This paper (2022) uses the Ising model of brain phase transition to assess fMRI BOLD data from a study in which LSD was administered (n=15). Several different concepts from statistical physics were applied to the fMRI data to analyze individualized Ising temperature increases under the influence of LSD and placebo, showing that LSD ingestion shifts the system away from the critical point between paramagnetic and ferromagnetic phases to a more disordered state. Overall, findings suggest that LSD increases the complexity of brain dynamics.

## Abstract

*“A topic of growing interest in computational neuroscience is the discovery of fundamental principles underlying global dynamics and self-organization of the brain. Statistical physics offers powerful tools to analyze complex many-body systems and has delivered computational models such as the generalized Ising model, one of the simplest known systems displaying phase transitions. Prior work has shown that different brain states may be mapped into pairwise maximum entropy Ising models at varying distances from the critical temperature. Here, we use this framework to analyze resting state fMRI BOLD data collected in an earlier study from fifteen subjects in a control condition (placebo) and during psychedelic ingestion (LSD). We first parcellate the data in AAL space and, to address the limited quantity of data available from typical fMRI BOLD studies, we create an “archetype” pairwise maximum entropy or Ising model representative of the entire dataset. The archetype model is then personalized for each individual and condition through an adjustment of the system temperature. We analyze the resulting set of temperatures to show first that, at the group level and in both conditions, the model is near criticality but situated in the paramagnetic phase. Second, the individualized Ising temperature increases significantly under the effects of LSD compared with the placebo condition (p = 9 x 10 ^{-5}). That is, LSD ingestion shifts the system away from the critical point between paramagnetic and ferromagnetic phases to a more disordered state. Next, we estimate the Lempel-Ziv-Welch (LZW) complexity of the binarized BOLD data (flattened along the spatial dimension first) for each participant and condition and of the synthetic data generated with the individualized model using the Metropolis algorithm. We find that the LZW complexity computed directly from experimental data reveals a weak statistical relationship with the condition (p = 0.04 one-tailed Wilcoxon test), and none with Ising temperature (r(13) = 0.13, p = 0.65), presumably due to the short length of the time series and group size. However, the LZW complexity computed using synthetic data from the personalized archetype model correlates strongly with individualized temperature (p = 2.7 x 10^{-6}) and hence with condition (p = 9 x 10^{-5}, one-tailed Wilcoxon test). We then explored the estimation of algorithmic complexity using the block decomposition method (BDM). The BDM complexity of the experimental data displayed a robust correlation with Ising temperature (r(13) = 0.56, p = 0.03) and a weak but significant correlation with condition (p = 0.04, one-tailed Wilcoxon test). We also calculated the BDM complexity of the synthetic data generated by the model, and it correlates strongly with Ising temperature (r(13) = 0.97, p = 8.9 x 10^{-10}) and hence condition (p = 2 x 10^{-4}, one-tailed Wilcoxon test). This study suggests, together with prior ones, that the effects of LSD increase the complexity of brain dynamics. In agreement with earlier work using the Ising formalism, we find the brain state in the placebo condition is already above the critical point, with LSD resulting in a shift away from criticality into a more disordered state.”*

**Authors: **Giulio Ruffini, Giada Damiani, Diego Lozano-Soldevilla, Nikolas Deco, Fernando Rosas, Narasis Kiani, Adrian Ponce-Alvarez, Morten L. Kringelback, Robin L. Carhart-Harris & Gustavo L. Deco

## Summary

Abstract

Statistical physics offers powerful tools to analyze complex many-body systems, such as the generalized Ising model, and we use this framework to analyze resting state fMRI BOLD data collected in an earlier study from fifteen subjects in a control condition and during psychedelic ingestion. We estimate the Lempel-Ziv-Welch complexity of binarized BOLD data for each participant and condition. The complexity of synthetic data generated with the personalized archetype model correlates strongly with individualized temperature and hence with condition. We calculated the block decomposition method (BDM) complexity of the experimental data and found that it correlates strongly with Ising temperature and hence condition. This study suggests that LSD increases the complexity of brain dynamics, shifting the state away from criticality into a more disordered state.

Author summary

We study brain dynamics under LSD using the Ising model formalism and algorithmic complexity using Lempel-Ziv and the Block Decomposition methods. We find that the effects of LSD translate into increased BOLD signal complexity and Ising temperature, in agreement with earlier findings and predictions from existing theories.

Introduction 1

The complex features displayed by the emergent neural dynamics of the human brain are reminiscent of behaviors of paradigmatic systems in statistical physics, and are particularly interesting at critical points where order and disorder are balanced, giving rise to a rich repertoire of short- and long-range correlations.

Theoretical work indicates that complex systems display special features at critical points, and that the brain’s inherent complexity is explained by criticality. Furthermore, it has been found that systems in nature often tune themselves into a critical state.

