Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunctions being the primary lens. If adjunctions are so important in mathematics, then perhaps they will isolate concepts of some importance in the empirical sciences. But the applications of adjunctions have been hampered by an overly restrictive formulation that avoids heteromorphisms or hets. By reformulating an adjunction using hets, it is split into two parts, a left and a right semiadjunction. Semiadjunctions (...) (essentially a formulation of universal mapping properties using hets) can then be combined in a new way to define the notion of a brain functor that provides an abstract model of the intentionality of perception and action (as opposed to the passive reception of sense-data or the reflex generation of behavior). (shrink)

Ever since Thomas Malthus used simple mathematics to model the evolution of food production vs the increasing human population of planet Earth and declared that we would very soon eat ourselves out of house and home, politicians have been falling hook, line and sinker for mathematical models. This paper examines how and why the mathematical modelling of complex systems has proved unreasonably effective in convincing mathematically illiterate politicians to take practical measures that have turned out to be disastrous. (...) It then suggests that mathematical modelling of the human brain is not a good strategy to pursue. (shrink)

Mathematical models of tumour invasion appear as interesting tools for connecting the information extracted from medical imaging techniques and the large amount of data collected at the cellular and molecular levels. Most of the recent studies have used stochastic models of cell translocation for the comparison of computer simulations with histological solid tumour sections in order to discriminate and characterise expansive growth and active cell movements during host tissue invasion. This paper describes how a deterministic approach based on reaction-diffusion (...) models and their generalisation in the mechano-chemical framework developed in the study of biological morphogenesis can be an alternative for analysing tumour morphological patterns. We support these considerations by reviewing two studies. In the first example, successful comparison of simulated brain tumour growth with a time sequence of computerised tomography (CT) scans leads to a quantification of the clinical parameters describing the invasion process and the therapy. The second example considers minimal hypotheses relating cell motility and cell traction forces. Using this model, we can simulate the bifurcation from an homogeneous distribution of cells at the tumour surface toward a nonhomogeneous density pattern which could characterise a pre-invasive stage at the tumour-host tissue interface. (shrink)

This article argues that many, if not most, behavior descriptions and sequencing are in essence an interpretation of signs, and are evaluated as sequences of signs by researchers. Thus, narrative analysis, as developed by Barthes and others, seems best suited to be used in behavioral/biosemiotic studies rather than mathematical modeling, and is very similar to some classic ethology methods. As our brain interprets behaviors as signs and attributes meaning to them, narrative analysis seems more suitable than mathematical (...) modeling to describe and study behavior. Mathematical models are, on many occasions, extremely reductionist and simplifying because of computational and/or numerical representation limitations that lead to errors and straitjacketing interpretations of reality. Since actual animals are not as optimal in real life as our mathematical models, here it is proposed that we should consider logical/verbal models and semantic interpretations as equally or even better suited for behavioral analysis, and refrain from enforcing mathematical modeling as the only way to study and understand biological problems, especially those of a behavioral and biosemiotic nature. (shrink)

Computational neuroscientists not only employ computer models and simulations in studying brain functions. They also view the modeled nervous system itself as computing. What does it mean to say that the brain computes? And what is the utility of the ‘brain-as-computer’ assumption in studying brain functions? In previous work, I have argued that a structural conception of computation is not adequate to address these questions. Here I outline an alternative conception of computation, which I call the (...) analog-model. The term ‘analog-model’ does not mean continuous, non-discrete or non-digital. It means that the functional performance of the system simulates mathematical relations in some other system, between what is being represented. The brain-as-computer view is invoked to demonstrate that the internal cellular activity is appropriate for the pertinent information-processing task.Keywords: Computation; Computational neuroscience; Analog computers; Representation; Simulation. (shrink)

We build a topological model, based on intuitionistic logic, for multi-agent biological systems (such as _Physarum polycephalum_, bacterial colonies or any other swarm), reacting to external nourishment stimuli. Our construction follows the topological description of brain activity, where particles (neurons) are activated by an external environment, represented by a topological space _X_ with an open cover \(\{U_i:i\in I\}\). The brain builds the model of this external space via the nerve (trace) of a topological space _X_. Here the body (...) of _Physarum polycephalum_ or a swarm made of networks of tubular structures represents a nerve (trace) of _X_ also which means that _Physarum polycephalum_ or a swarm gains orientation in the space of external stimuli even in the absence of any neural system. The logic of living organisms is based on open subsets of _X_ and thus can be represented by Heyting algebra (i.e. intuitionistically). We also consider the generalisation of the nerve construction to a categorical context, where the category is determined by the network structures of multi-agent biological system. This model can be generalised up to simulating the behaviour of any swarm by means of intuitionistic logic. (shrink)

Shepard's article presents an impressive application of the mathematics of symmetry to the understanding of motion. However, there are basic psychological phenomena that the model does not handle well. These include the importance of the orientations of rotational motions to salient reference systems for the understanding of the motions. An alternative model of the understanding of rotations is sketched. [Shepard].

