Recurrent fractal neural networks: a strategy for the exchange of local and global information processing in the brain

Fractals in Neuroanatomy and Basic Neurosciences: An Overview

The introduction of fractal geometry to the neurosciences has been a major paradigm shift over the last decades as it has helped overcome approximations and limitations that occur when Euclidean and reductionist approaches are used to analyze neurons or the entire brain. Fractal geometry allows for quantitative analysis and description of the geometric complexity of the brain, from its single units to the neuronal networks.As illustrated in the second section of this book, fractal analysis provides a quantitative tool for the study of the morphology of brain cells (i.e., neurons and microglia) and its components (e.g., dendritic trees, synapses), as well as the brain structure itself (cortex, functional modules, neuronal networks). The self-similar logic which generates and shapes the different hierarchical systems of the brain and even some structures related to its "container," that is, the cranial sutures on the skull, is widely discussed in the following chapters, with a link between the applications of fractal analysis to the neuroanatomy and basic neurosciences to the clinical applications discussed in the third section.

The fractal brain: scale-invariance in structure and dynamics

The past 40 years have witnessed extensive research on fractal structure and scale-free dynamics in the brain. Although considerable progress has been made, a comprehensive picture has yet to emerge, and needs further linking to a mechanistic account of brain function. Here, we review these concepts, connecting observations across different levels of organization, from both a structural and functional perspective. We argue that, paradoxically, the level of cortical circuits is the least understood from a structural point of view and perhaps the best studied from a dynamical one. We further link observations about scale-freeness and fractality with evidence that the environment provides constraints that may explain the usefulness of fractal structure and scale-free dynamics in the brain. Moreover, we discuss evidence that behavior exhibits scale-free properties, likely emerging from similarly organized brain dynamics, enabling an organism to thrive in an environment that shares the same organizational principles. Finally, we review the sparse evidence for and try to speculate on the functional consequences of fractality and scale-freeness for brain computation. These properties may endow the brain with computational capabilities that transcend current models of neural computation and could hold the key to unraveling how the brain constructs percepts and generates behavior.

Emergence of lump-like solitonic waves in Heimburg-Jackson biomembranes and nerves fractal model

The aim of this study is to extend the soliton propagation model in biomembranes and nerves constructed by Heimburg and Jackson for the case of fractal dimensions. Our analyses are based on the product-like fractal measure concept introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media and electromagnetic fields. The mathematical model presented in the paper is formulated for only a part of a single nerve cell (an axon). The analytical and numerical envelop soliton of this equation are reported. The results obtained prove the emergence of lump-type solitonic waves in nerves and biomembranes. In particular, these waves decay algebraically to the background wave in space direction. This scenario is viewed as a particular class of rational localized waves which are solutions of the integrable Ishimori I equation and the (2 + 1) Kadomtsev-Petviashvili I equation. The effects of fractal dimensions are discussed and were found to be significant to some extents.

Differentiation of athlete's heart and hypertrophic cardiomyopathy by the fractal dimension of left ventricular trabeculae

BACKGROUND

Differentiation between exercise induced adaptive myocardial hypertrophy (athlete's heart) and hypertrophic cardiomyopathy (HCM) is currently based on echocardiographic and cardiac magnetic resonance (CMR) criteria, but these may be insufficient in patients with subtle phenotype expression. This study aimed to assess whether left ventricular (LV) fractal pattern could permit to differentiate athlete's heart from HCM.

METHODS

We recruited retrospectively 61 elite marathon runners, 67 patients with HCM, and 33 healthy subjects. A CMR study was performed in all subjects and the LV trabeculae fractal dimension (FD) was measured in end-diastolic frames of each short-axis cine sequence. For group comparison, the ratio of maximal myocardial wall thickness (mMWT)/indexed LV end-diastolic volume (LVED) was determined.

RESULTS

As compared with athletes, patients with HCM had significantly (p < 0.001) greater FD in the LV basal (1.30 ± 0.07 vs. 1.23 ± 0.05) and apical (1.38 ± 0.06 vs. 1.30 ± 0.07) regions and in the whole heart (1.34 ± 0.05 vs. 1.27 ± 0.05). FD increased with age, left atrial area and indexed left ventricular mass (p < 0.05 for all) and correlated negatively with LV and RV end-diastolic volumes (p < 0.05 each). The addition of whole heart FD to the ratio of maximal myocardial wall thickness/indexed LVEDV lead to an improvement in the ability to discriminate HCM with a net reclassification index (NRI) of 71%.

CONCLUSIONS

The FD regional distribution of the LV trabeculae differentiates patients with athlete's heart from patients with HCM. The addition of whole heart FD to the mMWT/indexed LVEDV ratio improves the predictive capacity of the model to differentiate both entities.