This paper aims to leverage tools from the statistical physics literature to study the effects of LSD on the brain, to better understand and predict the outcomes of psychedelic interventions. Studies on functional neuroimaging regarding LSD effects have shown initial evidence of the mechanistic alterations on brain dynamics at the network level, with an increase of brain entropy, global functional integration, and more flexible brain dynamics.

We leverage maximum entropy models to describe the coordinated activity of various brain areas as captured by BOLD signals, and to observe differences with respect to the model’s temperature: a low temperature amplifies distinctions between energy levels and hence a higher dissimilarity between the likelihood of occupancy of different states.

We used data from 15 participants who underwent ECG, urine and blood tests, a psychiatric test and drug historical. They were excluded if they had a history of psychiatric diseases, family history of psychotic disorders, experience with psychedelics, pregnancy, or problems with alcohol. Participants attended twice the laboratory (once per condition), with a separation of 2 weeks. They were given 75 mg of LSD via a 10ml solution proportioned over two minutes with an infusion of saline afterward, and then were suggested to lie relaxed with closed eyes inside the MRI scanner.

Parcellated data series 90

The fMRI BOLD data were processed using FSL tools, and the region-averaged BOLD time series were generated.

Data binarization 95

To build an Ising model from the data, we transform the fMRI BOLD data series into a binary format and assign each data point a value of +1 or -1 depending on its value.

Lempel-Ziv complexity 102

Lempel-Ziv (LZ) algorithms are adaptive dictionary compression algorithms that parse a string into words and use increasingly long reappearing words to construct a dictionary.

We can approximate the description length of a sequence encoded by LZ by the number of words seen times the number of bits needed to identify a word. LZW provides an upper bound on algorithmic complexity, but it fails to effectively compress random-looking data generated by simple, but highly recursive programs, e.g., an image of the Mandelbrot set (deep programs). Despite this, LZW can be useful to study the complexity of data in an entropic sense.

Two associated metrics derived from LZ compression are commonly used: c(n) and lLZ . lLZ is more closely related to Kolmogorov complexity or description length, and can be normalized by dividing description length by the original string length. To calculate LZ complexity, we concatenated data from all subjects, conditions, and sessions, flattened the data along the spatial dimension first and then time, and compressed the flat binarized data with the LZ algorithm. We saved the output “archetype dictionary” and used it to compress data from all subjects.

Block decomposition method 149

The Block Decomposition Method (BDM) combines the action of two universally used complexity measures at different scales by dividing data into smaller pieces. It is used to account for both statistical regularities and algorithmic ones, and complements the use of lossless compression algorithms.

Ising model 161

Abstract frameworks from statistical physics can shed light on understanding emerging phenomena in large networks, such as phase transitions, in which the number of degrees of freedom goes to infinity and the system goes from order to disorder.

In this context, the 172 spins of neurons, columns, or brain regions are modeled by pair interactions, and the 173 emerging statistical properties of large networks of these elements are studied under 174 different conditions. The prototypical simplest system in this context is the classical 2D Ising model. The 2D Ising model used here allows for arbitrary pair interactions, and is given by the energy of the lattice of N spins.

Maximum entropy principle derivation of Ising model 194

The Ising model can be seen to arise naturally from the Maximum Entropy Principle, which states that the probability of finding a given spin configuration can be inferred from a simplicity criterion.

We can adapt the archetype model for each subject and condition by changing the model temperature T = 1/ , that is, by writing 232 with a fixed point at h = hp .

Metropolis algorithm and Ising model observables 236

The Metropolis algorithm is used to compute the observables of the Ising model built from the experimental data. The main global observable is the lattice average magnetization M over the spin lattice S , 247 and lattice energy, where recall N is the number of nodes in the lattice.

We can extend the definition of global susceptibility to local susceptibility by taking the standard deviation computation over the ensemble.

In order to smooth the results and deal with slow lattice fluctuations at cold temperatures in finite size systems, we work with modified variables and Metropolis generated data to compute the magnetization and susceptibility statistics.

Statistical analysis 267

We performed a paired Wilcoxon test to compare the results of the repeated measures within-subject design and a non-parametric test to determine the cause of the observed increase in temperature under LSD relative to placebo.

We performed a permutation test on the mean temperature difference, and a paired one-tailed Wilcoxon test with the LZW and BDM complexity estimates. The level of statistical significance was estimated by the fraction of t-values derived for every random shuffle that were higher than observed t-values.

Archetype connectivity 287

We estimated the parameters h and J for the archetype model using fMRI data from all subjects in both conditions, and checked for changes due to the effect of LSD. The LSD archetype model had slightly weaker weights and less variance.

Archetype phase diagrams 301

We calculated the critical temperature of the archetype model and found that it is well below T = 1, the nominal temperature of the model. The peaks of susceptibility, heat capacity and LZW variance occur at approximately the same temperature, while BDM complexity variance peaks later.

Individual temperature shift with LSD 317

We found that temperatures in the LSD state were substantially higher than in placebo, with a difference of 6.7% (mean standard deviation) between conditions and a statistically significant increase of 1.35 (Cohen’s d = 1.35).