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 as initial conditions, and is ‘processed’ as solutions of its 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 into a set of stable complementary and mutually exclusive eigenfunctions or ‘categories'. The characteristics of this cognitive set depend on the operator involved and the hierarchical level where the abstraction takes place. Depending on the context, the transition from one state to another occurs in such a cognitive operator. The chaotic itinerancy may play a crucial role for this process. In this paper we model the dynamics which may underlie such a cognitive process and the role of the thalamo-cortical pacemaker of the brain. In order to model them, conceptualization by the notion of ‘attractor ruin’ in high-dimensional dynamical systems is necessary. (shrink)

Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens. The central feature of an adjunction is what might be called "internalization through a universal" based on universal mapping properties. A recently developed "heteromorphic" theory (...) of adjoint functors allows the concepts to be more easily applied empirically. This suggests a conceptual structure, albeit abstract, to model a range of selectionist mechanisms (e.g., in evolution and in the immune system). Closely related to adjoints is the notion of a "brain functor" which abstractly models structures of cognition and action (e.g., the generative grammar view of language). (shrink)

From the theoretical discussions, transdisciplinarity starts to have practical consequences in the development of programs that include consortia of universities, bringing together a large variety of professionnals who set ambitious goals, such as the Human Genome Project in the past decade, and also the Human Brain Project for this decade. We intend to present an approach in the spirit of the new paradigms of knowledge in the Human Brain Project generous program started earlier this year in Europe. A (...) possible transdisciplinary approach on brain functions and structure is to bring a valuable, significant and innovative, unconventional contribution to the theory of multi-scale topics of the HBP project. More precisely, we intend to develop the subproject Mathematical and Theoretical Foundations of Brain Research within the HBP project. Our approach aims at exploring brain function from the perspective of the theories which influenced the last decade researches in Physics and Mathematics, such as the fractal theory, chaos and the dynamics of the nonlinear systems, in order to formulate new theories regarding brain mechanisms. We bring together models and new theories based on a principled approach to nonlinear reality. We wish to develop a new paradigm for assessing brain function, the theory of complex systems, which requires a revolutionary attitude, by rethinking how brain works and it is biologically structured. By this approach, we intend to study not only the corpuscular, but also the wave part of the matter. (shrink)

Mathematical models are potentially as useful for culture as for evolution, but cultural models must have different designs from genetic models. Social sciences must borrow from biology the idea of modeling, rather than the structure of models, because copying the product is fundamentally different from copying the design. Transfer of most cultural information from brains to artificial media increases the differences between cultural and biological information. (Published Online November 9 2006).

Metaphors and models involve correspondences between events in separate domains. They differ in the form and precision of how the correspondences are expressed. Examples include correspondences between phylogenic and ontogenic selection, and wave and particle metaphors of the mathematics of quantum physics. An implication is that the target article's metaphors of resistance to change may have heuristic advantages over those of momentum.

The ability to improve in speed and accuracy as a result of repeating some task is an important hallmark of intelligent biological systems. Although gradual behavioral improvements from practice have been modeled in spiking neural networks, few such models have attempted to explain cognitive development of a task as complex as addition. In this work, we model the progression from a counting-based strategy for addition to a recall-based strategy. The model consists of two networks working in parallel: a slower basal (...) ganglia loop and a faster cortical network. The slow network methodically computes the count from one digit given another, corresponding to the addition of two digits, whereas the fast network gradually “memorizes” the output from the slow network. The faster network eventually learns how to add the same digits that initially drove the behavior of the slower network. Performance of this model is demonstrated by simulating a fully spiking neural network that includes basal ganglia, thalamus, and various cortical areas. Consequently, the model incorporates various neuroanatomical data, in terms of brain areas used for calculation and makes psychologically testable predictions related to frequency of rehearsal. Furthermore, the model replicates developmental progression through addition strategies in terms of reaction times and accuracy, and naturally explains observed symptoms of dyscalculia. (shrink)

Over the last decade, fully distributed models have become dominant in connectionist psychological modelling, whereas the virtues of localist models have been underestimated. This target article illustrates some of the benefits of localist modelling. Localist models are characterized by the presence of localist representations rather than the absence of distributed representations. A generalized localist model is proposed that exhibits many of the properties of fully distributed models. It can be applied to a number of problems that are difficult for fully (...) distributed models, and its applicability can be extended through comparisons with a number of classic mathematical models of behaviour. There are reasons why localist models have been underused, though these often misconstrue the localist position. In particular, many conclusions about connectionist representation, based on neuroscientific observation, can be called into question. There are still some problems inherent in the application of fully distributed systems and some inadequacies in proposed solutions to these problems. In the domain of psychological modelling, localist modelling is to be preferred. Key Words: choice; competition; connectionist modelling; consolidation; distributed; localist; neural networks; reaction-time. (shrink)

Joshua Greene has argued that the empirical findings of cognitive science have implications for ethics. In particular, he has argued (1) that people’s deontological judgments in response to trolley problems are strongly influenced by at least one morally irrelevant factor, personal force, and are therefore at least somewhat unreliable, and (2) that we ought to trust our consequentialist judgments more than our deontological judgments when making decisions about unfamiliar moral problems. While many cognitive scientists have rejected Greene’s dual-process theory of (...) moral judgment on empirical grounds, philosophers have mostly taken issue with his normative assertions. For the most part, these two discussions have occurred separately. The current analysis aims to remedy this situation by philosophically analyzing the implications of moral dilemma research using the CNI model of moral decision-making – a formalized, mathematical model that decomposes three distinct aspects of moral-dilemma judgments. In particular, we show how research guided by the CNI model reveals significant conceptual, empirical, and theoretical problems with Greene’s dual-process theory, thereby questioning the foundations of his normative conclusions. (shrink)

