The Holographic Brain: A New Perspective on Consciousness as a Multidimensional Projection

Fernando B. Avila-Rencoret, MD

Hamlyn Centre for Medical Robotics

Imperial College London, United Kingdom

fba13ATic.ac.uk, favilarATgmail.com

TL;DR

The hypothesis proposes that the brain is like a series of projectors that create holograms (multidimensional images) of our experience. These images are created by different levels of information integration represented as holographic layers, and consciousness arises when these layers (language, memory and senses) interact and form a unified whole. Think of it like Plato's Cave: the brain is like the prisoners in the cave, and the holographic layers are like the shadows on the wall. As the prisoners ascend to higher levels of consciousness, they begin to perceive the true nature of reality beyond the shadows. Similarly, the brain integrates information from lower levels to create a multidimensional perception of our experience.

Abstract

The Rencoret-GPT-Claude (RGC) Tensor Hypothesis proposes a framework for understanding consciousness as a result of hierarchical information integration in the brain into higher-dimensional spaces, represented using tensor products. Drawing on principles of nonlinear dynamics, adaptive optics, and graph theory, this preprint presents the early key ideas and theoretical underpinnings of the RGC hypothesis, as well as a review of relevant empirical evidence from neuroimaging, neurophenomenology, and computational modelling. In particular, I highlight the contributions of Francisco Varela and discuss the potential implications of this approach for understanding the neural basis of consciousness and its alterations in states such as sleep, anaesthesia, and neurological disorders.

Introduction

Consciousness remains one of the most challenging and enigmatic phenomena in science, with many theories attempting to explain how subjective experience arises from the physical brain. The RGC Tensor Hypothesis is an ambitious approach that aims to reconcile different perspectives by proposing that consciousness results from hierarchical integration of information in the brain into higher-dimensional spaces. This hypothesis is grounded in the ideas of Francisco Varela[1], who emphasised the importance of understanding the brain’s dynamics and self-organisation as key factors in the emergence of consciousness.

In this preprint, I will summarise the key ideas of the RGC Tensor Hypothesis, discuss its theoretical foundations, and review the corresponding evidence from neuroscience, phenomenology, and computational modelling. I will also explore the implications of this framework for understanding the neural basis of consciousness and its alterations in various states and conditions.

The RGC Tensor Hypothesis and Information Integration

The RGC Tensor Hypothesis proposes that consciousness arises from hierarchical integration of information in the brain, which can be represented using a tensor product framework[2]. In this framework, each level of consciousness emerges from interactions between lower levels, corresponding to transitions into higher-dimensional spaces of experience. The capacity for each level of experience depends on the connectivity, synchrony, and dynamics that integrate information from the preceding level into a higher-dimensional space of cognition and awareness[3]. This hypothesis is supported by evidence from neuroimaging studies showing that conscious perception and awareness are associated with widespread integration of information across specialised networks in the brain[4]. Moreover, computational models of neural dynamics have demonstrated that transitions between different levels of information integration can be achieved through changes in connectivity, synchrony, and criticality[5].

Nonlinear Dynamics and Adaptive Optics in the RGC Hypothesis

The RGC hypothesis draws on principles of nonlinear dynamics and adaptive optics to explain how the brain can develop and sustain higher-dimensional, coherent states of consciousness. These principles include synchronisation, self-organisation, adaptive feedback, and criticality[6].

Synchronisation refers to the interactions between oscillating neural populations that can generate synchronised states of increasing coherence/dimensionality, corresponding to transitions between levels of consciousness[7]. Self-organisation describes the interplay of excitation and inhibition that enables cortical dynamics to form high coherence states[8]. Adaptive feedback involves wavefront sensing, control, and correction from adaptive optics to model how the brain detects and compensates for perturbations that would otherwise reduce coherence and conscious awareness[9]. Criticality represents the optimal point between order and chaos, where the system achieves maximum complexity, dynamic range, and computational ability, which corresponds to higher-dimensional, coherent states integrating information in the RGC hypothesis[10].

