A joint interspike interval difference stochastic spike train analysis: Detecting local trends in the temporal firing patterns of single neurons

Abstract

We introduce a stochastic spike train analysis method called joint interspike interval difference (JISID) analysis. By design, this method detects changes in firing interspike intervals (ISIs), called local trends, within a 4-spike pattern in a spike train. This analysis classifies 4-spike patterns that have similar incremental changes. It characterizes the higher-order serial dependence in spike firing relative to changes in the firing history. Mathematically, this spike train analysis describes the statistical joint distribution of consecutive changes in ISIs, from which the serial dependence of the changes in higher-order intervals can be determined. It is similar to the joint interspike interval (JISI) analysis, except that the joint distribution of consecutive ISI differences (ISIDs) is quantified. The graphical location of points in the JISID scatter plot reveals the local trends in firing (i.e., monotonically increasing, monotonically decreasing, or transitional firing). The trajectory of these points in the serial-JISID plot traces the time evolution of these trends represented by a 5-spike pattern, while points in the JISID scatter plot represent trends of a 4-spike pattern. We provide complete theoretical interpretations of the JISID analysis. We also demonstrate that this method indeed identifies firing trends in both simulated spike trains and spike trains recorded from cultured neurons.

Full text

Abstract. We introduce a stochastic spike train analysis
method called joint interspike interval dierence (JISID)
analysis. By design, this method detects changes in ®ring
interspike intervals (ISIs), called local trends, within a 4-
spike pattern in a spike train. This analysis classi®es 4-
spike patterns that have similar incremental changes. It
characterizes the high er-order serial dependence in spike
®ring relative to changes in the ®ring history. Mathe-
matically, this spike train analysis describes the statis-
tical joint distribution of consecutive changes in ISIs,
from which the serial dependence of the changes in
higher-order intervals can be determined. It is similar to
the joint interspike interval (JISI) analysis, except that
the joint distribution of consecutive ISI dierences
(ISIDs) is quanti®ed. The graphical location of points
in the JISID scatter plot reveals the local trends in ®ring
(i.e., monotonically increasing, monotonically decreas-
ing, or transitional ®ring). The trajectory of these points
in the serial-JISID plot traces the time evolution of these
trends represented by a 5-spike pattern, while points in
the JISID scatter plot represent trends of a 4-spike
pattern. We provide complete theoretical interpretations
of the JISID analysis. We also demonstrate that this
method indeed identi®es ®ring trends in both simulated
spike trains and spike trains recorded from cultured
neurons.
1 Introduction
We introd uce a stochastic spike train analysis technique
to detect and characterize the sequential changes in a
series of spike ®rings generated by individual neurons.
Spike ®rings are traditionally represented by a time-
series of action potentials called spike trains. A spike
train can be treated as a point process (Perkel et al.
1967a, b) based on the classical theories of stochastic
point processes (Correia and Landolt 1977; Cox and
Lewis 1966; Gerstein and Kiang 1960; Gray 1967;
Moore et al. 1966, 1970; Tuckwell 1988; Yang and Chen
1978). The ®ring characteristics of a neuron can be
described by the tempor al pattern of spike occurrences,
which is quanti®ed by the time sequence of spike
occurrences (see Sect. 3.1). Although a temporal pattern
can include an y number of spikes, in this paper we
examine patterns that are limited to a sequence of less
than six consecutive spikes. Speci®cally, we characterize
the changes within the temporal patterns, which we call
`local trends.' Other long-term trends can be analyzed
using many of the traditional time-series analyses (e.g.,
Brillinger 1975; Gotton 1981; Sugihara 1994).
We are interested in analyzing the local trends within
a short sequence of spikes for the following reasons.
Speci®c temporal patterns of a neuron may encode sig-
ni®cant information (Segundo et al. 1963; Sherry and
Klemm 1982, 1984; Terasawa et al. 1989; Tsukada et al.
1975, 1982, 1983). Dierent sequences of spike ®rings
may represent dierent encoding schemes, which may
re¯ect changes in the dynamics of a neuron and/or
changes in the signal content encoded by the ®ring
patterns. Therefore, analysis of the local trends can
potentially provide insight into the encoding properties
of neurons in the central nervous system.
Limiting the analysis to short patterns may reveal the
serial relationship between consecutive spike ®rings from
which the statistical dependence of a stochastic process
can be established. If a neuron obeys a simple Markov
process in which the occurrence of the next event is de-
pendent only on the present state and is independent of
all of its past states, then the serial dependence rela-
tionship of two consecutive intervals is sucient to
describe the neuron's ®ring. The analysis of a 3-spike
pattern (which includes two interspike intervals, ISIs) is
adequate. If, however, the neuron follows a higher-order
dependence process, then the serial dependence between
more than two consecutive intervals is required. In this
Biol. Cybern. 80, 309±326 (1999)
A joint interspike interval dierence stochastic spike train analysis:
detecting local trends in the temporal ®ring patterns of single neurons
Michelle A. Fitzurka
1
, David C. Tam
2
1
Department of Medical Physics, University of Wisconsin, Madison, WI 53705, USA
2
Center for Network Neuroscience and Department of Biological Sciences, University of North Texas, Denton, TX 76203, USA
Received: 13 May 1997 / Accepted in revised form: 9 December 1998
Correspondence to: D.C. Tam, Department of Biological Sciences,
University of North Texas, P. O. Box 305220, Denton, TX 76203±
5220, USA
(e-mail: dtam@unt.edu,
Tel.: +1-940-565-3261, Fax: +1-940-565-4136)