A number of conditioning experiments utilize food as a reward. Hunger is considered to be a critical factor governing the animal's behavior in these experiments. Despite its significance, most theories of animal conditioning fail to take hunger into consideration while analyzing the behavioral data. In this paper, we analyze the neuroscientific data supporting the hypothesis that hunger and food consumption affect the brain's dopamine system, which in turn governs the animal's behavior. According to this hypothesis, chronic hunger results in (...) a decrease in the extra-cellular dopamine levels in the animal's brain. This decrease is believed to trigger a series of reactions that increase the responsivity of the phasic dopamine system. A direct consequence of this is an increased vigor of all dopamine-dependent behaviors. The level of extra-cellular dopamine is also modulated by the process of food consumption via the neurotransmitter Acetylcholine. Food consumption raises the dopamine above the baseline level. This rise depends on the animals' hunger, which when satisfied increases the level of Acetylcholine, which causes the dopamine level to fall back to the baseline. Thus, extra-cellular dopamine governs the response vigor, with an increase in dopamine resulting in a more vigorous response. This paper makes two primary contributions. Firstly, we present an abstract mathematical model based on the above hypothesis. Our mathematical model is able to provide a simple explanation for a number of behavioral findings. Another contribution of this paper is the development of a neurocomputational Leabra model of dopamine and acetylcholine activity in the basal ganglia to incorporate hunger and satiation. The experimental results obtained from this neural model are also largely in agreement with behavioral findings. (shrink)

The target article draws on evolutionary theory to formulate a biosocial model of sex differences in quantitative abilities. Unfortunately, the data do not support some of the crucial hypotheses. The male advantage in geometry is not appreciably greater than the male advantagi in algebra, and the greater male variability in mathematics cited by Gear is not cross-culturally invariant.

Neural organization describes an approach to analyzing neural function in anatomically defined subsystems in the brain, the hippocampus, cerebellum, sensory systems, thalamus, basal ganglia, and cerebral cortex, combining information on neurocircuitry with mathematical models that link structure with function. It is an up-to-date source on the major schemes and background for neural modeling of the central nervous system and is combined with a Web site that includes tutorials and on-line modeling possibilities.

Not only does Hoffman claim that we do not see reality as it is, but that unperceived brains, trees and moons do not exist. His “interface theory of perception” is a peculiar blend of metaphorical ontology (objects are icons, space-time is a desktop) and mathematical modelling (the game-theoretical argument that fitness trumps truth. Conflating abstractions with concrete experience, evolution is used to refute everything (including evolution itself. Hoffman’s sweeping iconoclasm then lands where it took off: addressing the problem of (...) consciousness. After arguing against reality, he will tell us what it is. (shrink)

Much of our understanding of human thinking is based on probabilistic models. This innovative book by Jerome R. Busemeyer and Peter D. Bruza argues that, actually, the underlying mathematical structures from quantum theory provide a much better account of human thinking than traditional models. They introduce the foundations for modelling probabilistic-dynamic systems using two aspects of quantum theory. The first, 'contextuality', is a way to understand interference effects found with inferences and decisions under conditions of uncertainty. The second, 'quantum (...) entanglement', allows cognitive phenomena to be modeled in non-reductionist ways. Employing these principles drawn from quantum theory allows us to view human cognition and decision in a totally new light. Introducing the basic principles in an easy-to-follow way, this book does not assume a physics background or a quantum brain and comes complete with a tutorial and fully worked-out applications in important areas of cognition and decision. (shrink)

This article takes as its point of departure the view that the discovery of the mathematical formalism of quantum mechanics (QM) was not merely a discovery of a new mathematical way of dealing with physical, and specifically quantum, processes in nature. It was also a discovery of a general mathematical formalism (in part discovered in mathematics itself earlier), which, supplemented by certain additional rules, consistently described the processing of incomplete information about certain events and contexts in which (...) these events occur. This article proposes a quantum-like (QL) model of the functioning of the brain based on the (Hilbert-space) formalism of quantum mechanics, but now used as part of a QL mathematical model of neural processes in the brain, rather than for describing quantum physical processes. This model is, thus, fundamentally different from the (reductionist) quantum model of the brain and consciousness, according to which cognition arises by virtue of physical quantum processes in the brain. In the present view, the brain is an advanced biological system that developed the ability to create a QL representation of contexts, which, thus, allows one to describe a significant part of its functioning by the QM mathematical formalism. The possibility of such a description has nothing to do with the constitution and workings of the brain as a quantum system (composed of photons, electrons, protons, and so forth). The QL model offered here is based instead on conventional neurophysiological model of the functioning of the brain, even though the brain, the article suggests, does use the QL rule (given by von Neumann trace formula, used in QM) for the calculation of approximate averages for mental functions. The QL model developed in this article has a temporal basis, based on a (hypothetical) argument that cognitive processes are based on at least two time scales: a (very fine) subcognitive one and a (much coarser) cognitive one. (shrink)

Integrated Information Theory is one of the leading models of consciousness. It aims to describe both the quality and quantity of the conscious experience of a physical system, such as the brain, in a particular state. In this contribution, we propound the mathematical structure of the theory, separating the essentials from auxiliary formal tools. We provide a definition of a generalized IIT which has IIT 3.0 of Tononi et al., as well as the Quantum IIT introduced by Zanardi (...) et al. as special cases. This provides an axiomatic definition of the theory which may serve as the starting point for future formal investigations and as an introduction suitable for researchers with a formal background. (shrink)

