Stochastic models for neuronal firing

lSS S. K. Srinivasan, G. Rajamannar and A. Rangan: Stochastic Models for Neuronal Firing Kybernetik

24. McKean, T. A., Poppele, R. E., Rosenthal, N. P., Terzuolo, C. A. : The biologically relevant parameter in nerve impulse trains. Kybernetik 6, 168--170 (1970).

25. Murakami, M., Shigematsu, Y. : Duality of conduction mechanism in bipolar cells of the frog retina. Vision Res. 1O, 1--10 (1970).

26. Rall, W. : Branching dendritic trees and motoneuron mem- brane resistivity. Exp. Neurol. 1, 491--527 (1959).

27. - - Distinguishing theoretical synaptic potentials computed for different soma-dendritic distributions of synaptic in- put. J. Neurophysiol. 30, 1138--1168 (1967}.

28. Rushton, W. A. H. : The Ferrier lecture: Visual adaptation. Prec. roy. Soc. B 162, 20--46 (1965).

29. Rodieck, R.W. : Quantitative analysis of cat retinal ganglion cell response to visual stimuli. Vision Res. 5, 583--601 (1965).

30. Sakmann, B., Creutzfeldt, O.D.: Scotopic and mesopic light adaption in the cat's retina. Pfliigers Arch. 313, 168--185 (1969).

31. Singer, W., Creutzfeldt, O.D.: Reciprocal lateral inhibition of on-and off-center neurones in the lateral geniculate body of the cat. Exp. Brain Res. 10, 311--330 (1970).

32. Stevens, S. S. : On the 9sychophysical law. Psychol. Rev. 64, 153--181 (1957).

33. Varjh, D.: t~rber nichtlineare Analogschaltungen zur Simu- lierung biologischer Adaptationsvorg~nge. Progress in Brain Research-Vol. 17 (1965).

34. Westheimer, G. : Spatial interaction in the human retina during scotopic vision. J. Physiol. (Lond.) 181, 881--894 (1965).

35. Werblin, F. S., Dowling, J. E. : Organization of the retina of the mudpuppy necturus maculosus. Intracellular recording. J. Neurophysiol. 28, 339--355 (1969).

36. Wita, C. W., Briistle, R., Freund, H.-J. : Zeitliche Aspekte der neuronalen Informationsfibertragung im Sehsystem. Herbsttagg der Deutschen Physiol. Gesellschaft in Er- langen 1970.

37. Wiesel, T.: Recording inhibition and excitation in the cat's retinal ganglion cells with intracellular electrodes, Nature (Lond.) 18a, 264---265 (1959).

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Dr. A. Kern Inst. f. lnformations-Verarbeitung der Fraunhofer- Gesellsehaft BRD-7500 Karlsruhe-Waldstadt Breslauer Str. 48 Deutschland

Dr. H. Scheich Department of Neurosciences University of California San Diego, La Jells, Calif. 92037 U.S.A.

Stochastic Models for Neuronal Firing

S . K . SRINIVASAN, G. RAJAMANNAR and A. RANaAN Department of Mathematics, Indian Institute of Technology, Madras-36, India

Received October 2, 1970

Summary. The mechanism of formation of neuronal spike trains on the basis of selective interaction between two processes called excitatory and inhibitory processes, is studied. The techniques of stationary point processes are used to study the delay and deletion models proposed by Ten Hoopen and Reuver. These models are further generalised by associating a random life time to the inhibitory events. The probability frequency function governing the interval between two consecutive response yielding excitatory events, is obtained for these models.

1. Introduction

Neurophysiological models have received consider- able a t ten t ion in the pas t few years and several stochastic models have been proposed to explain the mechanism of neuronal spike trains. Quite recently, interval his tograms of spontaneous active lateral gen- ieulate neurons were obta ined experimental ly by Bishop, Levick and Williams (1964). Motivated by their experimental work, Ten Hoopen and Reuver (1965 and 1966} have studied the selective interaction of two independent recurrent point processes.