case, an analysis will be needed that examines patterns
that include more than three spikes (i.e., a 4-or-more
spike pattern). Although a matrix equation for
computing the higher-order transition probabilities from
lower-order ones can be obtained from the Chapman-
Kolmogorov equation for a Markov chain, we are
particularly interested in the extent of the dependence of
an unknown process on its speci®c history. The extent of
this dependence can be quanti®ed by the higher-order
transition probabilities (or the conditional probabilities
on its past series of histories), especially when the higher-
order serial dependence is nonlinear. In contrast to the
transition probabilities quanti®ed by most traditional
serial dependence analyses, we characterize the higher-
order dependence by the changes in conditional proba-
bilities.
Classically, a point process, such as a spike train, can
be analyzed using random variables such as the nth spike
occurrence at time, t
n
, or the nth interspike interval, s
n
.
An ISI is de®ned as the time dierence between con-
secutive spikes, s
n
 t
n
ÿ t
nÿ1
. A higher-order ISI is de-
®ned as the time interval that spans multiple ISIs.
Univariate statistics can be obtained based on this ®rst-
order ISI and/or other higher-order ISIs, such as auto-
correlation in which all orders of ISIs are included in the
analysis (Perkel et al. 1967a).
Bivariate statistics are often used to establish the re-
lationship between two random variables (such as s
n
and
s
m
) to determine serial dependence. Classical stochastic
spike train analyses exploring the relationship between
consecutive ®rst-order ISIs (s
n
and s
n1
) are called joint-
ISI (JISI) plots (Rodieck et al. 1962; Schulman and
Thorson 1964; Segundo et al. 1966). The stati stical dis-
tribution of this ®rst-order ISI pair (s
n
, s
n1
) represents a
two-dimensional ISI statistical distribution, or a joint-
ISI distribution. This JISI analysis establishes the serial
correlation of adjacent points in the time-series from
which the joint prob ability of serial ®rings can be
obtained by statistically estimating the transition
probabilities. Recently, the same analysis has been
re-examined in terms of deterministic chaos theories of
nonlinear dynam ics in which the JISI plots are referred
to as `interval return maps' (Nomura et al. 1994; Seg-
undo et al. 1994; e.g., Selz and Mandell 1992; Smith
1992). The joint correlation analysis can be extended to
correlate all orders of ISIs, called recurrence plot anal-
ysis (Aihara 1994; Eckmann et al. 1987; Kaluzny and
Tarnecki 1993; Mayer-Kress and Hubler 1989; She-
lhamer 1997). Note that these analyses can be applied to
the same set of data with dierent interpretations, either
in terms of stochastic theories or in terms of nonlinear
dynamics in relation to deterministic chaos theories.
Regardless of the dierences in interpretations, these
bivariate correlation analyses essentially correlate two
shifted, dependent, random variables.
In this paper, we design and describe an analysis that
reveals the sequential changes in ®ring intervals within a
short pattern of consecutive spikes rather than analyzing
the pattern itself. Instead of characterizing the past ®ring
history in terms of the ISIs, s
n
, (as in most time-series
analyses), we use the interspike-interval dierences (IS-
IDs), Ds
n
, as the random variable (Fitzurka 1996).
An ISID is de®ned as the time dierence between con-
secutive ISIs (i.e., Ds
n
 s
n
ÿ s
nÿ1
). This ISID variable is
chosen speci®cally because it quanti®es changes in the
consecutive ISIs; positive for increasing ISIs and negative
for decreasing ISIs. By determining the serial dependence
relationship between these consecutive ISIDs, the local
trend that we seek will ultimately be revealed. W e call this
technique joint-ISID (JISID) analysis (Fitzurka and Tam
1994) since it provides a joint probability density func-
tion (pdf) of the ISID pair (Ds
n
, Ds
n1
) from which the
changes in conditional pdf and transition probability
matrix of a Markov process or other dependent process
can be obtained. A similar analysis called `®rst-order
variability diagram' was employed by Babloyantz and
Maurer (1996) and Maurer et al. (1997) to study the
chaotic dynamics of the periodicity in cardiac rhythms.
They examined the variability of deviations from regu-
larity of the periodic pacemaker activity in the arrhythmic
heart, which provided some indications of the determin-
istic nature of these cycles not revealed by traditional
Fourier analyses (e.g., Rosenberg et al. 1989). In contrast,
we derived the JISID analysis from stochastic process
theories, quantifying the serial transition probabilities
based on the changes of a neuron's past ®ring histories
(Fitzurka and Tam 1995a). The statistical distributions
are speci®cally sought so that the unknown pdf of a
neuron recorded experimentally can be obtained statis-
tically, while such statistics are not sought by other sim-
ilar nonlinear analyses (Babloyantz and Destexhe 198 8).
The most important aspect of this JISID analysis is
that it is designed to capture ®ring patterns that have
similar incremental changes, independent of the time-
scales of the ISIs (or ®ring rate), so that they can be
classi®ed as having the same tendency for changes, or
similar trends. We shall show that the trends identi®ed by
this analysis can capture various permutations of 4-spike
patterns, including periodic and aperiodic, regular and
irregular, ran dom and nonrandom patterns. Thus, our
analysis is not necessarily limited to identifying the peri-
odicity of repeated patterns or the variability of devia-
tions from the fun damental period of cyclical patterns.
As a ®nal note, this JISID analysis is limited to de-
tecting the serial correlation of spike ®ring within a
single spike train in an individual neuron. Analyses de-
tecting the serial correlation (or temporal correlation)
between the spikes of dierent neurons can be found
elsewhere (e.g., Fitzurka and Tam 1997; Gerstein and
Perkel 1972; Perkel et al. 1967b, 1975; Tam et al. 1988).
2 Methods
As discussed, this JISID analysis is similar to the
traditional JISI analysis, except that ISIDs are used as
the random variables instead of ISIs. Speci®cally, the
serial correlation between two consecutive ISIDs (span-
ning three consecutive ISIs) is examined and quanti®ed
(Fitzurka and Tam 1995b), instead of between two
consecutive ISIs (Rodieck et al. 1962). In this way, the
serially dependent ®ring relationship spanning three (as
310

