Biol. Cybernetics 17, 183--t93 (1975) 9 by Springer-Verlag 1975
Some Views on the Role of Noise in "Self"-Organizing Systems J o h n S. Nicolis Department of Electrical Engineering Univ. of Patras, Patras, Greece E. P r o t o n o t a r i o s and E. Lianos Department of Electrical Engineering, National Technical Univ. of Athens, Athens Greece Received: April 30, 1974
Abstract This paper deals with information transfer from the environment and "self'-organization in open, nonlinear systems far from thermodynamic equilibrium - in the presence of either non-stationary phase jitter noise, or amplitude stationary noise. By "self'-organization we mean here the progressive formation within the system of sequential, ordered (coherent) relationships between appropriate dynamical variables-like for example, the phase differences between the oscillating components of the system. We take up (in Section II) the classical Laser as a specific example and examine in detail the influence of phase jitter noise in the mode (phase) locking process. We find - as expected - that phase fluctuations in the cavity cause degradation of the coherent behaviour (i.e. increase the entropy) of the system - which, however, levels off, or saturates with time. Further (in Section III) we examine systems where the number of self-sustained oscillating components may vary with time in such a way that the maximum entropy of the system increases faster than the overall instantaneous entropy. We put forth the hypothesis that in such cases - because of the increase of the redundancy the system gets organized not just in spite of, but merely because of the presence of Noise. Possible applications in biological systems (especially concerning a model of cerebral organization) are briefly discussed. It is understood here, that the system has to display some 9 preliminary dynamical structure before the organizing procedure takes over. What happens afterwards is the subject of this paper.
I. Introduction and General Formulation of the Problem
Self-organizin9 systems are hierarchical structures; when we deal with them we quite often are confronted with almost-finished products i.e. an "already" sufficiently organized system (like a fertilized egg for example). So, the best we can h o p e to do is to elaborate on the "last" steps of evolution. Incidentally, this is not that b a d : K n o w l e d g e of the system at the higher levels of the hierarchy is perhaps the prerequisite to correctly estimate the relevance of averaging processes we perform when we apply statistical mechanics in order to find (going backwards) links between higher and lower levels of description. O u r discussion in this paper will concern such partially organized systems. N o w a b o u t
the noise we m a k e the following assumptions: Alt h o u g h (strong) noise causes both amplitude and phase (jitter) modulation, we are considering here (low level) noise as either associated with the r a n d o m variables of the system with respect to which organization occurs or not. In the first category (Section II), we classify the n o n - s t a t i o n a r y phase jitter noise, while the organizing process has to do with phase locking. In the second category (Section III) belongs the amplitude (quasi-stationary) noise. After these preliminary remarks let us proceed. We consider an open non-conservative system consisting of a n u m b e r n(t) of nonlinear interacting c o m p o n e n t s ("reactants"). These components can be imagined as self-sustained oscillators (either of a H a m i l t o n i a n or non conservative type), lumped or distributed (normal modes). Each active c o m p o n e n t of the system is represented by a r a n d o m p h a s o r (i.e. a n o n h a r m o n i c oscillation in a "pool" of weak additive noise :)
Z~(t) = A~(t)e J ~ t ) where: q~(t) = cozt+ q~z(t), characterised by a continuous joint probability density distribution function (p.d.f): 0z(Az, q)~;t). Dynamically the system Z is modeled by a set of n(t) coupled nonlinear differential (rate) equations, which in the absence of time-varying spatial differentiation (V2zi = 0 ) read:
dz~ dt - L ( z l , z2 ...z~... z,~0
dzz - L ( z l , z2 ... z~... dt
z,~3
d zn(t) dt - f,(,)(z 1, Zz ... zz ... z,(,))
(1)
184
and unless otherwise stated it is assumed that:
dn(t) OA k dt ~ O t -
'
.~i
'
0
Oq)~ ,gt
By expressing the dynamical behaviour of the system as (1), we essentially imply that all activity in it originated as noise which has been more or less constrained at an early age through an interplay between the controlling environment and the system's parameters and boundary conditions. The noise has not been wholly constrained however; some of it (what we call additive noise in this paper) escapes control and gives rise to an element of arbitrariness or flexibility, or even (it depends on how you look at it) "originality" in the system's behaviour. Since linearization is not allowed in our problem because the system Z is considered far from equilibrium, the solution of (1) is prohibitively difficult (and eventually unnecessary) when n(t) is sufficiently large. In order to' follow the behaviour of the system in time, what is important to derive in a situation like this is not the analytical expressions of the coupled stochastic processes Z~ but rather their joined probability density distribution: Q(zl, z2 ... z,(t); t). This could be achieved in principle by deriving from (l) and subsequently solving under given initial and boundary conditions, the corresponding n(t)+ 1-dimensional Fokker-Planck equation (Lindsey, 1972), which describes the diffusion in time of the joint p.d.f. ~(zl, Zz 9.. z,(t); t). 1 From this distribution one can derive the "instantaneous" (conditional) entropy of the system: S(t) = c~ + c2 ~ O(z~, z2 ... z,~,; t)
9logz~(Zl, z 2 ... z,(t); t)dv
(2)
where: dv = dza. dz2 ... dz,(t), the integration is extended over the volume v in the functional space, zl,z2, ... z,l~) become eventually statistically correlated as a result of the organizing process going on, and el, c2 are constants9 One has to point out immediately of course that for a given dynamical system Z there exist many"entropies" since the entropy is a property of the state, i.e. a probability density distribution with respect to specific variables. Depending on the relevance of the variables chosen for each model the calculated entropy may or may not represent the degree of disorganization of the ~ system (see also next sections). We now proceed in the formulation of our problem as follows: Our open system Z interacts energetically with the environment U (Fig. 1) and concequently the 1 In the l i m i t of w e a k noise a r o u n d a s t e a d y state.
