The time-rescaling theorem and its application to neural spike train data analysis.

Abstract

Measuring agreement between a statistical model and a spike train data series, that is, evaluating goodness of fit, is crucial for establishing the model's validity prior to using it to make inferences about a particular neural system. Assessing goodness-of-fit is a challenging problem for point process neural spike train models, especially for histogram-based models such as perstimulus time histograms (PSTH) and rate functions estimated by spike train smoothing. The time-rescaling theorem is a well-known result in probability theory, which states that any point process with an integrable conditional intensity function may be transformed into a Poisson process with unit rate. We describe how the theorem may be used to develop goodness-of-fit tests for both parametric and histogram-based point process models of neural spike trains. We apply these tests in two examples: a comparison of PSTH, inhomogeneous Poisson, and inhomogeneous Markov interval models of neural spike trains from the supplementary eye field of a macque monkey and a comparison of temporal and spatial smoothers, inhomogeneous Poisson, inhomogeneous gamma, and inhomogeneous inverse gaussian models of rat hippocampal place cell spiking activity. To help make the logic behind the time-rescaling theorem more accessible to researchers in neuroscience, we present a proof using only elementary probability theory arguments. We also show how the theorem may be used to simulate a general point process model of a spike train. Our paradigm makes it possible to compare parametric and histogram-based neural spike train models directly. These results suggest that the time-rescaling theorem can be a valuable tool for neural spike train data analysis.

Full text

Page 1

NOTE
Communicated by Jonathan Victor
The Time-Rescaling Theorem and Its Application to Neural
Spike Train Data Analysis
Emery N. Brown
brown@srlb.mgh.harvard.edu
Neuroscience Statistics Research Laboratory, Department of Anesthesia and Critical
Care, Massachusetts General Hospital, Boston, MA 02114, U.S.A., and Division of
Health Sciences and Technology, Harvard Medical School/Massachusetts Institute of
Technology,Cambridge, MA 02139,U.S.A.
Riccardo Barbieri
barbieri@srlb.mgh.harvard.edu
Neuroscience Statistics Research Laboratory, Department of Anesthesia and Critical
Care, MassachusettsGeneral Hospital,Boston, MA 02114,U.S.A.
Val erie Ventura
vventura@stat.cmv.edu
Robert E. Kass
kass@stat.cmu.edu
Department of Statistics, Carnegie Mellon University, Center for the Neural Basis of
Cognition, Pittsburg,PA 15213,U.S.A.
Loren M. Frank
loren@neurostat.mgh.harvard.edu
Neuroscience Statistics Research Laboratory, Department of Anesthesia and Critical
Care, MassachusettsGeneral Hospital,Boston, MA 02114,U.S.A.
Measuring agreement between a statistical model and a spike train data
series, that is, evaluating goodness of ?t, is crucial for establishing the
model’s validity prior to using it to make inferences about a particular
neural system. Assessing goodness-of-?t is a challenging problem for
point process neural spike train models, especially for histogram-based
models such as perstimulus time histograms (PSTH) and rate functions
estimated by spike trainsmoothing. The time-rescalingtheorem is a well-
known result in probability theory, which states that any point process
withanintegrableconditionalintensityfunctionmaybetransformedinto
a Poisson process with unit rate. We describe how the theorem may be
used to develop goodness-of-?t tests for both parametric and histogram-
based point process models of neural spike trains.We apply these testsin
two examples:a comparison of PSTH, inhomogeneous Poisson,and inho-
mogeneous Markov interval models of neural spike trains from the sup-
Neural Computation 14, 325–346 (2001)
c °2001 Massachusetts Institute of Technology