This paper considers the way mathematical and computational models are used in network neuroscience to deliver mechanistic explanations. Two case studies are considered: Recent work on klinotaxis by Caenorhabditis elegans, and a longstanding research effort on the network basis of schizophrenia in humans. These case studies illustrate the various ways in which network, simulation and dynamical models contribute to the aim of representing and understanding network mechanisms in the brain, and thus, of delivering mechanistic explanations. After outlining this (...) mechanistic construal of network neuroscience, two concerns are addressed. In response to the concern that functional network models are non-explanatory, it is argued that functional network models are in fact explanatory mechanism sketches. In response to the concern that models which emphasize a network’s organization over its composition do not explain mechanistically, it is argued that this emphasis is both appropriate and consistent with the principles of mechanistic explanation. What emerges is an improved understanding of the ways in which mathematical and computational models are deployed in network neuroscience, as well as an improved conception of mechanistic explanation in general. (shrink)

The nature of attention is one of the oldest and most central problems in psychology. A huge amount of research has been produced on this subject in the last half century, especially on attention in the visual modality, but a general explanation has remained elusive. Many still view attention research as a field that is fundamentally fragmented. This book takes a different perspective and presents a unified theory of visual attention: the TVA model. The TVA model explains the many aspects (...) of visual attention by just two mechanisms for selection of information: filtering and pigeonholing. These mechanisms are described in a set of simple equations, which allow TVA to mathematically model a large number of classical results in the attention literature. The theory explains psychological and neuroscientific findings by the same equations; TVA is a complete theory of visual attention, linking mind and brain. Aimed at advanced students and professional researchers, Principles of Visual Attention contains a detailed review of the most important research done on attention in vision, spanning cognitive psychology, brain imaging, patient studies, and recordings from single cells in the visual cortex. The book explains the TVA model and shows how it accounts for attentional effects observed across all the research areas described. Principles of Visual Attention offers a uniquely integrated view on a central topic in cognitive neuroscience. (shrink)

A novel theory of reflective consciousness, will and self is presented, based on modeling each of these entities using self-referential mathematical structures called hypersets. Pattern theory is used to argue that these exotic mathematical structures may meaningfully be considered as parts of the minds of physical systems, even finite computational systems. The hyperset models presented are hypothesized to occur as patterns within the "moving bubble of attention" of the human brain and any roughly human-mind-like AI system. These (...) ideas appear to be compatible with both panpsychist and materialist views of consciousness, and probably other views as well. Their relationship with the CogPrime AI design and its implementation in the OpenCog software framework is elucidated in detail. (shrink)

Features of consciousness difficult to understand in terms of conventional neuroscience have evoked application of quantum theory, which describes the fundamental behavior of matter and energy. In this paper we propose that aspects of quantum theory (e.g. quantum coherence) and of a newly proposed physical phenomenon of quantum wave function "self-collapse"(objective reduction: OR -Penrose, 1994) are essential for consciousness, and occur in cytoskeletal microtubules and other structures within each of the brain's neurons. The particular characteristics of microtubules suitable for (...) quantum effects include their crystal-like lattice structure, hollow inner core, organization of cell function and capacity for information processing. We envisage that conformational states of microtubule subunits (tubulins) are coupled to internal quantum events, and cooperatively interact (compute) with other tubulins. We further assume that macroscopic coherent superposition of quantum-coupled tubulin conformational states occurs throughout significant brain volumes and provides the global binding essential to consciousness. We equate the emergence of the microtubule quantum coherence with pre-conscious processing which grows (for up to 500 milliseconds) until the mass-energy difference among the separated states of tubulins reaches a threshold related to quantum gravity. According to the arguments for OR put forth in Penrose (1994), superpositioned states each have their own space-time geometries. When the degree of coherent mass-energy difference leads to sufficient separation of space-time geometry, the system must choose and decay (reduce, collapse) to a single universe state. In this way, a transient superposition of slightly differing space-time geometries persists until an abrupt quantum classical reduction occurs. Unlike the random, "subjective reduction"( SR, or R) of standard quantum theory caused by observation or environmental entanglement, the OR we propose in microtubules is a self-collapse and it results in particular patterns of microtubule-tubulin conformational states that regulate neuronal activities including synaptic functions. (shrink)

Gliomas are diffuse and invasive brain tumors with the nefarious ability to evade even seemingly draconian treatment measures. Here we introduce a simple mathematical model for drug delivery of chemotherapeutic agents to treat such a tumor. The model predicts that heterogeneity in drug delivery related to variability in vascular density throughout the brain results in an apparent tumor reduction based on imaging studies despite continual spread beyond the resolution of the imaging modality. We discuss a clinical example (...) for which the model-predicted scenario is relevant. The analysis and results suggest an explanation for the clinical problem of the long-standing confounding observation of shrinkage of the lesion in certain areas of the brain with continued growth in other areas. (shrink)