Ceramide and S1P Signaling in Embryonic Stem Cell Differentiation

Bioactive sphingolipids are important regulators for stem cell survival and differentiation. Most recently, we have coined the term "morphogenetic lipids" for sphingolipids that regulate stem cells during embryonic and postnatal development. The sphingolipid ceramide and its derivative, sphingosine-1-phosphate (S1P), can act synergistically as well as antagonistically on embryonic stem (ES) cell differentiation. We show here simple as well as state-of-the-art methods to analyze sphingolipids in differentiating ES cells and discuss new protocols to use ceramide and S1P analogs for the guided differentiation of mouse ES cells toward neuronal and glial lineage.

Ordering Dynamics in Neuron Activity Pattern Model: An Insight to Brain Functionality

We study the domain ordering kinetics in d = 2 ferromagnets which corresponds to populated neuron activities with both long-ranged interactions, V(r) ∼ r⁻ⁿ and short-ranged interactions. We present the results from comprehensive Monte Carlo (MC) simulations for the nonconserved Ising model with n ≥ 2, interaction range considering near and far neighbors. Our model results could represent the long-ranged neuron kinetics (n ≤ 4) in consistent with the same dynamical behaviour of short-ranged case (n ≥ 4) at far below and near criticality. We found that emergence of fast and slow kinetics of long and short ranged case could imitate the formation of connections among near and distant neurons. The calculated characteristic length scale in long-ranged interaction is found to be n independent (L(t) ∼ t^1/(n−2)), whereas short-ranged interaction follows L(t) ∼ t^1/2 law and approximately preserve universality in domain kinetics. Further, we did the comparative study of phase ordering near the critical temperature which follows different behaviours of domain ordering near and far critical temperature but follows universal scaling law.

Multifractal analysis of information processing in hippocampal neural ensembles during working memory under Δ⁹-tetrahydrocannabinol administration

BACKGROUND

Multifractal analysis quantifies the time-scale-invariant properties in data by describing the structure of variability over time. By applying this analysis to hippocampal interspike interval sequences recorded during performance of a working memory task, a measure of long-range temporal correlations and multifractal dynamics can reveal single neuron correlates of information processing.

NEW METHOD

Wavelet leaders-based multifractal analysis (WLMA) was applied to hippocampal interspike intervals recorded during a working memory task. WLMA can be used to identify neurons likely to exhibit information processing relevant to operation of brain-computer interfaces and nonlinear neuronal models.

RESULTS

Neurons involved in memory processing ("Functional Cell Types" or FCTs) showed a greater degree of multifractal firing properties than neurons without task-relevant firing characteristics. In addition, previously unidentified FCTs were revealed because multifractal analysis suggested further functional classification. The cannabinoid type-1 receptor (CB1R) partial agonist, tetrahydrocannabinol (THC), selectively reduced multifractal dynamics in FCT neurons compared to non-FCT neurons.

COMPARISON WITH EXISTING METHODS

WLMA is an objective tool for quantifying the memory-correlated complexity represented by FCTs that reveals additional information compared to classification of FCTs using traditional z-scores to identify neuronal correlates of behavioral events.

CONCLUSION

z-Score-based FCT classification provides limited information about the dynamical range of neuronal activity characterized by WLMA. Increased complexity, as measured with multifractal analysis, may be a marker of functional involvement in memory processing. The level of multifractal attributes can be used to differentially emphasize neural signals to improve computational models and algorithms underlying brain-computer interfaces.

Letting the brain speak for itself

Metaphors of Computation and Information tended to detract attention from the intrinsic modes of neural system functions, uncontaminated by the observer's role in collection, and interpretation of experimental data. Recognizing the self-referential mode of function, and the propensity for self-organization to critical states requires a fundamentally new orientation, based on Complex System Dynamics as non-ergodic, non-stationary processes with inverse-power-law statistical distributions. Accordingly, local cooperative processes, intrinsic to neural structures, and of fractal nature, call for applying Fractional Calculus and models of Random Walks with long-term memory in Theoretical Neuroscience studies.

Introduction to the fractality principle of consciousness and the sentyon postulate

Recently, consciousness research has gained much attention. Indeed, the question at stake is significant: why is the brain not just a computing device, but generates a perception from within? Ambitious endeavors trying to simulate the entire human brain assume that the algorithm will do the trick: as soon as we assemble the brain in a computer and increase the number of operations per time, consciousness will emerge by itself. I disagree with this simplistic representation. My argument emerges from the "atomism paradox": the irreducible space of the consciously perceived world, the endospace is incompatible with the reducible and decomposable architecture of the brain or a computer. I will first discuss the fundamental challenges in current consciousness models and then propose a new model based on the fractality principle: "the whole is in each of its parts". This new model copes with the atomism paradox by implementing an iterative mapping of information from higher order brain structures to smaller scales on the cellular and molecular level, which I will refer to as "fractalization". This information fractalization gives rise to a new form of matter that is conscious ("bright matter"). Bright matter is composed of conscious particles or units named "sentyons". The internal fractality of these sentyons will close a loop (the "psychic loop") in a recurrent fractal neural network (RFNN) that allows for continuous and complete information transformation and sharing between higher order brain structures and the endpoint substrate of consciousness at the molecular level.

