A Maximum Entropy Model Applied to Spatial and Temporal Correlations from Cortical NetworksIn Vitro

Abstract

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 Ϯ 7% (mean Ϯ 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.

Neuronal couplings between retinal ganglion cells inferred by efficient inverse statistical physics methods

“…Existing algorithms for inverse Ising problems are based on time-consuming learning schemes (11)(12). It is possible, however, to drastically improve inverse Ising methods if one takes advantage of the fact that the neurons are more often silent than active.…”

“…Visual stimuli could be adjusted during experiments to explore particular aspects of retinal function. This possibility is not restricted only to multielectrode recordings in vertebrate retina: The same approach could be also used to analyze the activity of other neuronal systems, such as the recordings from slices of vertebrate cortex (12). where the parameters hi and Jij are called, respectively, effective fields and couplings; the value of F [{hi},{Jij}], called free energy in statistical physics, is such that the probabilities of the 2 N configurations sum up to 1.…”

Mismatched Decoding in the Brain

“…Below, we discuss differences between the present and previous studies using the maximum entropy approach (Schneidman et al, 2006;Shlens et al, 2006;Tang et al, 2008); limitations and extensions of the methodology used in this work; and future directions, which concern animal behavior experiments (Stopfer et al, 1997;Ishikane et al, 2005).…”

“…Previous studies using the maximum entropy approach (Schneidman et al, 2006;Shlens et al, 2006;Tang et al, 2008) emphasized the discrepancy between the independent model and actual probability distribution. That is, their results show that there are significant correlations in large neural populations.…”

Thermodynamics and signatures of criticality in a network of neurons

“…The maximum entropy approach to networks of neurons has been explored, in several different systems, for nearly a decade (14)(15)(16)(17)(18)(19)(20)(21)(22)(23), and there have been parallel efforts to use this approach in other biological contexts (24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35). Recently, we have used the maximum entropy method to build models for the activity of up to N = 120 neurons in the experiments described above (11); see Fig.…”