Aonan Tang

A maximum entropy model applied to spatial and temporal correlations from cortical networks in vitro.

Multineuron firing patterns are often observed, yet are predicted to be rare by models that assume independent firing. To explain these correlated network states, two groups recently applied a second-order maximum entropy model that used only observed firing rates and pairwise interactions as parameters (Schneidman et al., 2006; Shlens et al., 2006). Interestingly, with these minimal assumptions they predicted 90–99% of network correlations. If generally applicable, this approach could vastly simplify analyses of complex networks. However, this initial work was done largely on retinal tissue, and its applicability to cortical circuits is mostly unknown. This work also did not address the temporal evolution of correlated states. To investigate these issues, we applied the model to multielectrode data containing spontaneous spikes or local field potentials from cortical slices and cultures. The model worked slightly less well in cortex than in retina, accounting for 88 ± 7% (mean ± SD) of network correlations. In addition, in 8 of 13 preparations, the observed sequences of correlated states were significantly longer than predicted by concatenating states from the model. This suggested that temporal dependencies are a common feature of cortical network activity, and should be considered in future models. We found a significant relationship between strong pairwise temporal correlations and observed sequence length, suggesting that pairwise temporal correlations may allow the model to be extended into the temporal domain. We conclude that although a second-order maximum entropy model successfully predicts correlated states in cortical networks, it should be extended to account for temporal correlations observed between states.

Maximum Entropy Approaches to Living Neural Networks

Understanding how ensembles of neurons collectively interact will be a key step in developing a mechanistic theory of cognitive processes. Recent progress in multineuron recording and analysis techniques has generated tremendous excitement over the physiology of living neural networks. One of the key developments driving this interest is a new class of models based on the principle of maximum entropy. Maximum entropy models have been reported to account for spatial correlation structure in ensembles of neurons recorded from several different types of data. Importantly, these models require only information about the firing rates of individual neurons and their pairwise correlations. If this approach is generally applicable, it would drastically simplify the problem of understanding how neural networks behave. Given the interest in this method, several groups now have worked to extend maximum entropy models to account for temporal correlations. Here, we review how maximum entropy models have been applied to neuronal ensemble data to account for spatial and temporal correlations. We also discuss criticisms of the maximum entropy approach that argue that it is not generally applicable to larger ensembles of neurons. We conclude that future maximum entropy models will need to address three issues: temporal correlations, higher-order correlations, and larger ensemble sizes. Finally, we provide a brief list of topics for future research.

A second-order maximum entropy model predicts correlated network states, but not their evolution over time

Highly correlated network states are often seen in multielectrode data, yet are predicted to be rare by independent models. What can account for the abundance of these multi-neuron firing patterns? Recent work [1,2] has shown that it is possible to predict over 90% of highly correlated network states, even when correlations between neuron pairs are weak. To make these predictions, both groups used a maximum entropy model that fit only the firing rates and the pairwise correlations (a second-order maximum entropy model), and which was maximally uncommitted about all other model features. This new work raises several questions. First, how general are these results? Both previous reports largely used retinal data. Could this maximum entropy approach also succeed when applied to cortical slices? Although the original model explained correlations among spikes, could it also be used to explain the abundance of correlated states of local field potentials (LFPs)? A second issue concerns the abundance of correlated states over time. Can a second-order maximum entropy model predict sequences of correlated states? To examine the generality of this approach, we applied a second-order maximum entropy model to a variety of in vitro cortical networks, including acute slices from rat (n = 3) and human epileptic tissue (n = 1), as well as organotypic (n = 3) and dissociated cultures (n = 3) from rat. We explored its effectiveness in predicting correlated states of both spikes and LFPs at one time point. On average, the model accounted for 90 ± 6% (mean ± s.d.) of the available multi-information, in good agreement with previous studies. In all cases, the maximum entropy model significantly outperformed an independent model, demonstrating its effectiveness in explaining correlated states in cortical spikes and LFPs at one time point. We also explored how well the maximum entropy model predicted sequences of correlated states over time. Here, the model often failed to account for the observed sequence lengths. In 8/10 preparations, the distribution of observed sequences was significantly longer (p ≤ 0.003). We conclude that a second-order maximum entropy model can predict correlated states, but not their evolution over time. This suggests that higher-order maximum entropy models incorporating temporal interactions will be needed to account for the sequences of correlated states that are often observed in the data.