Integrated Information Theory (IIT) posits that integrated information ( Φ ) represents the quantity of a conscious experience. Here, the generalized Ising model was used to calculate Φ as a function of temperature in toy models of fully connected neural networks. A Monte–Carlo simulation was run on 159 normalized, random, positively weighted networks analogous to small five-node excitatory neural network motifs. Integrated information generated by this sample of small Ising models was measured across model parameter spaces. It was observed that integrated information, as an order parameter, underwent a phase transition at the critical point in the model. This critical point was demarcated by the peak of the generalized susceptibility (or variance in configuration due to temperature) of integrated information. At this critical point, integrated information was maximally receptive and responsive to perturbations of its own states. The results of this study provide evidence that Φ can capture integrated information in an empirical dataset, and display critical behavior acting as an order parameter from the generalized Ising model.

There is accumulating evidence that spontaneous fluctuations of the brain are sustained by a structural architecture of axonal fiber bundles. Various models have been used to investigate this structure–function relationship. In this work, we implemented the Ising model using the number of fibers between each pair of brain regions as input. The output of the Ising model simulations on a structural connectome was then compared with empirical functional connectivity data. A simpler two-dimensional classical Ising model was used as the baseline model for comparison purpose. Thermodynamic properties, such as the magnetic susceptibility and the specific heat, illustrated a phase transition from an ordered phase to a disordered phase at the critical temperature. Despite the differences between the two models, the lattice Ising model and the Ising model implemented on a structural connectome (the generalized Ising model) exhibited similar patterns of global properties. To study the behavior of the generalized Ising model around criticality, calculation of the dimensionality and critical exponents was performed for the first time, by introducing a new concept of distance based on structural connectivity. Same value inside the fitting error was found for the dimensionality in both models suggesting similar behavior of the models around criticality.

Using the critical Ising model of the brain, integrated information as a measure of consciousness is measured in toy models of generic neural networks. Monte Carlo simulations are run on 159 random weighted networks analogous to small 5-node neural network motifs. The integrated information generated by this sample of small Ising models is measured across the model parameter space. It is observed that integrated information, as a type of order parameter not unlike a concept like magnetism, undergoes a phase transition at the critical point in the model. This critical point is demarcated by the peaks of the generalized susceptibility of integrated information, a point where the 'consciousness' of the system is maximally susceptible to perturbations and on the boundary between an ordered and disordered form. This study adds further evidence to support that the emergence of consciousness coincides with the more universal patterns of self-organized criticality, evolution, the emergence of complexity, and the integration of complex systems. Author summaryUnderstanding consciousness through a scientific and mathematical language is slowly coming into reach and so testing and grounding these emerging ideas onto empirical observations and known systems is a first step to properly framing this ancient problem. This paper in particular explores the Integrated Information Theory of Consciousness framed within the physics of the Ising model to understand how and when consciousness, or integrated information, can arise in simple dynamical systems. The emergence of consciousness is treated like the emergence of other classical macroscopic observables in physics such as magnetism and understood as a dynamical phase of matter. Our findings show that the sensitivity of consciousness in a complex system is maximized when the system is undergoing a phase transition, also known as a critical point. This result, combined with a body of evidence highlighting the privelaged state of critical systems suggests that, like many other complex phenomenon, consciousness may simply follow from/emerge out of the tendency of a system to self-organize to criticality. January 4, 2019 1/12illustrate the control exhibited by the choice of connectivity onto the order parameters 127 across the uniformly sampled random networks, whereas the susceptibilities quantify the 128 mean fluctuations of those order parameters within each random network, averaged 129 across all random networks. These summary statistics give first-order insights into the 130 diagnosis and control of simple neural networks. We note that at the critical 131 temperature, denoted roughly by the peaks of χ, the susceptibility of Φ, χ Φ also peaks. 132 When looking at Φ across different simulations, σ 2 J (Φ), we observe that there seems to 133 be two transition points. One transition point at low temperatures leading into a 134 plateau region followed by a second transition close to the classical critical point where 135 January 4, 2019 4/12

Criticality is thought to be crucial for complex systems to adapt at the boundary between regimes with different dynamics, where the system may transition from one phase to another. Numerous systems, from sandpiles to gene regulatory networks to swarms to human brains, seem to work towards preserving a precarious balance right at their critical point. Understanding criticality therefore seems strongly related to a broad, fundamental theory for the physics of life as it could be, which still lacks a clear description of how life can arise and maintain itself in complex systems. In order to investigate this crucial question, we model populations of Ising agents competing for resources in a simple 2D environment subject to an evolutionary algorithm. We then compare its evolutionary dynamics under different experimental conditions. We demonstrate the utility that arises at a critical state and contrast it with the behaviors and dynamics that arise far from criticality. The results show compelling evidence that not only is a critical state remarkable in its ability to adapt and find solutions to the environment, but the evolving parameters in the agents tend to flow towards criticality if starting from a supercritical regime. We present simulations showing that a system in a supercritical state will tend to self-organize towards criticality, in contrast to a subcritical state, which remains subcritical though it is still capable of adapting and increasing its fitness.

It has long been hypothesized that operating close to the critical state is beneficial for natural and artificial systems. We test this hypothesis by evolving foraging agents controlled by neural networks that can change the system's dynamical regime throughout evolution. Surprisingly, we find that all populations, regardless of their initial regime, evolve to be subcritical in simple tasks and even strongly subcritical populations can reach comparable performance. We hypothesize that the moderately subcritical regime combines the benefits of generalizability and adaptability brought by closeness to criticality with the stability of the dynamics characteristic for subcritical systems. By a resilience analysis, we find that initially critical agents maintain their fitness level even under environmental changes and degrade slowly with increasing perturbation strength. On the other hand, subcritical agents originally evolved to the same fitness, were often rendered utterly inadequate and degraded faster. We conclude that although the subcritical regime is preferable for a simple task, the optimal deviation from criticality depends on the task difficulty: for harder tasks, agents evolve closer to criticality. Furthermore, subcritical populations cannot find the path to decrease their distance to criticality. In summary, our study suggests that initializing models near criticality is important to find an optimal and flexible solution.