opposed to two) consecutive intervals can be statistically
quanti®ed.
2.1 Theoretical descript ion
Let a spike train, xt (Fig. 1) with a total of N spikes be
represented by an n-tuple, t
1
; t
2
; ...; t
n
, where t
n
denotes the time of occurrence of the nth spike. An n-
tuple is an ordered list of n variables. Alternatively, the
spike train can be de®ned as
xt
X
nN
n1
dt ÿ t
n
1
where dt denotes the Dirac delta function. The ®rst-
order ISI is de®ned as the time dierence between two
consecutive spikes. Note that there are two ISIs adjacent
to a reference spike. The ISI preceding the nth reference
spike (the pre-ISI, s
n
) is de®ned as
s
n
 t
n
ÿ t
nÿ1
2
(see Fig. 1A) and the ISI succeeding the nth reference
spike (the post-ISI, s
n1
) is de®ned as
s
n1
 t
n1
ÿ t
n
3
Generalizing this, the jth order ISI with respect to the
nth spike is de®ned as
s
j
n
 t
n
ÿ t
nÿj
4
Similarly, an ISID is de®ned as the dierenc e between
consecutive ISIs (see Fig. 1B). The ISID preceding the
nth reference spike (the pre-ISID, Ds
n
) is de®ned as
Ds
n
 s
n
ÿ s
nÿ1
 t
n
ÿ 2t
nÿ1
 t
nÿ2
5
which re¯ects the dierence between the nth and
n ÿ 1th ISI. Note that the time interval spanned by
the ISID includes three spikes (see Fig. 1B), thus
encapsulating a 3-spike pattern represented by a 3-tuple,
t
nÿ2
; t
nÿ1
; t
n
. The ISID subsequent to the nth reference
spike (the post-ISID, Ds
n1
) is de®ned as
Ds
n1
 s
n1
ÿ s
n
 t
n1
ÿ 2t
n
 t
nÿ1
6
Although the ISID spans two ISIs, (5) is not the same as
the second-order ISI (Ds
n
6 s
2
n
 t
n
ÿ t
nÿ2
). Thus, the
statistical analysis using this random variable, Ds
n
, is not
a simple extension of the JISI analysis using the second-
order ISIs, s
2
n
, as the random variable.
Statistics using the ISID, Ds
n
, as the random variable
are given below. The ISID distribu tion, denoted by
IDs, is represented by
IDs
X
N
n3
dDs
n
ÿ Ds
X
N
n3
dt
n
ÿ 2t
nÿ1
 t
nÿ2
ÿ Ds
7
This statistical distribution provides an indicator of the
changes in ISIs with respect to the reference spike, with
positive ISIDs indicating a lengthening in the next ISI,
and negative ISIDs representing a shortening in the next
ISI relative to the previous ISI. This simple univariate
ISID statistic essentially characterizes the local trend of
a 3-spike pattern.
The ®ring trend analysis to be introduced in this pa-
per goes beyond this simple statistical analysis of ISIDs.
The focus of the analysis is on quantifying the serial
relationship of the 4-spike patterns evolving over time.
This 4-spike trend is characterized by correlating two
consecutive 3-spike patterns (sequentially shifted by a
spike relative to each other). Speci®cally, the serial de-
pendence of ISIDs is characterized by a statistical JISID
distribution, similar to the characterization of the serial
dependence of ISIs as quanti®ed by the statistical JISI
distribution in the JISI analysis. The JISID distribution,
JDs
1
; Ds
2
, is given by
JDs
1
; Ds
2

X
N
n4
dDs
n
ÿ Ds
1
dDs
n1
ÿ Ds
2
8
Fig. 1. Schematic diagram of a spike train showing
the times of occurrence of the spikes in relation to (A)
the interspike intervals (ISIs) and (B) the interspike
interval dierences (ISIDs)
311

A JISID `scatter plot' can be constructed by plotting
Ds
n
; Ds
n1
 as a point representing the x; y coordinate
in the JISID plot for all spikes (8n) in the spike train.
That is, a scatter plot is produced by plotting
Ds
n
; Ds
n1
 formed by the pre- and post-ISIDs relative
to the nth reference spike for each spike used in turn as
the refe rence spike (see Fig. 1B). Thus, each spike is
represented by a point in the JISID plot, and the loca l
trend is identi®ed by the quadrant in which the point is
located (see Sect. 3 for details). The statistical distribu-
tion of these points gives the joint distribution of (8)
sought in the an alysis for revealing whet her there is any
serial dependence between the consecutive ISIDs (i.e.,
with respect to consecutive changes in spike ®ri ng
intervals). Normalizing (8) gives a statistical estimation
of the joint pdf,
jDs
1
; Ds
2

P
N
n4
dDs
n
ÿ Ds
1
dDs
n1
ÿ Ds
2

N ÿ 4
9
and the conditional pdf representing the transition
probability of the ISIDs is given by
pDs
2
jDs
1

pDs
1
\ Ds
2

pDs
1


P
N
n4
dDs
n
ÿ Ds
1
dDs
n1
ÿ Ds
2

P
N
n4
dDs
n
ÿ Ds
1

10
Although this conditional pdf is informative in provid-
ing the transition probability (matrix) for a serially
dependent process, most similar analyses in the litera-
ture plot the joint pdf instead of the conditional pdf.
Traditionally, the joint probabilities are also given in a
transition matrix form, while the joint distribution of the
individual points are plotted in the JISI scatter plots
(Rodieck et al. 1962). The main advantage of plotting
the joint pdf scatter plot is to provide visualization of the
exact x; y coordinate of the points in the plot for
identifying the trends of the 4-spike patterns (see Sect. 3).
In addition to the scatter plot, a trajectory plot
tracing the sequential points in the JISID analysis can be
constructed by connecting the points between the nth
and n  1th reference spikes, as in other interval return
maps. This reveals explicitly the time evolution sequence
of JISID pairs extending the analysis to establishing the
relationship of a 5-spike sequence. We call this the `se-
rial-JISID plot' (Fitzurka and Tam 1995a).
2.2 Simulation and experimental methods
We will ®rst analyze simulated spike trains generated
with speci®c known spike-generating pdf's speci®cally to
illustrate that the analysis indeed detects and identi®es
these speci®c trends. The spike trains of all simulated
neurons (A-G, I-J) are generated with Gaussian-varied
ISIs, with the exception of the Poisson neuron (H). The
Gaussian variance at each spike is proportional to the
value of the ISI preceding that spike for neurons A-G,
and constant for neurons I and J. The analyses are then
applied to spike trains recorded from biological neurons
whose ®ring trends are not known a priori so that the
results of these analyses can be compared and demon-
strated to reveal and identify the unknown trends.
The biological neurons used in the analyses were
cultured spinal cord neurons. Networks of neurons ob-
tained from mouse embryos were cultured on multi-
microelectrode plates according to the methodology
described by Gross (1979) and Gross et al. (1977). Si-
multaneous extracellular recordings of action potentials
were made from these neurons. The spike trains were
digitized by a bank of digital signal processors (Plexon,
Dallas) sampling at 40 KHz. These cultured neurons
often ®re spontaneously, though sometimes bicuculline
was added to the network to disinhibit the activity. The
data presented in this paper were recorded after the
network was treated with bicuculline (60 ll, 1 mM).
3 Theoretical interpretations
3.1 De®nitions of spike ®ring patterns and local trends
A `temporal n-spike pattern' is de®ned by an ordered-list
of the spike occurrence times, t
1
; t
2
; ...; t
n
. This same
pattern can also be represented by an ordered-list of
ISIs, s
2
; s
3
; ...; s
n
, called the `ISI pattern.' This ISI
pattern encapsulates all similar spike trains that are
shifted in time by k (= constant), i.e., t
1
 k; t
2
k; ...; t
n
 k. It provides a time-shift invariant repre-
sentation of the spike pattern independent of the actual
spike occurrence times with a reduction of variables.
Mathematically, all similar time-shifted spike train s,
t
1
 k; t
2
 k; ...; t
n
 k, will map into a single ISI
pattern, s
2
; s
3
; ...; s
n
. Similarly, the ordered-list of the
ISIDs, Ds
3
; Ds
4
; ...; Ds
n
, en capsulates all ISI trends
that have similar incremental changes, i.e.,
s
2
 k; s
3
 k; ...; s
n
 k, with k being a constant. All
similar increment-invariant ISI patterns, s
2
 k;
s
3
 k; ...; s
n
 k, will map into a single ISID trend,
Ds
3
; Ds
4
; ...; Ds
n
.
Alternatively, the ISI pattern, s
2
; s
3
; ...; s
n
, can be
re-written as
s
2
; s
3
; ...; s
n
s
1
 Ds
2
; s
2
 Ds
3
; ...; s
nÿ1
 Ds
n

