fractal networks



Richard Dryden

16 January 1996

Abstract: It is proposed that consciousness does not emerge from a single level of biological organization (for example: from computational activity at the synaptic level in networks of neurons), but is a consequence of interdependent modelling activities by networks at different levels of organization including the molecular, organelle, and cellular levels, in some way entrained to produce consciousness. Fractal stacking and intercommunication of networks at different levels is proposed as a substrate that may be required for consciousness, either natural or machine-based. Adoption of this conceptual starting point may overcome some of the difficulties encountered when reductionist strategies are applied to the study of consciousness.


Contemporary discussions of consciousness tend to draw upon the most deeply held features of our individual world-views, be it quantum theory, information theory, chaos theory, parapsychology, or spiritualism, in the hope that some combination of these will provide enlightenment. There is clearly a problem of self-reference here in that consciousness is both the tool and the object (Miller, 1962):

consciousness is trying to understand itself, and has no reserve of modelling capacity to aid simplification or to step back and view itself from a more sophisticated metasystem. If we try to model consciousness by making simplifying assumptions or using reductive methodologies, it is in danger of slipping away in the process - consciousness seems to be a product of ‘wholeness’ rather than 'partness'. However, that is not to say that any attempt to understand consciousness is without merit or that progress cannot be made - there are sufficient regularities in the observable universe to allow us to extrapolate from things we can know more readily to those which are less accessible, such as consciousness.

In the proposal that follows, I shall combine ideas from general systems theory and neural network theory to prepare a conceptual framework from which to view consciousness. Essentially, I shall suggest that consciousness does not emerge from activities at a single level of biological organization (for example: from computational activity at the level of synaptic interconnection in networks of neurons), but is a consequence of interdependent modelling activities by networks at different levels of organization, in some way intermittently entrained to produce consciousness.

General systems theory

I have found the ideas incorporated into general systems theory (GST) very helpful in developing an understanding of the biological realm. GST identifies recurring patterns of organization in systems of different types. There are several variants of this theory, but I shall take the concepts developed by Bertalanfly (1968) as a starting point. Central to GST is the concept of partially-bounded open systems interacting with their environments by way of inputs and outputs, as illustrated in Fig. la. A selected part of the environment 'flows through' an open system over time, being changed by it - 'transformed' - in the process. Although some system boundaries have an observable reality, for example: the outer membrane of a cell, we must also appreciate the role of an observer in 'drawing' boundaries: this separation of our world into systems is still a reductionist approach.

Open systems are made of interdependent parts, each of which can be thought of as an open system in its own right. Open systems at a given level of organization can interact with each other to produce systems that 'emerge' at the next higher level of organization: for example, atoms and molecules interact to form cells, cells interact to form multicellular organisms, and the organisms of some species interact to form societies. Within a given level of organization, there can be recognizable patterns of interaction such as hierarchy.

For me, the strength of GST lies in the recognition of patterns that recur in systems at different levels, and I find it provides a framework for incorporating reductionist scientific observations into a more holistic worldview. For this reason, I feel that GST can contribute to the debate about consciousness.

From my studies of embryos and other biological systems, I would add another feature to this generalized conception of an open system: a capacity for the system to 'model' its environment in some way. That is, over a period of time the system develops a changing representation of its environment through its two-way interactions and as a consequence of its internal organization and functions.

I have chosen the word 'model' for this discussion in the belief that it has a more neutral feel to it than the more commonly-used alternatives such as 'consciousness' and 'awareness' (see Velmans, 1995, for a discussion of the difficulties in using these latter terms). No attempt will be made to pin down the exact meaning of 'model' at this stage; rather, it will be used to stand for representational activities in general, sentient or otherwise. Therefore, at the human level, 'model' would include both conscious and subconscious activities. For the sake of simplicity, I have placed the 'model' within the boundary of the open system in Fig. la (‘m’) - this is not necessarily to imply a physical presence within the system, simply a close interrelationship between modelling activities and the other systemic structures and processes at that level of organization.

