# GENERAL THEORIES OF THE BRAIN

**Supramolecular phrenology: not just spikes!**

__How does central nervous system process information__? Current theories are based on two tenets: a) Information is transmitted by action potentials, the language by which neurons communicate with each other – and b) homogeneous neuronal assemblies of cortical circuits operate on these neuronal messages where the operations are characterized by the intrinsic connectivity among neuronal populations. In this view, the size and time course of any spike is stereotypic and the information is restricted to the temporal sequence of the spikes; namely, the “neural code”. However, an increasing amount of novel data point towards an alternative hypothesis: a) __The role of neural code in information processing is overemphasized__. Instead of simply passing messages, action potentials play a role in dynamic coordination at multiple spatial and temporal scales, establishing network interactions across several levels of a hierarchical modular architecture, modulating and regulating the propagation of neuronal messages. b) Information is processed at all levels of neuronal infrastructure from macromolecules to population dynamics. For example, intra-neuronal (changes in protein conformation, concentration and synthesis) and extra-neuronal factors (extracellular proteolysis, substrate patterning, myelin plasticity, microbes, metabolic status) can have a profound effect on neuronal computations. This means molecular message passing may have cognitive connotations. This essay introduces the concept of “supramolecular chemistry”, involving __the storage of information at the molecular level and its retrieval, transfer and processing at the supramolecular level__, through transitory non-covalent molecular processes that are self-organized, self-assembled and dynamic. Finally, we note that the cortex comprises extremely heterogeneous cells, with distinct regional variations, macromolecular assembly, receptor repertoire and intrinsic microcircuitry. This suggests that __every neuron (or group of neurons) embodies ____different molecular information__ that hands an operational effect on neuronal computation. PDF

**Voronoi slices in histological brain tissues: every neuron is different from another**

We provide a novel, fast and cheap method for the morphological evaluation of simple 2-D images taken from histological samples. This method, based on computational geometry, leads to a novel kind of “tessellation” of every type of biological picture, in order to locate the zones equipped with the highest functional activity. As an example, we apply the technique to the evaluation of histological images from brain sections and demonstrate that the __cortical layers, rather than being a canonical assembly of homogeneous cells as usually believed, display scattered neuronal micro-clusters__ equipped with higher activity than the surrounding ones. PDF

**Neural Gauge theory: paper with the mighty Karl Friston and Biswa Sengupta**

Given the amount of knowledge and data accruing in the neurosciences, is it time to formulate a general principle for neuronal dynamics that holds at evolutionary, developmental, and perceptual timescales? In this paper, we propose that the brain (and other self-organized biological systems) can be characterized via the mathematical apparatus of a gauge theory. The picture that emerges from this approach suggests that __any biological system__ (from a neuron to an organism) __can be cast as resolving uncertainty about its external milieu, either by changing its internal states or its relationship to the environmen__t. Using formal arguments, we show that __a gauge theory for neuronal dynamics—based on approximate Bayesian inference__—has the potential to shed new light on phenomena that have thus far eluded a formal description, such as __attention and the link between action and perception__.

**Neural Gauge theory: novel insights in a book chapter (Springer)**

Recent advances in neuroscience highlight the complexity of the central nervous system (CNS) and call for general, multidisciplinary theoretical approaches. The aim of this chapter is to assess highly organized biological systems, in particular the CNS, via the physical and mathematical procedures of gauge theory – and to provide quantitative methods for experimental assessment. We first describe the nature of a gauge theory in physics, in a language addressed to an interdisciplinary audience. Then we examine __the possibility that brain activity is driven by one or more continuous forces, called gauge fields__, originating inside or outside the CNS. In particular, we go through the idea of symmetries, which is the cornerstone of gauge theories, and illustrate examples of possible gauge fields in the CNS. A deeper knowledge of gauge theories may lead to novel approaches to (self) organized biological systems, improve our understanding of brain activity and disease, and pave the way to innovative therapeutic interventions. PDF