Physics says that it cannot deal with the mind-brain problem, because it does not deal in subjectivities, and mind is subjective. However, biologists still claim to seek a material basis for subjective mental processes, which would thereby render them objective. Something is clearly wrong here. I claim that what is wrong is the adoption of too narrow a view of what constitutes objectivity, especially in identifying it with what a machine can do. I approach the problem in the light (...) of two cognate circumstances: the measurement problem in quantum physics, and the objectivity of standard mathematics, even though most of it is beyond the reach of machines. I argue that the only resolution to such problems is in the recognition that closed loops of causation are objective; i.e. legitimate objects of scientific scrutiny. These are explicitly forbidden in any machine or mechanism. A material system which contains such loops is called complex. Such complex systems thus must possess nonsimulable models; i.e. models which contain impredicativities or self-references which cannot be removed, or faithfully mapped into a single coherent syntactic time-frame. I consider a few of the consequences of the above, in the context of thus redrawing the boundary between subject and object. (shrink)

There are two main paradigms for brain-related science, with different implications for brain-focused intervention or advancement. The paradigm of homeostasis (“stability through constancy,” Walter Cannon), originating from laboratory-based experimental physiology pioneered by Claude Bernard, shows that living systems tend to maintain system functionality in the direction of constancy (or similitude). The aim of physiology is elucidate the factors that maintain homeostasis, and therapeutics aim to correct abnormal factor functions. The homeostasis paradigm does not formally recognize influences outside its (...) controlled experimental frames and it is variable in its modelling of neural contributions. The paradigm of allostatic orchestration (PAO) extends the principle of allostasis (“stability through change”) as originally put forth by Peter Sterling. The PAO originates from an evolutionary perspective and recognizes that biological set points change in anticipation of changing environments. The brain is the organ of central command, orchestrating cross-system operations to support optimal behavior at the level of the whole organism. Alternative views of blood pressure regulation and posttraumatic stress disorder illustrate differences between the paradigms. For the PAO, complexities of top-down neural effects and environmental context are foundational (not to be “factored out”), and anticipatory regulation is the principle of their interface. The allostatic state represents the integrated totality of brain-body interactions. Health itself is an allostatic state of optimal anticipatory oscillation, hypothesized to relate to the state of criticality, a mathematical point of poise between phases, on the border between order and disorder (or the “edge of chaos”). Diseases are allostatic states of impaired anticipatory oscillations, demonstrated as rigidifications of set points across the brain and body (disease comorbidity). Conciliation of the paradigms is possible, with “reactive homeostasis” resolved as an illusion stemming from anticipation of environmental monotony. Considerations are presented with respect to implications of the two paradigms for brain-focused intervention or advancement; the hypothesis that the state of criticality is a vehicle for evolutionary processes; concordance with a philosophy of freedom based on ethical individualism as well as self-creativity, non-obsolescence, empowerment, and citizenship; and concluding reflections on the science and ethics of the placebo, and the potential for virtuous cycles of brain-Anthropocene interactions. (shrink)

Philosophy of science has characterized scientific knowledge as fundamentally propositional . This account leads to an inability to recognize and articulate the significant role of non-propositional, visual representation in the practice of science. Toward the development of a more productive framework for understanding visual representation in science, the present study critiques the standard philosophical view, reviews the literature on visual representation in science, and examines the scientific case of neuroscience. Specifically, the study looks at current research known as "functional mapping (...) of the human brain" which uses neuroimaging technologies such as positron emission tomography in the localization of particular functions in the human brain. This case study suggests that, contrary to previous accounts in philosophy of science, visual representation and propositional representation work together as parallel, interacting, interwoven practices in the discovery/construction of scientific knowledge. Visual representations are not reduced to--translated into--linguistic and mathematical propositions, but instead are used in their multidimensional forms. The present study focuses on the external, materialized visual representations used in the practice of science, as opposed to internal, mental images that might be studied by cognitive psychology. Several hypotheses are proposed which describe aspects of the role of these visual representations in the practice of science. These include the following: a primary function of visual representation is the representation of structure which is made possible by the use of actual space in the process of visual representation; visual comparisons among visual representations of real objects and processes, visual representations of theoretical models, and visual representations of integrations of the two are used to assess the fit of the theoretical models to the real objects and processes; the construction of visual prototypes or prototypical visual representations of objects of scientific research are used to categorize and stabilize--hold stationary for the purposes of scientific examination--such objects; and the construction of material models make it possible for scientists to interact with and/or manipulate materialized structures as representations of real systems. (shrink)

Although the idea that cognitive structure changes as we learn is welcome, a variety of mathematical structures are needed to model the neural and cognitive processes involved. A specific example of bodily-kinaesthetic intelligence is given, building on a formalism given elsewhere. As the structure of cognition changes, previous learning can become tacit, adding to the complexity of cognition and its modeling.