Although at the human level conscious experience gives us direct access to modelling processes, the suggestion that we should extend the idea of modelling capacities to systems at other levels may appear particularly speculative. However, evidence in support of this view is beginning to accumulate, and recently there have been several publications proposing that cells, organelles, and even molecules have the capacity for computational and representational processes as a consequence of their organization (eg: Albrecht-Buehler, 1985; Hameroff, 1994; Bray, 1995).

This suggestion that open systems in general, regardless of level, might possess a faculty for modelling echoes panpsychism - the idea that all matter contains some quality that may be called 'mind' or 'consciousness'. Panpsychism has often been classified along with vitalism as being both unscientific and unnecessary (see for example Popper and Eccles, 1990), although panexperientialism and panpsychism have recently reappeared as discussion topics (de Quincey, 1994; Seager, 1995). However, it will be suggested below that modelling can be achieved by processes familiar to contemporary science, specifically those occurring in natural and artificial neural networks, and do not require that we formulate additional properties of matter. This is not to say that the experiential aspect of consciousness will necessarily emerge from the modelling processes being proposed here - rather that we need a suitable conceptual framework from which to begin to tackle the ‘hard question' of consciousness delineated by Chalmers (1995).

An internal model would impart a degree of autonomy to a system over its environment, or at least that would be the impression given to an observer, since stimulus-response loops will be buffered and modified by the computational activities of the model. The result will be that the system's responses show variation over time: the response elicited by a particular input pattern may not be the same when the same input is tried again after the system has gained other experiences in the interim. De Quincey (1994) summarises it like this: "A 'compound individual' is a hierarchical society of suborganisms each of which has its own level of experience and capacity for self-determination (for instance, an animal compounded of living cells, or a cell composed of organic molecules)" (page 223).

If this is the case, it allows the possibility for more subtle behaviour of individual systems as they interact with their environments and each other. In the case of a developing embryo, we can envisage the growing community of cells forming a network of social interactions (Dryden, 1991), each cell continuously harmonizing its intrinsic drives and gene selection with the environmental cues impinging upon it. At the same time, the embryo as a whole seems to have an identity or model, since perturbations to the normal flow of development can be coped with and responded to, even though particular cells are lost or damaged. Thus, when open systems interact socially, it seems that a new potential for creativity is produced.

Limitation of GST

There is, however, a shortcoming in this way of looking at the world and the systems within it. The stratification of the observed world by GST into levels of organization seems innocuous enough in the sense that it recognizes the apparent ‘wholeness' of a molecule, cell, or person, and identifies repeating patterns at each level, but the problem then arises of understanding how events at one level impinge upon events at another - for example, do events at a higher level determine what happens at lower levels (‘top-down' causation), or do events at lower levels dictate what happens at higher levels ('bottom-up' causation)? Or can both occur? Interestingly, scientific disciplines seem to be stratified in a similar way, each focusing on a particular level of organization, and a similar problem arises: a discipline that works well within one level (eg: cytology) may not have an obvious explanatory value for a discipline at the next level 'up' (eg: psychology). It is as if we have a good ‘within level' science but have need of a better 'between level' science (interscience?) to link the levels. Given that we consider the universe to be an interconnected whole, and science to be a consistent methodology for learning about it, this is quite surprising, and gives us cause to re-examine the way we observe the world and make subdivisions. Part of our current difficulty in understanding consciousness may result from inappropriate separations being made.

The problem of linkage also applies to the proposal that models may exist in systems at different levels: how would those models interact? (See Fig. 1 b.) For example, how would models within cells possibly interact with or in some other way contribute to consciousness at the human level? In the context of panpsychism, Seager (1995) calls this the 'combination problem': "how the myriad elements of 'atomic consciousness' can be combined into a new, complex and rich consciousness such as we possess" (page 280). Reductionism is an approach which requires dismantling a system and studying its parts, or studying the behaviour of a system in an artificially simplified and controlled environment. However, consciousness appears to be a property emerging from an intact system, from ‘wholeness' rather than 'partness', and may not be reducible in a conventional sense without risking losing the very insight being sought.