-
d Se > 0
= "ataxia,,
d Se < 0
= "order,,
Fig, i. Information interaction between the environment (2') and a system (X) composed of non-linear mutual interacting oscillators
differential of the entropy of the system dS(t) can be written (Glansdorff and Prigogine, 1971) as: dS(t) = dSi + dSe where dS~ is the differential of the internally produced entropy (dSi >0 always) and dSe is the differential of the entropy of the inflowing energy from the environment Z: The sign of the term dSe is undetermined. In the trivial case dSe > 0 i.e. disorder or "ataxia" is entering the system from outside and this accelerates the process of disorder by moving the system toward the thermodynamic equilibrium faster9 In the case dSe < 0 order or "negentropy" enters from theenvironment. It is understood of course that it is the system Z itself which "decides" about the sign of dSe. In other words information transfer is taking place between a series of events in Z' and another series of events in Z if there is some degree o f - not necessarily causal - correspondance between them. Two possibilities exist now: Either [dSe[ < dS~: as a result the degradation of the system will slow down and perhaps level off in time, or [dS~[ > dSv In this latter case dS(t)< 0 i.e. the entropy of the system will decrease at the expense of the environment 9 Such a possibility may very well occur in a "large enough" isolated "world" containing Z ' + Z: Within such a world we may witness ordered structures (Z) take shape without the world's overall evolution ceasing to comply with the second law of thermodynamics. Consider for instance the example of the crystallization in a saturated solution or, even more appropriately the case of growth of a population of bacteria in a "isolated" system consisting of water and nutritient such as glucose plus some mineral salts that can effectively be used by the metabolic mechanisms of the living cells. Such examples are however slightly irrelevant to the present situation: they either refer to structures near equilibrium (first example above) or they preassume the existence of highly organized, coherent structures far from equilibrium. In this work
185
we start however, with a given open poorly-organized system (Z) far from equilibrium and are being examining the condition under which we can make it more coherent (i.e. increase the number of ordered relationships between appropriate dynamical variables e.g. qh) by absorbing information from the environment in the presence of additive fluctuations existing between the components of the system. Under these circumstances and for a fixed number of active components n it appears that the non-stationary (phase jitter) noise (see Section II) will cause deterioration i.e. it will always overtake the whatever organizing processes which was acting on the phases of the oscillators. So the condition ]dSe] > dSi cannot be fulfilled at least in this case. After all the Fokker-Planck equation is a diffusion equation so what is predicted is a spreading rather then "shrinking" of the p.d.f. ~(qo1, rp2 ...qo,; t) with time and consequently the entropy of the system S(t) will increase (again for small deviations from the steady state). Nevertheless the organization of the system which can properly be defined in terms of its Redundancy R (expressing intercomponent (q~,, q0~)correlation) may increase even if the entropy S(t) goes up provided that the maximum entropy of the system
s(t)
increases faster than S(t) with time. Since R = 1 - - -
S max
and Sm.x = c3 + e4 log2n(t) it is obvious that: -OR ~ - >_0 _ if: S
OSmax
Ot
>__-Smax
8S
Ot
or
d
~
a~
d
[ l n S m j > ~ [IDS]. at
Of course for an invariant number of active components n = const the system Z can never comply with this requirement. The only possibility for organization is that new degrees of freedom continuously enter the organizing process under an instantaneous overall entropy growing slower than log 2 n(0. By what mechanism such a thing could possibly be realized? We elaborate somewhat on this in Section III; let us discuss for the moment two possible cases where this process might perhaps occur. 1. During the embryogenesis of a biological organism the Redundancy must increase (Fig. 2) although it is known that during this particular period of development the internal entropy variation of the system (AS), is increasing very fast. This has been indeed shown by Trintcher (1965) who gives experimental evidence that the entropy production dSi/dt per unit mass and unit time (in cal/gram hour) increases during the first (embryogenetic) period of the dif-
R
i
i
l P,
"O
a:
th process ~ t
Fig. 2. Suggested variation of Redundancy during the life cycle of a self-organizing system
ferentiating organism, then passes through a maximum and finally decreases to a steady state value with the attainment of the organisms adult state. This suggests for the entropy (A S)i a "sigmoid"- like type of variation:
(AS)i ~
1
1 + exp [ - a(t - to)]
i.e. a very steeply rising
front. Here the increase of redundancy is accomplished, we believe, through "fast" increase of n(t) i.e. cell division and multiplication. On the other hand during the adult life of the organism cells keep dividing but do not normaly multiply. The value of 8R/St must therefore be maintained in the region of "zero plus" while the system rides its "limit cycle" being in a dynamical steady state. [in case of ~R/Ot~>O (R--, 1) abnormally tight organization or "petrification" would follow which makes the system unable to display adaptive behaviour; for OR/St< 0(R~0) a "death process" is initiated.] 2. Our second example has to do with the electrical activity of the brain as related to changing physiological states: Intracellular records have revealed spontaneous wave like activity with amplitudes in the scale of 5-20 mV in many cerebral structures (Elul, 1966, 1968; Adey, t968). These intracellular recordings show that even in the absence of propagated nerve impulses, rhythmic noise - like slow potential changes occur due to (subthreshold) fluctuations of the cells membrane resting potential. Power density distributions in these records resemble closely spectra from similar analyses of gross Electroencephalographic records (EEG) from the same cortical regions: This means that the EEG can be considered as the (scalar) projection of a vector superposition of n(t) phasors (Fig. 3) with central frequencies c~a in the neighborhood of ~ 10Hz - each representing the individual neuron's fluctuating postsynaptic membrane
186
o Fig. 3. Diagram of n-interacting random phas0rs
potential. Now, as has been pointed out, many times by workers in this field [see Elul (1968) for example], the spontaneous electrical activity of the brain still remains one of the most elusive aspects of cerebral functions: It is a (non-stationary, beyond time intervals of a few seconds) continuous wave activity of variable amplitude and frequency with inconstant phase relations (~pz- (p~ 4= const) between its constituents phasors zx; its overall character is quite similar to random noise (although increasing skewness of the p.d.f, of the EEG composite phasor during performance of mental tasks (Elul, 1966, 1968) indicates progressive (linear) statistical coupling between the individual neuronal phasors). And yet the behavioural aspects of the brain operation are quite orderly. So, unless the electrical activity of the brain is simply concommitant with behaviour, there is a possibility that unlike man-made communication systems in which noise is an encumbrance, the brain is a system which may use noise as an inevitable and "beneficial" factor. It is known of course that neuron cells do not divide in the brain beyond a rather early age of the organism so one could not justify any increase of brain organization on that basis. There is, however, the possibility (and this is our hypothesis) of a number of phasors in the system being "dormant" for given periods of time, getting excited whenever the fulfilment of the condition ~R/Ot > 0 requires action. To avoid any misunderstanding we point out that just activating above threshold the previously "dormant" nonlinear oscillators is not enough; the procedure has to take place under at least constant entropy S(t) i.e. the newly activated degrees of freedom must be subsequently subjected to the same "locking process" already acting on the "old" members of the system. This locking process is achieved through information flowing from the environment. We will try to qualify the above statement in the subsequent sections.
In Section II we examine the influence of phasejitter noise in the efficiency of mode locking process in a multimode Laser with a fixed number n of normal modes. This is an example where the organization of the system may be theoretically "perfect" (R = 1) in the absence of noise but deterioration is initiated when phase jitter develops in the cavity. In Section III we examine the possibility of achieving amelioration of the Redundancy in the system (Z) by allowing the number of self-sustained oscillators n(t) to vary with time as a result of switching on and off caused by noise catalytic action. With the aid of such a model we may approach an understanding of how very noisy biological systems - like the brain - can display orderly behaviour.
II. The Influence of Non-Stationary Phase Jitter Noise in Laser Mode Locking
We first review briefly the Laser operation and the mode locking process in the absence of additive noise (Borenstein and Lamb, 1972; Harris and McDuff, 1965). We consider an one dimensional cavity resonator filled with an active material which in the classical approach is being considered as an ensemble of classical nonlinear oscillators homogeneously distributed in the cavity and activated by a classical (external pump) electromagnetic field. For specific combinations of the values of the parameters of the system the excited non-linear oscillators transfer on the average energy to the normal modes in the cavity. The activated (axial) normal modes form two categories: Those "hard-driven" which behave as self-sustained oscillators (with a quality factor "Q" beyond a certain critical "threshold") and the rest which being dynamically unstable amount to joule losses or thermal noise in the cavity. So, through the above classical non-linear interactions we establish within the cavity the existance of loosely coupled self-sustained normal modes. The "output" of such a multimode laser is not however coherent (since the amplitudes and phases of the individual "free-running" modes change independently with time), and differs little from random noise. In order to make the sum of these modes coherent we have to operate from outside ke. to provide the system with an organizing perturbation process which will cause the establishment of ordered (time independent) relationships between the successive phase differences viz. (p~+l(t)- (p~(t)= const, or, will "mode lock" the laser.