Page 2

326E. N. Brown, R. Barbieri, V. Ventura, R. E. Kass, and L. M. Frank
plementary eye ?eld of a macque monkey and a comparison of temporal
and spatialsmoothers, inhomogeneous Poisson, inhomogeneous gamma,
and inhomogeneous inverse gaussian models of rat hippocampal place
cellspiking activity.To helpmake the logic behind the time-rescalingthe-
orem more accessible to researchers in neuroscience, we present a proof
using only elementary probability theory arguments. We also show how
the theorem may be used to simulate a general point process model of a
spike train. Our paradigm makes it possible to compare parametric and
histogram-based neural spike train models directly. These results sug-
gest that the time-rescaling theorem can be a valuable tool for neural
spike train data analysis.
1 Introduction
Thedevelopmentofstatisticalmodelsthataccuratelydescribethestochastic
structure of neural spike trains is a growing area of quantitative research in
neuroscience. Evaluating model goodness of ?t—that is, measuring quan-
titatively the agreement between a proposed model and a spike train data
series—is crucial for establishing the model’s validity prior to using it to
makeinferencesabout theneural system being studied.Assessing goodness
of?tforpointneural spike train modelsisamorechallenging problemthan
for models ofcontinuous-valued processes. This is because typical distance
discrepancymeasuresappliedincontinuousdataanalyses, suchastheaver-
age sum ofsquared deviations between recordeddata values and estimated
values from the model, cannot be directlycomputed for pointprocess data.
Goodness-of-?tassessments areevenmorechallengingforhistogram-based
models, such as peristimulus time histograms (PSTH) and rate functions
estimated by spike train smoothing, because the probability assumptions
needed to evaluate model properties are often implicit. Berman (1983) and
Ogata (1988) developed transformations that, under a given model,convert
pointprocesseslikespike trains into continuous measures in orderto assess
model goodness of ?t. One of the theoretical results used to construct these
transformations is the time-rescaling theorem.
A form of the time-rescaling theorem is well known in elementary prob-
ability theory. It states that any inhomogeneous Poisson process may be
rescaledortransformedintoahomogeneousPoissonprocesswithaunitrate
(Taylor&Karlin,1994). Theinversetransformationisastandard methodfor
simulating an inhomogeneous Poisson process from a constant rate (homo-
geneous) Poisson process. Meyer (1969) and Papangelou (1972) established
thegeneral time-rescalingtheorem,whichstates that any pointprocesswith
an integrable rate function may be rescaled into a Poisson process with a
unit rate. Berman and Ogata derived their transformations by applying the
general form of the theorem. While the more general time-rescaling theo-
rem is well known among researchers in point process theory (Br emaud,
1981;Jacobsen, 1982;Daley & Vere-Jones, 1988;Ogata, 1988; Karr, 1991), the

Page 3

The Time-Rescaling Theorem327
theorem is less familiar to neuroscience researchers. The technical nature of
the proof, which relies on the martingale representation of a point process,
may have prevented its signi?cance from being more broadly appreciated.
The time-rescaling theorem has important theoretical and practical im-
plicationsfor application ofpoint process models in neural spike train data
analysis. To help make this result more accessible to researchers in neu-
roscience, we present a proof that uses only elementary probability theory
arguments. Wedescribehowthetheorem maybeusedtodevelopgoodness-
of-?t tests for both parametric and histogram-based point process models
of neural spike trains. We apply these tests in two examples: a compar-
ison of PSTH, inhomogeneous Poisson, and inhomogeneous Markov in-
terval models of neural spike trains from the supplementary eye ?eld of
a macque monkey and a comparison of temporal and spatial smoothers,
inhomogeneous Poisson, inhomogeneous gamma, and inhomogeneous in-
verse gaussian models of rat hippocampal place cell spiking activity. We
also demonstrate how the time-rescaling theorem may be used to simulate
a general point process.
2 Theory
2.1 The Conditional Intensity Function and the Joint Probability Den-
sity of the Spike Train. De?ne an interval (0, T], and let 0 < u1 < u2 <
, .. ., < un¡1 < un· T be a set of event (spike) times from a point process.
Fort 2 (0, T], let N(t) be the sample path ofthe associated countingprocess.
The sample path is a right continuous function that jumps 1 at the event
times and is constant otherwise (Snyder & Miller, 1991). In this way, N(t)
counts the number and location of spikes in the interval (0, t]. Therefore, it
contains all the information in the sequence of events or spike times. For
t 2 (0, T], we de?ne the conditional or stochastic intensity function as
Pr(N(t C Dt) ¡N(t) D 1 | Ht)
Dt
where HtD f0 < u1 < u2, .. ., uN(t) < tg is the history of the process up to
timetand uN(t)isthe timeofthe last spikepriorto t.Ifthe pointprocessisan
inhomogeneous Poisson process, then l(t | Ht) D l(t) is simplythe Poisson
rate function. Otherwise, it depends on the history of the process. Hence,
the conditional intensity generalizes the de?nition of the Poisson rate. It is
well known that l(t | Ht) can be de?ned in terms of the event (spike) time
probability density, f(t | Ht), as
l(t | Ht) D lim
Dt!0
,
(2.1)
l(t | Ht) D
f (t | Ht)
uN(t)f(u | Ht) du
1 ¡Rt
,
(2.2)
for t > uN(t) (Daley & Vere-Jones, 1988; Barbieri, Quirk, Frank, Wilson, &
Brown, 2001). We may gain insight into equation 2.2 and the meaning of the