Neural networks

Although GST stumbles at this point, it is possible to make progress with the idea of interacting models at different levels by bringing in and modifying the concept of neural networks.

The term 'neural network' is rather loosely used and may refer either to a network of biological neurons forming part of a nervous system (natural neural network) or a computer simulation of a network composed of interconnected units with neuron-like properties (artificial neural network). In both cases, each node in the network has one or several inputs of variable 'strength' (usually both excitatory and inhibitory) and summates the inputs, giving rise to an output or outputs when a stimulus threshold is crossed. (For a review of neural networks and a discussion of some of their limitations, see Crick, 1989.)

Artificial neural networks are often simulated with three layers of units: an input layer, an intermediate 'hidden' layer, and an output layer (Fig. 1 c, lower diagram). The connections between units are changed during periods of 'training' or 'learning': connections that contribute towards correct behaviour become strengthened, while connections that contribute to aberrant behaviour are given less weight, with the result that a distributed 'memory' of the task is established across the network in the pattern of different-strength connections. A trained network contains information about associations, categories, and algorithms in its pattern of connection strengths and operational rules, and in the sense used above, therefore embodies a 'model' of its task.

The neural network concept provides an explanation for modelling capacities in networks of interconnected units, and is playing a significant role in developing our understanding of biological systems. As noted above, the concept can be applied not only to studies of neuronal assemblies, but also to individual cells and parts of cells such as organelles and macromolecules (Albrecht-Buehler, 1985; Hameroff, 1994; Bray, 1995).

Combining GST and the neural network concept

The 'units' which form the nodes of a given network share most if not all of the properties already delineated for open systems: inputs, some kind of inner transformation or function, outputs, and interactions with other units. Therefore, we can redraw a neural network with open systems as the units (Fig. 1c, upper diagram).

But by building on our experience with GST, we can take the analysis further. If we are considering a natural neural network, the units are the neurons. Each neuron is a complex and living assemblage of interacting parts, and those parts can also be viewed as open systems. So we could model the neuron itself as a network with, for example, its organelles forming the units. Similarly, we could model organelles such as the mitochondria as networks, with their constituent molecules forming the units. This process of identifying network characteristics could probably be extended further to include molecules such as proteins which are known to respond to environmental cues. We begin to see an interconnected pattern of networks within networks. This interpretation is summarised in Fig. 2.

Looked at in this way, biological organization appears as nested sets of networks, with the units in a network at one level being networks in their own right at the next level down, and the units at that lower level are also formed from networks, and so on. A suitable term for this arrangement would be fractal network, since it captures the quality of self-similarity or recurring patterns at different levels (ie: nets-within-nets). The term 'fractal' is being used here in a slightly different way from Merrill and Port (1991) and Globus (1992), who describe fractal network patterns while considering a single level of organization rather than linking different levels, although clearly these two uses of the term 'fractal' are complementary. Hameroff (1994) also hinted at the pattern of organization outlined above: "the cytoskeleton within each of the brain's neurons could be viewed as a 'fractal-like' subdimension in a hierarchy of adaptive networks" (page 114).


Consciousness is widely believed to be the result of physiological processes in the brain: the 'stream of consciousness' perhaps being a product of changing patterns of nerve impulses flowing through an incredibly complex network of neurons communicating by way of synapses. By this view, consciousness 'emerges' from computational brain activities at the neuronal-synaptic level. However, not everyone is convinced that this approach will ever provide a sufficient explanation of consciousness, particularly its experiential aspects. As Hameroff (1994) points out: "the mechanism of consciousness may depend on an understanding of the organization of adaptive ('cognitive') functions within living cells" (page 97). Others propose the need for intervention by non-computational processes (Globus, 1992; Penrose, 1994), or even the introduction of novel properties of matter or information at a fundamental level of scientific description (Chalmers, 1995; Seager, 1995).