**Minimum frustration network in the brain: when evolution dictates our thoughts**

The minimum frustration principle is a computational approach which states that, in the long timescales of evolution, proteins’ free-energy decreases more than expected by thermodynamic constraints as their aminoacids assume conformations progressively closer to the lowest energetic state. Here we show that this general principle, borrowed from protein folding dynamics, can be fruitfully applied to nervous function too. Highlighting the foremost role of energetic requirements, macromolecular dynamics, and, above all, intertwined timescales in brain activity, the minimum frustration principle elucidates a wide range of mental processes, from sensations to memory retrieval. __Brain functions are compared to trajectories which, in long nervous timescales, are attracted towards the low-energy bottom of funnel-like structures __characterized both by robustness and plasticity. We discuss how the principle, as derived explicitly from evolution and selection of a funneling structure from microdynamics of contacts, is __different from other brain models__ equipped with energy landscapes, such as the Bayesian and free-energy principle and the Hopfield networks. In sum, we make available a novel approach to brain function cast in a biologically informed fashion, with the potential to be operationalized and assessed empirically. PDF

**MultiD Pandemonium architecture: the winner takes all also in nervous multidimensions**

A novel demon-based architecture is introduced to elucidate brain functions such as __pattern recognition during human perception__ and mental interpretation of visual scenes. Starting from the topological concepts of invariance and persistence, we introduce a __Selfridge pandemonium__ variant of brain activity that takes into account a novel feature, namely, demons that recognize short straight-line segments, curved lines and scene shapes, such as shape interior, density and texture. __L____ow-level representations of objects can be mapped to higher-level views (our mental interpretations__): a series of transformations can be gradually applied to a pattern in a visual scene, without affecting its invariant properties. This makes it possible to construct a symbolic multi-dimensional representation of the environment. These representations can be projected continuously to an object that we have seen and continue to see, thanks to the mapping from shapes in our memory to shapes in Euclidean space. Although perceived shapes are 3-dimensional (plus time), the evaluation of shape features (volume, colour, contour, closeness, texture, and so on) leads to n-dimensional brain landscapes. Here we discuss the advantages of our parallel, hierarchical model in pattern recognition, computer vision and biological nervous system’s evolution. PDF

**Brain functions duality: towards a single nervous activity encompassing all the mental faculties**

Tozzi A, Peters JF. 2019. The common features of different brain activities. Neurosci Lett, 692 (23): 41-46. https://doi.org/10.1016/j.neulet.2018.10.054.

The term “brain activity” refers to a wide range of mental faculties that can be assessed either on anatomical/functional micro-, meso- and macro- spatiotemporal scales of observation, or on intertwined cortical levels with mutual interactions. Our aim is to show that every brain activity encompassed in a given anatomical or functional level necessarily displays a counterpart in others, i.e., they are “dual”. “Duality” refers to the case where two seemingly different physical systems turn out to be equivalent. We describe a method, based on novel topological findings, that makes different manifolds (standing for different brain activities) able to scatter, collide and combine, in order that they merge, condense and stitch together in a quantifiable way. We develop a computational tool which explains how, despite their local cortical functional differences, __all mental processes__, from perception to emotions, from cognition to mind wandering, __may be reduced to a single, general brain activity that takes place in dimensions higher __than the classical three-dimensional plus time. This framework permits a topological duality among different brain activities and neuro-techniques, because it holds for all the types of spatio-temporal nervous functions, independent of their cortical location, inter- and intra-level relationships, strength, magnitude and boundaries. PDF

**Plasma-like collisionless nervous trajectories: long-range correlations in the brain**

Tozzi A, Peters JF, Deli E. 2018. Towards plasma-like collisionless trajectories in the brain. Neurosci Lett.662:105-109. https://doi.org/10.1016/j.neulet.2017.10.016.

