppressed. However, any thresholding functional may be used with
the coupled Poisson process, including a manual threshold if that
is believed to be best. In addition, other, more robust functionals,
may also be proposed in future.
Finally, we note that we have assumed that it does not matter
whether an extra-cellular cell recording fires positively or negatively. For example, in the spike train of Fig. 5(d) both positive or
negative spikes which exceed the top-hat threshold are considered
to be neuronal firing events. However, if the polarity of an action
potential is considered to be important then that can be easily built
into the thresholding functional.
2.3. Estimation of neuronal firing rates
Thus far, we have been able to argue that neuronal firings are
instances of discrete events which occur in clusters. So it is natural
to use the coupled Poisson process to model the inter-arrivals in
Fig. 5. We can see that for long periods, the firing rates will also
be quite large, because of the sheer number of neurons firing in
the vicinity of the STN (or rather, in the vicinity of the measure-
Signal power Noise power
.
(18)
It is measured in decibels (or dB). If the noise is assumed to be
Gaussian, then the noise power can be estimated from the variance.
In this case, noise power is much bigger than the power (mean
squared value) of the neuronal spike signals and hence we obtain
a negative SNR (in dB). Alternatively, one could measure SNR via
amplitude ratios, however, in this case, SNR would be positive only
at the instant an action potential exceeds the background noise
activity.
Because of the high level of noise, in some cases, the probability
density function is tri-modal rather than bi-modal. This behavior
is not encapsulated by the coupled Poisson process and so in order
to deal with this, we propose two heuristics which can be used to
carry out the peak picking estimation process described in Section
1.2:
Heuristic 1: After generating the histogram of the log-intervals,
select the two largest peaks in the histogram, assign 1 to the left-
2 to the
most (i.e., the faster of the two) peak location, assign p
ˆ 1 and ˆ 2 using
rightmost peak location and finally, solve for p̂, (13).
Heuristic 2: If a tri-modal distribution is exhibited in the histogram of the log-intervals, select the biggest peak and then check
to see which of the remaining peaks will, when combined with the
biggest, return the smallest mean-squared error (MSE) between
the model (10) and the estimated log tail probability given the
2
intervals. Then, assign 1 to the leftmost peak location, assign p
ˆ 1 and ˆ2
to the rightmost peak location and finally, solve for p̂, using (13).
3. Results and discussion
The 247 MERs were obtained and processed as described in Section 2. We have seen throughout this paper an example of one such
MER in Fig. 3. A histogram of the log intervals for this MER is shown
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Fig. 7. Negative peaks retained. (a) A simplified ideal profile of an action potential and its first derivative. The action potential on the left relates to an idealised intra-cellular
recording from a single neuron and its first derivative on the right is more like an extra-cellular action potential recorded by MER in vivo, (b) an MER is a sum of action
potentials. It is the result of many neurons firing in the vicinity of the STN and illustrates both constructive and destructive interference, (c) another section of the signal in
Fig. 3 where it can be seen that certain negative peaks are retained as a result of top-hat (i.e., top- and bottom-hat) filtering and the application of step 3.
in Fig. 8(a). The fY (y) curve in (12) before it is normalised by the
sample size is also shown.
After normalising by sample size, the fY (y) curve is given in
Fig. 8(b). Here, the leftmost peak corresponds to the clustered or
‘intra-burst’ neural activity in the STN while the rightmost peak
corresponds to the slower ‘inter-burst’ activity. To emphasize this
more strongly, refer to Fig. 9 where the spike train in Fig. 5(d) (the
entire spike train, not just the section shown in Fig. 5(d)) is divided
into its slower ‘inter-burst’ and faster ‘intra-burst’ neural activity.
It can be seen that the logical OR of the ‘intra-burst’ and ‘interburst’ APs in Fig. 9(a) and (b), respectively, is the original spike
train in Fig. 5(d). The spikes in Fig. 9(b) correspond to the end of
an inter-arrival whose length is less than or equal to 2.5 ms (the
value obtained from the tail probability plot in Fig. 10). It is clear
from this figure that there are 12 inter-arrivals that are shorter than
2.5 ms. This is consistent with the number of inter-arrivals shown
in the histogram of Fig. 8(a) that are less than log(0.0025) ≈ −6.
