John, S. Nicolis

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 preliminary dynamical structure before the organizing procedure takes over. What happens afterwards is the subject of this paper.

Markov chains are used to characterize a Dynamical system after it has reached the chaotic regime when certain external parameters have passed specific critical values. For a quantitative treatment of this stochastic behavior one needs an invariant measure. This measure then depends on the external parameters. We propose an emperical method to construct from first principles an invariant measure for the particular case of the triangular map.

This paper considers a non-conservative pendulum-type non-linear oscillator which emulates some behavioural features of systems displaying adaptive response, or biological rhythms, under the influence of an environmental excitation ('Zeitgeber') of ...

The two-slit delayed experiment is re-examined from the standpoint of nonlinear dynamics in the presence of multiple attractors and fractal basin boundaries. It is suggested that the results may be interpreted as the response of the underlying system to a temporary switch of one control parameter, rather than as a retroaction between this system and the observer.

A given but otherwise random environmental time series impinging on the input of a certain biological processor passes through with overwhelming probability practically undetected. A very small percentage of environmental stimuli, though, is «captured» by the processor's nonlinear dissipative operator asinitial conditions, that is, assolutions of the processor's dissipative dynamics. The processor, then, is in such cases instrumental in compressing or abstracting those stimuli, thereby making the external world to collapse from a previous regime of a «pure state» of suspended animation onto a set of stable eigenfunctions or «categories»—chaotic strange attractors. The characteristics of this cognitive set depend on the operator involved and the hierarchical level where the abstraction takes place. In this paper we model the physics of such a cognitive process and the role that the thalamo-cortical pacemaker of the (human) brain plays in both stimulating the individual attractors and permutating them on a time division multiplexing basis. A synthesis of the Markovian processes taking place within each individual attractor-memory and the Markovian or semi-Markovian process involving the intermittent jumping among the different attractor-memories are discussed. Una data ma casuale serie temporale ambientale che influisce sull'immissione di un determinato processore biologico passa con schiacciante probabilità praticamente inosservata. Una piccolissima percentuale di stimoli ambientali, tuttavia, è catturata dall'operatore dissipativo non lineare del processore comecondizioni iniziali, cioè comesoluzioni della dinamica dissipativa del processore. Il processore è quindi, in tali casi, strumentale nel comprimere o estrarre quegli stimoli, facendo perciò collassare il mondo esterno da un precedente regime di «stato puro» ad animazione sospesa ad un gruppo di autofunzioni stabili o «categorie»—attrattori caotici strani. Le caratteristiche di questo gruppo cognitivo dipendono dall'operatore coinvolto e dal livello gerarchico in cui l'estrazione ha luogo. In questo lavoro si modella la fisica di tale processo cognitivo e il ruolo che lo stimolatore talamo-corticale del cervello (umano) ha sia nello stimolare gli attrattori individuali che nel permutarli su una base che manda segnali multiplex a divisione temporale. Si discute una sintesi dei processi markoviani che hanno luogo all'interno di ogni attrattore-memoria individuale e il processo markoviano o semimarkoviano che interessa il salto intermittente tra i diversi attrattori-memoria. Временная последовательность на входе некоторого биологического процессора может иметь практически недетектируемую вероятность. Очень небольшой процент стимулов окружающей среды «захватывается» нелинейным диссипативным оператором процессора как начальные условия, т.е. представляется как решение диссипативпой динамики процессора. Затем процессор обрабатывает внешние стимулы, вынуждая внешний мир коллапсировать из предыдущего режима «чистого состояния» приостановленного оживления в систему собственных функций или «категорий». Характеристики зтой познавательной системы зависят от используемого оператора и иерархического уровня, на котором происходит абстрагирование. В зтой статье мы моделируем физику такого познавательного процесса. Обсуждается синтез марковских процессов, происходящий внутри отдельного злемента «атрактор-память», и марковский или полумарковский процесс, включающий пережежающиеся прыжки между различными злементами «атракторпамять».

""The interactions of self-organizing systems possessing at least two hierarchical levels with the environment are dealt with. The dynamical deliberations taking place at the lower level Q of the organism are modelled by a finite state controlled Markov chain. The transitional probability matrix is parametrized on control variables which are related to the probabilities of “pay off” in an underlying two-agent “game” described in the paper. The environment is envisaged as another Markov chain with fixed transitional probabilities. The higher hierarchical level W plays the role of a controller which receives from the lower level Q collective properties which measure (a) the percentage of occupancy of a “homeostatic” state and (b) the cross-correlation between the dynamical processes at the level Q and the environment. The design of the controller is set up as a two-objective control problem having to do with the maximization of the long term average of an appropriately weighted sum of the conflicting terms: probability of the homeostatic state and cross-correlation with the environment. Optimal strategies are found through computing simulation which determine the efferent controls precipitated from the level W to the level Q. The relevance of the model to a number of problems related to the adaptability of hierarchical structures is discussed. Finally some extensions of the present work to do with the communication process between two multi-hierarchical systems are mentioned. ""

The interaction between two self-organising systems, each possessing two hierarchical levels, is dealt with. The communication process is pursued as a bidirectional information transfer. The criterion for the selection of the intrasystem control algorithm(s) is based on the maximisation of the 'Joint Figure of Merit' related to the long-term average of an appropriately weighted sum of conflicting terms: Probabilities of homeostases versus adequate matching with the partner.

