Serotonergic Psychedelics LSD & Psilocybin Increase the Fractal Dimension of...

Serotonergic Psychedelics LSD & Psilocybin Increase the Fractal Dimension of Cortical Brain Activity in Spatial and Temporal Domains

1 Introduction

Since the turn of the century, there has been a renewal of interest in the science of serotonergic psychedelic drugs (LSD, psilocybin, mescaline, etc.), both in terms of possible medical applications of these drugs [1, 2], and what they might tell us about the relationship between activity in the brain and the phenomenological perception of consciousness [3, 4]. For those interested in the relationship between activity in the brain and consciousness, psychedelic drugs are particularly useful, as volunteers under the influence of a psychedelic are still able to report the nature of their experience and recall it even after returning to normal consciousness. This contrasts favourably with the other class of drugs commonly used to explore consciousness: anaesthetics, which by the very nature of their effects, make it difficult to gather first-person experiential data from a volunteer [5]. The subjective experience of the psychedelic state is associated with radical alterations to both internal and external senses, including visual distortions, vivid, complex closed-eye imagery, alterations to the sense of self, emotional extremes of euphoria and anxiety, and in extreme cases, psychosis-like effects [6]. The psychedelic experience can also have profound personal, and even spiritual or religious character [7, 8], which has made them central to the religious practices of many cultures around the world [9]. In this way, the study of the psychedelic state can inform not just the question of why consciousness emerges, but also the origins of some of the most quintessentially human psychological experiences.

Neuroimaging studies using fMRI and MEG have suggested that the experiential qualities of the psychedelic state can be explained, in part, by the effects these drugs have on the entropy of brain activity: a theory known as the Entropic Brain Hypothesis (EBH) [10, 4]. The EBH posits that during normal waking consciousness, activity in the brain is near, but slightly below, a critical zone between order and disorder, and that under the influence of psychedelic drugs the entropy of brain activity increases, bringing the system closer to the zone of criticality. In this context, ‘criticality’ can be thought of as similar to a phase-transition between two qualitatively different states: the sub-critical state, which is comparatively inflexible, highly ordered and displays low entropy, while the super-critical state may be highly entropic, flexible, and disorganized (this recalls a canonical model of critical processes, the Ising Model, where the critical temperature divides distinct phases, one where the magnetic spins are all aligned, and another where the spins are distributed chaotically, for review see [11]). The EBH is related to a larger theory of consciousness, known as Integrated Information Theory (IIT), which posits that consciousness is an emergent property of the integration of information in the brain [12, 13, 14] and that this mathematical formalism is categorically isomorphic to consciousness itself [15].

While it is currently impossible to directly measure the entropy of all of the activity in the whole of the brain, or the amount of information integration, there is much interest in using mathematical analysis of neuroimaging data to estimate the complexity of activity in the brain and relate that to consciousness. Studies with psilocybin have found that the patterns of functional connectivity in the brain undergo dramatic reorganization, characterized primarily by the rapid emergence and dissolution of unstable communities of interacting brain regions that do not occur in normal waking consciousness [16]. Similarly, under psilocybin, the repertoire of possible states functional connectivity networks can occupy is increased, which is interpreted as an increase in the entropy of the entire system [17]. Work on other psychedelics with pharmacology related to psilocybin has found similar results: under the influence of Ayahuasca, a psychedelic brew indigenous to the Amazon, the Shannon entropy of the degree distribution of functional connectivity networks is increased relative to normal consciousness [18] (encouragingly, the opposite effect has been shown under the conditions of sedation with propofol [19]). Analysis of MEG data from volunteers under the influence of lysergic acid diethylamide (LSD) has been shown an increase in the Lempel-Ziv complexity of the signals, which is thought to reflect increased complexity of activity in the brain [20]. LSD has also been recently shown to alter the connectome harmonics of brain networks, in a manner that suggests an increase in the complexity of network harmonics describing brain activity [21]. For a comprehensive review of the current state of psychedelic research into the EBH see The Entropic Brain - Revisited [4].

While a core element of the EBH is the theory that the psychedelic experience moves the brain closer to the zone of criticality, many of the measures that have been tested so far do not address the phenomena of criticality directly. These measures usually test where the brain falls on a unidimensional axis of order vs. randomness. Lempel-Ziv complexity [20], nodal entropy [18, 19] and the entropy of possible states [17], all describe a movement towards increased randomness and disorder, which is consistent with the entropic predictions of the EBH, but not necessarily informative about the relative proximity to the zone of criticality. In these analyses, a completely random system would score maximally high on complexity (for instance a completely random time-series would have a normalized Lempel-Ziv score of unity, which is the upper bound of the measure) however it is nearly impossible to imagine how a living brain could output totally random data, and such a brain would most likely not be conscious. While these analyses are interesting and have clearly been fruitful, they paint a limited picture of the brain as a complex system, and don’t directly test the central thesis of the EBH. To date, the only study that has directly addressed the criticality aspect of the EBH is the study of LSD and connectome harmonics [21], although other studies have found evidence of scale-free, power-law behaviour generally thought to be indicative of critical phenomena [22]. To address the relative lack of studies testing criticality directly, in this paper, we propose the fractal dimension of brain activity as a novel measure of complexity that provides insights into the criticality of the psychedelic state, as well as providing a measure of ‘complexity’ that is related to, but distinct from, the entropic measures described above.