They have assumed t h a t the pr imary process called exci ta tory is a s ta t ionary renewal point process charaeterised b y the probabi l i ty f requency function r (t). The secondary process called the inhibi tory process also consists of a series of events governed by another s ta t ionary renewal point process characterised by the probabi l i ty f requency funct ion y)(t). Each secondary event annihilates the next p r imary event. I f there are two or more inhibi tory events wi thout a p r imary event, only one subsequent p r imary event is deleted. E v e r y undeleted event gives rise to a response. Ten Hoopen

and Reuver (1965) obtained an explicit expression for p (t), the probabil i ty f requency function governing the intervals between two successive registered events when one of the two processes is a Poisson process. This model has fur ther been extended and higher order characteristics of the ou tpu t process have been obtained by Srinivasan and Ra jamanna r (1970a, 1970b) for the most general case when both inhibi tory and exci ta tory events consti tute two independent general point processes.

I n the case of nerve cells it is generally accepted tha t the undeleted exci ta tory events get stored and a response occurs after the cumulat ive storage of a certain number of events. Ten Hoopen and Reuver (1967) have proposed a modified version of their earlier models by insisting tha t the n- th undeleted event (the undeleted events no t being intercepted b y any inhibi tory event) gives rise to a response. I f in the course of format ion of exci ta tory events, one inhibi tory arrives then the accumulated excitatories are deleted, accumulat ion start ing anew thereafter. On the basis of these assumptions, Ten t Ioopen and Reuver have considered two models known as "de le t ion" and "de- l a y " models and obtained p(t), the interval distribu- t ion between two responses using certain auxil iary functions. As is well known, the interval distr ibution does not fully characterise a Point process when the responses consti tute a non-Markovian, non-renewal process. I t would be worthwhile to obtain other statistical features like the moments of the number of events of the ou tpu t process in a given interval of t ime as well as correlations of these events. I n this

8. Bd., He/t 5, 1971 S.K. Srinivasan, G. Rajamannar and A. Rangan: Stochastic Models for Neuronal Firing 189

paper we wish to present more general results by deriving an explicit expression for h~(t), the product density (see for example Rice, 1944 and Ramakrishnan, 1950) of degree one of responses. This in turn will enable us to determine the power spectrum of the responses which can be experimentally measured.

The layout of the paper is as follows. In Section 2 we derive the product density of degree one of responses and also the higher order characteristics for the deletion model proposed by Ten Hoopen and Reuver. Sections 3.1 and 3.2 deal with two models when the inhibitories have a random life t ime during which alone they are effective. The final section is devoted to the delay model�9

2. Deletion Model We assume tha t the excitatory process is a sta-

t ionary renewal point process which conaia~ of a series of events charaeterised by the probabil i ty frequency function r (t) adding a unit i tem at each event. The stored unit effects are cleared with the occurranee of an event of the inhibitory process which is also an independent stat ionary renewal point process with the interarrival distribution characterised by ~(t). A response is said to occur as soon as the stored items at ta in a particular integer value n. After a response a clearance takes place and addition starts anew. The main contribution of Ten Hoopen and Reuver in this problem is the identification of the probabili ty frequency function of the interval between two consecutive responses in terms of ~b and ~. However this result although verifiable experimentally is not suf- ficient to predict the mean number of responses in a certain interval. To do so we have to obtain a function similar to the renewal density. This is precisely the product density of degree one of the responses. To achieve this we proceed as follows.

We observe tha t the set of events tha t lead to a response between t and t-t-dt may be put into two mutual ly exclusive classes according as an inhibitory event occurs between 0 and t or not. The former class can further be subdivided into two other mutual ly exclusive classes according as to whether the last inhibitory in the inter~al (0, t) is preceded by an excitatory or not. Thus we obtain

c o

h~ Ct)= g'~ (t) f p~(u) du t

t u

-4- f F(u) Z(t --u) f g~(v) dv 0 0

t

. f r162 t u t

A- fF(u) g(t--u) du f g~(v)dv f dp(x--v)dx 0 0 st t (2.1)

t t

+ f ~(u) z (t - u) d u f r (v) r ~ (t - v) d v 0 u

t t

-t- f F(u) X (t - - u) d u f ~b (v) d v O it

t

�9 f r ( t -w) dw, ~J

co

g it) ~- ~ (u) du, (2.2) t

where g~ (t) is the product density of degree one of a renewal process whose interval distribution is the n-fold convolute of ~b (t); F(u) is the product density of degree one of inhibitory occuring at u, starting with a response a t u ~ 0; pi (t) denotes the probabil i ty frequency function of the interval between a response and the next inhibitory event. ~.bn(t) is the n-fold convolute of r (t). F (u) satisfies the equation

u

F(u) =p~(u)+ f pqt)h(u--t)dt (2.3) 0

where flit) is the product density of degree one of an inhibitory event given tha t an inhibitory event has occured a t t = 0, whose transform solution is given by