A chaotic dynamics model of creating Markovian strings of symbols as well as sequences of words is presented, and its possible relevance to Zipf’s law is discussed. Comparison from contemporary Greek prose as well as nucleotide sequences (strings) of mRNA Polymerase III and embryonic cDNA displaying a quasi-Zipf’s law behaviour is made. The order of generated Markovian strings as well as the information transferred between two symbolst steps apart (the «transinformation») is also estimated. Si presenta un modello di dinamica caotica di creazione di stringhe markoviane di simboli, nonché di sequenze di parole e la sua attinenza alla legge di Zipf. Si fa un confronto dalla prosa greca contemporanea e da sequenze (stringhe) di nucleotidi di mRNA polimerasi III e cDNA embrionico che mostra un comportamento della legge di quasi-Zipf. Si calcola anche l’ordine delle stringhe markoviane generate e l’informazione trasferita tra due simboli spostati di passit (la «transinformazione»). Предлагается модель хаотической динамики для образующихся Марковских цепочек символов, а также последовательностей слов. Обсуждается связь этой модели с законом Эипфа. Проводится сравнение с современнной греческой прозой, а также с последовательностями (цепочками) нуклеотидов mRNA полимеразы III и эмбриональных cDNA, обнаруживающих поведение, соответствующее закону квази-Эипфа. Определяется порядок образующихся Марковских цепочек, а также информация передаваемая на расстоянииt шагов между двумя символами.

In a well-known collection of his essays in cognitive psychology Miller (The Psychology of Communication. Penguin, 1974) describes in detail a number of experiments aiming at a determination of the limits (if any) of the human brain in processing information. He concludes that the 'channel capacity' of human subjects does not exceed a few bits
or that the number of categories of (one-dimensional) stimuli from which unambiguous judgment can be made are of the order of 'seven plus or minus two'. This 'magic number' holds also, Miller found, for the number of random digits a person can correctly recall on a row and also the number of sentences that can be inserted inside a sentence in a natural language and still be read through without confusion. In this paper we propose a dynamical model of information processing by a self-organizing system which is based on the possible use of strange attractors as cognitive devices. It comes as an amusing surprise to find that such a model can, among other things, reproduce the 'magic number seven plus-minus two' and also its variance in a
number of cases and provide a theoretical justification for them. This justification is based on the optimum length of a code which maximizes the dynamic storing capacity for the strings of digits constituting the set of external stimuli. This provides a mechanism for the fact that the 'human channel', which is so narrow and so noisy (of the order of just a few bits per second or a few bits per category) possesses the ability of squeezing or 'compressing' practically an unlimited number of bits per  symbol -there by giving rise to a phenomenal memory.

"The two-slit delayed experiment is re-examined from the standpoint of nonlinear dynamics in the presence of multiple attractors and fractal basin boundaries. It is suggested that the results may be interpreted as the response of the underlying system to a temporary switch of one control parameter, rather than as a retroaction between this system and the observer.
"

"The interactions of self-organizing systems possessing at least two hierarchical levels with the environment are dealt with. The dynamical deliberations taking place at the lower level Q of the organism are modelled by a finite state controlled Markov chain. The transitional probability matrix is parametrized on control variables which are related to the probabilities of “pay off” in an underlying two-agent “game” described in the paper.
The environment is envisaged as another Markov chain with fixed
transitional probabilities. The higher hierarchical level W plays the role of a controller which receives from the lower level Q collective properties which measure (a) the percentage of occupancy of a “homeostatic” state and (b) the cross-correlation between the dynamical processes at the level
Q and the environment.
The design of the controller is set up as a two-objective control problem having to do with the maximization of the long term average of an appropriately weighted sum of the conflicting terms: probability of the homeostatic state and cross-correlation with the environment. Optimal strategies are found through computing simulation which determine the efferent controls precipitated from the level W to the level Q. The relevance of the model to a number of problems related to the adaptability of hierarchical structures is discussed. Finally some extensions of the present work to do with the communication process between two multi-hierarchical systems are mentioned.
"