Stochastic model of the overdispersion in the place cell
discharge
Petr La´nsky´
a,*, Jean Vaillant
baInstitute of Physiology,Academy of Sciences of the Czech Republic,Videnska1083,142 20Prague4,Czech Republic bDepartment of Mathematics and Computer Sciences,Uni6ersity of Antilles-Guyane,97159Pointe-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.com/locate/biosystems
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 durspik-ing passes through the firspik-ing 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:lansky@biomed.cas.cz (P. La´nsky´).
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). Sponta(time-homoge-neous 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 posiaddi-tion, 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
nota-tion. A trajectory P={Pt; t]0} of the move-ment can be defined as a time function (a
stochas-tic process) giving the position x at time t;
xX¦R2, where X is a subset of R2 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 sdescrib-ingle space cell is defined in the following way: let us have a family with
respect to x of time-continuous stochastic
pro-cesses L={L(x, t), xX, t]0} — a random
field. The firing at locationx1=(x11, x21) at time
t1 has a Poissonian character with intensityL(x1,
t1). Thus, for any subset BP, where P denotes
the set of observable (Borel) subsets ofX, and any
path P, the mean number of spikes fired in B
conditional onL and the path P is
E(N(B)L,P)=
&
T
0
IBSPt(x)L(x,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 E(N(B)), the
mean with respect toLof the integral in Eq. (2.1)
has to be calculated. Then, the mean number of
spikes during the time period [0, T) is
E(N(B))=
&
T
0
IBSPt(x)E(L(x,t))dt, (2.2)
where E(L(x, 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,
E(NT)=R0Tl(t)dt (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,
Var(N(B)L)=E(N(B)L). (2.3)
Substituting Eq. (2.3) into well known formula (e.g. Rao, 1968) relating conditional and uncondi-tional variance, we obtain
Var(N(B))=E(Var(N(B)L))+Var(E(N(B)L)).
(2.4) Finally, for the unconditional variance yields
Var(N(B))=E(N(B)) 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,
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+
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 intensityL(x,t).
Due to the constant conditions during the
ex-periment, we may assume that for any fixed xX,
the stochastic process L(x, ·), is a second order
E(L(x,t))=mxB , (2.8)
wheremx]0 is a continuous function onXwhich
ensures its integrability. Let us denote Var(L(x,t))=sx
2B (2.9)
and the correlation function
Corr(L(x,t),L(x%,t%))=f(x,x%,t−t%). (2.10)
Substituting Eq. (2.8) into Eq. (2.1), the mean of the number of counts is
E(N(B))=
&
Taking some additional assumptions about the
character of the random intensity L(x, t) we can
include into the model the biological properties of
the neurons. For example, the intensityL 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 hasmexcitatory inputs with
inten-sities w1, … ,wm and if all of them are active it
fires in accordance with the Poisson process with
intensity l1w1+· · ·+wm. However, if a
frac-tion of sizekBmof 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 l2w1+ · · ·+wm−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
The estimation of functions mx and sx
2 in the
above introduced version of the model, in which the location is a continuously changing variable, presents a hardly solvable problem. Therefore a discretization of the sample space has to be per-formed. This discretization has its counterpart in usual treatment of experimentally obtained data. There, the arena in which the animal moves is divided into discrete pixels in which the sojourn times and spike counts are recorded (Fenton and Muller, 1998). The observed data for a given box
i are formed by a sequence of couples
(ti1,ni1), … , (tik
i,niki), where tij is the duration of
j-th stay and nij is the number of spikes fired
during that stay (realization of a random variable
Nij) andki is the total number of visits in box i.
We also consider the entire spaceXdivided into
disjoint subspaces (boxes) denoted B1, … ,Bm,
such that the processesL(x, ·) can be replaced by
L1={L1}t]0, … ,Lm={Lm}t]0. It means that
the boxes have to be sufficiently small to permit
replacement of different L(x, ·) by a single
stochastic process Li, in other words, having
al-most identical behavior of the intensity function within a box. Simultaneously, the boxes have to be sufficiently large to obtain enough experimen-tal data permitting a reliable estimation of the parameters of the intensities. There are several possibilities to define the intensity processes for the box. For example, we may assume that each
of Liis the average random intensity of
underly-ing Poisson process in the box Bi,
Li(t)=
1
b(Bi)
&
BiL(x,t)dx, (3.1)
where b is the measure (Borel) of the boxBi.
The following assumption depends on the time-correlation structure of the intensity processes. If these processes are stationary and if the
correla-tion-time is short and/or the time periods spent in
a given box are separated by sufficiently long periods of absence, we can approximate the
be-havior of the processLiby a sequence of
indepen-dent and iindepen-dentically distributed random variables
Lij(j=1, …) wherej denotes thej-th visit of the
animal in the given box. Formally, the above assumptions can be written in the form
E(Lij)=mi, (3.2)
SinceNij(the counts of spikes in i-th box during
j-th visit) follow Poisson distribution with random
parameter Lijtij, Eqs. (3.2) and (3.3) imply
E(Nij)=mitij (3.5)
A more detailed knowledge of distributional
properties of Lij will provide us with a deeper
insight into the model. Without it only the method of moments for estimation of the parame-ters can be used, while knowing the distribution
of Lij a more efficient estimation procedure like
the maximum likelihood (ML) method can be applied.