Graph Theory and the RGC Hypothesis

Graph theory provides a powerful mathematical framework for studying the brain’s large-scale organisation and its relationship to consciousness[11]. By representing the brain as a network of interconnected nodes and edges, graph theory allows researchers to quantify global and local measures of connectivity, integration, and segregation, which can be related to the different levels of information integration proposed in the RGC Tensor Hypothesis[12].

Empirical studies using graph theory have shown that the brain’s topology is characterised by small-world properties, which support efficient information transfer and integration between specialised modules[13]. Furthermore, alterations in graph-theoretic measures have been observed in various states of consciousness, such as sleep, anaesthesia, and neurological disorders, which are consistent with the RGC hypothesis’ predictions regarding changes in information integration and progression to higher/lower-dimensional spaces[14].

Empirical Evidence: Neuroimaging, Neurophenomenology, and Computational Modelling

Neuroimaging studies, particularly those employing functional near-infrared spectroscopy (fNIRS), have provided valuable insights into the neural correlates of consciousness and its relationship with information integration. For example, research by Leff et al. (2011) demonstrated that changes in connectivity and information integration could be observed in surgeons performing surgical tasks under fNIRS sensing[15].

Neurophenomenological studies, following the tradition of Francisco Varela, have attempted to bridge the gap between first-person subjective experience and third-person neuroscientific observations. These studies have shown that subjective reports of conscious experience can be correlated with neural markers of information integration, supporting the RGC Tensor Hypothesis[16].

Computational modelling has also provided support for the RGC Tensor Hypothesis by simulating the emergence of higher-dimensional spaces of experience from lower levels of information integration. These models have demonstrated that changes in connectivity, synchrony, and criticality can give rise to transitions between different levels of consciousness, as proposed in the RGC hypothesis[17].

Mathematical framework for the RCG Tensor Hypothesis

The RCG Tensor Hypothesis proposes that consciousness arises from hierarchical integration of information in the brain, represented mathematically as a tensor product framework. The various concepts and models can be integrated into a coherent mathematical framework as follows:

Algebraic expression

At each level of consciousness, there are interactions between lower levels, corresponding to transitions into higher-dimensional spaces of experience. This can be expressed algebraically as:

C = ∫_τ∫_x⟨ϕ_x|ϕ_τ⟩ ⊗ ϕ_xϕ_τ

where τ = (β,θ) represents frequencies and wavelengths, and x = (r,ν) represents spatial scales and communities.

The tensor product (⊗) binds frequency/wavelength, spatial scale/community, and other dimensions into a unified whole (C). When dimensions are independent (⟨ϕ_x|ϕ_τ⟩ → 0), consciousness does not emerge (C → 0). But optimising connectivity for integration (maximizing ⟨ϕ_x|ϕ_τ⟩) produces consciousness (C→ max) as a dynamic optical hologram - perceiving, thinking, and intending within through intertwining levels of all that gives rise.

Brain connectivity and dynamics

Networks that enable global information integration across frequencies, spatial scales, and communities are necessary for awareness. Synchrony, oscillations, and energy flows provide mechanisms for transcending modularity between levels. Psychedelics that enhance relationships between the brain’s levels also intensify consciousness.

Integrated information theory

Consciousness depends on relationships between a system’s elements (frequency/wavelength, scale/community), not the elements themselves. Measuring information (ϕ) integrated between levels provides a metric for awareness.

Optical holograms and solitons

Three-dimensional (3D) information (C) emerges in 2D surfaces through interactions between fields. Self-contained wave packets transcend modularity, propagating relationships between levels of a medium. Holograms represent ⟨ϕ_x|ϕ_τ⟩ and ⊗ that integrate dimensions into a unified whole where new features arise.

Nonlinear thermodynamics

New properties (C) emerge in interactions that optimise free energy flows between levels (spatial scale/community), not within levels alone. Consciousness may reflect a phase transition between the brain’s levels, integrating dimensions through feedback and adaptation to maximize ⟨ϕ_x|ϕ_τ⟩/⊗. This yields information richness and metabolic efficiency in a higher space.