11
so that the trends encapsulated by Ds
3
; Ds
4
; ...; Ds
n

are all similar ISI patterns represented by the n ÿ 1-
tuple, s
1
 Ds
2
 k; s
2
 Ds
3
 k; ...; s
nÿ1
 Ds
n
 k
with k as a constant. This shows that ISI patterns
having similar incremental changes are considered as
having a similar trend according to our de®nitions.
Applying these de®nitions to 4-spike patterns, the
relationship between the pre- and post-ISIDs [plotted as
a point, x; yDs
n
; Ds
n1
, graphically in the JISID
plot] is essentially a 2-tuple representation of the
ISID trend, Ds
n
; Ds
n1
. This x; y point in the
JISID plot also maps to all points, x; y; z
312

s
nÿ1
 k; s
n
 k; s
n1
 k, with similar incremental
changes in a 3-dimensional JISI plot. As explained
above, this 3-tuple ISI pattern, s
nÿ1
 k; s
n
 k; s
n1
 k,
also maps into any 4-spike patterns, t
nÿ2
 k; t
nÿ1
 k;
t
n
 k; t
n1
 k, that are independent of the absolute
spike occurrence times. Thus, a point in the conven-
tional JISI plot describes a 3-spike pattern, while a point
in the JISID plot describes a 4-spike trend. Although
unconventional and sometimes restrictive with these
de®nitions, we call these similar incremental changes in
intervals within the pattern a `trend,' conforming to the
tuple notation of spike patterns.
Because there are many similar 4-spike patterns rep-
resented by an ISI 3-tuple, s
nÿ1
 k; s
n
 k; s
n1
 k,
that correspond to the same ISID pair, Ds
n
; Ds
n1
 in
the JISID plot, classi®cation of similar spike patterns
into the same trend becomes automatic by this many-to-
one mapping of points from the JISI plot onto the same
point in the JISID plot. Th e advantages of this many-to-
one mapping of similar spike patterns to a ®ring trend
are given as follows. First, similar spike patterns are
implicitly classi®ed as the same ®ring trend mathemati-
cally and graphically. Second, the 4-spike patterns are
captured by a simpler 2-tuple representation of ISIDs
rather than a 3-tuple representation of ISIs or 4-tuple
representation of spike times. Third, the graphical rep-
resentation dimension of a 4-spike pattern is reduced
from 3D (three dimensions in the JISI plot) to 2D (two
dimensions in the JISID plot).
3.2 Interpretation of dierent regions
of the JISID scatter plot
The JISID plot, which we introduce here, can be divided
into four quadrants separated by the axes, each repre-
senting di erent 4-spike ®ring trends (see Fig. 2B). The
relationship between the speci®c changes in consecutive
ISIs spanning 4 spikes are identi®ed by the insets in
Fig. 2B. Points lying within the ®rst quadrant (i.e.,
Ds
n
> 0andDs
n1
> 0) show that the neuron ®res with
monotonically increasing ISIs (i.e., s
nÿ1
< s
n
< s
n1
),
indicating a `monotonically increasing trend.' Those in
the third quadrant (i .e., Ds
n
< 0 and Ds
n1
< 0) denote
monotonically decreasing ISIs (i.e., s
nÿ1
> s
n
> s
n1
),
implying a `monotonically decreasing trend.' The second
quadrant (i.e., Ds
n
< 0 and Ds
n1
> 0) indicates a
transition in the ISIs, changing from long to short to
long (i.e., s
nÿ1
> s
n
< s
n1
), whereas the fourth quad-
rant (i.e., Ds
n
> 0 and Ds
n1
< 0) indicates the opposite
transition (i.e., s
nÿ1
< s
n
> s
n1
). These are called `tran-
sitional ®ring trends.' [Note that these `transiti onal'
®ring trends refer to the reversal of monotonic trends in
this paper, which is distinguished from the `transition'
probability of a Markov process].
At the origin (Ds
n
 0 and Ds
n1
 0), the ISIs re-
main constant (i.e., s
nÿ1
 s
n
 s
n1
), indicating a
`constant trend.' Points falling along the axes represent
`ramp ®ring trends.' Two types of ramp trends are found
along the x-axis. Points lying along the x-axis represent
a ramp trend with s
nÿ1
< s
n
 s
n1
(increasing ramp
trend II), while points lying along the ÿx-axis signify a
ramp trend with s
nÿ1
> s
n
 s
n1
(decreasing ramp
trend I). Similarly, two other types of ramp trends are
found along the y-axis. Along the y-axis (Ds
n
 0) the
ramp trends revealed are s
nÿ1
 s
n
< s
n1
for the y
axis (increasing ramp trend I), and s
nÿ1
 s
n
> s
n1
for
the ÿy-axis (decreasing ramp trend II).
In summary, each point in this JISID scatter plot
corresponds to two adjacent interval dierence pairs,
Ds
n
; Ds
n1
, or three adjacent intervals, s
nÿ1
; s
n
; s
n1
,
or four adjacent spikes, t
nÿ2
; t
nÿ1
; t
n
; t
n1
; whereas each
point in the traditional JISI scatter plot (see Fig. 2 A)
represents only two adjacent intervals s
n
; s
n1
, or three
adjacent spikes, t
nÿ1
; t
n
; t
n1
.
3.3 Interpretation of the serial-JISID scatter plot
The evolution of the changes in ®ring, or ®ring trends,
can be further deduced from the serial-JISID plots.
These consecutive changes in ISIDs are represented by
the sequence of quadrants the sequential points traverse.
The serial dependence of these changes is determined by
visualizing within which quadrants sequential points are
located with respect to each other, and the angle of the
line connecting the sequent ial points.
4 Simulation and experimental results
4.1 Identi®cation of unique trends based on quadrant
A spi ke train with a repeated `triangularly modulated'
sequence is simulated for neuron A, whose ISIs increase
monotonically by a constant amount then decrease
monotonically by the same constant amount. It is
immediately evident that this neuron ®red with four
distinct trends, as seen by clusters in each quadra nt
(Fig. 3A). The clusters in quadrant s 1 and 3 represent
the monotonically increasing and decreasing ®ring
trends, respectively, while the clusters in quadrants 2
and 4 represent the long-short-long and short-long-short
transition ®ring trends (refer to Fig. 2B).
Comparing the JISI (Fig. 3B) analysis to the JISID
(Fig. 3A) analysis for this neuron (cf. Fig. 2A for in-
terpretation), there are eight distinct clusters found as
opposed to only four. Thus, there are eight distinct 3-
spike patterns found by JISI analysis, yet only four
distinct trends found by JISID analysis, because similar
incremental patterns are classi®ed as the same trend. For
example, the four clusters (a)±(d) above the diagonal in
the JISI plot represent the four proportional patterns
that are captured by a single trend in the cluster (a) in
quadrant 1 (monotonically increasing trend quadrant) in
the JISID plot.
In constrast to neuron A, neuron B (which ®res with a
triangularly modulated `sawtooth-shaped' repeated
monotonically increasing sequence) has one cluster in
quadrant 1 (monotonically increasing trend quadrant)
but no clust er in quadrant 3 (monotonically decreasing
trend quadrant) of the JISID plot (Fig. 3C). Comparing
313

