This paper was adapted from prior independent work started during my time as San Francisco State University for the purpose of publication. I would like to thank my family here and abroad for their support, Jennifer O’Meara, Zoe Martell, David C. Noelle, and the many colleagues, friends, instructors, and professors who provided feedback and critiques while this paper was under development. If there are any questions regarding this paper, I can be reached at amunoz-valverde@ucmerced.edu. A more detailed version is availabe upon request
I propose that Representation is neurally instantiated by n-dimensional simplexes extant in the brain, posit the existence of a Representational Space, and elucidate the consequences of this proposal. Representation is fundamental to any high-level account of mental phenomena and is especially crucial in an understanding of perceptual and phenomenological processing. Recent experiments under the Blue Brain and Human Brain Projects in the E.U. have shown that there are informational structures neurally instantiated within the brain, n-dimensional simplexes, which fulfil the necessary criteria to be the neurological structures that give rise to representation. I demonstrate that the dynamics and structure of simplices are likely adequate to explain the many confounding aspects of representation and propose that the neural instantiation of these simplexes can solve the binding problem. (This proposal is more thouroughly examined in a paper on mental imagery)
Keywords:Representation, n-dimensional, simplex, render, neural, instantiation, correlates, cliques, cavities, topology, bound, perception, mathematical, cognitive science, theory, computational cognitive neuroscience, directed hypergraphs, propagations through directed hypergraph networks, graph-like propagations
The proposition that representation in the brain is instantiated by n-dimensional simplexes is founded on the work presented in the research article published in Frontiers in Computational Neuroscience, titled “Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function”. This article serves as novel and compelling empirical evidence on the extance, form, function, and dynamics of n-dimensional simplexes in the brain. It is therefore worthwhile to review that article and its claims.
In order to better understand the work in that paper, I must review the definition of the terms simplex and n-dimensional. N-dimensional just means that the object in questions has an arbitrary number of dimensions. Dimensions here are defined as informational dimensions, where a network has a number of nodes which are all to all connected and does not refer to spatial dimensions. A simplex is an all-to-all connected network or an n-dimensional triangle. Clique and simplex are interchangeable terms, and a simplex within a larger simplex is referred to as a sub-simplex
In the Reimann, et al. paper, the team used prior work where they sliced rat neocortex very finely, imaged those slices, and digitally reconstructed the slices into a functional rat neo-cortex simulation, to analyze the reconstructed simulations’ activation patterns using topological tools. They turned their neocortex model into a topologically analyzable model by turning each chemical synapse into an edge and each neuron into a vertex. The resulting model was then analyzed using the elementary tools of algebraic topology, which is very broadly the study of shapes.
Their analysis uncovered a large count and variety of directed cliques and cavities, (which are simply complete and incomplete simplexes respectively) as well as subtle but significant differences in reconstructions from different animals, implying individual variations were reflected in the reconstructions. These cliques and cavities were far more numerous in the reconstructed neocortex than in control models of directed networks. Critically, this was the first time that high dimensional cliques and cavities were found in any network, organic or artificial.
I argue that their study provides an elegant and clean mathematically based model for the analysis of local and global neural activity and provides a proper framework and conception by which to model representational work. While the authors note that they believe simplicial structure and dynamics have much to do with processing, “We conjecture that a stimulus may be processed by binding neurons into cliques of increasingly higher dimension, as a specific class of cell assemblies, possibly to represent features of the stimulus…”, they do not fully elucidate how simplexes process information. I argue that representation as a product of simplicial dynamics provides that elucidation. As such I will refer often to this work due to its description of simplex dynamics, which I posit corelate strongly with representational dynamics.
In this section I review some of the attributes and propagation dynamics of simplices and how they possibly relate to representational phenomena.
Simplexes have facets or faces made up of smaller simplexes. For example, a 3-dimensional simplex is made up of four 2 dimensional simplexes which each share edges (or connections) with each other. The compositionality of simplices from sub-simplices alone accounts for a myriad of representational phenomena. In the same way that simplices are often composed of other sub-simplices, representations are very often made up of multiple other sub-representations.
