Evolution in Associative Systems

October 26, 2011 § Leave a comment

When did you have the last thought

of which you think that it was a really novel one?

The fact is that this is happening probably just right now, according to neuroscientist Gerald Edelman. Edelman argues that this is a direct consequence of an ‘unlabeled world’. He denies that the instructionist concept (now better known as computationalism) can solve the puzzle that brain-bearing organisms can behave adaptively. He writes [1]:

To survive in its eco-niche, an organism must either inherit or create criteria that enable it to partition the world into perceptual categories according to its adaptive needs. Even after that partition occurs as a result of experience, the world remains to some extent an unlabeled place full of novelty.

Even the very basic standard process of categorizing perceived signals thus has to be inventive all the time.Edelman is convinced, like us, that a standard computer program as well as the whole class of rule-based systems can not be inventive. What is needed is a framework that is able to genuinely create and establish novelty on its own. The candidate principle for him (and us alike) is evolution. Edelman proposes a model of group selection on the level of neurons.

It [m: this theory] argues that the ability of organisms to categorize an unlabeled world and behave in an adaptive fashion arises not from instruction or information transfer, but from processes of selection upon variation.

He continues to explain this idea in the following figure.

The most important part for the process of categorization and building up internal “representations” is in the bottom row, to which he comments:

Reentry. Binding of functionally segregated maps occurs in time through parallel selection and the correlation of the various maps’ neuronal groups. This process provides a fundamental basis for perceptual categorization. Dots at the ends of some of the active reciprocal connections indicate parallel and more or less simultaneous strengthening of synapses facilitating certain reentrant paths. Synaptic strengthening (or weakening) can occur in both the intrinsic and extrinsic reentrant connections of each map.

Re-entrant processes are indeed very important in order to get things clear in a group of units of an associative arrangement. Usually, modeling in associative systems leads to smooth arrangements, where a large number of conflicting activation is present. The boundaries between concepts remain unclear. A first-step solution is provided by reentrant signalling within the associative structure; Edelman formulates:

Such reentrant activity is constructive: because of its reciprocal and recursive properties and its parallel structure, reentry leads to new neuronal responses and it can resolve conflicts arising between the synaptic activities of different mapped areas […]. It should be sharply differentiated from feedback. Feedback is concerned with error correction and defined inputs and outputs, whereas reentry has no necessary preferred direction and no predefined input or output function.

According to Edelman, this reentrant mapping triggers a selection process. He proposed this theory of selection on the neural level that works as a standard process in the working brain the first time back in 1978 [2], and in an extended version in 1987 [3]. Edelman proposals for the linkage between processes on the neural level, perceptual categorization and even higher “functions” of the brain are highly plausible.

Yet, throughout his work, he uses just a rather intuitive notion of evolution. Well, one can imagine fuzzily how the creative play between variation and selection is going on according to Edelman’s scheme. Yet, he does not indicate exactly why this process should be called “evolutionary.” After all, not every selective process is already an evolutionary process. His whole theory hence suffers from a proper adaptation of the theory of evolution into the domain of neuroscience.

Evolutionary theory (ET) is a well-defined theoretical framework that “explains” the variety of living organisms on earth. The modern, and extended, version of the ET, developed by Ernst Mayr, Dobzhansky, Lorenz, Maynard-Smith, Richard Dawkins, Manfred Eigen and Peter Schuster among many others, incorporates genetics, mathematical models of population dynamics and mathematical game theory; it also uses insights from complex systems theory.

Despite some mathematical work about the dynamics of selective processes in natural evolution, a generalizing formalization of evolutionary theory or generalized evolutionary processes itself is still missing (as far as I can tell), which I find quite astonishing. Of course, one can find a lot of mathematical looking papers, there is a whole journal (J.theor.Biol.) hosting such. Yet, we are convinced that a generalization of evolutionary processes based on formal representation should not be limited to just representing population dynamics by some formulas bringing in some notion of probability, but at the same time keeping the evolutionary vocabulary of unchanged, i.e. still talking about species, mutations, pheno-genotypes, gene frequencies and so on. These are biological terms. Even if those terms are represented using the formalism of probability theory, the result is still a representation of biological structures. It is an abundant misunderstanding in theoretical work about evolution that such endeavors represent a generalization, but in fact, no generalization has been achieved. As a consequence, a transfer into questions about cultural evolution is inappropriate.