~(a) (2 .4)*

To obtain p~(t), we observe tha t the origin is not an arbi trary point for the inhibitory process in view of the fact tha t the event a t t -~ 0 is a response, thereby implying tha t the previous inhibitory event occurs prior to n prior excitatory events. I f the earlier inhibitory occurs a t t ime x units before the origin,

1 [this event occurs with probabil i ty ~'(0) in view

of stat ionari ty where ~r/, (0) is the derivative of the Laplace transform of ~p (x) a t s = 0] a response occurs a t t=- 0 with probabil i ty G[ix ) w h e r e ~ i x ) stands for the modified product density of degree one of a renewal process whose interval distribution is the n-fold convolute of ~b(t). Thus the joint probabil i ty tha t the excitatory event a t t : 0 yields a response and the next inhibitory event occurs between t and t~-dt, t being measured from the origin is given by

co

i f ~'(o) ~ ( t+x) O~(x)dx. 0

(2.5)

From the above joint probability, we can obtain p~ (t) by dividing it by the probabil i ty tha t a response occurs a t t = 0. Thus p~ (t) is given by

f ~(t+ x)~(x) dx p~(t) = o (2.6)

c o

f Z(x) O~(x) d~ 0

O~ (t) is given by

t

o~(t) = f v~(u) r 0

t (2.7) + f O~(u)r

0

where Vp(u} denotes the forward recurrence time of the excitatory process. Now if N ( 0 represents the number of responses over a finite interval (0, t), then

0

* Throughout this paper we use ~(8) and krt(s) to denote the Laplace transforms of ~(t) and ~(t) respectively.

190 S.K. Srinivasan, G. Rajamannar and A. Rangan: Stochastic Models for Neuronal Firing Kybernetik

I n view of the s t a t i ona ry na tu r e of the process, ha(t ) has a l imi t as t t ends to in f in i ty and is g iven by

1 h m h a (t) = k - - ~'(0) ~'(0)

t---> O~

(2.s)

+ f z (u)du f dv f r (v -- x) dx 0 u 0

�9 .fCn-~(x-- w)gT(w)dw . 0

F r o m general considera t ions of the t heo ry of po in t processes, i t is easy to observe t h a t the cons tan t k is r e l a t ed to p (t), the p r o b a b i l i t y f requency funct ion of t he in t e rva l be tween two consecut ive responses b y

i k =

f tp(t) dt 0

The power spec t rum R(o)) of the responses when t h e y

a re fed into a l inear f i l ter can be calculated. ~( to) which can expe r imen ta l l y be measured in t e rms of t he Fou r i e r t r ans fo rm of ha (t) and 0 (t) [see for example Sr in ivasan , 1968] where

o(t) = tt---> ~ h,~(ttQ ) =kha(t) t~--* ~ (2.9)

t~ - - t l ~ t is g iven b y

wi th

and

oo

(~o) = f r(t)e i~ dt (2.11) 0

~ (o~) = f ha (t) e~ ~t dt. (2.12) 0

I f we assume t h a t the exc i t a to ry events are gove rned b y the p r o b a b i l i t y f requency funct ion ~(t)-~t~e-. t, then we have

Cn (t) - - / ~ e- t ' t (~ t ) - - x (2.13) (n -- 1)!