187 One very efficient way to achieve this is the following: (Harris and Mc Duff, 1965). We introduce an intracavity phase perturbation at a frequency co,. which is approximately the frequency of the (axial) mode interval A f2. The effect of this intracavity phase perturbation is to frequency modulate the previously free running Laser modes i.e. to cause each one of these modes to become the center frequency of an FM signal9 While the free-running modes experience their gain from independent coordinates of the emission spectrum of the active material, the competing FM oscillations to a large extend "see" the same spectrum so they are "fed" from the same (i.e. the entire) source and not - as previously - from different parts of this source9 The competing FM oscillations are thus much more tightly coupled than were the previous free-running Laser modes. The result of this competition is that the strongest of the F M oscillations - usually the oscillation whose carrier is at the center of the emission spectrum of the working material, is able to completely quench the competing weaker oscillations and establish the steady state regime in which q~s+1 - r = const for all 2 between 1 and n where n is the number of locked modes9 Assuming small perturbations we write the coupled equations between each normal mode and those two in its immediate vicinity:
Oq)x _ 2A co) A s Ot 9{As+ ~cos(cps+l - r
~?As &
=
6c rbl = Aco - 4 L - {cos~01 + cos(c#1 - %)} At
_
6c 4L At {sinrpl + sin(tpl - %)}
(4)
where ~bo= ~(t). The spectral density of the noise is given, No Watt/Hz. In the following we intend to study the evolution of the phase difference ~b= (Pl - % in time and particularly to calculate for different sets of parameters: {Aco, c~= c3c/4L} of the system the p.d.f. Q(@; t)the variance o-g(t) and the entropy of the system S(t). It can be easily deduced from (4) that the corresponding Fokker-Planck equation has the form: OP(COo,~0t; t) at
c~ [(A co + at/(q~o,qO 1)p(Cpo , @1 ;t)] Ocpl +
N 0 02p(q~O, ~0l; t) 2
where:
tic 2L + As_ lcos(rpz- % _ i) }
condition i.e. only these three modes are considered self-sustained oscillators. We normalize the phases % and ~o1 to q~2 by taking q~2 const = 0 and we consider Ao = A2 = A1/2. The phase Cpo(t) is taken to perform a brownian motion i.e.: dqb/dt - ~(t) is an additive white Gaussian noise process. With these assumptions Eq. (3) become:
t/(Cpo, qh) = cos (opt - %) - cosqh
(3)
6c 2L
9 {As+ t sin(q~s+ 1 - ~oa)- As- 1 sin(q~s - q~s- 1)}. The steady state solutions of the above equations give the mode locking regime9 In these equations 2Aco = (2s - cos where g?s is the vacuum eigen frequency and cox is the eigen frequency of the same 2th normal mode in the presence of the active material filling the cavity: cos=Oo+2co,, where: f2o is the central mode frequency and co,,,in the external perturbation frequency. L is the geometrical length of the cavity, c is the velocity of light and 6 is the coupling coefficient between the 2 'h normal mode and its neighbouring modes 2 - 1 and 2 + 1. In order to make things easier in the example that follows, we are going to choose an active material with a narrow emission band such that only three normal modes (Ao, (Po), (At, opt), (A2, cP2) will satisfy the threshold
= 2 s i n ( opt
~ - ) sinqb2
Introducing the dimensionless time:
t' = at and the parameters: Am "normalized mistuning" 7 -- - - - , a "noise to signal ratio"
x = No 2a"
We rewrite the FP equation as follows: Op(cpo, ~0i ; t') Ot' 9 [(7 +
0 Oq)t c.00 ) p ( % .
Ol; t')3 +
OZp(~Oo,~ot ; t')
188
A(LI
J
No
NO-
a
10 ~
V I
I
50 Fig. 4 a
"
I
t 100
I
50
.
t
100
Fig. 4 b
S 3,8 I Z~=
3,3
0
N
NO =1@0
2,8
:a_~_
O,E
l
50
I
t 100
J
0
Fig. 4c
50
J
t 100
Fig. 4 d Fig. 4. The Entropy as a function of time (arbitrary units)
The p.d.fi p(~, t) of the phase difference: ~b= ~o1 -q~0 can in principle be deduced from the solution of FP equation as: p(~; t) = ~ p(q%, r + {#; t)dq~o -oo
and the entropy of the system:
S(t) = -
~ P(q~, t)log2P(~, t)dq~. -oo
Since, however, the FD equation cannot be solved analytically and its numerical integration is exceedingly difficult we resort to a Monte-Carlo type method. (See flow chart diagram in the appendix.) Some results of the computing simulation concerning the entropy S(t) are displayed in Figs. 4 for the indicated sets of variables; it is apparent that for weak jitter noise (i.e. limited time intervals) the disorganization of the mode locking process induced by phase jitter levels off with time.
189
f
Q
N No
r
a
No
I
50
t
I
I
100
50 Fig. 4 f
Fig. 4 e
llL Organization Through Stationary Amplitude Noise We now consider a system consisting of n(t) random phasors: z~ (Fig. 3) with fixed frequencies ~ very close together, and amplitudes A~(t) and phases opt(t) random processes. Let us briefly consider, first what may happen for n = const, and in the absence of noise i.e. in case where zz are deterministic non-linear coupled oscillators with independently varying amplitudes and phases between oscillators. Since the frequencies coz are very close together the system can find itself either in the regime of frequency entrainment (e.g. Nicolis et al., 1973) where all oscillators operate in synchronythereby implying functional homogeneization of the system, or, in the regime of phase locking q)z+x- (Px - c o n s t - which implies a sequential process or a differentiated state of function (and suggests the possibility for information storage via the above sequential phase relations). Now, in the general case, with za random phasors we are faced with the problem of estimating the Redundancy of the system as a function of time. Assuming that the amplitudes Az(t) of the phasors are statistically independent from the phases q~(t) and also vary slowly with time we can express the instantaneous entropy of the system as:
r
t 100
where ff(q01,q~2...q~.(t); t) is the joint p.d.f, for the phases of the system. In general the above expression is of academic interest due to the impossibility of calculating the joint p.d.f. In case, however, the probability density distribution of the resultant vector Q(R, O) (Fig. 3) can be deduced explicitly - during time intervals of slow variations - we can heuristically substitute S(t) by the expression:
~(O)log2Q(O)dO --o0
and the Redundancy of the system takes the form:
O(O)log20(O)dO R(t)= l -
-oo log2n(t)
where:
Q(O) = ~ o(R, O)dR. O'
We shall now assume that each phasor z among n independent units of the system, is a hard-nonlinear oscillator in a pool of ambient noise and obeys a dynamical stochastic equation of the form: 5z - 2~(1 - 4az2z + 8flz4~)kz + co2zz = ~(t)
-co -~3
l~
-oo
Q(q~ q~2 ... (PzcP,(o; t) . drPl dq~2 ... d(Pn(t)
(5)
where ~(t) is a stationary random process with zero mean value and power spectral density N (watts/Hz).
190
We are interested in the probabilities of temporal excitation and quenching (interruptions) of selfsustained oscillators like the one above contained in the system Z - under the influence of surrounding noise ~(t). A study of the system described by Eq. (5) has been carried out by Stratonovich (1965) who showed that under the influence of noise an initially non-excited or "dormant" hard oscillator can be asynchronously activated (Minorski, 1962) inspite of a stable steady state at equilibrium (i.e. at rest), and conversely the noise may have the opposite effect of asynchronously quenching existing stable oscillations thereby pushing back the "alive" oscillator into its stable non-excited state. If the (mean) time that the individual oscillator of the ensemble Z is in the (stable) excited state is greater than the (mean) time that the oscillator is in the "dormant" (stable) state, then n(t) will increase, and so there is a possibility for increasing the redundancy of the system l strictly "by noise". The rates of excitation "birth rate" K and quenching "death rate" K1 have been calculated by Stratonovich (1965) as:
K = 2eR ~ 2 ~ g - e f(R) and:
eR
f ( R ) - f(R1)
where:
R=(-a-~'f/2'2fl
R i = ( a + ] / / a12/ 2- 42ffll) u = -- (1 -- 3aR 2 + 5fiR 4)
u 1 = (1 - 3aR~ + 5fin~)
R2 f(R) = e
f(R1)=e
2
aR + fl -6-
Ri2
aR + fl R~ )
2
and N is the spectral density of the surrounding noise in watts/Hz. So the ratio of (mean) times during which the oscillator(s) can be in the non-excited or the excited state is given by:
Substituting the above parameters we find after some algebra the result: Kx
K
(a 2 - 4fl) 1/4
21/
where: 8
C = 1 ~ - (a + ~ a 2 - 4fl) {1
Va~4fl
(a+ ~ } .
For c~, fl of the same order of magnitude (strong hard non-linearity) we may have either: C > 0 or C < 0; in a~
the special case of a>flV2, C,,, ~ - .