neuron C (simulated with a repeated triangularly mod-
ulated monotonically decreasing sequence) with ne uron
B, we see that a cluster is found in the monotonically
decreasing quadrant 3 (Fig. 3E) as opposed to the
monotonically increasing quadrant 1 (see Fig. 3C),
which is as expected.
In summary, distinct trends are revealed by the char-
acteristic features in the JISID plots. First, the transi-
tional trends for these neurons are dierent, i.e., those for
neurons B and C show more abrupt asymmetrical tran-
sitions than A. This can be deduced by comparing the
proximity of the clusters to the axes in quadrants 2 and 4.
Fig. 2. Interpretation diagrams for the joint-ISI (JISI) (A) and joint-ISID (JISID) (B) scatter plots indicating the ®ring patterns that correspond
to dierent regions on the plot as depicted by the insets
314

Fig. 3. JISI and JISID scatter plots for neuron A ®ring with triangularly modulated spike patterns (A, B), for neuron B ®ring with repeated
increasing triangular sawtooth spike patterns (C, D), and for neuron C ®ring with repeated decreasing sawtooth spike patterns (E, F). A segment
of the spike train is displayed above the JISID plot for each neuron. Example spike patterns corresponding to the clusters are shown as insets.
[The histograms underneath and to the left of the JISID and JISI scatter plots represent the ISID and ISI histograms, using a 5 ms bin-width]
315