p (t) t he p r o b a b i l i t y f requency funct ion governing the in t e rva l be tween two successive responses is g iven by

oo

p ( t ) = Cn(t) f p~(u)du t (2.14)

t

+ f q(u)z(t--u)r 0

where q (0 d t denotes the p r o b a b i l i t y t h a t an inh ib i to ry even t occurs be tween t and t+d t with no response u p t o t, g iven t h a t a response has oceured a t t = O. W e observe t h a t q (t) sat isfies the equa t ion

q( t )= p~(t) f r f q(u)~v(t-- u)du t 0 (2,15)

c o

�9 f r

As a typ ica l example when n = 2 and when bo th the exci ta tor ies and inhibi tor ies cons t i tu te two in- dependen t Poisson processes with p a r a m e t e r s / x and we have for the Laplace t r ans fo rm solut ion of p (t) as

~ (2.16) p*(s) -- (~+s)2 + ~s "

Thus p (t) is given by #2

p(t)-- i-ff~T~## [e-t",-e -to,] (2.17)

where ( ~ + 2 # ) - V~2+4~#

QI= 2 and

~2----- 2

W e can also ob ta in an expression for p2(tl, t2), (t 2 > tl) the p robab i l i t y f requency funct ion governing two successive in te rva ls sepera ted b y responses b y classifying the events into four mu tua l ly exclusive classes as follows:

(i) The responses a t t 1 and t 2 are f irst and second responses af ter t--~ 0,

(ii) The response a t t 1 is t he f i rs t response af ter t--~--O,the response a t t~being the second or subsequent response af te r t 1 .

Off) The response a t t 1 is a second or subsequent response counted f rom t ~ 0 while the response a t t 2 is the n e x t response subsequent to the one a t t 1 .

(iv) The response a t t 1 is a second or subsequent response counted f rom t = 0 as also the response a t t 2 which is also a second or subsequent response counted f rom t ~ t I .

Hence we get oo

p~(tl, t~) = Cn(tl) ~n(t2 --ti) f pi(u)du t~

t~

+ r f pqu)z(t2 -u ) r --u)du tl

r o~

+ r f p'(u)du f 4~.(v)dv tl u - - t ~

ts

�9 f Q (w - u) Z (t~ - - w) 0 . (t 2 - - w) dw u

tL

+ f q(u)~n(t~--u)z(t~--u)d?.(t~--tt)du (2.18) O

t~ t~

+ f q(u) r ~(v-- u) 0 ta cQ

�9 f r (t) d t Z (t~_ - - v) dp~ (Q - - v) dv v - - t ~

ta ta oo

+ f q(u) r u)duf W(v-- u)duf r dt 0 tz V - - t l

ta

�9 f Q ( w - v) g (t z - - w ) On (t2 - - w) dw v

where Q(t)dt denotes the p robab i l i t y t h a t an inhib- i t o r y even t occurs be tween t and t+d t with no response up to t condi t ional upon an inh ib i to ry even t a t t - - 0. Q (t) satisfies the equa t ion

cQ t

q(t)-=~p(t) f dp,,(u)du-4- f QCu)~(t --u)du t 0 (2.19)

c o

�9 f ~ . (~)d~. t - - U

8. Bd., Heft 5, 1971 S.K. Srinivasan, G. Rajamannar and A. Rangan: Stochastic Models for Neuronal Firing 191

From (2.19) and (2.16) we can get an explicit expression for p2(t:, t~). In a similar manner we can also obtain the second order characteristic for any given form of r (t).

3. Inhibitories With Random Li/e Time

Most of the models in the theory of neuronal network assume tha t an inhibitory event can annihi- late the next p r imary event irrespective of the t ime of its arrival. This assumption is not very realistic and infact it m a y be worthwhile to introduce a criterion for the effectiveness of the inhibitory pulses as a function of t ime [see for example Ten Hoopen, 1966]. In this section we introduce a random life t ime for the inhibitories so tha t the inhibitories become ineffective after some t ime after their arrival. Thus we assume tha t the inhibitories have a life t ime distributed with a common probabil i ty frequency function fl(v). We assume as before tha t n undeleted excitatory events are required for a response. An inhibitory is deleted and its life t ime becomes extinct with the arrival of an excitatory or an inhibitory during its life time. In the former case the particular event itself is not registered whereas in the lat ter case the life t ime of the latter inhibitory alone is in oper- ation. We further insist tha t ff an inhibitory arrives when the accumulation of excitatory events is in progress, both the llfe t ime of the inhibitory event and the accumulated excitatory events are wiped out, accumulation starting anew. A typical realization when n ~ 2 taking into account all the above assumptions is presented graphically in Fig. 1.

