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Criticality Maximizes Complexity in Neural Tissue

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ABSTRACT

The analysis of neural systems leverages tools from many different fields. Drawing on techniques from the study of critical phenomena in statistical mechanics, several studies have reported signatures of criticality in neural systems, including power-law distributions, shape collapses, and optimized quantities under tuning. Independently, neural complexity—an information theoretic measure—has been introduced in an effort to quantify the strength of correlations across multiple scales in a neural system. This measure represents an important tool in complex systems research because it allows for the quantification of the complexity of a neural system. In this analysis, we studied the relationships between neural complexity and criticality in neural culture data. We analyzed neural avalanches in 435 recordings from dissociated hippocampal cultures produced from rats, as well as neural avalanches from a cortical branching model. We utilized recently developed maximum likelihood estimation power-law fitting methods that account for doubly truncated power-laws, an automated shape collapse algorithm, and neural complexity and branching ratio calculation methods that account for sub-sampling, all of which are implemented in the freely available Neural Complexity and Criticality MATLAB toolbox. We found evidence that neural systems operate at or near a critical point and that neural complexity is optimized in these neural systems at or near the critical point. Surprisingly, we found evidence that complexity in neural systems is dependent upon avalanche profiles and neuron firing rate, but not precise spiking relationships between neurons. In order to facilitate future research, we made all of the culture data utilized in this analysis freely available online.

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Neural complexity and criticality are not trivially related. (A) Bethe Lattice or branching model. Activity spread from layer to layer starting at the leftmost neuron. The likelihood to transmit activity along a connection was ptrans = 0.5, yielding a critical model system. (B) The avalanche sizes in the model were power-law distributed, as expected. (C) The avalanches exhibited shape collapse, as expected. (D–F) Analysis of small avalanches (stopped on or before the 8th layer). (D) The small avalanches exhibited power-law distributed sizes and non-zero complexity. (E) Randomizing the neuron identities (“neuron shuffling”) at each time bin preserved avalanche size distribution, but removed complexity. (F) Randomizing the order of the network states (“Poisson” randomization applied to whole network states) preserved complexity, but disrupted the avalanche size distribution.
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Figure 5: Neural complexity and criticality are not trivially related. (A) Bethe Lattice or branching model. Activity spread from layer to layer starting at the leftmost neuron. The likelihood to transmit activity along a connection was ptrans = 0.5, yielding a critical model system. (B) The avalanche sizes in the model were power-law distributed, as expected. (C) The avalanches exhibited shape collapse, as expected. (D–F) Analysis of small avalanches (stopped on or before the 8th layer). (D) The small avalanches exhibited power-law distributed sizes and non-zero complexity. (E) Randomizing the neuron identities (“neuron shuffling”) at each time bin preserved avalanche size distribution, but removed complexity. (F) Randomizing the order of the network states (“Poisson” randomization applied to whole network states) preserved complexity, but disrupted the avalanche size distribution.

Mentions: To ensure that complexity and criticality were not trivially related, we used a simple branching or Bethe Lattice model tuned to the critical point (see Section 2.4, Figure 5A). Using the full model, we found the power-law distributed avalanche sizes (Figure 5B) and shape collapse (Figure 5C). We then limited our analysis to only the avalanches that stopped on or before the 8th layer (“small avalanches”). We performed this truncation because the number of neurons grows exponentially with network layer. This growth severely impacted sub-sampling in the complexity calculation. In the original small avalanches, the sizes of the avalanches were power-law distributed and a non-zero complexity was observed (Figure 5D). We then performed two types of randomizations. First, we randomized the neuron identities in the avalanches (neuron shuffling). This operation preserved the avalanches and their power-law size distributions, but removed the complexity (Figure 5E). Conversely, when we randomized the order of the network states, but preserved neuron spiking states at each individual time bin (Poisson randomization for the whole network state), the complexity was preserved, but the distribution of avalanche size changed dramatically (Figure 5F). This difference in behavior under randomization highlights the differences between criticality and complexity analyses. Complexity only focuses on instantaneous correlations between specific neurons, whereas criticality analyses only focus on the time order of the total number of active neurons.