As a consequence, we can adopt the idea of evolution only vaguely and imprecise into other domains outside of biology (culture, economy, general systems theory). The task of generalizing natural evolution can not be solved keeping the terms of biology. We do not have to describe biological mechanisms using the generalizing language of mathematics, instead we have to generalize the theory itself, that is, we have to abstract its most basic terms. This does not mean that we must dismiss all the mechanisms invented by natural evolution each and forever. But they can not be elements of a general theory of evolution.

This is what the rest of this paper is about. As a result, we will have a formulation of evolutionary theory at our disposal that is not only much more general than the biological version of it. It also can be transferred readily to any other scientific domain without using metaphors or introducing theoretical shortfalls. Examples for such domains are economy, social sciences, or urbanism.

Formalizing the Theory of Evolution

From a bird’s view one could say that evolution is the historical creation of information through complexity. The concept of evolution has been conceived primarily as concept from biology for more than 150 years. Almost since its first coherent formulation by Darwin (2003) there also have always been attempts to apply it to the progression of human culture. Generally, those attempts have been refuted, because a more or less direct transfer implicitly denies cultural achievements. Thus, we have to reconstruct a more general form of evolutionary theory in order to import it to a theory about the change of cities. Unfortunately, we can provide only a brief outline of the reformulation here.

In biology, evolutionary theory has been revised or extended several times. Never­theless, its core can be still compressed to the following two propositions (the plus sign does not mean arithmetic addition here), which express the basic elements of this theory (you may find more complicated looking versions here [4], but they all boil down to the following):

Evolution = Variation + Heredity + Selection      (1)

Fitness = number of offsprings in secondary filial generation F2      (2)

The concept of fitness is the core of the operationalization of evolutionary theory. As a measure it is only meaningful within a system of competing species. There are, of course, a lot of side conditions one have to be aware of and the mechanisms regarding the single terms of this equation are still under investigation. These equations reflect the characteristics of biological matter, i.e. genes and physiology making up a body, which is immersed in a population of bodies, similar (within a species) and different ones (competing species). A brain, or even groups of neurons do not have such a structure, thus we have to extract the abstract structure from the equation above in order to harvest the benefit.

Fortunately, this is quite easy. Our key element is a probabilized version of memory, where memory is a possibly structured random process that renders bits of information unreliable up to their complete deletion. This concept is very different from the concept of memes, which refers “directly” to the metaphor of the gene. Neuronal maps are yet not defined by sth discrete like genes. In neuronal maps—or any other non-biological entity that one wants to view “under the light of evolution”—the mechanisms of a transfer of information is completely different and totally incommensurable as compared to a biological entity like a “species.” Above all, there is no such thing like a “neuronal code” or even a “neuronal language,” as we know not only today. The same conclusion can be drawn from Wittgenstein’s philosophical work. The concept of the “meme” is ill-designed from ground-up.

We start by conceiving here­di­ty and selection as (abstract) memories. Heredity is obviously a highly accurate long-term memory, actualized as lasting, replicable structures, called DNA, albeit the DNA is not the only lasting memory in eukaryotic organisms. Nowadays it is well established that there is an epigenetic pathway (external to DNA/RNA) for passing information between generations. Obviously, epigenetic transmission of information is less stable than DNA-based transmission. After a few generations the trace is lost. Nevertheless, it is a strong memory, potentially very important to explain evolutionary phenomena.

Selection on the other hand is just the inverse of forgetting, or the difference established by it. Yet, if we try not to get trapped by the stickiness of words, we can clearly recognize that selection is just a particular memory function, especially as selection is an effect on the level of  populations.

Finally, variation can be con­ceptualized as a randomness operator acting on those two kinds of memories. We may call it impreciseness, influencing of the capability to evolve, speed of evolution, in any case we have some kind of transformation of some parts of the content of the memories, for whatsoever reason or mechanism.