I I ] I I I I I I I Exciia~ories

InbibiSortes 7"'-I I I

I I I I I Responses

Fig. 1

I t is interesting to observe tha t although a life t ime is postulated for an inhibitory, a nonzero life t ime is realised only when it is not preceded by excitatories in accumulation. When excitatories are in accumulation, the inhibitory deletes the accumulated excitatory events and its life t ime gets terminated instantaneously. Thus in this model any particular inhibitory exercises control over the excitatories in one of the two mutual ly exclusive ways. Under these modifications we wish to derive the interval distribution between two consecutive responses. The problem is extremely difficult when both the ex- eitatories and inhibitories form renewal point processes. We first consider the case when both the excitatories and inhibitories arrive according to a Pois- son process and then generalise the results to the situation when the inkibitories are distributed according to a renewal point process.

3.1. Model I First let us consider the special case when both

the inhibitory events and excitatory events arrive according to Poisson process with parameters/u and respectively and the inhibitories have a random life t ime % the ]fie times of different inhibitories being identically and independently distributed with a probability frequency function fl (~).

We classify the inhibitories into two mutual ly exclusive types; type 0 inhibitory and type 1 inhibitory. Type 0 inhibitory is an inhibitory whose life t ime is terminated instantaneously a t its arrival which means tha t accumulation of excitatory events has been in progress a t tha t instant. Type 1 inhibitory is an inhibitory which has a non-zero life time.

To obtain p(t), we observe tha t the set of events tha t lead to a response may be put into two mutual ly exclusive classes according as an inhibitory arrives between 0 and t or not. The former class can further be subdivided into two other mutual ly exclusive classes according as whether the last inhibitory in the interval (0, t) is of type 0 or type 1. Thus p(t) is given by

t p (t) = e - ~ r (t) + ; re (u) Z (t - u) r (t - u) d u

0 t t - - u

+ f F:(u)z(t--u)du f fl(t) d~ 0 0

(3.1.1) " { , ! ~ (v -- u) r (t --v) dv

u + r }

+ f r 1 6 2

where r = F e - ' t , ~p(t) = 2 e -~t

q ~ . ( t ) - ( ,~ - 1): '

~(t)dt is the probabil i ty of one inhibitory of type 0 between t and t + d t with no response in (0, t) given tha t a response has occured a t t----0 and ~(t)dt is the probabil i ty of inhibitory of type 1 between t and t + d t with no response in (0, t) given tha t a response has occured at t ~ 0. /70 (t) and ~ ( t ) satisfy the equa- ~ o n s ,

t co

Fo(t ) =v/(t) f dp(tldT f r 0 t - - v t t co

+ fFo(u)~p(t--u)duf r f r ds 0 u t - - v

t t - - u

-I- f ~ (u)~p (t --u)du f fl (v) dr (3.1.2) 0 0

f u + �9 t co "/[ (v- )dvs (w-v)ew f e,

v t - - w

+ f r u)dv f d?,~_:(s)ds , u + 7: t - - v

oo t

(t) -=~p (t) f r (t) d t § f ~ (u) v 2 (t -- u) d u t 0

c o t Oo

�9 f r f~(u)v2(t--u)du f r t - - u 0 t - - u

t t - , , t o . ~ . o . j

+ f ~(u)v/(t--u)du f fl(~ld~ o 0

u + v c~

�9 f r r u t - - v

192 S. K. Srinivasan, G. Rajamarmar and A. Rangan: Stochastic Models for Neuronal Firing K ybernetik

An interesting special case arises when n = 2. In this case it is interesting to note tha t an excitatory suffers both forward and backward inhibition symmetrically. We obtain the transform solution of the interval distribution between two consecutive responses as

p* (s) = ~=~'~(s) ~2 ( ~ + ~ + 0 2 + ( ~ + ~ + 0 =

+/~2 F~* (s)/~*(~ + ~ + *) (~. + # + s)= (3.1.4) 1 ~2fl*)A + p + 8)