Fractals are ubiquitous in nature and dramatic visualizations of colourful constructs like the Mandelbrot set have even permeated popular culture [23]. Psychedelic culture in particular shows a strong affinity for fractal patterns, as much of the imagery experienced under the influence of psychedelics is described as fractal in character. Fractals are defined by the property of having a non-integer dimension, which can be naively thought of as how ‘rough’ or ‘complex’ the shape in question is, or slightly more formally, the extent to which it maintains symmetry across different scales [24]. This is commonly known as ‘self-similarity,’ and can be intuitively understood as the invariance of appearance across scales: for example, the pattern of small creeks flowing together can resemble the pattern of large rivers carrying orders of magnitude more water [25]. In systems that display self-organizing criticality, as the system naturally evolves towards a critical point, its spatial structure will tend to take on increasingly fractal character that can be described in terms of fractal dimension [26, 27, 28], and in systems which can be ‘tuned’ to a critical state (such as the Ising model, which has been explored as a model of critical brain activity [29, 30, 31]), fractal structures emerge near the critical point [32]. If, under the influence of a psychedelic, the brain is moving closer towards a state of criticality, as the EBH posits, then we might expect any fractal character in brain activity to become more pronounced. There is some evidence of a symmetrical effect when consciousness is lost: in states of sleep and drug-induced anaesthesia, the fractal dimension of brain activity drops significantly, with the exception of REM sleep, during which the fractal dimension rises again [33, 34]. As REM sleep is the state of sleep when the greatest quantity of phenomonological experience takes place (in the form of dreams), this suggests that the fractal dimension of brain activity is related to the ‘quantity’ of experiential consciousness available to an individual. Similarly, in rats, during ketamine-induced anaesthesia the fractal dimension of brain activity is significantly higher in key-brain regions associated with consciousness when compared with anaesthesia induced by other drugs [35], and as ketamine is known to induce vivid, dream-like states of consciousness at high doses [36], which comports with the REM sleep finding.

There has been considerable interest in applying techniques of fractal analysis to questions in neuroscience and considerable evidence has mounted that both the physical structure of the brain itself, and the patterns of activity measured by neuroimaging paradigms display pronounced fractal character [37, 38]. Changes to the fractal dimension of brain structures are associated with changes in cognition and clinically significant diagnosis, such as schizophrenia and obsessive-compulsive disorder [39], intelligence [40], Alzheimer’s disease [41], and ageing [42]. There is some preliminary evidence that cortical functional connectivity networks display fractal character, both during rest and tasks [43] and that this fractal character plays an important role in regulating how information is propagated through the brain [44].

While fractal dimension is usually though to encode complexity in terms of self-similarity rather than entropy directly, there is a connection between the two values: fractal dimension is related to Renyi entropy, which is itself a generalization of the classical measure of Shannon entropy [45]. Computational models have shown that as the fractal dimension of a shape rises, so does the associated Renyi entropy [46]. Another measure, the information dimension, relates the fractal dimension to the information content of a fractal at different scales [47, 48]. Based on these findings, and the results reported by Bak et al., (1987), we propose that the fractal dimension is a natural metric by which to test the EBH, for several reasons. First, unlike other metrics of entropy, fractals are intimately related to the phenomena of criticality, which is predicted to be significant for consciousness, and the fractal dimension encodes information relevant to a system’s evolution towards criticality. Second, in this context, they are a novel method of describing the behaviour of the brain as a complex system and so give information beyond the axis of order versus randomness. Finally, despite the differences between the measure of fractal dimension and classical entropy, the two are related in some fundamental ways. The fractal dimension sits at a sweet spot of not being so radical that it cannot be related to previous results, while still being novel enough to open the door to new and informative avenues of study.

To test the relationship between the fractal dimension of activity of brain and consciousness, we used fMRI data from subjects under the influence of either LSD or psilocybin, provided by the Psychedelic Research Group at Imperial College London. From this data, we created 1000-node functional connectivity networks and performed a network-specific variation of the box-counting algorithm [49] to extract the fractal dimension. We also used a second measure, the Higuchi fractal dimension [50], to test the temporal fractal dimension of BOLD time-series. These two measures capture two axes on which the complexity of brain activity might be measured: spacial (network fractal dimension) and temporal (Higuchi fractal dimension). If the psychedelic state is associated with a movement towards a critical zone associated with increased fractal character, we would expect to see this when examined on multiple measures, and so these two measures serve as internal validation for each-other. While the network fractal dimension is not spacial in the way, for example, a 2-dimensional box-counting analysis of activity at the cortical surface would be, it does return insight into how information processing may be distributed across multiple, spatially distinct brain regions.