Now we can reform­ulate all ingredients of the classical and basic evolutionary theory in terms of probabilistic memory.

Lemma 1: An organism is a device which “exists as” and which can maintain an assemblage M, called a probabilistic memory configuration. M consists of different kinds of memories mi, each of different duration and resolution. “Probabilistic” here means that the parameters, such as duration or resolution, are not determined by some fixed value, but by probabilistic densities. From the theoretical perspective, these densities may be interpreted as outcomes of random processes; from the perspective of scientific model identification, the “probabilistic turn” allows to change from question “what is it?” to the question about mechanisms and their orchestration.

Any embedding “evolutionary” process picks at least two different “memories” from that ensemble. We will return to this threshold condition about the number of memories in a moment. If the durations are sufficiently different, that organism as defined in (3) will be a “subject” of evolution, i.e. a “species” embedded in contingent historical constraints. For principle reasons, a species should not be defined by an identity relation, as it is done in evolutionary biology. In biology, a species is defined in a positive definite manner: if there is a DNA configuration that in principle is provable unique, then this DNA configuration is a species. Another individual has to have the same configuration (here is the identity relation), in order to establish / belong to that assumed species. This way of defining a species is deeply unsuitable, not only in biology (see chapter about logics, where we will deal with the issue of “identity”).

Instead, in the next lemma we define it as a limit process, which expresses the compatibility of two different memory configurations.

Lemma 2: In a population of size n, we define the limes of the probability for a unification of two different sets of probabilistic memory configurations M (e.g. biological organisms, structures of cultural ensembles) under conditions of potential interaction. If the unification of two memory configurations can not take place in a sustainable manner, or the probability for such unification tends towards zero, the two configurations can be conceived as “separated & competing probabilistic sustainability programs,” i.e. a quasi-species. A (quasi-)species [Q] can be defined by the limit value 0 of the following limes process:

If the memory configurations can not overlap, we may call them a species. Yet, even in biology it is well known that this separation is never complete, A salient example are ducks, of which many “species” could interbreed. Yet, they rarely do. The main reason are different and incompatible rituals before mating. In other words, the incompatibility is in the realm of information. Yet, sometimes it happens though, because some bytes in the ritual overlap.

Species, in the classical as well as in the probabilistic framework, do not only maintain memories in their inside. Any multi-particle system may establish some memory as a phenomenon on the level of the collection / population. The information stored therein may well-be significant or even essential for the persistence or stability of the whole (quasi-)species.

Lemma 3: Given a population of organisms, the combination (synchronic union) of at least two cross-linked memories M with different duration d and, optionally, of different temporal resolution r, modified by an operator for synchronous randomness, results in an evolution, if most of the individuals contain such a memory structure.

Evolution results in the creation of stable memory configurations, which are able to maintain themselves across the boundary of the individual. A plethora of mechanisms can be observed in the case of biological evolution. Yet, for our abstraction here, only the effect is interesting.

Just one question is remaining to be answered: Why should be there at least two different kinds of memories in the population of entities in order to allow for evolution? Of course, in an organism, even in the most simplest ones, we find many more levels and kinds of memories than just two. Yet, two memories are presumably the lower threshold. The reason is the mirrored in two conflicting requirements: First, it is necessary for the entity to maintain long-term stability, which includes reproducible structural information. This memory needs to be a private memory, i.e. accessible only to the hosting individual itself. Secondly, the generation of variability requires a second memory outside of and largely independent from the first one. This second memory may be actualized as a “private” memory or as a memory shared between individuals.

Already on the level of genomes we find two levels of memory. In bacteria, we find the plasmid. In eukaryotes (higher organisms), we find diploidy, with the interesting case of reweighing diploidy and haploidy in moss. Diploidy opens the possibility for a whole new game of (sexual) recombination without endangering the stability of the memory on the lower level. The further progression of memory types then is the population with its network effects, social systems for stabilizing amorphous network effects, and finally individual cognitive capabilities. Each of those additional level are themselves not simple, but already multi-layered instances of memory, with a sheer explosion of levels for cognition. Humans even learnt to externalize cognition into culture, beginning with language, simple drawings, writing, cities, books, and finally digital engines. These few and indeed coarse examples may already be sufficient to indicate the possibilities of the reformulated theory of evolution by means of probabilistic memory.