Plasma studies depict collisionless, collective movements of charged particles. In touch with these concepts, originally developed by the far-flung branch of high energy physics, here __we evaluate the role of collective behaviors and long-range functional couplings of charged particles in brain dynamics__. We build a novel, empirically testable, brain model which takes into account collisionless movements of charged particles in a system, the brain, equipped with oscillations. The model is cast in a mathematical fashion with the potential of being operationalized, because it can be assessed in terms of McKean-Vlasov equations, derived from the classical Vlasov equations for plasma. A plasma-like brain also elucidates cortical phase transitions in the context of a brain at the edge of chaos, describing the required order parameters. In sum, showing how the brain might exhibit plasma-like features, we go through the concept of holistic behavior of nervous functions. PDF

**How and why the microcolumn resembles a fullerenic structure and gives rise to a barcode**

Tozzi A, Peters JF, Ori O. 2017. Fullerenic-topological tools for honeycomb nanomechanics. Towards a fullerenic approach to brain functions. Fullerenes, Nanotubes and Carbon nanostructures.25 (4): 282-288. https://dx.doi.org/10.1080/1536383X.2017.1283618.

Fullerenic structures equipped with Stone-Wales transformations have been successfully utilized in the study of macromolecular assemblies. Here we show that this approach could be useful in the assessment of issues from a far-flung research area, i.e., neuroscience. Indeed, __the basic morphological and functional unit of the brain, called the human microcolumn__, is a tubular structure that can be flattened in the guise of a fullerene-like two-dimensional lattice. We describe this procedure in order to build a fullerene-like microcolumn, in which neuronal firing and electric signal propagation are assessed in terms of topological neural network modifications, instead of the canonical logic circuits. Every node stands for a neuron, where neural computations take place. This means that __nervous activity, other than logic circuits, could instead depend on topological transformations__ and symmetry constraints dictated by Stone-Wales transformations occurring in the upper cortical layers. A two-dimensional fullerene-like lattice not only simulates the real microcolumn’s microcircuitry, but also makes it possible to build artificial networks equipped with robustness, plasticity and fastness. In this note, electric signal propagation is investigated in terms of pure topological modifications of the neural honeycomb network. PDF

Tozzi A, Peters JF, Ori O. 2017. Cracking the barcode of fullerene-like cortical microcolumns. Neurosci Letters,644, 100–106. https://dx.doi.org/10.1016/j.neulet.2017.02.064.

Artificial neural systems and nervous graph theoretical analysis rely upon the stance that the neural code is embodied in logic circuits, e.g., spatio-temporal sequences of ON/OFF spiking neurons. Nevertheless, this assumption does not fully explain complex brain functions. Here we show how __nervous activity, other than logic circuits, could instead depend on topological transformations and symmetry constraints occurring at the__ __micro-level of the cortical microcolumn__, i.e., the embryological, anatomical and functional basic unit of the brain. Tubular microcolumns can be flattened in fullerene-like two-dimensional lattices, equipped with about 80 nodes standing for pyramidal neurons where neural computations take place. We show how the __countless possible combinations of activated neurons embedded in the lattice resemble a barcode__. Despite the fact that further experimental verification is required in order to validate our claim, different assemblies of firing neurons might have the appearance of diverse codes, each one responsible for a single mental activity. A two-dimensional fullerene-like lattice, grounded on simple topological changes standing for pyramidal neurons’ activation, not just displays analogies with the real microcolumn’s microcircuitry and the neural connectome, but also the potential for the manufacture of plastic, robust and fast artificial networks in robotic forms of full-fledged neural systems.

**Sprott equations in the brain: a single equation describes linear and nonlinear nervous dynamics**

Peters JF, Tozzi A, Deli E. 2017. Towards Equations for Brain Dynamics and the Concept of Extended Connectome. SF J Neuro Sci 1:1.

__The brain is a system at the edge of chaos equipped with nonlinear dynamics__ and functional energetic landscapes. However, so far no connection has been found between the electric activities of the brain and the physiological repertoire of behavior. Recent work suggests the integrated nature of __information processing in the brain not only via synaptic connectivity__, but a wholesome physical organization, which takes the shape of a toroidal particle trajectories, chaotic attractors or standing waves. The characterization of brain activities concerning the type of attractors or the trajectories of the nervous phase space is also missing. Starting from a system governed by differential equations in which a dissipative strange attractor coexists with an invariant conservative torus, __we developed a 3D model of brain phase space__ which has the potential to be operationalized and assessed empirically. We achieved a system displaying either a torus or a strange attractor, depending just on the initial conditions. Further, the system generates a funnel-like attractor equipped with a fractal structure. Changes in three brain phase parameters lead to modifications in the funnel’s breadth or in torus/attractor superimposition. We have found that the higher frequencies of evoked activities are more deterministic because the greater funnel breadth reduces the degrees of freedom. Thus, evoked activities are more deterministic. In contrast, the resting state corresponds to lower frequencies and represents greater degrees of freedom, which engender daydreaming, mind wandering and other liberal, often arbitrary mental associations. Our model __connects the physiological manifestations of consciousness __with the electric activities of the brain and it also powerfully explains the differences in motivation between evoked and resting activities based on energy use. This idea might point to the origin of a large repertoire of brain functions, such as sensations/perceptions, memory and self-generated thoughts. PDF

