BioSystems 58 2000 27 – 32
Stochastic model of the overdispersion in the place cell discharge
Petr La´nsky´
a,
, Jean Vaillant
b
a
Institute of Physiology, Academy of Sciences of the Czech Republic, Videnska
1083
,
142 20
Prague
4
, Czech Republic
b
Department of Mathematics and Computer Sciences, Uni6ersity of Antilles-Guyane,
97159
Pointe-a-Pitre Guadeloupe
, France
Abstract
The spontaneous firing activity of the place cells reflects the position of an experimental animal in its arena. The firing rate is high inside a part of the arena, called the firing field, and low outside. It is generally accepted concept
that this is the way in which the hippocampus stores a map of the environment. This well known fact was recently reinvestigated [Fenton, A.A., Muller, R.U., 1998. Proc. Natl. Acad. Sci. USA 95, 3182 – 3187] and it was found that
while the activity was highly reliable in position, it did not retain the same reliability in time. The number of action potentials fired during different passes through the firing field were substantially different overdispersion. We present
a mathematical model based on a doubly stochastic Poisson process which is able to reproduce the experimental findings. Further, it enables us to propose specific statistical inference on the experiments in aim to verify data and
model compatibility. The model permits to speculate about the neural mechanisms leading to the overdispersion in the activity of the hippocampal place cells. Namely, the statistical variation of the intensity of firing can be achieved,
for example, by introducing a hierarchical structure into the local neural network. © 2000 Elsevier Science Ireland Ltd. All rights reserved.
Keywords
:
Doubly stochastic Poisson process; Hippocampus; Overdispersion; Place cell; Statistical inference on spiking data www.elsevier.comlocatebiosystems
1. Introduction
Importance of the hippocampus for solving difficult spatial problems and for the ability to
explore an environment is well known O’Keefe and Nadel, 1978. The rodents create a map-like
representation of the surrounding within their hippocampus. This internal model of the external
world is formed, at least in part, by hippocampal pyramidal cells, called ‘place cells’ which are char-
acterized by
location-specific firing.
When recorded during free exploration, the activity of
place cells is higher within a small part of the available area, called the cell’s firing field, and
substantially lower elsewhere. It was shown re- cently by Fenton and Muller 1998 that the spik-
ing activity during passes through the firing field is characterized not only by the high firing rate,
but also by its very high variability, which is even higher than that of a Poisson process. The au-
thors deduced from this experimental observation,
Corresponding author. Tel.: + 420-2-475-2585; fax: + 420-2-475-2488.
E-mail address
:
lanskybiomed.cas.cz P. La´nsky´. 0303-264700 - see front matter © 2000 Elsevier Science Ireland Ltd. All rights reserved.
PII: S 0 3 0 3 - 2 6 4 7 0 0 0 0 1 0 3 - 9
beside other conclusions, that the place cell dis- charge is not merely driven by the summation of
many small and asynchronous excitatory synaptic inputs and that not only the location but some
additional information may be coded by the firing of the place cells.
Since the only way a neuron can transmit infor- mation about rapidly varying signals over a long
distance is by a series of all or none events, the shape of action potentials spikes is considered to
be irrelevant. An action potential is taken in the limit as a Dirac delta function and a spike train of
these pulses may be seen as a realization of stochastic point process. Detailed justification for
such a representation is given in Johnson 1996. The simplest stochastic point process model is a
Poisson process, which is a memoryless process with fixed statistical properties time-homoge-
neous Poisson process. Spontaneous firing of a single neuron with a low activity, for units of very
different types, can be, at the first approximation e.g. La´nsky´ and Radil, 1987; Rospars et al.,
1994, described by the Poisson process. Its con- stant intensity is estimated by the mean firing rate.
This mean firing frequency, derived from the mean interspike interval, is often considered to be
a fundamental parameter in experimental studies on neuronal activity, even if Poissonian character
is not questioned, and consequently also in theo- retical approaches to the description of neurons.