There exists ample evidence that poverty in and of itself is a negative factor for health in general (Esposito, 2016) and mental health in particular (Rogers & Pilgrim, 2003). From shorter life spans to increased stress, people in poverty and near it are the most vulnerable and therefore the most exploited subsect of people in this and nearly all nations.
A classical example used in philosophy of mind to demonstrate representational coherence is the example of Unicorns. Very succinctly, if I take a person who has never been taught what a unicorn is, but has seen horses and horns, I can still give them a very good concept of what a unicorn looks like by asking them to mentally attach their mental representation of a horn to the horse’s forehead. The question of how the mental attachment is accomplished is easily answered if representations are simplicial. When that person is imagining a unicorn for the first time, they are simply activating a representation of a horse, activating a representation of a horn, and then activating a pathway in-between those two representations in such a manner that they cohere into a new simplex, that of a unicorn. Even the question of how and why the horn simplex attaches at the forehead and not elsewhere can be answered if we recall that the representation of a horse is also made up of sub-simplexes, which the horn simplex could variably attach to. In the same manner that we could ask the person to attach the horn to the horses head we could ask them to attach it to the horse’s nose or knee. If I were to ask the person to do so they would just attach that horn simplex to the horse nose or knee sub-simplex instead of the horse forehead sub-simplex.
Neural simplexes can be parts of multiple other larger simplexes. Those larger simplexes have connections in and out of their own. When I gave the unicorn example, I did not mention the magical qualities that typically accompany the representation of unicorn. If I were to ask a person from a north American cultural background what attributes are associated with unicorns, I’d likely receive the response of magic. However, there is no such connotation or association with either horns or horses. So, why does such a connotation arise? The connotations which arise from the combination of horse and horn into unicorn are likely due to a slightly separate mechanism of threshold dependent firing. It is possible and indeed likely that the tangential associations that result from unicorn are due to a representational or simplicial “module” that has inputs from both horn and horse simplexes. If both are activated then the combined inputs are sufficient to activate the magical module, enabling the perception of a magical aspect of the unicorn representation. While this is highly speculatory, what I are demonstrating is not so much that the intonation of magic as related to unicorns must work in this way, but rather I am demonstrating that the simplicial model is versatile enough to accommodate that connotation and answer other such questions about representational dynamics.
As for contextual priming, afferent excitation can of course affect which representations are selected. When I prime an individual with the term “Halloween”, Jack is easily interpreted as the name of an individual. When I prime the same person with the term “Bottle of”, I no longer expect that person to respond that Jack is a person, but instead a drink. This context dependence is accounted for and arises naturally from the simplex model. What occurs when priming happens is that there are representations which do not fully activate others but still increase the propensity of their activation. So, when I prime the individual with “Halloween” what I am doing is activating one representation, “Halloween”, which has afferent connections, but not complete simplicial binding, to the representation of Jack the Halloween character. When I prime the person with “bottle of”, I am doing the same for their representation of Jack Daniels brand alcohol. The same process can be used to mechanistically explain why ambiguous terms, similar sounding words, and even homophones can be perceived in massively different ways given the correct suggestion or context. The words that sound like “raise” can be distinguished in verbal communication effortlessly due to the context surrounding them. The terms, rays, raze, and raise all sound the same, but each has a different meaning (and rays has more than one meaning itself! Rays of sunshine and manta rays are spelled and sound the same).
Again, what is important here is that these examples are all computable and this account holds a lot of explanatory power stemming from the relatively sparse propositions I’ve outlined above.
After stimulating the simulation outlined above, it was found that when activity was correlated, active cliques formed cavities which grew in dimensionality. This was repeated in the same circuit with differing stimuli and then in other simulations. It was found that while different patterns occurred, the same temporal progression, where correlated activity between increasingly higher dimensional cliques formed and then fell apart, was generalized, was found in all the simulated circuits. Neurons were also found to take part in multiple simplexes.