If we are to take into account the levels of organization that underpin the synaptic level of brain activity, we need some effective way of linking activities at the different levels. In the hypothesis outlined above, the behaviour of an open system at any given level of organization is considered to be the result of social interactions between partly autonomous components which are each capable of modelling aspects of their environments. Modelling is considered to be a fundamental property of open systems, allowing them to interact in subtle and creative ways. It is suggested that the modelling and social activities at each level are integrated to produce the emergent properties recognized at higher levels, including consciousness.

Interaction, together with the presence of individual capacities for modelling, can be a source of new complexity in open systems. Structures, specializations, processes, and institutions emerge in a way that presumably would not be possible without interaction between individual members with social potentials. Communities of cells build organisms; communities of ants build complex anthills. In the human social context we are familiar with systems of government, justice, education, health care and so on which arise as emergent properties of individual social interactions. If we form a conceptual link between these creative activities of social systems and the adaptive behaviour of neural networks, then we begin to have a better understanding of emergent properties, including consciousness.

To bridge between levels, I have suggested that it is helpful to integrate the open system concepts of GST with the concept of neural networks, giving rise to a fractal arrangement of adaptive nets. This approach has two advantages:

bullet the proposed network organization of open systems would be the source of a capacity to learn and adapt - this property of network organization would lend weight to the observation that open systems seem to have an ability to 'model' their situations
bullet in the conceptual framework proposed here, the networks at all levels are clearly interconnected (each network is a unit at the next level up) to form a continuous organizational structure: there is no between-level linkage problem of the sort generated by standard GST.

The fractal network concept provides the potential for more 'depth' when discussing consciousness in the sense that it links activities at more than one level of organization. However, by simply adding more layers of networks we are not necessarily going to feel much closer to answering the 'hard problem' of consciousness (Chalmers, 1995: what is the source of the experiential aspect of consciousness?), since doubts have been expressed about the ability of monolevel networks to exhibit understanding of their computational activities (eg: Searle, 1990), let alone have subjective experiences.

Nonetheless, the interconnected nature of the hypothesized fractal network may help us to understand how proposed lower-level influences might interact with consciousness. For example, there is considerable discussion about the possible involvement of quantum coherence in consciousness. Penrose (1994) suggests that "coherence could be part of what is needed for consciousness" (page 408), and with Hameroff (1994) believes that cytoplasmic microtubules might provide a suitable location for a 'dithering' between quantum and classical realms. Seager (1995) agrees with Penrose that "only coherent multiparticle systems will preserve the peculiar quantum mechanical properties that underlie the appropriate 'summation rules"', and continues "only systems that can maintain quantum coherence will permit 'psychic combination' so that complex states of consciousness will be associated only with such systems" (Page 285).

The fractal network concept appears to accommodate the possibility of between-level coupling, but at this stage it is not clear how this might be achieved or maintained. The concept has the advantage that all parts of the network remain interconnected and therefore represent a single entity, so although at the synaptic level of brain function there may be shifting patterns of activity, there could still be continuity and intercommunication in a more global sense at sublevels, perhaps allowing extensive coherence.

There is reason then to believe that conscious experience requires some form of 'entrainment' or resonant coupling of activities at different levels. Since conscious alertness cannot be sustained continuously for long periods; and is punctuated by periods of reduced alertness including sleep, this may be an indication of a periodic need to uncouple activities at different levels, perhaps to allow restorative processes to be carried out. In the absence of coherence, it could be envisaged that computational processes would still be possible within a given level, but perhaps without the full experiential accompaniment.

Although speculative, the fractal network hypothesis is open to testing by currently-available methods. Fractal networks are amenable to computer simulation (Merrill and Port, 1991), giving an insight into their computational properties. The effects of blocking communication between networks at different levels of organization could then be studied - this may illuminate the suggestion that anaesthesia operates by blocking normal microtubular action and thus uncoupling quantum coherence effects (Hameroff, 1994). It would also be interesting to test the suggestion that modelling at the molecular and organelle levels influences the modelling capabilities of individual cells: extrapolation from these lower and less complex levels might help us to better understand the emergence of human consciousness.


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