Criticality Maximizes Complexity in Neural Tissue
Neural complexity and criticality are not trivially related. (A) Bethe Lattice or branching model. Activity spread from layer to layer starting at the leftmost neuron. The likelihood to transmit activity along a connection was ptrans = 0.5, yielding a critical model system. (B) The avalanche sizes in the model were power-law distributed, as expected. (C) The avalanches exhibited shape collapse, as expected. (D–F) Analysis of small avalanches (stopped on or before the 8th layer). (D) The small avalanches exhibited power-law distributed sizes and non-zero complexity. (E) Randomizing the neuron identities (“neuron shuffling”) at each time bin preserved avalanche size distribution, but removed complexity. (F) Randomizing the order of the network states (“Poisson” randomization applied to whole network states) preserved complexity, but disrupted the avalanche size distribution.
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Figure 5: Neural complexity and criticality are not trivially related. (A) Bethe Lattice or branching model. Activity spread from layer to layer starting at the leftmost neuron. The likelihood to transmit activity along a connection was ptrans = 0.5, yielding a critical model system. (B) The avalanche sizes in the model were power-law distributed, as expected. (C) The avalanches exhibited shape collapse, as expected. (D–F) Analysis of small avalanches (stopped on or before the 8th layer). (D) The small avalanches exhibited power-law distributed sizes and non-zero complexity. (E) Randomizing the neuron identities (“neuron shuffling”) at each time bin preserved avalanche size distribution, but removed complexity. (F) Randomizing the order of the network states (“Poisson” randomization applied to whole network states) preserved complexity, but disrupted the avalanche size distribution.
Mentions: To ensure that complexity and criticality were not trivially related, we used a simple branching or Bethe Lattice model tuned to the critical point (see Section 2.4, Figure 5A). Using the full model, we found the power-law distributed avalanche sizes (Figure 5B) and shape collapse (Figure 5C). We then limited our analysis to only the avalanches that stopped on or before the 8th layer (“small avalanches”). We performed this truncation because the number of neurons grows exponentially with network layer. This growth severely impacted sub-sampling in the complexity calculation. In the original small avalanches, the sizes of the avalanches were power-law distributed and a non-zero complexity was observed (Figure 5D). We then performed two types of randomizations. First, we randomized the neuron identities in the avalanches (neuron shuffling). This operation preserved the avalanches and their power-law size distributions, but removed the complexity (Figure 5E). Conversely, when we randomized the order of the network states, but preserved neuron spiking states at each individual time bin (Poisson randomization for the whole network state), the complexity was preserved, but the distribution of avalanche size changed dramatically (Figure 5F). This difference in behavior under randomization highlights the differences between criticality and complexity analyses. Complexity only focuses on instantaneous correlations between specific neurons, whereas criticality analyses only focus on the time order of the total number of active neurons.

View Article: PubMed Central - PubMed

ABSTRACT

The analysis of neural systems leverages tools from many different fields. Drawing on techniques from the study of critical phenomena in statistical mechanics, several studies have reported signatures of criticality in neural systems, including power-law distributions, shape collapses, and optimized quantities under tuning. Independently, neural complexity—an information theoretic measure—has been introduced in an effort to quantify the strength of correlations across multiple scales in a neural system. This measure represents an important tool in complex systems research because it allows for the quantification of the complexity of a neural system. In this analysis, we studied the relationships between neural complexity and criticality in neural culture data. We analyzed neural avalanches in 435 recordings from dissociated hippocampal cultures produced from rats, as well as neural avalanches from a cortical branching model. We utilized recently developed maximum likelihood estimation power-law fitting methods that account for doubly truncated power-laws, an automated shape collapse algorithm, and neural complexity and branching ratio calculation methods that account for sub-sampling, all of which are implemented in the freely available Neural Complexity and Criticality MATLAB toolbox. We found evidence that neural systems operate at or near a critical point and that neural complexity is optimized in these neural systems at or near the critical point. Surprisingly, we found evidence that complexity in neural systems is dependent upon avalanche profiles and neuron firing rate, but not precise spiking relationships between neurons. In order to facilitate future research, we made all of the culture data utilized in this analysis freely available online.

No MeSH data available.


Related in: MedlinePlus