The present article addresses the logic of the sphere, or the Sphere Model of Consciousness developed by Patrizio Paoletti over three decades of research. M.E.D. Ed., 2002; Flussi, territori, luogo II. M.E.D. Ed., 2002; Fare il punto nave. M.E.D. Ed., 2005; In: Proceedings conference at Leslie and Susan Gonda Multidisciplinary Brain Research Center. Bar Ilan University. Faculty of Neuroscience, Israel, 2007; Osservazione—Quaderni di Pedagogia per il Terzo Millennio, Ed. 3P, 2011; Mediazione—Quaderni di Pedagogia per il Terzo Millennio, Ed. 3P, (...) 2011). The SMC model has been experimentally applied in the educational field and lies at the base of neuroscientific and psychoeducational research conducted by the Paoletti Foundation. In recent years, it has been studied by several researchers in the field of neuroscience. Following a logical-mathematical introduction regarding the properties of the spherical shape, we illustrate for the first time the structuring of the model and its neural foundations. We emphasize the central space of the sphere, defined by the geometric model and highlight its descriptive and heuristic properties in relation to the study of consciousness. Finally, we present educational applications of the model, particularly with respect to the center of the sphere, defined in the current context as the “place of pre-existence.”. (shrink)

We consider consciousness attributed to systems in space-time which can be purely physical without biological background and focus on the mathematical understanding of the phenomenon. It is shown that the set theory based on sets in the foundations of mathematics, when switched to set theory based on ZFC models, is a very promising mathematical tool in explaining the brain/mind complex and the emergence of consciousness in natural and artificial systems. We formalise consciousness-supporting systems in physical space-time, but (...) this is localised in open domains of spatial regions and the result of this process is a family of different ZFC models. Random forcing, as in set theory, corresponds precisely to the random influence on the system of external stimuli, and the principles of reflection of set theory explain the conscious internal reaction of the system. We also develop the conscious Turing machines which have their external ZFC environment and the dynamics is encoded in the random forcing changing models of ZFC in which Turing machines with oracles are formulated. The construction is applied to cooperating families of conscious agents which, due to the reflection principle, can be reduced to the implementation of certain concurrent games with different levels of self-reflection. (shrink)

There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms that parses an adjunction into two separate parts. Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category (...) theory and is focused on the interpretation and application of the mathematical concepts. (shrink)

Choice Outstanding Academic Title, 1996. In hundreds of articles by experts from around the world, and in overviews and "road maps" prepared by the editor, The Handbook of Brain Theory and Neural Networkscharts the immense progress made in recent years in many specific areas related to two great questions: How does the brain work? and How can we build intelligent machines? While many books have appeared on limited aspects of one subfield or another of brain theory and (...) neural networks, the Handbookcovers the entire sweep of topics—from detailed models of single neurons, analyses of a wide variety of biological neural networks, and connectionist studies of psychology and language, to mathematical analyses of a variety of abstract neural networks, and technological applications of adaptive, artificial neural networks. The excitement, and the frustration, of these topics is that they span such a broad range of disciplines including mathematics, statistical physics and chemistry, neurology and neurobiology, and computer science and electrical engineering as well as cognitive psychology, artificial intelligence, and philosophy. Thus, much effort has gone into making the Handbookaccessible to readers with varied backgrounds while still providing a clear view of much of the recent, specialized research in specific topics. The heart of the book, part III, comprises of 267 original articles by leaders in the various fields, arranged alphabetically by title. Parts I and II, written by the editor, are designed to help readers orient themselves to this vast range of material. Part I, Background, introduces several basic neural models, explains how the present study of brain theory and neural networks integrates brain theory, artificial intelligence, and cognitive psychology, and provides a tutorial on the concepts essential for understanding neural networks as dynamic, adaptive systems. Part II, Road Maps, provides entry into the many articles of part III through an introductory "Meta-Map" and twenty-three road maps, each of which tours all the Part III articles on the chosen theme. (shrink)

We affirm the dynamical systems approach taken by Feldman and Levin, but argue that a more mathematically rigorous and standard exposition of the model according to dynamical systems theory would greatly increase readability and testability. Such an explication would also have heuristic value, suggesting new variations of the model. We present one such variant, a new solution to the redundancy problem.

The interaction between syntax and its semantics is one which has been well studied in categorical logic. The results of this particular study are employed to understand how the brain is able to create meanings. To emphasize the toy character of the proposed model, we prefer to speak of the homunculus brain rather than the brain per se. The homunculus brain consists of neurons, each of which is modeled by a category, and axons between neurons, which (...) are modeled by functors between the corresponding neuron-categories. Each neuron has its own program enabling its working, i.e. a theory of this neuron. In analogy to what is known from categorical logic, we postulate the existence of a pair of adjoint functors, called Lang and Syn, from a category, now called BRAIN, of categories, to a category, now called MIND, of theories. Our homunculus is a kind of “mathematical robot”, the neuronal architecture of which is not important. Its only aim is to provide us with the opportunity to study how such a simple brain-like structure could “create meanings” and perform abstraction operations out of its purely syntactic program. The pair of adjoint functors Lang and Syn model the mutual dependencies between the syntactical structure of a given theory of MIND and the internal logic of its semantics given by a category of BRAIN. In this way, a formal language and its meanings are interwoven with each other in a manner corresponding to the adjointness of the functors Lang and Syn. Higher cognitive functions of abstraction and realization of concepts are also modelled by a corresponding pair of adjoint functors. The categories BRAIN and MIND interact with each other with their entire structures and, at the same time, these very structures are shaped by this interaction. (shrink)

Network neuroscience provides a systems approach to the study of the brain and enables the examination of interactions measured at different temporal and spatial scales. We review current methods to quantify the structure of brain networks and compare that structure across different clinical cohorts, cognitive states, and subjects. We further introduce the emerging mathematical concept of multilayer networks and describe the advantages of this approach to model changing brain dynamics over time. We conclude by offering several (...) concrete examples of how multilayer network approaches to neuroimaging data provide novel insights into brain structure and evolving function. (shrink)