Lemma 4: Finally we can define fitness. First we set a term for the risk of getting extinct. A quasi-species Q disappears, if the probability to find it in a distribution approaches zero, and the closer that probability approaches to zero, the higher is that risk. If that quasi-species “is able” to deal with that risk, this mapping should be zero. The fitness of the quasi-species as an informational setup,then can be expressed again as a probability, namely that this capability to deal with the risk is >0. Taken all together we get:

This formulation of fitness has several advantages as compared to classical version that is in use in biology. First, any threat of circularity is abandoned. It has often be said that survival of the fittest simply means “survivor of the survivor.” Albeit this accuse does not respect the aspect of mutuality in evolutionary processes, it nevertheless poses a difficulty for traditional evolutionary theories. Secondly, the measurement of fitness itself becomes more clear. We don’t have to count the number of F2 offsprings, nor have we rely to any arbitrary measure of this kind at all. Fertility, and even differential fertility is rarely linked to the probability of getting extinct. Often we can measure just that a population is shrinking, for instance due to climate change: it may proof absolute impossible to find or even quantify something like a differential fertility, especially by resorting to fertility towards F2. Which “species” should be compared for that? Impossible knowledge about the future would be necessary for that, creating a circularity in reasoning. Thirdly, we can set up stochastic simulations about populations much more easily and else on a more abstract level, which is supportive for increasing the results of proposals about fitness. The traditional setup of fitness is only feasible for direct competition. Yet, even Darwin himself expressed doubts about such an assumption.

Synopsis and Similar Work

Our first achievement is clearly the replacement of biological terms by abstract concepts. On the achieved level of abstraction, all the mechanisms that can be observed in (or hypothesized about) natural evolution are not really interesting. Of course, “nature” has been quite inventive during the last 3.6 billion years, not only regarding the form of life, but also with regard to the mechanisms of evolution. There are plenty of them. But what is the analogon of diploidy in culture? Or even that of gene? Or where is the implementation of the genotype-phenotype separation? Claiming the existence of memes [5] as such an analogon does not help much, because the mechanism are vastly different between society and biological cells. It is one of the lesson that can be learnt in biology, to ask about the mechanism [e.g. 6]

In our perspective, it is nonsense to keep biological terms, rewrite them with a bit of math, and then to impose them to explanatory schemes about the evolution of culture. The result is simply nonsense. It is simply a categorical mistake to claim that cultural evolution is Darwininan, as Richerson and Boyd [7] meant.

The situation resulting from the theoretical gap that we tried to fill here is disorder, at least. In their excellent paper, Lipo and Madsen [8] diagnose rightly that there has been no proper transfer of evolutionary theory from biology to anthropology. Yet, they stick to biological terms, too.

As we already mentioned above that the problem of generalizing natural evolution can not be solved keeping the terms of biology. We do not have to describe biological mechanisms using the generalizing language of mathematics, instead we have to generalize the theory itself, that is, we have to find abstractions for its most basic terms.

Precisely this we did. Notably, we need not to refer to entities like genes anymore, and we also could remove the notions of selection, variation or heredity. These terms from the theory of natural selection are all special instances of abstract probabilistic memory processes. There is no categorical difference any more between Darwinian, Lamarckian or neutral evolution. If there is a discourse about evolutionary phenomena would mention those, or if such a discourse would invoke notions of phenotype, species, etc., we could be absolutely sure that the respective argument is not general enough for a general theory of evolution.

Darwinian, Lamarckian or neutral evolution differ just in the way the memory processes are instantiated. Of course, it has to be expected that natural selection (we already remarked that before) will “create and establish” a lot of different mechanisms that create better adaptivity of the evolutionary process itself [9]. So, if we are too close to our subject (the evolutionary phenomenon), we just arrive at the opinion that there are these different types of evolution.