4. A simple example
To illustrate the above derived results, we con-sidered only one box neglecting, at this stage, all the problems related to the random movement of the animal in the arena and its random sojourn times in different boxes. Also for the stochastic intensity we selected a very simple mechanism, an
alternating Poisson process with intensitymtaking
only two levels — high,l1, and low,l2. Therefore
the firings alternate in two Poissonian regimes
with intensity l1 or l2 each of them being
inde-pendently and exponentially distributed with
parameter m. A network model leading to this
kind of activity of the place cell is sketched in Section 2. To estimate the Fano factor for the
model neuron we divided the activity into m
equally long intervals of lengthTand counted the
spikes in them, n1, … ,nm. In equilibrium, for the
mean of the intensity process holds E(L)=(l1+
l2)/2 (we have to realize that high and low periods
have the same mean duration 1/m) and thus the
mean of the observed counts is
E(Ni)=l1+l2
2 T. (4.1)
After some calculations we can show that covari-ance function of the intensity process in the steady-state is
Cov(L(t),L(t%))=(l1−l2)
2
4 exp(−2mt−t%).
and for the Fano factor holds
We may assume that parameters T and m are
relatively stable, however, from Eq. (4.4) can be seen that, as expected, with increasing the
differ-ence between l1 and l2 the Fano factor can be
adjusted to any experimentally obtained value.
Let us finally illustrate the role ofTand mainly
of m in formula (4.4). If the period spent in the
box is large (T ), which in reality is a random
variable, then the Fano factor becomes
indepen-dent on T and tends to a constant value,
F1+ (l1−l2)
2
2(l1+l2)m
. (4.5)
On the other hand if the sojourn time in the box
is relatively very short (T ), then the Fano
factor tends to one and the activity is not distin-guishable from a Poisson process. The same effect
can be observed for fast alternating intensity L
(m ). Again, we have F1 and the process is
asymptotically Poissonian. For slowly alternating
intensity (m0), we have
in which the overdispersion still depends on the sojourn time. From the limiting behaviour with
respect tomandT, we can see the inverse role of
these parameters, namely apparent in formulas (4.5) and (4.6).
Acknowledgements
This work was supported by Academy of
Sci-ences Grant No. A7011712/1997, by Grant
MSMT VS96086, and by joint cooperation pro-ject Barrande between France and the Czech Re-public. The paper was completed during the visit of its first author at the Department of Mathe-matics and Computer Sciences of the University of Antilles — Guyane and he thanks for their warm hospitality.
References
Adrian, E.D., 1928. The Basis of Sensation: The Action of the Sense Organs. WW Norton, New York.
Blum, K.I., Abbott, L.F., 1996. A model of spatial map formation in the hippocampus of the rat. Neural Comput. 8, 85 – 93.
Fano, U., 1947. Ionization yield radiations. II. The fluctua-tions of the number of ions. Phys. Rev. 72, 26 – 29. Fenton, A.A., Muller, R.U., 1998. Place cell discharge is
extremely variable during individual passes of the rat through the firing field. Proc. Natl. Acad. Sci. USA 95, 3182 – 3187.
Geisler, W.S., Albrecht, D.G., Salvi, R.J., Saunders, S.S., 1991. Discrimination performance of single neurons: rate and temporal-pattern information. J. Neurophysiol. 66, 334 – 362.
Gummer, A.W., 1991. Probability density function of succes-sive intervals of a nonhomogenous Poisson process under low-frequency conditions. Biol. Cybern. 65, 23 – 30. Johnson, D.H., 1996. Point process models of single-neuron
discharges. J. Comput. Neurosci. 3, 275 – 300.
Kano, S., Shigenaga, S., 1982. Modeling of nervous systems doubly stochastic Poisson processes. Bull. Info. Cybern. 20, 42 – 54.
La´nsky´, P., Radil, T., 1987. Statistical inference on sponta-neous neuronal discharge patterns. Biol. Cybern. 55, 299 – 311.
La´nsky´, P., Sato, S., 1999. The stochastic diffusion models of nerve membrane depolarization and interspike interval generation. J. Peripheral Nerv. Syst. 4, 27 – 42.
Lowen, S.B., Teich, M.C., 1991. Doubly stochastic Poisson point process driven by fractal shot noise. Phys. Rev. A 43, 4192 – 4215.
O’Keefe, J., Nadel, L., 1978. The Hippocampus as a Cognitive Map. Clarendon Press, Oxford.
Rao, C.R., 1968. Linear Statistical Inference and Its Applica-tions. Wiley, New York.
Rieke, F., Warland, D., de Ruyter van Steveninck, R.R., Bialek, W., 1997. Spikes: Exploring the Neural Code. MIT Press, Cambridge.
Redish, A.D., Touretzky, D.S., 1998. The role of the hippocampus in solving the Morris water maze. Neural Comput. 10, 73 – 111.
Rospars, J.-P., Lansky, P., Vaillant, J., Duchamp-Viret, P., Duchamp, A., 1994. Spontaneous activity of first- and second-order neurons in the olfactory system. Brain Res. 662, 31 – 44.
Shimokawa, T., Rogel, A., Pakdaman, K., Sato, S., 1999. Stochastic resonance and spike-timing precision in an ensemble of leaky integrate and fire neuron models. Phys. Rev. E 95, 3461 – 3470.
Teich, M.C., Heneghan, C., Lowen, S.B., Ozaki, T., Kaplan, E., 1997. Fractal character of the neural spike train in the visual system of the cat. J. Opt. Soc. Am. A 14, 529 – 546. Vaillant, J., La´nsky´, P., 2000. Multidimensional counting pro-cesses and evoked neuronal activity. IMA J. Math. Appl. Med. Biol. 17, 53 – 74.
