The parameters and their estimation

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

The estimation of functions m x and s x 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 t i1, n i1 , … , t ik i , n ik i , where t ij is the duration of j-th stay and n ij is the number of spikes fired during that stay realization of a random variable N ij and k i is the total number of visits in box i. We also consider the entire space X divided into disjoint subspaces boxes denoted B 1 , … , B m , such that the processes Lx, · can be replaced by a m-dimensional vector of random processes L 1 = {L 1 } t ] 0 , … , L m = {L m } t ] 0 . It means that the boxes have to be sufficiently small to permit replacement of different Lx, · by a single stochastic process L i , 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 L i is the average random intensity of underly- ing Poisson process in the box B i , L i t = 1 bB i B i Lx, tdx, 3.1 where b is the measure Borel of the box B i . 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 andor the time periods spent in a given box are separated by sufficiently long periods of absence, we can approximate the be- havior of the process L i by a sequence of indepen- dent and identically distributed random variables L ij j = 1, … where j denotes the j-th visit of the animal in the given box. Formally, the above assumptions can be written in the form EL ij = m i , 3.2 VarL ij = s i 2 3.3 and CovL ij L ik = 0 for j k. 3.4 Since N ij the counts of spikes in i-th box during j-th visit follow Poisson distribution with random parameter L ij t ij , Eqs. 3.2 and 3.3 imply EN ij = m i t ij 3.5 and VarN ij = m i t ij + s i 2 t ij 2 . 3.6 A more detailed knowledge of distributional properties of L ij 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 L ij a more efficient estimation procedure like the maximum likelihood ML method can be applied.

4. A simple example