So the above
results indicate that for given parameters a, fl, e the mean time during which the oscillator is in the nonexcited state versus the corresponding time that the oscillator is alive varies as ] / ~ e + wv. We can always choose the parameters a, fl, e so that for a given (moderate) level of noise N to have K1/K < 1 i.e. "birth" rate achieve a " d e a t h " rate > 1, so the oscillator(s) spend
more time in the excited state and concequently the number of excited oscillators within a given time interval At increases. We call n(t) the number of oscillators which are "alive" at a given moment and nl (t) the corresponding number of oscillators that are quenched (or rather quiescent). We assume an overall constant number of oscillators in the system i.e. n + nl = no = const. Each oscillator of the species n that "dies" becomes an oscillator of the species nl and vice versa. The rate equations for n and n~ assume the form: d n / d t = K l n l - K n and dnl/dt=Kn-Kln~ from which the closure condition follows nl+n = const. The steady states of the above rate equations are those points for which: dn/dt and dnUdt simultaneously vanish. Every point on the line Kx nt - Kn = 0 on the plane (n, nx) will have this property. So for fixed overall number of oscillators no the steady state point will be at the intersection of the straight line n + nt = no and Kin i - K n = O . Since Ka + K > 0 the steady state will be stable. If neq and n~oq are the steady state numbers of excited and non-excited oscillators we will have neq + nloq = no K and: Kneq-K~nloq=Oand calling K--T= z we find:
f(R1)
K i _ e N V ulN K ~Ri e
]//N eC/N
neq--
Tn 0 z-l-1 'nl~q-
no zd-I
"
191
For the cases of interest where z > 1 neq > n t oq and we see that the steady state numbers of excited vs "dormant" oscillators does not depend on the individual rate constants but only their ratio "c. We also note that these steady state populations depend on the overall number of the oscillators in the system. So the maximum entropy of the system at the steady state will be given as: Smax~l~176176176
1+ - t ) "
We therefore deduce that a "resonably" intense noisy environment (of amplitude stationary noise) may indeed excercise "beneficial" effects (that is act as a more or less "nutricious" environment where oscillators literally "feed on noise"), by increasing the redundancy of the system, under the condition of course that: dn(t) d
Appendix The Computer Simulation
dt
dt (lnS) ~ - n(t~-
a phase-locking organizing process. On the other hand amplitude noise may be beneficiary since it may trigger a birth and death process in the dynamics (of the degrees of freedom) of the system. The combination of both types of noise in a self-organizing system may lead to an increase of redundancy in spite of the fact that the entropy of the system increases. Let us finally emphasize that our attempt to apply the above model in the temporal organization of the brain has so far no direct experimental support and may very well be wrong. It would be, therefore, worthwhile to investigate experimentally the behaviour of switching "on" and "off' (synaptic and genetic) phenomena in cortical and thalamic centers as adaptive responses to changing physiological and environmental conditions (see "note added in p r o o f " at the end of the paper).
(6)
On the other hand, very intense noise acts as a "poisonous" environment by increasing the ratio of the death v.s. the birth process in the population of oscillators of the system S. It remains to add a few words about the origin of the necessary organizing process (or pacemaker) which will exploit the created reduntant degrees of freedom and lead the system to Eq. (6). Let us refer specifically to the case of brain organization we mentioned earlier in the introduction. At present such a pace maker has not yet been discovered in the brain. Hypotheses have been forwarded about possible candidates such as the so called "facultative pacemaker theory" of Anderson and Anderssen (1968), who suggested that the thalamus should contain the cortex's pacemaker by showing that all major thalamic nuclei have the ability to produce rhythmic activity and to control an appropriate part of the cortex. This suggestion however is still under investigation.
1V. Conclusions
Since the Fokker-Planck-Kolmogorov equation cannot be solved analytically, and since even the numerical integration is exceedingly difficult, we have resorted to a Monte-Carlo type method in order to determine the evolution in time of the probability density function p(~; t) of the phase difference:
9 (t) = x~(t) - xl(t).
The dynamicalequations (4) can he put in the form of an ito stochastic differentialequationas follows: dx(t) =f[x(t)] dt + gdw(t) where the state vector:
I potfl Lx3(t)J kA~(t)J f o r t > t o w i t h i n i t i a l state
x(to)= xo;f[x(t)]
is a three-dimensional
vector whose components are zero-memory non-linear transformations of x(t).
[f3(x)J Lf3(xl,x2,x3)J q is also a three dimensional constant vector:
q=
[ql] q2
Lq3A In this paper we have considered the influence of two distinct types of noise in self-organizing systems: phase jitter, or Nonlinear Multiplicative noise (Section II), and amplitude stationary noise (Section III). We have shown that phase jitter noise is always harmful at least when bearing specificity e.g. acting on
W(t) is a one-dimensional Wiener-Levy process (brownian motion) describing the phase jitter. The finite difference equation corresponding, for computer simulation purposes, to the ito stochastic differential equation is: x(n + 1) = x(n) +f[ x(n)] A t + g A W (n)
192
C ,,
Start
)
S
Read data Define constants
I
(Re) Set initial conditions
t
, t 1
Geussian distribution routine
Uniform random Numbers generator (0 TO1) Subroutine
Generate d wl (n)
Find next step x(n+l)=x(n}+F[x(n)]Z~t+dwnl-Compute ~ ( n ) = x 2 (n)-xl (n) _- No Compute ~'~ (n), .,~'~Z(n) Store data for histograms (Probability density)
"~ Yes
No
Compute entropy(s) and variance (o,2)
9,-
t
(
stop
)
Fiot histograms (Probability density of ~(n) from 1000 members of the ensemble) for time instance every 80z~t
Scheme 1. Computer p r o g r a m m e flow chart
193 where the index n corresponds to time t = n A t ( t = 0 for n = 0 , A t = uniform time increment) and A W(n) are independent random increments with Gaussian distribution zero mean, and variance No A t. The flow-chart attached herein indicates the sequence of the program we have used for calculating the entropy S(t).
Acknowledoement. The authors thank Miss Alinda Macrycosta for the typing of the manuscript.
References Adey, W.R.: In: The Mind: Biological approaches to its function Coming, Balaban (Eds.), New York, Wiley Interscience 1968 Andersen, P, Andersson, S: Physiological basis of the Alpha rhythm, New York: Appleton Century Crofts 1968 Borenstein, M., Lamb,W., Jr.: Phys. Rev. Gen. Phys. 5, 1298-1311 (1972) Elul, R.: In: Progress in Biomedical Engineering 131-150, Fogel, L (Ed.) Washington D.C.: Spartan Books 1966 EluI, R.: Data aquisition and Processing in Biology and Medicine p.p. 99-t 14. Enslein (Ed.) Oxford: Pergamon, 1968
Harris, S., McDuff, O.: IEEE J. Quantum Electronics, QE-1, 245-262 (1965) Glandsdorff, P., Prigogine, I.: Thermodynamics Theory of Structure, Stability and Fluctuations, New York: Wiley 197l Lindsey, W.C.: Synchronization systems in communication and control, Prentice Hall 1972 Minorski, N.: Non-linear Oscillations. Princeton: Van Nostrand 1962 Nicolis, J.S., Galanos, G., Protonotarios, M.: Int. J. Control 18, 1009-1027 (1973) Stratonovich, R.L.: In: Non-linear transformation of Stochastic Processes. Kuznetsov, P.I,, Stratonovich, R.L., Tikhonov, V.I; (Eds.). London: Pergamon 1965 Trincher, K. S.: Biology and Information, N.Y.: Consultans Bureau 1965 Prof. Dr. J. S. Nicolis University of Patras School of Engineering Patras, Greece
Note added in proof: Cumulative experiments recently reviewed by Hyd6n (1969) seem to encourage the above hypothesis. Hyd6n and his group found that learning experiments in rats for example had been consistently accompanied by a relative RNA increase per neuron in the learning cortex by 60-100 %. This indicates a stimulation of the genome i.e. the RNA response can be interpreted as reflecting an activation of hitherto silent gene areas in brain cells. Such activation of (in general repressed) genetic loops (switches) can be caused by ionic fluctuations penetrating the genome of the neurons and may be due to either external stimulation (e.g. Microwave radiation, Spiegel and Joines, 1973), or to coupling with the evergoing noisy electrical wave activity of the post-synaptic membrane potentials. Additional References: Hyd6n, H.: Biochemical approaches to learning and memory. In: Beyond ReductionisrrL Koestler, Smythies (Eds.) London: Hutchinson 1969 Spiegel, R.J., Joines, W.T.: Bull. Math. Biol. 35, 591-605 (1973)
Some Views on the Role of Noise in "Self"-Organizing Systems J o h n S. Nicolis Department of Electrical Engineering Univ. of Patras, Patras, Greece E. P r o t o n o t a r i o s and E. Lianos Department of Electrical Engineering, National Technical Univ. of Athens, Athens Greece Received: April 30, 1974
Abstract This paper deals with information transfer from the environment and "self'-organization in open, nonlinear systems far from thermodynamic equilibrium - in the presence of either non-stationary phase jitter noise, or amplitude stationary noise. By "self'-organization we mean here the progressive formation within the system of sequential, ordered (coherent) relationships between appropriate dynamical variables-like for example, the phase differences between the oscillating components of the system. We take up (in Section II) the classical Laser as a specific example and examine in detail the influence of phase jitter noise in the mode (phase) locking process. We find - as expected - that phase fluctuations in the cavity cause degradation of the coherent behaviour (i.e. increase the entropy) of the system - which, however, levels off, or saturates with time. Further (in Section III) we examine systems where the number of self-sustained oscillating components may vary with time in such a way that the maximum entropy of the system increases faster than the overall instantaneous entropy. We put forth the hypothesis that in such cases - because of the increase of the redundancy the system gets organized not just in spite of, but merely because of the presence of Noise. Possible applications in biological systems (especially concerning a model of cerebral organization) are briefly discussed. It is understood here, that the system has to display some 9 preliminary dynamical structure before the organizing procedure takes over. What happens afterwards is the subject of this paper.
