Maximum entropy models of neuronal populations at and off criticality

Maximum entropy models of neuronal populations at and off criticality investigate whether static maximum entropy (ME) models can distinguish between dynamical states like criticality and supercriticality, which are otherwise difficult to separate using only avalanche statistics. This work addresses this by analyzing spontaneous activity in organotypic rat cortex slice cultures under baseline, hypoexcitable (subcritical), and drug-induced hyperexcitable (supercritical) conditions, finding that while static ME models correctly identify subcritical states as lacking thermodynamic hints of criticality, they fail to discriminate between dynamical criticality and supercriticality.

Experimental Setup and Classification

The study utilized spontaneous activity from organotypic rat cortex slice cultures maintained on a planar microelectrode array (MEA). Neuronal populations were classified into three dynamic regimes based on their avalanche size and duration distributions: critical, subcritical, and supercritical. Critical cultures exhibited power-law behaviors in avalanche statistics. Subcritical cultures displayed exponential size and duration distributions, while supercritical cultures showed a pronounced increase in large avalanches of the order of the system size. These dynamical classifications were confirmed using an interpretable neural network model that could be tuned to these states by adjusting a single parameter.

Maximum Entropy Modeling Framework

The core methodology involves constructing a probability distribution, PMEM(σ), that maximizes entropy subject to constraints derived from measured statistics, such as individual firing rates and pairwise correlations. The resulting distribution is mathematically equivalent to the Boltzmann distribution with temperature T=1. This framework allows for the analysis of thermodynamic quantities like specific heat and susceptibility. The method was implemented using a gradient-descent algorithm called Boltzmann Machine (BM) learning to minimize the Kullback-Leibler divergence between empirical data distributions and the inferred ME distributions.

Distinguishing Dynamical States via ME Signatures

The researchers found that static ME models inferred from critical cultures showed signatures of criticality in thermodynamic quantities, e.g., specific heat. However, these signatures were also present and equally strong in models inferred from supercritical cultures despite their altered dynamics and poor functional performance. Conversely, ME models inferred from subcritical cultures do not show thermodynamic hints of criticality. This indicates that static ME models can correctly distinguish subcritical systems from critical/supercritical systems based on avalanche metrics.

Inferred Ising-like Models and Parameter Analysis

The ME distributions were mapped onto a generalized Ising-like model defined by a Hamiltonian H(σ). The inferred parameters—local fields (hi), interaction constants (Jij), and potentials (VK)—were learned using the BM algorithm. For the IF network, the local fields hi shifted from strongly negative values toward less negative and positive values as systems are driven from the subcritical to the critical and supercritical state. Notably, VK were not negligible in critical and supercritical states, showing a stronger increase in large avalanches in the supercritical regime compared to criticality.

Thermodynamic Signatures Across States

The thermodynamic response functions, specifically specific heat Cv/N and intensive susceptibility χ/N, were analyzed as functions of the temperature parameter T. In both critical and supercritical systems, these quantities show pronounced maxima near the effective temperature T = 1. These maxima grow superlinearly with system size N in IF networks and with the number of sampled electrodes in cultures. However, for subcritical systems, Cv and χ do not show such evidence of criticality, indicating that static ME models correctly distinguish between these states based on both dynamical metrics and thermodynamic properties.

Conclusion on Criticality Assessment

The study concludes that while static ME models are effective at distinguishing subcritical from critical/supercritical systems based on avalanche dynamics, they may not be able to discriminate between avalanche criticality and supercriticality, suggesting that dynamics is relevant to capture the supercritical behavior and distinguish it from criticality. Therefore, caution is advised when drawing conclusions about criticality or deviations from it solely based on static ME modeling approaches that do not constrain dynamical properties. The authors suggest that ME approaches incorporating dynamic information may be more suitable for distinguishing between critical and supercritical states.

Key Findings Enumerated:

1. Static ME models inferred from critical cultures show thermodynamic signatures of criticality (e.g., specific heat).

2. These signatures are also present in models inferred from supercritical cultures, despite altered dynamics.

3. ME models inferred from subcritical cultures do not show thermodynamic hints of criticality.

4. The static ME models correctly distinguish subcritical systems from critical/supercritical ones based on avalanche metrics (e.g., power-law vs exponential distributions).

5. Static ME models may fail to discriminate between dynamical criticality and supercriticality, as dynamics are relevant for capturing supercritical behavior.

6. K-pairwise Ising models inferred from critical and supercritical systems exhibit pronounced maxima in Cv and χ near T=1, suggesting they operate at or close to a critical point in the thermodynamic limit.

7. Subcritical systems (both numerical and experimental) do not show clear maxima in Cv and χ, indicating they are not operating at criticality according to these thermodynamic measures.

