Mathematical Consciousness Science

Mathematical Consciousness Science (MCS) is an interdisciplinary field in the intersection between the scientific study of consciousness and applied mathematics. Many mathematicians have taken an interest in consciousness over the centuries including René Descartes, Gottfried Wilhelm Leibniz, Alfred North Whitehead, Bertrand Russell, Alan Turing and Roger Penrose, to name a few, whilst others have developed areas of mathematics that are finding new applications in MCS including Thomas Bayes, Ludwig Boltzmann, Andrey Markov and Claude Shannon.

The term Mathematical Consciousness Science began to be used and recognized from around 2018 onward following a rapid increase in the development of new mathematical and/or computational models and formal theories of consciousness that began from around 2005. Many researchers in the MCS research community anticipate that mathematical approaches are needed to tackle challenges such as: the Hard problem of consciousness, which is the problem of explaining why and how we have phenomenal experience; explaining how consciousness relates to the physical domain, particularly regarding the brain and artificial systems; and, fundamentally, the many questions involving consciousness and Quantum Mechanics. Such challenges are at the heart of MCS but any research concerning consciousness, some aspect of consciousness or some issue involving consciousness, where mathematically or computationally formulated models or theories play a central role, fall within the field’s scope. MCS also exists to complement the work of researchers working in the wider field of the scientific study of consciousness which intersects with Neuroscience, Philosophy and Experimental Psychology, for example.

Mathematical and computational models and theories of consciousness

Archetypal models

Being hypotheses about consciousness and its relation to the physical domain, the archetypal model of consciousness arguably has three parts, namely, a mathematical model of salient aspects of the physical system (e.g. circuit models, network models, joint probability distributions etc), a mathematical model for aspects of conscious experience (e.g. topological spaces, metric spaces, matrices of relationships, categories, intensity scales etc) and some form of arguably innate mapping between the physical domain model and the consciousness domain model (e.g. a homomorphism, limit or optimal boundary point, functor, scalar function etc). The models make various predictions about, for example, phenomenal perception, the relational content of consciousness, the level and intensity of consciousness, attention, and the unity (and disunity) of consciousness within and between systems. The physical domain models and consciousness domain models are also of interest in their own right and some researchers in MCS focus on the development of these models.

Examples include:

• Integrated Information Theory (IIT) models the physical system with Markov processes obtained from a circuit model. The consciousness domain model involves several outputs of the IIT algorithm but most notably includes a non-negative scalar function ${\textstyle \Phi }$, related to the intrinsic irreducibility of a network, that, according to the theory, measures level of consciousness. The algorithm, mapping between the physical domain model and the consciousness domain model, has causation at its heart. Therefore, a key modelling assumption in IIT is that a system’s consciousness is directly related to its causal properties.
• Expected Float Entropy Minimisation (EFE) and its extension to model unity, models the physical system with a joint probability distribution that represents the systems bias to being in certain states over other states. The consciousness domain model involves a hierarchy of relational models in the form of matrices of real valued parameters in the range ${\textstyle [0,1]}$. According to the theory, the relational models provide an interpretation of system status that gives the relational content of the associated experience. The theory’s extension to model unity then deals with integration. The mapping between the physical domain model and the consciousness domain model involves the minimisation of expected float entropy so that the resulting relational model gives the minimum expected entropy interpretation of system states. Therefore, two key modelling assumptions in EFE minimisation are that consciousness is a minimum entropy interpretation of system states and that the structural content of consciousness comes from the correlations and relationships encoded in the bias of a system.

Computational models

At some level of abstraction, computational models in MCS are typically models of the brain together with hypotheses that relate them in some way to consciousness. The hypothetical relation to consciousness may depend on properties of the model that are a-priori present by design or are discovered, or confirmed to exist, experimentally. Physical systems used to instantiate computational models vary widely but usually involves Neuromorphic Computing at some level of abstraction. Computational models can also generate synthetic data for use in other models of consciousness such as archetypal models. The development of computational models in MCS has a significant overlap with Artificial Consciousness (AC) research.

Examples include:

• Cortex-like complex systems of networks-of-networks have been used to model the cortex and involve spiking nodes, that model neurons, and real time lags that model signal transmission delays. The system is a network of sparsely connected modules where each module is a network of densely connected nodes. Instantiation of the model on the SpiNNaker supercomputer revealed a wide set of latent, internal, common dynamical modes of operational behaviour. The hypothesis relating the model to consciousness is that the observed modes are candidates for sensations, feelings and moods in conscious systems. SpiNNaker is being used as one component of the neuromorphic computing platform for the Human Brain Project.
• The Conscious Turing Machine (CTM) is a computational model that formalizes the Global Workspace Theory (GWT). GWT postulates the existence of a type of short term working memory in the brain to which various subsystems may gain access and influence the contents of. The hypothesis relating the model to consciousness is that it is the content of this working memory that we are conscious of. The CTM formalization helps to remove ambiguity and allows GWT to be instantiated in order to obtain experimental results.

Higher level models

There are a number of models in MCS that propose a higher level viewpoint of conscious systems than archetypal models.

Examples include:

• The Free Energy Principal (FEP) is a model for how living and non-living systems remain in non-equilibrium steady-states by restricting themselves to a limited number of states. The model is closely connected to autopoiesis and provides a principle by which systems may create an internal model of the outside environment in order to maintain their own integrity. The minimisation of free energy is formally related to variational Bayesian methods and was originally introduced as an explanation for embodied perception in neuroscience.

Theories involving Quantum Mechanics

Ever since the 1930s, when Schrödinger and Einstein discussed the Schrödinger’s cat thought experiment, there has been the idea that somehow consciousness may have something to do with Quantum Mechanics (QM). Accordingly MCS includes a number of research directions involving QM. The two main possibilities appear to be that either some kind of consciousness-like property of certain configurations of matter collapses wave functions (a refinement of the role of the observer in the Schrödinger’s cat thought experiment) or that wave function collapse plays a part in how consciousness happens. Researchers in MCS continue to narrow down the theoretical possibilities.

Examples include:

• Orchestrated Objective Reduction (Orch OR) is a theory of consciousness that attributes non-computable quantum processing as the basis of consciousness. It involves an objective process of wave function collapse based on conservation of energy by way of the proposition that if a particle with mass is in a superposition then it ought to gives rise to an effective self-energy due to the resulting superposition of distorted spacetime. The theory further proposes that quantum processing may occur within structures called microtubules in the dendrites of neurons and that the objective wave function collapse selects states neither randomly nor algorithmically and plays a part in how consciousness arises from the system.
• Proto-consciousness induced quantum collapse is a refinement of the idea that the act of conscious observation collapses wave functions. Archetypal models of consciousness tend to predict the weak presence of some aspects of consciousness even for some small simple systems. In principle, these quantities can be incorporated into stochastic differential equations that reduce to the Schrödinger equation in the quantum regime and give classical behaviour for macroscopic objects. At intermediate scales, consciousness-like properties (proto-consciousness) can then have noticeable effects on the predictions of such collapse models which can then be tested experimentally. Examples include Quantum Integrated Information (QII) induced quantum collapse which extends Integrated Information Theory. Model Unity, in the Expected Float Entropy minimisation model of consciousness, has also been suggested for use in collapse models. More fundamentally, it has been shown that if consciousness does have any forward influence on the physical domain then this will at least manifest itself as an influence on quantum collapse.