I. Introduction and General Formulation of the Problem
Self-organizin9 systems are hierarchical structures; when we deal with them we quite often are confronted with almost-finished products i.e. an "already" sufficiently organized system (like a fertilized egg for example). So, the best we can h o p e to do is to elaborate on the "last" steps of evolution. Incidentally, this is not that b a d : K n o w l e d g e of the system at the higher levels of the hierarchy is perhaps the prerequisite to correctly estimate the relevance of averaging processes we perform when we apply statistical mechanics in order to find (going backwards) links between higher and lower levels of description. O u r discussion in this paper will concern such partially organized systems. N o w a b o u t
the noise we m a k e the following assumptions: Alt h o u g h (strong) noise causes both amplitude and phase (jitter) modulation, we are considering here (low level) noise as either associated with the r a n d o m variables of the system with respect to which organization occurs or not. In the first category (Section II), we classify the n o n - s t a t i o n a r y phase jitter noise, while the organizing process has to do with phase locking. In the second category (Section III) belongs the amplitude (quasi-stationary) noise. After these preliminary remarks let us proceed. We consider an open non-conservative system consisting of a n u m b e r n(t) of nonlinear interacting c o m p o n e n t s ("reactants"). These components can be imagined as self-sustained oscillators (either of a H a m i l t o n i a n or non conservative type), lumped or distributed (normal modes). Each active c o m p o n e n t of the system is represented by a r a n d o m p h a s o r (i.e. a n o n h a r m o n i c oscillation in a "pool" of weak additive noise :)
Z~(t) = A~(t)e J ~ t ) where: q~(t) = cozt+ q~z(t), characterised by a continuous joint probability density distribution function (p.d.f): 0z(Az, q)~;t). Dynamically the system Z is modeled by a set of n(t) coupled nonlinear differential (rate) equations, which in the absence of time-varying spatial differentiation (V2zi = 0 ) read:
dz~ dt - L ( z l , z2 ...z~... z,~0
dzz - L ( z l , z2 ... z~... dt
z,~3
d zn(t) dt - f,(,)(z 1, Zz ... zz ... z,(,))
(1)
184
and unless otherwise stated it is assumed that:
dn(t) OA k dt ~ O t -
'
.~i
'
0
Oq)~ ,gt
By expressing the dynamical behaviour of the system as (1), we essentially imply that all activity in it originated as noise which has been more or less constrained at an early age through an interplay between the controlling environment and the system's parameters and boundary conditions. The noise has not been wholly constrained however; some of it (what we call additive noise in this paper) escapes control and gives rise to an element of arbitrariness or flexibility, or even (it depends on how you look at it) "originality" in the system's behaviour. Since linearization is not allowed in our problem because the system Z is considered far from equilibrium, the solution of (1) is prohibitively difficult (and eventually unnecessary) when n(t) is sufficiently large. In order to' follow the behaviour of the system in time, what is important to derive in a situation like this is not the analytical expressions of the coupled stochastic processes Z~ but rather their joined probability density distribution: Q(zl, z2 ... z,(t); t). This could be achieved in principle by deriving from (l) and subsequently solving under given initial and boundary conditions, the corresponding n(t)+ 1-dimensional Fokker-Planck equation (Lindsey, 1972), which describes the diffusion in time of the joint p.d.f. ~(zl, Zz 9.. z,(t); t). 1 From this distribution one can derive the "instantaneous" (conditional) entropy of the system: S(t) = c~ + c2 ~ O(z~, z2 ... z,~,; t)
9logz~(Zl, z 2 ... z,(t); t)dv
(2)
where: dv = dza. dz2 ... dz,(t), the integration is extended over the volume v in the functional space, zl,z2, ... z,l~) become eventually statistically correlated as a result of the organizing process going on, and el, c2 are constants9 One has to point out immediately of course that for a given dynamical system Z there exist many"entropies" since the entropy is a property of the state, i.e. a probability density distribution with respect to specific variables. Depending on the relevance of the variables chosen for each model the calculated entropy may or may not represent the degree of disorganization of the ~ system (see also next sections). We now proceed in the formulation of our problem as follows: Our open system Z interacts energetically with the environment U (Fig. 1) and concequently the 1 In the l i m i t of w e a k noise a r o u n d a s t e a d y state.