Second, it can be noted that some neurons do not nec-
essarily ®re with all four trends cla ssi®ed by the quad-
rants. Third, the JISID analysis did indeed identify and
classify similar incremental patterns as the same trend for
all these neurons. In particular, the similar proportional
4-spike patterns that belong to the same monotonic
trends are identi®ed by a single cluster in the JISID plot
that corresponds to multiple clusters aligned parallel to
the diagonal in the corresponding JISI plots.
When these bivariate joint interval (JISI and JISID)
analyses are compared with the univariate (ISI) analysis,
the distributions of points in the JISI scatter plots are
not symmetrical about the central diagonal (see Fig. 3D,
F); however, the pre-ISI and post-ISI histograms are the
same, as expected by de®nition. In fact, these ISI his-
tograms can be considered as histograms constructed by
collapsing (summing) all points in the JISI plot onto the
x-ory-axis, respectively. Furthermore, the ISID histo-
grams alone do not reveal the speci®c 4-spike trends that
JISID plots reveal, similar to the fact that ISI histograms
do not uncover the speci®c 3-spike patterns that JISI
plots uncover. Finally, the ISI distributions (see Fig. 3B,
D, F) for all three neurons are similar (and, in fact,
almost identical for B and C), yet the JISI and JISID
distributions are distinctly dierent. This shows that
bivariate analyses (such as JISI and JISID) are essential
in distinguishing the dierences in trends and patterns
unique to each neuron.
4.2 Detection of multiple trends
in sinusoidally modulated ®ring
The JISID plot (Fig. 4A) of neuron D (®ring with
sinusoidally varying ISIs) shows that there are points in
all four quadrants. All four main types of ®ring trends
are represented. For comparison, the clusters for the
sinusoidally modulated neuron, D, outline an elliptical
ring (Fig. 4A,C), whereas the clusters for the triangu-
larly modulated ®ring neuron, A, outline the corners of a
rectangle (Fig. 3B). This reveals the dierences between
speci®c abrupt and smooth trends, as in the triangular
and sinusoidal cases, res pectively.
Figure 4C, D displays the JISID and JISI histograms
representing the 3D topographical pro®le of the joint
pdf's. The transition probabilities at various intervals
are represented by the peaks of the histograms. The
transition probability is zero except for the time intervals
outlined by the elliptical `ring' for both JISI and JISID
pdf's (Fig. 4C,D). This is consistent with the fact that
dierentiating a sine function gives a cosine function,
and that the sin e and cosine functions are essentially
sinusoidally modulating waveforms (bo th represented as
an ellipse in a phase plane). Thus, JISID analysis is
analogous to the `dierent iation' of the JISI analysis. It
is clear that the size of the `spread' of points about each
cluster simply re¯ects the proportional Gaussian vari-
ance used at each ®ring interval and does not necessarily
imply higher ®ring probabilities. The merging of the
clusters can be considered more an advantage than a
disadvantage because the joint pdf with a merged ellip-
tical outline is characteristic of sinusoidally modulated
®ring trends. This illustrates that cluster analysis is not
needed to identify and/or separate the individual clusters
in the JISI or JISID scatter plots.
4.3 Detection of transitional trends in burst ®ring
The JISID scatter plot for a bursting neuron, D
(Fig. 5A), reveals the burst-onset trend by cluster (d)
along the )x-axes; the intrabur st (constant) trend by
cluster (a); the burst-oset trend by cluster (b) along the
y-axis; and the transitional trend from end-of-burst to
beginning-of-next-burst by cluster (c) along the antidi-
agonal ()45° line). These precise ®ring trends (burst-
onset, burst-oset, and transitional trends) are clearly
revealed by the speci®c quadrants and axes in which the
clusters are located in the JISID plot. The symmetry
along the antidiagonal implies that this neuron enters
into a burst-®ring sequence the same way as it termi-
nates its burst-®ring.
Comparing the JISI plot (Fig. 5B) with the JISID
scatter plot (Fig. 5A), incidentally, there are more clus-
ters found in the latter (i.e., four clusters vs three). This
is because there are more types of changes in ®ring than
the types of ®ring patterns for this neuron. It is also
important to recognize the correspondence between the
trajectory lines in a serial-JISI plot (Fig. 5D) and the
clusters in a JISID scatter plot (Fig. 5A). For example,
the vertical trajectory from (a) to (b) in the serial-JISI
plot (Fig. 5D) represents the same 4-spike trend as the
cluster (b) in the JISID scatter plot (Fig. 5A). The trend
unidenti®ed in the JISI plot is the constant trend found
explicitly in cluster (a) in the JISID scatter plot
(Fig. 5A). This degeneracy is due to the fact that no
distinct trajectory lines are drawn when the spike pat-
terns remain relatively constant within cluster (a) in the
JISI plot (Fig. 5D).
Finally, the serial-JISID plot (Fig. 5C) can provide
an additional level of description to the JISID plot
similarly to the serial-JISI plot. By tracing sequential
points, the serial-JISID plot reveals the sequence in
which the 4-spike trends are changing. Thus, 5-spike
trends can be recovered from the trajectory trace in the
serial-JISID plot, as indicated by the 5-spike insets in the
®gure. It is immediately evident that an outline of a
`southeasterly pointing arrowhead' characterizes the
four dierent 5-spike trends revealed by the serial-JISID
analysis. This arrowhead trajectory is characteristic of
burst-®ring trends traced in a clockwise fashion as
shown in Fig. 5C. It will be shown below that this ar-
rowhead trajectory is also a characteristic of other ir-
regular, nonperiodic burs t-patterns.
4.4 Detection of transitional trends
in arrhythmic pacemaker ®ring
Next, we will use our JISID analysis to determine the
trends characteristic of a pacemaker neuron, F, that
occasionally skips periods (25% failure probability).
316

Fig. 4. JISID (A) and JISI (B) scatter plots for the sinusoidally modulated ®ring neuron D. Three-dimensional JISID (C) and JISI (D)
histograms show the joint (transition) probabilities of 4-spike ®ring trends and 3-spike ®ring patterns, respectively
317

[This arrhythmicity is sometimes called `skipping' by
Longtin and Hinter (1996) and Segundo et al. (1991).] A
central cluster at the origin in the JISID plot (Fig. 6A)
indicates a constant pacemaking trend. The other
secondary outlying clusters at integral multiples of ISIDs
reveal the various trends resulting from the skipped
Fig. 5. JISID (A) and JISI (B) scatter plots for the burst ®ring neuron E. Serial-JISID (C) and serial-JISI (D) scatter plots for a short segment of
the bursting data. [Arrows shown in the pre-ISI and post-ISI histograms (in B) indicate that the amplitude of the histogram bar extends beyond
the current scale of the graph] [Arrows shown in the serial-JISID and serial JISI plots (in C and D) indicate the clockwise direction of sequential
evolution of points]
318

periods. This characterizes the trends for an arrhythmic
pacemaker neuron since there is a predominance of a
fundamental ISI with only sporadic `drop-outs' in ®ring,
creating other sporadically changing trend s. Another
interesting feature is the transi tional trend characteristic
of a burst-like ®ring neuron uncovered by the clusters
lying within a `southeasterly pointing arrowhead' for this
arrhythmic pacemaker neuron F (cf. Figs. 5A and 6A).
Figure 6C,D shows the JISID and JISI scatter plots for
a biological neuron, X. The clusters of points for this
Fig. 6. JISID (A) and JISI (B) scatter plots for the arrhythmic pacemaker neuron F. JISID (C) and JISI (D) scatter plots for biological neuron X
319