+~3F~* (s) 2(~+-~,+~) 0s~

1 afl*(k +/~ + s) [ ( k + p + s ) ~ ds

where

F~* (0 - ~t (8) (3.1.5) r (') and

Fo* ( s ) = F~* (s){A + ~ + s 1- - @*(k +~s +0 la } --1

r (s) = (X + # + s)

r O) = [ + 8) 2 ( k + F + s ) (3.1.6)

~/~*(k+~+6 ( k + / ~ + s ) - - / ~ + ( ~ + ~ + 0 /z as

k# ~ 8fl*(k+#+s) ~#

�9 f l * ( k + # + s ) - x t , " ~t~*(x+v+s)

@*(a+~,+,).l + k/~ ~ ~s ]"

We further observe tha t when the inhibitories are assumed to have a zero life t ime in this model, tha t is they are effective only a t the instant a t which they arrive, so tha t fl(~) = ~ (~), we identify Eq. (2.16) with Eq. (3.1.4).

3.2. Model I I In the present model let us deal with a more

generalised situation when the inhibitories are gov- rened b y a s ta t ionary renewal point process ~v (t) while the exci tatory events are governed by a Poisson process with parameter F. The interval distribution between two responses in this case is given by

cr t

p (t) = f p* (t) d t r (t) + f Fo (u) Z (t - - u) d~. (t --u) du 0

t t - - u

+ fFx(u) z ( t - -u )du f f l (~ )d~ o o (3.2.1)

�9 { f?(v- -u)r

z~+ v } + f d~(v--u)C.(t--v)dv .

~t

(3.1.2) and (3.1.3) hold good with ~p(t) in their first term on the right hand side being replaced by p~(t) where p~'(t) is the probabil i ty frequency function of the interval between a response and the next in-

Co

hibitory event and Z (t) = f ~p (u)du. t

Using the same arguments as in the derivation of (2.6) incorporating the random life t ime of the inhibitory and invoking the stationarity of the process so tha t the probabili ty of an inhibitory of type O at

t ime x units before the origin is i~ and of ~"(0)

type 1 is q such that 1. ~'(o) .P + q = Thus we obtain

pi(t ) = ~(t) (3.2.2)

where

~ ( 0 =

and

0

p f v,(t--x)a'~(-x)gx v,'(o) --00

q )axffl(,)a, f --00 0

I X + ~

" l / r x)G[(--v)dv

0

+ f x-F v

. o }

x + ~ v

0 p f ] Z (-- x) G'~ (-- x) d x ~'(o)

- - o o

D - - x ,

q f ~ x f~(~)d~ ( - - x) dx ~Y'(0)

- - c o 0

f ~ + T 0

"/f x)o (-v)av + f v) ( x x + ' r

0

�9 x)dv + f e ( v - x)d, x + v

" / ,n - l (w- -v )G~(- -w) dw} .

Since the process is stationary, p~ (t) is a genuine probability frequency function. Using the condition co

f pi(~) dt -~ 1, the constant p can be evaluated. 0

4. Delay Model In this case the same propositions hold as the

deletion model without life t ime for the inhibiteries except tha t the inhibitory event induces a sequence of excitatory events and stops the sequence initiated by the preceding inhibitory event. I t is assumed tha t the interval between an inhibitory event and the next excitatory event as well as tha t between two consecutive excitatory events is given by ~b(t). Using the same arguments as before we find p(t) the interval distribution between two consecutive responses is given by

O0"

p(,) = r f p*(u)au t t (4.1)

+ fq (u )Z ( t - u ) r ( t - - u ) d u 0

where q(t)dt is the probabili ty tha t starting with a response at t = 0, an inhibitory event occurs between t and t+dt with no response upto t. q(t) satisfies the

8. Bd., Heft 5, 1971 G.D. McCann and S. F. Foster: Binocular Interactions of Motion Detection ~'ibers 1 9 3

equat ion c o

q(t) =pqt) f r t ( 4 . 2 )

t "4- f q (u) g (t -- u) du f r dv

0 t - - u

pl (t) can be calculated using the same arguments as before:

t f g'~(x)~p(t-k x) dx

p~(t)~ o (4.3)

f 9~(x) X(x) dx 0

Eqs. (4.1), (4.2) and (4.3) provide a complete solution to the problem of interval distr ibution between two responses. If, on the other hand we wish to obtain the p roduc t densi ty of degree one of responses, i t can be obtained directly. We observe tha t the set of events which leads to a response between t and t A-dt m a y be either the first or subsequent set of n exci ta tory events after the response at t = 0, wi thout the inter- ception of any inhibi tory event in between or the first or subsequent set of n events generated by the last inhibi tory event in the interval (0, t). Thus we have