-
d Se > 0
= "ataxia,,
d Se < 0
= "order,,
Fig, i. Information interaction between the environment (2') and a system (X) composed of non-linear mutual interacting oscillators
differential of the entropy of the system dS(t) can be written (Glansdorff and Prigogine, 1971) as: dS(t) = dSi + dSe where dS~ is the differential of the internally produced entropy (dSi >0 always) and dSe is the differential of the entropy of the inflowing energy from the environment Z: The sign of the term dSe is undetermined. In the trivial case dSe > 0 i.e. disorder or "ataxia" is entering the system from outside and this accelerates the process of disorder by moving the system toward the thermodynamic equilibrium faster9 In the case dSe < 0 order or "negentropy" enters from theenvironment. It is understood of course that it is the system Z itself which "decides" about the sign of dSe. In other words information transfer is taking place between a series of events in Z' and another series of events in Z if there is some degree o f - not necessarily causal - correspondance between them. Two possibilities exist now: Either [dSe[ < dS~: as a result the degradation of the system will slow down and perhaps level off in time, or [dS~[ > dSv In this latter case dS(t)< 0 i.e. the entropy of the system will decrease at the expense of the environment 9 Such a possibility may very well occur in a "large enough" isolated "world" containing Z ' + Z: Within such a world we may witness ordered structures (Z) take shape without the world's overall evolution ceasing to comply with the second law of thermodynamics. Consider for instance the example of the crystallization in a saturated solution or, even more appropriately the case of growth of a population of bacteria in a "isolated" system consisting of water and nutritient such as glucose plus some mineral salts that can effectively be used by the metabolic mechanisms of the living cells. Such examples are however slightly irrelevant to the present situation: they either refer to structures near equilibrium (first example above) or they preassume the existence of highly organized, coherent structures far from equilibrium. In this work
185
we start however, with a given open poorly-organized system (Z) far from equilibrium and are being examining the condition under which we can make it more coherent (i.e. increase the number of ordered relationships between appropriate dynamical variables e.g. qh) by absorbing information from the environment in the presence of additive fluctuations existing between the components of the system. Under these circumstances and for a fixed number of active components n it appears that the non-stationary (phase jitter) noise (see Section II) will cause deterioration i.e. it will always overtake the whatever organizing processes which was acting on the phases of the oscillators. So the condition ]dSe] > dSi cannot be fulfilled at least in this case. After all the Fokker-Planck equation is a diffusion equation so what is predicted is a spreading rather then "shrinking" of the p.d.f. ~(qo1, rp2 ...qo,; t) with time and consequently the entropy of the system S(t) will increase (again for small deviations from the steady state). Nevertheless the organization of the system which can properly be defined in terms of its Redundancy R (expressing intercomponent (q~,, q0~)correlation) may increase even if the entropy S(t) goes up provided that the maximum entropy of the system
s(t)
increases faster than S(t) with time. Since R = 1 - - -
S max
and Sm.x = c3 + e4 log2n(t) it is obvious that: -OR ~ - >_0 _ if: S
OSmax
Ot
>__-Smax
8S
Ot
or
d
~
a~
d
[ l n S m j > ~ [IDS]. at
Of course for an invariant number of active components n = const the system Z can never comply with this requirement. The only possibility for organization is that new degrees of freedom continuously enter the organizing process under an instantaneous overall entropy growing slower than log 2 n(0. By what mechanism such a thing could possibly be realized? We elaborate somewhat on this in Section III; let us discuss for the moment two possible cases where this process might perhaps occur. 1. During the embryogenesis of a biological organism the Redundancy must increase (Fig. 2) although it is known that during this particular period of development the internal entropy variation of the system (AS), is increasing very fast. This has been indeed shown by Trintcher (1965) who gives experimental evidence that the entropy production dSi/dt per unit mass and unit time (in cal/gram hour) increases during the first (embryogenetic) period of the dif-
R
i
i
l P,
"O
a:
th process ~ t
Fig. 2. Suggested variation of Redundancy during the life cycle of a self-organizing system
ferentiating organism, then passes through a maximum and finally decreases to a steady state value with the attainment of the organisms adult state. This suggests for the entropy (A S)i a "sigmoid"- like type of variation:
(AS)i ~
1
1 + exp [ - a(t - to)]
i.e. a very steeply rising
front. Here the increase of redundancy is accomplished, we believe, through "fast" increase of n(t) i.e. cell division and multiplication. On the other hand during the adult life of the organism cells keep dividing but do not normaly multiply. The value of 8R/St must therefore be maintained in the region of "zero plus" while the system rides its "limit cycle" being in a dynamical steady state. [in case of ~R/Ot~>O (R--, 1) abnormally tight organization or "petrification" would follow which makes the system unable to display adaptive behaviour; for OR/St< 0(R~0) a "death process" is initiated.] 2. Our second example has to do with the electrical activity of the brain as related to changing physiological states: Intracellular records have revealed spontaneous wave like activity with amplitudes in the scale of 5-20 mV in many cerebral structures (Elul, 1966, 1968; Adey, t968). These intracellular recordings show that even in the absence of propagated nerve impulses, rhythmic noise - like slow potential changes occur due to (subthreshold) fluctuations of the cells membrane resting potential. Power density distributions in these records resemble closely spectra from similar analyses of gross Electroencephalographic records (EEG) from the same cortical regions: This means that the EEG can be considered as the (scalar) projection of a vector superposition of n(t) phasors (Fig. 3) with central frequencies c~a in the neighborhood of ~ 10Hz - each representing the individual neuron's fluctuating postsynaptic membrane
186
o Fig. 3. Diagram of n-interacting random phas0rs
potential. Now, as has been pointed out, many times by workers in this field [see Elul (1968) for example], the spontaneous electrical activity of the brain still remains one of the most elusive aspects of cerebral functions: It is a (non-stationary, beyond time intervals of a few seconds) continuous wave activity of variable amplitude and frequency with inconstant phase relations (~pz- (p~ 4= const) between its constituents phasors zx; its overall character is quite similar to random noise (although increasing skewness of the p.d.f, of the EEG composite phasor during performance of mental tasks (Elul, 1966, 1968) indicates progressive (linear) statistical coupling between the individual neuronal phasors). And yet the behavioural aspects of the brain operation are quite orderly. So, unless the electrical activity of the brain is simply concommitant with behaviour, there is a possibility that unlike man-made communication systems in which noise is an encumbrance, the brain is a system which may use noise as an inevitable and "beneficial" factor. It is known of course that neuron cells do not divide in the brain beyond a rather early age of the organism so one could not justify any increase of brain organization on that basis. There is, however, the possibility (and this is our hypothesis) of a number of phasors in the system being "dormant" for given periods of time, getting excited whenever the fulfilment of the condition ~R/Ot > 0 requires action. To avoid any misunderstanding we point out that just activating above threshold the previously "dormant" nonlinear oscillators is not enough; the procedure has to take place under at least constant entropy S(t) i.e. the newly activated degrees of freedom must be subsequently subjected to the same "locking process" already acting on the "old" members of the system. This locking process is achieved through information flowing from the environment. We will try to qualify the above statement in the subsequent sections.
In Section II we examine the influence of phasejitter noise in the efficiency of mode locking process in a multimode Laser with a fixed number n of normal modes. This is an example where the organization of the system may be theoretically "perfect" (R = 1) in the absence of noise but deterioration is initiated when phase jitter develops in the cavity. In Section III we examine the possibility of achieving amelioration of the Redundancy in the system (Z) by allowing the number of self-sustained oscillators n(t) to vary with time as a result of switching on and off caused by noise catalytic action. With the aid of such a model we may approach an understanding of how very noisy biological systems - like the brain - can display orderly behaviour.
II. The Influence of Non-Stationary Phase Jitter Noise in Laser Mode Locking
We first review briefly the Laser operation and the mode locking process in the absence of additive noise (Borenstein and Lamb, 1972; Harris and McDuff, 1965). We consider an one dimensional cavity resonator filled with an active material which in the classical approach is being considered as an ensemble of classical nonlinear oscillators homogeneously distributed in the cavity and activated by a classical (external pump) electromagnetic field. For specific combinations of the values of the parameters of the system the excited non-linear oscillators transfer on the average energy to the normal modes in the cavity. The activated (axial) normal modes form two categories: Those "hard-driven" which behave as self-sustained oscillators (with a quality factor "Q" beyond a certain critical "threshold") and the rest which being dynamically unstable amount to joule losses or thermal noise in the cavity. So, through the above classical non-linear interactions we establish within the cavity the existance of loosely coupled self-sustained normal modes. The "output" of such a multimode laser is not however coherent (since the amplitudes and phases of the individual "free-running" modes change independently with time), and differs little from random noise. In order to make the sum of these modes coherent we have to operate from outside ke. to provide the system with an organizing perturbation process which will cause the establishment of ordered (time independent) relationships between the successive phase differences viz. (p~+l(t)- (p~(t)= const, or, will "mode lock" the laser.