neuron are similar to neuron F (cf. Fig. 6A,C). This shows
that the JISID analysis does indeed reveal the transitional
trends that are characteristic of an arrhythmic pacemaker
neuron, not only in a simulated neuron but also in a real
biological neuron, without any a priori knowledge of the
®ring trends regarding rhythmicity.
4.5 Detection of transitional trends
in alternating biperiodic ®ring
A biperiodic neuron, G, is simulated to alternate
between two Gaussian-varied periods with occasional
drop-outs (25% failure rate) in ®ring. It is immediately
evident from the JISID plot (Fig. 7A) that the majority
of points fall within two clusters in quadrants 2 and 4,
i.e., the long-short-long transition and the short-long-
short transition quadrants. This suggests that the neuron
tends to ®re with transitional trends most of the time,
revealing the two alternating trends. The secondary
clusters seen at the origin and on the x-and)y-axes are
a result of the neuron failing to follow strictly alternat-
ing ®ring, consistent with the simulation pa rameters.
[Note that this type of biperiodic ®ring trend is not
necessarily the same as `doublet' ®ring because a neuron
can ®re doublets periodically (as in this example) or it
can ®re doublets randomly (which is not biperiodic). A
dierent spike train analysis can be used to extract the
doublet ®rings characteristics in relation to temporal
integration (Tam 1998). It will be shown below (see Se ct.
4.7, Fig. 9) that this JISID analysis can actually be used
to distinguish spike trends that are repeated either
periodically or randomly .]
Comparing neurons E and G, it is obvious from the
ISI histograms (see Figs. 5B and 7B) that both neurons
®red primarily with a short period and occasionally a
long period, yet these histograms do not distinguish the
dierence between the burst ®ring (neuron E) and the
alternating biperiodic ®ring (neuron G), nor do they
distinguish the dierent transitional trends uncovered by
the JISID analyses. Furthermore, clusters located in the
`southeasterly pointing arrowhead', characteristic of the
bursting neuron E (Fig. 5A), are not found at the cor-
responding locations for neuron G (Fig. 7A) in the JI-
SID plots. The JISID analysis clearly distinguishes the
transitional trends between a bursting neuron and an
alternating biperiodic neuron. This illustrates that
higher-order interval serial correlation analyses, such as
the JISID analysis, are needed to quantify higher-order
®ring characteristics.
Compared wi th the cultured neuron, Y, in the same
network (Fig. 7C,D), clusters are found similar to neu-
ron G. Note that although these two neurons ®re with
spike patterns in dierent time-scales, the dierence in
time-scales of spike patterns (and mean ®ring rates) be-
tween these two neurons does not preclude the ability of
the JISID analysis to extract the biperiodic trends with
occasional drop-outs for both neurons. This demon-
strates the robustness of the JISID an alysis in identifying
local trends independent of the speci®c time-scale.
4.6 Characteristics of random ®ring
Next, we will examine the characteristics of a neuron, H,
®ring randomly with a Poisson process to provide the
baseline (control) pdf needed for comparison between a
neuron that ®res randomly from a neuron that does not.
The JISI scatter plot for this neuron (Fig. 8B) follows a
2D negative exponential distribution with dead-time in
both the s
n
and the s
n1
axis directions. This is consistent
with the fact that if a neuron ®res with an indepe ndent
process, the JISI histogram (or joint pdf) is equal to the
product of the pre-ISI and post-IS I histograms. This
analysis also reveals Poisson-like ®ring trends in another
spinal cord neuron, Z, (Fig. 8C ,D) recorded in the same
cultured network as neurons X and Y, even though we
included signi®cantly fewer spikes in the analyses for the
biological neurons.
The JISID analysis for this neuron (Fig. 8A) reveals
that most of the points are found to fall close to the
origin, indicating that most of the time there is little
change in consecutive ®ring intervals (refer to Fig. 2B).
It is important to point out that the distribution of
points does not resemble any negative exponential
pro®les symmetrical about the x- and y-axes, as if the
JISI plot (Fig. B) were reproduced in all four quad-
rants. Instead, the topographical pro®le of the JISID
plot outlines a south-easterly pointing arrowhead
(Fig. 8A). This is consistent with the fact that a Poisson
neuron's ®ring often resembles a `burstlike' pattern
since the probability of a short ISI occurrence is much
greater than a longer ISI occurrence (see ISI histogram
of Fig. 8B), producing a pseudo-burstlike pattern. Note
also that the refractory period is re¯ected in the ab-
sence of points close to the axes in the JISI plot, but no
such absence is found at the origin in the JISID plot
because the dead-time is theoretically `subtracted' in the
ISIDs.
This `arrowhead' distribution for a Poisson neuron
further illustrates that JISID analysis is not merely a
simple collapse of a 3D JISI plot (not shown) into a 2D
JISID plot. By de®nition, the joint pdf of a 3D JISI plot
should be equal to the product of three iden tical pdf's of
the ISI plots for an independent process. However, the
distribution in the JISID plot for a Poisson process
(Fig. 8A) does not resemble the product of the pre- and
post-ISID distributions (Fig. 8A) as if the pre- and post-
ISIDs are independent, nor does it resemble the sym-
metrical distribution about the x- and y-axes, as if the
JISI plot (Fig. 8B) were reproduced in all four quad-
rants. This is because the pre- and post-ISIDs are not
independent variables (as discussed in Sect. 1) by de®-
nition from (5) and (6). Thus, the JISID analysis is not
equivalent to a 3D JISI analysis projected onto a 2D
plane.
4.7 Detection of the same trends independent
of whether they repeat randomly or periodically
Lastly, we will illustrate the ability of the JISID analysis
to detect the same ®ring trends independent of whether
320

they repeat randomly or periodically. This is of partic-
ular importance in detecting and identifying any arbi-
trary spike patterns that may be encoded by neurons.
We simulated an arbitrary sequence of spikes represent-
ing a particular 4-spike pattern encoded by neurons I
and J. The Gaussian-modulated 4-spike pattern selected
is represented by the mean ISI 3-tuple (19, 31, 73 ms),
with prime numbers for the ISIs to avoid periodicity
Fig. 7. JISID (A) and JISI (B) scatter plots for the alternating biperiodic neuron G. JISID (C) and JISI (D) scatter plots for biological neuron Y
321

within the pattern itself. Both neurons I and J ®re with
this same 4-spike pattern, except that neuro n I repeats
this pattern periodically with a mean interburst period of
179 ms (another prime number), while neuron J repeats
this pattern randomly.
The JISID plots (Fig. 9A,C) for these neurons clearly
show a centrally located cluster (a) that corresponds to
the trend of the 4-spike burst pattern for both neurons as
predicted. The outlying clusters (b), (c), and (d) in
Fig. 9A represent the other repeated trends trailing and
Fig. 8. JISID (A) and JISI (B) scatter plots for the Poisson neuron H. JISID (C) and JISI (D) scatter plots for biological neuron Z
322

leading to the burst pattern, consistent with the fact that
neuron I repeats this pattern periodically. Compared
with Fig. 9C, three diuse bands of points are found
surrounding the central cluster (a) in Fig. 9C, indicative
of the variety of trends leading to or trailing this re-
peated 4-spike pattern. It is important to notice that
points are not distributed uniformly throughout the JI-
SID plot (Fig. 9C) nor dispersed with a distribution
Fig. 9. JISID (A) and JISI (B) scatter plots for neuron I ®ring with a periodically repeated 4-spike burst pattern. JISID (C) and JISI (D) scatter
plots for neuron J ®ring with the same 4-spike pattern as neuron I except that the patterns are repeated randomly
323