Directed cliques demonstrate informative flow at smaller scales I.E. locally, and cavities demonstrate flow across the entirety of the network. There were more activating neurons than inhibitory neurons, as such inhibitory neurons are left out of the discussion for now (although I hypothesize the function primarily in the lateral inhibition between incompatible concepts or those trained in the network to be mutually exclusive generally).
As the authors note, simplexes attach to and activate larger simplexes which activate other simplexes and so on and so forth until the pattern of activation dissipates or turns into noise. This sort of dynamic of spreading activation in a non-linear (but still organized) fashion is able to be tied to a phenomenologically accessible attribute of representation. Representations in the mind flow into each other, meld, and exhibit temporality. The formation and decay of high dimensional simplexes along with an account of neuronal fatigue can easily account for the temporality of representations. For example, the representation of the decomposition of an apple over time can be modeled as a simplex evolving from a bud to ripe apple to a dying apple to a dead apple in a single large pattern of simplex activations. The flow and temporality of these concepts are therefore consequences of the same attributes of simplexes and neurons respectively. One may try to keep a pattern of activation or simplex in one’s mind for a long time, but it is very difficult to do so. One could posit that one could keep thinking about the interior of their home for hours, however this is also accounted for, and indeed predicted, by this account. The flow of simplex propagations and the ability of neurons to fatigue and recover would imply that cyclical representations will be easier to continue representing than static representations, and at least phenomenologically this seems to be the case, so mentally wandering about your home would be an easier task than keeping the details of an image vivid and constant in your brain (a note, I more thoroughly outline mental imagery in another paper, contact me for that paper).
In the Reimann paper simplexes of up to 11 dimensions were found, with the researchers noting that there could be larger simplexes. What this sort of multidimensionality enables is the possibility of multiple versions of the same representation to be bound together and represented as a concept while retaining the ability to have different sub-simplices activated in a systemic way. Again, an example will help illustrate. Take our representation of a house. When I mentally rotate my house I can begin at the front door, take a look around the sides, the backyard, and then imagine the entryway into my home and what awaits me inside. All of those concepts you understand to be your home, although those areas have very different uses and look very different. However, your brain coheres them as one thing and is able to represent multiple smaller parts of your home as you select them. When you rotate your home, an attentional modality in your brain is increasing the input to a segment of the larger home simplex corresponding to your home. As you rotate your home and manipulate it to see, say the backyard, you are switching that input to another sub-simplex of the greater simplex house, the backyard simplex. You may ask the question of why you cannot represent both at the same time. It is likely, even probable, that lateral inhibition helps us distinguish between concepts (more on that when I talk about learning). So there is lateral inhibition between those concepts, stopping you from representing both at the same time. Note that I have not strayed from computable and biologically plausible and extant concepts. Inhibition is achieved by neurotransmitters like GABA and again, a great number of simplicial neural structures have been found in the brain.
Using the simplexes as established in this paper, there is a clear line from simplex to learning. Let’s say I want to teach someone something quickly and well. I could teach them a whole new structure from the bottom up, or I could relate what I want to teach to what the person already knows. The simplex representation model would lead us to believe that cohering our desired concept to an extant concept would be easier to do then the opposite, and indeed that is what I find in the literature. Alternatively, If I were to want to know how to distinguish between two bugs, it makes sense that I would want to know what makes them different. And as I got comfortable with the distinction it may very well become automatic for me to distinguish between the two bugs, or as I have experienced recently, between two different but very similar fiber counts in fabrics of the same material. That distinction, again very easily, can be attributed to lateral inhibition between the two concepts which is neurally and computationally executable. As I learn the difference, I will begin to build up that lateral inhibition to the point where effort on my part is not necessary, when the representation comes up, I just represent one thing as bug type one and the other as bug type three, or one sheet as 800 thread count and another as 1000 thread count.
The authors state that when directionality is lost a great deal of information is lost as well, they state that a clique of four neurons, if directionality is ignored, leaves us with a single solution, however, when direction is taken into account, there are 3^6 possible configurations. A cyclical clique always decays into directed cliques with the same dimension or smaller, and a single connection between two neurons forms a 2 clique. A cyclical three clique decomposes into three two cliques.