1 h (t) -= g'~ (t) f p~ (u) du t ( 4 . 4 )

t -[- f F ( u ) x ( t - - u ) g ' ~ ( t - - u ) d u

0

where F (u ) and Z(t) are obtained f rom Eqs. (2.2) and (2.3). The limiting form of this as t tends to infinity is given by

c o

Lt /h ( t ) ---- - - k~(0~- X(u)g~(u)du. (4.5) t - - ~ r 0

From (4.4) and (4.5) we can obtain the power spect rum of the responses.

We finally observe tha t higher order p roduc t den- sities can also be obtained in the present case b y making use of the fact t h a t the point of occurrence of inhibi tory event is also a renewal point for the ex- c i ta tory process as well as for the responses.

Re]erences Bishop, P. 0., Levick, W.R., Williams, W.O.: Statistical

analysis of the dark discharge of lateral geniculate neurons. J. Physiol. (Lend.) 170, 598 (1964).

Ramakrishnan, A.: Stochastic processes relating to particles distributed in a continuous infinity of states. Prec. Camb. Phil. Soc. 46, 595 (1950).

Rice, S . 0 . : Mathematical analysis of random noise. Bell. Sys. Tech. J. 24, 46 (1945).

Srinivasan, S. K. : Stochastic theory and cascade processes. New York: American Elsevier Publishing Company 1969.

- - Symposia in mathematics and theoretical physics, eel. 9. New York: Plenum Press 1966.

- - Rajamannar, G.: Selective interaction between two independent stationary recurrent point processes. J. appl. Prob. 7, 476 (1970).

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Prof. S. K. Srinivasan G. Rajamarmar A. Rangan Department of Mathematics Indian Inst. of Technology Madras 36, India

Binocular Interactions of Motion Detection Fibers in the Optic Lobes of Flies* GILBERT D. MCCAN~ a n d SUSAN F. FOSTER

Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California, U.S.A.

Received December 2, 1970

Zusammenlassu~. Nervenfasern, die die optischen Gan- glien beider Kopfseiten yon Musca und Phaen6~/a verbinden und die selektiv auf Musterbewegungen antworten, zeigen binoculare Wechselwirkungen. Details dieser Wecheelwirkun- gen und der Ort an dam sic in den optisehen Ganglien statt- finden wurden ermittelt. Die Bedeutung dieser Einheiten fiir die direkte Steuerung yon Wendereaktionen beim Flug wurde ebenfalls untersucht.

Introduction Studies have been in progress to determine the

response characteristics of a group of neural fibers t h a t pass between the two optic lobes th rough the midbrain of the housefly M ~ c a domestica and the blowfly, Plw~nioia sericata (Bishop et al., MeCann and Dill). These respond to the full visual field of their ipsilateral eye and selectively detect mot ion in four directions. These four directions are approximate ly as

* This research was supported by the National Institutes Health, United States Public Health Service Grant NB 03627.

follows: (1) horizontal ly inward toward the vertical midline axis of the head, (2) horizontal ly outward, (3) downward and (4) upward. Fur thermore , these units process small field mot ion informat ion in such a manner as to add all contributions of mot ion in their selected direction and to subt rac t all i n~v idua l com- ponents in the opposite direction. They ignore contri- but ions of mot ion perpendicular to their directions (MeCann and Dill).

Such a uni t produces a m a x i m u m response when all mot ion in the field of one eye is in the same pre- ferred direction and has a minimal dis turbance f rom small field mot ion in the opposite direction. I t would thus appear t h a t the p r imary funct ion of these units might be to evaluate the mot ion of the insect relative to its static surround.

This investigation is concerned with the correlation between these un i t responses and flight reactions. Ex- periments on the response properties of neural uni ts

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