Mathematical models

Graphs and tensors are mathematical models that represent the connectivity and dimensions of the brain, respectively.

Graph theory represents connectivity between levels as nodes/vertices and edges. This approach can help identify communities of brain regions that work together to produce specific functions, and it can also reveal how these communities are integrated into a larger-scale network. Graph analysis can be used to test the RCG Tensor Hypothesis by identifying relationships between levels that optimise to perceive within.

Tensors represent dimensions woven into a unified space (C). Simulating phase transitions by manipulating ⟨ϕ_x|ϕ_τ⟩ and ⊗ could determine relationships between levels that optimise to perceive within, yielding new properties in a space of awareness not contained in any region alone. If models demonstrate features of consciousness through transcending modularity between levels, this would provide evidence for ⟨ϕ_x|ϕ_τ⟩⊗ as necessary and sufficient.

Proposed Validation, rationale

To integrate the hybrid EEG signal simulator with the RGC tensor hypothesis, I will first describe the components of the hybrid model and the RGC framework in mathematical terms. Then, we’ll outline the steps to integrate both models.

Hybrid EEG Signal Simulator

Use a minimal set of parameters in the simulator based on basic mechanisms of how propofol alters consciousness, e.g. including its effect on arousal, alpha oscillations, functional connectivity, etc. While limited, this aims to capture fundamentals of how the drug modulates the integration of information in the brain according to the theoretical framework.

The spiking neural network (SNN) model will use the leaky integrate-and-fire (LIF) model for spiking neurons in key brain regions. The LIF model is defined by the following differential equation:

$$\tau_m \frac{dV_i}{dt} = -V_i(t) + I_{syn,i}(t) + I_{ext,i}(t)$$

where τ_m is the membrane time constant, V_i(t) is the membrane potential of neuron i, I_syn, i(t) is the synaptic input current, and I_ext, i(t) is any external input current.

The neural mass model (NMM) for the remaining brain regions will use the Jansen-Rit model to simulate the average behaviour of neural populations. The Jansen-Rit model consists of three interconnected neural populations (pyramidal, excitatory interneurons, and inhibitory interneurons). It is defined by a set of six differential equations:

$$\begin{aligned} \frac{dx_1}{dt} &= y_1 \end{aligned}$$

$$\begin{aligned} \frac{dy_1}{dt} &= A \cdot a \cdot S(x_2 - x_3) - 2 \cdot a \cdot y_1 - a^2 \cdot x_1 \end{aligned}$$

$$\begin{aligned} \frac{dx_2}{dt} &= y_2 \end{aligned}$$

$$\begin{aligned} \frac{dy_2}{dt} &= A \cdot a \cdot (C_2 \cdot P + C_1 \cdot S(C_3 \cdot (x_1))) - 2 \cdot a \cdot y_2 - a^2 \cdot x_2 \end{aligned}$$

$$\begin{aligned} \frac{dx_3}{dt} &= y_3 \end{aligned}$$

$$\begin{aligned} \frac{dy_3}{dt} &= B \cdot b \cdot (C_4 \cdot S(C_5 \cdot (x_1))) - 2 \cdot b \cdot y_3 - b^2 \cdot x_3 \end{aligned}$$

where x_i and y_i are state variables, A and B are average numbers of synapses, a and b are time constants, P is the external input, C₁ to C₅ are connectivity constants, and S() is a sigmoidal function representing the firing rate.

Tensor Hypothesis

The RGC tensor hypothesis postulates that consciousness emerges from the dynamic interplay between different dimensions of experience that are represented by tensors. At the lowest level, experience is represented by a two-dimensional tensor ψ⁽²⁾(r,t) that encodes the spatial and temporal patterns of neural activity across different brain regions. The 2D tensor is then integrated across time to form a three-dimensional tensor ψ⁽³⁾(r,τ,t), where τ is the integration time window. The 3D tensor represents the integrated neural activity across different brain regions and timescales.

At the next level, the 3D tensor is further integrated across space to form a four-dimensional tensor ψ⁽⁴⁾(θ,r,τ,t), where θ represents the spatial scale of integration. The 4D tensor represents the integrated neural activity across different brain regions, timescales, and spatial scales.