similar to that of the Poisson neuron H (see Fig. 8A),
even though neuron J randomly repeats the 4-spike
pattern. Most importantly, based on the JISID plot, we
can infer that it ®res most often with a single repeated
trend as indicated by the central peak cluster (a), and
that this spike sequence repeats randomly as indicated
by the radiating bands, unlike neuron and (cf. Figs. 8A
and 9A).
Lastly, clusters (and bands) of points are often found
to `cross' the quadrant boundaries. For instance, cluster
(a), representing the 4-spike pattern (19, 31, 73 ms),
crosses the boundary of quadrants 1 and 2 for neurons I
and J (Figs. 9A,C). Based on the theoretical analyses
described above (refer to Fig. 2B), this `switching' in
trend actually represents a `gradual ' transition from one
type of trend to another, rather than an `abrupt' tran-
sition . In fact, if the Gaussian variance of the ISIs for
neuron I is reduced, the clusters would not cross the
quadrant boundaries. This demonstrates that quadrant
boundary crossing is not necessarily indicative of abrupt
switching or chaos, because the underlying spike gen-
eration process is still the same.
5 Discussion
We have demonstrated that the JISID analysis intro-
duced in this pa per can extract local trends in ®ring both
theoretically and experimentally. This method is shown
to extract local ®ring trends de®ned by sequential
changes in ISIs. Th is local trend provides a description
of the serial dependence of spike ®ring. We have
extended the traditional JISI serial dependence analysis
to include changes within these intervals. Using this
JISID analysis allows us to capture dierent incremental
changes in spike patterns that have similar trends. The
local trend, as de®ned and discussed in this paper,
characterizes the spike patterns that include fewer than
six spikes.
One of the advantages of this analysis is the graphical
identi®cation of the trends by the quadrants. The nature
of the trend, representing how the spike ®rings are
changing, is explicitly represented in the graph. Speci®c
trends are identi®ed by the location of points in a
quadrant, while others are revealed by the proxi mity of
points to the axis or origin.
Another advantage of this analysis is that an addi-
tional order of the prior history is captured implicitly.
Although this analysis is not the only correlation anal-
ysis that correlates higher-order ISIs, it does dier fun-
damentally in many respects. First, while most other
correlation analyses characterize the dependence rela-
tionship in terms of the spike occurrence intervals, the
JISID analysis describes the changes in these time in-
tervals. The changes in the past history that aect the
spike ®ring are captured by two consecutive ISIDs (i.e.,
three consecutive ISIs). Second, while autocorrelation
recurrence analysis describes all higher-order ISIs (Ka-
luzny and Tarnecki 1993; Shelhamer 1997), JISID
analysis extracts only the lower-order ISIs. We speci®-
cally select the lowest-order ISIs for the analysis because
we want to ®nd out whether the lowest-order ISIs are
sucient to quantify the sequential changes in spike
®ring. Furthermore, the pdf's of the lower-order ISIs are
often `buried' by the pdf's of the higher-order ISIs in
autocorrelograms when all orders of ISIs are plotted on
the same graph. Finally, this analysis is dierent from
the one that displays the jth order ISI dierence at each
nth spike relative to the shifted n  jth spike (Segundo
et al. 1995).
Although this JISID analysis plots the serial rela-
tionships of the lower-order ISIs, the relationship in the
next higher order can also be revealed by connecting
sequential points in the JISID plot as is traditionally
done with other return map analyses. The resulting
`trajectory' re¯ects the evolution of trends and identi®es
the sequential trends by tracing the dierent quadrants
the points traverse (i.e., it shows how trends can change
from one type to another). A higher-order serial de-
pendence, if it exists, can be deduced by these trajectory
plots.
Another advantage is the reduction of dimensions
(from 3D to 2D) needed to represent these sequential
changes or ®ring trends, or a reduction from the 3-tuple
s
nÿ1
; s
n
; s
n1
 to a 2-tuple Ds
n
; Ds
n1
 representation.
While a 3D JISI return map has been used to represent
the 3-tuple, this JISID analysis re-maps the same in-
formation onto a 2D representation dierently. As
noted in Sect. 4.6, such a mapping is not a simple, direct
topographical projection from a 3D space onto a 2D
plane.
An added advantage is that this 2-tuple measure
(JISID pair) quanti®ed by the JISID analysis represents
relative changes explicitly (regardless of the time-scale of
the ISIs). Since a single point in the JIS ID plot can map
into multiple points in the 3D JISI return map, a unique
trend identi®ed by the JISID analysis cannot be equiv-
alently represented by the 3D JISI analysis. The inter-
pretation is that a unique trend identi®ed by a point in
the JISID analysis corresponds to a set of similar in-
crementally changing ®ring patterns, s
nÿ1
 k; s
n
 k;
s
n1
 k, in the 3D JISI analysis. Most importantly, the
classi®cation of similar patterns into the same trend al-
lows us to compare changes in patterns that are in dif-
ferent time-scales. This time-invariant method of
detection of spike trends is dierent from the other time-
invariant spike trend detection methods using a spike
index employed by Tam (1996).
As a consequence of this many-to-one mapping from
a 3-tuple to a 2-tuple representation (or equivalently,
from 3D to 2D), multiple spike patterns are mapped into
the same trend. This degeneracy precludes identi®cation
of the exact spike patterns that correspond to the same
trend. However, this information representing the orig-
inal ISI pattern can be recovered with a related analysis
called `®rst order ISID phase plane analysis' (Fitzurka
and Tam 1996a), where the speci®c relation between the
ISIs and ISIDs are retrieved by plotting a given ISID
with respect to its associated ISIs. Furthermore, another
higher-order serial dependence can be revealed by the
`second-order ISID phase plane analysis' (Fitzurka and
Tam 1996b).
324

Finally, the trends revealed within the spike patterns
by this JISID analysis are not limited to periodi c or
quasi-periodic, regular or irregular, chaotic or noncha-
otic patterns. Any 4-spike pattern, in any combination
or permutation, whether periodic or random, can be
classi®ed into speci®c trends because these trends are
identi®ed simply by the quadrant in which the point is
located. The most signi®cant contribution of this anal-
ysis is that it can be applied universally to classify any
type of 4-spike ®ring trend regardless of its periodicity or
randomness of the trends and patterns, because no a
priori assumptions were made about the linearity or
nonlinearity of the underlying process that generates the
particular spike patterns.
Acknowledgements. This research was supported by the Oce of
Naval Research (ONR grant numbers N00014-93-1-0135 and
N00014-94-1-0686) and the Faculty Research Grant from the
University of North Texas to D.C.T.
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