Signatures of brain criticality unveiled by maximum entropy analysis across cortical states

Entropy of Neuronal Spike Patterns

Neuronal spike patterns are the fundamental units of neural communication in the brain, which is still not fully understood. Entropy measures offer a quantitative framework to assess the variability and information content of these spike patterns. By quantifying the uncertainty and informational content of neuronal patterns, entropy measures provide insights into neural coding strategies, synaptic plasticity, network dynamics, and cognitive processes. Here, we review basic entropy metrics and then we provide examples of recent advancements in using entropy as a tool to improve our understanding of neuronal processing. It focuses especially on studies on critical dynamics in neural networks and the relation of entropy to predictive coding and cortical communication. We highlight the necessity of expanding entropy measures from single neurons to encompass multi-neuronal activity patterns, as cortical circuits communicate through coordinated spatiotemporal activity patterns, called neuronal packets. We discuss how the sequential and partially stereotypical nature of neuronal packets influences the entropy of cortical communication. Stereotypy reduces entropy by enhancing reliability and predictability in neural signaling, while variability within packets increases entropy, allowing for greater information capacity. This balance between stereotypy and variability supports both robustness and flexibility in cortical information processing. We also review challenges in applying entropy to analyze such spatiotemporal neuronal spike patterns, notably, the "curse of dimensionality" in estimating entropy for high-dimensional neuronal data. Finally, we discuss strategies to overcome these challenges, including dimensionality reduction techniques, advanced entropy estimators, sparse coding schemes, and the integration of machine learning approaches. Thus, this work summarizes the most recent developments on how entropy measures contribute to our understanding of principles underlying neural coding.

Maximum-entropy-based metrics for quantifying critical dynamics in spiking neuron data

An important working hypothesis to investigate brain activity is whether it operates in a critical regime. Recently, maximum-entropy phenomenological models have emerged as an alternative way of identifying critical behavior in neuronal data sets. In the present paper, we investigate the signatures of criticality from a firing rate-based maximum-entropy approach on data sets generated by computational models, and we compare them to experimental results. We found that the maximum entropy approach consistently identifies critical behavior around the phase transition in models and rules out criticality in models without phase transition. The maximum-entropy-model results are compatible with results for cortical data from urethane-anesthetized rats data, providing further support for criticality in the brain.

State-dependent complexity of the local field potential in the primary visual cortex

The local field potential (LFP) is as a measure of the combined activity of neurons within a region of brain tissue. While biophysical modeling schemes for LFP in cortical circuits are well established, there is a paramount lack of understanding regarding the LFP properties along the states assumed in cortical circuits over long periods. Here we use a symbolic information approach to determine the statistical complexity based on Jensen disequilibrium measure and Shannon entropy of LFP data recorded from the primary visual cortex (V1) of urethane-anesthetized rats and freely moving mice. Using these information quantifiers, we find consistent relations between LFP recordings and measures of cortical states at the neuronal level. More specifically, we show that LFP's statistical complexity is sensitive to cortical state (characterized by spiking variability), as well as to cortical layer. In addition, we apply these quantifiers to characterize behavioral states of freely moving mice, where we find indirect relations between such states and spiking variability.

Ising-like model replicating time-averaged spiking behaviour of in vitro neuronal networks

We analyze time-averaged experimental data from in vitro activities of neuronal networks. Through a Pairwise Maximum-Entropy method, we identify through an inverse binary Ising-like model the local fields and interaction couplings which best reproduce the average activities of each neuron as well as the statistical correlations between the activities of each pair of neurons in the system. The specific information about the type of neurons is mainly stored in the local fields, while a symmetric distribution of interaction constants seems generic. Our findings demonstrate that, despite not being directly incorporated into the inference approach, the experimentally observed correlations among groups of three neurons are accurately captured by the derived Ising-like model. Within the context of the thermodynamic analogy inherent to the Ising-like models developed in this study, our findings additionally indicate that these models demonstrate characteristics of second-order phase transitions between ferromagnetic and paramagnetic states at temperatures above, but close to, unity. Considering that the operating temperature utilized in the Maximum-Entropy method isT o = 1 , this observation further expands the thermodynamic conceptual parallelism postulated in this work for the manifestation of criticality in neuronal network behavior.