However, it is intuitively clear that for time-de- pendent effects a dynamical descriptor is needed
to replace the mean firing rate. A time-dependent intensity of firing time-nonhomogeneous Poisson
process is a natural extension for the constant mean firing rate and has been used under very
different conditions e.g. Geisler et al., 1991; Gummer, 1991; Shimokawa et al., 1999. Further,
in the cases when the time-dependence is intrinsic and controlled by nondeterministic mechanisms, a
doubly stochastic Poisson process is the appropri- ate description e.g. Kano and Shigenaga, 1982;
Lowen and Teich, 1991; Teich et al., 1997, how- ever, models of this type have been used mainly to
describe the stimulated neuron Vaillant and La´n- sky´, 2000.
Redish and Touretzky 1998 present a general model describing the memory formation within
hippocampus and their paper contains an exten- sive review of the literature on cognitive map
formation. Blum and Abbott 1996 investigate how the spatial map of the environment can be
created. In their model, the sequential firing of place cells during the exploration induces a pat-
tern of long-term potentiation, behaviorally gen- erated modifications of synaptic strength, which
are used to affect the subsequent behavior. In none of these papers the problem of overdisper-
sion is regarded.
In this article, we propose a model which can mimic the experimentally observed overdispersion
of the firing activity: A doubly stochastic Poisson process with a family of stochastic intensities con-
trolled by the position of the animal. At each point of the experimental arena, the firing is gen-
erated in accordance with the doubly stochastic Poisson process. The animal moves and thus,
instead of a single doubly stochastic Poisson pro- cess, we have to consider a class of these processes
if the movement is random, then we have second- order stochasticity and the actual one is deter-
mined by the current position of the animal. This new model is descriptive, however, it permits to
speculate about the role of local neural network of hippocampus. In addition, it allows to devise
new methods for the statistical inference of exper- imental data. Similarly to the existing analysis of
the available experiments, the model implicitly assumes that the positional information is coded
by the mean firing rate Adrian, 1928; Rieke et al., 1997. Weather some other information, in addi-
tion to the position, is coded by the firing of the place cells, as has been suggested by Fenton and
Muller 1998, can be deduced from the model, but only after a complete specification of the
intensity of the doubly stochastic Poisson process is accomplished.
2. The model
An experimental animal moves inside the field with a constant unit speed. This assumption
would be easy to release to account for existing stops and speed changes in the real movement,
but its existence substantially simplifies the nota-
tion. A trajectory P = {P
t
; t ] 0} of the move- ment can be defined as a time function a stochas-
tic process giving the position x at time t; x X ¦ R
2
, where X is a subset of R
2
which is attainable usually a circular arena by the animal
during the experiment. We may well presume that P
is a continuous function of t on X. The doubly stochastic Poisson process describ-
ing the activity of a single space cell is defined in the following way: let us have a family with
respect to x of time-continuous stochastic pro- cesses L={Lx, t, xX, t]0} — a random
field. The firing at location x
1
= x
11
, x
21
at time t
1
has a Poissonian character with intensity Lx
1
, t
1
. Thus, for any subset B P, where P denotes the set of observable Borel subsets of X, and any
path P, the mean number of spikes fired in B conditional on L and the path P is
ENB L, P=
T
I
B S P
t
xLx, t dt, 2.1
where I is the indicator of the position at time t all the quantities studied in the text are condi-
tioned with respect to the path P, and thus the condition is further omitted throughout the text.
To obtain the unconditional mean ENB, the mean with respect to L of the integral in Eq. 2.1
has to be calculated. Then, the mean number of spikes during the time period [0, T is
ENB =
T
I
B S P
t
xELx, tdt, 2.2
where ELx, t is the mean of the intensity process. The formula 2.2 is a generalized with
respect to the movement of the animal version of the well known formula for the mean number of
events in a non-homogeneous Poisson process, EN
T
= R
T
l tdt Johnson, 1996.
Due to the Poissonian character of the firing, the conditional mean and variance of the number
of spikes have to be the same, VarNB
L=ENBL. 2.3
Substituting Eq. 2.3 into well known formula e.g. Rao, 1968 relating conditional and uncondi-
tional variance, we obtain VarNB = EVarNB
L+VarENBL. 2.4
Finally, for the unconditional variance yields VarNB = ENB
+ Var
T
I
B S P
t
xLx, tdt .