187 One very efficient way to achieve this is the following: (Harris and Mc Duff, 1965). We introduce an intracavity phase perturbation at a frequency co,. which is approximately the frequency of the (axial) mode interval A f2. The effect of this intracavity phase perturbation is to frequency modulate the previously free running Laser modes i.e. to cause each one of these modes to become the center frequency of an FM signal9 While the free-running modes experience their gain from independent coordinates of the emission spectrum of the active material, the competing FM oscillations to a large extend "see" the same spectrum so they are "fed" from the same (i.e. the entire) source and not - as previously - from different parts of this source9 The competing FM oscillations are thus much more tightly coupled than were the previous free-running Laser modes. The result of this competition is that the strongest of the F M oscillations - usually the oscillation whose carrier is at the center of the emission spectrum of the working material, is able to completely quench the competing weaker oscillations and establish the steady state regime in which q~s+1 - r = const for all 2 between 1 and n where n is the number of locked modes9 Assuming small perturbations we write the coupled equations between each normal mode and those two in its immediate vicinity:
Oq)x _ 2A co) A s Ot 9{As+ ~cos(cps+l - r
~?As &
=
6c rbl = Aco - 4 L - {cos~01 + cos(c#1 - %)} At
_
6c 4L At {sinrpl + sin(tpl - %)}
(4)
where ~bo= ~(t). The spectral density of the noise is given, No Watt/Hz. In the following we intend to study the evolution of the phase difference ~b= (Pl - % in time and particularly to calculate for different sets of parameters: {Aco, c~= c3c/4L} of the system the p.d.f. Q(@; t)the variance o-g(t) and the entropy of the system S(t). It can be easily deduced from (4) that the corresponding Fokker-Planck equation has the form: OP(COo,~0t; t) at
c~ [(A co + at/(q~o,qO 1)p(Cpo , @1 ;t)] Ocpl +
N 0 02p(q~O, ~0l; t) 2
where:
tic 2L + As_ lcos(rpz- % _ i) }
condition i.e. only these three modes are considered self-sustained oscillators. We normalize the phases % and ~o1 to q~2 by taking q~2 const = 0 and we consider Ao = A2 = A1/2. The phase Cpo(t) is taken to perform a brownian motion i.e.: dqb/dt - ~(t) is an additive white Gaussian noise process. With these assumptions Eq. (3) become:
t/(Cpo, qh) = cos (opt - %) - cosqh
(3)
6c 2L
9 {As+ t sin(q~s+ 1 - ~oa)- As- 1 sin(q~s - q~s- 1)}. The steady state solutions of the above equations give the mode locking regime9 In these equations 2Aco = (2s - cos where g?s is the vacuum eigen frequency and cox is the eigen frequency of the same 2th normal mode in the presence of the active material filling the cavity: cos=Oo+2co,, where: f2o is the central mode frequency and co,,,in the external perturbation frequency. L is the geometrical length of the cavity, c is the velocity of light and 6 is the coupling coefficient between the 2 'h normal mode and its neighbouring modes 2 - 1 and 2 + 1. In order to make things easier in the example that follows, we are going to choose an active material with a narrow emission band such that only three normal modes (Ao, (Po), (At, opt), (A2, cP2) will satisfy the threshold
= 2 s i n ( opt
~ - ) sinqb2
Introducing the dimensionless time:
t' = at and the parameters: Am "normalized mistuning" 7 -- - - - , a "noise to signal ratio"
x = No 2a"
We rewrite the FP equation as follows: Op(cpo, ~0i ; t') Ot' 9 [(7 +
0 Oq)t c.00 ) p ( % .
Ol; t')3 +
OZp(~Oo,~ot ; t')
188
A(LI
J
No
NO-
a
10 ~
V I
I
50 Fig. 4 a
"
I
t 100
I
50
.
t
100
Fig. 4 b
S 3,8 I Z~=
3,3
0
N
NO =1@0
2,8
:a_~_
O,E
l
50
I
t 100
J
0
Fig. 4c
50
J
t 100
Fig. 4 d Fig. 4. The Entropy as a function of time (arbitrary units)
The p.d.fi p(~, t) of the phase difference: ~b= ~o1 -q~0 can in principle be deduced from the solution of FP equation as: p(~; t) = ~ p(q%, r + {#; t)dq~o -oo
and the entropy of the system:
S(t) = -
~ P(q~, t)log2P(~, t)dq~. -oo
Since, however, the FD equation cannot be solved analytically and its numerical integration is exceedingly difficult we resort to a Monte-Carlo type method. (See flow chart diagram in the appendix.) Some results of the computing simulation concerning the entropy S(t) are displayed in Figs. 4 for the indicated sets of variables; it is apparent that for weak jitter noise (i.e. limited time intervals) the disorganization of the mode locking process induced by phase jitter levels off with time.
189
f
Q
N No
r
a
No
I
50
t
I
I
100
50 Fig. 4 f
Fig. 4 e
llL Organization Through Stationary Amplitude Noise We now consider a system consisting of n(t) random phasors: z~ (Fig. 3) with fixed frequencies ~ very close together, and amplitudes A~(t) and phases opt(t) random processes. Let us briefly consider, first what may happen for n = const, and in the absence of noise i.e. in case where zz are deterministic non-linear coupled oscillators with independently varying amplitudes and phases between oscillators. Since the frequencies coz are very close together the system can find itself either in the regime of frequency entrainment (e.g. Nicolis et al., 1973) where all oscillators operate in synchronythereby implying functional homogeneization of the system, or, in the regime of phase locking q)z+x- (Px - c o n s t - which implies a sequential process or a differentiated state of function (and suggests the possibility for information storage via the above sequential phase relations). Now, in the general case, with za random phasors we are faced with the problem of estimating the Redundancy of the system as a function of time. Assuming that the amplitudes Az(t) of the phasors are statistically independent from the phases q~(t) and also vary slowly with time we can express the instantaneous entropy of the system as:
r
t 100
where ff(q01,q~2...q~.(t); t) is the joint p.d.f, for the phases of the system. In general the above expression is of academic interest due to the impossibility of calculating the joint p.d.f. In case, however, the probability density distribution of the resultant vector Q(R, O) (Fig. 3) can be deduced explicitly - during time intervals of slow variations - we can heuristically substitute S(t) by the expression:
~(O)log2Q(O)dO --o0
and the Redundancy of the system takes the form:
O(O)log20(O)dO R(t)= l -
-oo log2n(t)
where:
Q(O) = ~ o(R, O)dR. O'
We shall now assume that each phasor z among n independent units of the system, is a hard-nonlinear oscillator in a pool of ambient noise and obeys a dynamical stochastic equation of the form: 5z - 2~(1 - 4az2z + 8flz4~)kz + co2zz = ~(t)
-co -~3
l~
-oo
Q(q~ q~2 ... (PzcP,(o; t) . drPl dq~2 ... d(Pn(t)
(5)
where ~(t) is a stationary random process with zero mean value and power spectral density N (watts/Hz).
190
We are interested in the probabilities of temporal excitation and quenching (interruptions) of selfsustained oscillators like the one above contained in the system Z - under the influence of surrounding noise ~(t). A study of the system described by Eq. (5) has been carried out by Stratonovich (1965) who showed that under the influence of noise an initially non-excited or "dormant" hard oscillator can be asynchronously activated (Minorski, 1962) inspite of a stable steady state at equilibrium (i.e. at rest), and conversely the noise may have the opposite effect of asynchronously quenching existing stable oscillations thereby pushing back the "alive" oscillator into its stable non-excited state. If the (mean) time that the individual oscillator of the ensemble Z is in the (stable) excited state is greater than the (mean) time that the oscillator is in the "dormant" (stable) state, then n(t) will increase, and so there is a possibility for increasing the redundancy of the system l strictly "by noise". The rates of excitation "birth rate" K and quenching "death rate" K1 have been calculated by Stratonovich (1965) as:
K = 2eR ~ 2 ~ g - e f(R) and:
eR
f ( R ) - f(R1)
where:
R=(-a-~'f/2'2fl
R i = ( a + ] / / a12/ 2- 42ffll) u = -- (1 -- 3aR 2 + 5fiR 4)
u 1 = (1 - 3aR~ + 5fin~)
R2 f(R) = e
f(R1)=e
2
aR + fl -6-
Ri2
aR + fl R~ )
2
and N is the spectral density of the surrounding noise in watts/Hz. So the ratio of (mean) times during which the oscillator(s) can be in the non-excited or the excited state is given by:
Substituting the above parameters we find after some algebra the result: Kx
K
(a 2 - 4fl) 1/4
21/
where: 8
C = 1 ~ - (a + ~ a 2 - 4fl) {1
Va~4fl
(a+ ~ } .
For c~, fl of the same order of magnitude (strong hard non-linearity) we may have either: C > 0 or C < 0; in a~
the special case of a>flV2, C,,, ~ - .
