We can see that an FA causes an increase of 0(M ’*') to _2
E(r ), but a decrease of 0(M ) for higher order coefficients, an
—1 —2
FB causes an increase of 0(M ) to E(r ) and an increase of 0(M ) for higher order coefficientsswhile an FD causes a decrease of
_2
0(M ) for all E(r ). The variance is affected by an amount which _2
is only 0(M ), and so the standard deviation for all types of —J5
contamination is approximately M . This suggests that the SCG is insensitive to small amounts of contamination, as any deviation in the mean is swamped by the variance, although the first-order coefficient for false additions is more sensitive than other coefficients. This conclusion is further illustrated by the expressions for the power which show that as M increases the
probability of not making a type II error (i.e. of correctly detecting contamination) approaches the probability of making a type I error
(i.e. of concluding that there is contamination when in fact there is not).
We have found from simulations that the presence of
contamination may cause striking changes to some of the SCO’s, but not beyond the variability that is already inherent for the pure process. Given the SCG of such a realization one would usually not be able to distinguish whether or how it was contaminated, although one is better able to identify the type of contamination, given also the SCG for the pure process on which it is based.
This work with gamma distributed intervals suggests that the correlation structure of renewal processes can be investigated
adequately through the SCG without undue concern over low intensity contamination. However, other approaches to their analysis may be preferable. It is intuitive that counting properties will be less
5 9. s e n s i t i v e t o t h e t y p e s o f c o n t a m i n a t i o n we h a v e d i s c u s s e d t h a n a r e i n t e r v a l p r o p e r t i e s , w h i c h i m p l i e s t h a t t h e p e r i o d o g r a m o f t h e c o u n t i n g p r o c e s s i s t h e a p p r o p r i a t e s t a t i s t i c f o r s t u d y i n g t h e s t r u c t u r e o f a p u r e p r o c e s s w h i c h i s s u s p e c t e d t o b e c o n t a m i n a t e d . I f , on t h e o t h e r h a n d , one w i s h e s t o i d e n t i f y a n d s t u d y t h e c o n t a m i n a t i o n , t h e n i t i s s t r a i g h t f o r w a r d (when t h e p u r e p r o c e s s i s r e n e w a l ) t o w r i t e down a n d a n a l y s e t h e l i k e l i h o o d .
6o.
CHAPTER b
SELECTIVE INTERACTION MODELS k.l INTRODUCTION
The motivation for this chapter again comes from the field of neurophysiology; we discuss selective interaction models, a class of stochastic models which has been developed to describe the spontaneous behaviour of single neurons. We first introduce aspects of the
physiological background at the naive and abstracted level at which mathematical treatment is possible. More detailed accounts are given by Fienberg (197^) and Sampath and Srinivasan (1977)- A nerve cell
can be thought of as a conducting fluid contained in a leaky membrane; this is fed from other parts of the nervous system by fibres
conducting electrical signals which cause ionic transfer across the membrane. Neural discharges or spikes, which are essentially
instantaneous, are monitored by inserting an electrode into the cell. The membrane potential may be altered in the following ways. Input spikes change the potential by a discrete quantity; they are referred to as excitations or inhibitions if the change in potential is
positive or negative, respectively. In the absence of stimulation the potential decays towards a rest level and when the accumulation of stimulation attains a threshold level the neuron discharges and the potential returns to the rest level. After a discharge there is a refractory period during which the neuron does not respond to stimulation.
Stochastic models based on this simplified understanding of neural activity fall into two groups according to whether the magnitude of the input stimulation is small in relation to the
6i. equation may be used to describe the membrane potential, or
otherwise, in which case the discrete nature of the inputs is important. Selective interaction models fall into the latter category. They
arose from the observation by Bishop, Levick and Williams (1964)
of distinctive multimodal histograms for the times between discharges, in which up to nine peaks of decreasing magnitude were located at multiples of the smallest (non-zero) peak (as well as a large peak at the origin due to bursts of subsidiary discharge associated with each main spike). Similar observations were made by Rose citi.'L (1967), while other authors have reported multimodal histograms with different characteristics. As we have seen in Chapter 2 , the thinning of a regular point process may result in a multimodal interevent p.d.f. and Bishop, Levick and Williams hypothesized that these data came from an excitatory process which had been thinned by an independent
inhibitory process.