We suppose that the formulas above are the first available and proper proposal of a theoretical structure that allows a transfer of evolutionary theory into anthropology (or into the realm of machine-based epistemology) and the question about cultural evolution that satisfy the following conditions:

  • – it is theoretically sound;
  • – it does not reduce culture to biology, even not implicitly;
  • – the core ideas of evolutionary theories remain intact;
  • – the theory can be used to derive proper models (check the chapter about theory of theory!)



The achieved advantages are completely due to the complete move to probability functions. (Here, and about probabilistic modeling  in general, I owe much to Albert Tarantola’s work) In our perspective, even species “are” just distributions, but not representations of a book full with genetic code, presumed to be unique and well-defined. These entities (formerly: “species”) need not even be sharply dis­tinctive! A similar proposal has been made by evolutionary biologist Manfred Eigen in 1977 [10], who coined the term “quasi-species.” For our context, these entities can be anything, the only requirement is that this “anything” could be conceived as a stacked collection of memories (biologically: “organism”) in the sense given above. Note, that this conceptuali­zation of evolution allows for Darwinian as well as for Lamarckian processes, for indivi­dual-based selection as well as for group selection, for horizontal (by vectors) as well as for vertical transfer of information (through inheritance). Thus we may conclude that it is truly a generalization of the biological notion of evolution—at first. We suggest that it is an appropriate generalization of any evolutionary process.

The proposed memory dynamics—which we could call generalized evolutionary theory—is closely linked to the con­cept of com­plexity and information, and probably not complete without them. Even without them it seems, however, that the should be well possible to build operable models of memory stacks, in other words, to design the capability for evolutionary change apriori. The free parameters for a prospective “memory design” that will exhibit properties of an evolution are the number of memory layers, their mutual dynamics and the inner properties / mechanisms of the memories.

We may conclude that we now have a concept of evolution at our disposal that we can apply to domains outside the natural history of biological species, without committing naive equivalence enforcements. We achieved this by a mapping of the biological version of evolutionary theory onto the framework of probabilistic memories As you can see, we successfully replaced the notion of “species” by a much more abstract concept. This abstraction is key for the correct, non-metaphorical transfer of the theory into other domains.

From a philosophical perspective the proposed approach bears an interesting point. In medieval philosophy (scholastics), the concept of “species” was not associated with the classification of organisms. Yet, the question was, first, how could it be possible to recognize things as belonging together and then second, how could it be possible to have a unique name for it. The expectation was that the universe (as a work of God) should not be unordered or unstable. Note that we find prototypical identity thinking here. A reminiscent to that can be traced even nowadays regarding the “representationalist fallacy,” and also in the more intense forms of realism. Our hope is that the probabilization of species and the structure of evolution helps to overcome those naive stances.

It is finally a complete adoption of the “informational turn” into evolutionary biology. As long as species and evolution as the most central concepts in biology are not defined in terms of probability densities, biology can’t deal with aspects of information in a “natural” manner, thus remaining in the materialist corner of science.

Evolution and Self-Organizing Maps

Interestingly, Edelman who writes about the brain on the level of neurons also uses the concept of “maps,” established by groups of neurons. The role of these maps in Edelman’s theory is the coupling from perceptual “input” to the decision towards the action then ultimately taken. Of course, one has to include self-directed restructuring, whether implicit or explicit, of the cognitive apparatus into account, to. Not all actions need to result in motor action. In Edelman’s model, reentrant processes occurring in these maps are responsible for perceptual categorization. We will demonstrate in another chapter how processes around a SOM can be arranged such that they start to categorize and to distill crisp, if not to say “idealistic” representations from their observational data.

Anyway, we still need to clarify where and how (abstract) evolutionary processes can take place in, by or with a Self-Organizing Map, or a group thereof.

In a collection, if not to say population of SOM we may refer to an individual SOM as individual entity only, if they do not share too much actual memory with the other instances. If we’d instantiate these individual entities as standard SOM, we would have the situation that our entities comprise only a single type of memory (see Lemma 1). Fortunately, there are some different methods we could apply in order to add further levels of memory to render single entities into “individuals.”