A note, multidimensional refers to informational multidimensionality, not spatial multidimensionality, this must be noted as there appears to be some confusion on the subject (Tozzi, 2017).
Fig. 4 denotes a likely model for the recognition of words from phonemes and consequent activation of representation given the successful recognition of the word, along with the inhibitory properties of whole word recognition when a word (Jameson) is constructed from another word (Jam). The representation of jam will not be activated as the activation of the relate word Jameson inhibits it, ensuring the correct information gets through.
The representation of jam is posited to be simplicial in nature, with sub-simplices making up representative portions down to simplices which constitute the basis of the given experience. So, the simplex for jam may include the simplicial memory of the texture and reaction of jam physically to a spoon, the representation of sugar, the hardness of a jar, etc. So, the simplex directs the activation of differing modalities which jointly recreate a representation of jam. Put another way, the simplex simply directs towards other simplexes which constitute the experience of jam. This formulation also accounts for the binding problem, as the binding occurs temporally and though simplex binding.
I also propose that the outside world is not rendered directly, but rather pulls upon recalled constructed simplex representations to recreate a useful account of the world. Imagine a walk through our campus. I could create a new representation of the trees every time one enters our visual field but doing so would be massively cognitively tasking. It is much more likely that upon recognition of a tree our brain simply generates a representation of a tree from representative parts it already has. This has important implications, as it implies that I navigate a world that is not only constructed but is constructed from prior experience.
Context dependent learning and recall are also more easily described under this model. The chemical bath from the limbic system can increase the probability of activation of simplexes formed under similar contexts. This also accounts for differences in behavior and cognition when in emotional states.
I account for symbolic gesture comprehension. There is research showing that symbolic gestures and spoken language are processed by the same neural system (Xu, Gannon, Emmorey, Smith, & Braun, 2009), and separate research shows that speech like cerebral activity is found in profoundly deaf people (Perirro, et al., 2000), implying again a shared system. Under our framework this is trivial and expected, as I expect the recognition to lead to the same representative structures regardless of the origin of recognition.
A complete and accurate description or model of the mind should be able to account for just about any mental phenomena, it should be generalizable, and it is apparent that this account of mental phenomena can account for most if not all of what the brain does.
Look around in your room and identify an object. That object has a color, a shape in three dimensions, and a surrounding. At the same time your foot may be falling asleep, and you may be listening to music. The binding problem asks how color, shape, surrounding, and sensory input are all combined to present your conscious self with a coherent scene. Possible solutions include spatial and temporal binding. Spatial binding is where a detected object’s location on a mental map can correspond with a detected color’s location, and the binding occurs due to the perception of the two stimuli at the same place. Temporal binding is when things are perceived at the same time are bound to be the same object, however empirical evidence for both has been mixed (Holcombe, 2009).
More recent attempts at solving the binding problem (Chella, Frixione, & Lieto, 2018) have included a concept of conceptual spaces, which do utilize multidimensional representations as described by the Reimann et. al. paper outlined below, but do not do much work positing how such a representation would function, and merely connect it to the conceptual spaces concept, which was extant prior to the Reimann paper.
Chella, Frixione, and Lieto also mention diagrammatic models of representation which posit that representations spatially resemble the objects they represent. HoIver, it also states that such posits are lacking in a general theory and that while there is a certain intuitive appeal, there is no common or Ill-understood theoretical framework.
Chella, Frixione, and Lieto then go on to name three systems, SOAR, ACT-R, and Sigma, which are part of their so called “Standard Model of Mind”, with criticism of the SOAR and ACT-R systems, and a description of some deficits of Sigma. The three closely related systems are mixes of several systems, which again serve rather as black boxes, and the authors seemingly support picture like visual representations, the type which they stated had no well understood theoretical framework.
I hope to have outlined a novel and useful account of representation representational processing. Other work along these lines is available upon request, mainly on mental imagery, the binding problem, conscious report, and a more in depth outline of representational processing using the roughly equivalent language of directed hypergraphs.
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