Making Sense of Inner Sense 'Terra cognita' is terra incognita. It is difficult to find someone not taken abackand fascinated by the incomprehensible but indisputable fact: there are material systems which are aware of themselves. Consciousness is self-cognizing code. During homo sapiens's relentness and often frustrated search for self-understanding various theories of consciousness have been and continue to be proposed. However, it remains unclear whether and at what level the problems of consciousness and intelligent thought can be resolved. Science's greatest (...) challenge is to answer the fundamental question: what precisely does a cognitive state amount to in physical terms? Albert Einstein insisted that the fundamental ideas of science are essentially simple and can be expressed in a language comprehensible to everyone. When one thinks about the complexities which present themselves in modern physics and even more so in the physics of life, one may wonder whether Einstein really meant what he said. Are we to consider the fundamental problem of the mind, whose understanding seems to lie outside the limits of the mind, to be essentially simple too? Knowledge is neither automatic nor universally deductive. Great new ideas are typically counterintuitive and outrageous, and connecting them by simple logical steps to existing knowledge is often a hard undertaking. The notion of a tensor was needed to provide the general theory of relativity; the notion of entropy had to be developed before we could get full insight into the laws of thermodynamics; the notice of information bit is crucial for communication theory, just as the concept of a Turing machine is instrumental in the deep understanding of a computer. To understand something, consciousness must reach an adequate intellectual level, even more so in order to understand itself. Reality is full of unending mysteries, the true explanation of which requires very technical knowledge, often involving notions not given directly to intuition. Even though the entire content and the results of this study are contained in the eight pages of the mathematical abstract, it would be unrealistic and impractical to suggest that anyone can gain full insight into the theory that presented here after just reading abstract. In our quest for knowledge we are exploring the remotest areas of the macrocosm and probing the invisible particles of the microcosm, from tiny neutrinos and strange quarks to black holes and the Big Bang. But the greatest mystery is very close to home: the greatest mystery is human consciousness. The question before us is whether the logical brain has evolved to a conceptual level where it is able to understand itself. (shrink)

The principles of sexual selection were used as an organizing framework for interpreting cross-national patterns of sex differences in mathematical abilities. Cross-national studies suggest that there are no sex differences in biologically primary mathematical abilities, that is, for those mathematical abilities that are found in all cultures as well as in nonhuman primates, and show moderate heritability estimates. Sex differences in several biologically secondary mathematical domains are found throughout the industrialized world. In particular, males consistently outperform (...) females in the solving of mathematical word problems and geometry. Sexual selection and any associated proximate mechanisms influence these sex differences in mathematical performance indirectly. First, sexual selection resulted in greater elaboration in males than in females of the neurocognitive systems that support navigation in three-dimensional space. Knowledge implicit in these systems reflects an understanding of basic Euclidean geometry, and may thus be one source of the male advantage in geometry. Males also use more readily than females these spatial systems in problem-solving situations, which provides them with an advantage in solving word problems and geometry. In addition, sex differences in social styles and interests, which also appear to be related in part to sexual selection, result in sex differences in engagement iii mathematics-related activities, thus further increasing the male advantage in certain mathematical domains. A model that integrates these biological influences with sociocultural influences on the sex differences in mathematical performance is presented in this article. (shrink)