Page 4

328E. N. Brown, R. Barbieri, V. Ventura, R. E. Kass, and L. M. Frank
conditional intensity function by computing explicitly the probability that
given Ht, a spike ukoccurs in [t, t C Dt) where k D N(t) C 1. To do this, we
note that the events fN(t C Dt) ¡ N(t) D 1 | Htg and fuk2 [t, t C Dt) | Htg
are equivalent and that therefore,
Pr(N(t C Dt) ¡N(t) D 1 | Ht) D Pr(uk2 [t, t C Dt) | uk> t, Ht)
DPr(uk2 [t, t C Dt) | Ht)
Pr(uk> t | Ht)
RtCDt
f(t | Ht)Dt
1 ¡Rt
D l(t | Ht)Dt.
D
t
f(u | Ht) du
uN(t)f (u | Ht) du
1 ¡Rt
¼
uN(t)f (u | Ht) du
(2.3)
Dividing by Dt and taking the limit gives
lim
Dt!0
Pr(uk2 [t, t C Dt) | Ht)
Dt
D
f(t | Ht)
uN(t)f (u | Ht) du
D l(t | Ht),
1 ¡Rt
(2.4)
which is equation 2.1. Therefore, l(t | Ht)Dt is the probability of a spike in
[t, t C Dt) when there is history dependence in the spike train. In survival
analysis, the conditional intensity is termed the hazard function because in
thiscase,l(t | Ht)Dtmeasuresthe probabilityofafailureordeathin[t, tCDt)
given that the process has survived up to time t (Kalb?eisch & Prentice,
1980).
Because we would like to applythe time-rescaling theorem to spiketrain
data series, we require the joint probability density of exactly n event times
in (0, T].Thisjointprobabilitydensity is(Daley &Vere-Jones, 1988;Barbieri,
Quirk, et al., 2001)
f(u1, u2, . .., un\ N(T) D n)
D f (u1, u2, ... , un\ unC1> T)
D f (u1, u2, ... , un\ N(un) D n) Pr(unC1> T | u1, u2, . .. , un)
n Y
(
un
D
kD1
l(uk| Huk) exp
?
¡
Zuk
uk¡1
l(u | Hu) du
¼
¢ exp
¡
ZT
l(u | Hu) du
)
,
(2.5)

Page 5

The Time-Rescaling Theorem329
where
f(u1, u2, . .. , un\ N(un) D n)
n Y
D
kD1
l(uk| Huk) exp
?
¡
Zuk
uk¡1
l(u | Hu) du
¼
(2.6)
Pr(unC1> T | u1, u2, . .. , un) D exp
(
¡
ZT
un
l(u | Hu) du
)
,
(2.7)
and u0D 0. Equation 2.6 is the joint probability density of exactly n events
in (0, un], whereas equation 2.7 is the probability that the n C 1st event
occurs after T. The conditional intensity function provides a succinct way
to represent the joint probability density of the spike times. We can now
state and prove the time-rescaling theorem.
Time-Rescaling Theorem.
from a point process with a conditional intensity function l(t | Ht) satisfying
0 < l(t | Ht) for all t 2 (0, T]. De?ne the transformation
Zuk
for k D 1, .. ., n, and assume L (t) < 1 with probability one for all t 2 (0, T].
Then the L (uk)’s are a Poisson process with unit rate.
Let 0 < u1< u2<, ... , < un< T bearealization
L (uk) D
0
l(u | Hu) du,
(2.8)
Proof.
RT
with mean one. Because the tktransformation is one-to-one and tnC1 > tT
if and only if unC1> T, the joint probability density of the tk’s is
f(t1, t2, .. ., tn\ tnC1> tT)
D f(t1, .. ., tn) Pr(tnC1> tT| t1, .. ., tn).
We evaluate each of the two terms on the right side of equation 2.9. The
following two events are equivalent:
Let tk
D
L (uk) ¡ L (uk¡1) for k
D
1, .. ., n and set tT
D
unl(u | Hu) du. To establish the result, it suf?ces to show that the tks
are independent and identically distributed exponential random variables
(2.9)
ftnC1 > tT| t1, ... , tng D funC1 > T | u1, u2, . .., ung.
Hence
(2.10)
Pr(tnC1> tT| t1, t2, . .. , tn) D Pr(unC1> T | u1, u2, ... , un)
D exp
(
¡
ZT
un
l(u | Hun) du
)
D expf¡tTg,
(2.11)