From Fig. 8(b), it is possible to use Heuristic 2 above in conjunction with the peak picking estimation process to estimate the
ˆ 1 = 839.6622 Hz and
parameters p, 1 , 2 . This yields p̂ = 0.2519, ˆ 2 = 380.2739 Hz. Fig. 10 shows the log tail probabilities estimated
from the spike train in Fig. 5(d) (i.e., the entire spike train, not just
the section shown in Fig. 5(d)) and the logarithm of (10), which
ˆ 1 = 839.6622 Hz
is displayed as the solid line using p̂ = 0.2519, ˆ 2 = 380.2739 Hz. It should be noted that the firing rates estiand mated by 1 and 2 relate to ‘intra-burst’ and ‘inter-burst’ firing
rates (respectively) of the group of neurons in close proximity to
the MER probe tip. Therefore, the estimated firing rates are much
higher than would be observed for each single neuron in the STN.
In the last set of experiments, we ran the algorithm over all
of our 247 MERs using both the basic and top-hat thresholding
functionals. For each MER, we obtained estimates for p, 1 and
2 using Heuristic 2 above and used these to generate the theoretical log tail probability (solid line in Fig. 10). We also used the
spike train of the MERs to estimate the log tail probability given the
inter-arrivals (dotted line in Fig. 10). The mean squared error (MSE)
between the theoretical log tail probability and the estimated log
tail probability are shown in Fig. 11 for all the MERs (solid line). We
compare this to the MSE for each MER assuming a single Poisson
process is the correct model of the inter-arrival times (i.e., estimate
only a single by calculating the mean of the inter-arrival times).
This is illustrated by the dashed line in Fig. 11. We further compare these to what the MSE for each MER would be if we fitted
a straight line to the estimated log tail probability (and of course,
using only the estimated slope of that line to estimate ). This is
illustrated by the dotted line in Fig. 11. Performing this last step
E.E. Zelniker et al. / Journal of Neuroscience Methods 174 (2008) 281–291
Fig. 8. Histogram of the log intervals. (a) The fY (y) curve in (12) before it is normalised by the sample size, (b) The fY (y) curve in Eq. (12) which estimates the pdf
of Y.
obviously biases the estimation of to that which best fits the log
tail probability in a mean squared sense. However, in most cases
the log tail probability estimated from the coupled Poisson process is closer to that from the biased single Poisson process than
that from the unbiased single Poisson process. In addition, both the
coupled and biased Poisson models produce lines that are below
that of the unbiased Poisson process. The mean MSEs over all MERs
are summarised in Table 1. The mean MSEs are lower for the tophat filter than the basic threshold which suggests that the top-hat
filter with Heuristic 2 performs better than the basic threshold with
Heuristic 2.
It is also worth noting that out of the 247 MERs analysed using
the top-hat threshold, 85 had bimodal distributions, indicating that
the coupled Poison process is suitable and the rest were tri-modal
indicating that they would require the use of Heuristic 2 outlined in
Section 2.3 to adequately fit the coupled Poisson model to the MER
data. However, upon observing the histograms it would appear that
the vast majority of tri-modal distributions are really bimodal as the
third mode is due to noise or was a point of inflection on one of the
major modes.
We will end this section with a discussion as to why only some of
the distributions were bi-modal. Firstly, we have observed a number of mis-detections. This is purely to do with the spike detection
289
Fig. 9. The spike train in Fig. 5(d) (the entire spike train, not just the section shown
in Fig. 5(d)) is here divided into its slower ‘inter-burst’ and faster ‘intra-burst’ neural
activity, i.e., the logical OR of the ‘inter-burst’ and ‘intra-burst’ spike trains in Fig. 9(a)
and (b) results in the spike train in Fig. 5d). (a) Spikes relating to intra-burst activity
of the MER in Fig. 3, (b) the initial intra-burst spikes after an inter-burst period of
the MER in Fig. 3.
process, i.e., the thresholding functional F. It is not 100% accurate, however, we have shown that our top-hat filter is better than
existing technique of manual thresholding, which is often considered to be the ‘gold’ standard in a clinical setting. There are no
existing thresholding methods that are 100% accurate. There will
always be noise, especially with in vivo measurements. Therefore,
having an analysis methodology, such as the couple Poisson process, that is immune to a degree of mis-detections is clearly of
importance.