**The Bloch theorem correlates high- and low- brain frequencies**

Brain electric activity exhibits two important features: oscillations with different timescales, characterized by diverse functional and psychological outcomes, and a temporal power law distribution. In order to further investigate __the relationships between low- and high- frequency spikes__ in the brain, we used a variant of the Borsuk-Ulam theorem which states that, when we assess the nervous activity as embedded in a sphere equipped with a fractal dimension, we achieve two antipodal points with similar features (the slow and fast, scale-free oscillations). We demonstrate that slow and fast nervous oscillations mirror each other over time via a sinusoid relationship and provide, through the Bloch theorem from solid-state physics, the possible equation which links the two timescale activities. We show that, based on topological findings, __nervous activities occurring in micro-levels are projected to single activities at meso- and macro-levels__. This means that brain functions assessed at the higher scale of the whole brain necessarily display a counterpart in the lower ones, and vice versa. Our topological approach makes it possible to assess brain functions both based on entropy, and in the general terms of particle trajectories taking place on donut-like manifolds. Condensed brain activities might give rise to ideas and concepts by combination of different functional and anatomical levels. Furthermore, cognitive phenomena, as well as social activity can be described by the laws of quantum mechanics; memories and decisions exhibit holographic organization. In physics, the term duality refers to a case where two seemingly different systems turn out to be equivalent. This topological duality holds for all the types of spatio-temporal brain activities, independent of their inter- and intra-level relationships, strength, magnitude and boundaries, allowing us to connect the physiological manifestations of consciousness to the electric activities of the brain. PDF

**The unexpected occurrence of J-functions & complex functions in the brain**

The modular function j, central in the assessment of abstract mathematical problems, describes elliptic, intertwined trajectories that move in the planes of both real and complex numbers. Recent clues suggest that the j-function might display a physical counterpart, equipped with a quantifiable real component and a hidden imaginary one, currently undetectable by our senses and instruments. Here we evaluate whether the real part of the modular function can be spotted in the electric activity of the human brain. We assessed EEGs from five healthy males, eyes-closed and resting state, and superimposed the electric traces with the bidimensional curves predicted by the j-function. We found that the two trajectories matched in more than 85% of cases, independent from the subtending electric rhythm and the electrode location. Therefore, the real part of the j-function’s peculiar wave is ubiquitously endowed all over normal EEGs paths. We discuss the implications of such correlation in neuroscience and neurology, highlighting how the j-function might stand for the one of the basic oscillations of the brain, and how __the still unexplored imaginary part might underlie several physiological and pathological __nervous features. PDF

**Mirzakhani’s hyperbolic spaces are also in the brain**

Biological activities, including cellular metabolic pathways, protein folding and brain function, __can be described in terms of curved trajectories in hyperbolic spaces__ which are constrained by energetic requirements. Here, starting from theorems recently-developed by a deceased Field Medal young mathematician, we show how it is feasible to find and quantify the shortest, energy-sparing functional trajectories taking place in nervous systems’ concave phase spaces extracted from real EEG traces. This allows neuroscientists to focus their studies on the few, most prominent functional EEG’s paths and loops able to explain, elucidate and experimentally assess the rather elusive mental activity. PDF

**Can the nervous connectome be embedeed in hyperbolic spaces?**

As recently suggested, the brain activity displays multidimensional features). Recent work has shown that __the appropriate isometric space for embedding complex networks__ (and in particular the neural multidimensional ones, such as the human connectome) __is not the ﬂat Euclidean space, but a negatively curved hyperbolic space__. Indeed, hyperbolic space has the property that __power-law degree distributions__, strong clustering and hierarchical community structure __emerge naturally when random graphs are embedded in hyperbolic space__. It is therefore logical to exploit the structure of the hyperbolic space for useful embeddings of complex networks. It has been demonstrated that, when applied to the task of classifying vertices of complex networks, hyperbolic space embeddings signiﬁcantly outperform embeddings in Euclidean space.