Critical dynamics arise during structured information presentation within embodied in vitro neuronal networks

Understanding how brains process information is an incredibly difficult task. Amongst the metrics characterising information processing in the brain, observations of dynamic near-critical states have generated significant interest. However, theoretical and experimental limitations associated with human and animal models have precluded a definite answer about when and why neural criticality arises with links from attention, to cognition, and even to consciousness. To explore this topic, we used an in vitro neural network of cortical neurons that was trained to play a simplified game of 'Pong' to demonstrate Synthetic Biological Intelligence (SBI). We demonstrate that critical dynamics emerge when neural networks receive task-related structured sensory input, reorganizing the system to a near-critical state. Additionally, better task performance correlated with proximity to critical dynamics. However, criticality alone is insufficient for a neuronal network to demonstrate learning in the absence of additional information regarding the consequences of previous actions. These findings offer compelling support that neural criticality arises as a base feature of incoming structured information processing without the need for higher order cognition.

Heterogeneity across neural populations: Its significance for the dynamics and functions of neural circuits

Neural populations show patterns of synchronous activity, as they share common correlated inputs. Neurons in the cortex that are connected by strong synapses cause rapid firing explosions. In addition, areas that are connected by weaker synapses have a slower dynamics and they can contribute to asymmetries in the input distributions. The aim of this work is to develop a neural model to investigate how the heterogeneities in the synaptic input distributions affect different levels of organizational activity in the brain dynamics. We analytically show how small changes in the correlation inputs can cause large changes in the interactions of the outputs that lead to a phase transition, demonstrating that a simple variation in the direction of a biased skewed distribution in the neuronal inputs can generate a transition of states in the firing rate, passing from spontaneous silence ("down state") to an absolute spiking activity ("up state"). We present an exact quantification of the dynamics of the output variables, showing that when considering a biased skewed distribution in the inputs of neuronal population, the critical point is not in an asynchronous or synchronous state but rather at an intermediate value.

Subsampled Directed-Percolation Models Explain Scaling Relations Experimentally Observed in the Brain

Recent experimental results on spike avalanches measured in the urethane-anesthetized rat cortex have revealed scaling relations that indicate a phase transition at a specific level of cortical firing rate variability. The scaling relations point to critical exponents whose values differ from those of a branching process, which has been the canonical model employed to understand brain criticality. This suggested that a different model, with a different phase transition, might be required to explain the data. Here we show that this is not necessarily the case. By employing two different models belonging to the same universality class as the branching process (mean-field directed percolation) and treating the simulation data exactly like experimental data, we reproduce most of the experimental results. We find that subsampling the model and adjusting the time bin used to define avalanches (as done with experimental data) are sufficient ingredients to change the apparent exponents of the critical point. Moreover, experimental data is only reproduced within a very narrow range in parameter space around the phase transition.

Statistical complexity is maximized close to criticality in cortical dynamics

Complex systems are typically characterized as an intermediate situation between a complete regular structure and a random system. Brain signals can be studied as a striking example of such systems: cortical states can range from highly synchronous and ordered neuronal activity (with higher spiking variability) to desynchronized and disordered regimes (with lower spiking variability). It has been recently shown, by testing independent signatures of criticality, that a phase transition occurs in a cortical state of intermediate spiking variability. Here we use a symbolic information approach to show that, despite the monotonical increase of the Shannon entropy between ordered and disordered regimes, we can determine an intermediate state of maximum complexity based on the Jensen disequilibrium measure. More specifically, we show that statistical complexity is maximized close to criticality for cortical spiking data of urethane-anesthetized rats, as well as for a network model of excitable elements that presents a critical point of a nonequilibrium phase transition.