So the above
results indicate that for given parameters a, fl, e the mean time during which the oscillator is in the nonexcited state versus the corresponding time that the oscillator is alive varies as ] / ~ e + wv. We can always choose the parameters a, fl, e so that for a given (moderate) level of noise N to have K1/K < 1 i.e. "birth" rate achieve a " d e a t h " rate > 1, so the oscillator(s) spend
more time in the excited state and concequently the number of excited oscillators within a given time interval At increases. We call n(t) the number of oscillators which are "alive" at a given moment and nl (t) the corresponding number of oscillators that are quenched (or rather quiescent). We assume an overall constant number of oscillators in the system i.e. n + nl = no = const. Each oscillator of the species n that "dies" becomes an oscillator of the species nl and vice versa. The rate equations for n and n~ assume the form: d n / d t = K l n l - K n and dnl/dt=Kn-Kln~ from which the closure condition follows nl+n = const. The steady states of the above rate equations are those points for which: dn/dt and dnUdt simultaneously vanish. Every point on the line Kx nt - Kn = 0 on the plane (n, nx) will have this property. So for fixed overall number of oscillators no the steady state point will be at the intersection of the straight line n + nt = no and Kin i - K n = O . Since Ka + K > 0 the steady state will be stable. If neq and n~oq are the steady state numbers of excited and non-excited oscillators we will have neq + nloq = no K and: Kneq-K~nloq=Oand calling K--T= z we find:
f(R1)
K i _ e N V ulN K ~Ri e
]//N eC/N
neq--
Tn 0 z-l-1 'nl~q-
no zd-I
"
191
For the cases of interest where z > 1 neq > n t oq and we see that the steady state numbers of excited vs "dormant" oscillators does not depend on the individual rate constants but only their ratio "c. We also note that these steady state populations depend on the overall number of the oscillators in the system. So the maximum entropy of the system at the steady state will be given as: Smax~l~176176176
1+ - t ) "
We therefore deduce that a "resonably" intense noisy environment (of amplitude stationary noise) may indeed excercise "beneficial" effects (that is act as a more or less "nutricious" environment where oscillators literally "feed on noise"), by increasing the redundancy of the system, under the condition of course that: dn(t) d
Appendix The Computer Simulation
dt
dt (lnS) ~ - n(t~-
a phase-locking organizing process. On the other hand amplitude noise may be beneficiary since it may trigger a birth and death process in the dynamics (of the degrees of freedom) of the system. The combination of both types of noise in a self-organizing system may lead to an increase of redundancy in spite of the fact that the entropy of the system increases. Let us finally emphasize that our attempt to apply the above model in the temporal organization of the brain has so far no direct experimental support and may very well be wrong. It would be, therefore, worthwhile to investigate experimentally the behaviour of switching "on" and "off' (synaptic and genetic) phenomena in cortical and thalamic centers as adaptive responses to changing physiological and environmental conditions (see "note added in p r o o f " at the end of the paper).
(6)
On the other hand, very intense noise acts as a "poisonous" environment by increasing the ratio of the death v.s. the birth process in the population of oscillators of the system S. It remains to add a few words about the origin of the necessary organizing process (or pacemaker) which will exploit the created reduntant degrees of freedom and lead the system to Eq. (6). Let us refer specifically to the case of brain organization we mentioned earlier in the introduction. At present such a pace maker has not yet been discovered in the brain. Hypotheses have been forwarded about possible candidates such as the so called "facultative pacemaker theory" of Anderson and Anderssen (1968), who suggested that the thalamus should contain the cortex's pacemaker by showing that all major thalamic nuclei have the ability to produce rhythmic activity and to control an appropriate part of the cortex. This suggestion however is still under investigation.
1V. Conclusions
Since the Fokker-Planck-Kolmogorov equation cannot be solved analytically, and since even the numerical integration is exceedingly difficult, we have resorted to a Monte-Carlo type method in order to determine the evolution in time of the probability density function p(~; t) of the phase difference:
9 (t) = x~(t) - xl(t).
The dynamicalequations (4) can he put in the form of an ito stochastic differentialequationas follows: dx(t) =f[x(t)] dt + gdw(t) where the state vector:
I potfl Lx3(t)J kA~(t)J f o r t > t o w i t h i n i t i a l state
x(to)= xo;f[x(t)]
is a three-dimensional
vector whose components are zero-memory non-linear transformations of x(t).
[f3(x)J Lf3(xl,x2,x3)J q is also a three dimensional constant vector:
q=
[ql] q2
Lq3A In this paper we have considered the influence of two distinct types of noise in self-organizing systems: phase jitter, or Nonlinear Multiplicative noise (Section II), and amplitude stationary noise (Section III). We have shown that phase jitter noise is always harmful at least when bearing specificity e.g. acting on
W(t) is a one-dimensional Wiener-Levy process (brownian motion) describing the phase jitter. The finite difference equation corresponding, for computer simulation purposes, to the ito stochastic differential equation is: x(n + 1) = x(n) +f[ x(n)] A t + g A W (n)
192
C ,,
Start
)
S
Read data Define constants
I
(Re) Set initial conditions
t
, t 1
Geussian distribution routine
Uniform random Numbers generator (0 TO1) Subroutine
Generate d wl (n)
Find next step x(n+l)=x(n}+F[x(n)]Z~t+dwnl-Compute ~ ( n ) = x 2 (n)-xl (n) _- No Compute ~'~ (n), .,~'~Z(n) Store data for histograms (Probability density)
"~ Yes
No
Compute entropy(s) and variance (o,2)
9,-
t
(
stop
)
Fiot histograms (Probability density of ~(n) from 1000 members of the ensemble) for time instance every 80z~t
Scheme 1. Computer p r o g r a m m e flow chart
193 where the index n corresponds to time t = n A t ( t = 0 for n = 0 , A t = uniform time increment) and A W(n) are independent random increments with Gaussian distribution zero mean, and variance No A t. The flow-chart attached herein indicates the sequence of the program we have used for calculating the entropy S(t).
Acknowledoement. The authors thank Miss Alinda Macrycosta for the typing of the manuscript.
References Adey, W.R.: In: The Mind: Biological approaches to its function Coming, Balaban (Eds.), New York, Wiley Interscience 1968 Andersen, P, Andersson, S: Physiological basis of the Alpha rhythm, New York: Appleton Century Crofts 1968 Borenstein, M., Lamb,W., Jr.: Phys. Rev. Gen. Phys. 5, 1298-1311 (1972) Elul, R.: In: Progress in Biomedical Engineering 131-150, Fogel, L (Ed.) Washington D.C.: Spartan Books 1966 EluI, R.: Data aquisition and Processing in Biology and Medicine p.p. 99-t 14. Enslein (Ed.) Oxford: Pergamon, 1968
Harris, S., McDuff, O.: IEEE J. Quantum Electronics, QE-1, 245-262 (1965) Glandsdorff, P., Prigogine, I.: Thermodynamics Theory of Structure, Stability and Fluctuations, New York: Wiley 197l Lindsey, W.C.: Synchronization systems in communication and control, Prentice Hall 1972 Minorski, N.: Non-linear Oscillations. Princeton: Van Nostrand 1962 Nicolis, J.S., Galanos, G., Protonotarios, M.: Int. J. Control 18, 1009-1027 (1973) Stratonovich, R.L.: In: Non-linear transformation of Stochastic Processes. Kuznetsov, P.I,, Stratonovich, R.L., Tikhonov, V.I; (Eds.). London: Pergamon 1965 Trincher, K. S.: Biology and Information, N.Y.: Consultans Bureau 1965 Prof. Dr. J. S. Nicolis University of Patras School of Engineering Patras, Greece
Note added in proof: Cumulative experiments recently reviewed by Hyd6n (1969) seem to encourage the above hypothesis. Hyd6n and his group found that learning experiments in rats for example had been consistently accompanied by a relative RNA increase per neuron in the learning cortex by 60-100 %. This indicates a stimulation of the genome i.e. the RNA response can be interpreted as reflecting an activation of hitherto silent gene areas in brain cells. Such activation of (in general repressed) genetic loops (switches) can be caused by ionic fluctuations penetrating the genome of the neurons and may be due to either external stimulation (e.g. Microwave radiation, Spiegel and Joines, 1973), or to coupling with the evergoing noisy electrical wave activity of the post-synaptic membrane potentials. Additional References: Hyd6n, H.: Biochemical approaches to learning and memory. In: Beyond ReductionisrrL Koestler, Smythies (Eds.) London: Hutchinson 1969 Spiegel, R.J., Joines, W.T.: Bull. Math. Biol. 35, 591-605 (1973)