LZW and BDM complexity from data and model (synthetic 330 data) 331

We extracted features related to the algorithmic complexity of the archetype model from the empirical fMRI data and the synthetic data generated with the Metropolis algorithm. These features showed a similar trend to the one obtained with the Ising temperatures.

The mean and median LZW complexity shift with LSD were 0.008 and 0.005, with a standard deviation of 0.013. The mean and median BDM complexity shift with LSD were 338 and 298, with a standard deviation of 803.6.

The increase in LZW and BDM complexity in the LSD state was statistically significant. The correlation between the features extracted from the data and the model was strong, and the difference in LZW and BDM complexity correlated strongly with the delta Ising temperature.

Participants in the LSD trial answered two types of questionnaires to give feedback on their psychedelic experience. The delta LZW from experimental data does not display a significant relationship with the delta Ising temperature or the delta LZW complexity from synthetic data.

Fig 5 shows that the complexity estimates of LZW and BDM are shifted with LSD using synthetic data.

After each scan, the authors used a button-press and a digital display screen to report their altered states of consciousness (ASCs). The authors reported that elementary imagery, complex imagery, and audio-video synesthesia were the most important ASCs. We calculated the questionnaire scores difference between the LSD and placebo sessions for each category, and checked the Pearson correlation between the scores and the features extracted from the data.

No strong statistical relationships were seen between subjective reports and the Ising features, but trends were observed between complex imagery and Ising temperatures and elementary imagery and LZW complexity. The VAS-style ratings showed positive trends for Ising temperatures and BDM complexity, but negative trends for LZW complexity.

We found it especially useful to generate a global archetype model assimilating the data from all participants and conditions. This model generates behavior as a function of temperature similar to the standard, nearest-neighbor Ising model, with a clearly defined critical point.

We found a very strong correlation between individualized temperature and condition. The individualized temperatures derived from fMRI BOLD data almost uniformly increased with the LSD condition relative to placebo, and both the archetype temperature and the individualized temperatures were found to be above the critical point of the model.

Figure 3 (bottom right) shows that BDM complexity increases monotonically with temperature, but the slope of the increase is correlated with model parameters and describing the simulation condition. This means that BDM can be used to get more insight from empirical data and design better simulations.

Our results are consistent with the notion that psychedelics drive brain dynamics into a more disordered state, away from criticality, and are explained by the free-energy principle.

Psychedelics are believed to begin with stimulation of a specific serotonin receptor subtype (5-HT2AR) on cortical layer V pyramidal neurons, which results in increased neuronal excitability and dysregulated cortical population-level activity. This disrupts the regular functioning of high-level system properties. In [68], we hypothesized that increased apparent complexity of spontaneous activity would follow from finer grain functional architecture, and in [71], we observed that this was indeed the case. The present study extends on previous work by incorporating algorithmically-oriented methods such as BDM.

In line with previous work using similar methods, we found that the resting brain in the placebo condition is already above the critical point, and that psychedelics move dynamics into a more disordered state. However, the relation between these metrics and subjective experience is weak.

1 Conclusions 499

We used the Ising model formalism to study criticality features of brain dynamics under LSD. We found that the system in the placebo condition was already in the paramagnetic phase, above the critical point, with LSD resulting in a shift away from criticality into a more disordered state. This study used the Ising framework to fit individual data, and found that LSD shifts system dynamics closer to criticality. However, further work is needed to reconcile these results.

We checked the correlation between features extracted from data in Figure 6.

correlation with metrics 524

The ASC questionnaire measures experience of unity, spiritual experience, blissful state, insightfulness, disembodiment, impaired control and cognition, anxiety, complex imagery, elementary imagery, audio-visual synesthesiae.

D Comparison with 5-HT2A receptors map 525

The LSD effects seem to be related to the density of the 5-HT2A receptors, so we checked the correlation between the changes in the archetype parameters obtained separately from the LSD and placebo data and the 5-HT2A receptor density.

### Find this paper

LSD-induced increase of Ising temperature and algorithmic complexity of brain dynamics

https://doi.org/10.1371/journal.pcbi.1010811

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### Study details

**Compounds studied**

LSD

**Topics studied**

Neuroscience

**Study characteristics**

Re-analysis
Theory Building
Bio/Neuro

**Participants**

15
Humans

### Authors

Authors associated with this publication with profiles on Blossom

**Robin Carhart-Harris**

Dr. Robin Carhart-Harris is the Founding Director of the Neuroscape Psychedelics Division at UCSF. Previously he led the Psychedelic group at Imperial College London.

### Linked Research Papers

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**The entropic brain: a theory of conscious states informed by neuroimaging research with psychedelic drugs**

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**[RETRACTED] A mechanistic model of the neural entropy increase elicited by psychedelic drugs**

This whole-brain simulation study (DMF model) reproduced increased entropy under psychedelics, but also found this to be different (higher/lower) in various parts of the brain, this was correlated with the connections between brain regions, not serotonin (5-HT2a) receptor density.