Secondly, because of the high level of noise, in some cases, we
have observed the probability density function to have more than
two modes. This does not necessarily mean that the actual results
fail to fit the theoretical model. The extra mode is often due to the
fact that the thresholding functional is not robust enough. Clearly,
the coupled Poisson process will perform better with a more robust
thresholding functional. But our results indicate that the current
thresholding functionals are sufficient. It is also possible to argue
that it might be better to use a multi Poisson process instead of
just a coupled Poisson process to account for these extra modes.
This would involve using a more complicated Markov Chain and
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E.E. Zelniker et al. / Journal of Neuroscience Methods 174 (2008) 281–291
Fig. 10. Estimated log tail probability (dotted line) of the inter-arrival times from
the spike train data in Fig. 5(d) (note that the entire spike train, not just the section
shown in Fig. 5(d) was used to generate the tail probability) and the theoretical log
ˆ 1 = 839.6622 Hz and
tail probability (solid line) that corresponds to p̂ = 0.2519, ˆ 2 = 380.2739 Hz.
this has a well established theory in the mathematical literature
(e.g., see Durbin et al., 1999; Rabiner, 2000; Ephraim and Merhav,
2002). In fact, the three-state model in Fig. 2 is a discrete two-stage
Markov Chain. If we were to use a multi Poisson process, an important question would arise: specifically, what is the physiological
interpretation of this model? inter- and intra-burst activity will not
make sense anymore in this setting. So, is this a worthwhile model
to pursue? The reason we opted for the coupled Poisson process is
because scientists in this field already understand the concept of
inter- and intra-burst activity. Furthermore, with the coupled Poisson process, we are estimating only 3 parameters, i.e., 1 , 2 and
p. If we were to use a Markov Chain, we would need to estimate
Table 1
Mean MSE’s
Single mean
Coupled Poisson
Biased 1st order fit
Mean MSE basic threshold
Mean MSE top-hat threshold
1.9906
0.5739
0.2607
1.5903
0.5136
0.0802
Fig. 11. Mean squared errors of each of the 247 MER traces. (a) Basic threshold, (b)
top-hat filter.
many more parameters given the same amount of information and
this inevitably means that each parameter will be estimated less
robustly.
4. Conclusions
In this paper, we have demonstrated that that the coupled Poisson process, proposed by Zhou et al. (1998), is an appropriate model
for estimating neuronal firing rates obtained via MERs from the STN
during DBS surgery. We have proposed a novel ‘top-hat’ filter to
adaptively threshold the MER data and obtain a list of inter-arrival
times (the spike train). Also, the performance of the proposed
thresholding functional was demonstrated to be superior to a conventional threshold. In addition, two heuristics were proposed to
estimate inter-burst and intra-burst firing rates via a peak picking
estimation process. Results on 247 MERs from ten subjects have
shown good correspondence with the coupled Poisson process in
a probabilistic sense. In addition, we obtained, with the log tail
plots that, for our particular application, long (inter-burst) intervals are typically > 2.5 ms while short (intra-burst) intervals are
typically < 2.5 ms. Although we have shown that the coupled Poisson process model appears to work well globally, it is independent
of the spike detection method and, more robust thresholding func-
E.E. Zelniker et al. / Journal of Neuroscience Methods 174 (2008) 281–291
tionals may be proposed that improve the peak picking estimation
procedure.
Although there are both false alarms and false misses, it is important to note that our method builds a statistical model (a PDF)
and detects the peaks in this, therefore, a ‘certain number’ of false
alarms and false misses are not going to affect the location of these
peaks too much and assuming the false alarm and miss rates are
approximately equal it won’t affect their relative probability. That
is, we have a model that is robust to false alarms and misses as these
will never be zero.
Acknowledgments
The authors would like to thank Dr Terry Coyne, Dr Felicity Sinclair, St. Andrew’s War Memorial Hospital, the Wesley Hospital and
Kim Parton from Medtronic for their involvement in the project,
providing MER data and instrumentation.
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