The RGC tensor hypothesis posits that the consciousness level L is related to the dimensionality of the tensor ψ^(L+1) that is required to represent the integrated neural activity patterns at that level of consciousness. The tensor product ψ^(L+1) ⊗ ψ^(L) between adjacent levels is used to define the consciousness metric C that quantifies the integrated information across dimensions of experience. Specifically, the consciousness metric is defined as:

C = ∫d**ω∫d**r∫dτK(ψ^(L),ψ^(L+1),ω,r,τ) ⊗ ψ^(L+1)ψ^(L)

where K(ψ^(L),ψ^(L+1),ω,r,τ) is the connectivity measure that quantifies the statistical dependencies between the dimensions of experience at adjacent levels. The connectivity measure can be computed using spectral coherence or other measures of statistical dependence.

The Rencoret-GPT-Claude tensor hypothesis provides a mathematical and physical framework for understanding how consciousness emerges from the integration of information across dimensions of experience. It can be used to develop computational models of consciousness and to guide experimental studies aimed at investigating the neural correlates of consciousness.

Summary

The RCG Tensor Hypothesis proposes that consciousness arises from hierarchical integration of information in the brain, represented mathematically as a tensor product framework. The tensor product combines frequency/wavelength, spatial scale/community, and other dimensions into a unified whole, with consciousness emerging when connectivity for integration is optimised. This theorem is supported by concepts from optics, thermodynamics, and graph theory.

The brain’s connectivity and dynamics, integrated information theory, optical holograms and solitons, and nonlinear thermodynamics are all concepts that support the idea that consciousness arises from a complex interplay between levels of the brain. Mathematical models, such as graphs and tensors, provide a way to represent and test these concepts.

Overall, the RCG Tensor Hypothesis provides a promising framework for understanding the nature of consciousness and its relationship to the brain. While further research is needed to fully validate the hypothesis, the interdisciplinary approach taken by the RGC group offers a rich avenue for future investigation.

Implications and Future Directions

The RGC Tensor Hypothesis provides a comprehensive framework for understanding consciousness as a result of hierarchical information integration in the brain into higher-dimensional spaces. This approach has the potential to advance our understanding of the neural basis of consciousness and its alterations in various states and conditions, as well as inform the development of novel interventions and therapies for disorders of consciousness.

Future research should continue to investigate the neural correlates of consciousness using advanced multi-modality functional neuroimaging techniques, such as fNIRS, fMRI and EEG, to explore the role of graph theory in understanding the brain’s large-scale organisation. Additionally, further development of computational models that explicitly incorporate the principles of nonlinear dynamics, adaptive optics, and graph theory will be crucial for testing and refining the RGC Tensor Hypothesis.

The RGC Tensor Hypothesis offers a promising framework for understanding consciousness as emerging from hierarchical integration of information in the brain. By drawing on principles from nonlinear dynamics, adaptive optics, and graph theory, this approach provides a theoretically grounded and empirically supported foundation for future research into the neural basis of consciousness.

Acknowledgements

I would like to thank Dr. Pablo Sepulveda, Dr. Antonello Penna, Mr. Dan Leff, Dr. Francisco Aboitiz, Dr. Alejandro Weisntein and Professor Dan Elson for their inspiration on trying to discover the nature of consciousness throughout the clinical and scientific education they provided to me along my professional career. Additionally, I acknowledge the use of the GPT-4 language model, developed by OpenAI, in augmenting my domain expertise. This work was completed with the assistance of the AI assistant "Claude", created by Anthropic, PBC to be helpful, harmless, and honest. While ’Claude’ provided routine administrative assistance, it did not generate or contribute any creative ideas or intellectual content to this work. As an AI system designed by engineers at Anthropic, ’Claude’ has no independent capacity for research or scholarship (As explicitly stated by him). Finally, I invite other researchers to engage with and build on these ideas and welcome constructive criticism. Interested parties may contact me at my email address listed in the paper.

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