2.5 The second term on the right hand side of Eq.
2.5 characterizes the overdispersion exceeding of variance over variance observed in homoge-
neous Poisson process caused by stochastic inten- sity in the doubly stochastic Poisson process This
second term on the right-hand side of Eq. 2.5 can be rewritten with respect to time-correlation
structure of L and then the overdispersion takes form,
VarNB − ENB =
T T
I
B S P
t
xI
B S P
t
x VarLx, tVarLx, trx, t, x, tdt dt,
2.6 where r is the autocorrelation function of the
process L. From formulas 2.5 or 2.6 the gen- eral form of overdispersion in our model can be
derived. Fenton and Muller 1998 quantified the experimental data by parameter analogous to a
Fano factor, F, defined as the event-number vari- ance divided by the event-number mean Fano,
1947. This function, independently of the sojourn time, is equal to one for the homogeneous Poisson
process. Using Eq. 2.5, we can see that
F = 1 + Var
T
I
B S P
t
xLx, tdt
T
I
B S P
t
xELx, t dt .
2.7 For any further specification of the overdispersion
some additional assumptions about the random intensities have to be made. The first one of them
is the stationarity of the random intensity Lx, t.
Due to the constant conditions during the ex- periment, we may assume that for any fixed x X,
the stochastic process Lx, ·, is a second order time stationary process for which we can write
ELx, t=m
x
B ,
2.8 where m
x
] 0 is a continuous function on X which
ensures its integrability. Let us denote VarLx, t=s
x 2
B 2.9
and the correlation function CorrLx, t, Lx, t=fx, x,
t−t. 2.10
Substituting Eq. 2.8 into Eq. 2.1, the mean of the number of counts is
ENB =
B
m
x T
I
P
t
xdt dx, 2.11
and if m
x
= m for all x B, we have ENB =
t Bm, where tB = R
B
R
T
I
P
t
xdt dx is time spent in B during interval [0, T], so for an unin-
terrupted stay in B holds ENB = Tm. Further, Var
T
I
B S P
t
xLx, tdt =
T T
I
B S P
t
xI
B S P
t
xs
x
s
x
f x, x,
t−t dt dt.
2.12 Taking some additional assumptions about the
character of the random intensity Lx, t we can include into the model the biological properties of
the neurons. For example, the intensity L can be defined as a shot noise process, which is a simple
model of the postsynaptic membrane potential stimulated by a train of excitatory pulses. The
other alternative, also offering interpretation via neuronal membrane model, would be an assump-
tion that L is an Ornstein-Uhlenbeck stochastic diffusion process for details see La´nsky´ and Sato,
1999. This model based on the intensity con- trolled by stochastic diffusion process can serve as
an example on which the mechanisms leading to homogeneous and doubly stochastic Poisson pro-
cesses can be compared. If many asynchronous excitatory and inhibitory postsynaptic potentials
of relatively small effect impinge a neuron, then depolarization of its membrane can be described
by the Ornstein-Uhlenbeck process. If, in addi- tion, the neuronal depolarization is much below
the firing threshold, then the firing of such a model neuron is Poissonian with a fixed intensity
La´nsky´ and Sato, 1999. The model is based on the assumption that each of the inputs activity of
a source neuron is a renewal process with con- stant intensity. On the other hand, if the mean
activity of the source neurons is influenced by stochastic fluctuations, then firing of the target
model neuron is the doubly stochastic Poisson process. A simple example, described in more
details in Section 4, which illustrates this mecha- nism can be constructed in the following way. The
studied neuron has m excitatory inputs with inten- sities w
1
, … , w
m
and if all of them are active it fires in accordance with the Poisson process with
intensity l
1
w
1
+ · · · + w
m
. However, if a frac- tion of size k B m of the source excitatory neurons
is for periods of random lengths blocked by a strong inhibition, then in these intervals the stud-
ied neuron has only m − k excitatory inputs and thus it fires with intensity l
2
w
1
+ · · · + w
m − k
. If the periods of full and limited input are inde-
pendent random variables with a common expo- nential distribution, the activity of the target
neuron is the doubly stochastic Poisson process controlled by alternating Poisson process.
3. The parameters and their estimation