**Elliptic curves in the brain: a feasible conceptual revolution**

The Diffusion Tensor Imaging and Diffusion Tensor Tractography of neural projections performed by Alizadeh et al. are elliptic curves, i.e., they can be abstractly described in terms of two-dimensional paths without cusps or intersections. These kind of cubic equations’ curves are embedded in an algebraic two-dimensional finite field, accurately defined and quantified in terms of points and numbers (both integers and rational). The same type of elliptic curve can be found when examining the wavefronts of EEG and fMRI patterns. What elliptic curves bring on the table, when assessing of brain functions? In our case, __elliptic curves__ (standing for anatomical neural projections detected by tractographic techniques) __lie inside a finite field (the brain) which can be subdivided in numbered zones__ (characterized by integer and rational numbers) and assessed through algebraic weapons, number theory, complex analysis, algebraic geometry and representation theory. Here we provide a few examples. Elliptic curves are equipped with __symmetries (they are abelian__, in technical terms), apparently hidden at first sight. This allows to compare anatomical/functional neuronal features with matching description which are located in far-flung brain areas. Further, it is noteworthy that __half of the elliptic curves displays a low amount of rational numbers, while the other half an infinite number__. In our operational terms, this means that __half of the nervous patterns are continuous, while half are discontinuous__ and proceed in temporal/spatial quantized steps. The last, but not the least, elliptic curve is a a type of cubic curve whose solutions are confined to a region of space that is topologically equivalent to a torus. This means that anatomical and functional nervous trajectories can be __assessed in the easily manageable terms of trajectories on a torus. __ PDF

**Constraints dictated by MAST-1: when topology dictates fetal development**

Tozzi A. 2019. MAST1: macroscopic local perturbations in brain growth. (electronic response to: Tripathy R, Leca I, van Dijk T, Weiss J, van Bon BW, et al. 2018. Mutations in MAST1 Cause Mega-Corpus-Callosum Syndrome with Cerebellar Hypoplasia and Cortical Malformations.Neuron, 100(6):1354-1368.e5. doi: 10.1016/j.neuron.2018.10.044).

**1368.e5. doi: 10.1016/j.neuron.2018.10.044).**

Tripathy et al. (2018) report that Mast1, espressed just in postmitotic neuronal dendritic and axonal compartments, is associated with the microtubule cytoskeleton in a MAP-dependent manner. *M*ice with __Mast1____ microdeletions display peculiar macroscopic features, such as enlarged corpus callosum and smaller cerebellum, in absence of megaloencephaly__. These findings let us to hypothesize that __the macroscopic growth of the brain tissue is regulated by physical constraints__: keeping __invariant the brain size__ (i.e., in absence of megaloencephaly), __the central nervous tissue of animals harboring Mast1 microdeletions undergoes a general rearrangement__. In physical/mathematical terms, a three-dimensional lattice (standing for the whole brain mass) harbors vectors and tensors which product must be held constant.

__When a lattice perturbation occurs (as in the case of cytoskeleton genetic Mast1 alterations), one of the tensors modifies__. In order to keep invariant the tensor product, another tensor need to vary: in simpler words,

__the fact that more axons cross__the midline in

*Mast1*Leu278 del mice

__means that the size of other structures (in this case, the cerebellum) must decrease__. In touch with this observation, Tripathy et al. (2018) report that, in animals harboring

*Mast1*microdeletions, we find that the PI3K/AKT3/mTOR pathway is unperturbed, whereas Mast2 and Mast3 levels are diminished, indicative of a dominant-negative mode of action.

**Exagonal sphapes are not just in cortical microcolumns, but also in the neurons of the myenteric plexus: **