Remember that we will need at least two levels of memory in order to satisfy the minimal condition for evolutionary processes. Evolutionary processes inside the “machine” are in turn mandatory  for the capability to find new solutions that are not pre-programmed in any sense beyond a certain abstract materiality (see the chapter about associativity). One of these possibilities is the regulation of the connectivity in the SOM, that is, to allow for a symmetry-break with regard to the spread of information. In the standard SOM this spread s fully symmetric: it follows the pattern of a disc (radial symmetry) and it is the same decay function everywhere in the SOM. Another possibility to introduce a further level of memory is provided by the 2-layer approach as realized in the WebSom. Notably, the WebSom draws heavily on a probabilization of the input. Else, pullulation and some other principles of growth naturally lead to a multi-memory entity, in other words to an “individual.”

The next level of integration is the quasi-species (Lemma 2). The differentiation of the population of SOM into quasi-species need not be implemented as such, if we apply the more advanced growing schemes, i.e. plant-like or animal-like growth. Separate entities will emerge as a consequence of growth. Differentiation will of course not happen if we apply crystal-like growth scheme. The growth of coordinated swarms on the other do not provide sufficient stability for learning.

The connectivity between individuals will organize itself appropriately and quasi-species will appear quite naturally as a result of the process itself, if we allow for two principles. First, their should be a signal horizon within the population. Second, SOM individuals should communicate not on the data level, but on the “behavioral” level. Interacting on the “data level” would enforce homogeneity among the SOM entities in many cases.

In order to start the evolutionary process we need just to organize some kind of signal horizon, or, resulting in almost the same effects, some kind of scarcity (time, energy). Any kind of activity, whether on the level of the population or on the level of individual SOMs, then has to be stopped before its analytical end. Given the vast number of possible models given a properties vector of length n>100(0)+, learning and exploring the parameter space anyway will never be complete. The appropriate means to counteract that combinatorial explosion (see the chapter about modeling) is… evolution! On the individual level animals are able to optimize their search behavior very close to theoretical expectations. Davies [10] proposed the so-called “marginal value theorem” to organize (and to predict) the optimal amount of time to search for something under conditions of ignorance.

Finally, lemma 4 about fitness would wait for discussion. Yet, for now we are not interested in measuring the fitness of classes of SOMs in our SOM world, though this could change, of course.

(The thoughts about formalization of evolutionary theory have been partially published in another context in [12].)

  • [1] Gerald M. Edelman (1993), Neural Darwinism: Selection and Reentrant Signaling in Higher Brain Function. Neuron, 10(2): 115-125.
  • [2] Gerald M. Edelman, Group selection and phasic re-entrant signalling: a theory of higher brain function. In: G. M. Edelman and V. B. Mountcastle (eds.), The Mindful Brain. MIT Press, 1978.
  • [3] Gerald M. Edelman, Neural Darwinism: the theory of neuronal group selection. Basic Books, 1987.
  • [4] Richard Lewontin, The Genetic Basis of Evolutionary Change. Columbia University Press, New York 1974
  • [5] Susan Blackmore
  • [6] Paul Patrick Bateson (ed.), The Development and integration of behaviour. 1991
  • [7 ] Peter J. Richerson, Robert Boyd, Evolution: The Darwinian Theory of Social Change, in: Schelkle w., Krauth w.H., Kohli M., Ewarts G. (eds.), Paradigms of Social Change: Modernization, Development, Transformation, Evolution, Campus Verlag, Frankfurt 2000.
  • [8] Carl P. Lipo, Mark E. Madsen (1999), The Evolutionary Biology of Ourselves: Unit Requirements and Organizational Change in United States History. arXiv:adap-org/9901001v1
  • [9] Wolfgang Wagner, Evolution der Evolutionsfähigkeit. In: Dress, A., Hendrichs, H., G. Küppers: Selbstorganisation. Die Entstehung von Ordnung in Natur und Gesellschaft. München 1986.
  • [10] Manfred Eigen, …
  • [11] Davies, the marginal value theorem, in Krebs Davies. Ecology.
  • [12] Klaus Wassermann (2011), Sema Città: Deriving Elements for an applicable City Theory. in: Conf.Proc. 29th eCAADe, RESPECTING FRAGILE PLACES, University of Ljubljana (Slovenia), 21-24 September 2011, pp.134-142.


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