This interpretation was formulated as a point process model by ten Hoopen and Reuver (1965) and has since been diversified in a number of ways, which we now describe. Two independent point processes arrive at a neuron : a point of one of these (the inhibitory process, I)
deletes ensuing points of the other (the excitatory process, E) in a manner prescribed by the model, resulting in an observable output process of undeleted excitations referred to as the response process, R. Points of E, I and R will be referred to as E-, I- and R-events, respectively. In the original formulation of the model by Ten Hoopen and Reuver an I-event deletes the next E-event if and only if there is no intervening I-event. We refer to this as the basic deletion mechanism. In particular, when I is a renewal process and E
is a Poisson process we refer to R as the renewal inhibited Poisson (r.i.P.) process; similarly we refer to the Poisson inhibited renewal (P.i.r.) process and the renewal inhibited renewal (r.i.r.)
62. process.
We now note some of the directions in which the model has been generalized to account for well-known characteristics of neural behaviour. Coleman and Gastwirth
(1969)
supposed that a (random)deadtime was associated with each inhibition, during which it was able to delete excitations. This is a suitable means of imitating the decay of membrane potential for a discrete model and it is also associated with the idea of refractoriness. Ten Hoopen and Reuver
(1967t1) and Hochman and Fienberg (1971) introduced spatial and
temporal summation to the selective interaction model by supposing that an accumulation of excitation could produce a response provided that no inhibition arrived during the process of accumulation. Räde (1972a)
supposed that each I-event could delete a (random) number of excitations. Ten Hoopen and Reuver
(1 9 6 7 t,
1968) proposed models in which I and E were dependent, events of one type triggering a- sequence of events of the other type.These writers oonfined their interest to the case in which either E or I is a renewal process and the other is a Poisson process, and derived expressions for the interresponse p.d.f. However, it is not generally the case that R is a renewal process and so the
interval p.d.f. does not give a complete probabilistic description of R. Srinivasan and Rajamannar (1970) derived the second order product
density for the r.i.r. process; Lawrance (1970b,1971a,b) performed a comprehensive second order analysis for the r.i.P. process, which he (1970a,1979) generalized in part to the selective interaction of two general point processes according to the basic deletion mechanism.
Ten Hoopen and Reuver
(1965)
, ten Hoopen(1966b)
and Fienberg and Hochman (1972) demonstrated the ability of these models to63. produce multimodal interval p.d.f.’s. In this context it is important to have analytical results when E is an underdispersed renewal
process for the basic deletion mechanism and when E is a Poisson process and the summation of excitation is required to produce a
response; this is particularly so when R is not a renewal process since then interval thinning is distinguishable from the point
thinning described in Chapter 2. It is of course also important to know that these models are capable of generating response processes which are not multimodal since they will then have a much broader
range of application. Many pertinent comments are made by Fienberg (197*0 and Sampath and Srinivasan (1977) in evaluating the significance of various aspects of the model.
In this chapter our purpose is to develop more fully some
techniques which enable us to derive simply and explicitly analytical expressions for the second order properties of the broad class of selective interaction models for which I-events are regeneration
points for the bivariate point process (l,R). We are therefore mainly concerned with the case that I is a renewal process and E is a Poisson process and we discuss this within the framework of
Berman (1978). A further emphasis is to specify carefully the
categories by which the variations of the basic deletion mechanism can be defined; it is felt that this provides a unified and comprehensive framework for discerning those elements of the model which are
relevant in particular applications.
In U.2 we consider the properties of regenerative multivariate point processes, among which the embedded renewal process is
particularly relevant to the study of selective interaction models. In 4.3 we recall the results for the basic deletion mechanism. We obtain simplified expressions for the second order counting properties of the r.i.P. process and discuss the similarity of the P.i.r.