Fractals in the nervous system: conceptual implications for theoretical neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power-law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review.

The systems view in addiction research

The wealth of information accessible on the molecular level after completion of sequencing of the human and other genomes and in conjunction with new high-throughput technologies like microarrays has paved the way for research on whole systems rather than on single components. Here we describe exemplarily the construction of a rather complex molecular network involved in alcohol addiction by using information from DNA-microarray studies in alcohol-dependent animals. In this network, haemoglobin downregulation in different parts of the brain reward system plays a central role in affecting synaptic plasticity, circadian rhythmicity and opioid receptors. Furthermore, we discuss the dynamic aspect of biological systems with respect to repeated and intermittent drug intake. This aspect can best be captured by the allostatic model on the molecular level. Using a molecular oscillator model where levels of oscillations are changed by repetitive drug administration, changes in set point adjustment are described that underlay allostatic shifts in drug reinforcement processes.

Emotional task-dependent low-frequency fluctuations and methylphenidate: Wavelet scaling analysis of 1/f-type fluctuations in fMRI of the cerebellar vermis

Ion channel currents, neural firing patterns, and brain BOLD signals display 1/f-type fluctuations or fractal properties in time. By design, fMRI methods attempt to minimize the contribution of variance from low-frequency physiological 1/f-noise. New fMRI methods are described to visualize and measure 1/f-type BOLD fluctuations in volunteers recalling affectively neutral or emotional memories or meditating (i.e., attending to breathing) then retrospectively rating emotional content. A wavelet scaling exponent (alpha) was used to characterize signals from 0.015625 to 0.5Hz in cerebellar lobules VIII to X of the vermis (posterior inferior vermis; PIV), a region coordinating balance, eye tracking, locomotion, and vascular tone, and a possible site of pathology in attention deficit hyperactivity disorder (ADHD).

RESULTS

Changes in alpha and emotional measures were correlated in PIV voxels (r = 0.622, d.f .= 14, P < 0.0005), but not other regions examined. In contrast, conventional means and standard deviations of PIV voxels were unchanged. Methylphenidate, shown to decrease slow oscillations in rodent basal ganglia [Ruskin DN, Bergstrom DA, Shenker A, Freeman LE, Baek D, Walters JR. Drugs used in the treatment of attention-deficit/hyperactivity disorder affect postsynaptic firing rate and oscillation without preferential dopamine autoreceptor action. Biol Psychiatry 2001;49:340-50.], abolished task-dependent alpha changes in the PIV of an adult with ADHD. Wavelet analysis of long BOLD time series appears well suited to fractal physiology and studies of pharmacologically modulated cerebellar-thalamic-cortical function in ADHD or other psychiatric disorders.

Single-neuron theory of consciousness

By most accounts, the mind arises from the integrated activity of large populations of neurons distributed across multiple brain regions. A contrasting model is presented in the present paper that places the mind/brain interface not at the whole brain level but at the level of single neurons. Specifically, it is proposed that each neuron in the nervous system is independently conscious, with conscious content corresponding to the spatial pattern of a portion of that neuron's dendritic electrical activity. For most neurons, such as those in the hypothalamus or posterior sensory cortices, the conscious activity would be assumed to be simple and unable to directly affect the organism's macroscopic conscious behavior. For a subpopulation of layer 5 pyramidal neurons in the lateral prefrontal cortices, however, an arrangement is proposed to be present such that, at any given moment: (i) the spatial pattern of electrical activity in a portion of the dendritic tree of each neuron in the subpopulation individually manifests a complexity and diversity sufficient to account for the complexity and diversity of conscious experience; (ii) the dendritic trees of the neurons in the subpopulation all contain similar spatial electrical patterns; (iii) the spatial electrical pattern in the dendritic tree of each neuron interacts non-linearly with the remaining ambient dendritic electrical activity to determine the neuron's overall axonal response; (iv) the dendritic spatial pattern is reexpressed at the population level by the spatial pattern exhibited by a synchronously firing subgroup of the conscious neurons, thereby providing a mechanism by which conscious activity at the neuronal level can influence overall behavior. The resulting scheme is one in which conscious behavior appears to be the product of a single macroscopic mind, but is actually the integrated output of a chorus of minds, each associated with a different neuron.