As a fastidious and diligent researcher, I have analyzed the provided scientific paper, Maximum entropy models of neuronal populations at and off criticality. The core findings revolve around bridging the gap between dynamical measures of criticality (neuronal avalanches) and static thermodynamic signatures (maxima in specific heat and susceptibility) derived from Maximum Entropy (ME) models.

Here are the specific improvements to AI systems that can be made, categorized by capability:

)1. Enhanced State Classification and Robustness

The paper demonstrates that static ME models can correctly distinguish between subcritical, critical, and supercritical states based on avalanche dynamics (power-law scaling vs. exponential decay). However, it notes a limitation: they may not be able to discriminate between avalanche criticality and supercriticality.

• The AI system can be improved by integrating the ME model's thermodynamic signatures (maxima in specific heat/susceptibility near T=1) as a secondary, complementary feature for state classification.

• It can perform more robust distance to criticality measurements by analyzing both dynamical measures (avalanche scaling) and thermodynamic signatures simultaneously, potentially resolving ambiguities where pure dynamics fail (e.g., distinguishing between critical and supercritical states).

)2. Optimized Model Inference via Machine Learning

The paper details a Gradient-Descent method called Boltzmann Machine (BM) learning to infer the parameters of the K-pairwise Ising model from empirical data by minimizing Kullback-Leibler divergence.

• The AI system can be improved by implementing this BM learning framework directly into its training pipeline. Instead of simply fitting a pre-defined model, the AI can iteratively adjust its internal parameters (representing synaptic strengths, local fields, and coupling constants) to best match observed neural activity statistics (firing rates, pairwise correlations, synchrony).

• This allows the AI to learn the underlying effective thermodynamic Hamiltonian that governs its own behavior.

)3. Improved Feature Extraction for Complex Dynamics

The paper shows that specific features of the inferred Ising model—such as the distribution of local fields (hi), interaction constants (Jij), and synchrony probability P(K)—are highly sensitive indicators of network state transitions.

• The AI system can be improved by using these learned parameters as high-fidelity internal representations. Instead of relying solely on raw spike data, the AI should maintain and monitor distributions of these derived quantities (e.g., tracking the shift in the distribution of local fields hi or the change in P(K)).

• This allows for real-time monitoring of a system's approach to criticality by observing shifts in these thermodynamic-like features, providing a richer diagnostic than just avalanche counts.

)4. Predictive Capability and Dynamic Range Estimation

The paper shows that the Ising models (especially at critical/supercritical points) predict higher-order correlations (three-point functions Tijk) more accurately than simpler models, and that criticality maximizes dynamic range (stimulus response).

• The AI system can be improved by using the full K-pairwise Ising model structure to perform predictive tasks related to its own operation. For example, if the AI is a control system, it could use the predicted Tijk correlations to estimate its sensitivity or dynamic range—how effectively it can process external stimuli before entering an unstable regime.

• This allows for proactive control adjustments based on inferred thermodynamic stability rather than reactive responses to observed outputs alone.

The improved AI system would be a sophisticated, adaptive neural network capable of:

1. Identifying its current operational state (subcritical, critical, or supercritical) with high confidence by cross-referencing both its output statistics (avalanche patterns) and its internal thermodynamic signatures (susceptibility/specific heat proxies).

2. Self-tuning its underlying connectivity/parameters using a Boltzmann Machine learning algorithm to optimize performance against target data distributions.

3. Operating within a learned thermodynamic landscape where it can predict its response characteristics (like dynamic range) based on its current state, enabling more efficient and robust decision-making under varying conditions.

Empirical evidence of scaling behaviors in neuronal avalanches suggests that neuronal populations in the brain operate near criticality. Departure from scaling in neuronal avalanches has been used as a measure of distance to criticality and linked to brain disorders. A distinct line of evidence for brain criticality has come from thermodynamic signatures in maximum entropy (ME) models. Both of these approaches have been widely applied to the analysis of neuronal data. However, the relationship between deviations from avalanche criticality and thermodynamics of ME models of neuronal populations remains poorly understood. To address this question, we study spontaneous activity of organotypic rat cortex slice cultures in physiological and drug-induced hypo- or hyper-excitable conditions, which are classified as critical, subcritical and supercritical based on avalanche dynamics. We find that static ME models inferred from critical cultures show signatures of criticality in thermodynamic quantities, e.g. specific heat. However, such signatures are also present and equally strong in models inferred from supercritical cultures -- despite their altered dynamics and poor functional performance. On the contrary, ME models inferred from subcritical cultures do not show thermodynamic hints of criticality. Importantly, we confirm these results using an interpretable neural network model that can be tuned to and away from avalanche criticality. Our findings indicate that static maximum entropy models, although not constraining dynamical features, correctly distinguish subcritical from critical/supercritical systems. However, they may not be able to discriminate between avalanche criticality and supercriticality, suggesting that dynamics is relevant to capture the supercritical behavior and distinguish it from criticality.