The scientific study of consciousness, also referred to as consciousness science, is a young scientific field devoted to understanding how conscious experiences and the brain relate. It comprises a host of theories, experiments, and analyses that aim to investigate the problem of consciousness empirically, theoretically, and conceptually. This thesis addresses some of the questions that arise in these investigations from a formal and mathematical perspective. These questions concern theories of consciousness, experimental paradigms, methodology, and artificial consciousness. -/- Regarding (...) theories of consciousness, the thesis contributes to the understanding of the mathematical structure that some of the formal theories in the field propose. The work presented here targets the theory of consciousness known as Integrated Information Theory (IIT) and the neuroscientific theory known as Predictive Processing Theory or Free Energy Principle in its Active Inference form (AI-PP). The thesis provides axiomatic definitions of the mathematical structures that constitute these theories, and uses these definitions to address some of the open questions surrounding the theories. For AI-PP, this includes a rigorous derivation of the formula for Active Inference via Free Energy minimisation and a proof of compositionality of Free Energy. For IIT, this includes resolutions of some of the criticisms of IIT's formal scope and applications, but also the identification of new issues that concern the formalism and its derivation. When possible, the definitions are provided in the mathematical framework of category theory. -/- Regarding experiments, the thesis addresses the main paradigm for testing and falsifying theories of consciousness currently applied in the field. This paradigm consists of comparing the conscious experience that a theory predicts with the conscious experience that is inferred from behavioural data or report by use of measures of consciousness. The thesis provides a formal model of this paradigm and shows that under a certain condition—if inference and prediction are independent—, any minimally informative theory of consciousness can always be falsified. This is deeply problematic since the field’s reliance on report or behaviour to infer conscious experiences, in conjunction with the general structure of most contemporary theories of consciousness, implies such independence. This observation provides the exact formal underpinning of the well-known unfolding argument. The thesis analyses the origin of the problem and identifies precisely which changes are required to avoid this problem in future research. The thesis furthermore shows that the problem of falsifying theories of consciousness, and of empirical comparisons of theories of consciousness more generally, follows from a pervasive closure paradigm in consciousness science, that consists of taking a neuroscientific account of the brain as input to a theory of consciousness, so as to explain what consciousness is, without allowing for modifications or adaptations of the neuroscientific account that would accommodate consciousness as part of the brain's functioning. As is shown in the thesis, this paradigm has implications that point to a fundamental need of revision. -/- Regarding methodological and conceptual questions, the thesis contributes to the foundations of structural research in consciousness science. Structural research aims to use mathematical structures or mathematical spaces, instead of verbal descriptions or simple categorisations, to represent conscious experiences scientifically, for example when building theories of consciousness, or when exploring new empirical avenues to measure consciousness. Despite considerable advances in this realm, there was, prior to this thesis, no explicit definition of what a mathematical structure of conscious experience should be; that is, how the attribution of mathematical structure to conscious experiences should be systematically understood. Perhaps the most important contribution of this thesis to the field is to propose such a definition. The definition, a structural concept, extends existing approaches wherever available, and provides a basis for developing a common formal language to study consciousness, bridging developments as far apart as psychophysics and phenomenology. In addition, and independently of this proposal, the thesis offers a critical analysis of which metaphysical premises need to be presumed in structural research, whether the use of particular formal tools (such as structure-preserving mappings or homomorphisms) is justified, and how structural theories of consciousness could otherwise be built in the first place. An attempt to expand the results from consciousness to more general problems in philosophy of science is made in the context of the well-known Newman problem. -/- Regarding the question of artificial consciousness—can AI feel?—, the thesis contributes two results that take the form of no-go theorems, as known from physics. The first no-go theorem shows that if consciousness is relevant for the temporal evolution of a system's states—if it is dynamically relevant—, then contemporary AI systems cannot be conscious. That is because AI systems run on CPUs, GPUs, TPUs or other processors which have been designed and verified to adhere to computational dynamics that systematically preclude or suppress deviations. The second no-go theorem is situated in the context of computational functionalism, a view which posits that consciousness is a computation. The theorem shows that if computational functionalism holds true, consciousness cannot be a Turing computation. Rather, it must be a novel type of computation that has recently been proposed by Geoffrey Hinton, called mortal computation. -/- This thesis is part of a global effort to pioneer a mathematical perspective in consciousness science, now called Mathematical Consciousness Science. The hope behind the research carried out in this PhD is to illustrate the power and usefulness of mathematical approaches in different areas of consciousness science, and in doing so, to lay the foundations for future mathematical work that complements and supports empirical and theoretical work in the further development of this exciting field. (shrink)

Effective conditioning requires a correlation between the experimenter's definition of a response and an organism's, but an animal's perception of its behavior differs from ours. These experiments explore various definitions of the response, using the slopes of learning curves to infer which comes closest to the organism's definition. The resulting exponentially weighted moving average provides a model of memory that is used to ground a quantitative theory of reinforcement. The theory assumes that: incentives excite behavior and focus the excitement on (...) responses that are contemporaneous in memory. The correlation between the organism's memory and the behavior measured by the experimenter is given by coupling coefficients, which are derived for various schedules of reinforcement. The coupling coefficients for simple schedules may be concatenated to predict the effects of complex schedules. The coefficients are inserted into a generic model of arousal and temporal constraint to predict response rates under any scheduling arrangement. The theory posits a response-indexed decay of memory, not a time-indexed one. It requires that incentives displace memory for the responses that occur before them, and may truncate the representation of the response that brings them about. As a contiguity-weighted correlation model, it bridges opposing views of the reinforcement process. By placing the short-term memory of behavior in so central a role, it provides a behavioral account of a key cognitive process. (shrink)

As a full-blown research topic, numerical cognition is investigated by a variety of disciplines including cognitive science, developmental and educational psychology, linguistics, anthropology and, more recently, biology and neuroscience. However, despite the great progress achieved by such a broad and diversified scientific inquiry, we are still lacking a comprehensive theory that could explain how numerical concepts are learned by the human brain. In this perspective, I argue that computer simulation should have a primary role in filling this gap because (...) it allows identifying the finer-grained computational mechanisms underlying complex behavior and cognition. Modeling efforts will be most effective if carried out at cross-disciplinary intersections, as attested by the recent success in simulating human cognition using techniques developed in the fields of artificial intelligence and machine learning. In this respect, deep learning models have provided valuable insights into our most basic quantification abilities, showing how numerosity perception could emerge in multi-layered neural networks that learn the statistical structure of their visual environment. Nevertheless, this modeling approach has not yet scaled to more sophisticated cognitive skills that are foundational to higher-level mathematical thinking, such as those involving the use of symbolic numbers and arithmetic principles. I will discuss promising directions to push deep learning into this uncharted territory. If successful, such endeavor would allow simulating the acquisition of numerical concepts in its full complexity, guiding empirical investigation on the richest soil and possibly offering far-reaching implications for educational practice. (shrink)

Plamondon & Alimi's derivation of the kinematic model is mathematically flawed. By simply naming a particular parameter combination the model fails to explicate the sources of variability. As a result, the model cannot distinguish between various sources of error, such as those resulting from task demands and those resulting from movement execution.

The authors' proposal for the evolutionary origins of historical myths does not hold up to scrutiny, as illustrated by a simple mathematical model. Group-level explanations, such as defining the conditions for in-group membership, are dismissed by the authors but are far more plausible, as illustrated by the ongoing war in Ukraine.